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Parametrization of all stabilizing controllers of nonlinear systems
SYSTUtS & CONTROL LLrTTI[ItS ELSEVIER
Systems & Control Letters 29 (1997) 207-213
Parametrization of all stabilizing controllers of nonlinear systems J u n - i c h i I m u r a a,*, T s u n e o Y o s h i k a w a b a Division of Machine Design Engineering, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 739, Japan b Department of Mechanical Engineering, Faculty of Engineering, Kyoto University, Kyoto 606, Japan Received 20 October 1995; revised 13 September 1996
Abstract
This paper develops a new approach to the parametrization of all asymptotically stabilizing controllers of nonlinear systems including time-varying ones, both for state feedback and output feedback. Assumptions of stabilizability and a kind of detectability are requirer. The approach is very direct and simple, and is not based on the so-called factorization and on the H~ framework of nonlinear systems which are used in previous papers. A key of the derivation of the parametrization in this paper is to find an explicit state space representation of free functions, such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems.
Keywords: Nonlinear systems; Parametrization; Asymptotic stabilization; Output feedback
1. Introduction
It is one of the fundamental problems to parameterize all stabilizing controllers of nonlinear systems. Most of the previous papers on this topic are based on operator theory, i.e., the coprime factorization approach [1,2, 4-6, 9, 14], similarly to the case of linear systems. On the other hand, very recently, Lu [7] has developed a state space approach to the controller parametrization problem, which is not based on coprime factorizations. He has derived a parametrization of all smooth state feedback controllers that asymptotically stabilize nonlinear systems under the assumption of stabilizability. His approach is a natural nonlinear extension of the state space approach to the controller parametrization problem of linear systems, which is based on a kind of Ho~ framework (i.e. the LFT framework). Therefore in his approach, there is rLo computational problem of * Corresponding author. Fax: +81-824-22-7193; E-mail: imura@ mec.hiroshima-u.ac.jp.
how to get a coprime factorization of nonlinear systems in state space description. In the case of output feedback, Lu also has given a parametrized controller that asymptotically stabilizes the system. However, it has not been shown there that the derived parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems. His approach developed in the case of state feedback may not be extendable to the case of output feedback. In this paper, we develop a new direct approach to derive a parametrization of all asymptotically stabilizing controllers of nonlinear systems in the case of output feedback as well as state feedback. It is not based on the so-called factorization approach and on the Ho~ framework approach. The assumptions are stabilizability and a kind of detectability. A key in our approach is to find an explicit state space representation of free dynamics, such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems. This makes the proof for the parametrization much simpler, and surprisingly, also allows us to treat the general case
0167-6911/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S01 67-691 1(96)00065-5
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Imura. T. Yoshikawa/Systems & ControlLetters 29 (1997) 207 213
including time-varying systems in both cases of state and output feedback. We use the following notation: ~" denotes an n-dimensional Euclidean space and the Euclidean norm is denoted by I1" II, A real-valued function if(.): ~ ~ ~ is said to belong to the class K (or ~9 E K ) if it is continuous and strictly increasing with 440) = 0.
2. Parametrization of all stabilizing controllers via state feedback In this section, we give the parametrization of all C 1 feedback stabilizing state feedback controllers of nonlinear systems. The problem to be considered here is as follows.
Problem 2.1. Consider the following nonlinear system:
2 = f ( x ) + 9(x)u, x(to) = x0,
{ ~ = f ( x ) - s(x - ~) + 9(x)u, Ks
0 u
fq(q, ~,x), k(~) + hq(q,~,x), ~(to) = ~o, q(to) = qo,
where (a) 2 = f ( x ) + g(x)k(x) is asymptotically stable at the origin and k( O) = O, (b) ~ = s(~) is asymptotically stable at the origin and s is C 1, (c) 0 = fq(q, 0, 0) is asymptotically stable at the origin,
(d) fq(q,x,x) = fq(q,0,0), (e) hq(q,x,x) = hq(q,O,O), (f) fq and hq are C 1 and hq(O, O, O) = O. Proof. (Sufficiency) Letting e = x - ~, the closed loop system is given by
(1)
= f ( x ) + g(x)k(x)
where x E g~" is the state, u E ~m is the input, and to is the initial time. f ( . ) : ~" ~ ~" and 9(') : ~" R "x'~ are C 1 known functions with f ( 0 ) = 0. For the above system, consider the state feedback controller Kx defined by
~ = fc(x~,x), Kx [ u = hc(xc,x),
for the system (1) is given by
Xc(t0) =xc0,
(2)
where fc(', ") : ~n~ x R" ~ ~"o and he(., .) : ~"~ x ~n ~ ~m are C 1, f c ( 0 , 0 ) = 0, h c ( 0 , 0 ) = 0, and x~0 is the initial state of the controller. Then find all C 1 state feedback controllers of Kx that asymptotically stabilize the system given by (1). Please note that Kx expresses a static state feedback controller when nc = 0.
Definition 2.1. A pair ( f , 9) is said to be asymptotically stabilizable, if there exists a C 1 function k(.) : ~n ~ ~m such that 2 = f ( x ) + 9(x)k(x) is asymptotically stable at the origin and k(0) = 0. Then the following result is obtained.
