Dynamic High-Gain Scaling - Semantic Scholar
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 12, DECEMBER 2004
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Dynamic High-Gain Scaling: State and Output Feedback With Application to Systems With ISS Appended Dynamics Driven by All States Prashanth Krishnamurthy and Farshad Khorrami
Abstract—In this paper, we propose a dynamic high-gain scaling technique and solutions to coupled Lyapunov equations leading to results on state-feedback, output-feedback, and input-to-state stable (ISS) appended dynamics with nonzero gains from all states and input. The observer and controller designs have a dual architecture and utilize a single dynamic scaling. A novel procedure for designing the dynamics of the high-gain parameter is introduced based on choosing a Lyapunov function whose derivative is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The system is allowed to contain uncertain terms dependent on all states and uncertain appended ISS dynamics with nonlinear gains from all system states and input. In contrast, previous results require uncertainties to be bounded by a function of the output and require the appended dynamics to be ISS with respect to the output, i.e., require the gains from other states and the input to be zero. The generated control laws have an algebraically simple structure and the associated Lyapunov functions have a simple quadratic form with a scaling. The design is based on the solution of two pairs of coupled Lyapunov equations for which a constructive procedure is provided. The proposed observer/controller structure provides a globally asymptotically stabilizing output-feedback solution for the benchmark open problem proposed in our earlier work with the provision that a magnitude bound on the unknown parameter be given. Index Terms—Adaptive, high-gain, input-to-state stable (ISS), nonlinear, output feedback, scaling.
I. INTRODUCTION
T
HE class of systems considered in this paper is the uncertain generalized output-feedback canonical form coupled to appended dynamics having nonzero gains from all states
.. . Manuscript received July 31, 2002; revised July 31, 2003, June 8, 2004, and September 2, 2004. Recommended by Associate Editor S. Ge. This work was supported in part by the National Science Foundation under Grant ECS-9977693. Earlier versions of this paper were presented in parts at the 2002 IEEE Conference on Decision and Control, Las Vegas, NV, and the 2003 American Control Conference, Denver, CO. The authors are with the Control/Robotics Research Laboratory (CRRL), Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]; khorrami@ smart.poly.edu). Digital Object Identifier 10.1109/TAC.2004.839235
.. .
.. .
.. . (1) is the state, is the output, is the input, and is the state of the appended dynamics. , , , are known scalar real-valued continand , , and uous functions of their arguments. , , , are time-varying scalar real-valued uncertain functions.1 is the relative degree of the system. It is assumed that sufficient conditions (e.g., local Lipschitz property) on and needed for local existence and uniqueness of solutions of and the (1) are satisfied. We consider the state-feedback output-feedback problems for the system (when the dynamics are not present) and the output-feedback problem with the appended dynamics. Among the control design methodologies in the literature for various classes of nonlinear systems (see [1]–[3] and the references therein), backstepping and its robust and adaptive variants are particularly suited for systems of a lower triangular depends on (strict feedback) form wherein the dynamics of . While (1) is not, in general, in lower triangular , no dynamics), the form in the state-feedback case ( state-feedback control design for (1) can be carried out using a variant of robust backstepping under certain assumptions on and . , no dynamics) case, various In the output-feedback ( approaches have been proposed under different sets of assumptions. In [4]–[7], solutions were proposed assuming existence of control Lyapunov functions with certain properties. In [4], where
= 1 . . . +1
= 1 ...
1q , i ; ; s , and , i ; ; n, can depend on all the states and the input. However, q and are shown in (1) to be functions only of subsets of the state to emphasize the state dependence of the bounds to be introduced on these functions.
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weak linear detectability and the existence of a control Lyapunov function which satisfies certain growth conditions in the unmeasured state variables for the state-feedback stabilization problem were required. [5]–[7] address the problem under the assumption of the existence of control Lyapunov functions of a specific where is the unstructure ( measured state component) for the state-feedback stabilization problem and the output-injection problem for the dual system. Assuming that are known and linear in the unmeasured , state components, i.e., the generalized output-feedback canonical form [8], [9] is obtained which reduces to the standard output-feedback canonical , are conform [10]–[12] under the assumption that stants, i.e., all nonlinearities are only output-dependent. This re’s are quirement was relaxed in [13] to the assumption that bounded. This restriction was removed in [14] using an observer with gains generated through a matrix difof order ferential Riccati equation. In [9], assuming that a constant positive–definite matrix can be found to satisfy a certain inequality, and the oba solution was proposed of dynamic order server structure recovered the standard linear reduced-order observer when specialized to the case of linear systems. It was shown in [9] that a sufficient condition for the existence of a matrix satisfying the required inequality is the cascading upper diagonal dominance (CUDD) condition requiring the upper dito be larger than and (to agonal terms within constant factors). Adaptive extensions of the results in [9] and [14] were considered in [15] assuming that parametric uncertainties appear in output-dependent terms. In [16], assuming for and that , , that are incrementally linearly bounded in unmeasured states, the matrix differential Riccati equation in [14] was collapsed to a scalar differential Riccati equation driven by . High-gain techniques both for controller and observer design have been investigated in the literature. The well-known adaptive high-gain controller given in its basic form by , is applicable to minimum-phase systems with relative-degree one (see [17]–[19] and the references therein). Static high-gain scaling based observers [20], [21] which introduce with a constant provide semiglobal observer gains solutions. The observer analysis utilizes scaled observer errors (or ) with being the estimation error of the state. A global high-gain observer and controller with constant were considered in [22] for systems of form (1) gains without the appended dynamics, with with known constant , , , and . A global high-gain observer and controller with gains being powers of were proposed in [23] for linear systems with appended stable nonlinear zero dynamics and inputmatched nonlinearities. The dynamic high-gain scaling based observer in [16] differs from previous results in two features. First, the dynamics of are a scalar differential Riccati equation driven by guaranteeing boundedness of if remains ). bounded (which is not guaranteed by the dynamics This feature is important since the Lyapunov functions used in most high-gain designs are weak in the sense that the boundedness of the Lyapunov function does not directly imply boundedness of the high-gain parameter . Instead, the boundedness
of the high-gain parameter has to be inferred from the boundedness of the Lyapunov function and the dynamics of the high-gain parameter. In classical dynamic high-gain designs where , the boundedness of the high-gain parameter renorm of the output signal which might not hold quires finite under additive perturbations. A second feature in [16] is that an is introduced in the scaled observer additional scaling error definition. The parameter is chosen to be a positive constant large enough to decouple a pair of coupled Lyapunov equations. In [24], it was shown that the pair of coupled Lyapunov equations can be solved simultaneously without requiring the additional scaling and that the introduction of the additional scaling provides a weaker observer error convergence guarantee . It was also shown in [24] that a negative can be used to improve the convergence guarantee. In this paper, we propose a new paradigm for dynamic highgain scaling based observer and controller design. The basic design philosophy is to first attain a high-gain observer and controller (which incorporates an adaptation parameter in the adaptive case) to stabilize the nominal system obtained by omit, dynamics, the inverse dynamics ting the , with states , and the uncertain functions , . This nominal system is a chain of nonlinear integrators (2) The high-gain scaling asymptotically achieves an approximation of the system (1) by the nominal system (2) when the highgain scaling parameter is increased [24]. Thus, as shown in [24], the high-gain scaling induces a CUDD [9] condition by amplifying the upper diagonal terms. The controller design freedoms and the dynamics of the high-gain parameter (and the adaptation parameter in the adaptive case) are then chosen to achieve closed-loop stability. The Lyapunov function used is the sum of scaled Lyapunov functions associated with appended dynamics and inverse dynamics, and scaled quadratic functions of observer errors, , observer estimates (with a nonlinear function of ), and adaptation error. The structure of the Lyapunov function is chosen such that the derivative of the Lyapunov function is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The dynamics of are designed so that is large until becomes large. This is a new design philosophy for dynamics of the high-gain parameter and unlike [16], the dynamics of the high-gain scaling parameter are not required to be of the form of a Riccati equation. However, the boundedness of is ensured by boundedness of the output, the parameter estimate, and the derivative of the parameter estimate, which in turn is inferred from boundedness of the Lyapunov function. The controller and the observer in the output-feedback case have a dual architecture and utilize a single dynamic scaling. Unlike [16], an additional scaling can not be incorporated to decouple the pairs of coupled Lyapunov equations that arise in the design since such an additional scaling with a positive appears to be an obstacle in a dual observer/controller design due to the weaker observer error convergence guarantee obtained. In the designs in this paper, two pairs of coupled Lyapunov equations appear that extend the ones in [24] and the solutions are presented in the Appendix.
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In contrast to previous results, the system (1) is allowed to contain uncertain functions dependent on all the states and the input and also input-to-state stable (ISS) [25] appended dynamics driven by all the states and the input. The previously available results require that: 1) unknown parameters appear multiplied with output nonlinearities or uncertain terms can be bounded as products of unknown parameters and output nonlinearities; and 2) the appended ISS dynamics have nonzero gain only from the output , i.e., a Lyapunov function for the appended dynamics exists such that the derivative of the function of Lyapunov function is the difference of a class and a class function of [15], [26], [27]. These assumptions are essentially due to the fact that, in backstepping based can be efficiently assigned, the designs, while the gain to same is not true for the gains to the other states which appear in the Lyapunov function and its derivative in increasingly complicated combinations. Both these requirements are relaxed using the dynamic high-gain scaling technique advocated in this paper. The basic controller design technique is introduced with a state-feedback design for system (1) without the appended dynamics in Section II. The dual high-gain observer/controller design for the output-feedback case for system (1) without the appended dynamics is presented in Section III. In Section IV, the advocated design methodology is applied to adaptive output-feedback for systems with the appended ISS dynamics . II. STATE-FEEDBACK CONTROL The dynamics (1) without the appended written as2
dynamics can be
(3) where, for notational simplicity, we have introduced the dummy variables and for . A. Assumptions
with being a dummy variable are bounded-inputbounded-state (BIBS) stable with the states and the inputs where5 . Assumption A3: The functions , , can be bounded as (5) for all
and where and , , , are known continuous nonnegative func, , , and tions. Positive constants , , exist such that for all
(6) exist such that Assumption A4: Positive constants , , . Assumption A5: A continuous nonnegative function exists such that . Remark 1: For simplicity, it is assumed that the same function serves in the bounding of the terms on the left-hand side of (5). To remove possible conservativeness of the resulting design, different ’s can be utilized in (5). The design in that case can be carried out along the same lines as presented here. Furcan be taken to be a function of both and the thermore, time as long as it is bounded uniformly in time as a function of . The bounds in (5) can be generalized by allowing additive positive constants. In this case, the proposed design provides practical stabilization results. Thus, practical tracking can be achieved by the approach presented here by converting the tracking problem into a stabilization problem using the change and , , of coordinates is a bounded reference trajectory (with bounded where first derivative). B. Controller Design
as output), controlAssumption A1: Observability (with lability, and uniform relative degree of (3), i.e., a constant exists such that3 , , and for all . Assumption A2: The dynamics [the inverse dynamics4 of (3)]
Define (7) is a design freedom to be chosen later.6 where is a dynamic high-gain parameter whose dynamics will also be chosen later. The dynamics chosen for will ensure that for all time . The dynamics of are given by7
(4) 2While can depend on all states and the input, a triangular state dependence is shown in (3) to highlight the structure of the bounds in Assumptions A3 and A3’. 3j j denotes the Euclidean norm of a vector . In particular, if is a scalar, j j denotes its absolute value. 4The inverse dynamics are the dynamics of the states of the zero dynamics, i.e., x ; ;x , when viewed as driven by the states x ; ;x and the output-zeroing input v u y x = y . The zero dynamics are obtained when v ; ; v are zero in (4).
7=[
...
]
= + ...
()
( ... ()
)
(8) 5In the output-feedback case, the required assumption on the inverse dynamics is Assumption A2 with y being the empty vector. 6This form of is picked to ensure certain bounds as will be seen later in the stability analysis. 7For notational convenience, we drop the arguments of functions whenever no confusion will result.
