Adaptive Neural Control for a Class of Nonlinear Systems With ...
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Adaptive Neural Control for a Class of Nonlinear Systems With Uncertain Hysteresis Inputs and Time-Varying State Delays Beibei Ren, Student Member, IEEE, Shuzhi Sam Ge, Fellow, IEEE, Tong Heng Lee, Member, IEEE, and Chun-Yi Su, Senior Member, IEEE Abstract—In this paper, adaptive variable structure neural control is investigated for a class of nonlinear systems under the effects of time-varying state delays and uncertain hysteresis inputs. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov–Krasovskii functionals in the design, and the effect of the uncertain hysteresis with the Prandtl–Ishlinskii (PI) model representation is also mitigated using the proposed control. By utilizing the integral-type Lyapunov function, the closed-loop control system is proved to be semiglobally uniformly ultimately bounded (SGUUB). Extensive simulation results demonstrate the effectiveness of the proposed approach. Index Terms—Neural networks (NNs), Prandtl–Ishlinskii (PI) hysteresis model, time-varying delays, variable structure control.
I. INTRODUCTION
I
N recent years, control of nonlinear systems preceded by unknown hysteresis nonlinearities has received a great deal of attention, since hysteresis nonlinearities are common in many industrial processes, especially in position control of smart material-based actuators, including piezoceramics and shape memory alloys. Control of a system with hysteresis nonlinearities is challenging, because they are nondifferentiable nonlinearities and severely limit system performance by giving rise to undesirable inaccuracy or oscillations, and even lead to closed-loop instability [1]. Furthermore, due to the nonsmooth characteristics of hysteresis nonlinearities, traditional control methods are inadequate in dealing with the effects of unknown hysteresis. Therefore, advanced control techniques to mitigate the effects of hysteresis have been called upon and have been studied for decades. In [1], adaptive control with an adaptive hysteresis inverse was presented for plants with unknown parameterized hysteresis. Robust control was developed by combining the inverse
Manuscript received June 11, 2008; revised December 03, 2008; accepted February 07, 2009. First published May 27, 2009; current version published July 09, 2009. This work was supported in part by A*STAR SERC Singapore under Grant 052 101 0097. B. Ren, S. S. Ge, and T. H. Lee are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]; [email protected]; eleleeth@nus. edu.sg). C.-Y. Su is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC H3G 1M8 Canada (e-mail: cysu@alcor. concordia.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNN.2009.2016959
compensation for a novel dynamic hysteresis model in magnetostrictive actuators in [2]. In [3], robust adaptive control was investigated for a class of nonlinear system with unknown backlash-like hysteresis, for which, adaptive backstepping control was designed in [4]. Apart from the above hysteresis models, there exist many other hysteresis models in the literature, since hysteresis is a very complex phenomenon. Interested readers can refer to [5] for a review of the hysteresis models. For different kinds of hysteresis models, different compensation methods should be adopted. As such, it is challenging to fuse those hysteresis models with the available control techniques. It appears that the Prandtl–Ishlinskii (PI) hysteresis model, which is a subclass of Preisach-type model, can be explored in connection with the existing robust adaptive control methods. In [6] and [7], adaptive variable structure control and adaptive backstepping methods were proposed, respectively, for a class of continuous-time nonlinear dynamic systems preceded by hysteresis nonlinearity with the PI hysteresis model representation. However, since the nonlinear functions in most of the above works were assumed to be known, it is therefore of interest to develop methods to deal with unknown nonlinearities, so as to enlarge the class of applicable systems. Other than hysteresis, time delay is another problem that is often encountered in physical systems, for example, in the turbojet engines, aircraft systems, microwave oscillators, nuclear reactors, rolling mills, chemical processes, and hydraulic systems, among others [8]. The existence of time delays in a system frequently becomes a source of instability, and may degrade the control performance. The control of the time-delay systems is challenging since they involve infinite-dimensional functional differential equations, which are more difficult to handle than finite-dimensional ordinary differential equations [9]. To guarantee the stability of time-delay systems, a number of different approaches have been proposed [10]. Lyapunov–Krasovskii functionals [11], combined with the linear matrix inequality (LMI) technique, have been used to establish a framework for the stability and control of time-delay systems [12]–[15]. In [16], Lyapunov–Krasovskii functionals were used with backstepping for a class of single-input–single-output (SISO) nonlinear time-delay systems with a “triangular structure,” which was later commented that it could not be “constructively obtained” in [17]. The need for knowledge of system nonlinearities was removed with the use of adaptive neural network (NN) control in [18], which was extended to a class of multiple-input–multiple-output (MIMO) nonlinear systems in block-triangular form with unknown state
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REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR SYSTEMS
delays [19]. Apart from the Lyapunov–Krasovskii method, the Lyapunov–Razumikhin technique has also been investigated for linear time-delay systems [20], as well as for nonlinear time-delay systems [21], [22]. Although there are some works that deal with hysteresis, or time delay, individually, the combined problem, despite its practical relevance, is largely open in the literature to the best of our knowledge, with the exception of [23], in which turning cutting systems were modeled as plants containing linearly parameterized nonlinearities, backlash hysteresis, and known constant time delay. Motivated by [23], in this paper, we make several technical contributions as follows. First, we remove the restriction of linearly parameterized nonlinear systems considered in [23], and tackle a larger and more complex class of nonlinear systems with unknown nonlinearities, for which direct approximation-based control using NNs is adopted due to their universal approximation capabilities. Second, we consider nonlinear systems that are preceded by uncertain hysteresis inputs in the PI form, which is more complex than the backlash type, but can capture the hysteresis phenomenon more accurately. We fuse the PI hysteresis with adaptive neural control to the reduce the effects of uncertain hysteresis. Third, we relax the assumption of known constant time delay considered in [23], to unknown time-varying delay in our paper, for which Lyapunov–Krasovskii functionals are used to compensate. The organization of this paper is as follows. The problem formulation and preliminaries are given in Section II. In Section III, adaptive variable structure neural control is developed for a class of single-input–single-output (SISO) time-varying state delay systems with hysteresis by utilizing an integral-type Lyapunov function first, which is extended to MIMO systems later. Results of extensive simulation studies are shown to demonstrate the effectiveness of the approach in Section IV, followed by conclusion in Section V. II. PROBLEM FORMULATION AND PRELIMINARIES A. Prandtl–Ishlinskii Hysteresis Model
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Fig. 1. Hysteresis curves given by u (t) =
and are the output and the input of where is a given density funcPI hysteresis model, respectively; tion, satisfying with ; and is subinterknown as the play operator. In addition, there are vals, and the function is monotone on each of the subintervals . The density function vanishes for large values of . As such, it is reasonable to choose a large enough constant such that the given density function vanishes, despite is commonly chosen as the upper limit of the fact that integration in the literature. For the detailed description about the PI hysteresis model, the interested readers can refer to [6], [7], and references therein. As an illustration, Fig. 1 shows generated by (1), with , , and the input , . This numerical result shows that the PI hysteresis model (1) indeed generates hysteresis curves. B. Problem Formulation Consider the following class of uncertain MIMO nonconsisting of interconnected subsystems in a linear system Brunovsky form with time-varying state delays and uncertain PI hysteresis inputs:
First, we briefly introduce the PI hysteresis model which is adopted in this paper. According to [6] and [7], the PI hysteresis model with a play operator can be represented as follows: (1)
(3)
where th subsystem, and
for
and (2)
p v (t) 0 d [v ](t).
, ; are the delay-free state variables of the with
with ; denotes the th subsystem output; and are are the unknown differenunknown continuous functions; are the smooth and bounded initial tiable control gains; functions; , are unknown time-varying state delays, and as will be defined later is a known positive constant; , and is the input of the th subsystem and the output of the th hysteresis.
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Substituting the PI hysteresis model (1) into the plant (3), we obtain the integrated system
(4)
Our control objective is to track the specified desired trajectory to a small neighborhood of zero with the output , while ensuring that all the signals in the corresponding closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Remark 1: Although it appears possible to rewrite (4) into the nonaffine form , it still cannot be handled by the method proposed by [24], in which implicit function theorem was adopted to handle the nonaffine problem. The reason is that if we want to apply implicit function theorem to a function, one requirement is that the first-order derivative of the function is not equal to zero. However, due to the nonsmooth characteristransformed from (4) is tics of hysteresis, the function nondifferentiable and thus does not satisfy the conditions of applying implicit function theorem. Therefore, we need to seek for new solutions in this paper. in (4) is an integral function Remark 2: Noticing that of control input signal , which needs to be designed later, we cannot assume is bounded before we prove the boundedness of control input , even if the output of PI model is bounded for bounded input. Therefore, standard robust adaptive control used for dealing with bounded disturbance cannot be applied here. To solve this problem, we will develop the comprehensive control in the subsequent Section III. and , Assumption 1: There exist two positive constants , . such that Remark 3: Assumption 1 implies that smooth functions are either strictly positive or strictly negative, which is being bounded away from zero is the reasonable because in (4), which is necessary controllable condition of system in most control schemes [25], [26]. Without loss of generality, , . In addition, the we will assume that and need not be known, as they are used in the constants stability analysis only. and its time Assumption 2: The desired trajectory th-order remain bounded, i.e., derivatives up to the with known compact set . Assumption 3: The unknown time-varying state delays satisfy the following inequalities: (5) with known constants and . and Assumption 4: There exist known constants , such that , and for all , . Remark 4: It is reasonable to set an upper bound for the den, based on its properties that with sity function .