+g(x){k(x - e) - k(x) + hq(q,x - e,x)}, 0 = fq(q,x - e,x), = s(e). Conditions ( a ) - ( f ) imply that this closed loop system is asymptotically stable at the origin [15]. (Necessity) Suppose that a state feedback controller given by
f ~o L(xc,x), =
Kx [ u =hc(xc,x),
xc(to) =x~o
asymptotically stabilizes the system (1), where fe(O, O) = 0 and he(O, O) = O. This means that the closed loop system
2 = f ( x ) + g(x)h~(xc,x), xc = f~(xc,x)
is asymptotically stable at the origin. Thus by constructing appropriate functions fq and hq satisfying the conditions ( c ) - ( f ) , we have only to show Ks = Kx. So let q g ql + x - ~ and
fq(q,e,x)
= -f01) - s(x - ¢) + 9(rl)he(q2,rl)] f~(q2, q)
Theorem 2.1. Suppose that (f, g) is asymptotically stabilizable. Then a parametr&ation of all asymptotically stabilizing controllers via C 1 state feedback
(3)
J
(4) h q ( q , ¢ , x ) = - k ( q l ) + h c ( q 2 , q),
(5)
J.-i. Imura, T Yoshikawal Systems & Control Letters 29 (1997) 207-213
w h e r e q = [q~, T q2T ]T • Then we can show that the above
functions satisfy the conditions (c)-(f), noting that the closed loop system given by (3) is asymptotically stable at the origin. In addition, if we set ql(t0) = ~0, then ql(t) - ~(t). Therefore, the controller Ks under ql(t0) = Go and q2(to) =x¢0 with (4) and (5) is equivalent to the system 02 = f c ( q 2 , x ) , u = hc(qz,x),
q2(i!o) -- xco
which is the controller Kx. This completes the proof. [] The proof developed in Theorem 2.1 is quite different from that by Lu [7]. In addition, even in the linear case, such a proof has not appeared so far. This proof is quite simple and direct in the sense that we do not need to use the so-called factorization approach, albeit it is important. A key of the proof is the direct derivation of (4) and (5). The first element of the vector field fq in (4) is for the generation of ~ in Ks, which is used to cancel out the effect by the dynamics on ~ in Ks, and the second element is for the generation ofx~ under the conditions ~(t0 ) = q 1(to) and xc(t0 ) = q2(t0 ), which is used to express any stabilizing controller. In this way, the construction offq and hq given by (4) and (5) is very intuitive in the proof of the necessity. In addition, the introduction of the variable r/plays an important role to guarantee that (4) and (5) satisfy the conditions (c)-(f). This technique of the proof allows us to treat the output tbedback case and even the case of the general systems including the time-varying case as we will see later. Remark 2.1. In Theorem 2.1, it is assumed that ( f , g ) is asymptotically stabilizable. Whenever a C 1 dynamic state feedback controller exists, does there exist a C 1 static state feedback controller? For this question, we do not have a complete solution yet. However, we can show that if there exists a C 1 dynamic state feedback controller Kx that asymptotically stabiliTes the system (1) in the sense that there exists a Lyapunov function V(-) : ~+,c ~ showing the asymptotic stability of the closed loop system and the V satisfies rank 02 V/Ox~(O) = n¢, then there exists a C ~ static state feedback controller u = k(x) thai: asymptotically stabilizes the system (1) and k ( 0 ) = 0 under rank g ( 0 ) = m. (The proof is a simple extension of Theorem 3 by Tsinias [13].) In this result, the additional condi-
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tion (i.e., the rank condition of V) is still needed together with the asymptotic stability of the closed loop system by the dynamic controller. It is an interesting open problem to solve the above question completely. Next, we will show the other two kinds of parametrization of all stabilizing state feedback controllers of nonlinear systems. This result can be proven by the same technique as Theorem 2.1. Corollary 2.1. The other two kinds of parametriza-
tion of all stabilizing state feedback controllers of nonlinear systems are given by K[ and K[' as follows:
(i) { ~ = f ( ~ ) + g(~)k(~) - g(x)k(x) + g(x)u, K,s
0 = fq(q, ¢,x), u -- k(x) + hq(q, ~,x), ¢(to) = ~o,
q(to) = qo,
where (a) ~ = f ( x ) + g(x)k(x) is asymptotically stable at the origin and k( O) = O, (b) O = fq(q,0,0) is asymptotically stable at the origin, (c) fq(q,x,x) = fq(q,0,0), (d) hq(q,x,x) = hq(q,O,O), (e) fq and hq are C 1 and hq(O, O, O) = O.
(ii) ~0 = fq(q),
K[' (u = k(x) + hq(q), q(to)-- go, where (a) at the (b) (c)
k = f ( x ) + g(x)k(x) is asymptotically stable origin and k(O) = O, q = f q ( q ) is asymptotically stable at the origin, fq and hq are C 1 and hq(O) = O.
Proof. The proof is quite similar to that of Theorem 2.1. So we will show the equations which correspond to (4) and (5) only. In the case of (i), we can choose
fq(q, ~,x)
:[f(ql)+g(ql)k(ql)+g(rl){-k(rl) .-he(q2, r/)} ] fc(q2, r/) (6)
210
J.-i. Imura, T. Yoshikawal Systems & Control Letters 29 (1997) 207-213
I
Y-x
11
Fig. 1. Feedback system.
hq(q, ~,x) = -k(t/) + hc(q2, t/),
(7)
where r / = ql + x - ~. In the case of (ii), we can choose
fq(q)= [ f(ql)+O(ql)hc(q2'ql) ] fc(q2, ql )
(8)
hq(q) = - k ( q I ) ÷ hc(q2, qt )"
(9)
[]
This characterization by K~ corresponds to y = PQr in the case of linear systems , where P is a stable plant transfer function and Q is a free stable proper transfer function. Nonlinear extensions similar to the above example are discussed in [1,7]. In the case of Ks, we can easily see that the same characterization of the closed loop system from r to y as the case of Ks~ will be obtained. In the case of Ks", on the other hand, it does not seem easy to take account of the desired value r in the closed loop, since the vector field of K~~ does not include the state of the plant. The above feedback system is one example, and the advantage of each parametrization may depend on the kind of control problems studied.