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where denotes the partial derivative of with respect to its argument evaluated at . The control input is designed as
C. Stability Analysis A Lyapunov function9 is introduced as (14)
(9) with
being design functions. The dynamics of under the control law (9) can be written as (10)
where
is the
matrix with
element
where is a symmetric positive–definite matrix to be defined later. Note that since contains an term which is not scaled implies boundedness of . The dyby , boundedness of namics of will be designed such that boundedness of is guaranteed by boundedness of . Boundedness of all closed-loop states then follows from boundedness of and . The presence term in is made possible by the design of an unscaled freedom which provides a crossterm between and . Differentiating (14)
(11) with zeros elsewhere, and8 (15)
(12) Using Assumption A3
(13) Remark 2: Note the simplicity of the control law (9). The design freedoms in the controller (i.e., the parameters and functions in the controller that are free to be picked by the designer) , , and the dynamics of the high-gain scaling paare rameter . The construction in the Appendix provides a possible choice of explicitly. The first term in the bracket IN , a state of the in(9) simply cancels out the term involving verse dynamics, which appears matched with the control input. The remaining terms can be interpreted as a “linear” feedback with gains dependent both on the plant state and of , are the controller state . Noting from (7) that , simply scaled versions of , and is a scaled version of incorporating an extra term dependent on , the control law can be interpreted as essentially having three components: a term to cancel the state of the inverse dynamics that appears matched with the control input, a nonlinear -dependent term, and a state-dependent linear feedback of . As will be seen in the stability analysis below, a Lyapunov function quadratic (with the scaling ) can be used to demonin , strate closed-loop stability. These two features of the control design methodology, i.e., a simple form of the control law involving the three components highlighted above and an essentially quadratic Lyapunov function are retained in the outputfeedback case in Section III and in the case of output-feedback with ISS appended dynamics in Section IV.
2
notation diag(T ; . . . ; T ) denotes an m m diagonal matrix with ) denotes an diagonal elements T ; . . . ; T . Also, lowerdiag(T ; . . . ; T m m matrix with the lower diagonal entries being T ; . . . ; T and zeros elsewhere. 8The
2
(16) If
, (16) yields
(17) Using Assumptions A1, A3, and A4, along with (17) and ap, pealing to Theorem A2 in the Appendix with10 , , , , , , , and , a symmetric positive–definite matrix , positive constants , , and , and continuous functions are found such that11
(18) (19) for all , , and such that . The proof of Theorem A1 along with the duality transformation in the proof of Theorem A2 provides a constructive procedure to find , , , , and . Note that the inequality (18) requires the while the inequality (19) which property involves only constant matrices is true irrespective of the values 9By a Lyapunov function, we mean a differentiable function V ( ) that satisfies V (0) = 0 and V ( ) > 0 for all = 0 in the domain of definition. Note that by the dynamics to be introduced for r , we have r (t) 1 for all time t. 10I denotes the identity matrix of appropriate dimensions. 11The positive term on the right-hand side of (18) is generated by the term 0(x )[ + ] x =r in the bound (17).
6
j jj j
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of the states. However, by Theorem A2, the following inequality which is weaker than (18) is valid for all values of the states (20) Using straightforward algebra,12
(26) with , , and being nonnegative design parameters, the inequalities (15) and (19)–(21) imply that (27)
(21) where is a positive design parameter. picked to be a continuous function such that13
is
From (25) and (27), we see that can be made nonpositive by choosing either large (i.e., ) or large (i.e., ). This implies that to stabilize the system, the dynamics of should be chosen such that is large till becomes large, i.e., should be made to increase to a large enough value fast enough. This is achieved by choosing the dynamics of to be of the form (28)
(22) where is a positive function of bounded below in magnitude by a positive constant. If satisfies
with being a nonnegative constant smaller than and being a nonnegative continuous function such that the following properties are true: a) b)
(23) c)
(24) we have, using (15), (18), and (21)
(25) On the other hand, for any value of
satisfies
is a square symmetric matrix, (P ) denotes its largest eigenvalue. is continuous and bounded below in magnitude by a positive constant, it assumes the same sign for all arguments. 12If
P
, if
13Since
a nonnegative function such that satisfying the holds inequality the inequality holds.
(29) Remark 3: Property b) can be satisfied by picking such for with being any positive constant that yielding . Since for all , and such that property c) is easily satisfied by picking . Property b) is required to infer boundedness while the third property ensures of from boundedness of that for all time . In the special case in which depends only on , the function can be reduced to the and the properties above form can be satisfied by picking, for instance, to be identically unity and , resulting in the dynamics (30) The dynamics (30) which has the form of a scalar Riccati equation was proposed in [16] in the context of high-gain based observer design. The advantage of the general dynamics (28) is is allowed to be an arbithat is allowed to be zero and trary function of all the states while still retaining the crucial property (whose significance will be apparent in the following stability analysis) that the boundedness of follows from the alone. This freedom in allows Assumpboundedness of which are functions of the tion A3 to include functions entire system state . The fact that can be taken to be zero is
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vital in the output-feedback design with ISS appended dynamics in Section IV. With the dynamics of given by (28), using (25) and (27), with chosen smaller than , for any value of and with , the Lyapunov inequality
A. Assumptions Assumption A3 : A nonnegative continuous function and functions , , are known such , , and all that for all real numbers
(31) is nonincreasing and is, hence, holds. This implies that bounded on the maximal interval of existence of solutions . Boundedness of implies boundedness of and . The second property of listed in (29) implies that boundedness of follows from boundedness of and property for all . Boundedness c) in (29) implies that . Invoking the of and implies boundedness of BIBS assumption on inverse dynamics in Assumption A2, are also bounded. Thus, all closed-loop signals are bounded. Hence, solutions do not escape in finite time ). Furthermore, from (31), and , and hence, (i.e., go to zero asymptotically as . The foregoing Lyapunov analysis provides Theorem 1. Theorem 1: Under Assumptions A1–A5, given any initial for the plant state and for the conconditions troller state, the closed-loop system formed by (3), (9), and (28) and the possesses a unique solution on the time interval following properties hold. 1) All closed-loop signals are bounded and , . 2) If Assumption A2 is strengthened to include the minimum-phase condition that the inverse dynamics (4) is , asymptotically stable with then , . If the inverse dynamics are further required to be ISS, then with the dynamics of as in (30), the equilibrium is globally asymptotically stable where . The proof of the second statement in Theorem 1 is straightforward using composite Lyapunov functions similar to the analysis in Section IV. Under the minimum-phase assumption on the inverse dynamics, by continuity of and , it follows from the dynamics (28) that asymptotically converges to . By the set the properties (29), this set contains at least one value in the . In the special case when satisfies the interval Riccati equation (30), this set consists of the single point . III. OUTPUT-FEEDBACK CONTROL
(32) Assumption A4 : Positive constants
and
exist such that
(33) (34) Assumptions A3 and A4 are contained in Assumptions A3 and A4 , respectively, with the additional requirements in Assumptions A3 and A4 being imposed to handle the output-feedback case. The inequality (33) which is identical to Assumption A4 is required in the controller design and its dual (34) is required in the observer design. Assumption A5 is is then always satisfied in the output-feedback case since a function of only. The output-feedback solution below is based on Assumptions A1, A2, A3 , and A4 . and be the matrices deRemark 4: Let fined in (11) and (12), respectively. Using Assumptions A1 and A4 , and appealing to Theorem A2 in the Appendix with , , an arbitrary vector in , , , , , , , , and , a symmetric positive–definite matrix , positive constants , , and , and continuous are found such that functions
In this section, we consider the output-feedback control of system (3) with being the output.14 In this case, we need to strengthen Assumptions A3 and A4 as follows. (35) for all
. Let
14The
design for the more general class (1) which includes appended ISS dynamics is considered in Section IV.
(36)
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and let
be the
matrix with
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entry
B. Observer Design A reduced-order observer with states given by16
is
(37) with zeros everywhere else. Using Assumptions A1 and A4 , , and applying Theorem A1 in the Appendix with , an arbitrary vector in , with being a positive constant, , , , , , , , and , a symmetric positive definite matrix , positive constants , , and , and continuous functions are found such that15
(41) where is a time-varying signal whose dynamics will be deare design functions of of the form signed later, and
(42) (38)
for all where . Furthermore, by are linear constant-coefficient combiTheorem A1, nations of . Hence, using Assumption A4 , a positive constant exists such that
being functions chosen as in Remark 4. The rewith duced-order observer in (41) is similar, in structure, to the observer in our earlier work [9] with modifications to incorporate the dynamic high-gain scaling introduced in the spirit of [16]. Defining the observer errors as (43)
(39)
the observer error dynamics can be written in the form
Remark 5: A particular case in which Assumption A3 is satare of the form isfied is if (44) with being a dummy variable. The scaled observer errors are now defined as (40) (45) where and are known matrix functions of of dimenand , respectively, and sions is a vector of time-varying uncertain parameters with a known magnitude bound. The assumptions in [16] and [22] are obtained as special cases of Assumption A3 . In [16], it is assumed that . In [22] (where it is assumed that ), is required with to be bounded as being a constant. Furthermore, both [16] and [22] assume that , , a special case of Assumption A4 .
0j
j
~
j
j
j
0j
j 0
. The dynamics of the scaled
(46) where, for notational convenience, we have introduced
(47)
j 0
A1 implies that (38) is satisfied with [ I C C ] on the right hand side of the first inequality in (38). Since is bounded below by , I I with = .
15Theorem
with the dummy variable observer errors are given by
f
16For
simplicity of notation, we introduce the dummy variables x ^ = 0.
=g
=
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The dynamics (46) can be written in the matrix form as (48) where
, , and
with and
are defined in (36) and (37), respectively. If the dynamics of are chosen as in (28) with being identi, with being a positive cally unity and a continuous positive function, it can be constant and seen that the Lyapunov function with chosen as in Remark 4 satisfies
(51) The control law is picked to be of the form
(52)
(49) is negative if is In the absence of , the derivative of . If Assumption A3 is strengthened chosen smaller than to the requirement (the case in [16]) that and , then . Hence, . renders Choosing negative. Thus, if the states remain bounded (which implies remains bounded), the observer errors go to zero asymptotically. With the more general bound in Assumption A3 , the observer design renders the scaled observer errors ISS with respect to and the scaled observer estimates , . The closed-loop stability will be inferred in Section III-D through a composite Lyapunov analysis in conjunction with the controller designed in Section III-C. C. Controller Design Introduce the signals
,
, defined as
(50) where The dynamics of
where are functions chosen as in Remark 4. Note that (52) is essentially the same as (9) with unavailable states , (51) can be replaced by estimates. Defining written in matrix form as (53) where
with , , , and and are matrices defined in (11) and (12), respectively. Remark 6: Note that the observer and controller designs in Sections III-B and III-C depend only on the nominal system (2) and the nominal inverse dynamics system ,
(54) The choice of , also depends only on the nominal systems. The robustness to uncertain terms and appended dynamics will be achieved through the design function and the dynamics of . If the inverse dynamics system is stable, the observer (41) and the controller (52) globally asymptotically stabilize the nominal system (2) if is picked to be an appropriate positive constant and is suitably chosen. For sim. In this case, and plicity, assume for now that can be chosen to be positive constants such that
is a design freedom to be chosen later. are (55) Global asymptotic stability of the nominal system can be verified using the Lyapunov function (56) where and are positive–definite matrices as defined in Remark 4 and is a positive constant larger than . In this case, and are zero,
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, and
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. Differen-
Using
tiating (56), (57) In Section III-D, it is shown that stability of the overall system (3), (41), and (52) can be ensured by appropriately choosing and the dynamics of the high-gain the design function parameter .
(62) , we have
and the decomposition
D. Stability Analysis Closed-loop stability can be analyzed using the Lyapunov function (58) where is a positive constant. Differentiating (58)
(63) Choosing
to satisfy (64)
and using (39), we have (59) and the dyWe need to pick the function namics of to make negative definite. To achieve this, we derive bounds below for various terms in (59). , can be overbounded as Using Assumption A3 , if
(65) Similarly
(66) The function
is picked such that
(60) Therefore
(67) (61)
with
being a positive function of
.
