An Adaptive Output Feedback Control ... - Semantic Scholar
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
TuP06-3
An Adaptive Output Feedback Control Methodology for Non-Minimum Phase Systems Naira Hovakimyan 1 , Bong-Jun Yang2 , Anthony J. Calise School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332
limited to minimum phase systems. Output feedback control of non-minimum phase nonlinear systems remains to be one of the challenging problems in control theory. Ref. [14] addresses this problem and proposes a tool for robust semiglobal practical stabilization. The authors are not aware of any results on adaptive output feedback control of non-minimum phase, uncertain nonlinear systems.
Abstract A method of output feedback design of an adaptive controller is presented that can be used to augment a fixed-gain linear controller. The key feature is that it is applicable to non-minimum phase nonlinear systems, having both parametric uncertainty and unmodeled dynamics. Ultimate boundedness of error signals can be shown using Lyapunov’s direct method. An example is provided to illustrate the effectiveness of the approach.
The reported results on adaptive control have been mostly relying on assumptions like affinity in the control and/or the uncertain parameters, thus limiting the domain of their applicability. More recently, research has focused on relaxing these assumptions by incorporating neural networks (NNs) in the design [15–19]. The role of the NN in these approaches is to model and cancel the effect of the uncertainty in the dynamics. However these approaches are limited to systems with known dimension, and therefore are not robust to unmodeled dynamics. In an adaptive output feedback setting, robustness to unmodeled dynamics can be achieved precluding the use of a state observer in the feedback path. Our recent results show two different ways of achieving adaptive control without the use of a state observer: one is through incorporating a strictly positive real filter in the adaptive element [20], the other is based on observing the tracking error dynamics [21]. Both of these output feedback control approaches are based on augmenting an inverting control design with a NN that reconstructs the unknown dynamics from finite history of available measurements [22]. Despite its popularity for nonlinear control design, feedback linearization can only be applied to minimum phase systems. Moreover, there exist many controllers that are gain scheduled and are not based on inverting designs. From a practical perspective, it is highly desirable to augment these controllers with an adaptive element. The methodology in [23] provides an approach to augmenting a linear controller that relies on definition of error dynamics for which the stability analysis contained in [20, 21] is applicable. This required that the adaptive element be based on feedback inversion. Consequently, this approach to augmentation is limited to minimum phase systems.
1 Introduction One of the most important problems in control theory is that of controlling an uncertain system in order to have its output track a given reference signal. Recent fundamental results include methods for robust design based on the theory of differential games and L2 -gain analysis [1, 2], geometric methods [3], methods for adaptive design of observers and output feedback controllers [4], methods for employing high gain observers [5], backstepping algorithms [6], the notion of input-to-state stability [7], methods for decentralized stabilization of interconnected systems [8], to name just a few. A good survey for constructive nonlinear control can be found in a recent paper of Kokotovic et al. [9]. A common goal for these research efforts is the development of systematic design methods for observing system states and/or controlling system outputs in the presence of structured uncertainties, such as parameter variations, and unstructured uncertainties, such as unmodeled dynamics and disturbances. While for linear systems stabilization and tracking can always be achieved by output feedback via standard methods, like pole-placement, separation principle, LQR or H∞ [10], for nonlinear systems this is not an obvious task. Most of the nonlinear control methodologies, robust or adaptive, impose the assumption on asymptotic stability of the zero-dynamics [6, 11–13], and thus are 1 Research Scientist II, Senior Member IEEE, e-mail: [email protected] 2 Graduate Research Assistant 3 Professor, IEEE Member
0-7803-7516-5/02/$17.00 ©2002 IEEE
3
949
Here we revisit the problem formulation in [23] and develop a methodology that does not rely on feedback inversion and is applicable to non-minimum phase systems. We make use of a linear design model and a linear controller that satisfies a tracking performance specification in the absence of modeling error. To achieve the tracking goal for the nonlinear system, the linear controller is augmented with a neural network (NN) that operates on a finite history of the available measurements [22]. The adaptive laws are written in terms of the output of a linear observer for the nominal system’s error dynamics as in [21]. The basic assumption is that the relative degree of the regulated output variable of the non-minimum phase system is the same as that of the linear model used in the design of the linear controller. In other words, the relative degree of the regulated output variable is known, but the dimension of the plant dynamics is unknown. Another requirement is that the zero-dynamics of the linear model represent the internal dynamics of the non-minimum phase system to sufficient accuracy. The error dynamics are shown to have similar features as the one in [21], thus the proof of ultimate boundedness follows directly. The benefits of the approach are that it is robust to both parametric uncertainties and unmodeled dynamics, and that it is applicable to non-minimum phase systems.
