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The almost disturbance decoupling problem with internal stability for linear systems subject to input saturation—state feedback case
Pergamon
0005-1098(95)00176-X
Auromalrcu, Vol. 32, No. 4, pp. 6lY-624, 19% Copyright 0 1996 Elsevm Swnce Ltd Printed in Great Britain. All nghts reserved 0005.lOYS/Yh $ls.oo + 0.00
Brief Paper
The Almost Disturbance
Decoupling
Problem with
Internal Stability for Linear Systems Subject to Input Saturation-State
ZONGLI Key Words-Linear
Feedback Case*
LIN,‘( AL1 SABERIS
and ANDREW
R. TEEL§
systems subject to input saturation; almost disturbance decoupling,
understood
(see e.g. Weiland and Willems. 1989: Ozcetin et of ADDPMS was also extended to some nonlinear systems having a certain strong relative degree (Saberi and Sannuti, 1988). In this paper, we attempt to extend this notion to linear systems subject to input saturation. The recent renewed interest in linear systems subject to input saturation is mainly because of the wide recognition of the inherent constraints on the control input. This renewed interest has led to many interesting results on such systems. Most of these results, however, pertain only to issues related to global and semi-global internal stabilization of such systems. In regard to global stabilization, it was shown in Sontag and Sussmann (1990) that a linear system subject to input saturation can be globally asymptotically stabilized if and only if the system in the absence of saturation is asymptotically null controllable with bounded contro1s.n Moreover, it was also shown (see Fuller, 1969; Sussmann and Yang, 1991) that, in general global asymptotic stabilization of linear systems subject to input saturation cannot be achieved by using linear feedback laws and that nonlinear feedback laws should be used for this purpose. A nested feedback design technique for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in Tee1 (1992) and completed in Sontag and Yang (1991) and Sussmann ef al. (1994). The notion of semi-global stabilization of linear systems subject to input saturation was first introduced in Lin and Saberi (1993a) (and in Lin and Saberi (1995a) for the discrete-time counterpart). The semi-globa< framework for stabilization requires feedback laws that yield a closed-loop system that has an asymptotically stable equilibrium whose domain of attraction includes an a priori given (arbitrarily large) bounded set. In Lin and Saberi (1993a, 1995a), it was shown that, both for discrete time and continuous time, one can achieve semi-global stabilization of linear asymptotically null controllable with bounded controls systems subject to input saturation using linear feedback laws. A low-gain design technique based on the eigenstructure assignment was also proposed to construct semi-globally stabilizing controllers. These low-gain control laws were constructed in such a way that the control input does not saturate for any a priori given arbitrarily large bounded set of initial conditions. Utilizing the H2 and H, optimal control theory, alternative ARE-based approaches for designing semi-globally stabilizing low-gain feedback laws were later proposed independently in Lin et al. (1996) and Tee1 (1995). In a recent paper (Lin and Saberi, 1995b; see also Lin and
Abstract-We consider the almost disturbance decoupling problem (ADDP) and/or almost D-bounded disturbance decoupling problem ((ADDP),) with internal stability for linear systems subject to input saturation and input-additive disturbance via linear static state feedback. We show that the almost D-bounded disturbance decoupling problem with local asymptotic stability ((ADDP/LAS),) is always solvable via linear static state feedback as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the open-loop system are, and the locations of these poles play a role only in the solution of (ADDP/GAS),, (ADDP/SGAS), or ADDP/GAS where semi-global or global asymptotic stability is required.
al. 1993). The notion
I. Introduction
The problem of disturbance decoupling or almost disturbance decoupling has a vast history, occupying a central part of classical as well as modern control theory. Several important problems, such as robust control, decentralized control, non-interacting control, model reference and tracking control, can be recast as an almost disturbance decoupling problem. Regardless of from where the problem arises. the basic almost disturbance decoupling problem is to find an output feedback control law such that in the closed-loop system the disturbances are quenched, say in the &-gain sense. up to any prespecified degree of accuracy while maintaining internal stability. In the linear system setting. the above problem is labeled by Willems (1981, 1982) as ADDPMS (the almost disturbance decoupling problem with measurement feedback and internal stability). In the case that, instead of a measurement feedback, a state feedback is used, the above problem is termed the ADDPS (the almost disturbance decoupling problem with internal stability). The ADDPMS for linear systems is now very well *Received 24 January 1994; revised 23 September 1994; revised 25 April 1995: received in final form 5 October 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Basar. Corresponding author Professor Zongli Lin. Tel. + 1516 632 9344: Fax + 1 516 632 8490: E-mail [email protected]. t Department of Applied Mathematics and Statistics, SUNY at Stony Brook. Stony Brook, NY 11794-3600, U.S.A. $ School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752. U.S.A. $,Department of Electrical Engineering, University of Minnesota, 4-174 EE/CS Building, 200 Union Street SE, Minneapolis, MN 55455. U.S.A.
n A linear system is asymptotically null controllable with bounded controls if and only if it is stabilizable and all the poles of the open-loop system are in the closed left half-plane. 619
620
Brief Papers p t
[ 1. =] and any D > 0, L;(D) denotes the set of all x E L; such that IIxJ(,>,5 D.
Q+-j
Saturation I
H I
Z:
Linear Syst,em p L
I
i =f(x, d).
X E R”,
d E R”‘,
(1)
we make the following definitions.
_______________L________________________.___~_.__
Fig. I. Linear system subject to input saturation.