C. RBFNN Approximation In control engineering, radial basis function neural network (RBFNN) has been successfully used as a linearly parameterized function approximator to achieve various objectives, such as modeling, identification, and feedback linearization, by virtue of its the universal approximation capabilities, learning and adaptation, and parallel distributed structures [27], [28]. In this paper, the following RBFNN [29] is used to approximate : the continuous function (6) where the input vector
, weight vector , with the NN node number ; and , with being chosen as the commonly used Gaussian functions, which have the form
(7) is the center of the receptive where field and is the width of the Gaussian function. It has been proven that network (6) can approximate any conto any arbitrary tinuous function over a compact set accuracy as (8) where is ideal NN weights, and is the NN approximation error. such Assumption 5: There exist ideal constant weights with constant for all . Morethat over, is bounded by on the compact set . are “artificial” quantities that are reThe ideal weights quired for analytical purposes. According to the discussion in is defined as follows: [30],
which is unknown and needs to be estimated in control design. be the estimate of and the weight estimation error Let be . Remark 5: Although RBFNN is employed in our control design, it can be replaced by other linearly parameterized function approximators such as high-order NNs, fuzzy systems, polynomials, splines and wavelet networks without difficulty. For a unified framework of different approximation structures in adaptive approximation-based control, interested readers can refer to [31]. D. Preliminaries The following lemma will be used for control design and system stability analysis in the remainder of this paper. Lemma 1 [32]: For any continuous function satisfying , where , there exist positive
REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR SYSTEMS
smooth functions such that fying
satis-
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Substituting (13) into (12) leads to
(9)
(14)
Remark 6: According to Lemma 1, the unknown continuous functions of delayed states in (4), , satisfy the inequality
where . Define the following integral Lyapunov function candidate, which was first proposed in [33] to avoid control singularity:
(10) with being positive continuous functions, . In this paper, we consider the special case whereby the are known. As for the case of unbounding functions known bounding functions, interested readers can refer to [19]. , denotes the 2-norm, Throughout this paper, and and denote the smallest and largest eigenvalues of a square matrix , respectively. The following func, with a positive constant , is introduced for the purtion pose of the control design: if if
(11)
(15)
, and . Then, can be rewritten as the following form by using the first mean value theorem for integrals: where
According to Assumption 1, is positive definite with respect to with respect to time , we obtain
, it is clear that . Differentiating
III. CONTROL DESIGN AND STABILITY ANALYSIS In this section, we will carry out adaptive NN control design in (4) to achieve stable output tracking. In order for system to illustrate the design methodology clearly, the SISO case (i.e., ) is discussed first, which is generalized to the MIMO ) subsequently. For both cases, the closedcase (i.e., loop system will be proved to be SGUUB by Lyapunov stability analysis. The following definitions and notations are used throughout and as the control design and stability analysis. Define
(16)
Due to
and , it is shown that
(17) and the filtered tracking error
as Substituting (14) and (17) into (16) results in (12)
where are chosen such that the polynomial is a Hurwitz polynomial. A. Adaptive NN Control for SISO Case For the SISO case where in the following form:
(18)
, system (4) can be rewritten Using (10) and Young’s inequality, (18) becomes (13)
(19)
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where
Define the following compact sets: (24) (25) (26) is a sufficiently large compact set satisfying defined later in Theorem 1, and is a positive design constant that can be chosen arbitrarily small and “ ” in (26) is used to denote the complement set of set . In addition, it has been shown that is a compact set in [34]. be the approximation of the function , Let . Then, using RBFNN defined in (23), on the compact set as discussed in Section II-C, we have where
with . To overcome the design difficulties from the unknown in (19), the following Lyatime-varying delays punov–Krasovskii functional can be considered [34]:
(27) where the approximation error satisfies with positive constant , . For (22), we design a control law as follows:
(20) (28) The time derivative of
can be expressed as follows: (29) (21)
which can be used to cancel the time delay term on the righthand side of (19), thus circumvent the design difficulty due to the unknown time-varying delay , without introducing any additional uncertainties to the system. For concise notation, the will be omitted whenever the time variables and time-varying delay term is eliminated, in the remainder of this paper. Combining (19) and (21), we obtain (22)
(30) where is defined in (11), is the estimate of the , is a positive constant, and is density function chosen as (31)
as a positive constant specified by the designer. with The adaptation laws are designed as follows:
where
(32) (23)
if
in (23) contains the term , which is not well defined at and may lead to the controller singularity problem, if we utilize to construct the control law. As such, care must be taken to guarantee the boundedness of the control as discussed in [34]. It is noted that the controller singularity takes place at , where the control objective is supposed to the point be achieved. From a practical point of view, once the system reaches its origin, no control action should be taken for less is hard to detect owing to the power consumption. As existence of measurement noise, it is more practical to relax our control objective of convergence to a “ball” rather than the origin.
if
Remark 7: Note that
(33) with , , , and being strictly positive constants. Remark 8: The term in (29) is used to cancel the effect . Unlike traditional caused by the hysteresis term is either robust adaptive controller designs, where assumed to be bounded by a constant or a known function, here is presented as an integral function of control input signal , and there are no assumptions on its boundis not a edness. Considering that the density function function of time, it can be treated as a “parameter” of the hysteresis model and adaption law can be developed to obtain
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an estimate of it. This is crucial for the success of the adaption law design [6]. Remark 9: From (28) and (29), we notice that both sides of depends on as (28) contain the control signal , because can be seen from (29). This is known as the fixed-point problem, where the solvability of can be proved following the proof of Theorem 1.4 about the existence of the hysteresis inverse operator in [35]. Since it is difficult to obtain the explicit solution for from (28), we introduce several possible implementation directly from (28). One is the methods instead of solving time-scale separation approach, recently proposed in [36]: the is a solution of a “fast” dynamical equacontrol signal tion, which means the dynamics of the controller is faster than that of the system plant. Thus, time-scale separation is achieved between the system plant and the controller dynamics using the singular perturbation theory. Second method is adopting the numerical implementation of the inverse hysteresis operator as in [35], where a real-time inverse feedforward control was designed for piezoelectric actuators. In this paper, we introduce a to small delay to evaluate the input: at time , we use in (29) for suitably small , such that in (28) compute , , . The limitabecomes a function of , tion of this method is that its accuracy depends on the choice of . The effects of the variations of will be investigated later in the simulation part in Section IV. Theorem 1: Consider the closed-loop system consisting of the plant (13), the control laws (28) and (29), and adaptation laws (32) and (33). Under Assumptions 1–4, given some ini, , belong to , we can conclude tial conditions that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop defined by system will remain in the compact set
Proof: The method of proof is generally similar to that in our previous works [37], [38], although the details of analysis are different and more complex, due to the presence of time delay and hysteresis in the system. In this proof, we will show , on which the NN approximation is that for a compact set valid, there exist some control parameters and a nonempty initial , such that as long as the initial conditions start compact set , the states and the weights will remain in the conservative in , and finally converge to the compact set . compact set . The proof Both of them belong to the chosen compact set includes two steps, and one could see the whole picture at the end of the proof of Step 2. Step 1: Suppose that both the states and the weights belong , i.e., , , on which NN approxito mation (27) is valid. Consider the following Lyapunov function candidate:
(34)
Considering the adaptive neural control laws and adaptation laws from (28)–(33), the stability analysis is carried out in the following two regions, respectively. , then . Noting (27) • Region 1: If and submitting (28) into (40), we have
with
where entiating
and with respect to time
(38) . Differ-
(39) Substituting (22) into (39) leads to
(40)
(35)
(41) Using Young’s inequality, we have (36) and the tracking error will converge to a neighborhood of zero. In addition, the states and the weights in the closed-loop system defined by will eventually converge to the compact set
(42) Substituting (30), (32), and (42) into (41) leads to
(37) (43) where
.
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For the third term in (43), by completion of squares, we have
b)
, which is the complement set of in , i.e., . In this case, from (33), we have
(44) (49)
For the last two terms in (43), using (2) and (29), we have Substituting (49) into (45), we have
(50)
Combining (45), (48), and (50), we know that
(51) By completion of squares, we have (52) Integrating both sides of (52) over
results in
(45) (53) According to (33), the adaptation law for the estimate of density function is divided into two cases, due to the different regions which belongs to. Therefore, we also need to consider two cases for the analysis of (45). . a) According to (33), we have
According to Assumption 4, we know that Therefore
.
(46) (47) Substituting (46) and (47) into (45), we have
(54) Substituting (44), (51), and (54) into (43), we have
(48) (55)
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where
Multiplying (55) by have
and integrating over
, we
(56) . Therefore, according to
where
in (38), the definition of and . • Region 2: If , then . In this , , , case, the control signal , i.e., all the signals are kept bounded. . From (12), Define we know that 1) there is a state–space representation for , i.e., with mapping , , being a and such stable matrix; 2) there are positive constants that , and 3) the solution of is
Accordingly, it follows that
(57) Noting
and
Fig. 2. Compact sets.