In the linear case, it is well known that a parametrization of all stabilizing controllers for stable systems is given by
3. Extension to other cases
K = (I + Q p ) - I Q ,
3.1. Output feedback case
(10)
where P and Q express a transfer function of a stable plant and any stable proper transfer function, respectively. We can show that the parametrization of K~ is a nonlinear analog by simple computation. Lu [7] has given only the case Ks" as a parametrization of all C 1 state feedback controllers by using a framework of nonlinear H ~ control problems. On the other hand, our technique allows us to derive Ks~ and Ks" as well as Ks. Note that there is some difference between Ks~ (and also Ks) and K~~. In fact, the vector field of the controller Ks~(and Ks) includes the state of the plant, i.e., x, while in K~~it does not. Such a difference will be clearer when we consider the following feedback system shown in Fig. 1, where the plant is given by (1) and y = x and the controller is given by K~ with the input r, in other words,
{
~ = f ( ~ ) + g(~)k(¢) - g(x)k(x) + g(x)u, 4 = fq(q, ~,x - r), u = k(x) + hq(q,~,x - r), ~(to) = ~o, q(to) = qo,
where r is the desired value and, k, fq, and hq satisfy the conditions ( a ) - ( e ) in Corollary 2.1. Let us consider the case of ~0 =x0. Then the map of this feedback system from r to y is expressed by
Jc = f ( x ) + g(x)k(x) + g(x)hq(q,x,x - r), (1 = fq(q,x,x - r), y=-x, x(to)=xo, q(to)=qo.
Compared with Lu's technique, our approach allows us to give a parametrization of the output feedback case as well as the state feedback case. Consider the system given by (1) and the output
y=h(x),
(11)
where h(.) : ~ ~ ~P is a C 1 function, and h ( 0 ) = 0. The problem to be considered here is as follows. Problem 3.1. Consider the output feedback controller
Ky defined by ~Xc = fc(Xc, Y) hc(xe, y),
Ky t u =
xe(t0) =xeo
(12)
where f~(., .) : ~nc x ~p____~nc and he(', ') : Nnc x RP--~ R 'n are C 1, fc(0, 0) = 0, he(0, 0) -- 0, and xe0 is the initial state of the controller. Then find all output feedback controllers of Ky that asymptotically stabilize the system given by (1) and (11 ). Definition 3.1 (Vidyasagar [16]). A pair ( f , h ) is said to be weakly detectable, if there exist C 1 functions I ( . ) : R n x R n ~ R n, V(.): R n x ~n R ", and functions ~, fl, and ~ of class K such that, for all x and ~ in a neighborhood of the
J.-i. Imura, T. YoshikawalSystems & Control Letters 29 (1997) 207-213
origin,
(llx -
11)<
11),
8V -ffX-X{f(x) + 9(x)u} + ~-~{f(~) + g(~)u + l(~,h(x))}
<-
(llx- 11),
l(O,O)-- o.
Then the following result is obtained. Theorem3.1. Consider the system 9iven by (1) and (11 ). Suppose that (f, O) is asymptotically stabilizable and (f,h) is weakly detectable. Then a parametrization of all asymptotically stabilizin9 controllers via C 1 output feedback for the system of (1) and (11 ) is given .by { ~ = f ( ~ ) + g ( ~ ) u + l(~,y), Ko
@= fq(q' ~' Y)' u = k(~) + hq(q,, ~, y), ((to) = ¢0, q(to) = qo,
where (a))~ = f ( x ) + 9(x)k(x) is asymptotically stable at the origin and k( O), = O, (b) l(. ) satisfies the condition in Definition 3.1 and l(x,h(x)) = O, (c) @= fq(q,0,0) is asymptotically stable at the origin, (d) fq(q,x,h(x)) = fq(q, 0,0), (e) hq(q,x,h(x)) --=hq(q,O,O), (f) fq and hq are (;1 and hq(0, 0, 0) = 0. Proof. It directly follows from Theorem 3.1 in [16] that the closed loop system is asymptotically stable. The proof of the necessity is quite similar to the proof of the state feedback case, by choosing fq(q, ~, Y) = [f(q,)+g(q,)hc(q2,h(ql)+y-h(~))+l(~,Y)]
211
Lu [7] has also shown that the controller parametrized similarly to Ko in Theorem 3.1 asymptotically stabilizes the closed loop system under the assumption of (f,o)-stabilizability and (f,h)weak detectability. However, his technique used in the case of state feedback may not be applied to show that the parametrized controller expresses all stabilizing controllers in the output feedback case. On the other hand, Theorem 3.1 shows that the obtained parametrization expresses all asymptotically stabilizing controllers even in the output feedback case, by using the same technique as for the state feedback case in Section 2, under the assumption of (f,9)-stabilizability and (f,h)-weak detectability.
3.2. General nonlinear time-varyin9 system case Our technique developed in Sections 2 and 3.1, in addition, can be applied to the case of general nonlinear time-varying systems. In other words, we can give a parametrization of all asymptotically stabilizing controllers for general nonlinear time-varying systems. Here we will concentrate on the state feedback case. The output feedback case can also be treated in a similar way. Let us consider the general nonlinear system given by
k = f(x,u,t),
x(to)=xo,
(15)
where x E ~n is the state, u E R m is the input, and to is the initial time. f ( . , . , .) : ~n × ~m × ~ ~ ~n is a C l known function with f ( 0 , 0, t) = 0. In addition, the function f is assumed to exist in a neighborhood Uz of the origin and sup sup c3f(x,u, t) t zEuz ~z
< oo,
where z = [x u] T. For simplicity, we call hereafter the function that satisfies the above condition the D.B. function, since the derivatives of the function is bounded for any t. Then we consider the following problem.
f c ( q 2 , i i ( q l ) + y -- h(~))
(13)
Problem 3.2 (State feedback case). Let us consider the time-varying state feedback controller given by
hq(q,~,y) ------ k ( q l ) + h c ( q 2 , h ( q l ) + y -
h(¢)).