Sabcri, 1993b) we have introduced yet another design technique, the so-called low-and-high gain design technique, for a chain of integrators subject to input saturation. This design technique was basically conceived for semi-glovaf control problems beyond stabilization, and was related to the performance issues such as semi-global stabilization with enhanced utilization of the available control capacity and semi-global practical stabilization in the presence of input-additive external disturbance. This design technique was later successfully applied to general linear asymptotically null controllable with bounded controls systems subject to input saturation (Saberi er al., lYY6). As a natural development over the Internal stabilization, Liu PI al. (1993) recently studied the input/output stabilizability of linear systems subject to input saturation. Their results show that, under the assumptions that all the poles of the open-loop system are in the closed left half-plane, with those on the jw axis simple, and that the system in the absence of input saturation is stabilizable and detectable, the linear system subject to input saturation is simultaneously finite-gain &-stabilizable and globally asymptotically stabilizable. Using the low-and-high gain design technique proposed in Lin and Saberi (1993b. 1995b) and Saberi et al. (1996), we showed in Lin et al. (1995) that, under the mild condition that the external input is uniformly bounded, the simultaneous L, stabilization and internal stabilization is always achievable via linear static state feedback as long as the system m the absence of input saturation is stabilizable. no matter where the poles of the open-loop system are, and the locations of these poles play a role only when global or semi-global asymptotic stabilization is required. In this paper, we further exploit the ahote-mentioned low-and-high gain design technique to study the almost disturbance decoupling problem with internal stahility for linear systems subject to input saturation and input-additive disturbance. Our main results show that the almost D-bounded disturbance decoupling problem with local asymptotic stability is always solvable as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the open-loop system are. and the locations of these poles play a role only when WC need to solve the (ADDP/SGAS), (almost D-bounded disturbance decoupling problem with semi-global asymptotic stability), (ADDP/GAS), (almost D-bounded disturbance decoupling problem with global asymptotic stability) or the ADDP/GAS (almost disturbance decoupling problem with global asymptotic stability). In comparison with Lin ef al. (199.5). where, under the same conditions, the bounded input L, stabilization and the internal stabilization were achieved simultaneously, the new results of this paper show that the L, gain of the resulting finite-gain &-stable closed-loop system can be made arbitrarily small. This paper is organized as follows. In Section 2, we define the problems at hand. The solutions to these problems are given in Section 3. Finally, we make brief concluding remarks in Section 4. 2. Preliminaries and problem sralemenl Nut&on. For x e W”, lix,I denotes the Euclidean norm of X. For any p E [l, z), we denote by L; the set of all measurable functions x(.): [O,~)+W” such that J, IJx(t)//Pdt
Defnirion
is
I,,,-stable
2.1: cp shility.
For any p E 11, a], the system Z if, given any d E I,,, and x(0) = 0, x E L;
Defirurion 2.2: D-bounded input LV stability. For any p E Cl, 0~1and any D >O, the system 2 is D-bounded inpur L,,,-slab& if, given any d E L;(D), and x(O) = 0, x E L; Defnirion 2.3: Finire guin L, stability. For any p E [l, DC],the system Z is finite gain L,-stable if it is &,-stable and in addition there exists a yr, >O such that
/l-Ul~.~5 r, II&,
t’d E LT.
Further. the smallest such y,, is called the L, gain of the system 1. Definition 2.4: D-bounded inpur finite gain L, stability. For any p E [l. m] and any D > 0, the system X is D-bounded input finite-gain L,-stable if it is D-bounded input &-stable and in addition there exists a yo,r > 0 such that
lld~, 5 yap lidll,~p Vd E 4W). Further. the smallest such Y~,~ is called the D-bounded input L, guin of the system We now consider a linear system subject saturation and input-additive disturbance d, I$:
i=Ax+Bu(u+d),
XEW,
UEW”,
to input dEWm, (2)
where (A, B) is stabilizable, and K [W”‘-r[Wmis a saturation function defined as follows. Definition 2.5. A function LT:R”’ ---fR”’ is called a sarurarion f&ction if
(L) a(u) IS decentralized,
i.e.
r da
i
4s [ii) u, is locally Lipschitz; (iii) SC,(S) > 0 whenever s f 0;
Remark 2.1. Graphically, the assumption on u is that each u, is in the first and third quadrants and there exist a A > 0 and a k >O such that the nonlinearity lies in the region between the vertical axis and the graph (s, k sata (s)), where sat,(s) = sgn (s) min (IsI, A}. i.e.
s[g,(as) - k sat& (s)] ~(1
Wo 2 1.
(4)
Moreover, for simplicity hut without loss of generality, throughout this paper, we shall assume that k = 1. Remark 2.2. Since each o, is locally Lipschitz, given a A > 0, there exists a continuous, nondecreasing function L: R, e W. such that, for each i = 1.. , m,
lu,(si + 4) ~ uii(s,)l~ L(ld;l) ldil Ws, E
{sE
R : 1s1I A}, Vd, E F8.
IF, (-‘I
We next pose the following three problems. Definition 2.6: Almost D-bounded disturbance decoupling problem wirh local asymptotic stabi& via linear static state
Brief Papers ((ADDP/LAS)o). For any given p E [l, m] and any D > 0, the (ADDP/LAS), for the system Z”, is defined as follows. For any a priori given (arbitrarily small) n >O, find, if possible, a linear state feedback law u = -Fx such that
feedback
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable; (ii) the closed-loop system is D-bounded input finite-gain &-stable, and its D-bounded input Lr gain from d to x is less than or equal to TJ,i.e.
