School of Electrical Engineering and Computer Science, Washington State ... system which is subject to "input saturation", the global stabilization can be solved ...
Systems & Control Letters 21 (1993) 225-239 North-Holland
225
Semi-global exponential stabilization of linear systems subject to "input saturation" via linear feedbacks Zongli Lin and Ali Saberi School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, USA
Received 11 November 1992 Revised 16 February 1993 Abstract: It is known that a linear time-invariant systemsubject to "input saturation" can be globallyasymptoticallystabilized if it has
no eigenvalues with positive real parts. It is also shown by Fuller (1977) and Sussmann and Yang (1991) that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws. In this paper we show the existence of linear state feedbackand/or output feedbackcontrol laws for semi-globalexponential stabilization rather than global asymptoticstabilization of such systems. We explicitly construct linear static state feedback laws and/or linear dynamicoutput feedback laws that semi-globally exponentially stabilize the given system. Our results complement the "negativeresult" of Fuller (1977) and Sussmann and Yang(1991). Keywords." Semi-globalexponential stabilization; bounded control; input saturation.
1. Introduction In this paper we focus on the problem of stabilization of a general linear system which is subject to "input saturation". This problem has a rich history (see, for example, [2]), and much of the literature of the 1950s and 1960s on the absolute stability was motivated by this problem (see, for example, [1, 6, 7]). A recent result due to Songtag and Sussmann [10] shows that only linear stabilizable systems having no open-loop eigenvalues with positive real parts can be globally asymptotically stabilized by a bounded control. Another interesting aspect of this problem was shown by Fuller [4] and more recently by Sussmann and Yang [12]. They showed that for a system of a chain of integrators of length n where n > 3 and which is subject to input saturation, there does not exist a linear control law that globally asymptotically stabilizes the given system. The implication of the results of [I0, 4, 12] is straightforward and can be concluded as follows: Given a linear system which is subject to "input saturation", the global stabilization can be solved if and only if all the eigenvalues of the given system are in the closed left half plane, and, even then, in general one must use nonlinear control and only very simple cases can be handled via linear feedback control laws. (An interesting nonlinear control law of nested saturation type for global asymptotical stabilization of a chain of integrators subject to input saturation was proposed in [13], which was later extended by [11].) In this paper we present a result that complements the "negative results" of [4, 12]. We consider semi-global exponential stabilization of general linear systems subject to "input saturation" and show that, in contrast to the case of global asymptotic stabilization, one can semi-globally exponentially stabilize such systems via linear feedback laws. Here by semi-global exponential stabilization, as usual, we mean local exponential stabilization of the system such that the domain of attraction of the closed-loop system contains an a priori given bounded set. (For a precise definition of semi-global exponential stabilization, see
Correspondence to. Z. Lin, School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA
99164-2752, USA. 0167-6911/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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Definitions 2.1 and 2.2.) Relaxing the requirement of global stabilization to that of semi-global stabilization, from an engineering point of view, makes sense, since in general a plant's model is usually valid in some region of the state space. Such a relaxation gives us simple linear control laws and a stronger stability property for the closed-loop system, that is, the exponential stability of the closed-loop system, rather than asymptotical stability. This paper is organized as follows. We formulate our problem in Section 2. Section 3 deals with the state feedback case, while the output feedback case is dealt with in Section 4. Section 5 draws the conclusions of our current work.
2. Problem statement
We consider a linear system subject to "input saturation" described by 5c = Ax + /3Oh(U),
(2.1)
y = Cx,
(2.2)
where x 6 ~R" is the state, u ~ 9~m is the control input to the saturator, y ~ 9~p is the measurement output, and ah(S) is a bounded vector function defined as O h (S) = ['O'h 1(S 1 ), 0"h2($2) . . . . .