ment variables, f and h are sufficiently smooth partially known functions, f (0) = 0, h(0) = 0, and r is the relative degree of the system. Moreover, n need not be known. The output y can be equivalently expressed as y = cTm ξ, where cTm = [ 1 0 · · · 0 ] ∈ R1×r , ξ T , [ x1 · · · xr ], ξ ∈ Rr . After a suitable change of coordinates, the linear plant model can also be put in normal form [14]: z˙ l =F0 z l + g 0 xl1 x˙ l1 =xl2 .. . x˙ lr−1 =xlr x˙ lr =hT0 z l + a1 xl1 + · · · + ar xlr + bu yl =xl1 where z l ∈ Rm−r are the states of the zero dynamics, and m ≤ n is the dimension of the plant model. Consider the following stabilizing linear controller for the dynamics in (2): x˙ c = Ac xc + bc (yc − y) where xc ∈ Rnc . The plant model in (2), when regulated by (3), constitutes a “reference model”. Defining ξ Tl , [ xl1 · · · xlr ] and denoting xcl the states of the controller in (3) when applied to (2), i.e. when y is replaced by yl in (3), the “reference model” can be written as: Am − bm dc cTm Bz bm cTc ξl ξ˙ l z˙ l = g 0 cTm F0 0 zl T xcl −bc cm 0 Ac x˙ cl | {z } | {z } | {z } ¯ A
˙l x
¯r b
yl
Consider an observable and stabilizable nonlinear system in normal form [3]: f (z o , x1 , · · · , xr )
x˙ 1
=
x2 .. .
x˙ r−1 x˙ r
= =
xr h(z o , x1 , · · · , xr , u)
y
=
x1
xl
bm dc + 0 yc bc | {z }
2 Problem Formulation
=
(3)
uec = cTc xc + dc (yc − y) ,
The paper is organized as follows. In Section 2 we define the system dynamics, the linear model used for the design of the linear controller, the linear control law and state the problem formulation. Section 3 addresses the design of the adaptive element and derivation of the error dynamics. In Section 4 we describe a modification to the adaptive element that can be used to improve the tracking performance. An illustrative example is given in Section 5. Conclusions are in Section 6. Throughout the manuscript bold symbols are used for column vectors, small letters for scalars, capitol letters for matrices.
z˙ o
(2)
=
Am =
Bz =
where z o ∈ Rn−r are the states of the internal dynamics, u ∈ R1 and y ∈ R1 are control and measure-
950
0 0 .. .
1 0 .. .
0 1 .. .
··· ··· .. .
a1
a2
···
ar
cm
|
where
(1)
0 {z
ξl 0 zl , } x cl
£
¯T c y
0 0 .. . hT0
¤
r×(m−r)
,
(4)
, r×r
bm =
0 0 .. . b
(5)
r×1
where ∆1 (z 0 , ξ, u) = 1b [h(z 0 , ξ, u)−hT0 z 1 −a1 x1 −· · ·− ar xr − bu] can be viewed as the portion of the modeling error that lies in the range space of the control. Augment the linear controller in (3) with an adaptive signal: u = uec − uN N , (12)
The dynamics in (4) can be written in the following compact form: ¯ r yc ¯ l+b x˙ l =Ax yl =¯ cTy xl ,
(6)
where xl ∈ Rm+nc , and A¯ is Hurwitz. The objective is to augment the linear control law uec in (3) with an adaptive element so that when applied to the system (1) the output y tracks yc . In what follows, we will derive error dynamics for the signal yl − y and show that it is similar to the one in [21, 24], and hence is ultimately bounded. Since yl tracks yc asymptotically, boundedness of yl − y ensures that y tracks yc with bounded errors.
where uN N is designed to approximately cancel ∆1 . Notice from (12) and (11) that ∆1 depends on uN N through u, and that the role of uN N is to cancel ∆1 . The following assumption introduces a sufficient condition for the existence and uniqueness of a solution for uN N . Assumption 2 The mapping uN N 7→ ∆1 is a contraction.
3 The Adaptive Element and the Error Dynamics
Assumption 1 is equivalent to [20]: ³ ∂h ´ sign(b) = sign ∂u ¯ ∂h ¯ ¯ ¯ ¯ ¯/2 < |b| < ∞ . ∂u
For the derivation of the error dynamics we need the following assumption, establishing the relationship between the system (1) and the linear model (2). Assumption 1 The z 0 dynamics in (1) can be rearranged as: z˙ 1 z˙ 2
=
f 1 (z 1 , z 2 , ξ), z 1 ∈ Rm−r
(7)
=
f 2 (z 1 , z 2 , ξ), z 2 ∈ Rn−m ,
(8)
(13)
These conditions imply that control reversal is not permitted, and that there is a lower bound on the estimate of control effectiveness, b. A single hidden layer neural network (SHLNN) is used to approximate ∆1 in (11). We first recall the main result from [22, 26].
where the zero solution of z˙ 2 = f 2 (0, z 2 , 0) is asymptotically stable, and the dynamics in (7) can be written as:
Theorem 1 For arbitrary ²∗ > 0, there exist bounded constant weights M , N such that:
z˙ 1 = f 1 (z 1 , z 2 , ξ) = F0 z 1 + g 0 y + ∆2 (z 1 , z 2 , ξ) , (9) where ∆2 = f 1 (z 1 , z 2 , ξ) − F0 z 1 − g 0 y can be viewed as modeling error in the dynamics of z 1 , satisfying the following upper bound [25]:
∆1 = M T σ(N T η) + ε(η), kε(η)k ≤ ²∗ ,
(14)
where ε(η) is the NN reconstruction error and η is the network input vector
(10) x ∈ Ωx ⊂ Rm+nc £ ¤T with known γ1 , γ2 > 0, and x , ξ T z T1 xTc . k∆2 k ≤ γ1 kxk + γ2 ,
η(t) = [ 1
Remark 1 Assumption 1 basically states the relationship between the nonlinear system in (1) and the linear model in (2), used for the design of the linear controller. In case m = n there are no z 2 dynamics, and the relationship in (9) implies that the zero-dynamics in (2) represent the internal dynamics of (1) up to the accuracy of ∆2 . If m < n, Assumption 1 does not preclude stable dynamics in (1) that are not modeled in (2), like that of z 2 .