Definition 2.7: Almost D-bounded disturbance decoupling problem with semi-global asymptotic stability via linear static state feedback ((ADDP/SGAS)p). For any given p E [l, m] and any D > 0, the (ADDP/SGAS), for the system Z”, is
defined as follows. For any a priori given (arbitrarily large) bounded set WC W” and anv a nriori eiven (arbitrarilv small) number n > 0, find, if possible, a liiear state feedback law u = -Px such that
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stabel with W contained in its domain of attraction; (ii) the closed-loop system is D-bounded input finite-gain L,,-stable, and its D-bounded input L, gain from d to x is less than or equal to n, i.e. II~IIL, 5 1) lIdIlL,
Vd E L,“(D).
eigenvalues of A are in the open left half plane. Then the (ADDPIGAS), is solvable for any p E (1, ~1. Moreover, if tr is globally Lipschitz then the ADDP/GAS is solvable for any P E (1, ml. Remark
solvability 3.1. The (ADDP/SGAS)e, (ADDP/GAS), p = 1 remains unknown.
the system 2; is defined as follows. For any a priori given (arbitrarily small) number 7) >O, find, if possible, a linear state feedback law u = -Fx such that (i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; (ii) the closed-loop system is finite gain L,-stable, and its L, gain from d to x is less than or equal to n, i.e.
Proof
of Theorem 3.1. We prove this theorem by first explicitly constructing a family of state feedback laws u = -F(p)x, parameterized in p, and then showing that, for any given n > 0, there exists a p* such that, for each p 2 p*, each p E (1, m], and each D >O the closed-loop system is D-bounded input finite-gain L,-stable, with its D-bounded input finite-gain Lr gain from d to x less than or equal to 7, and, in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable. This family of state feedback laws takes the form
U = -(l + p)B’Px,
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; (ii) the closed-loop system is D-bounded input finite-gain L,-stable, and its D-bounded input Lr gain from d to x is less than or equal to TJ,i.e.
3. Main results In this section, we first present our main results in three theorems and then give proofs for them.
A’P+PA-2PBB’P+Q=O
i = Ax + Ba(-(1
(7)
+ pr)B’Px
+ d).
(8)
v = X’PX.
(9)
Let c be such that x E L,(c) implies that (IB’Pxll c A, where L,(c) is a level set defined as L,(c) = {x E Iw”:V(x) 5 c}. We first consider part (i) of the (ADDP/LAS), problem: local asymptotic stability when d -0. In this case, the evaluation of the derivative of V along the trajectories of the closed-loop system, using Remark 2.1, gives for all x E L,(c), ri = -x’Qx + 2x’PB[a(-(1 = x’Qx - 2 2 5
+ p)B’Px)
+ B’Px]
vJu,((l + p)v,) - sata (vi)]
-x'Qx,
(10)
where we have defined v E R”’ by v = -B’Px. This shows that the equilibrium x = 0 of the closed-loop system (8) with d = 0 is locally asymptotically stable for all p 20. It remains to show that there exists a p* > 0 such that, for all p s p*, the closed-loop system (8) is also D-bounded input finite-gain L,-stable, with its D-bounded Lp gain from d to x less than or equal to 7. To this end, we evaluate the derivative of this V along the trajectories of the system (8) yielding, for all x E L,(c), V = -x’Qx = -x’Qx
+ 2w’PB[u(-(l f - 22
r=l
p)B’Px
+ d) + B’Px]
~,[a~((1 + p)v, + d,) - sat, (vi)].
(11)
Recalling Remarks 2.1 and 2.2, we have bv,I 2 IW 3 -~,[a~((1 + ph + 4 - sat, (v,)l 50, lbvil 5 Id,13 - vibdU + phi + 4) - u,,(v,)+ ai(s) - satA(vi)1
Theorem
3.1: (ADDP/LAS)o. Consider the system Zd, Then, given any D > 0, the (ADDP/LAS), is solvable for
bii(U + phi + 4) - u,(v,)l
ml.
Theorem 3.2: (ADDPISGAS),. Consider the system Zd, and assume that all the eigenvalue of A are in the closed left half-plane. Then, given any D >O, the (ADDP/SGAS)n is solvable for any p E (1, a]. 3.3: (ADDPICAS),
system
to the
and where Q is any positive definite matrix. With this family of state feedback laws, the closed-loop system takes the form
11~11~~ 5 17l141~p Vd E L,“(D).
Theorem
(6)
solution
,=I
as follows. For any a priori given (arbitrarily small) n >O, find, if possible, a linear state feedback law u = -Fx such that
the
p _z 0,
where P is the unique positive-definite algebraic Riccati equation
Vd E L;.
Definition 2.9: Almost D-bounded disturbance decoupling problem with global asymptotic stability via linear static state feedback ((ADDP/GAS)o). For any given p E [l, m] and any D > 0, the (ADDPIGAS), for the system Z”, is defined
any p E (1,
of (ADDP/LAS),, and ADDP/GAS for
We now pick a Lyapunov function
Definition 28: Almost disturbance decoupling problem with global asymptotic stability via linear static state feedback (ADDPICAS). For any p E [l, m], the ADDP/SGAS for
lI~ll~.~5 r) Ildll~,
621
Xd, any
and ADDPICAS.
any D >O.
Assume
that
Consider all the
+
IIB’PII’-vL(2D)
/(x/I’--B(dil’+B,
where 0 E (0,l) is such that 1 + 195 p. Hence we conclude that (12)
622
Brief Papers
for some positive Choose
constants
(Y, and ,f$ independent
(
I,; :
from p
This completes
our proof.