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Using (35), (38), (61), (63), (65), (66), and (67), (59) may be rewritten as
(68) where
(69) From (68), it is seen that the inequality
(70) is achieved under either of the following two cases: 1) and where ; 2) and where ; are nonnegative design parameters. As in where , , and Section II, the interpretation is that can be made nonpositive by choosing either or large, i.e., to stabilize the system, should be made to increase to a large enough value fast enough. This is achieved by designing the dynamics of to be of the form (28) with having the properties (29). In the output-feedback case considered in this section, is a function of only and and can be taken to be of the form so that, as noted in Remark 3, can be taken to be identically unity and picking , , the dynamics (30) are obtained. Choosing to be a positive constant small enough such that (71) (70) reduces to (72) From (72), the boundedness of on the maximal interval of can be inferred. This implies the existence of solutions boundedness of , , and . The boundedness of implies the boundedness of . Invoking the BIBS Assumption A2 on the inverse dynamics, all the closed-loop signals are seen to be . Using bounded. Hence, solutions exist for all time and (72), it can be seen that , , and tend to zero asymptotically. , it follows that , Noting from (42) that also go to zero asymptotically. The preceding Lyapunov analysis provides Theorem 2. Theorem 2: Under Assumptions A1, A2, A3 , and A4 , given for the plant state and any initial conditions
for the controller state with , the closed-loop system formed by (3), (41), (52), and (28) possesses a unique solution and the following properties hold. on the time interval , 1) All closed-loop signals are bounded, , and , . 2) If Assumption A2 is strengthened to include the minimum-phase condition that the inverse dynamics (4) is , asymptotically stable with , , then , . If the inand verse dynamics are further required to be ISS, then as in (30), the equilibrium with the dynamics of is globally . asymptotically stable where The design procedure is summarized as follows. , , and symmetric positive1) Obtain definite matrices and to satisfy (35) and (38). small enough to satisfy (71) and pick 2) Pick to satisfy (67). Pick nonnegative constants and . 3) Implement the dynamics of in (28). 4) Implement the observer dynamics (41). as in (50). The control law is given by 5) Define (52). E. Illustrative Examples Example 1: In [15], we proposed the third-order system
(73) is as a benchmark open problem. In (73), the state, is the input, is the output, and is a constant unknown parameter. System (73) is of a very simple form containing a single nonlinearity and a single unknown parameter. If any of the components of the single nonlinearity are dropped, the solution can be obtained using available techniques. If is known, [16], [24] provide solutions. If is removed, the is removed, the system is in standard system is linear. If output-feedback canonical form. However, hitherto, there has been no control design approach which can provide a global asymptotic output-feedback result for this system even with the knowledge of an a priori magnitude bound on . The technique proposed in this paper solves this problem utilizing such a mag. Another interesting observation is the fact nitude bound that the following system obtained from (73) by replacing with can not be stabilized by continuous dynamic output feedback as may be shown by an argument similar to [28]
(74) System (73) belongs to the class of systems (3) with , , , , ,
,
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and . Assumption A1 is satisfied with . Assumptions A2 and A4 are trivially satisfied. Assumption A3 is satand . The isfied with coupled Lyapunov inequalities (35) and (38) are satisfied by , , , , choosing, for instance, , and
. In this case,
,
, , , , and . Thus, and . A reduced-order observer is constructed for this system as
. Assumption A1 is satisfied with inverse dynamics
. The (84)
are BIBS stable with state and inputs and . Hence, Assumption A2 is satisfied. Assumption A3 is satisfied with and . Assumption A4 is satisfied with . In this case
(75) Defining (85)
(76) the control law for is picked as chosen to be , (71) is satisfied with , , and , , , and
. is . Picking
and (35) and (38) are satisfied with
. Picking are obtained as
(77)
(78)
(86)
(79) Picking
, the dynamics of
Note that this choice of , , and satisfies (39) with . The observer and controller are given by
are given by (80)
Global asymptotic stability can be demonstrated using the Lyapunov function (81) Differentiating (81) (82) (87) , all the closed-loop states (except ) go to zero Thus, as asymptotically. Example 2: Consider the fourth-order system
where namics of
,
. The dy-
are as given in (28). IV. ADAPTIVE OUTPUT-FEEDBACK WITH APPENDED ISS DYNAMICS
(83) System (83) belongs to the class of systems (3) with , , , , , , , , and
, ,
In this section, we consider the application of the dynamic high-gain scaling technique to the output-feedback control design for system (1) which includes the appended which are driven by all states and dynamics , input. The design in this section is based on Assumptions A1, A2 , A3 , A4 , and A6.
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A. Assumptions
, are class functions, , , are continuous nonnegative functions, and is an unknown nonnegative constant. Furthermore, nonnegative constants and exist such that17 for all , where
Assumption A2 : The inverse dynamics of (1) with being a dummy variable
,
(95)
(88) are ISS with state and input function satisfying
with ISS Lyapunov
(89)
The local order estimates
, , and
as
hold.18 can be Remark 7: Additive class functions of allowed in (93) and (94). This case can be transformed back to the case considered here (under mild local-order restrictions) by and forming a new considering as the collection whose derivative satisfies (93) and (94). Lyapunov function This is easily seen from the fact that a cascade of ISS systems is ISS. B. Observer and Controller Design
where is a class function, and , are nonnegative continuous functions, and is an unknown nonnegand ative constant. Furthermore, nonnegative constants exist such that for all
Assumption A3 : Nonnegative continuous functions and , are known such that for all , , , , and
(90)
The reduced-order observer dynamics are given by (41) with , . Defining the observer errors as in (43) and the scaled observer errors as in (45), the dynamics of the scaled are given by (48) where observer error vector and are defined in (36) and (37), respectively, and
,
(96)
Introduce
with
,
, defined as
(97)
(91) with being an unknown nonnegative constant. , subsystems are Assumption A6: The , ISS with ISS Lyapunov functions satisfying
where is a dynamic adaptation parameter and is a design freedom to be chosen later. The control law is picked to be of the form (52) with being functions of chosen as in Remark 4. The dynamics of are given by (53) with and defined in (11) and (12), respectively, , , and
(92)
(93)
(98)
3
for i = s + 1 is a dummy variable identically equal to zero. this is not the case, practical stabilization results can be obtained [15], [26] by upper bounding to within additive constants the functions 3 , 3 , and by functions that satisfy the aforementioned local order estimates. 17
(94)
18If
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C. Stability Analysis The inverse dynamics with state the form (88) with
where tion A2
are of
(99)
(103)
. Hence, using Assump-
As in Section III-D, upper bounds are obtained for the terms in as (102) and (103)using Assumption A3 and
(104)
(100) Consider the observer and controller Lyapunov functions defined as (101) and are picked as in Remark 4. The dynamics of where the high-gain parameter and the adaptation parameter will be designed such that is larger than 1, is positive, and and are monotonically nondecreasing. will be designed such (and hence ) is positive. Differentiating that and defined in (101) and using (35) and (38)
(102)
(105)
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(106)
(113) where
(107)
(114)
(108)
(115)
(116)
(109)
(117) (118)
(110)
(111) where is a positive constant chosen to satisfy (64). Defining a composite Lyapunov function
(119) (112)
The term involving
in (113) can be upper bounded as
and using (100)–(111) (120) where
(121)
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By Assumption A6, and as . Hence, , , and as . Hence, a continuous nonnegative function exists . Using a reasoning similar to such that that used in the proof of [29, Th. 2], it is seen that the local order as implies the existence estimate of a new Lyapunov function , class functions and , and a nonnegative continuous function such that
The composite Lyapunov function for the overall system is defined as
where parameter estimator dynamics are designed as
(125) . The
(122) with
as , independent of , and as . Hence, a nonnegative continuous exists such that . function Using Assumption A6 ,
(126)
is initialized to be positive. By (126), so that remains positive for all time. Using (113), (122)–(124), and (126), and differentiating (125)
(123) for
. Also
(127) where
(128)
(129)
(124)
(130) (131)
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(132) The design freedom and by defining
is picked to dominate the functions
(133) being a positive function of lower bounded by a with positive constant. As in Sections II and III, it is seen from (127) that is negative if either the high-gain parameter or its derivative is large (compared to appropriate functions of , , and ). This is a consequence of introducing the scalings in the Lyapunov function definitions (112) and (125). This suggests that the dynamics of should be chosen such that is large until becomes large, thus ensuring that is negative for all time. Accordingly, the dynamics of are designed as
(134)
where
(135)
(136)
and is a nonnegative continuous function satisfying the prop, and being functions of , , and erties (29) with , and a function of , , , and . As noted in Remark 3, the property (b) in (29) can be satisfied by picking such that for with being any positive constant so that . Properties (a) and (c) in (29) are trivially satisfied. By (134), is monotonically nondefor all time. Furthermore, remains creasing so that bounded if , , and remain bounded. From (126), is a , and that is guaranteed to be bounded function of , , if , , , and are bounded. Hence, boundedness of follows from boundedness of implying that boundedness of follows from boundedness of . Considering the two cases a) (which implies that ); b) ; it is seen from (127), (133), (135), and (136) that in both cases
(137)
Hence, is a nonincreasing function of time implying that , , , and are bounded on the maximal interval of existence of . Boundedness of also follows from boundsolutions edness of . Thus, all closed-loop states are seen to remain and, hence, . Furthermore, from bounded on (137), , , , , go to zero as . This , , , and the conimplies that trol input asymptotically go to zero, i.e., all closed-loop states . The prop(except and ) go to zero asymptotically as erties of the closed-loop system are summarized in Theorem 3. Theorem 3: Under Assumptions A1, A2 , A3 , A4 , and A6, , ) for the plant state and given any initial conditions ( ( , , ) for the controller state with , the closed-loop system formed by (1), (41), (52), and (134) pos. Furthersesses a unique solution on the time interval more, all closed-loop signals are bounded, , , , , , and , . can not be incorpoRemark 8: Note that the term rated into the dynamics (134) since this would generate terms and , in involving through (127) which cannot be handled for any nonzero and the stabilizing terms in (127) which feature , . This is a direct consequence of the fact that the inverse dynamics and the appended dynamics are simply driven by the states according to the Lyapunov bounds in Assumptions A2 and A6 and are not directly influenced by the control input. Since is constrained to be zero, the stabilization of (i.e., the crucial property that boundedness of follows from boundedness of ) relies on the presence of design freedom .
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D. Illustrative Example
V. CONCLUSION
Consider the system
(138) where are uncertain parameters. Global output-feedback control for this system cannot be achieved through any technique previously available in the literature. Specifically, the coupling with the appended dynamics , , and which have nonzero gains from the unmeasured states, , , and , and the uncertain parameters which appear multiplied with unmeasured states cannot be handled. and . System (138) is of the form (1) with Assume that a magnitude bound on and is available. Magnitude bounds for , , and are not necessary. The unknown parameter appearing in the bounds in Assumptions A2 , A3 , and A6 is defined as . Assumption A1 is satisfied with . The inverse dynamics system given by
(139) is ISS with ISS Lyapunov function which satisfies (89) with , , , , and thus verifying Assumption A2 . Assumption A3 holds , with , and , . Assumption A4 is sat. The subsystems with states , , isfied with are ISS with ISS Lyapunov functions satisfying , (92), (93), and (94) with , , , and . The upper diagonal terms and of system (138) are the same as in Example 2 in Sec, , , and are as in (85). tion III-E so that matrices The coupled Lyapunov inequalities (35) and (38) are satisfied with , , , , , , , , , , , , , and shown in (86). The observer and controller are given by (87) replaced by . The funcwith tion is picked to satisfy (133). The dynamics of the adaptation parameter and the high-gain parameter are given in (126) and (134), respectively.