¯ Td (t) u
¯ Td (t) ]T , y
kηk ≤ η ∗
¯ Td (t) = [u(t) u
u(t − d)
¯ Td (t) = [y(t) y
y(t − d) · · · y(t − (n1 − 1)d)]
· · · u(t − (n1 − r − 1)d)]
T
T
(15) with n1 ≥ n and d > 0, σ being a vector of squashth σ(·), £ing functions ¤ £ Tits i¤ element being defined like T σ(N η) i = σ (N η)i . The adaptive signal uN N is designed as:
The last equation of the system dynamics in (1) can be put in the following form:
ˆ T σ(N ˆ T η) , uN N = M
h(z 0 , x1 , · · · , xr , u) = hT0 z 1 +a1 x1 +· · ·+ar xr +b(u+∆1 ) (11)
ˆ and N ˆ are estimates of M and N to be where M adapted online.
951
(16)
a linear bound in the error norm. This type of upper bound can always be encompassed by the stability proof laid out in [21] at the price of having larger ultimate bounds. Due to space limitations, this is not elaborated here.
Using (11), (8), and (9), the nonlinear system in (1) under the regulation of (12) along with (3) can be written as: x˙ = z˙ 2 = y = in which
¯ b
=
bm 0 , 0
¯r yc − bu ¯ NN + ∆ ¯ +b Ax f 2 (z 2 , z 1 , ξ) ¯y x , c
(17)
∆1 ∆ = ∆2 , 0
Remark 2 If the systems in (1) and (2) are minimum phase, Assumption 1 is not required. The linear controller in (3) can be designed to stabilize only the ξ l dynamics in (2), thus having the closed loop system in (4) without z l states. The error vector in (18) can be defined without the z l −z 1 states, and thus the unmatched uncertainty ∆2 will not be present in the error dynamics. Then the overall stability analysis framework will be exactly the same as in [21].
0 ∆1 = · · · . b∆1
With the following definition of the error vector E T , [ (ξ l − ξ)T
(z l − z 1 )T
(xcl − xc )T ] (18)
following (6) and (17), the error dynamics can be expressed as: ¯ N N − ∆1 ) − B∆2 ˙ =AE ¯ + b(u E ¯ , z =CE
4 Reducing the Ultimate Bound In many realistic applications it is of great interest to achieve tracking performance with the smallest possible ultimate bound. For non-minimum phase systems this is hard to ensure even for linear systems. Towards this end, depending upon the particular application the control design in (12) can be augmented by an additional linear compensator. Consider the following augmentation of the controller in (12):
(19)
where z represents the signals available for feedback: · ¸ · ¸ yl − y cm 0 z= = E, (20) xcl − xc 0 I {z } | ¯ C
£
¤
u ¯ = u + udc ,
0 I 0 , I ∈ R(m−r)×(m−r) . The and B T = weight adaptation laws are similar to the ones proposed in [21]. Introduce the following linear error observer for the dynamics in (19): ˆ + K(z − z ˆ˙ = A¯E ˆ) E ˆ, ˆ = C¯ E z
where u has been defined in (12), and udc is a linear dynamic compensator, defined as: x˙ dc = Adc xdc + Bdc z
(24)
udc = cTdc xdc + ddc z .
(21)
With u ¯ defined in (23), the dynamics in (17) can be re-written:
where K is chosen to make A¯ − K C¯ stable, and the following adaptive laws: ¯ + kM ˆ TPb ˆ] ˆ˙ = − F [(σ ˆ T η)E ˆ −σ M ˆ0N ¯M ˆ b ˆ Tσ ˆ˙ = − G[η EP ˆ] , N ˆ 0 + kN
(23)
¯r yc − b(u ¯ N N + udc ) + ∆ ¯ +b x˙ =Ax y =¯ cy x z˙ 2 =f 2 (z 2 , z 1 , ξ) .
(22)
in which F, G > 0 are positive definite adaptation gain ˆ η), σ ˆ , σ(N matrices, k > 0 is a constant, σ ˆ 0 is the Jacobian computed at the estimates. From this point on the stability analysis laid out in [21,24] can be followed to prove ultimate boundedness of error signals in (19). However, it is important to point out the difference between the error dynamics in [21, 24] and the one in (19). The error dynamics in [21, 24] has only matched uncertainty, i.e. the forcing term therein has the form ¯ N N − ∆1 ). In (19), due to the non-minimum phase b(u nature of the nonlinear system, the error dynamics also involve the states of the internal dynamics, which results in unmatched uncertainty ∆2 . However, Assumption 1 implies that the unmatched uncertainty satisfies
(25)
Applying the controller in (24) to dynamics in (25), and comparing it with the reference model of (6) leads to the following redefined error dynamics: ¸ ¸· ¸ · · ¯ dc C¯ bc ¯ dc ˙ E A¯ + bd E = xdc Bdc C¯ Adc x˙ dc {z } | {z } | L
+
·
¯ b 0
¸
(uN N − ∆1 ) −
Ea
·
B 0
¸
∆2 ,
that can be re-written: · ¸ · ¸ ¯ b B ˙ E a = LE a + (uN N − ∆1 ) − ∆2 . (26) 0 0
952
The additional controller in (24) is an observer-based design: the state feedback gain is found using LQR design, and the observer part of the design is carried out using a Loop Transfer Recovery (LTR) method [27]. The error observer is built so that its poles are 3 times faster than those of the error dynamics.