Proof ojf Theorem 3.2. We prove this theorem by first explicitly constructing a family of state feedback laws (I : ~F(F, p)x, parameterized in F and p, and then showing that. for any a priori given (arbitrarily large) bounded set W and any a priori given (arbitrarily small) number n > 0, there exists an P* >O and for each F E (0, ~$1, there exists a P*(F) >O such that, for each p t (1, m], each D >O and each 61‘p*(c). E E (0. F*], the closed-loop system is D-bounded input finite-gain L,,-stable. with its D-bounded input L,, gain from rl to x less than or equal to n, and, in the absence of the distrubance d. its equilibrium x = 0 is locally asymptotically stable. with -ti. contained in its domain of attraction. This family of state feedback laws takes the form
,,‘(:rl_l .:_. )“Y
Then L, (c) is an invariant set for all p > p:‘. Letting W y V (’ “)“, it follows from (12) that
which in turn shows that. for x f 0.
I, = For some poaitivc constants cy? and /!I: independent of p. As will be seen shortlv. WC work with W because its behavior can be compared with that of a stable linear system through standard comparison theorems (Walter, 1970). A similar reasoning was also used in Vidyasagar (1993. pp. 2866289). Recall that d E L;:’ for p F (I. x]. This means that. if p E (1. x).
1I’ Il4II;:’ or. if ,’
SC1,
(15)
x
ll(/ll,
7
css,,“up
0
i/l/ii ‘. x
(I +p)B’P(c)x.
where P(c) is the unique algebraic Riccati equation A’P(F)
+
e>o.
positive-definite
P(F)A -. 2f(r)BB’f(~)
pzo.
(23)
solution
to the
+ FQ = 0,
(24)
where Q is any positive-detinitic matrix. Before moving on. let us recall a property on P(E) from Lin et ul. (1996) in a lemma. For completeness, we also include the proof of this lemma. Lrrnma 3.1. Assume that (A, H) is stabilizable and that A has all its eigenvalues in the closed left half-plane. Then, for all F j 0. there exists a unique matrix P(F) > 0 that solves the ARE (24). Moreover.
116)
hm P(F)
= 0.
(25)
* 4,
It is then [I t (1. x).
clear
that
11~111’ ‘I’ F L,,.,,
, ,,,. Furthermore.
11 f’rrwf ctf f~cwmu 3.1. The cxistencc of a unique positivedefinite solution P(F) for all F >O has been established in Willems (1971). The same paper established that for F = 0 there is a unique solution P(0) = 0 for which A - BB’P(0) has all eigcnvalues in the closed left half-plane. Continuity of the solution of the algebraic Riccati equation for e =O (in other words that P(F) + P(0) = 0 as F+O) has been established in Trentelman (1987). n
LJsing standard comparison thcorcms on (14). find that W t I-,, ,, ,,), and furthermore that
WC easily
With the family of state feedback closed-loop system takes the form .\: = n.u + Btr( mm(1 + P)B’P(F)X WC now pick the Lyapunov
,’
Finally.
a
,;(,,N;;
/i.Yll.-_tulW”’ for some have
positive
constant
0
the
(26)
(27)
(1%
and let ( bt) be such that
(20)
Such a c exists. since lim, +(,P(F) = 0 by Lemma 3.1 and ‘N‘is bounded. Let F* t (0. I] be such that, for each F E (0, F*]. .t F Lv(c’) implies that l~B’P(f),r;~ 51, where the level set f.,,(c) is defined as L, (c) = {I t KY’: V(x) 5 c}. The existence of such an F* is again due to the fact that lim,_,,, P(F) = 0. ‘The evaluation of the derivative of V along the trajectories of the closed-loop system in the absence of d. using Remark 2. I. shows that. for all .\- F L, (c).
1
that U’ x~V”
recalling
H
+ d).
(23)
function
v =.r’P(e).<
g
laws
“I“, WC‘have It”
I independent
of 11. Hence
WC
which shows that
=. for some poaitivc constant (\, independent of I’. This shows the finite-gain stability of the closed-loop system. Choosing ,’
we have
max{p;,(~)““‘\.
F.V
Qr .
(28)
whcrc we have defined u t W”’ hy LI= pB’P(f)x. The relation (2X) shows that, for all I‘ FI (0, F*] and p 20, the equilibrium x = 0 of the closed-loop system (26) with d - 0 is locallv asymptotically stable. with ‘Urc L,(c) contained in its domam of attraction. It remains to show that. for each F t (0, F*], them exists a /I.~(?.) ,t) such that, for all p ?p*(e), F c (0. F*], the closed-loop system (26) is D-bounded input finite-gain I.,,-stable. with its D-hounded input L,, gain from rl to x less
623
Brief Papers than or equal to TJ.This can be shown in a similar way as in 0 the proof of Theorem 3.1.
Proof of Theorem 3.3. Again, we prove this theorem by first explicitly constructing a family of state feedback laws u = -F(p)x, parameterized in p, and then showing that, for any given 77>O, there exists a p* such that, for all p >p* and for each p E (1, m] and each D >O, the closed-loop system is D-bounded input finite-gain &-stable (finite-gain Lp stable if (Tis globally Lipschitz), with its D-bounded input L,, gain (L, gain if g is globally Lipschitz) from d to x less that or equal to ?, and, in the absence if the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable. This family of state feedback laws takes the form ll = -pB’Px, p 2 0, (29) where P is the unique Lyapunov equation
positive-definite
solution
of the
A’P+PA+Q=O
(30)
and Q is any positive-definite matrix. With this family of state feedback laws, the closed-loop system takes the form 1= Ax + Bv( -pB’Px
+ d).
(31)
We now pick a Lyapunov function v = X’PX.