In this paper, we have proposed a dynamic high-gain scaling based control technique. A new paradigm for design of dynamics of the high-gain parameter was introduced based on the choice of a Lyapunov function guaranteeing a negative derivative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). Therefore, the dynamics of the high-gain parameter are designed such that the derivative is large until becomes large enough. This approach can be thought of as a technique to expand semiglobal high-gain designs into global ones by using a dynamic high-gain parameter which is made to increase to the required value fast enough. It appears that further extensions to classes of systems for which only semiglobal results are currently available might be feasible with this approach. We applied a variant of the advocated approach to low gain state and output feedback for feedforward systems in [30] and [31]. The designs presented in this paper are applicable to systems with uncertain terms involving all the states and input and also ISS appended dynamics driven by all the states and input. Unknown parameters were allowed in the bounds on the uncertain functions and the derivatives of the ISS Lyapunov functions of the appended dynamics. It was seen that the introduction of appended dynamics and complexity of bounds on the uncertain functions do not result in complexity of the observer and the controller but are rather handled through the dynamics of the high-gain parameter. Conversely, since the high-gain controller structure is “almost” linear with gains being state-dependent, further restriction on the system dynamics may further simplify the control structure degenerating in the limiting case to a purely linear controller. Based on the results of this paper and its earlier versions [32], [33], this was pursued in [34]. Another application of our proposed observer/controller structure is a globally asymptotically stabilizing output-feedback controller for the benchmark open problem proposed in our earlier work [15] with the assumption that a magnitude bound on the unknown parameter is available. However, the original problem posed in [15] to stabilize (73) with being a completely unknown (no known magnitude bound) parameter remains open. APPENDIX Theorem A1: Let be an matrix for any in some with entries on the upper diagonal and design set freedoms on the first column, i.e.,
(140) with zeros everywhere else. Let be an matrix for any and any . Let . Let be the vector . Let be any given function. Let , , , , positive constants , , and exist such that
(141)
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(142) (143) for all
and all where , , are nonnegative functions of . be independent of its argument. Let the sign of each , functions , and positive Then, a matrix constants , , and exist to satisfy
any given positive constant. Then, positive constants smaller in magnitude than , positive constants , , and a positive constant exist such that functions (144) is satisfied with shown in (146). Proof: Consider the dynamical system with and the change of coordi. The inverse transformation is given by19 nates , with . The dynamics of can be written as
(144) (145) and with each being for all chosen to be a linear constant-coefficient combination of , , and . Theorem A1 is proved through Lemmas A1 and A2. In Lemma A1, a family of matrices that satisfy (145) is identified. In Lemma A2, it is shown that there is a matrix in this family that also satisfies (144). Lemma A1: Let be as defined in Theorem A1 and let
(148)
where (142)
are dummy variables. Using
(149)
Picking
,
to satisfy
(146) , being constants. Assume that with and . Then, a (143) is satisfied with positive constants positive constant exists such that if are smaller in magnitude than , then, for any given positive constants , positive constants and exist such that (145) holds. Proof: The lemma follows by continuity since (145) since, by asssumpholds when belong to compact sets in (0, ). tion, the elements of can be decomposed as where The matrix and . . Hence, , a (possibly conservative) Noting that estimate for can be written using the diagonal dominance condition for as
(150) we obtain, via algebraic manipulations similar to those used in the proof of [9, Lemma 1], for
(147) Note that the choice of depends only on the elements of while the constants and depend also on . be as defined in Theorem Lemma A2: Let , , , and A1 and let inequalities (141) and (142) hold with some positive , and , , . Let be constants ,
(151) 19We
n.
follow the convention that
f
= 0 and
f
= 1 if n
>
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where for
where
is any constant smaller than and to satisfy
, and choosing
(156)
(157) with
being any positive constant, we have
(158) (152) with
, is a function of only
and
. Note that .
Since
(153) Using (151) and (153)
Arbitrary positive constants and are picked and are computed recursively using (156). is given by (157) and are found recursively using , , and , (150). Noting that . (144) is satisfied with as in Proof of Theorem A1: Picking , (155), the inequality (145) is satisfied by Lemma A1, since , . Choosing , to satisfy as in (150) and (157), the (156), and picking , inequality (158) is satisfied implying that (144) is also satisfied. be an matrix for any in some Theorem A2: Let with entries on the upper diagonal and design set on the last row, i.e., freedoms
(159) with
zeros
everywhere else. be an and any . Let , , positive constants , , and exist such that
Let matrix for any ,
. Let ,
(154)
(160)
(155)
(161) (162)
Choosing20
20Note that, by assumption,
'
assumes the same sign for all arguments.
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for all
and all where , , are nonnegative functions of . be independent of its argument. Let the sign of each , functions , and positive Then, a matrix exist to satisfy constants , , and
(163) (164) and with each being for all chosen to be a linear constant-coefficient combination of . matrix with Proof of Theorem A2: Let be the 1’s on the antidiagonal, i.e., with zeros everywhere else. Premultiplication and postmultiplication of a matrix by reverses the order of the rows and columns, reand , spectively. Defining and are of the form considered in Theorem A1 with , , and . Let where and are diagonal matrices with each diagonal element taking values matrix with element in { 1, 1} and is the being if and zero otherwise. Note that , , satisfy the assumptions of Theorem A1. Hence, apand , a matrix , functions plying Theorem A1 with , and positive constants , , and are found such that (144) and (145) are satisfied for all and with , , and replaced by , , and , respectively. Furthermore, do not depend on and and are linear constant-coefficient combi. Letting nations of and , (144) and (145) yield
(165) (166) Let an 1 if
and where denotes diagonal matrix with the diagonal entry being and 1 otherwise. Then,
from (161). Hence, using (165) and (166), (163) and (164) are , , and obtained with . Remark 9: Theorem A1 provides sufficient conditions for robust solvability of the pair of coupled Lyapunov equations (144) and (145). Assumptions (141) and (142) impose a CUDD [9] condition on and . It is shown in [24] that in the case where is zero, the conditions (141) and (143) are necessary and sufficient for satisfying (144) and (145). Theorem A1 is stated in the observer context with Theorem A2 being its dual in the controller context. REFERENCES [1] R. Marino and P. Toméi, Nonlinear Control Design. Geometric, Adaptive, Robust. Upper Saddle River, NJ: Prentice-Hall, 1995.
[2] M. Krstic´ , I. Kanellakopoulos, and P. V. Kokotovic´ , Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [3] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 2003. [4] L. Praly, “Lyapunov design of a dynamic output feedback for systems linear in their unmeasured state components,” in Proc. IFAC Nonlinear Control Systems Design Symp., Bordeaux, France, June 1992, pp. 63–68. [5] J.-B. Pomet, R. M. Hirshorn, and W. A. Cebuhar, “Dynamic output feedback regulation for a class of nonlinear systems,” Math. Control, Signals, Syst., vol. 6, pp. 106–124, 1993. [6] S. Battilotti, “Global output regulation and disturbance attenuation with global stability via measurement feedback for a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 315–327, Mar. 1996. , “A note on reduced order stabilizing output feedback controllers,” [7] Syst. Control Lett., vol. 30, no. 2–3, pp. 71–81, 1997. [8] P. Krishnamurthy, F. Khorrami, and Z. P. Jiang, “Global output feedback tracking for nonlinear systems in generalized output canonical form,” in Proc. Amer. Control Conf., Arlington, VA, June 2001, pp. 4241–4246. [9] , “Global output feedback tracking for nonlinear systems in generalized output-feedback canonical form,” IEEE Trans. Automat. Contr., vol. 47, pp. 814–819, May 2002. [10] D. Bestle and M. Zeitz, “Canonical form design for nonlinear observers with linearizable error dynamics,” Int. J. Control, vol. 23, pp. 419–431, 1981. [11] A. J. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, no. 1, pp. 47–52, 1983. [12] R. Marino and P. Toméi, “Global adaptive output feedback control of nonlinear systems, part I: linear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 17–32, Jan. 1993. [13] G. Besancon, “State affine systems and observer-based control,” in Proc. IFAC Nonlinear Control Systems Design Symp., Enschede, The Netherlands, July 1998, pp. 399–404. [14] L. Praly and I. Kanellakopoulos, “Output feedback asymptotic stabilization for triangular systems linear in the unmeasured state components,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, Dec. 2000, pp. 2466–2471. [15] P. Krishnamurthy and F. Khorrami, “Robust adaptive control for nonlinear systems in generalized output-feedback canonical form,” Int. J. Adapt. Control Signal Processing, vol. 17, no. 4, pp. 285–311, May 2003. [16] L. Praly, “Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate,” in Proc. IEEE Conf. Decision and Control, Orlando, FL, Dec. 2001, pp. 3808–3813. [17] H. K. Khalil and A. Saberi, “Adaptive stabilization of a class of nonlinear systems using high-gain feedback,” IEEE Trans. Automat. Control., vol. AC-32, pp. 1031–1035, Nov. 1987. [18] A. Ilchmann, “High-gain adaptive control: an overview,” in Proc. IEE Colloq. Adaptive Control, London, U.K., June 1996, pp. 1–4. [19] A. Ilchmann and E. P. Ryan, “On gain adaptation in adaptive control,” IEEE Trans. Automat. Contr., vol. 48, pp. 895–899, May 2003. [20] A. R. Teel and L. Praly, “Global stabilizability and observability imply semi-global stabilizability by output feedback,” Syst. Control Lett., vol. 22, no. 5, pp. 313–325, 1994. [21] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Automat. Contr., vol. 41, pp. 177–188, Feb. 1996. [22] C. Qian and W. Lin, “Output feedback control of a class of nonlinear systems: A nonseparation principle paradigm,” IEEE Trans. Automat. Contr., vol. 47, pp. 1710–1715, Oct. 2002. [23] E. Bullinger and F. Allgöwer, “Adaptive -tracking for nonlinear systems with higher relative degree,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, Dec. 2000, pp. 4771–4776. [24] P. Krishnamurthy, F. Khorrami, and R. S. Chandra, “Global high-gain-based observer and backstepping controller for generalized output-feedback canonical form,” IEEE Trans. Automat. Contr., vol. 48, pp. 2277–2284, Dec. 2003. [25] E. D. Sontag, “On the input-to-state stability property,” Eur. J. Control, vol. 1, pp. 24–36, 1995. [26] L. Praly and Z. P. Jiang, “Stabilization by output feedback for systems with ISS inverse dynamics,” Syst. Control Lett., vol. 21, no. 1, pp. 19–33, 1993. [27] Z. P. Jiang, A. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Math. Control, Signals, Syst., vol. 7, pp. 95–120, 1994. [28] F. Mazenc, L. Praly, and W. P. Dayawansa, “Global stabilization by output feedback: examples and counterexamples,” Syst. Control Lett., vol. 23, no. 2, pp. 119–125, 1994.
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[29] E. D. Sontag and A. Teel, “Changing supply functions in input/state stable systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1476–1478, Aug. 1995. [30] P. Krishnamurthy and F. Khorrami, “Global robust control of feedforward systems: state-feedback and output-feedback,” in Proc. IEEE Conf. Decision and Control, Maui, HI, Dec. 2003, pp. 6145–6150. [31] , “A high-gain scaling technique for adaptive output feedback control of feedforward systems,” IEEE Trans. Automat. Contr., vol. 49, pp. 2286–2292, Dec. 2004. , “Generalized adaptive output-feedback form with unknown pa[32] rameters multiplying high output relative-degree states,” in Proc. IEEE Conf. Decision and Control, Las Vegas, NV, Dec. 2002, pp. 1503–1508. [33] , “A dual high gain controller for the uncertain generalized outputfeedback canonical form with appended dynamics driven by all states,” in Proc. Amer. Control Conf., Denver, CO, June 2003, pp. 4766–4771. [34] L. Praly and Z. P. Jiang, “Linear output feedback with dynamic high gain for nonlinear systems,” Syst. Control Lett., vol. 53, no. 2hi, pp. 107–116, 2004.
Prashanth Krishnamurthy received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Chennai, India, in 1999, and the M.S. in electrical engineering from Polytechnic University, Brooklyn, NY, in 2002. He is currently working toward the Ph.D. degree at Polytechnic University. He is the coauthor of 30 journal and conference papers, and of Modeling and Adaptive Nonlinear Control of Electric Motors (New York: Springer-Verlag, 2003). His research interests include robust and adaptive nonlinear control with applications.
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Farshad Khorrami was born in Iran in 1962. He received the B.S. degrees in mathematics and electrical engineering, the M.S. degree in mathematics, and the Ph.D. degree in electrical engineering, all from The Ohio State University, Columbus, in 1982, 1984, 1984, and 1988, respectively. He is currently a Professor of electrical and computer engineering at Polytechnic University, Brooklyn, NY, where he started as an Assistant Professor in September 1988. His research interests include adaptive and nonlinear control, large scale systems and decentralized control, unmanned autonomous vehicles, smart structures, robotics and high-speed positioning applications, and microprocessor-based control and instrumentation. He has published more than 150 refereed journal and conference papers and currently holds ten U.S. patents, with two more pending. He has developed and directed the Control/Robotics Research Laboratory (CRRL) at Polytechnic University. He is also an author of a recently published book entitled Modeling and Adaptive Nonlinear Control of Electric Motors (New York: Springer-Verlag, 2003). Dr. Khorrami has served on the program committees of several conferences, and has been a Member of the Conference Editorial Board of the IEEE Control Systems Society.