Reference Model
+
yc
+
Exising Controller
Existing Controller
uec ( ym )
Plant Model
xcm uec(y)
xc
+
u
+
y
-~
Fig. 2 compares output responses subject to a square wave reference command.The nominal response does not track the reference command, while the augmented control law achieves a reasonable response. Fig. 3
y
u ad
uNN + + u dc ud SHL NN
True Plant
-
ym
yd
Linear Controller
Error Observer
xcm-xc
Output response Comparison
E 3
Figure 1: Output Feedback Augmentation 2
Notice that with the choice of design gains in (24) the eigenvalues of L can always be placed in the open lefthalf plane. The dynamics in (26) are similar to that in (19), except for the dimension of the error vector. Thus its stability analysis can be carried out along lines similar to those in [21, 24]. The conceptual layout for the overall controller design process is presented in Figure 1.
y
1
0
−1
−2
y
m
without NN with NN −3
5 Simulation Results
0
Consider the following nonlinear system in normal form:
10
20
30
40 50 time(sec)
60
70
80
90
Figure 2: Comparison of Step Responses
ξ˙ = −2ξ − 0.9z1 − 0.5z2 + u z˙1 = 0.9z1 + 0.1z1 z22 + 1.7ξ 3 z˙2 = −3z2 + z1 + ξ 2 y = ξ.
shows the NN output uN N and the uncertainties ∆1 and ∆2 . All of the responses are bounded.
(27)
2
modelling error
The linear model used for the design of the linear controller is given in the following normal form: ξ˙l = −2ξl − zl + u z˙l = zl + 2ξl y = ξl .
(28)
NN
0 −1 −2
The transfer function for the linear model is given by
∆ 1 u
1
0
10
20
30
40 50 time(sec)
60
70
80
2
90
∆
modelling error
2
s−1 y(s) = u(s) s(s + 1)
(29)
resulting in the following uncertainties ∆1 = 0.1z1 − 0.5z2 , ∆2 = 0.1z1 z22 − 0.1z1 − 0.3ξ.
1 0 −1 −2
The following linear controller has been designed for the linear model: s+1 uec (s) =− . yc (s) − y(s) s + 2.6
0
5
10
15
20 time(sec)
25
30
35
40
Figure 3: Modeling error and adaptive signal
(30)
953
[13] R. Marino and P. Tomei. Global adaptive output feedback control of nonlinear systems, part ii: Nonlinear parameterization. IEEE Trans. Autom. Contr., 38:33–48, 1993.
6 Conclusion This paper describes an approach for augmenting a linear controller with an output feedback adaptive element. The approach is applicable to non-minimum phase nonlinear systems. The key properties of the design are that only output variables are used, and the design is adaptive to both parametric errors and unmodelled/unmatched dynamics. The main assumptions are that the relative degree of the regulated variable is known, and the unmatched uncertainty in the error dynamics satisfies a conic sector bound.
[14] A. Isidori. A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback. IEEE Transactions on Automatic Control, 45(10):1817–1827, 2000. [15] K.S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, 1:4– 27, 1990. [16] Y. Kim and F.L. Lewis. High Level Feedback Control with Neural Networks. World Scientific, 1998.
References
[17] F. Lewis. Nonlinear network structures for feedback control. Asian Journal of Control, 1(4):205–228, 1999.
[1] T. Basar and B. Bernhard. H∞ -Optimal Control and Related Minimax Design Problems. Boston, MA: Birkhauser, 1990.
[18] S. Seshagiri and H.K. Khalil. Output feedback control of nonlinear systems using rbf neural networks. IEEE Trans. Neural Networks, 11(1):69–79, 2000.
[2] A.J. Van der Schaft. L2 -gain analysis of nonlinear systems and nonlinear H∞ control. IEEE Trans. Autom. Contr., 37(6):770–784, 1992. [3] A. Isidori. Nonlinear Control Systems. Springer, Berlin, 1995.
[19] G.A. Rovithakis. Robustifying nonlinear systems using high-order neural network controller. IEEE Transactions on Automatic Control, 44:102–108, 1999.
[4] R. Marino and P. Tomei. Nonlinear Control Design: Geometric, Adaptive, & Robust. Prentice Hall, New Jersey, 1995.
[20] A.J. Calise, N. Hovakimyan, and M. Idan. Adaptive output feedback control of nonlinear systems using neural networks. Automatica, 37:1201–1211, 2001.
[5] H.K. Khalil. Nonlinear Systems. Prentice Hall, New Jersey, 2002.
[21] N. Hovakimyan, F. Nardi, N. Kim, and A.J. Calise. Adaptive output feedback control of uncertain systems using single hidden layer neural networks. IEEE Transactions on Neural Networks, 2002, available from http://controls.ae.gatech.edu/papers.