Noting that (35) is identical to (12) in the proof of Theorem 3.1, the rest of the proof follows the same way as in 0 the proof of Theorem 3.1. Finally, we conclude this section with a remark regarding Theorem 3.3. Remark 3.2. It was shown in Liu er al. (1993) (see also Lin er al., 1995) that when the open-loop system has only simple jw poles and the matrix A is, without loss of generality,
skew-symmetric, the state feedback u = -pB’x achieves global asymptotic stabilization in the absence of the disturbance d and finite-gain Lp stabilization for any p > 0. One might naturally wonder if the same class of state feedback laws would achieve almost D-bounded disturbance decoupling with global asymptotic stability ((ADDP/GAS),) as p + 00. The following example shows that, in general, this is not the case. Example 3.1. Consider the following linear system subject to
input saturation: i=[_y
where (dl~ 4. The open-loop system has two poles at *j. Pick the family of state feedback laws as
(32)
The evaluation of this Laypunov function along the trajectories of the closed-loop system in the absence of the disturbance d, using Condition 3 of Definition 2.5, shows that
i]x+[:]sat,(u+d)
U = -pB’x = -p[l
11x, p > 1
(37)
Assuming that the saturation element is nonexistent in the closed-loop system, we calculated the impulse response from d to u as
ri = -x’Qx + Zx’PB[cr(-pB’Px)]
qq = _ P(P+ G=Q(p+q?)t
$77
= -x’Qx - 2 2 u;u,(pu;) i=l 5
_P(-P+m)
-x'Qx,
where we have defined v E KY”by v = -B’Px. The relation (33) shows that the equilibrium x = 0 of the closed-loop system (31) with d = 0 is globally asymptotically stable. It remains to show that there exists a p* > 0 such that, for all p 2 p*, the closed-loop system (31) is also D-bounded input finite-gain &-stable (finite-gain L, stable if (+ is globally Lipschitz), with its D-bounded L, gain (L, gain if (T is globally Lipschitz) from d to x less than or equal to 7. To this end, we evaluate the derivative of this V along the trajectories of the system (31), yielding li = -x’Qx + 2x’PBu( -pB’Px
e-(p-*)t
k$Fi
(33) It can be shown that
Ih(
dt 5 4,
which in turn shows that 11~+ dllL, 5 1, and hence the closed-loop system will operate linearly even in the presence of the saturation element. For the linear closed-loop system, the transfer function from d to x is given by H(s) =
+ d)
L1
s+l sz + 2ps + 1 s - 1
l
Hence = -x’Qx
- 2 2 u,u,(pv, + d,).
(34)
,=I
Recalling Remarks 2.1 and 2.2, we have IPU,/2 Id,13 -v,(Pu, IPU,I5 Id,13 -w,(/w = -4~,b,
which shows that, for a constant disturbance d, IdI5 4, the steady state of the state will remain a constant of the same magnitude.
+ d,) 50,
+ 4)
+ 4) - o,(O)]
~~~~l~-~[u,(pq +d,) - a,(O)]
I;
5 {
IIB'Pll'-e Ilx~\'-eld,I'+e
if 17is globally Lipschitz,
jj IIB’PII’-B L(2D)
Ilxll’~~eld,l’+e
where f3 E (0, 1) is such that 1 + f?sp, condition for the function (T. Hence, we conclude that
if lldll 5 D,
and 6 is a Lipschitz
4. Concluding remarks We have considered the almost disturbance decoupling problem (ADDP) and/or almost D-bounded disturbance decoupling problem (ADDP), with various internal stability for linear systems subject to input saturation and input-additive disturbance via linear static state feedback. Our main results show that the almost D-bounded disturbance decoupling problem with local asymptotic stability is always solvable as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the oper-loop system are, and the locations of these poles play a role only in the solution of (ADDP/SGAS),, (ADDP/GAS), of ADDP/GAS where semi-global or global asymptotic stability is required. Acknowledgement-The
for some positive constants (Y, and PI independent
of p.
(A.R.T.) was supported ECS-9309523.
research of one of the authors in part by the NSF under Grant
624
Brief Papers
References
Fuller, A. T. (1969). In-the-large stability of relay and saturating control systems with linear controller. Int. J. Control, 10,457-480.
Lin, Z. and A. Saberi (1993a). Semi-global exponential stabilization of linear systems subject to ‘input saturation’ via linear feedbacks. Syst. Control Left., 21, 225-239. Lin, Z. and A. Saberi (1993b). A low-and-high gain approach to semi-global stabilization and/or semi-global practical stabilization of a class of linear systems subject to input saturation via linear state and output feedback. In Proc. 32nd IEEE Conf on Decision and Control, San Antonio, TX, pp. 1820-1821. Lin, Z. and A. Saberi (1995a). Semi-global exponential stabilization of linear discrete-time systems subject to ‘input saturation’ via linear feedbacks. Sysf. Control Left., 24, 125-132. Lin, Z. and A. Saberi, (1995b). A semi-global low-and-high gain design technique for linear systems with input saturation-stabilization and disturbance rejection, hit. J. Robust and Nonlinear Control, 5,381-398.
Lin, Z.. A. Saberi and A. R. Tee1 (1995). Simultaneous L,-stabilization and internal stabilization of linear systems subject to input saturation-state feedback case. Sysl. Control Left., 25, 219-226.
Lin, Z., A. A. Stoorvogel and A. Saberi. (1996). Output regulation for linear systems subject to input saturation. Automatica, 32, 29-47.