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Dynamic High-Gain Scaling: State and Output Feedback With Application to Systems With ISS Appended Dynamics Driven by All States Prashanth Krishnamurthy and Farshad Khorrami
Abstract—In this paper, we propose a dynamic high-gain scaling technique and solutions to coupled Lyapunov equations leading to results on state-feedback, output-feedback, and input-to-state stable (ISS) appended dynamics with nonzero gains from all states and input. The observer and controller designs have a dual architecture and utilize a single dynamic scaling. A novel procedure for designing the dynamics of the high-gain parameter is introduced based on choosing a Lyapunov function whose derivative is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The system is allowed to contain uncertain terms dependent on all states and uncertain appended ISS dynamics with nonlinear gains from all system states and input. In contrast, previous results require uncertainties to be bounded by a function of the output and require the appended dynamics to be ISS with respect to the output, i.e., require the gains from other states and the input to be zero. The generated control laws have an algebraically simple structure and the associated Lyapunov functions have a simple quadratic form with a scaling. The design is based on the solution of two pairs of coupled Lyapunov equations for which a constructive procedure is provided. The proposed observer/controller structure provides a globally asymptotically stabilizing output-feedback solution for the benchmark open problem proposed in our earlier work with the provision that a magnitude bound on the unknown parameter be given. Index Terms—Adaptive, high-gain, input-to-state stable (ISS), nonlinear, output feedback, scaling.
I. INTRODUCTION
T
HE class of systems considered in this paper is the uncertain generalized output-feedback canonical form coupled to appended dynamics having nonzero gains from all states
.. . Manuscript received July 31, 2002; revised July 31, 2003, June 8, 2004, and September 2, 2004. Recommended by Associate Editor S. Ge. This work was supported in part by the National Science Foundation under Grant ECS-9977693. Earlier versions of this paper were presented in parts at the 2002 IEEE Conference on Decision and Control, Las Vegas, NV, and the 2003 American Control Conference, Denver, CO. The authors are with the Control/Robotics Research Laboratory (CRRL), Department of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: [email protected]; khorrami@ smart.poly.edu). Digital Object Identifier 10.1109/TAC.2004.839235
.. .
.. .
.. . (1) is the state, is the output, is the input, and is the state of the appended dynamics. , , , are known scalar real-valued continand , , and uous functions of their arguments. , , , are time-varying scalar real-valued uncertain functions.1 is the relative degree of the system. It is assumed that sufficient conditions (e.g., local Lipschitz property) on and needed for local existence and uniqueness of solutions of and the (1) are satisfied. We consider the state-feedback output-feedback problems for the system (when the dynamics are not present) and the output-feedback problem with the appended dynamics. Among the control design methodologies in the literature for various classes of nonlinear systems (see [1]–[3] and the references therein), backstepping and its robust and adaptive variants are particularly suited for systems of a lower triangular depends on (strict feedback) form wherein the dynamics of . While (1) is not, in general, in lower triangular , no dynamics), the form in the state-feedback case ( state-feedback control design for (1) can be carried out using a variant of robust backstepping under certain assumptions on and . , no dynamics) case, various In the output-feedback ( approaches have been proposed under different sets of assumptions. In [4]–[7], solutions were proposed assuming existence of control Lyapunov functions with certain properties. In [4], where
= 1 . . . +1
= 1 ...
1q , i ; ; s , and , i ; ; n, can depend on all the states and the input. However, q and are shown in (1) to be functions only of subsets of the state to emphasize the state dependence of the bounds to be introduced on these functions.
0018-9286/04$20.00 © 2004 IEEE
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weak linear detectability and the existence of a control Lyapunov function which satisfies certain growth conditions in the unmeasured state variables for the state-feedback stabilization problem were required. [5]–[7] address the problem under the assumption of the existence of control Lyapunov functions of a specific where is the unstructure ( measured state component) for the state-feedback stabilization problem and the output-injection problem for the dual system. Assuming that are known and linear in the unmeasured , state components, i.e., the generalized output-feedback canonical form [8], [9] is obtained which reduces to the standard output-feedback canonical , are conform [10]–[12] under the assumption that stants, i.e., all nonlinearities are only output-dependent. This re’s are quirement was relaxed in [13] to the assumption that bounded. This restriction was removed in [14] using an observer with gains generated through a matrix difof order ferential Riccati equation. In [9], assuming that a constant positive–definite matrix can be found to satisfy a certain inequality, and the oba solution was proposed of dynamic order server structure recovered the standard linear reduced-order observer when specialized to the case of linear systems. It was shown in [9] that a sufficient condition for the existence of a matrix satisfying the required inequality is the cascading upper diagonal dominance (CUDD) condition requiring the upper dito be larger than and (to agonal terms within constant factors). Adaptive extensions of the results in [9] and [14] were considered in [15] assuming that parametric uncertainties appear in output-dependent terms. In [16], assuming for and that , , that are incrementally linearly bounded in unmeasured states, the matrix differential Riccati equation in [14] was collapsed to a scalar differential Riccati equation driven by . High-gain techniques both for controller and observer design have been investigated in the literature. The well-known adaptive high-gain controller given in its basic form by , is applicable to minimum-phase systems with relative-degree one (see [17]–[19] and the references therein). Static high-gain scaling based observers [20], [21] which introduce with a constant provide semiglobal observer gains solutions. The observer analysis utilizes scaled observer errors (or ) with being the estimation error of the state. A global high-gain observer and controller with constant were considered in [22] for systems of form (1) gains without the appended dynamics, with with known constant , , , and . A global high-gain observer and controller with gains being powers of were proposed in [23] for linear systems with appended stable nonlinear zero dynamics and inputmatched nonlinearities. The dynamic high-gain scaling based observer in [16] differs from previous results in two features. First, the dynamics of are a scalar differential Riccati equation driven by guaranteeing boundedness of if remains ). bounded (which is not guaranteed by the dynamics This feature is important since the Lyapunov functions used in most high-gain designs are weak in the sense that the boundedness of the Lyapunov function does not directly imply boundedness of the high-gain parameter . Instead, the boundedness
of the high-gain parameter has to be inferred from the boundedness of the Lyapunov function and the dynamics of the high-gain parameter. In classical dynamic high-gain designs where , the boundedness of the high-gain parameter renorm of the output signal which might not hold quires finite under additive perturbations. A second feature in [16] is that an is introduced in the scaled observer additional scaling error definition. The parameter is chosen to be a positive constant large enough to decouple a pair of coupled Lyapunov equations. In [24], it was shown that the pair of coupled Lyapunov equations can be solved simultaneously without requiring the additional scaling and that the introduction of the additional scaling provides a weaker observer error convergence guarantee . It was also shown in [24] that a negative can be used to improve the convergence guarantee. In this paper, we propose a new paradigm for dynamic highgain scaling based observer and controller design. The basic design philosophy is to first attain a high-gain observer and controller (which incorporates an adaptation parameter in the adaptive case) to stabilize the nominal system obtained by omit, dynamics, the inverse dynamics ting the , with states , and the uncertain functions , . This nominal system is a chain of nonlinear integrators (2) The high-gain scaling asymptotically achieves an approximation of the system (1) by the nominal system (2) when the highgain scaling parameter is increased [24]. Thus, as shown in [24], the high-gain scaling induces a CUDD [9] condition by amplifying the upper diagonal terms. The controller design freedoms and the dynamics of the high-gain parameter (and the adaptation parameter in the adaptive case) are then chosen to achieve closed-loop stability. The Lyapunov function used is the sum of scaled Lyapunov functions associated with appended dynamics and inverse dynamics, and scaled quadratic functions of observer errors, , observer estimates (with a nonlinear function of ), and adaptation error. The structure of the Lyapunov function is chosen such that the derivative of the Lyapunov function is negative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). The dynamics of are designed so that is large until becomes large. This is a new design philosophy for dynamics of the high-gain parameter and unlike [16], the dynamics of the high-gain scaling parameter are not required to be of the form of a Riccati equation. However, the boundedness of is ensured by boundedness of the output, the parameter estimate, and the derivative of the parameter estimate, which in turn is inferred from boundedness of the Lyapunov function. The controller and the observer in the output-feedback case have a dual architecture and utilize a single dynamic scaling. Unlike [16], an additional scaling can not be incorporated to decouple the pairs of coupled Lyapunov equations that arise in the design since such an additional scaling with a positive appears to be an obstacle in a dual observer/controller design due to the weaker observer error convergence guarantee obtained. In the designs in this paper, two pairs of coupled Lyapunov equations appear that extend the ones in [24] and the solutions are presented in the Appendix.
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In contrast to previous results, the system (1) is allowed to contain uncertain functions dependent on all the states and the input and also input-to-state stable (ISS) [25] appended dynamics driven by all the states and the input. The previously available results require that: 1) unknown parameters appear multiplied with output nonlinearities or uncertain terms can be bounded as products of unknown parameters and output nonlinearities; and 2) the appended ISS dynamics have nonzero gain only from the output , i.e., a Lyapunov function for the appended dynamics exists such that the derivative of the function of Lyapunov function is the difference of a class and a class function of [15], [26], [27]. These assumptions are essentially due to the fact that, in backstepping based can be efficiently assigned, the designs, while the gain to same is not true for the gains to the other states which appear in the Lyapunov function and its derivative in increasingly complicated combinations. Both these requirements are relaxed using the dynamic high-gain scaling technique advocated in this paper. The basic controller design technique is introduced with a state-feedback design for system (1) without the appended dynamics in Section II. The dual high-gain observer/controller design for the output-feedback case for system (1) without the appended dynamics is presented in Section III. In Section IV, the advocated design methodology is applied to adaptive output-feedback for systems with the appended ISS dynamics . II. STATE-FEEDBACK CONTROL The dynamics (1) without the appended written as2
dynamics can be
(3) where, for notational simplicity, we have introduced the dummy variables and for . A. Assumptions
with being a dummy variable are bounded-inputbounded-state (BIBS) stable with the states and the inputs where5 . Assumption A3: The functions , , can be bounded as (5) for all
and where and , , , are known continuous nonnegative func, , , and tions. Positive constants , , exist such that for all
(6) exist such that Assumption A4: Positive constants , , . Assumption A5: A continuous nonnegative function exists such that . Remark 1: For simplicity, it is assumed that the same function serves in the bounding of the terms on the left-hand side of (5). To remove possible conservativeness of the resulting design, different ’s can be utilized in (5). The design in that case can be carried out along the same lines as presented here. Furcan be taken to be a function of both and the thermore, time as long as it is bounded uniformly in time as a function of . The bounds in (5) can be generalized by allowing additive positive constants. In this case, the proposed design provides practical stabilization results. Thus, practical tracking can be achieved by the approach presented here by converting the tracking problem into a stabilization problem using the change and , , of coordinates is a bounded reference trajectory (with bounded where first derivative). B. Controller Design
as output), controlAssumption A1: Observability (with lability, and uniform relative degree of (3), i.e., a constant exists such that3 , , and for all . Assumption A2: The dynamics [the inverse dynamics4 of (3)]
Define (7) is a design freedom to be chosen later.6 where is a dynamic high-gain parameter whose dynamics will also be chosen later. The dynamics chosen for will ensure that for all time . The dynamics of are given by7
(4) 2While can depend on all states and the input, a triangular state dependence is shown in (3) to highlight the structure of the bounds in Assumptions A3 and A3’. 3j j denotes the Euclidean norm of a vector . In particular, if is a scalar, j j denotes its absolute value. 4The inverse dynamics are the dynamics of the states of the zero dynamics, i.e., x ; ;x , when viewed as driven by the states x ; ;x and the output-zeroing input v u y x = y . The zero dynamics are obtained when v ; ; v are zero in (4).
7=[
...
]
= + ...
()
( ... ()
)
(8) 5In the output-feedback case, the required assumption on the inverse dynamics is Assumption A2 with y being the empty vector. 6This form of is picked to ensure certain bounds as will be seen later in the stability analysis. 7For notational convenience, we drop the arguments of functions whenever no confusion will result.