[6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic. Nonlinear and Adaptive Control Design. John Wiley & Sons, New York, 1995. [7] E.D. Sontag and Y. Wang. New characterizations of input-to-state stability. IEEE Trans. Autom. Contr., 41(9):1283 –1294, 1996.
[22] E. Lavretsky, N. Hovakimyan, and A.J. Calise. Reconstruction of continuous-time dynamics using delayed outputs and feedforward neural networks. IEEE Transactions on Automatic Control, Submitted.
[8] Z.P. Jiang. Decentralized and adaptive nonlinear tracking of large-scale systems via output feedback. IEEE Trans. Autom. Contr., 45(11):2122 –2128, 2000.
[23] A.J. Calise, B.J. Yang, and J.I. Craig. Augmentation of an existing linear controller with an adaptive element. American Control Conference, 2002.
[9] P. Kokotovic and M. Arcak. Constructive nonlinear control: A historical perspective. Automatica, 37:637–662, 2001.
[24] N. Hovakimyan and A.J. Calise. Adaptive output feedback control of uncertain multi-input multioutput systems using single hidden layer neural networks. American Control Conference, 2002.
[10] J.C. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis. State-space solutions to standard H2 and H∞ control problems. IEEE Transactions on Automatic Control, 34:831–847, 1989.
[25] Y.H. Chen. Design of robust controllers for uncertain dynamical systems. IEEE Transactionsons on Automatic Control, 33(5):487–491, 1988.
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TuP06-3
An Adaptive Output Feedback Control Methodology for Non-Minimum Phase Systems Naira Hovakimyan 1 , Bong-Jun Yang2 , Anthony J. Calise School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332
limited to minimum phase systems. Output feedback control of non-minimum phase nonlinear systems remains to be one of the challenging problems in control theory. Ref. [14] addresses this problem and proposes a tool for robust semiglobal practical stabilization. The authors are not aware of any results on adaptive output feedback control of non-minimum phase, uncertain nonlinear systems.
Abstract A method of output feedback design of an adaptive controller is presented that can be used to augment a fixed-gain linear controller. The key feature is that it is applicable to non-minimum phase nonlinear systems, having both parametric uncertainty and unmodeled dynamics. Ultimate boundedness of error signals can be shown using Lyapunov’s direct method. An example is provided to illustrate the effectiveness of the approach.
The reported results on adaptive control have been mostly relying on assumptions like affinity in the control and/or the uncertain parameters, thus limiting the domain of their applicability. More recently, research has focused on relaxing these assumptions by incorporating neural networks (NNs) in the design [15–19]. The role of the NN in these approaches is to model and cancel the effect of the uncertainty in the dynamics. However these approaches are limited to systems with known dimension, and therefore are not robust to unmodeled dynamics. In an adaptive output feedback setting, robustness to unmodeled dynamics can be achieved precluding the use of a state observer in the feedback path. Our recent results show two different ways of achieving adaptive control without the use of a state observer: one is through incorporating a strictly positive real filter in the adaptive element [20], the other is based on observing the tracking error dynamics [21]. Both of these output feedback control approaches are based on augmenting an inverting control design with a NN that reconstructs the unknown dynamics from finite history of available measurements [22]. Despite its popularity for nonlinear control design, feedback linearization can only be applied to minimum phase systems. Moreover, there exist many controllers that are gain scheduled and are not based on inverting designs. From a practical perspective, it is highly desirable to augment these controllers with an adaptive element. The methodology in [23] provides an approach to augmenting a linear controller that relies on definition of error dynamics for which the stability analysis contained in [20, 21] is applicable. This required that the adaptive element be based on feedback inversion. Consequently, this approach to augmentation is limited to minimum phase systems.
1 Introduction One of the most important problems in control theory is that of controlling an uncertain system in order to have its output track a given reference signal. Recent fundamental results include methods for robust design based on the theory of differential games and L2 -gain analysis [1, 2], geometric methods [3], methods for adaptive design of observers and output feedback controllers [4], methods for employing high gain observers [5], backstepping algorithms [6], the notion of input-to-state stability [7], methods for decentralized stabilization of interconnected systems [8], to name just a few. A good survey for constructive nonlinear control can be found in a recent paper of Kokotovic et al. [9]. A common goal for these research efforts is the development of systematic design methods for observing system states and/or controlling system outputs in the presence of structured uncertainties, such as parameter variations, and unstructured uncertainties, such as unmodeled dynamics and disturbances. While for linear systems stabilization and tracking can always be achieved by output feedback via standard methods, like pole-placement, separation principle, LQR or H∞ [10], for nonlinear systems this is not an obvious task. Most of the nonlinear control methodologies, robust or adaptive, impose the assumption on asymptotic stability of the zero-dynamics [6, 11–13], and thus are 1 Research Scientist II, Senior Member IEEE, e-mail: [email protected] 2 Graduate Research Assistant 3 Professor, IEEE Member
0-7803-7516-5/02/$17.00 ©2002 IEEE
3
949
Here we revisit the problem formulation in [23] and develop a methodology that does not rely on feedback inversion and is applicable to non-minimum phase systems. We make use of a linear design model and a linear controller that satisfies a tracking performance specification in the absence of modeling error. To achieve the tracking goal for the nonlinear system, the linear controller is augmented with a neural network (NN) that operates on a finite history of the available measurements [22]. The adaptive laws are written in terms of the output of a linear observer for the nominal system’s error dynamics as in [21]. The basic assumption is that the relative degree of the regulated output variable of the non-minimum phase system is the same as that of the linear model used in the design of the linear controller. In other words, the relative degree of the regulated output variable is known, but the dimension of the plant dynamics is unknown. Another requirement is that the zero-dynamics of the linear model represent the internal dynamics of the non-minimum phase system to sufficient accuracy. The error dynamics are shown to have similar features as the one in [21], thus the proof of ultimate boundedness follows directly. The benefits of the approach are that it is robust to both parametric uncertainties and unmodeled dynamics, and that it is applicable to non-minimum phase systems.