Liu, W., Y. Chitour and E. Sontag (1993). Remarks on linite gain stabilizability of linear systems subject to input saturation. In Proc. 32nd IEEE Conf on Decision and Control, San Antonio, TK. pp. 18081813. Ozcetin, H. K., A. Saberi and P. Sannuti (1993). Design for H, almost disturbance decoupling problem with internal stability via state or measurement feedback-singular perturbation approach. Int. J. Control, 55, 901-944. Saberi, A. and P. Sannuti (1988). Global stabilization with almost disturbance decoupling of a class of uncertain nonlinear systems. Int. J. Control, 45, 1655-1704. Saberi, A., Z. Lin and A. R. Tee1 (1996). Control of linear systems subject to input saturation, IEEE Trans. Autom. Control. to appear.
Sontag, E. D. and H. J. Sussmann (1990). Nonlinear output feedback design for linear systems with saturating controls. In Proc. 29th IEEE Conf on Decision and Control, Honolulu, HI, pp. 3414-3416 Sontag, E. D. and Y. Yang (1991). Global stabilization of linear systems with bounded feedback, Report 91-09, Rutgers Center for Systems and Control. Sussmann, H. J. and Yang, Y. (1991). On the stabilizability of multiple integrators by means of bounded feedback controls. In Proc. 30th IEEE Conf on Decision and Control, Brighton, U.K., pp. 70-72. Sussmann, H. J.. E. D. Sontag and Y. Yang, (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Autom. Control, AC-39, 241 I-2425.
Teel. A. R. (1992). Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst.Control Len., 18, 165-171. Teel, A. R. (1995). Semi-global stabilization of linear controllable systems with input nonlinearities. IEEE Trans. Autom. Control, AC-40, 96-100. Trentelman. H. L. (1987). Families of linear-quadratic problems: continuity properties. IEEE Trans. Autom. Control, AC-32, 323-329.
Vidyasagar, M. (1993). Nonlinear Systems Analysis, 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. Walter. W. (1970). Differential and Integral Inequalities. Springer-Verlag, Berlin. Weiland, S. and J. C. Willems (1989). Almost disturbance decoupling. IEEE Trans. Autom. Control, AC-34, 277-286. Willems, J. C. (1971). Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control, AC-16,621-634.
Willems. J. C. (1981). Almost invariant approach to high gain feedback design. controlled invariant subspaces. IEEE Control, AC%, 235-252. Willems, J. C. (1982). Almost invariant approach to high gain feedback design. conditionally invariant subspaces. IEEE Control, AC-27, 1071-1085.
subspaces: an Part I: almost Trans.
Aurom.
subspaces: an Part II: almost Trans. Autom.
0005-1098(95)00176-X
Auromalrcu, Vol. 32, No. 4, pp. 6lY-624, 19% Copyright 0 1996 Elsevm Swnce Ltd Printed in Great Britain. All nghts reserved 0005.lOYS/Yh $ls.oo + 0.00
Brief Paper
The Almost Disturbance
Decoupling
Problem with
Internal Stability for Linear Systems Subject to Input Saturation-State
ZONGLI Key Words-Linear
Feedback Case*
LIN,‘( AL1 SABERIS
and ANDREW
R. TEEL§
systems subject to input saturation; almost disturbance decoupling,
understood
(see e.g. Weiland and Willems. 1989: Ozcetin et of ADDPMS was also extended to some nonlinear systems having a certain strong relative degree (Saberi and Sannuti, 1988). In this paper, we attempt to extend this notion to linear systems subject to input saturation. The recent renewed interest in linear systems subject to input saturation is mainly because of the wide recognition of the inherent constraints on the control input. This renewed interest has led to many interesting results on such systems. Most of these results, however, pertain only to issues related to global and semi-global internal stabilization of such systems. In regard to global stabilization, it was shown in Sontag and Sussmann (1990) that a linear system subject to input saturation can be globally asymptotically stabilized if and only if the system in the absence of saturation is asymptotically null controllable with bounded contro1s.n Moreover, it was also shown (see Fuller, 1969; Sussmann and Yang, 1991) that, in general global asymptotic stabilization of linear systems subject to input saturation cannot be achieved by using linear feedback laws and that nonlinear feedback laws should be used for this purpose. A nested feedback design technique for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in Tee1 (1992) and completed in Sontag and Yang (1991) and Sussmann ef al. (1994). The notion of semi-global stabilization of linear systems subject to input saturation was first introduced in Lin and Saberi (1993a) (and in Lin and Saberi (1995a) for the discrete-time counterpart). The semi-globa< framework for stabilization requires feedback laws that yield a closed-loop system that has an asymptotically stable equilibrium whose domain of attraction includes an a priori given (arbitrarily large) bounded set. In Lin and Saberi (1993a, 1995a), it was shown that, both for discrete time and continuous time, one can achieve semi-global stabilization of linear asymptotically null controllable with bounded controls systems subject to input saturation using linear feedback laws. A low-gain design technique based on the eigenstructure assignment was also proposed to construct semi-globally stabilizing controllers. These low-gain control laws were constructed in such a way that the control input does not saturate for any a priori given arbitrarily large bounded set of initial conditions. Utilizing the H2 and H, optimal control theory, alternative ARE-based approaches for designing semi-globally stabilizing low-gain feedback laws were later proposed independently in Lin et al. (1996) and Tee1 (1995). In a recent paper (Lin and Saberi, 1995b; see also Lin and
Abstract-We consider the almost disturbance decoupling problem (ADDP) and/or almost D-bounded disturbance decoupling problem ((ADDP),) with internal stability for linear systems subject to input saturation and input-additive disturbance via linear static state feedback. We show that the almost D-bounded disturbance decoupling problem with local asymptotic stability ((ADDP/LAS),) is always solvable via linear static state feedback as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the open-loop system are, and the locations of these poles play a role only in the solution of (ADDP/GAS),, (ADDP/SGAS), or ADDP/GAS where semi-global or global asymptotic stability is required.