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where denotes the partial derivative of with respect to its argument evaluated at . The control input is designed as
C. Stability Analysis A Lyapunov function9 is introduced as (14)
(9) with
being design functions. The dynamics of under the control law (9) can be written as (10)
where
is the
matrix with
element
where is a symmetric positive–definite matrix to be defined later. Note that since contains an term which is not scaled implies boundedness of . The dyby , boundedness of namics of will be designed such that boundedness of is guaranteed by boundedness of . Boundedness of all closed-loop states then follows from boundedness of and . The presence term in is made possible by the design of an unscaled freedom which provides a crossterm between and . Differentiating (14)
(11) with zeros elsewhere, and8 (15)
(12) Using Assumption A3
(13) Remark 2: Note the simplicity of the control law (9). The design freedoms in the controller (i.e., the parameters and functions in the controller that are free to be picked by the designer) , , and the dynamics of the high-gain scaling paare rameter . The construction in the Appendix provides a possible choice of explicitly. The first term in the bracket IN , a state of the in(9) simply cancels out the term involving verse dynamics, which appears matched with the control input. The remaining terms can be interpreted as a “linear” feedback with gains dependent both on the plant state and of , are the controller state . Noting from (7) that , simply scaled versions of , and is a scaled version of incorporating an extra term dependent on , the control law can be interpreted as essentially having three components: a term to cancel the state of the inverse dynamics that appears matched with the control input, a nonlinear -dependent term, and a state-dependent linear feedback of . As will be seen in the stability analysis below, a Lyapunov function quadratic (with the scaling ) can be used to demonin , strate closed-loop stability. These two features of the control design methodology, i.e., a simple form of the control law involving the three components highlighted above and an essentially quadratic Lyapunov function are retained in the outputfeedback case in Section III and in the case of output-feedback with ISS appended dynamics in Section IV.
2
notation diag(T ; . . . ; T ) denotes an m m diagonal matrix with ) denotes an diagonal elements T ; . . . ; T . Also, lowerdiag(T ; . . . ; T m m matrix with the lower diagonal entries being T ; . . . ; T and zeros elsewhere. 8The
2
(16) If
, (16) yields
(17) Using Assumptions A1, A3, and A4, along with (17) and ap, pealing to Theorem A2 in the Appendix with10 , , , , , , , and , a symmetric positive–definite matrix , positive constants , , and , and continuous functions are found such that11
(18) (19) for all , , and such that . The proof of Theorem A1 along with the duality transformation in the proof of Theorem A2 provides a constructive procedure to find , , , , and . Note that the inequality (18) requires the while the inequality (19) which property involves only constant matrices is true irrespective of the values 9By a Lyapunov function, we mean a differentiable function V ( ) that satisfies V (0) = 0 and V ( ) > 0 for all = 0 in the domain of definition. Note that by the dynamics to be introduced for r , we have r (t) 1 for all time t. 10I denotes the identity matrix of appropriate dimensions. 11The positive term on the right-hand side of (18) is generated by the term 0(x )[ + ] x =r in the bound (17).
6
j jj j
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of the states. However, by Theorem A2, the following inequality which is weaker than (18) is valid for all values of the states (20) Using straightforward algebra,12
(26) with , , and being nonnegative design parameters, the inequalities (15) and (19)–(21) imply that (27)
(21) where is a positive design parameter. picked to be a continuous function such that13
is
From (25) and (27), we see that can be made nonpositive by choosing either large (i.e., ) or large (i.e., ). This implies that to stabilize the system, the dynamics of should be chosen such that is large till becomes large, i.e., should be made to increase to a large enough value fast enough. This is achieved by choosing the dynamics of to be of the form (28)
(22) where is a positive function of bounded below in magnitude by a positive constant. If satisfies
with being a nonnegative constant smaller than and being a nonnegative continuous function such that the following properties are true: a) b)
(23) c)
(24) we have, using (15), (18), and (21)
(25) On the other hand, for any value of
satisfies
is a square symmetric matrix, (P ) denotes its largest eigenvalue. is continuous and bounded below in magnitude by a positive constant, it assumes the same sign for all arguments. 12If
P
, if
13Since
a nonnegative function such that satisfying the holds inequality the inequality holds.
(29) Remark 3: Property b) can be satisfied by picking such for with being any positive constant that yielding . Since for all , and such that property c) is easily satisfied by picking . Property b) is required to infer boundedness while the third property ensures of from boundedness of that for all time . In the special case in which depends only on , the function can be reduced to the and the properties above form can be satisfied by picking, for instance, to be identically unity and , resulting in the dynamics (30) The dynamics (30) which has the form of a scalar Riccati equation was proposed in [16] in the context of high-gain based observer design. The advantage of the general dynamics (28) is is allowed to be an arbithat is allowed to be zero and trary function of all the states while still retaining the crucial property (whose significance will be apparent in the following stability analysis) that the boundedness of follows from the alone. This freedom in allows Assumpboundedness of which are functions of the tion A3 to include functions entire system state . The fact that can be taken to be zero is
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vital in the output-feedback design with ISS appended dynamics in Section IV. With the dynamics of given by (28), using (25) and (27), with chosen smaller than , for any value of and with , the Lyapunov inequality
A. Assumptions Assumption A3 : A nonnegative continuous function and functions , , are known such , , and all that for all real numbers
(31) is nonincreasing and is, hence, holds. This implies that bounded on the maximal interval of existence of solutions . Boundedness of implies boundedness of and . The second property of listed in (29) implies that boundedness of follows from boundedness of and property for all . Boundedness c) in (29) implies that . Invoking the of and implies boundedness of BIBS assumption on inverse dynamics in Assumption A2, are also bounded. Thus, all closed-loop signals are bounded. Hence, solutions do not escape in finite time ). Furthermore, from (31), and , and hence, (i.e., go to zero asymptotically as . The foregoing Lyapunov analysis provides Theorem 1. Theorem 1: Under Assumptions A1–A5, given any initial for the plant state and for the conconditions troller state, the closed-loop system formed by (3), (9), and (28) and the possesses a unique solution on the time interval following properties hold. 1) All closed-loop signals are bounded and , . 2) If Assumption A2 is strengthened to include the minimum-phase condition that the inverse dynamics (4) is , asymptotically stable with then , . If the inverse dynamics are further required to be ISS, then with the dynamics of as in (30), the equilibrium is globally asymptotically stable where . The proof of the second statement in Theorem 1 is straightforward using composite Lyapunov functions similar to the analysis in Section IV. Under the minimum-phase assumption on the inverse dynamics, by continuity of and , it follows from the dynamics (28) that asymptotically converges to . By the set the properties (29), this set contains at least one value in the . In the special case when satisfies the interval Riccati equation (30), this set consists of the single point . III. OUTPUT-FEEDBACK CONTROL
(32) Assumption A4 : Positive constants
and
exist such that
(33) (34) Assumptions A3 and A4 are contained in Assumptions A3 and A4 , respectively, with the additional requirements in Assumptions A3 and A4 being imposed to handle the output-feedback case. The inequality (33) which is identical to Assumption A4 is required in the controller design and its dual (34) is required in the observer design. Assumption A5 is is then always satisfied in the output-feedback case since a function of only. The output-feedback solution below is based on Assumptions A1, A2, A3 , and A4 . and be the matrices deRemark 4: Let fined in (11) and (12), respectively. Using Assumptions A1 and A4 , and appealing to Theorem A2 in the Appendix with , , an arbitrary vector in , , , , , , , , and , a symmetric positive–definite matrix , positive constants , , and , and continuous are found such that functions
In this section, we consider the output-feedback control of system (3) with being the output.14 In this case, we need to strengthen Assumptions A3 and A4 as follows. (35) for all
. Let
14The
design for the more general class (1) which includes appended ISS dynamics is considered in Section IV.
(36)
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and let
be the
matrix with
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entry
B. Observer Design A reduced-order observer with states given by16
is
(37) with zeros everywhere else. Using Assumptions A1 and A4 , , and applying Theorem A1 in the Appendix with , an arbitrary vector in , with being a positive constant, , , , , , , , and , a symmetric positive definite matrix , positive constants , , and , and continuous functions are found such that15
(41) where is a time-varying signal whose dynamics will be deare design functions of of the form signed later, and
(42) (38)
for all where . Furthermore, by are linear constant-coefficient combiTheorem A1, nations of . Hence, using Assumption A4 , a positive constant exists such that
being functions chosen as in Remark 4. The rewith duced-order observer in (41) is similar, in structure, to the observer in our earlier work [9] with modifications to incorporate the dynamic high-gain scaling introduced in the spirit of [16]. Defining the observer errors as (43)
(39)
the observer error dynamics can be written in the form
Remark 5: A particular case in which Assumption A3 is satare of the form isfied is if (44) with being a dummy variable. The scaled observer errors are now defined as (40) (45) where and are known matrix functions of of dimenand , respectively, and sions is a vector of time-varying uncertain parameters with a known magnitude bound. The assumptions in [16] and [22] are obtained as special cases of Assumption A3 . In [16], it is assumed that . In [22] (where it is assumed that ), is required with to be bounded as being a constant. Furthermore, both [16] and [22] assume that , , a special case of Assumption A4 .
0j
j
~
j
j
j
0j
j 0
. The dynamics of the scaled
(46) where, for notational convenience, we have introduced
(47)
j 0
A1 implies that (38) is satisfied with [ I C C ] on the right hand side of the first inequality in (38). Since is bounded below by , I I with = .
15Theorem
with the dummy variable observer errors are given by
f
16For
simplicity of notation, we introduce the dummy variables x ^ = 0.
=g
=
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The dynamics (46) can be written in the matrix form as (48) where
, , and
with and
are defined in (36) and (37), respectively. If the dynamics of are chosen as in (28) with being identi, with being a positive cally unity and a continuous positive function, it can be constant and seen that the Lyapunov function with chosen as in Remark 4 satisfies
(51) The control law is picked to be of the form
(52)
(49) is negative if is In the absence of , the derivative of . If Assumption A3 is strengthened chosen smaller than to the requirement (the case in [16]) that and , then . Hence, . renders Choosing negative. Thus, if the states remain bounded (which implies remains bounded), the observer errors go to zero asymptotically. With the more general bound in Assumption A3 , the observer design renders the scaled observer errors ISS with respect to and the scaled observer estimates , . The closed-loop stability will be inferred in Section III-D through a composite Lyapunov analysis in conjunction with the controller designed in Section III-C. C. Controller Design Introduce the signals
,
, defined as
(50) where The dynamics of
where are functions chosen as in Remark 4. Note that (52) is essentially the same as (9) with unavailable states , (51) can be replaced by estimates. Defining written in matrix form as (53) where
with , , , and and are matrices defined in (11) and (12), respectively. Remark 6: Note that the observer and controller designs in Sections III-B and III-C depend only on the nominal system (2) and the nominal inverse dynamics system ,
(54) The choice of , also depends only on the nominal systems. The robustness to uncertain terms and appended dynamics will be achieved through the design function and the dynamics of . If the inverse dynamics system is stable, the observer (41) and the controller (52) globally asymptotically stabilize the nominal system (2) if is picked to be an appropriate positive constant and is suitably chosen. For sim. In this case, and plicity, assume for now that can be chosen to be positive constants such that
is a design freedom to be chosen later. are (55) Global asymptotic stability of the nominal system can be verified using the Lyapunov function (56) where and are positive–definite matrices as defined in Remark 4 and is a positive constant larger than . In this case, and are zero,
KRISHNAMURTHY AND KHORRAMI: DYNAMIC HIGH-GAIN SCALING
, and
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. Differen-
Using
tiating (56), (57) In Section III-D, it is shown that stability of the overall system (3), (41), and (52) can be ensured by appropriately choosing and the dynamics of the high-gain the design function parameter .
(62) , we have
and the decomposition
D. Stability Analysis Closed-loop stability can be analyzed using the Lyapunov function (58) where is a positive constant. Differentiating (58)
(63) Choosing
to satisfy (64)
and using (39), we have (59) and the dyWe need to pick the function namics of to make negative definite. To achieve this, we derive bounds below for various terms in (59). , can be overbounded as Using Assumption A3 , if
(65) Similarly
(66) The function
is picked such that
(60) Therefore
(67) (61)
with
being a positive function of
.