[]
Kxt
(14)
( xc = fc(xc,x, t) u = hc(xc,x, t) xc( to ) = xco
J.-i. Imura, 72 YoshikawalSystems & Control Letters 29 (1997) 207-213
212
where Arc(., ., .) : N,c x ~ x R --+ ~nc and hc(', ', ") : ~nc x ~n x R --~ N m are the C 1 D.B. functions, f~(0, 0, t) = 0, he(0, 0, t) = 0, and Xc0 is the initial state of the controller. Then find all state feedback controllers of Kxt that uniformly asymptotically stabilize the system given by (15). In addition, we define the following stabilizability, which corresponds to the (f, g)-stabilizability. Definition 3.2. The system given by (15) is said to be asymptotically stabilizable via static state feedback, if there exists a function k(.,.) : R n × E -* Em such that (i) The system 2 = f(x,k(x,t),t) is uniformly asymptotically stable at the origin, (ii) k(0,t) = 0, (iii) k(x, t) is the C l D.B. function. Then we get the following result.
Suppose that the system given by (15) is asymptotically stabilizable via static state feedback. Then a parametrization of all C l state feedback controllers g s t that uniformly asymptotically stabilize the system (15) is given by T h e o r e m 3.2.
hq(q,~,x,t)=-k(ql,t)+hc(q2,q,t), where ~ = q l + x -
(17)
4. []
Finally, as an application we will show an interesting example of Theorem 3.2. We consider the driftless system £1 = Ul,
3f2 ~-- U2,
(18)
3C3 =X2U 1
which is called the chained system in the field o f n o n holonomic control systems [8] and has lately attracted considerable attention on stabilization. By Brockett's Theorem [3], it is well known that there is no C l timeinvariant state feedback controller that asymptotically stabilizes the system given by (18). Thus some continuous time-varying feedback controllers and some discontinuous controllers that asymptotically stabilize the system (18) have been developed by many researchers [8, 11, 12]. We will here concentrate on C 1 time-varying state feedback controllers and give a parametrization of all C 1 time-varying state feedback controllers that uniformly asymptotically stabilize (18). Pomet [10] has derived the following C 1 timevarying static state feedback controller that uniformly asymptotically stabilizes the system (18).
{ 4 = f ( x , u , t ) - s(x - ~), Kst
k(x, t)
q = fq(q, 4,x,t), u = k(~,t) + hq(q, ~,x,t), ~(to) = 40,
=[
x 2 s i n t - ( x l + x2cost) - - ( X 1 + X2 COS t)Xm COS t -- (XiX 2 + X3 )
q(to) = qo,
] '
(19)
where (a) 2 -- f(x, k(x, t), t) is uniformly asymptotically
stable at the origin, k(O,t) = O, and k(x,t) is the C 1 D.B. function, (b) ~ = s(~) is asymptotically stable at the origin, (c) q = fq(q, 0,0, t) is uniformly asymptotically stable at the origin, (d) fq(q,x,x,t) = fq(q,0,0, t), (e) hq(q,x,x,t) = hq(q,O,O,t), (f) fq and hq are the C 1 D.B. functions and hq(O,O,O,t) = O. Proof. The proof is similar to the proof of Theorem 3.1. The equations corresponding to (4) and (5) are given by fq(q, ~,x, t) =
! f(q, hc(q2,~l,t),t) - s(x - ~)] f¢ (q2, r/, t)
J (16)
where x -- [xl,x2] T. This controller satisfies conditions (i)-(iii) in Definition 3.2. So the system (18) is asymptotically stabilizable via C 1 time-varying static state feedback. Thus we can apply Theorem 3.2 to the case of the system (18) to get a parametrization of all uniformly asymptotically stabilizing controllers for the system (18). For example, one of the parametrizations is given by
/ ul
Kst 4= fq(q,~,x,t), u = k(~, t) + hq(q, ~,x, t), ~(to) = Go, q(to) = qo, where ( = [~1 ~2 ~3] T, k(., .) is given by (19), A is a stable matrix, and fq and hq are any functions satisfying the conditions ( c ) - ( f ) in Theorem 3.2.
J.-i. Imura, 12 YoshikawalSystems & Control Letters 29 (1997) 207-213
4. Conclusions In this paper, we hwge given a parametrization of all state and output feedback controllers that asymptotically stabilize nonlinear systems under the assumption of static state feedback stabilizability and a kind of detectability. The technique developed here is quite simple and direct, and allows us to treat even general timevarying nonlinear systems for both state and output feedback. In addition, the technique developed here can be applied to derive a parametrization of all globally asymptotically stabilizing controllers and (globally) exponentially stabilizing controllers. It is one of the future researches to extend our technique to the cases of continuous (non-smooth) and discontinuous feedback controllers.
Acknowledgements The authors would like to thank Professor T. Sugie and Professor A.J. van der Schaft for fruitful discussions.