ment variables, f and h are sufficiently smooth partially known functions, f (0) = 0, h(0) = 0, and r is the relative degree of the system. Moreover, n need not be known. The output y can be equivalently expressed as y = cTm ξ, where cTm = [ 1 0 · · · 0 ] ∈ R1×r , ξ T , [ x1 · · · xr ], ξ ∈ Rr . After a suitable change of coordinates, the linear plant model can also be put in normal form [14]: z˙ l =F0 z l + g 0 xl1 x˙ l1 =xl2 .. . x˙ lr−1 =xlr x˙ lr =hT0 z l + a1 xl1 + · · · + ar xlr + bu yl =xl1 where z l ∈ Rm−r are the states of the zero dynamics, and m ≤ n is the dimension of the plant model. Consider the following stabilizing linear controller for the dynamics in (2): x˙ c = Ac xc + bc (yc − y) where xc ∈ Rnc . The plant model in (2), when regulated by (3), constitutes a “reference model”. Defining ξ Tl , [ xl1 · · · xlr ] and denoting xcl the states of the controller in (3) when applied to (2), i.e. when y is replaced by yl in (3), the “reference model” can be written as: Am − bm dc cTm Bz bm cTc ξl ξ˙ l z˙ l = g 0 cTm F0 0 zl T xcl −bc cm 0 Ac x˙ cl | {z } | {z } | {z } ¯ A
˙l x
¯r b
yl
Consider an observable and stabilizable nonlinear system in normal form [3]: f (z o , x1 , · · · , xr )
x˙ 1
=
x2 .. .
x˙ r−1 x˙ r
= =
xr h(z o , x1 , · · · , xr , u)
y
=
x1
xl
bm dc + 0 yc bc | {z }
2 Problem Formulation
=
(3)
uec = cTc xc + dc (yc − y) ,
The paper is organized as follows. In Section 2 we define the system dynamics, the linear model used for the design of the linear controller, the linear control law and state the problem formulation. Section 3 addresses the design of the adaptive element and derivation of the error dynamics. In Section 4 we describe a modification to the adaptive element that can be used to improve the tracking performance. An illustrative example is given in Section 5. Conclusions are in Section 6. Throughout the manuscript bold symbols are used for column vectors, small letters for scalars, capitol letters for matrices.
z˙ o
(2)
=
Am =
Bz =
where z o ∈ Rn−r are the states of the internal dynamics, u ∈ R1 and y ∈ R1 are control and measure-
950
0 0 .. .
1 0 .. .
0 1 .. .
··· ··· .. .
a1
a2
···
ar
cm
|
where
(1)
0 {z
ξl 0 zl , } x cl
£
¯T c y
0 0 .. . hT0
¤
r×(m−r)
,
(4)
, r×r
bm =
0 0 .. . b
(5)
r×1
where ∆1 (z 0 , ξ, u) = 1b [h(z 0 , ξ, u)−hT0 z 1 −a1 x1 −· · ·− ar xr − bu] can be viewed as the portion of the modeling error that lies in the range space of the control. Augment the linear controller in (3) with an adaptive signal: u = uec − uN N , (12)
The dynamics in (4) can be written in the following compact form: ¯ r yc ¯ l+b x˙ l =Ax yl =¯ cTy xl ,
(6)
where xl ∈ Rm+nc , and A¯ is Hurwitz. The objective is to augment the linear control law uec in (3) with an adaptive element so that when applied to the system (1) the output y tracks yc . In what follows, we will derive error dynamics for the signal yl − y and show that it is similar to the one in [21, 24], and hence is ultimately bounded. Since yl tracks yc asymptotically, boundedness of yl − y ensures that y tracks yc with bounded errors.
where uN N is designed to approximately cancel ∆1 . Notice from (12) and (11) that ∆1 depends on uN N through u, and that the role of uN N is to cancel ∆1 . The following assumption introduces a sufficient condition for the existence and uniqueness of a solution for uN N . Assumption 2 The mapping uN N 7→ ∆1 is a contraction.