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Using (35), (38), (61), (63), (65), (66), and (67), (59) may be rewritten as
(68) where
(69) From (68), it is seen that the inequality
(70) is achieved under either of the following two cases: 1) and where ; 2) and where ; are nonnegative design parameters. As in where , , and Section II, the interpretation is that can be made nonpositive by choosing either or large, i.e., to stabilize the system, should be made to increase to a large enough value fast enough. This is achieved by designing the dynamics of to be of the form (28) with having the properties (29). In the output-feedback case considered in this section, is a function of only and and can be taken to be of the form so that, as noted in Remark 3, can be taken to be identically unity and picking , , the dynamics (30) are obtained. Choosing to be a positive constant small enough such that (71) (70) reduces to (72) From (72), the boundedness of on the maximal interval of can be inferred. This implies the existence of solutions boundedness of , , and . The boundedness of implies the boundedness of . Invoking the BIBS Assumption A2 on the inverse dynamics, all the closed-loop signals are seen to be . Using bounded. Hence, solutions exist for all time and (72), it can be seen that , , and tend to zero asymptotically. , it follows that , Noting from (42) that also go to zero asymptotically. The preceding Lyapunov analysis provides Theorem 2. Theorem 2: Under Assumptions A1, A2, A3 , and A4 , given for the plant state and any initial conditions
for the controller state with , the closed-loop system formed by (3), (41), (52), and (28) possesses a unique solution and the following properties hold. on the time interval , 1) All closed-loop signals are bounded, , and , . 2) If Assumption A2 is strengthened to include the minimum-phase condition that the inverse dynamics (4) is , asymptotically stable with , , then , . If the inand verse dynamics are further required to be ISS, then as in (30), the equilibrium with the dynamics of is globally . asymptotically stable where The design procedure is summarized as follows. , , and symmetric positive1) Obtain definite matrices and to satisfy (35) and (38). small enough to satisfy (71) and pick 2) Pick to satisfy (67). Pick nonnegative constants and . 3) Implement the dynamics of in (28). 4) Implement the observer dynamics (41). as in (50). The control law is given by 5) Define (52). E. Illustrative Examples Example 1: In [15], we proposed the third-order system
(73) is as a benchmark open problem. In (73), the state, is the input, is the output, and is a constant unknown parameter. System (73) is of a very simple form containing a single nonlinearity and a single unknown parameter. If any of the components of the single nonlinearity are dropped, the solution can be obtained using available techniques. If is known, [16], [24] provide solutions. If is removed, the is removed, the system is in standard system is linear. If output-feedback canonical form. However, hitherto, there has been no control design approach which can provide a global asymptotic output-feedback result for this system even with the knowledge of an a priori magnitude bound on . The technique proposed in this paper solves this problem utilizing such a mag. Another interesting observation is the fact nitude bound that the following system obtained from (73) by replacing with can not be stabilized by continuous dynamic output feedback as may be shown by an argument similar to [28]
(74) System (73) belongs to the class of systems (3) with , , , , ,
,
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and . Assumption A1 is satisfied with . Assumptions A2 and A4 are trivially satisfied. Assumption A3 is satand . The isfied with coupled Lyapunov inequalities (35) and (38) are satisfied by , , , , choosing, for instance, , and
. In this case,
,
, , , , and . Thus, and . A reduced-order observer is constructed for this system as
. Assumption A1 is satisfied with inverse dynamics
. The (84)
are BIBS stable with state and inputs and . Hence, Assumption A2 is satisfied. Assumption A3 is satisfied with and . Assumption A4 is satisfied with . In this case
(75) Defining (85)
(76) the control law for is picked as chosen to be , (71) is satisfied with , , and , , , and
. is . Picking
and (35) and (38) are satisfied with
. Picking are obtained as
(77)
(78)
(86)
(79) Picking
, the dynamics of
Note that this choice of , , and satisfies (39) with . The observer and controller are given by
are given by (80)
Global asymptotic stability can be demonstrated using the Lyapunov function (81) Differentiating (81) (82) (87) , all the closed-loop states (except ) go to zero Thus, as asymptotically. Example 2: Consider the fourth-order system
where namics of
,
. The dy-
are as given in (28). IV. ADAPTIVE OUTPUT-FEEDBACK WITH APPENDED ISS DYNAMICS
(83) System (83) belongs to the class of systems (3) with , , , , , , , , and
, ,
In this section, we consider the application of the dynamic high-gain scaling technique to the output-feedback control design for system (1) which includes the appended which are driven by all states and dynamics , input. The design in this section is based on Assumptions A1, A2 , A3 , A4 , and A6.
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A. Assumptions
, are class functions, , , are continuous nonnegative functions, and is an unknown nonnegative constant. Furthermore, nonnegative constants and exist such that17 for all , where
Assumption A2 : The inverse dynamics of (1) with being a dummy variable
,
(95)
(88) are ISS with state and input function satisfying
with ISS Lyapunov
(89)
The local order estimates
, , and
as
hold.18 can be Remark 7: Additive class functions of allowed in (93) and (94). This case can be transformed back to the case considered here (under mild local-order restrictions) by and forming a new considering as the collection whose derivative satisfies (93) and (94). Lyapunov function This is easily seen from the fact that a cascade of ISS systems is ISS. B. Observer and Controller Design
where is a class function, and , are nonnegative continuous functions, and is an unknown nonnegand ative constant. Furthermore, nonnegative constants exist such that for all
Assumption A3 : Nonnegative continuous functions and , are known such that for all , , , , and
(90)
The reduced-order observer dynamics are given by (41) with , . Defining the observer errors as in (43) and the scaled observer errors as in (45), the dynamics of the scaled are given by (48) where observer error vector and are defined in (36) and (37), respectively, and
,
(96)
Introduce
with
,
, defined as
(97)
(91) with being an unknown nonnegative constant. , subsystems are Assumption A6: The , ISS with ISS Lyapunov functions satisfying
where is a dynamic adaptation parameter and is a design freedom to be chosen later. The control law is picked to be of the form (52) with being functions of chosen as in Remark 4. The dynamics of are given by (53) with and defined in (11) and (12), respectively, , , and
(92)
(93)
(98)
3
for i = s + 1 is a dummy variable identically equal to zero. this is not the case, practical stabilization results can be obtained [15], [26] by upper bounding to within additive constants the functions 3 , 3 , and by functions that satisfy the aforementioned local order estimates. 17
(94)
18If
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C. Stability Analysis The inverse dynamics with state the form (88) with
where tion A2
are of
(99)
(103)
. Hence, using Assump-
As in Section III-D, upper bounds are obtained for the terms in as (102) and (103)using Assumption A3 and
(104)
(100) Consider the observer and controller Lyapunov functions defined as (101) and are picked as in Remark 4. The dynamics of where the high-gain parameter and the adaptation parameter will be designed such that is larger than 1, is positive, and and are monotonically nondecreasing. will be designed such (and hence ) is positive. Differentiating that and defined in (101) and using (35) and (38)
(102)
(105)
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(106)
(113) where
(107)
(114)
(108)
(115)
(116)
(109)
(117) (118)
(110)
(111) where is a positive constant chosen to satisfy (64). Defining a composite Lyapunov function
(119) (112)
The term involving
in (113) can be upper bounded as
and using (100)–(111) (120) where
(121)
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By Assumption A6, and as . Hence, , , and as . Hence, a continuous nonnegative function exists . Using a reasoning similar to such that that used in the proof of [29, Th. 2], it is seen that the local order as implies the existence estimate of a new Lyapunov function , class functions and , and a nonnegative continuous function such that
The composite Lyapunov function for the overall system is defined as
where parameter estimator dynamics are designed as
(125) . The
(122) with
as , independent of , and as . Hence, a nonnegative continuous exists such that . function Using Assumption A6 ,
(126)
is initialized to be positive. By (126), so that remains positive for all time. Using (113), (122)–(124), and (126), and differentiating (125)
(123) for
. Also
(127) where
(128)
(129)
(124)
(130) (131)
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(132) The design freedom and by defining
is picked to dominate the functions
(133) being a positive function of lower bounded by a with positive constant. As in Sections II and III, it is seen from (127) that is negative if either the high-gain parameter or its derivative is large (compared to appropriate functions of , , and ). This is a consequence of introducing the scalings in the Lyapunov function definitions (112) and (125). This suggests that the dynamics of should be chosen such that is large until becomes large, thus ensuring that is negative for all time. Accordingly, the dynamics of are designed as
(134)
where
(135)
(136)
and is a nonnegative continuous function satisfying the prop, and being functions of , , and erties (29) with , and a function of , , , and . As noted in Remark 3, the property (b) in (29) can be satisfied by picking such that for with being any positive constant so that . Properties (a) and (c) in (29) are trivially satisfied. By (134), is monotonically nondefor all time. Furthermore, remains creasing so that bounded if , , and remain bounded. From (126), is a , and that is guaranteed to be bounded function of , , if , , , and are bounded. Hence, boundedness of follows from boundedness of implying that boundedness of follows from boundedness of . Considering the two cases a) (which implies that ); b) ; it is seen from (127), (133), (135), and (136) that in both cases
(137)
Hence, is a nonincreasing function of time implying that , , , and are bounded on the maximal interval of existence of . Boundedness of also follows from boundsolutions edness of . Thus, all closed-loop states are seen to remain and, hence, . Furthermore, from bounded on (137), , , , , go to zero as . This , , , and the conimplies that trol input asymptotically go to zero, i.e., all closed-loop states . The prop(except and ) go to zero asymptotically as erties of the closed-loop system are summarized in Theorem 3. Theorem 3: Under Assumptions A1, A2 , A3 , A4 , and A6, , ) for the plant state and given any initial conditions ( ( , , ) for the controller state with , the closed-loop system formed by (1), (41), (52), and (134) pos. Furthersesses a unique solution on the time interval more, all closed-loop signals are bounded, , , , , , and , . can not be incorpoRemark 8: Note that the term rated into the dynamics (134) since this would generate terms and , in involving through (127) which cannot be handled for any nonzero and the stabilizing terms in (127) which feature , . This is a direct consequence of the fact that the inverse dynamics and the appended dynamics are simply driven by the states according to the Lyapunov bounds in Assumptions A2 and A6 and are not directly influenced by the control input. Since is constrained to be zero, the stabilization of (i.e., the crucial property that boundedness of follows from boundedness of ) relies on the presence of design freedom .
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D. Illustrative Example
V. CONCLUSION
Consider the system
(138) where are uncertain parameters. Global output-feedback control for this system cannot be achieved through any technique previously available in the literature. Specifically, the coupling with the appended dynamics , , and which have nonzero gains from the unmeasured states, , , and , and the uncertain parameters which appear multiplied with unmeasured states cannot be handled. and . System (138) is of the form (1) with Assume that a magnitude bound on and is available. Magnitude bounds for , , and are not necessary. The unknown parameter appearing in the bounds in Assumptions A2 , A3 , and A6 is defined as . Assumption A1 is satisfied with . The inverse dynamics system given by
(139) is ISS with ISS Lyapunov function which satisfies (89) with , , , , and thus verifying Assumption A2 . Assumption A3 holds , with , and , . Assumption A4 is sat. The subsystems with states , , isfied with are ISS with ISS Lyapunov functions satisfying , (92), (93), and (94) with , , , and . The upper diagonal terms and of system (138) are the same as in Example 2 in Sec, , , and are as in (85). tion III-E so that matrices The coupled Lyapunov inequalities (35) and (38) are satisfied with , , , , , , , , , , , , , and shown in (86). The observer and controller are given by (87) replaced by . The funcwith tion is picked to satisfy (133). The dynamics of the adaptation parameter and the high-gain parameter are given in (126) and (134), respectively.