, , such that some and From the definitions of
and . in (35) and (36) as follows
we can see that the values of and depend on the choice , , , and . Thereof the control parameters , there exist some control fore, for a given NN compact set parameters such that for a . Then, as the set of initial condiwe define the initial compact set such that . Therefore, for all tions that belong to , we have for . If and are larger than , this means that the initial . This conditions do not belong to a valid initial compact set completes the proof. B. Adaptive NN Control for MIMO Case
, we have
Substituting (57) into the above inequality leads to
Therefore, we can conclude that all the closed-loop signals are SGUUB for some initial conditions, and the tracking error will converge to a neighborhood of zero. Furthermore, from (56), we also can have
In the foregoing discussions, we design control for the SISO case by Lyapunov synthesis design, so as to elucidate the main ideas of our control design. In this section, we extend the previous result to the MIMO case (4). System (4) is block triangular with respect to inputs , as seen in the fact that nonlinearities only contain inputs from the preceding subsystems. This structure of interconnection facilitates systematic recursive design. Substituting (4) into (12) leads to (59)
(58) where . Therefore, and , as . Step 2: In this step, we prove that there exist some control , such that parameters and a nonempty initial compact set , the states and the as long as initial conditions belong in weights under the proposed control, for , will never es, which belongs to cape from the conservative compact set , as shown in Fig. 2. the chosen compact set in From the definition of the bounds of the compact sets (34) and in (37), we can see that for a given , there exist
. where Define the following integral Lyapunov function candidate: (60)
, and . Applying the first mean value theorem for integrals to (60), we have where
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which is positive definite with respect to due to Assumption . 1, with respect to time , we obtain Differentiating
By utilizing Young’s inequality, we obtain that
(63) and
(64) Substituting (63) and (64) into (62), we have
(65)
(61)
To overcome the design difficulties from the unknown in (65), the following time-varying delays Lyapunov–Krasovskii functional can be considered:
Using (10), after some manipulations, (61) becomes
(66) The time derivative of
can be expressed as follows:
(62) where (67) which can be used to cancel the time delay term on the righthand side of (65). Combining (65) and (67), we obtain (68) where
(69) with
.
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Define the following compact sets:
(70)
(71) (72)
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laws (79) and (80). Under Assumptions 1–4, given that some , belong in , we can conclude initial conditions that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop system will remain in the compact set defined by (81) with
(73) where
is a sufficiently large compact set satisfying defined in Theorem 2, and is a positive design constant that can be chosen arbitrarily small. The compact set is the complement set of set . be the approximation of the function , Let . Then, using RBFNN as defined in (69), on the compact set discussed in Section II-C, we have (74) satisfies where the approximation error with positive constant , . Similar to the procedures of Section III-A, we design the following control law for the system in (4):
and the tracking error will converge to a neighborhood of zero. In addition, the states and the weights in the closed-loop system will eventually converge to the compact set defined by (82)
(75)
where
(76) (77) is defined in (11); where density function ; parameter is chosen as and
is the estimate of the is any positive constant
Proof: The proof is built on that of Theorem 1, and for the conciseness, we will only outline the general approach without going into specific details. For the th subsystem, we design that takes into account the inputs from the preceding subsystems, i.e., . Suppose that both the states and the weights belong to , , , on which NN approximation (74) i.e., is valid. Consider the following Lyapunov function candidate:
(78) a positive constant specified by the designer. with The adaptation laws are chosen as
and with respect to time leads to
(79)
where entiating
(80)
Substituting (68) into (84) leads to
(83) . Differ-
if (84)
if
with , , , and being strictly positive constants. Based on the above design for control and adaptation laws, we are ready to establish the following result for the MIMO case. Theorem 2: Consider the closed-loop system consisting of the plant (4), the control laws (75) and (76), and adaptation
(85)
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Considering the adaptive neural control laws and adaptation laws from (75)–(80), the stability analysis is carried out in the following two regions, respectively. , then . Noting (74) • Region 1: If and submitting (75) into (85), we have
Multiplying (91) by have
Using Young’s inequality, we have (87) Substituting (77), (79), and (87) into (86) leads to
, we
(92) where
(86)
and integrating over
. Therefore,
and . , then . In this case, the • Region 2: If , , , control signal , i.e., all the signals are kept bounded. Similar to the discussion in Theorem 1, we can conclude that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop defined in (81), and system will remain in the compact set defined in (82). will eventually converge to the compact set This completes the proof. IV. SIMULATION STUDIES
(88) For the third term in (88), by completion of squares, we have (89) For the last two terms in (88), we can obtain the similar conclusions as (51) and (54)
In this section, results of extensive simulation studies are presented to demonstrate the effectiveness of the proposed adaptive NN approach to deal with uncertain nonlinear systems under the effects of time delay and hysteresis. For clear illustration, we consider first a simplified SISO plant with first-order dynamics, and study the tracking performance of the controller, as well as perform detailed analysis on the effects of control parameter variations. Subsequently, a MIMO plant consisting of two interconnected second-order subsystems is tackled, and the closed-loop properties and tracking behavior are investigated. A. SISO Case For the SISO case, we consider the following first-order scalar nonlinear system with hysteresis and state delay:
(93) (90) where is the plant output; is the plant input and the output of the PI hysteresis model as in (1): , with for , , ; and the time-varying delay and , , and . The objective is to design control such that the output can track the desired trajectory . We adopt the control law and adaption laws designed in Section III-A in the following:
Substituting (89) and (90) into (88), we have
(91) (94) where (95) (96)
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Fig. 3. Output tracking performance of SISO plant S .
Fig. 5. Tracking error comparison result of SISO plant v .
Fig. 4. Control signals of SISO plant S .
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S
with and without
Fig. 6. Learning behavior of NNs of SISO plant S .
(97) (98) if (99) if , and is the estimate of the density where function of . The input of the NNs is . Employing ten nodes for each input dimension, we end nodes for the network . The up with bounding function for the time-delay term is chosen as , and the following initial conditions and controller design parameters are adopted in the simulation: , , , , , , , , , , , . and The simulation results for SISO plant , as described in (93), are shown in Figs. 3–12. From Fig. 3, we can observe that good
tracking performance is achieved. At the same time, the boundedness of the control signals are shown in Fig. 4. It is noted that there is a large difference between and , indicating the significant hysteresis effect. In particular, we highlight the importance of the term in (94), which is used to mitigate the effect caused by the hysteresis term in PI hysteresis model , as discussed in Remark 8. The comparison of tracking errors in the presence and absence of is shown in Fig. 5, and it is seen that with , the tracking error resulting from hysteresis is attenuated accordingly. Figs. 6 and 7 show the nonlinear approximation capability of NNs and the norm of NN weights, respectively. The behavior of the estimate of the density function is also indicated in Fig. 8. To investigate the effects of the control parameters on the tracking performance, and to provide recommendations for their selection, we provide the following comparison results and in Figs. 9 and 10. First, as for the design constants shown in Fig. 9, the tracking error can be reduced by increasing the parameter . Second, from (99) and Fig. 10, we know that higher learning rate, i.e., increase of , results in better
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Fig. 7. Norm of NN weights of SISO plant
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S
.
Fig. 10. Tracking error comparison result of SISO plant
S
for different .
Fig. 8. Behavior of the estimate values of the density function p^(t; r). Fig. 11. Tracking error comparison result of SISO plant S for different .
Fig. 9. Tracking error comparison result of SISO plant
S
for different k .
to measurement noise, excitation of high-frequency unmodeled dynamics, as well as excessive control efforts. A similar tradeoff exists with regard to the parameter , which represents the , in (99). In learning rate of the density function estimate general, if is chosen to be too large, then the stability and the robustness of the system may be compromised in a similar way as high gain control. We need to mention that, due to the use of sign function , controllers (94) and (95) become discontinuous, which may excite unmodeled high-frequency plant dynamics and cause the chattering phenomenon. To avoid the undesired chattering phenomenon, we replace the sign function in the , which above control laws with the saturation function is defined as if
tracking performance. While the above results seem to indicate and should be large, caution must be exercised in the that choice of these parameters, due to the fact that there are some tradeoffs between the control performance and other issues. In particular, for the case of control gain , the price to be paid is the high gain control, which also can be seen from (94) and (96). Problems associated with high gain control include sensitivity
if
(100)
if where is a very small positive constant. Therefore, the different choices of also can affect the tracking performance, as shown in Fig. 11. The smaller , the closer is the saturation function
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Fig. 13. Output tracking performance of MIMO plant S .
1
()
Fig. 12. We introduce a small delay t into v t in calculating v (95) to implement the control v t (94) numerically as pointed in Remark 9. Different : is the t have different effects on the tracking performance, where T sampling time: (a) t T ; (b) t T. T ; (c) t T ; and (d) t
1
() 1=
1 =5
1 = 10
= 0 005 1 = 15
approximate to the sign function. As such, though the better tracking performance can be achieved with the smaller , the chattering phenomenon will become more serious, as a result, which degrades the performance finally. In addition, as discussed in Remark 9, we adopt a numerical to implement the conmethod by introducing a small delay trol in (94) instead of solving it directly. The choices of the delay affect the performance as shown in Fig. 12. With the increasing of , the performance becomes worse. In this paper, , where is the sampling time. we choose B. MIMO Case Consider the following MIMO nonlinear system consisting of two interconnected second-order subsystems with time delay and hysteresis:
(101)
where are the plant outputs, ; are the plant inputs and the outputs of the PI hysteresis model as in (1): with for , , and ; and the time-varying , , delays , and . The objective is to design control such that the output can track the desired trajectory , . The control law and adaption laws in (75)–(80) are adopted. The inputs of the NNs are and
Fig. 14. Control signals of MIMO plant S .