al. 1993). The notion
I. Introduction
The problem of disturbance decoupling or almost disturbance decoupling has a vast history, occupying a central part of classical as well as modern control theory. Several important problems, such as robust control, decentralized control, non-interacting control, model reference and tracking control, can be recast as an almost disturbance decoupling problem. Regardless of from where the problem arises. the basic almost disturbance decoupling problem is to find an output feedback control law such that in the closed-loop system the disturbances are quenched, say in the &-gain sense. up to any prespecified degree of accuracy while maintaining internal stability. In the linear system setting. the above problem is labeled by Willems (1981, 1982) as ADDPMS (the almost disturbance decoupling problem with measurement feedback and internal stability). In the case that, instead of a measurement feedback, a state feedback is used, the above problem is termed the ADDPS (the almost disturbance decoupling problem with internal stability). The ADDPMS for linear systems is now very well *Received 24 January 1994; revised 23 September 1994; revised 25 April 1995: received in final form 5 October 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor Tamer Basar. Corresponding author Professor Zongli Lin. Tel. + 1516 632 9344: Fax + 1 516 632 8490: E-mail [email protected]. t Department of Applied Mathematics and Statistics, SUNY at Stony Brook. Stony Brook, NY 11794-3600, U.S.A. $ School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752. U.S.A. $,Department of Electrical Engineering, University of Minnesota, 4-174 EE/CS Building, 200 Union Street SE, Minneapolis, MN 55455. U.S.A.
n A linear system is asymptotically null controllable with bounded controls if and only if it is stabilizable and all the poles of the open-loop system are in the closed left half-plane. 619
620
Brief Papers p t
[ 1. =] and any D > 0, L;(D) denotes the set of all x E L; such that IIxJ(,>,5 D.
Q+-j
Saturation I
H I
Z:
Linear Syst,em p L
I
i =f(x, d).
X E R”,
d E R”‘,
(1)
we make the following definitions.
_______________L________________________.___~_.__
Fig. I. Linear system subject to input saturation.
Sabcri, 1993b) we have introduced yet another design technique, the so-called low-and-high gain design technique, for a chain of integrators subject to input saturation. This design technique was basically conceived for semi-glovaf control problems beyond stabilization, and was related to the performance issues such as semi-global stabilization with enhanced utilization of the available control capacity and semi-global practical stabilization in the presence of input-additive external disturbance. This design technique was later successfully applied to general linear asymptotically null controllable with bounded controls systems subject to input saturation (Saberi er al., lYY6). As a natural development over the Internal stabilization, Liu PI al. (1993) recently studied the input/output stabilizability of linear systems subject to input saturation. Their results show that, under the assumptions that all the poles of the open-loop system are in the closed left half-plane, with those on the jw axis simple, and that the system in the absence of input saturation is stabilizable and detectable, the linear system subject to input saturation is simultaneously finite-gain &-stabilizable and globally asymptotically stabilizable. Using the low-and-high gain design technique proposed in Lin and Saberi (1993b. 1995b) and Saberi et al. (1996), we showed in Lin et al. (1995) that, under the mild condition that the external input is uniformly bounded, the simultaneous L, stabilization and internal stabilization is always achievable via linear static state feedback as long as the system m the absence of input saturation is stabilizable. no matter where the poles of the open-loop system are, and the locations of these poles play a role only when global or semi-global asymptotic stabilization is required. In this paper, we further exploit the ahote-mentioned low-and-high gain design technique to study the almost disturbance decoupling problem with internal stahility for linear systems subject to input saturation and input-additive disturbance. Our main results show that the almost D-bounded disturbance decoupling problem with local asymptotic stability is always solvable as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the open-loop system are. and the locations of these poles play a role only when WC need to solve the (ADDP/SGAS), (almost D-bounded disturbance decoupling problem with semi-global asymptotic stability), (ADDP/GAS), (almost D-bounded disturbance decoupling problem with global asymptotic stability) or the ADDP/GAS (almost disturbance decoupling problem with global asymptotic stability). In comparison with Lin ef al. (199.5). where, under the same conditions, the bounded input L, stabilization and the internal stabilization were achieved simultaneously, the new results of this paper show that the L, gain of the resulting finite-gain &-stable closed-loop system can be made arbitrarily small. This paper is organized as follows. In Section 2, we define the problems at hand. The solutions to these problems are given in Section 3. Finally, we make brief concluding remarks in Section 4. 2. Preliminaries and problem sralemenl Nut&on. For x e W”, lix,I denotes the Euclidean norm of X. For any p E [l, z), we denote by L; the set of all measurable functions x(.): [O,~)+W” such that J, IJx(t)//Pdt
Defnirion
is
I,,,-stable
2.1: cp shility.
For any p E 11, a], the system Z if, given any d E I,,, and x(0) = 0, x E L;
Defirurion 2.2: D-bounded input LV stability. For any p E Cl, 0~1and any D >O, the system 2 is D-bounded inpur L,,,-slab& if, given any d E L;(D), and x(O) = 0, x E L; Defnirion 2.3: Finire guin L, stability. For any p E [l, DC],the system Z is finite gain L,-stable if it is &,-stable and in addition there exists a yr, >O such that
/l-Ul~.~5 r, II&,
t’d E LT.