3 The Adaptive Element and the Error Dynamics
Assumption 1 is equivalent to [20]: ³ ∂h ´ sign(b) = sign ∂u ¯ ∂h ¯ ¯ ¯ ¯ ¯/2 < |b| < ∞ . ∂u
For the derivation of the error dynamics we need the following assumption, establishing the relationship between the system (1) and the linear model (2). Assumption 1 The z 0 dynamics in (1) can be rearranged as: z˙ 1 z˙ 2
=
f 1 (z 1 , z 2 , ξ), z 1 ∈ Rm−r
(7)
=
f 2 (z 1 , z 2 , ξ), z 2 ∈ Rn−m ,
(8)
(13)
These conditions imply that control reversal is not permitted, and that there is a lower bound on the estimate of control effectiveness, b. A single hidden layer neural network (SHLNN) is used to approximate ∆1 in (11). We first recall the main result from [22, 26].
where the zero solution of z˙ 2 = f 2 (0, z 2 , 0) is asymptotically stable, and the dynamics in (7) can be written as:
Theorem 1 For arbitrary ²∗ > 0, there exist bounded constant weights M , N such that:
z˙ 1 = f 1 (z 1 , z 2 , ξ) = F0 z 1 + g 0 y + ∆2 (z 1 , z 2 , ξ) , (9) where ∆2 = f 1 (z 1 , z 2 , ξ) − F0 z 1 − g 0 y can be viewed as modeling error in the dynamics of z 1 , satisfying the following upper bound [25]:
∆1 = M T σ(N T η) + ε(η), kε(η)k ≤ ²∗ ,
(14)
where ε(η) is the NN reconstruction error and η is the network input vector
(10) x ∈ Ωx ⊂ Rm+nc £ ¤T with known γ1 , γ2 > 0, and x , ξ T z T1 xTc . k∆2 k ≤ γ1 kxk + γ2 ,
η(t) = [ 1
Remark 1 Assumption 1 basically states the relationship between the nonlinear system in (1) and the linear model in (2), used for the design of the linear controller. In case m = n there are no z 2 dynamics, and the relationship in (9) implies that the zero-dynamics in (2) represent the internal dynamics of (1) up to the accuracy of ∆2 . If m < n, Assumption 1 does not preclude stable dynamics in (1) that are not modeled in (2), like that of z 2 .
¯ Td (t) u
¯ Td (t) ]T , y
kηk ≤ η ∗
¯ Td (t) = [u(t) u
u(t − d)
¯ Td (t) = [y(t) y
y(t − d) · · · y(t − (n1 − 1)d)]
· · · u(t − (n1 − r − 1)d)]
T
T
(15) with n1 ≥ n and d > 0, σ being a vector of squashth σ(·), £ing functions ¤ £ Tits i¤ element being defined like T σ(N η) i = σ (N η)i . The adaptive signal uN N is designed as:
The last equation of the system dynamics in (1) can be put in the following form:
ˆ T σ(N ˆ T η) , uN N = M
h(z 0 , x1 , · · · , xr , u) = hT0 z 1 +a1 x1 +· · ·+ar xr +b(u+∆1 ) (11)
ˆ and N ˆ are estimates of M and N to be where M adapted online.
951
(16)
a linear bound in the error norm. This type of upper bound can always be encompassed by the stability proof laid out in [21] at the price of having larger ultimate bounds. Due to space limitations, this is not elaborated here.
Using (11), (8), and (9), the nonlinear system in (1) under the regulation of (12) along with (3) can be written as: x˙ = z˙ 2 = y = in which
¯ b
=
bm 0 , 0
¯r yc − bu ¯ NN + ∆ ¯ +b Ax f 2 (z 2 , z 1 , ξ) ¯y x , c
(17)
∆1 ∆ = ∆2 , 0
Remark 2 If the systems in (1) and (2) are minimum phase, Assumption 1 is not required. The linear controller in (3) can be designed to stabilize only the ξ l dynamics in (2), thus having the closed loop system in (4) without z l states. The error vector in (18) can be defined without the z l −z 1 states, and thus the unmatched uncertainty ∆2 will not be present in the error dynamics. Then the overall stability analysis framework will be exactly the same as in [21].
0 ∆1 = · · · . b∆1
With the following definition of the error vector E T , [ (ξ l − ξ)T
(z l − z 1 )T
(xcl − xc )T ] (18)
following (6) and (17), the error dynamics can be expressed as: ¯ N N − ∆1 ) − B∆2 ˙ =AE ¯ + b(u E ¯ , z =CE
4 Reducing the Ultimate Bound In many realistic applications it is of great interest to achieve tracking performance with the smallest possible ultimate bound. For non-minimum phase systems this is hard to ensure even for linear systems. Towards this end, depending upon the particular application the control design in (12) can be augmented by an additional linear compensator. Consider the following augmentation of the controller in (12):
(19)
where z represents the signals available for feedback: · ¸ · ¸ yl − y cm 0 z= = E, (20) xcl − xc 0 I {z } | ¯ C
£
¤
u ¯ = u + udc ,
0 I 0 , I ∈ R(m−r)×(m−r) . The and B T = weight adaptation laws are similar to the ones proposed in [21]. Introduce the following linear error observer for the dynamics in (19): ˆ + K(z − z ˆ˙ = A¯E ˆ) E ˆ, ˆ = C¯ E z
where u has been defined in (12), and udc is a linear dynamic compensator, defined as: x˙ dc = Adc xdc + Bdc z
(24)
udc = cTdc xdc + ddc z .