, where and , . Employing three nodes nodes for the for each input dimension, we end up with , and nodes for the network . network The bounding functions for the time-delay term are chosen as and , and the following initial conditions and controller design parameters are , adopted in the simulation: , , , , , , , , , , , , , and . Simulation results for MIMO plant , as described in (101), are shown in Figs. 13–20. From Fig. 13, it is seen that good tracking performance is achieved despite large initial tracking and , and they converge to a small neighborhood errors of zero in a relatively short time. At the same time, it can be observed, in Figs. 14–16, that the control signals, norms of NN and remain bounded. Fig. 17 shows weights, and states and the nonlinear approximation capability of NNs
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Fig. 15. Norm of NN weights of MIMO plant S .
Fig. 16. Other states of MIMO plant S .
Fig. 17. Learning behavior of NNs of MIMO plant S .
. Similar relationships between variations of control parameters and effects on tracking performance, as shown for the SISO case, can be verified for the MIMO case as well in Figs. 18–20.
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Fig. 18. Tracking error comparison result of MIMO plant S for different k and k .
Fig. 19. Tracking error comparison result of MIMO plant S for different and .
Fig. 20. Tracking error comparison result of MIMO plant S for different .
V. CONCLUSION Adaptive variable structure neural control has been proposed for a class of uncertain MIMO nonlinear systems with
REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF NONLINEAR SYSTEMS
unknown state time-varying delays and PI hysteresis nonlinearities. The uncertainties from unknown time-varying delays have been compensated for through the use of appropriate Lyapunov–Krasovskii functionals. The effect of the unknown hysteresis with the PI models was also mitigated using the proposed control. The controller has been made to be free from singularity problem by utilizing integral Lyapunov function. Based on the principle of sliding-mode control, the developed controller can guarantee that all signals involved are SGUUB. Simulation results have verified the effectiveness of the proposed approach. As for future work, it would be interesting to extend the results reported here to deal with some applications involving hysteresis and time delays, such as smart material actuators.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for constructive comments that helped to improve the quality and presentation of this paper.
REFERENCES [1] G. Tao and P. V. Kokotovic, “Adaptive control of plants with unknown hysteresis,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [2] X. Tan and J. S. Baras, “Modeling and control of hysteresis in magnestrictive actuators,” Automatica, vol. 40, no. 9, pp. 1469–1480, 2004. [3] C. Y. Su, Y. Stepanenko, J. Svoboda, and T. P. Leung, “Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2427–2432, Dec. 2000. [4] J. Zhou, C. Y. Wen, and Y. Zhang, “Adaptive backstepping control design of a class of uncertain nonlinear systems with unknown backlash-like hysteresis,” IEEE Trans. Autom. Control, vol. 49, no. 10, pp. 1751–1757, Oct. 2004. [5] J. W. Macki, P. Nistri, and P. Zecca, “Mathematical models for hysteresis,” SIAM Rev., vol. 35, pp. 94–123, 1993. [6] C. Y. Su, Q. Wang, X. Chen, and S. Rakheja, “Adaptive variable structure control of a class of nonlinear systems with unknown Prandtl-Ishlinskii hysteresis,” IEEE Trans. Autom. Control, vol. 50, no. 12, pp. 2069–2074, Dec. 2005. [7] Q. Wang and C. Y. Su, “Robust adaptive control of a class of nonlinear systems including actuator hysteresis with Prandtl-Ishlinskii presentations,” Automatica, vol. 42, no. 5, pp. 859–867, 2006. [8] P. L. Liu and T. J. Su, “Robust stability of interval time-delay systems with delay-dependence,” Syst. Control Lett., vol. 33, no. 4, pp. 231–239, 1998. [9] L. Dugard and E. Veriest, Stability and Control of Time-Delay Systems. Berlin, Germany: Springer-Verlag, 1997. [10] J.-P. Richard, “Time-delay systems: An overview of some recent advances and open problems,” Automatica, vol. 39, pp. 1667–1694, 2003. [11] J. Hale, Theory of Functional Differential Equations, 2nd ed. New York: Springer-Verlag, 1977. [12] V. B. Kolmanovskii and J. Richard, “Stability of some linear systems with delays,” IEEE Trans. Autom. Control, vol. 44, no. 5, pp. 984–989, May 1999. [13] S.-L. Niculescu, Delay Effects on Stability: A Robust Control Approach. London, U.K.: Springer-Verlag, 2001. [14] K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, MA: Birkhäuser, 2003. [15] V. L. Kharitonov and D. Melchor-Aguilar, “Lyapunov-Krasovskii functionals for additional dynamics,” Int. J. Robust Nonlinear Control, vol. 13, pp. 796–804, 2003. [16] S. K. Nguang, “Robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 756–762, Apr. 2000.
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[17] S. Zhou, G. Feng, and S. K. Nguang, “Comments on robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 9, pp. 1586–1586, Sep. 2002. [18] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural network control of nonlinear systems with unknown time delays,” IEEE Trans. Autom. Control, vol. 48, no. 11, pp. 2004–2010, Nov. 2003. [19] S. S. Ge and K. P. Tee, “Approximation-based control of nonlinear MIMO time-delay systems,” Automatica, vol. 43, no. 6, pp. 31–43, 2007. [20] Y. Sun, J. Hsieh, and H. Yang, “On the stability of uncertain systems with multiple time-varying delays,” IEEE Trans. Autom. Control, vol. 42, no. 1, pp. 101–105, Jan. 1997. [21] B. Xu and Y. Liu, “An improved Razumikhin-type theorem and its applications,” IEEE Trans. Autom. Control, vol. 39, no. 4, pp. 839–841, Apr. 1994. [22] M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1048–1060, Jul. 2001. [23] J. Pan, C.-Y. Su, and Y. Stepanenko, “Modeling and robust adaptive control of metal cutting mechanical systems,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 1268–1273. [24] S. S. Ge, C. C. Hang, and T. Zhang, “Adaptive neural network control of nonlinear systems by state and output feedback,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 29, no. 6, pp. 818–828, Dec. 1999. [25] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [26] R. Sepulchre, M. Jankovic, and P. V. Kokotovic´, Constructive Nonlinear Control. London, U.K.: Springer-Verlag, 1997. [27] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [28] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [29] S. Haykin, Neural Networks: A Comprehensive Foundations, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [30] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Boston, MA: Kluwer, 2002. [31] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control. Hoboken, NJ: Wiley, 2006. [32] W. Lin and C. J. Qian, “Adaptive control of nonlinearly parameterized systems: The smooth feedback case,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1249–1266, Aug. 2002. [33] S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 741–747, 1999. [34] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 34, no. 1, pp. 499–516, Feb. 2004. [35] P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” Inst. Electr. Eng. Proc.—Control Theory Appl., vol. 148, no. 3, pp. 185–192, 2001. [36] N. Hovakimyan, E. Lavretsky, and A. Sasane, “Dynamic inversion for nonaffine-in-control systems via time-scale separation. Part I,” J. Dyn. Control Syst., vol. 13, no. 4, pp. 451–465, 2007. [37] T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, pp. 1021–1033, 2007. [38] S. S. Ge and C. Wang, “Adaptive neural network control of uncertain MIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 674–692, May 2004.
Beibei Ren (S’06) received the B.E. degree in mechanical and electronic engineering and the M.E. degree in automation from Xidian University, Xi’an, China, in 2001 and 2004, respectively. Currently, she is working towards the Ph.D. degree at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Her current research interests include neural networks, adaptive control, nonlinear control, and their applications.
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Shuzhi Sam Ge (S’90–M’92–SM’99–F’06) received the B.Sc. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1986 and the Ph.D. degree and DIC from Imperial College, London, U.K., in 1993. He is founding Director of Social Robotics Lab of Interactive Digital Media Institute and Professor of the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He has (co)authored three books and over 300 international journal and conference papers, and edited one book. His current research interests include social robotics, multimedia fusion, adaptive control, intelligent systems, and artificial intelligence. Dr. Ge is the founding Editor-in-Chief of the International Journal of Social Robotics (Springer-Verlag). He has served/been serving as an Associate Editor for a number of flagship journals and a Book Editor of the Taylor & Francis Automation and Control Engineering Series. At IEEE Control Systems Society, he served/serves as Vice President for Technical Activities (2009–2010) and Member of Board of Governors of IEEE Control Systems Society (2007–2009).