References [1] V. Anantharam and C.A. Desoer, On the stabilization of nonlinear systems, IEEE Trans. Automat. Control AC-29 (1984) 569-572. [2] J.A. Ball and M. Verma, Factorization and feedback stabilization for nonlinear systems, Systems Control Lett. 23 (1994) 187-196. [3] R.W. Brockett, Asymptotic stability and feedback stabilization, in: R.W. Br,ackett, R.S. Millman and H.J. Sussmann,
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eds., Differential Geometric Control Theory (Birkh~iuser, Basel, 1983) 181-191. [4] R.J.P. de Figueiredo and G. Chen, Nonlinear Feedback Control Systems: An Operator Theory Approach (Academic Press, New York, 1993). [5] C.A. Desoer and G. Kabuli, Right factorization of a class of time-varying nonlinear systems, IEEE Trans. Automat. Control AC-33 (1988) 755-757. [6] J. Hammer, Non-linear systems, stabilization, and coprimeness, Int. J. Control 42 (1985) 1-20. [7] W.-M. Lu, A state-space approach to parametrization of stabilizing controllers for nonlinear systems, IEEE Trans. Automat. Control 40 (1995) 1576-1588. [8] R.M. Murray and S.S. Sastry, Nonholonomic motion planning: steering using sinusoids, IEEE Trans. Automat. Control AC-38 (1993) 700-716. [9] A.D.B. Paice and J.B. Moore, On the Youla-Kucera parametrization for nonlinear systems, Systems Control Lett. 14 (1990) 121-129. [10] J.-P. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drit~, Systems Control Lett. 18 (1992) 147-158. [11] C. Samson, Control of chained systems application to path following and time-varying point-stabilization of mobile robots, IEEE Trans. Automat. Control 40 (1995) 64-77. [12] O.J. Sordalen and O. Egeland, Exponential stabilization of nonholonomic chained systems, 1EEE Trans. Automat. Control 40 (1995) 35-49. [13] J. Tsinias, Sufficient Lyapunov like conditions for stabilization, Math Control Signals Systems 2 (1989) 343-357. [14] M.S. Verma, Coprime fractional representation and stability of nonlinear feedback systems, Int. J. Control 48 (1988) 897-918. [15] M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interconnections: stability and stabilizability, IEEE Trans. Automat. Control 25 (1980) 773-779. [16] M. Vidyasagar, On the stabilization of nonlinear systems using state detection, 1EEE Trans. Automat. Control 25 (1980) 504-509.
Systems & Control Letters 29 (1997) 207-213
Parametrization of all stabilizing controllers of nonlinear systems J u n - i c h i I m u r a a,*, T s u n e o Y o s h i k a w a b a Division of Machine Design Engineering, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 739, Japan b Department of Mechanical Engineering, Faculty of Engineering, Kyoto University, Kyoto 606, Japan Received 20 October 1995; revised 13 September 1996
Abstract
This paper develops a new approach to the parametrization of all asymptotically stabilizing controllers of nonlinear systems including time-varying ones, both for state feedback and output feedback. Assumptions of stabilizability and a kind of detectability are requirer. The approach is very direct and simple, and is not based on the so-called factorization and on the H~ framework of nonlinear systems which are used in previous papers. A key of the derivation of the parametrization in this paper is to find an explicit state space representation of free functions, such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems.
Keywords: Nonlinear systems; Parametrization; Asymptotic stabilization; Output feedback
1. Introduction
It is one of the fundamental problems to parameterize all stabilizing controllers of nonlinear systems. Most of the previous papers on this topic are based on operator theory, i.e., the coprime factorization approach [1,2, 4-6, 9, 14], similarly to the case of linear systems. On the other hand, very recently, Lu [7] has developed a state space approach to the controller parametrization problem, which is not based on coprime factorizations. He has derived a parametrization of all smooth state feedback controllers that asymptotically stabilize nonlinear systems under the assumption of stabilizability. His approach is a natural nonlinear extension of the state space approach to the controller parametrization problem of linear systems, which is based on a kind of Ho~ framework (i.e. the LFT framework). Therefore in his approach, there is rLo computational problem of * Corresponding author. Fax: +81-824-22-7193; E-mail: imura@ mec.hiroshima-u.ac.jp.
how to get a coprime factorization of nonlinear systems in state space description. In the case of output feedback, Lu also has given a parametrized controller that asymptotically stabilizes the system. However, it has not been shown there that the derived parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems. His approach developed in the case of state feedback may not be extendable to the case of output feedback. In this paper, we develop a new direct approach to derive a parametrization of all asymptotically stabilizing controllers of nonlinear systems in the case of output feedback as well as state feedback. It is not based on the so-called factorization approach and on the Ho~ framework approach. The assumptions are stabilizability and a kind of detectability. A key in our approach is to find an explicit state space representation of free dynamics, such that the parametrized controller expresses all asymptotically stabilizing controllers of nonlinear systems. This makes the proof for the parametrization much simpler, and surprisingly, also allows us to treat the general case
0167-6911/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S01 67-691 1(96)00065-5
208
J.-L
Imura. T. Yoshikawa/Systems & ControlLetters 29 (1997) 207 213
including time-varying systems in both cases of state and output feedback. We use the following notation: ~" denotes an n-dimensional Euclidean space and the Euclidean norm is denoted by I1" II, A real-valued function if(.): ~ ~ ~ is said to belong to the class K (or ~9 E K ) if it is continuous and strictly increasing with 440) = 0.
2. Parametrization of all stabilizing controllers via state feedback In this section, we give the parametrization of all C 1 feedback stabilizing state feedback controllers of nonlinear systems. The problem to be considered here is as follows.
Problem 2.1. Consider the following nonlinear system:
2 = f ( x ) + 9(x)u, x(to) = x0,
{ ~ = f ( x ) - s(x - ~) + 9(x)u, Ks
0 u
fq(q, ~,x), k(~) + hq(q,~,x), ~(to) = ~o, q(to) = qo,
where (a) 2 = f ( x ) + g(x)k(x) is asymptotically stable at the origin and k( O) = O, (b) ~ = s(~) is asymptotically stable at the origin and s is C 1, (c) 0 = fq(q, 0, 0) is asymptotically stable at the origin,
(d) fq(q,x,x) = fq(q,0,0), (e) hq(q,x,x) = hq(q,O,O), (f) fq and hq are C 1 and hq(O, O, O) = O. Proof. (Sufficiency) Letting e = x - ~, the closed loop system is given by
(1)
= f ( x ) + g(x)k(x)
where x E g~" is the state, u E ~m is the input, and to is the initial time. f ( . ) : ~" ~ ~" and 9(') : ~" R "x'~ are C 1 known functions with f ( 0 ) = 0. For the above system, consider the state feedback controller Kx defined by
~ = fc(x~,x), Kx [ u = hc(xc,x),
for the system (1) is given by
Xc(t0) =xc0,
(2)
where fc(', ") : ~n~ x R" ~ ~"o and he(., .) : ~"~ x ~n ~ ~m are C 1, f c ( 0 , 0 ) = 0, h c ( 0 , 0 ) = 0, and x~0 is the initial state of the controller. Then find all C 1 state feedback controllers of Kx that asymptotically stabilize the system given by (1). Please note that Kx expresses a static state feedback controller when nc = 0.