ahm (Sin) ]'
with f ahi(Si)
= si hi
if Isll ~ hi if s i < - h i if s i > h i ,
(2.3)
where s = [sx, $2 . . . . . Sm] and h = [ h i , h2 . . . . . hm], hi > O. We make the following assumptions on the system (2.1)-(2.2): (A1) All the eigenvalues of A are located on the closed left half s-plane. (A2) The pair (A, 13) is stabilizable. (A3) The pair (A, C) is detectable. Remark 2.1. We should point out that assumptions A1 and A2 are equivalent to the notion of asymptotic null-controllability that was introduced in [8, 9]. Before stating the problem at hand, we have the following definitions. Definition 2.2 (Semi-global exponential stabilization via linear static state feedback). The system (2.1)-(2.2) is semi-globally exponentially stabilizable by linear static state feedback if, for any a priori given bounded set of initial conditions W = ~", there exists a state feedback law u = K x such that the equilibrium x = 0 of the closed-loop system is locally exponentially stable and W is contained in the domain of attraction of the equilibrium x = 0. Definition 2.3 (Semi-global exponential stabilization via linear dynamic output feedback of dynamical order l). The system (2.1)-(2.2) is semi-globally exponentially stabilizable by linear dynamic output feedback of dynamical order l if, for any a priori given bounded set W c 9~"+z, there exists a linear dynamic output feedback control law = AeonZ + BconY, z ~ } ~ t,
u =Cco, z + D¢o. y,
(2.4) (2.5)
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such that the equilibrium (0, 0) of the closed-loop system consisting of the system (2.1)-(2.2) and the controller (2.4)-(2.5) is locally exponentially stable and W is contained in the domain of attraction of the equilibrium (0, 0). The problem posed in this paper is to establish the semi-global exponential stabilization via linear static state feedback laws and/or linear dynamic output feedback laws for the system (2.1)-(2.2).
3. State feedback design
In this section we examine the problem of semi-global exponential stabilization via linear state feedback control of the asymptotic null-controllable linear systems. Our main result is given in the following theorem. Theorem 3,1. The linear system (2.1)-(2.2) satisfying assumptions A1 and A2 is semi-globally exponentially stabilizable via linear state feedback. Namely, for any a priori given (arbitrarily large) bounded set W and any (arbitrarily small) numbers hi > 0, i = 1 to m, there is a linear control law u = - K x such that (a) the equilibrium x = 0 of the closed-loop system is locally exponentially stable and (b) W is contained in the domain of attraction of the equilibrium x = O.
Before proving this theorem, let us digress to prove a lemma, which will play a fundamental role in our proof of Theorem 3.1. Lemma 3.2. Consider a linear single-input system in the controllable canonical form 2 = A x + Bu,
0]
where
A =
0 0
1 0
0 1
... ...
0 0
:
:
:
"..
:
0 --
0 an
--
a n - 1
--
0
...
a n - 2
•
•
0
1 •
--
al
B =
•
°
6 1
Assume that all the eigenvalues are in the closed left half s-plane. Let K(e) • ~ 1x n be the state feedback gain such that 2(A - BK(e)) = --e + 2(A). Then, there exists an ~* > 0 such that, for all 0 < e 0 and an ctc > 0, both independent of e, such that
II~e II -< ~ee-"ll ~(0)II, We then view ~e as an input signal to the dynamics of ~p. By L e m m a 3.2, there exists an e*, 0 < e* < min{1/2, a}, such that, for all 0 < e < e*, I l K p ( e ) e t a ' - ~ ' g ' t " q l < flpee-" for some tip > 0. We then have Ilgp(e)~pll < Ilgp(g)etAP-B'r'tO)'xp(0)ll + Kv(e)
p._ 0 is a tunable parameter to be chosen according to the a priori given b o u n d e d set W and the value ofh. Let the set W b e given by W = {x: I]x[[ < 1} and h = 0.5. Then an estimate o f f * is given by e* = 0.01. The simulation results are shown in Figure 1.
Z. Lin, A. Saberi / Semi-globalexponential stabilization
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5
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150
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60
80
100
120
140
160
g
Fig. 1. Example 3.1. e = 0.01. a) Xl; b) x2;
C)
X3; d) x4; e) xs; f) x6; g) u.