Tong Heng Lee (M’90) received the B.A. degree (with first class honors) in engineering tripos from Cambridge University, Cambridge, U.K., in 1980 and the Ph.D. degree in electrical engineering from Yale University, New Haven, CT, in 1987. Currently, he is Professor at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests are in the areas of adaptive systems, knowledge-based control, intelligent mechatronics, and computational intelligence. He has coauthored
three research monographs and holds four patents (two of which are in the technology area of adaptive systems, and the other two are in the area of intelligent mechatronics). Dr. Lee currently holds Associate Editor appointments in the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, Control Engineering Practice (an IFAC journal), the International Journal of Systems Science (Taylor and Francis, London, U.K.), and Mechatronics journal (Pergamon, Oxford, U.K.). He was the recipient of the Cambridge University Charles Baker Prize in Engineering and the 2004 ASCC (Melbourne) Best Industrial Control Application Paper Prize.
Chun-Yi Su (SM’99) received the Ph.D. degree in control theory and its applications from South China University of Technology, Guangzhou, China, in 1990. Currently, he is Professor and Concordia Research Chair at Concordia University, Montreal, QC, Canada. His current research interest is in control of systems involving hysteresis nonlinearities. Dr. Su has served on the editorial boards of several journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. He has also severed as an organizing committee member for many conferences.
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Adaptive Neural Control for a Class of Nonlinear Systems With Uncertain Hysteresis Inputs and Time-Varying State Delays Beibei Ren, Student Member, IEEE, Shuzhi Sam Ge, Fellow, IEEE, Tong Heng Lee, Member, IEEE, and Chun-Yi Su, Senior Member, IEEE Abstract—In this paper, adaptive variable structure neural control is investigated for a class of nonlinear systems under the effects of time-varying state delays and uncertain hysteresis inputs. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov–Krasovskii functionals in the design, and the effect of the uncertain hysteresis with the Prandtl–Ishlinskii (PI) model representation is also mitigated using the proposed control. By utilizing the integral-type Lyapunov function, the closed-loop control system is proved to be semiglobally uniformly ultimately bounded (SGUUB). Extensive simulation results demonstrate the effectiveness of the proposed approach. Index Terms—Neural networks (NNs), Prandtl–Ishlinskii (PI) hysteresis model, time-varying delays, variable structure control.
I. INTRODUCTION
I
N recent years, control of nonlinear systems preceded by unknown hysteresis nonlinearities has received a great deal of attention, since hysteresis nonlinearities are common in many industrial processes, especially in position control of smart material-based actuators, including piezoceramics and shape memory alloys. Control of a system with hysteresis nonlinearities is challenging, because they are nondifferentiable nonlinearities and severely limit system performance by giving rise to undesirable inaccuracy or oscillations, and even lead to closed-loop instability [1]. Furthermore, due to the nonsmooth characteristics of hysteresis nonlinearities, traditional control methods are inadequate in dealing with the effects of unknown hysteresis. Therefore, advanced control techniques to mitigate the effects of hysteresis have been called upon and have been studied for decades. In [1], adaptive control with an adaptive hysteresis inverse was presented for plants with unknown parameterized hysteresis. Robust control was developed by combining the inverse
Manuscript received June 11, 2008; revised December 03, 2008; accepted February 07, 2009. First published May 27, 2009; current version published July 09, 2009. This work was supported in part by A*STAR SERC Singapore under Grant 052 101 0097. B. Ren, S. S. Ge, and T. H. Lee are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]; [email protected]; eleleeth@nus. edu.sg). C.-Y. Su is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC H3G 1M8 Canada (e-mail: cysu@alcor. concordia.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNN.2009.2016959
compensation for a novel dynamic hysteresis model in magnetostrictive actuators in [2]. In [3], robust adaptive control was investigated for a class of nonlinear system with unknown backlash-like hysteresis, for which, adaptive backstepping control was designed in [4]. Apart from the above hysteresis models, there exist many other hysteresis models in the literature, since hysteresis is a very complex phenomenon. Interested readers can refer to [5] for a review of the hysteresis models. For different kinds of hysteresis models, different compensation methods should be adopted. As such, it is challenging to fuse those hysteresis models with the available control techniques. It appears that the Prandtl–Ishlinskii (PI) hysteresis model, which is a subclass of Preisach-type model, can be explored in connection with the existing robust adaptive control methods. In [6] and [7], adaptive variable structure control and adaptive backstepping methods were proposed, respectively, for a class of continuous-time nonlinear dynamic systems preceded by hysteresis nonlinearity with the PI hysteresis model representation. However, since the nonlinear functions in most of the above works were assumed to be known, it is therefore of interest to develop methods to deal with unknown nonlinearities, so as to enlarge the class of applicable systems. Other than hysteresis, time delay is another problem that is often encountered in physical systems, for example, in the turbojet engines, aircraft systems, microwave oscillators, nuclear reactors, rolling mills, chemical processes, and hydraulic systems, among others [8]. The existence of time delays in a system frequently becomes a source of instability, and may degrade the control performance. The control of the time-delay systems is challenging since they involve infinite-dimensional functional differential equations, which are more difficult to handle than finite-dimensional ordinary differential equations [9]. To guarantee the stability of time-delay systems, a number of different approaches have been proposed [10]. Lyapunov–Krasovskii functionals [11], combined with the linear matrix inequality (LMI) technique, have been used to establish a framework for the stability and control of time-delay systems [12]–[15]. In [16], Lyapunov–Krasovskii functionals were used with backstepping for a class of single-input–single-output (SISO) nonlinear time-delay systems with a “triangular structure,” which was later commented that it could not be “constructively obtained” in [17]. The need for knowledge of system nonlinearities was removed with the use of adaptive neural network (NN) control in [18], which was extended to a class of multiple-input–multiple-output (MIMO) nonlinear systems in block-triangular form with unknown state
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delays [19]. Apart from the Lyapunov–Krasovskii method, the Lyapunov–Razumikhin technique has also been investigated for linear time-delay systems [20], as well as for nonlinear time-delay systems [21], [22]. Although there are some works that deal with hysteresis, or time delay, individually, the combined problem, despite its practical relevance, is largely open in the literature to the best of our knowledge, with the exception of [23], in which turning cutting systems were modeled as plants containing linearly parameterized nonlinearities, backlash hysteresis, and known constant time delay. Motivated by [23], in this paper, we make several technical contributions as follows. First, we remove the restriction of linearly parameterized nonlinear systems considered in [23], and tackle a larger and more complex class of nonlinear systems with unknown nonlinearities, for which direct approximation-based control using NNs is adopted due to their universal approximation capabilities. Second, we consider nonlinear systems that are preceded by uncertain hysteresis inputs in the PI form, which is more complex than the backlash type, but can capture the hysteresis phenomenon more accurately. We fuse the PI hysteresis with adaptive neural control to the reduce the effects of uncertain hysteresis. Third, we relax the assumption of known constant time delay considered in [23], to unknown time-varying delay in our paper, for which Lyapunov–Krasovskii functionals are used to compensate. The organization of this paper is as follows. The problem formulation and preliminaries are given in Section II. In Section III, adaptive variable structure neural control is developed for a class of single-input–single-output (SISO) time-varying state delay systems with hysteresis by utilizing an integral-type Lyapunov function first, which is extended to MIMO systems later. Results of extensive simulation studies are shown to demonstrate the effectiveness of the approach in Section IV, followed by conclusion in Section V. II. PROBLEM FORMULATION AND PRELIMINARIES A. Prandtl–Ishlinskii Hysteresis Model
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Fig. 1. Hysteresis curves given by u (t) =
and are the output and the input of where is a given density funcPI hysteresis model, respectively; tion, satisfying with ; and is subinterknown as the play operator. In addition, there are vals, and the function is monotone on each of the subintervals . The density function vanishes for large values of . As such, it is reasonable to choose a large enough constant such that the given density function vanishes, despite is commonly chosen as the upper limit of the fact that integration in the literature. For the detailed description about the PI hysteresis model, the interested readers can refer to [6], [7], and references therein. As an illustration, Fig. 1 shows generated by (1), with , , and the input , . This numerical result shows that the PI hysteresis model (1) indeed generates hysteresis curves. B. Problem Formulation Consider the following class of uncertain MIMO nonconsisting of interconnected subsystems in a linear system Brunovsky form with time-varying state delays and uncertain PI hysteresis inputs:
First, we briefly introduce the PI hysteresis model which is adopted in this paper. According to [6] and [7], the PI hysteresis model with a play operator can be represented as follows: (1)
(3)
where th subsystem, and
for
and (2)
p v (t) 0 d [v ](t).
, ; are the delay-free state variables of the with
with ; denotes the th subsystem output; and are are the unknown differenunknown continuous functions; are the smooth and bounded initial tiable control gains; functions; , are unknown time-varying state delays, and as will be defined later is a known positive constant; , and is the input of the th subsystem and the output of the th hysteresis.