Definition 2.1. A pair ( f , 9) is said to be asymptotically stabilizable, if there exists a C 1 function k(.) : ~n ~ ~m such that 2 = f ( x ) + 9(x)k(x) is asymptotically stable at the origin and k(0) = 0. Then the following result is obtained.
+g(x){k(x - e) - k(x) + hq(q,x - e,x)}, 0 = fq(q,x - e,x), = s(e). Conditions ( a ) - ( f ) imply that this closed loop system is asymptotically stable at the origin [15]. (Necessity) Suppose that a state feedback controller given by
f ~o L(xc,x), =
Kx [ u =hc(xc,x),
xc(to) =x~o
asymptotically stabilizes the system (1), where fe(O, O) = 0 and he(O, O) = O. This means that the closed loop system
2 = f ( x ) + g(x)h~(xc,x), xc = f~(xc,x)
is asymptotically stable at the origin. Thus by constructing appropriate functions fq and hq satisfying the conditions ( c ) - ( f ) , we have only to show Ks = Kx. So let q g ql + x - ~ and
fq(q,e,x)
= -f01) - s(x - ¢) + 9(rl)he(q2,rl)] f~(q2, q)
Theorem 2.1. Suppose that (f, g) is asymptotically stabilizable. Then a parametr&ation of all asymptotically stabilizing controllers via C 1 state feedback
(3)
J
(4) h q ( q , ¢ , x ) = - k ( q l ) + h c ( q 2 , q),
(5)
J.-i. Imura, T Yoshikawal Systems & Control Letters 29 (1997) 207-213
w h e r e q = [q~, T q2T ]T • Then we can show that the above
functions satisfy the conditions (c)-(f), noting that the closed loop system given by (3) is asymptotically stable at the origin. In addition, if we set ql(t0) = ~0, then ql(t) - ~(t). Therefore, the controller Ks under ql(t0) = Go and q2(to) =x¢0 with (4) and (5) is equivalent to the system 02 = f c ( q 2 , x ) , u = hc(qz,x),
q2(i!o) -- xco
which is the controller Kx. This completes the proof. [] The proof developed in Theorem 2.1 is quite different from that by Lu [7]. In addition, even in the linear case, such a proof has not appeared so far. This proof is quite simple and direct in the sense that we do not need to use the so-called factorization approach, albeit it is important. A key of the proof is the direct derivation of (4) and (5). The first element of the vector field fq in (4) is for the generation of ~ in Ks, which is used to cancel out the effect by the dynamics on ~ in Ks, and the second element is for the generation ofx~ under the conditions ~(t0 ) = q 1(to) and xc(t0 ) = q2(t0 ), which is used to express any stabilizing controller. In this way, the construction offq and hq given by (4) and (5) is very intuitive in the proof of the necessity. In addition, the introduction of the variable r/plays an important role to guarantee that (4) and (5) satisfy the conditions (c)-(f). This technique of the proof allows us to treat the output tbedback case and even the case of the general systems including the time-varying case as we will see later. Remark 2.1. In Theorem 2.1, it is assumed that ( f , g ) is asymptotically stabilizable. Whenever a C 1 dynamic state feedback controller exists, does there exist a C 1 static state feedback controller? For this question, we do not have a complete solution yet. However, we can show that if there exists a C 1 dynamic state feedback controller Kx that asymptotically stabiliTes the system (1) in the sense that there exists a Lyapunov function V(-) : ~+,c ~ showing the asymptotic stability of the closed loop system and the V satisfies rank 02 V/Ox~(O) = n¢, then there exists a C ~ static state feedback controller u = k(x) thai: asymptotically stabilizes the system (1) and k ( 0 ) = 0 under rank g ( 0 ) = m. (The proof is a simple extension of Theorem 3 by Tsinias [13].) In this result, the additional condi-
209
tion (i.e., the rank condition of V) is still needed together with the asymptotic stability of the closed loop system by the dynamic controller. It is an interesting open problem to solve the above question completely. Next, we will show the other two kinds of parametrization of all stabilizing state feedback controllers of nonlinear systems. This result can be proven by the same technique as Theorem 2.1. Corollary 2.1. The other two kinds of parametriza-
tion of all stabilizing state feedback controllers of nonlinear systems are given by K[ and K[' as follows:
(i) { ~ = f ( ~ ) + g(~)k(~) - g(x)k(x) + g(x)u, K,s
0 = fq(q, ¢,x), u -- k(x) + hq(q, ~,x), ¢(to) = ~o,
q(to) = qo,
where (a) ~ = f ( x ) + g(x)k(x) is asymptotically stable at the origin and k( O) = O, (b) O = fq(q,0,0) is asymptotically stable at the origin, (c) fq(q,x,x) = fq(q,0,0), (d) hq(q,x,x) = hq(q,O,O), (e) fq and hq are C 1 and hq(O, O, O) = O.
(ii) ~0 = fq(q),
K[' (u = k(x) + hq(q), q(to)-- go, where (a) at the (b) (c)
k = f ( x ) + g(x)k(x) is asymptotically stable origin and k(O) = O, q = f q ( q ) is asymptotically stable at the origin, fq and hq are C 1 and hq(O) = O.