4. Output feedback design T h i s s e c t i o n d e a l s w i t h t h e p r o b l e m o f s e m i - g l o b a l e x p o n e n t i a l s t a b i l i z a t i o n via o u t p u t f e e d b a c k of d e t e c t a b l e a n d a s y m p t o t i c n u l l - c o n t r o l l a b l e systems. O u r m a i n result is g i v e n in the f o l l o w i n g t h e o r e m .
Theorem 4.1. Let assumptions A 1 - A 3 be satisfied. Then the system (2.1)-(2.2) is semi-globally exponentially stabilizable via linear output feedback of dynamical order n. Namely, for any a priori given (arbitrarily large) bounded set W c ~2n and any (arbitrarily small) numbers hi > O, i = 1 to m, there exists a linear output feedback law of dynamical order n such that
Z. Lin, A. Saberi / Semi-global exponential stabilization
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100
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200
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i
h
Fig. 2. Example 4.1. e = 0.01. a) xl; b) x2; c) x3; d) x4; e) x5; f) x6; g) :~1; h) :~2.
(a) The equilibrium (0, O) of the closed-loop system is locally exponentially stable; (b) W is contained in the domain of attraction of the equilibrium (0, 0).
P r o o f . W e first c o n s t r u c t a f a m i l y of d y n a m i c o u t p u t f e e d b a c k c o n t r o l l e r s p a r a m e t e r i z e d in e as follows:
,2 = A~ + Bah(--K(e)~) + L ( y - C~),
(4.1)
u = - K(e)~,
(4.2)
Z. Lin, A. Saberi / Semi-global exponential stabilization 20
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04
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o
20
40
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SO
100
120
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Fig. 3. Figure 2 continued, i) 23;J) ~4; k) 25; 1) 26; m) u.
where K ( e ) is as given in the proof of Theorem 3.1, and L is any matrix such that all the eigenvalues of A - L C are in the open left half s-plane. Following the same idea as in the proof of Theorem 3.1, we first consider the closed-loop system without saturation element. Namely, we set trh(u) = u in both the system (2.1)-(2.2) and the proposed controller (4.1)-(4.2). Then we show that for the given set W and h~ > 0, i = 1 to m, there exists an e* > 0 such that, for all i = 1 to m, Vee(0, e*]
lui(t)l~hi,
and
V[x(0)',2(0)']'eW.
(4.3)
where ug(t) is the ith element of u = - K ( e ) 2 . Having done this, the result of the theorem then follows readily since the closed-loop system with saturation element, namely, (2.1)-(2.2) and (4.1)-(4.2), remains linear for all I-x(0)', 2 ( 0 ) ' ] ' e W. To show this, we observe that the closed-loop system with ah(U) = U can be rewritten as 2 = (A -
BK(e))x
d = (A -
LC)e,
u = -- K ( e ) x
where e = x - 2.
+ BK(e)e,
+ K(e)e,
(4.4) (4.5) (4.6)
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The stability of the above system for all e > 0 follows from the separation principle. Next we need to show that for the given W a n d the given hi > 0, i = 1, 2 . . . . . m, there exists an e* > 0 such that (4.3) holds. To this end, as in the p r o o f of T h e o r e m 3.1, we write the system (4.4)-(4.5) as p
A i t Y t + A~,.,£,. + F i B K ( e ) e ,
xi = (Ai -- B i K i ( e ) ) ~ i +
i = 1,2 . . . . .
p,
(4.7)
l=i+l
xe = Aexe,
(4.8)
0 = (A - LC)e,
(4.9)
where F - 1 = I F ' 1 , Y'2 , • • • , F 'p ] ' " Viewing [~'~, e ' ] ' as a new ~ and using Observation 3.3, it can be shown, exactly as in the p r o o f of Theorem 3.1, that there exists an e* such that, for all i = 1 to m, lu:i(t)l