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Substituting the PI hysteresis model (1) into the plant (3), we obtain the integrated system
(4)
Our control objective is to track the specified desired trajectory to a small neighborhood of zero with the output , while ensuring that all the signals in the corresponding closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Remark 1: Although it appears possible to rewrite (4) into the nonaffine form , it still cannot be handled by the method proposed by [24], in which implicit function theorem was adopted to handle the nonaffine problem. The reason is that if we want to apply implicit function theorem to a function, one requirement is that the first-order derivative of the function is not equal to zero. However, due to the nonsmooth characteristransformed from (4) is tics of hysteresis, the function nondifferentiable and thus does not satisfy the conditions of applying implicit function theorem. Therefore, we need to seek for new solutions in this paper. in (4) is an integral function Remark 2: Noticing that of control input signal , which needs to be designed later, we cannot assume is bounded before we prove the boundedness of control input , even if the output of PI model is bounded for bounded input. Therefore, standard robust adaptive control used for dealing with bounded disturbance cannot be applied here. To solve this problem, we will develop the comprehensive control in the subsequent Section III. and , Assumption 1: There exist two positive constants , . such that Remark 3: Assumption 1 implies that smooth functions are either strictly positive or strictly negative, which is being bounded away from zero is the reasonable because in (4), which is necessary controllable condition of system in most control schemes [25], [26]. Without loss of generality, , . In addition, the we will assume that and need not be known, as they are used in the constants stability analysis only. and its time Assumption 2: The desired trajectory th-order remain bounded, i.e., derivatives up to the with known compact set . Assumption 3: The unknown time-varying state delays satisfy the following inequalities: (5) with known constants and . and Assumption 4: There exist known constants , such that , and for all , . Remark 4: It is reasonable to set an upper bound for the den, based on its properties that with sity function .
C. RBFNN Approximation In control engineering, radial basis function neural network (RBFNN) has been successfully used as a linearly parameterized function approximator to achieve various objectives, such as modeling, identification, and feedback linearization, by virtue of its the universal approximation capabilities, learning and adaptation, and parallel distributed structures [27], [28]. In this paper, the following RBFNN [29] is used to approximate : the continuous function (6) where the input vector
, weight vector , with the NN node number ; and , with being chosen as the commonly used Gaussian functions, which have the form
(7) is the center of the receptive where field and is the width of the Gaussian function. It has been proven that network (6) can approximate any conto any arbitrary tinuous function over a compact set accuracy as (8) where is ideal NN weights, and is the NN approximation error. such Assumption 5: There exist ideal constant weights with constant for all . Morethat over, is bounded by on the compact set . are “artificial” quantities that are reThe ideal weights quired for analytical purposes. According to the discussion in is defined as follows: [30],
which is unknown and needs to be estimated in control design. be the estimate of and the weight estimation error Let be . Remark 5: Although RBFNN is employed in our control design, it can be replaced by other linearly parameterized function approximators such as high-order NNs, fuzzy systems, polynomials, splines and wavelet networks without difficulty. For a unified framework of different approximation structures in adaptive approximation-based control, interested readers can refer to [31]. D. Preliminaries The following lemma will be used for control design and system stability analysis in the remainder of this paper. Lemma 1 [32]: For any continuous function satisfying , where , there exist positive
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smooth functions such that fying
satis-
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Substituting (13) into (12) leads to
(9)
(14)
Remark 6: According to Lemma 1, the unknown continuous functions of delayed states in (4), , satisfy the inequality
where . Define the following integral Lyapunov function candidate, which was first proposed in [33] to avoid control singularity:
(10) with being positive continuous functions, . In this paper, we consider the special case whereby the are known. As for the case of unbounding functions known bounding functions, interested readers can refer to [19]. , denotes the 2-norm, Throughout this paper, and and denote the smallest and largest eigenvalues of a square matrix , respectively. The following func, with a positive constant , is introduced for the purtion pose of the control design: if if
(11)
(15)
, and . Then, can be rewritten as the following form by using the first mean value theorem for integrals: where
According to Assumption 1, is positive definite with respect to with respect to time , we obtain
, it is clear that . Differentiating
III. CONTROL DESIGN AND STABILITY ANALYSIS In this section, we will carry out adaptive NN control design in (4) to achieve stable output tracking. In order for system to illustrate the design methodology clearly, the SISO case (i.e., ) is discussed first, which is generalized to the MIMO ) subsequently. For both cases, the closedcase (i.e., loop system will be proved to be SGUUB by Lyapunov stability analysis. The following definitions and notations are used throughout and as the control design and stability analysis. Define
(16)
Due to
and , it is shown that
(17) and the filtered tracking error
as Substituting (14) and (17) into (16) results in (12)
where are chosen such that the polynomial is a Hurwitz polynomial. A. Adaptive NN Control for SISO Case For the SISO case where in the following form:
(18)
, system (4) can be rewritten Using (10) and Young’s inequality, (18) becomes (13)
(19)
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where
Define the following compact sets: (24) (25) (26) is a sufficiently large compact set satisfying defined later in Theorem 1, and is a positive design constant that can be chosen arbitrarily small and “ ” in (26) is used to denote the complement set of set . In addition, it has been shown that is a compact set in [34]. be the approximation of the function , Let . Then, using RBFNN defined in (23), on the compact set as discussed in Section II-C, we have where
with . To overcome the design difficulties from the unknown in (19), the following Lyatime-varying delays punov–Krasovskii functional can be considered [34]:
(27) where the approximation error satisfies with positive constant , . For (22), we design a control law as follows:
(20) (28) The time derivative of
can be expressed as follows: (29) (21)
which can be used to cancel the time delay term on the righthand side of (19), thus circumvent the design difficulty due to the unknown time-varying delay , without introducing any additional uncertainties to the system. For concise notation, the will be omitted whenever the time variables and time-varying delay term is eliminated, in the remainder of this paper. Combining (19) and (21), we obtain (22)
(30) where is defined in (11), is the estimate of the , is a positive constant, and is density function chosen as (31)
as a positive constant specified by the designer. with The adaptation laws are designed as follows:
where
(32) (23)
if
in (23) contains the term , which is not well defined at and may lead to the controller singularity problem, if we utilize to construct the control law. As such, care must be taken to guarantee the boundedness of the control as discussed in [34]. It is noted that the controller singularity takes place at , where the control objective is supposed to the point be achieved. From a practical point of view, once the system reaches its origin, no control action should be taken for less is hard to detect owing to the power consumption. As existence of measurement noise, it is more practical to relax our control objective of convergence to a “ball” rather than the origin.
if
Remark 7: Note that
(33) with , , , and being strictly positive constants. Remark 8: The term in (29) is used to cancel the effect . Unlike traditional caused by the hysteresis term is either robust adaptive controller designs, where assumed to be bounded by a constant or a known function, here is presented as an integral function of control input signal , and there are no assumptions on its boundis not a edness. Considering that the density function function of time, it can be treated as a “parameter” of the hysteresis model and adaption law can be developed to obtain
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an estimate of it. This is crucial for the success of the adaption law design [6]. Remark 9: From (28) and (29), we notice that both sides of depends on as (28) contain the control signal , because can be seen from (29). This is known as the fixed-point problem, where the solvability of can be proved following the proof of Theorem 1.4 about the existence of the hysteresis inverse operator in [35]. Since it is difficult to obtain the explicit solution for from (28), we introduce several possible implementation directly from (28). One is the methods instead of solving time-scale separation approach, recently proposed in [36]: the is a solution of a “fast” dynamical equacontrol signal tion, which means the dynamics of the controller is faster than that of the system plant. Thus, time-scale separation is achieved between the system plant and the controller dynamics using the singular perturbation theory. Second method is adopting the numerical implementation of the inverse hysteresis operator as in [35], where a real-time inverse feedforward control was designed for piezoelectric actuators. In this paper, we introduce a to small delay to evaluate the input: at time , we use in (29) for suitably small , such that in (28) compute , , . The limitabecomes a function of , tion of this method is that its accuracy depends on the choice of . The effects of the variations of will be investigated later in the simulation part in Section IV. Theorem 1: Consider the closed-loop system consisting of the plant (13), the control laws (28) and (29), and adaptation laws (32) and (33). Under Assumptions 1–4, given some ini, , belong to , we can conclude tial conditions that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop defined by system will remain in the compact set
Proof: The method of proof is generally similar to that in our previous works [37], [38], although the details of analysis are different and more complex, due to the presence of time delay and hysteresis in the system. In this proof, we will show , on which the NN approximation is that for a compact set valid, there exist some control parameters and a nonempty initial , such that as long as the initial conditions start compact set , the states and the weights will remain in the conservative in , and finally converge to the compact set . compact set . The proof Both of them belong to the chosen compact set includes two steps, and one could see the whole picture at the end of the proof of Step 2. Step 1: Suppose that both the states and the weights belong , i.e., , , on which NN approxito mation (27) is valid. Consider the following Lyapunov function candidate:
(34)
Considering the adaptive neural control laws and adaptation laws from (28)–(33), the stability analysis is carried out in the following two regions, respectively. , then . Noting (27) • Region 1: If and submitting (28) into (40), we have
with
where entiating
and with respect to time
(38) . Differ-
(39) Substituting (22) into (39) leads to
(40)
(35)
(41) Using Young’s inequality, we have (36) and the tracking error will converge to a neighborhood of zero. In addition, the states and the weights in the closed-loop system defined by will eventually converge to the compact set
(42) Substituting (30), (32), and (42) into (41) leads to
(37) (43) where
.