Proof. The proof is quite similar to that of Theorem 2.1. So we will show the equations which correspond to (4) and (5) only. In the case of (i), we can choose
fq(q, ~,x)
:[f(ql)+g(ql)k(ql)+g(rl){-k(rl) .-he(q2, r/)} ] fc(q2, r/) (6)
210
J.-i. Imura, T. Yoshikawal Systems & Control Letters 29 (1997) 207-213
I
Y-x
11
Fig. 1. Feedback system.
hq(q, ~,x) = -k(t/) + hc(q2, t/),
(7)
where r / = ql + x - ~. In the case of (ii), we can choose
fq(q)= [ f(ql)+O(ql)hc(q2'ql) ] fc(q2, ql )
(8)
hq(q) = - k ( q I ) ÷ hc(q2, qt )"
(9)
[]
This characterization by K~ corresponds to y = PQr in the case of linear systems , where P is a stable plant transfer function and Q is a free stable proper transfer function. Nonlinear extensions similar to the above example are discussed in [1,7]. In the case of Ks, we can easily see that the same characterization of the closed loop system from r to y as the case of Ks~ will be obtained. In the case of Ks", on the other hand, it does not seem easy to take account of the desired value r in the closed loop, since the vector field of K~~ does not include the state of the plant. The above feedback system is one example, and the advantage of each parametrization may depend on the kind of control problems studied.
In the linear case, it is well known that a parametrization of all stabilizing controllers for stable systems is given by
3. Extension to other cases
K = (I + Q p ) - I Q ,
3.1. Output feedback case
(10)
where P and Q express a transfer function of a stable plant and any stable proper transfer function, respectively. We can show that the parametrization of K~ is a nonlinear analog by simple computation. Lu [7] has given only the case Ks" as a parametrization of all C 1 state feedback controllers by using a framework of nonlinear H ~ control problems. On the other hand, our technique allows us to derive Ks~ and Ks" as well as Ks. Note that there is some difference between Ks~ (and also Ks) and K~~. In fact, the vector field of the controller Ks~(and Ks) includes the state of the plant, i.e., x, while in K~~it does not. Such a difference will be clearer when we consider the following feedback system shown in Fig. 1, where the plant is given by (1) and y = x and the controller is given by K~ with the input r, in other words,
{
~ = f ( ~ ) + g(~)k(¢) - g(x)k(x) + g(x)u, 4 = fq(q, ~,x - r), u = k(x) + hq(q,~,x - r), ~(to) = ~o, q(to) = qo,
where r is the desired value and, k, fq, and hq satisfy the conditions ( a ) - ( e ) in Corollary 2.1. Let us consider the case of ~0 =x0. Then the map of this feedback system from r to y is expressed by
Jc = f ( x ) + g(x)k(x) + g(x)hq(q,x,x - r), (1 = fq(q,x,x - r), y=-x, x(to)=xo, q(to)=qo.
Compared with Lu's technique, our approach allows us to give a parametrization of the output feedback case as well as the state feedback case. Consider the system given by (1) and the output
y=h(x),
(11)
where h(.) : ~ ~ ~P is a C 1 function, and h ( 0 ) = 0. The problem to be considered here is as follows. Problem 3.1. Consider the output feedback controller
Ky defined by ~Xc = fc(Xc, Y) hc(xe, y),
Ky t u =
xe(t0) =xeo
(12)
where f~(., .) : ~nc x ~p____~nc and he(', ') : Nnc x RP--~ R 'n are C 1, fc(0, 0) = 0, he(0, 0) -- 0, and xe0 is the initial state of the controller. Then find all output feedback controllers of Ky that asymptotically stabilize the system given by (1) and (11 ). Definition 3.1 (Vidyasagar [16]). A pair ( f , h ) is said to be weakly detectable, if there exist C 1 functions I ( . ) : R n x R n ~ R n, V(.): R n x ~n R ", and functions ~, fl, and ~ of class K such that, for all x and ~ in a neighborhood of the
J.-i. Imura, T. YoshikawalSystems & Control Letters 29 (1997) 207-213
origin,
(llx -
11)<
11),
8V -ffX-X{f(x) + 9(x)u} + ~-~{f(~) + g(~)u + l(~,h(x))}
<-
(llx- 11),
l(O,O)-- o.
Then the following result is obtained. Theorem3.1. Consider the system 9iven by (1) and (11 ). Suppose that (f, O) is asymptotically stabilizable and (f,h) is weakly detectable. Then a parametrization of all asymptotically stabilizin9 controllers via C 1 output feedback for the system of (1) and (11 ) is given .by { ~ = f ( ~ ) + g ( ~ ) u + l(~,y), Ko
@= fq(q' ~' Y)' u = k(~) + hq(q,, ~, y), ((to) = ¢0, q(to) = qo,
where (a))~ = f ( x ) + 9(x)k(x) is asymptotically stable at the origin and k( O), = O, (b) l(. ) satisfies the condition in Definition 3.1 and l(x,h(x)) = O, (c) @= fq(q,0,0) is asymptotically stable at the origin, (d) fq(q,x,h(x)) = fq(q, 0,0), (e) hq(q,x,h(x)) --=hq(q,O,O), (f) fq and hq are (;1 and hq(0, 0, 0) = 0. Proof. It directly follows from Theorem 3.1 in [16] that the closed loop system is asymptotically stable. The proof of the necessity is quite similar to the proof of the state feedback case, by choosing fq(q, ~, Y) = [f(q,)+g(q,)hc(q2,h(ql)+y-h(~))+l(~,Y)]
211
Lu [7] has also shown that the controller parametrized similarly to Ko in Theorem 3.1 asymptotically stabilizes the closed loop system under the assumption of (f,o)-stabilizability and (f,h)weak detectability. However, his technique used in the case of state feedback may not be applied to show that the parametrized controller expresses all stabilizing controllers in the output feedback case. On the other hand, Theorem 3.1 shows that the obtained parametrization expresses all asymptotically stabilizing controllers even in the output feedback case, by using the same technique as for the state feedback case in Section 2, under the assumption of (f,9)-stabilizability and (f,h)-weak detectability.