In this paper, we have proposed a dynamic high-gain scaling based control technique. A new paradigm for design of dynamics of the high-gain parameter was introduced based on the choice of a Lyapunov function guaranteeing a negative derivative if either the high-gain parameter or its derivative is large enough (compared to functions of the states). Therefore, the dynamics of the high-gain parameter are designed such that the derivative is large until becomes large enough. This approach can be thought of as a technique to expand semiglobal high-gain designs into global ones by using a dynamic high-gain parameter which is made to increase to the required value fast enough. It appears that further extensions to classes of systems for which only semiglobal results are currently available might be feasible with this approach. We applied a variant of the advocated approach to low gain state and output feedback for feedforward systems in [30] and [31]. The designs presented in this paper are applicable to systems with uncertain terms involving all the states and input and also ISS appended dynamics driven by all the states and input. Unknown parameters were allowed in the bounds on the uncertain functions and the derivatives of the ISS Lyapunov functions of the appended dynamics. It was seen that the introduction of appended dynamics and complexity of bounds on the uncertain functions do not result in complexity of the observer and the controller but are rather handled through the dynamics of the high-gain parameter. Conversely, since the high-gain controller structure is “almost” linear with gains being state-dependent, further restriction on the system dynamics may further simplify the control structure degenerating in the limiting case to a purely linear controller. Based on the results of this paper and its earlier versions [32], [33], this was pursued in [34]. Another application of our proposed observer/controller structure is a globally asymptotically stabilizing output-feedback controller for the benchmark open problem proposed in our earlier work [15] with the assumption that a magnitude bound on the unknown parameter is available. However, the original problem posed in [15] to stabilize (73) with being a completely unknown (no known magnitude bound) parameter remains open. APPENDIX Theorem A1: Let be an matrix for any in some with entries on the upper diagonal and design set freedoms on the first column, i.e.,
(140) with zeros everywhere else. Let be an matrix for any and any . Let . Let be the vector . Let be any given function. Let , , , , positive constants , , and exist such that
(141)
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(142) (143) for all
and all where , , are nonnegative functions of . be independent of its argument. Let the sign of each , functions , and positive Then, a matrix constants , , and exist to satisfy
any given positive constant. Then, positive constants smaller in magnitude than , positive constants , , and a positive constant exist such that functions (144) is satisfied with shown in (146). Proof: Consider the dynamical system with and the change of coordi. The inverse transformation is given by19 nates , with . The dynamics of can be written as
(144) (145) and with each being for all chosen to be a linear constant-coefficient combination of , , and . Theorem A1 is proved through Lemmas A1 and A2. In Lemma A1, a family of matrices that satisfy (145) is identified. In Lemma A2, it is shown that there is a matrix in this family that also satisfies (144). Lemma A1: Let be as defined in Theorem A1 and let
(148)
where (142)
are dummy variables. Using
(149)
Picking
,
to satisfy
(146) , being constants. Assume that with and . Then, a (143) is satisfied with positive constants positive constant exists such that if are smaller in magnitude than , then, for any given positive constants , positive constants and exist such that (145) holds. Proof: The lemma follows by continuity since (145) since, by asssumpholds when belong to compact sets in (0, ). tion, the elements of can be decomposed as where The matrix and . . Hence, , a (possibly conservative) Noting that estimate for can be written using the diagonal dominance condition for as
(150) we obtain, via algebraic manipulations similar to those used in the proof of [9, Lemma 1], for
(147) Note that the choice of depends only on the elements of while the constants and depend also on . be as defined in Theorem Lemma A2: Let , , , and A1 and let inequalities (141) and (142) hold with some positive , and , , . Let be constants ,
(151) 19We
n.
follow the convention that
f
= 0 and
f
= 1 if n
>
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where for
where
is any constant smaller than and to satisfy
, and choosing
(156)
(157) with
being any positive constant, we have
(158) (152) with
, is a function of only
and
. Note that .
Since
(153) Using (151) and (153)
Arbitrary positive constants and are picked and are computed recursively using (156). is given by (157) and are found recursively using , , and , (150). Noting that . (144) is satisfied with as in Proof of Theorem A1: Picking , (155), the inequality (145) is satisfied by Lemma A1, since , . Choosing , to satisfy as in (150) and (157), the (156), and picking , inequality (158) is satisfied implying that (144) is also satisfied. be an matrix for any in some Theorem A2: Let with entries on the upper diagonal and design set on the last row, i.e., freedoms
(159) with
zeros
everywhere else. be an and any . Let , , positive constants , , and exist such that
Let matrix for any ,
. Let ,
(154)
(160)
(155)
(161) (162)
Choosing20
20Note that, by assumption,
'
assumes the same sign for all arguments.
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for all
and all where , , are nonnegative functions of . be independent of its argument. Let the sign of each , functions , and positive Then, a matrix exist to satisfy constants , , and
(163) (164) and with each being for all chosen to be a linear constant-coefficient combination of . matrix with Proof of Theorem A2: Let be the 1’s on the antidiagonal, i.e., with zeros everywhere else. Premultiplication and postmultiplication of a matrix by reverses the order of the rows and columns, reand , spectively. Defining and are of the form considered in Theorem A1 with , , and . Let where and are diagonal matrices with each diagonal element taking values matrix with element in { 1, 1} and is the being if and zero otherwise. Note that , , satisfy the assumptions of Theorem A1. Hence, apand , a matrix , functions plying Theorem A1 with , and positive constants , , and are found such that (144) and (145) are satisfied for all and with , , and replaced by , , and , respectively. Furthermore, do not depend on and and are linear constant-coefficient combi. Letting nations of and , (144) and (145) yield
(165) (166) Let an 1 if
and where denotes diagonal matrix with the diagonal entry being and 1 otherwise. Then,
from (161). Hence, using (165) and (166), (163) and (164) are , , and obtained with . Remark 9: Theorem A1 provides sufficient conditions for robust solvability of the pair of coupled Lyapunov equations (144) and (145). Assumptions (141) and (142) impose a CUDD [9] condition on and . It is shown in [24] that in the case where is zero, the conditions (141) and (143) are necessary and sufficient for satisfying (144) and (145). Theorem A1 is stated in the observer context with Theorem A2 being its dual in the controller context. REFERENCES [1] R. Marino and P. Toméi, Nonlinear Control Design. Geometric, Adaptive, Robust. Upper Saddle River, NJ: Prentice-Hall, 1995.
[2] M. Krstic´ , I. Kanellakopoulos, and P. V. Kokotovic´ , Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [3] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 2003. [4] L. Praly, “Lyapunov design of a dynamic output feedback for systems linear in their unmeasured state components,” in Proc. IFAC Nonlinear Control Systems Design Symp., Bordeaux, France, June 1992, pp. 63–68. [5] J.-B. Pomet, R. M. Hirshorn, and W. A. Cebuhar, “Dynamic output feedback regulation for a class of nonlinear systems,” Math. Control, Signals, Syst., vol. 6, pp. 106–124, 1993. [6] S. Battilotti, “Global output regulation and disturbance attenuation with global stability via measurement feedback for a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 315–327, Mar. 1996. , “A note on reduced order stabilizing output feedback controllers,” [7] Syst. Control Lett., vol. 30, no. 2–3, pp. 71–81, 1997. [8] P. Krishnamurthy, F. Khorrami, and Z. P. Jiang, “Global output feedback tracking for nonlinear systems in generalized output canonical form,” in Proc. Amer. Control Conf., Arlington, VA, June 2001, pp. 4241–4246. [9] , “Global output feedback tracking for nonlinear systems in generalized output-feedback canonical form,” IEEE Trans. Automat. Contr., vol. 47, pp. 814–819, May 2002. [10] D. Bestle and M. Zeitz, “Canonical form design for nonlinear observers with linearizable error dynamics,” Int. J. Control, vol. 23, pp. 419–431, 1981. [11] A. J. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, no. 1, pp. 47–52, 1983. [12] R. Marino and P. Toméi, “Global adaptive output feedback control of nonlinear systems, part I: linear parameterization,” IEEE Trans. Automat. Contr., vol. 38, pp. 17–32, Jan. 1993. [13] G. Besancon, “State affine systems and observer-based control,” in Proc. IFAC Nonlinear Control Systems Design Symp., Enschede, The Netherlands, July 1998, pp. 399–404. [14] L. Praly and I. Kanellakopoulos, “Output feedback asymptotic stabilization for triangular systems linear in the unmeasured state components,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, Dec. 2000, pp. 2466–2471. [15] P. Krishnamurthy and F. Khorrami, “Robust adaptive control for nonlinear systems in generalized output-feedback canonical form,” Int. J. Adapt. Control Signal Processing, vol. 17, no. 4, pp. 285–311, May 2003. [16] L. Praly, “Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate,” in Proc. IEEE Conf. Decision and Control, Orlando, FL, Dec. 2001, pp. 3808–3813. [17] H. K. Khalil and A. Saberi, “Adaptive stabilization of a class of nonlinear systems using high-gain feedback,” IEEE Trans. Automat. Control., vol. AC-32, pp. 1031–1035, Nov. 1987. [18] A. Ilchmann, “High-gain adaptive control: an overview,” in Proc. IEE Colloq. Adaptive Control, London, U.K., June 1996, pp. 1–4. [19] A. Ilchmann and E. P. Ryan, “On gain adaptation in adaptive control,” IEEE Trans. Automat. Contr., vol. 48, pp. 895–899, May 2003. [20] A. R. Teel and L. Praly, “Global stabilizability and observability imply semi-global stabilizability by output feedback,” Syst. Control Lett., vol. 22, no. 5, pp. 313–325, 1994. [21] H. K. Khalil, “Adaptive output feedback control of nonlinear systems represented by input-output models,” IEEE Trans. Automat. Contr., vol. 41, pp. 177–188, Feb. 1996. [22] C. Qian and W. Lin, “Output feedback control of a class of nonlinear systems: A nonseparation principle paradigm,” IEEE Trans. Automat. Contr., vol. 47, pp. 1710–1715, Oct. 2002. [23] E. Bullinger and F. Allgöwer, “Adaptive -tracking for nonlinear systems with higher relative degree,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, Dec. 2000, pp. 4771–4776. [24] P. Krishnamurthy, F. Khorrami, and R. S. Chandra, “Global high-gain-based observer and backstepping controller for generalized output-feedback canonical form,” IEEE Trans. Automat. Contr., vol. 48, pp. 2277–2284, Dec. 2003. [25] E. D. Sontag, “On the input-to-state stability property,” Eur. J. Control, vol. 1, pp. 24–36, 1995. [26] L. Praly and Z. P. Jiang, “Stabilization by output feedback for systems with ISS inverse dynamics,” Syst. Control Lett., vol. 21, no. 1, pp. 19–33, 1993. [27] Z. P. Jiang, A. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Math. Control, Signals, Syst., vol. 7, pp. 95–120, 1994. [28] F. Mazenc, L. Praly, and W. P. Dayawansa, “Global stabilization by output feedback: examples and counterexamples,” Syst. Control Lett., vol. 23, no. 2, pp. 119–125, 1994.
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[29] E. D. Sontag and A. Teel, “Changing supply functions in input/state stable systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1476–1478, Aug. 1995. [30] P. Krishnamurthy and F. Khorrami, “Global robust control of feedforward systems: state-feedback and output-feedback,” in Proc. IEEE Conf. Decision and Control, Maui, HI, Dec. 2003, pp. 6145–6150. [31] , “A high-gain scaling technique for adaptive output feedback control of feedforward systems,” IEEE Trans. Automat. Contr., vol. 49, pp. 2286–2292, Dec. 2004. , “Generalized adaptive output-feedback form with unknown pa[32] rameters multiplying high output relative-degree states,” in Proc. IEEE Conf. Decision and Control, Las Vegas, NV, Dec. 2002, pp. 1503–1508. [33] , “A dual high gain controller for the uncertain generalized outputfeedback canonical form with appended dynamics driven by all states,” in Proc. Amer. Control Conf., Denver, CO, June 2003, pp. 4766–4771. [34] L. Praly and Z. P. Jiang, “Linear output feedback with dynamic high gain for nonlinear systems,” Syst. Control Lett., vol. 53, no. 2hi, pp. 107–116, 2004.
Prashanth Krishnamurthy received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Chennai, India, in 1999, and the M.S. in electrical engineering from Polytechnic University, Brooklyn, NY, in 2002. He is currently working toward the Ph.D. degree at Polytechnic University. He is the coauthor of 30 journal and conference papers, and of Modeling and Adaptive Nonlinear Control of Electric Motors (New York: Springer-Verlag, 2003). His research interests include robust and adaptive nonlinear control with applications.
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Farshad Khorrami was born in Iran in 1962. He received the B.S. degrees in mathematics and electrical engineering, the M.S. degree in mathematics, and the Ph.D. degree in electrical engineering, all from The Ohio State University, Columbus, in 1982, 1984, 1984, and 1988, respectively. He is currently a Professor of electrical and computer engineering at Polytechnic University, Brooklyn, NY, where he started as an Assistant Professor in September 1988. His research interests include adaptive and nonlinear control, large scale systems and decentralized control, unmanned autonomous vehicles, smart structures, robotics and high-speed positioning applications, and microprocessor-based control and instrumentation. He has published more than 150 refereed journal and conference papers and currently holds ten U.S. patents, with two more pending. He has developed and directed the Control/Robotics Research Laboratory (CRRL) at Polytechnic University. He is also an author of a recently published book entitled Modeling and Adaptive Nonlinear Control of Electric Motors (New York: Springer-Verlag, 2003). Dr. Khorrami has served on the program committees of several conferences, and has been a Member of the Conference Editorial Board of the IEEE Control Systems Society.