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For the third term in (43), by completion of squares, we have
b)
, which is the complement set of in , i.e., . In this case, from (33), we have
(44) (49)
For the last two terms in (43), using (2) and (29), we have Substituting (49) into (45), we have
(50)
Combining (45), (48), and (50), we know that
(51) By completion of squares, we have (52) Integrating both sides of (52) over
results in
(45) (53) According to (33), the adaptation law for the estimate of density function is divided into two cases, due to the different regions which belongs to. Therefore, we also need to consider two cases for the analysis of (45). . a) According to (33), we have
According to Assumption 4, we know that Therefore
.
(46) (47) Substituting (46) and (47) into (45), we have
(54) Substituting (44), (51), and (54) into (43), we have
(48) (55)
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where
Multiplying (55) by have
and integrating over
, we
(56) . Therefore, according to
where
in (38), the definition of and . • Region 2: If , then . In this , , , case, the control signal , i.e., all the signals are kept bounded. . From (12), Define we know that 1) there is a state–space representation for , i.e., with mapping , , being a and such stable matrix; 2) there are positive constants that , and 3) the solution of is
Accordingly, it follows that
(57) Noting
and
Fig. 2. Compact sets.
, , such that some and From the definitions of
and . in (35) and (36) as follows
we can see that the values of and depend on the choice , , , and . Thereof the control parameters , there exist some control fore, for a given NN compact set parameters such that for a . Then, as the set of initial condiwe define the initial compact set such that . Therefore, for all tions that belong to , we have for . If and are larger than , this means that the initial . This conditions do not belong to a valid initial compact set completes the proof. B. Adaptive NN Control for MIMO Case
, we have
Substituting (57) into the above inequality leads to
Therefore, we can conclude that all the closed-loop signals are SGUUB for some initial conditions, and the tracking error will converge to a neighborhood of zero. Furthermore, from (56), we also can have
In the foregoing discussions, we design control for the SISO case by Lyapunov synthesis design, so as to elucidate the main ideas of our control design. In this section, we extend the previous result to the MIMO case (4). System (4) is block triangular with respect to inputs , as seen in the fact that nonlinearities only contain inputs from the preceding subsystems. This structure of interconnection facilitates systematic recursive design. Substituting (4) into (12) leads to (59)
(58) where . Therefore, and , as . Step 2: In this step, we prove that there exist some control , such that parameters and a nonempty initial compact set , the states and the as long as initial conditions belong in weights under the proposed control, for , will never es, which belongs to cape from the conservative compact set , as shown in Fig. 2. the chosen compact set in From the definition of the bounds of the compact sets (34) and in (37), we can see that for a given , there exist
. where Define the following integral Lyapunov function candidate: (60)
, and . Applying the first mean value theorem for integrals to (60), we have where
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which is positive definite with respect to due to Assumption . 1, with respect to time , we obtain Differentiating
By utilizing Young’s inequality, we obtain that
(63) and
(64) Substituting (63) and (64) into (62), we have
(65)
(61)
To overcome the design difficulties from the unknown in (65), the following time-varying delays Lyapunov–Krasovskii functional can be considered:
Using (10), after some manipulations, (61) becomes
(66) The time derivative of
can be expressed as follows:
(62) where (67) which can be used to cancel the time delay term on the righthand side of (65). Combining (65) and (67), we obtain (68) where
(69) with
.
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Define the following compact sets:
(70)
(71) (72)
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laws (79) and (80). Under Assumptions 1–4, given that some , belong in , we can conclude initial conditions that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop system will remain in the compact set defined by (81) with
(73) where
is a sufficiently large compact set satisfying defined in Theorem 2, and is a positive design constant that can be chosen arbitrarily small. The compact set is the complement set of set . be the approximation of the function , Let . Then, using RBFNN as defined in (69), on the compact set discussed in Section II-C, we have (74) satisfies where the approximation error with positive constant , . Similar to the procedures of Section III-A, we design the following control law for the system in (4):
and the tracking error will converge to a neighborhood of zero. In addition, the states and the weights in the closed-loop system will eventually converge to the compact set defined by (82)
(75)
where
(76) (77) is defined in (11); where density function ; parameter is chosen as and
is the estimate of the is any positive constant
Proof: The proof is built on that of Theorem 1, and for the conciseness, we will only outline the general approach without going into specific details. For the th subsystem, we design that takes into account the inputs from the preceding subsystems, i.e., . Suppose that both the states and the weights belong to , , , on which NN approximation (74) i.e., is valid. Consider the following Lyapunov function candidate:
(78) a positive constant specified by the designer. with The adaptation laws are chosen as
and with respect to time leads to
(79)
where entiating
(80)
Substituting (68) into (84) leads to
(83) . Differ-
if (84)
if
with , , , and being strictly positive constants. Based on the above design for control and adaptation laws, we are ready to establish the following result for the MIMO case. Theorem 2: Consider the closed-loop system consisting of the plant (4), the control laws (75) and (76), and adaptation
(85)
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Considering the adaptive neural control laws and adaptation laws from (75)–(80), the stability analysis is carried out in the following two regions, respectively. , then . Noting (74) • Region 1: If and submitting (75) into (85), we have
Multiplying (91) by have
Using Young’s inequality, we have (87) Substituting (77), (79), and (87) into (86) leads to
, we
(92) where
(86)
and integrating over
. Therefore,
and . , then . In this case, the • Region 2: If , , , control signal , i.e., all the signals are kept bounded. Similar to the discussion in Theorem 1, we can conclude that the overall closed-loop neural control system is SGUUB in the sense that all of the signals in the closed-loop system are bounded, i.e., the states and the weights in the closed-loop defined in (81), and system will remain in the compact set defined in (82). will eventually converge to the compact set This completes the proof. IV. SIMULATION STUDIES
(88) For the third term in (88), by completion of squares, we have (89) For the last two terms in (88), we can obtain the similar conclusions as (51) and (54)
In this section, results of extensive simulation studies are presented to demonstrate the effectiveness of the proposed adaptive NN approach to deal with uncertain nonlinear systems under the effects of time delay and hysteresis. For clear illustration, we consider first a simplified SISO plant with first-order dynamics, and study the tracking performance of the controller, as well as perform detailed analysis on the effects of control parameter variations. Subsequently, a MIMO plant consisting of two interconnected second-order subsystems is tackled, and the closed-loop properties and tracking behavior are investigated. A. SISO Case For the SISO case, we consider the following first-order scalar nonlinear system with hysteresis and state delay:
(93) (90) where is the plant output; is the plant input and the output of the PI hysteresis model as in (1): , with for , , ; and the time-varying delay and , , and . The objective is to design control such that the output can track the desired trajectory . We adopt the control law and adaption laws designed in Section III-A in the following:
Substituting (89) and (90) into (88), we have
(91) (94) where (95) (96)
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Fig. 3. Output tracking performance of SISO plant S .
Fig. 5. Tracking error comparison result of SISO plant v .
Fig. 4. Control signals of SISO plant S .
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S
with and without
Fig. 6. Learning behavior of NNs of SISO plant S .
(97) (98) if (99) if , and is the estimate of the density where function of . The input of the NNs is . Employing ten nodes for each input dimension, we end nodes for the network . The up with bounding function for the time-delay term is chosen as , and the following initial conditions and controller design parameters are adopted in the simulation: , , , , , , , , , , , . and The simulation results for SISO plant , as described in (93), are shown in Figs. 3–12. From Fig. 3, we can observe that good
tracking performance is achieved. At the same time, the boundedness of the control signals are shown in Fig. 4. It is noted that there is a large difference between and , indicating the significant hysteresis effect. In particular, we highlight the importance of the term in (94), which is used to mitigate the effect caused by the hysteresis term in PI hysteresis model , as discussed in Remark 8. The comparison of tracking errors in the presence and absence of is shown in Fig. 5, and it is seen that with , the tracking error resulting from hysteresis is attenuated accordingly. Figs. 6 and 7 show the nonlinear approximation capability of NNs and the norm of NN weights, respectively. The behavior of the estimate of the density function is also indicated in Fig. 8. To investigate the effects of the control parameters on the tracking performance, and to provide recommendations for their selection, we provide the following comparison results and in Figs. 9 and 10. First, as for the design constants shown in Fig. 9, the tracking error can be reduced by increasing the parameter . Second, from (99) and Fig. 10, we know that higher learning rate, i.e., increase of , results in better
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Fig. 7. Norm of NN weights of SISO plant
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S
.
Fig. 10. Tracking error comparison result of SISO plant
S
for different .
Fig. 8. Behavior of the estimate values of the density function p^(t; r). Fig. 11. Tracking error comparison result of SISO plant S for different .
Fig. 9. Tracking error comparison result of SISO plant
S
for different k .
to measurement noise, excitation of high-frequency unmodeled dynamics, as well as excessive control efforts. A similar tradeoff exists with regard to the parameter , which represents the , in (99). In learning rate of the density function estimate general, if is chosen to be too large, then the stability and the robustness of the system may be compromised in a similar way as high gain control. We need to mention that, due to the use of sign function , controllers (94) and (95) become discontinuous, which may excite unmodeled high-frequency plant dynamics and cause the chattering phenomenon. To avoid the undesired chattering phenomenon, we replace the sign function in the , which above control laws with the saturation function is defined as if
tracking performance. While the above results seem to indicate and should be large, caution must be exercised in the that choice of these parameters, due to the fact that there are some tradeoffs between the control performance and other issues. In particular, for the case of control gain , the price to be paid is the high gain control, which also can be seen from (94) and (96). Problems associated with high gain control include sensitivity
if
(100)
if where is a very small positive constant. Therefore, the different choices of also can affect the tracking performance, as shown in Fig. 11. The smaller , the closer is the saturation function
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Fig. 13. Output tracking performance of MIMO plant S .