3.2. General nonlinear time-varyin9 system case Our technique developed in Sections 2 and 3.1, in addition, can be applied to the case of general nonlinear time-varying systems. In other words, we can give a parametrization of all asymptotically stabilizing controllers for general nonlinear time-varying systems. Here we will concentrate on the state feedback case. The output feedback case can also be treated in a similar way. Let us consider the general nonlinear system given by
k = f(x,u,t),
x(to)=xo,
(15)
where x E ~n is the state, u E R m is the input, and to is the initial time. f ( . , . , .) : ~n × ~m × ~ ~ ~n is a C l known function with f ( 0 , 0, t) = 0. In addition, the function f is assumed to exist in a neighborhood Uz of the origin and sup sup c3f(x,u, t) t zEuz ~z
< oo,
where z = [x u] T. For simplicity, we call hereafter the function that satisfies the above condition the D.B. function, since the derivatives of the function is bounded for any t. Then we consider the following problem.
f c ( q 2 , i i ( q l ) + y -- h(~))
(13)
Problem 3.2 (State feedback case). Let us consider the time-varying state feedback controller given by
hq(q,~,y) ------ k ( q l ) + h c ( q 2 , h ( q l ) + y -
h(¢)).
[]
Kxt
(14)
( xc = fc(xc,x, t) u = hc(xc,x, t) xc( to ) = xco
J.-i. Imura, 72 YoshikawalSystems & Control Letters 29 (1997) 207-213
212
where Arc(., ., .) : N,c x ~ x R --+ ~nc and hc(', ', ") : ~nc x ~n x R --~ N m are the C 1 D.B. functions, f~(0, 0, t) = 0, he(0, 0, t) = 0, and Xc0 is the initial state of the controller. Then find all state feedback controllers of Kxt that uniformly asymptotically stabilize the system given by (15). In addition, we define the following stabilizability, which corresponds to the (f, g)-stabilizability. Definition 3.2. The system given by (15) is said to be asymptotically stabilizable via static state feedback, if there exists a function k(.,.) : R n × E -* Em such that (i) The system 2 = f(x,k(x,t),t) is uniformly asymptotically stable at the origin, (ii) k(0,t) = 0, (iii) k(x, t) is the C l D.B. function. Then we get the following result.
Suppose that the system given by (15) is asymptotically stabilizable via static state feedback. Then a parametrization of all C l state feedback controllers g s t that uniformly asymptotically stabilize the system (15) is given by T h e o r e m 3.2.
hq(q,~,x,t)=-k(ql,t)+hc(q2,q,t), where ~ = q l + x -
(17)
4. []
Finally, as an application we will show an interesting example of Theorem 3.2. We consider the driftless system £1 = Ul,
3f2 ~-- U2,
(18)
3C3 =X2U 1
which is called the chained system in the field o f n o n holonomic control systems [8] and has lately attracted considerable attention on stabilization. By Brockett's Theorem [3], it is well known that there is no C l timeinvariant state feedback controller that asymptotically stabilizes the system given by (18). Thus some continuous time-varying feedback controllers and some discontinuous controllers that asymptotically stabilize the system (18) have been developed by many researchers [8, 11, 12]. We will here concentrate on C 1 time-varying state feedback controllers and give a parametrization of all C 1 time-varying state feedback controllers that uniformly asymptotically stabilize (18). Pomet [10] has derived the following C 1 timevarying static state feedback controller that uniformly asymptotically stabilizes the system (18).
{ 4 = f ( x , u , t ) - s(x - ~), Kst
k(x, t)
q = fq(q, 4,x,t), u = k(~,t) + hq(q, ~,x,t), ~(to) = 40,
=[
x 2 s i n t - ( x l + x2cost) - - ( X 1 + X2 COS t)Xm COS t -- (XiX 2 + X3 )
q(to) = qo,
] '
(19)
where (a) 2 -- f(x, k(x, t), t) is uniformly asymptotically
stable at the origin, k(O,t) = O, and k(x,t) is the C 1 D.B. function, (b) ~ = s(~) is asymptotically stable at the origin, (c) q = fq(q, 0,0, t) is uniformly asymptotically stable at the origin, (d) fq(q,x,x,t) = fq(q,0,0, t), (e) hq(q,x,x,t) = hq(q,O,O,t), (f) fq and hq are the C 1 D.B. functions and hq(O,O,O,t) = O. Proof. The proof is similar to the proof of Theorem 3.1. The equations corresponding to (4) and (5) are given by fq(q, ~,x, t) =
! f(q, hc(q2,~l,t),t) - s(x - ~)] f¢ (q2, r/, t)
J (16)
where x -- [xl,x2] T. This controller satisfies conditions (i)-(iii) in Definition 3.2. So the system (18) is asymptotically stabilizable via C 1 time-varying static state feedback. Thus we can apply Theorem 3.2 to the case of the system (18) to get a parametrization of all uniformly asymptotically stabilizing controllers for the system (18). For example, one of the parametrizations is given by
/ ul
Kst 4= fq(q,~,x,t), u = k(~, t) + hq(q, ~,x, t), ~(to) = Go, q(to) = qo, where ( = [~1 ~2 ~3] T, k(., .) is given by (19), A is a stable matrix, and fq and hq are any functions satisfying the conditions ( c ) - ( f ) in Theorem 3.2.
J.-i. Imura, 12 YoshikawalSystems & Control Letters 29 (1997) 207-213
4. Conclusions In this paper, we hwge given a parametrization of all state and output feedback controllers that asymptotically stabilize nonlinear systems under the assumption of static state feedback stabilizability and a kind of detectability. The technique developed here is quite simple and direct, and allows us to treat even general timevarying nonlinear systems for both state and output feedback. In addition, the technique developed here can be applied to derive a parametrization of all globally asymptotically stabilizing controllers and (globally) exponentially stabilizing controllers. It is one of the future researches to extend our technique to the cases of continuous (non-smooth) and discontinuous feedback controllers.
Acknowledgements The authors would like to thank Professor T. Sugie and Professor A.J. van der Schaft for fruitful discussions.
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