(21)
With u ¯ defined in (23), the dynamics in (17) can be re-written:
where K is chosen to make A¯ − K C¯ stable, and the following adaptive laws: ¯ + kM ˆ TPb ˆ] ˆ˙ = − F [(σ ˆ T η)E ˆ −σ M ˆ0N ¯M ˆ b ˆ Tσ ˆ˙ = − G[η EP ˆ] , N ˆ 0 + kN
(23)
¯r yc − b(u ¯ N N + udc ) + ∆ ¯ +b x˙ =Ax y =¯ cy x z˙ 2 =f 2 (z 2 , z 1 , ξ) .
(22)
in which F, G > 0 are positive definite adaptation gain ˆ η), σ ˆ , σ(N matrices, k > 0 is a constant, σ ˆ 0 is the Jacobian computed at the estimates. From this point on the stability analysis laid out in [21,24] can be followed to prove ultimate boundedness of error signals in (19). However, it is important to point out the difference between the error dynamics in [21, 24] and the one in (19). The error dynamics in [21, 24] has only matched uncertainty, i.e. the forcing term therein has the form ¯ N N − ∆1 ). In (19), due to the non-minimum phase b(u nature of the nonlinear system, the error dynamics also involve the states of the internal dynamics, which results in unmatched uncertainty ∆2 . However, Assumption 1 implies that the unmatched uncertainty satisfies
(25)
Applying the controller in (24) to dynamics in (25), and comparing it with the reference model of (6) leads to the following redefined error dynamics: ¸ ¸· ¸ · · ¯ dc C¯ bc ¯ dc ˙ E A¯ + bd E = xdc Bdc C¯ Adc x˙ dc {z } | {z } | L
+
·
¯ b 0
¸
(uN N − ∆1 ) −
Ea
·
B 0
¸
∆2 ,
that can be re-written: · ¸ · ¸ ¯ b B ˙ E a = LE a + (uN N − ∆1 ) − ∆2 . (26) 0 0
952
The additional controller in (24) is an observer-based design: the state feedback gain is found using LQR design, and the observer part of the design is carried out using a Loop Transfer Recovery (LTR) method [27]. The error observer is built so that its poles are 3 times faster than those of the error dynamics.
Reference Model
+
yc
+
Exising Controller
Existing Controller
uec ( ym )
Plant Model
xcm uec(y)
xc
+
u
+
y
-~
Fig. 2 compares output responses subject to a square wave reference command.The nominal response does not track the reference command, while the augmented control law achieves a reasonable response. Fig. 3
y
u ad
uNN + + u dc ud SHL NN
True Plant
-
ym
yd
Linear Controller
Error Observer
xcm-xc
Output response Comparison
E 3
Figure 1: Output Feedback Augmentation 2
Notice that with the choice of design gains in (24) the eigenvalues of L can always be placed in the open lefthalf plane. The dynamics in (26) are similar to that in (19), except for the dimension of the error vector. Thus its stability analysis can be carried out along lines similar to those in [21, 24]. The conceptual layout for the overall controller design process is presented in Figure 1.
y
1
0
−1
−2
y
m
without NN with NN −3
5 Simulation Results
0
Consider the following nonlinear system in normal form:
10
20
30
40 50 time(sec)
60
70
80
90
Figure 2: Comparison of Step Responses
ξ˙ = −2ξ − 0.9z1 − 0.5z2 + u z˙1 = 0.9z1 + 0.1z1 z22 + 1.7ξ 3 z˙2 = −3z2 + z1 + ξ 2 y = ξ.
shows the NN output uN N and the uncertainties ∆1 and ∆2 . All of the responses are bounded.
(27)
2
modelling error
The linear model used for the design of the linear controller is given in the following normal form: ξ˙l = −2ξl − zl + u z˙l = zl + 2ξl y = ξl .
(28)
NN
0 −1 −2
The transfer function for the linear model is given by
∆ 1 u
1
0
10
20
30
40 50 time(sec)
60
70
80
2
90
∆
modelling error
2
s−1 y(s) = u(s) s(s + 1)
(29)
resulting in the following uncertainties ∆1 = 0.1z1 − 0.5z2 , ∆2 = 0.1z1 z22 − 0.1z1 − 0.3ξ.
1 0 −1 −2
The following linear controller has been designed for the linear model: s+1 uec (s) =− . yc (s) − y(s) s + 2.6
0
5
10
15
20 time(sec)
25
30
35
40
Figure 3: Modeling error and adaptive signal
(30)
953
[13] R. Marino and P. Tomei. Global adaptive output feedback control of nonlinear systems, part ii: Nonlinear parameterization. IEEE Trans. Autom. Contr., 38:33–48, 1993.
6 Conclusion This paper describes an approach for augmenting a linear controller with an output feedback adaptive element. The approach is applicable to non-minimum phase nonlinear systems. The key properties of the design are that only output variables are used, and the design is adaptive to both parametric errors and unmodelled/unmatched dynamics. The main assumptions are that the relative degree of the regulated variable is known, and the unmatched uncertainty in the error dynamics satisfies a conic sector bound.
[14] A. Isidori. A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback. IEEE Transactions on Automatic Control, 45(10):1817–1827, 2000. [15] K.S. Narendra and K. Parthasarathy. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks, 1:4– 27, 1990. [16] Y. Kim and F.L. Lewis. High Level Feedback Control with Neural Networks. World Scientific, 1998.
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