1
()
Fig. 12. We introduce a small delay t into v t in calculating v (95) to implement the control v t (94) numerically as pointed in Remark 9. Different : is the t have different effects on the tracking performance, where T sampling time: (a) t T ; (b) t T. T ; (c) t T ; and (d) t
1
() 1=
1 =5
1 = 10
= 0 005 1 = 15
approximate to the sign function. As such, though the better tracking performance can be achieved with the smaller , the chattering phenomenon will become more serious, as a result, which degrades the performance finally. In addition, as discussed in Remark 9, we adopt a numerical to implement the conmethod by introducing a small delay trol in (94) instead of solving it directly. The choices of the delay affect the performance as shown in Fig. 12. With the increasing of , the performance becomes worse. In this paper, , where is the sampling time. we choose B. MIMO Case Consider the following MIMO nonlinear system consisting of two interconnected second-order subsystems with time delay and hysteresis:
(101)
where are the plant outputs, ; are the plant inputs and the outputs of the PI hysteresis model as in (1): with for , , and ; and the time-varying , , delays , and . The objective is to design control such that the output can track the desired trajectory , . The control law and adaption laws in (75)–(80) are adopted. The inputs of the NNs are and
Fig. 14. Control signals of MIMO plant S .
, where and , . Employing three nodes nodes for the for each input dimension, we end up with , and nodes for the network . network The bounding functions for the time-delay term are chosen as and , and the following initial conditions and controller design parameters are , adopted in the simulation: , , , , , , , , , , , , , and . Simulation results for MIMO plant , as described in (101), are shown in Figs. 13–20. From Fig. 13, it is seen that good tracking performance is achieved despite large initial tracking and , and they converge to a small neighborhood errors of zero in a relatively short time. At the same time, it can be observed, in Figs. 14–16, that the control signals, norms of NN and remain bounded. Fig. 17 shows weights, and states and the nonlinear approximation capability of NNs
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Fig. 15. Norm of NN weights of MIMO plant S .
Fig. 16. Other states of MIMO plant S .
Fig. 17. Learning behavior of NNs of MIMO plant S .
. Similar relationships between variations of control parameters and effects on tracking performance, as shown for the SISO case, can be verified for the MIMO case as well in Figs. 18–20.
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Fig. 18. Tracking error comparison result of MIMO plant S for different k and k .
Fig. 19. Tracking error comparison result of MIMO plant S for different and .
Fig. 20. Tracking error comparison result of MIMO plant S for different .
V. CONCLUSION Adaptive variable structure neural control has been proposed for a class of uncertain MIMO nonlinear systems with
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unknown state time-varying delays and PI hysteresis nonlinearities. The uncertainties from unknown time-varying delays have been compensated for through the use of appropriate Lyapunov–Krasovskii functionals. The effect of the unknown hysteresis with the PI models was also mitigated using the proposed control. The controller has been made to be free from singularity problem by utilizing integral Lyapunov function. Based on the principle of sliding-mode control, the developed controller can guarantee that all signals involved are SGUUB. Simulation results have verified the effectiveness of the proposed approach. As for future work, it would be interesting to extend the results reported here to deal with some applications involving hysteresis and time delays, such as smart material actuators.
ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for constructive comments that helped to improve the quality and presentation of this paper.
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[17] S. Zhou, G. Feng, and S. K. Nguang, “Comments on robust stabilization of a class of time-delay nonlinear systems,” IEEE Trans. Autom. Control, vol. 47, no. 9, pp. 1586–1586, Sep. 2002. [18] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural network control of nonlinear systems with unknown time delays,” IEEE Trans. Autom. Control, vol. 48, no. 11, pp. 2004–2010, Nov. 2003. [19] S. S. Ge and K. P. Tee, “Approximation-based control of nonlinear MIMO time-delay systems,” Automatica, vol. 43, no. 6, pp. 31–43, 2007. [20] Y. Sun, J. Hsieh, and H. Yang, “On the stability of uncertain systems with multiple time-varying delays,” IEEE Trans. Autom. Control, vol. 42, no. 1, pp. 101–105, Jan. 1997. [21] B. Xu and Y. Liu, “An improved Razumikhin-type theorem and its applications,” IEEE Trans. Autom. Control, vol. 39, no. 4, pp. 839–841, Apr. 1994. [22] M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1048–1060, Jul. 2001. [23] J. Pan, C.-Y. Su, and Y. Stepanenko, “Modeling and robust adaptive control of metal cutting mechanical systems,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 1268–1273. [24] S. S. Ge, C. C. Hang, and T. Zhang, “Adaptive neural network control of nonlinear systems by state and output feedback,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 29, no. 6, pp. 818–828, Dec. 1999. [25] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [26] R. Sepulchre, M. Jankovic, and P. V. Kokotovic´, Constructive Nonlinear Control. London, U.K.: Springer-Verlag, 1997. [27] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992. [28] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [29] S. Haykin, Neural Networks: A Comprehensive Foundations, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [30] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Boston, MA: Kluwer, 2002. [31] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control. Hoboken, NJ: Wiley, 2006. [32] W. Lin and C. J. Qian, “Adaptive control of nonlinearly parameterized systems: The smooth feedback case,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1249–1266, Aug. 2002. [33] S. S. Ge, C. C. Hang, and T. Zhang, “A direct adaptive controller for dynamic systems with a class of nonlinear parameterizations,” Automatica, vol. 35, no. 4, pp. 741–747, 1999. [34] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,” IEEE Trans. Syst. Man Cybern. B, Cybern., vol. 34, no. 1, pp. 499–516, Feb. 2004. [35] P. Krejci and K. Kuhnen, “Inverse control of systems with hysteresis and creep,” Inst. Electr. Eng. Proc.—Control Theory Appl., vol. 148, no. 3, pp. 185–192, 2001. [36] N. Hovakimyan, E. Lavretsky, and A. Sasane, “Dynamic inversion for nonaffine-in-control systems via time-scale separation. Part I,” J. Dyn. Control Syst., vol. 13, no. 4, pp. 451–465, 2007. [37] T. P. Zhang and S. S. Ge, “Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs,” Automatica, vol. 43, pp. 1021–1033, 2007. [38] S. S. Ge and C. Wang, “Adaptive neural network control of uncertain MIMO nonlinear systems,” IEEE Trans. Neural Netw., vol. 15, no. 3, pp. 674–692, May 2004.
Beibei Ren (S’06) received the B.E. degree in mechanical and electronic engineering and the M.E. degree in automation from Xidian University, Xi’an, China, in 2001 and 2004, respectively. Currently, she is working towards the Ph.D. degree at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Her current research interests include neural networks, adaptive control, nonlinear control, and their applications.
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Shuzhi Sam Ge (S’90–M’92–SM’99–F’06) received the B.Sc. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1986 and the Ph.D. degree and DIC from Imperial College, London, U.K., in 1993. He is founding Director of Social Robotics Lab of Interactive Digital Media Institute and Professor of the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He has (co)authored three books and over 300 international journal and conference papers, and edited one book. His current research interests include social robotics, multimedia fusion, adaptive control, intelligent systems, and artificial intelligence. Dr. Ge is the founding Editor-in-Chief of the International Journal of Social Robotics (Springer-Verlag). He has served/been serving as an Associate Editor for a number of flagship journals and a Book Editor of the Taylor & Francis Automation and Control Engineering Series. At IEEE Control Systems Society, he served/serves as Vice President for Technical Activities (2009–2010) and Member of Board of Governors of IEEE Control Systems Society (2007–2009).
Tong Heng Lee (M’90) received the B.A. degree (with first class honors) in engineering tripos from Cambridge University, Cambridge, U.K., in 1980 and the Ph.D. degree in electrical engineering from Yale University, New Haven, CT, in 1987. Currently, he is Professor at the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests are in the areas of adaptive systems, knowledge-based control, intelligent mechatronics, and computational intelligence. He has coauthored
three research monographs and holds four patents (two of which are in the technology area of adaptive systems, and the other two are in the area of intelligent mechatronics). Dr. Lee currently holds Associate Editor appointments in the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, Control Engineering Practice (an IFAC journal), the International Journal of Systems Science (Taylor and Francis, London, U.K.), and Mechatronics journal (Pergamon, Oxford, U.K.). He was the recipient of the Cambridge University Charles Baker Prize in Engineering and the 2004 ASCC (Melbourne) Best Industrial Control Application Paper Prize.
Chun-Yi Su (SM’99) received the Ph.D. degree in control theory and its applications from South China University of Technology, Guangzhou, China, in 1990. Currently, he is Professor and Concordia Research Chair at Concordia University, Montreal, QC, Canada. His current research interest is in control of systems involving hysteresis nonlinearities. Dr. Su has served on the editorial boards of several journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. He has also severed as an organizing committee member for many conferences.