Further. the smallest such y,, is called the L, gain of the system 1. Definition 2.4: D-bounded inpur finite gain L, stability. For any p E [l. m] and any D > 0, the system X is D-bounded input finite-gain L,-stable if it is D-bounded input &-stable and in addition there exists a yo,r > 0 such that
lld~, 5 yap lidll,~p Vd E 4W). Further. the smallest such Y~,~ is called the D-bounded input L, guin of the system We now consider a linear system subject saturation and input-additive disturbance d, I$:
i=Ax+Bu(u+d),
XEW,
UEW”,
to input dEWm, (2)
where (A, B) is stabilizable, and K [W”‘-r[Wmis a saturation function defined as follows. Definition 2.5. A function LT:R”’ ---fR”’ is called a sarurarion f&ction if
(L) a(u) IS decentralized,
i.e.
r da
i
4s [ii) u, is locally Lipschitz; (iii) SC,(S) > 0 whenever s f 0;
Remark 2.1. Graphically, the assumption on u is that each u, is in the first and third quadrants and there exist a A > 0 and a k >O such that the nonlinearity lies in the region between the vertical axis and the graph (s, k sata (s)), where sat,(s) = sgn (s) min (IsI, A}. i.e.
s[g,(as) - k sat& (s)] ~(1
Wo 2 1.
(4)
Moreover, for simplicity hut without loss of generality, throughout this paper, we shall assume that k = 1. Remark 2.2. Since each o, is locally Lipschitz, given a A > 0, there exists a continuous, nondecreasing function L: R, e W. such that, for each i = 1.. , m,
lu,(si + 4) ~ uii(s,)l~ L(ld;l) ldil Ws, E
{sE
R : 1s1I A}, Vd, E F8.
IF, (-‘I
We next pose the following three problems. Definition 2.6: Almost D-bounded disturbance decoupling problem wirh local asymptotic stabi& via linear static state
Brief Papers ((ADDP/LAS)o). For any given p E [l, m] and any D > 0, the (ADDP/LAS), for the system Z”, is defined as follows. For any a priori given (arbitrarily small) n >O, find, if possible, a linear state feedback law u = -Fx such that
feedback
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable; (ii) the closed-loop system is D-bounded input finite-gain &-stable, and its D-bounded input Lr gain from d to x is less than or equal to TJ,i.e.
Definition 2.7: Almost D-bounded disturbance decoupling problem with semi-global asymptotic stability via linear static state feedback ((ADDP/SGAS)p). For any given p E [l, m] and any D > 0, the (ADDP/SGAS), for the system Z”, is
defined as follows. For any a priori given (arbitrarily large) bounded set WC W” and anv a nriori eiven (arbitrarilv small) number n > 0, find, if possible, a liiear state feedback law u = -Px such that
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stabel with W contained in its domain of attraction; (ii) the closed-loop system is D-bounded input finite-gain L,,-stable, and its D-bounded input L, gain from d to x is less than or equal to n, i.e. II~IIL, 5 1) lIdIlL,
Vd E L,“(D).
eigenvalues of A are in the open left half plane. Then the (ADDPIGAS), is solvable for any p E (1, ~1. Moreover, if tr is globally Lipschitz then the ADDP/GAS is solvable for any P E (1, ml. Remark
solvability 3.1. The (ADDP/SGAS)e, (ADDP/GAS), p = 1 remains unknown.
the system 2; is defined as follows. For any a priori given (arbitrarily small) number 7) >O, find, if possible, a linear state feedback law u = -Fx such that (i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; (ii) the closed-loop system is finite gain L,-stable, and its L, gain from d to x is less than or equal to n, i.e.
Proof
of Theorem 3.1. We prove this theorem by first explicitly constructing a family of state feedback laws u = -F(p)x, parameterized in p, and then showing that, for any given n > 0, there exists a p* such that, for each p 2 p*, each p E (1, m], and each D >O the closed-loop system is D-bounded input finite-gain L,-stable, with its D-bounded input finite-gain Lr gain from d to x less than or equal to 7, and, in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable. This family of state feedback laws takes the form
U = -(l + p)B’Px,
(i) in the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; (ii) the closed-loop system is D-bounded input finite-gain L,-stable, and its D-bounded input Lr gain from d to x is less than or equal to TJ,i.e.
3. Main results In this section, we first present our main results in three theorems and then give proofs for them.
A’P+PA-2PBB’P+Q=O
i = Ax + Ba(-(1
(7)
+ pr)B’Px
+ d).
(8)
v = X’PX.
(9)
Let c be such that x E L,(c) implies that (IB’Pxll c A, where L,(c) is a level set defined as L,(c) = {x E Iw”:V(x) 5 c}. We first consider part (i) of the (ADDP/LAS), problem: local asymptotic stability when d -0. In this case, the evaluation of the derivative of V along the trajectories of the closed-loop system, using Remark 2.1, gives for all x E L,(c), ri = -x’Qx + 2x’PB[a(-(1 = x’Qx - 2 2 5
+ p)B’Px)
+ B’Px]
vJu,((l + p)v,) - sata (vi)]
-x'Qx,
(10)
where we have defined v E R”’ by v = -B’Px. This shows that the equilibrium x = 0 of the closed-loop system (8) with d = 0 is locally asymptotically stable for all p 20. It remains to show that there exists a p* > 0 such that, for all p s p*, the closed-loop system (8) is also D-bounded input finite-gain L,-stable, with its D-bounded Lp gain from d to x less than or equal to 7. To this end, we evaluate the derivative of this V along the trajectories of the system (8) yielding, for all x E L,(c), V = -x’Qx = -x’Qx
+ 2w’PB[u(-(l f - 22
r=l
p)B’Px
+ d) + B’Px]
~,[a~((1 + p)v, + d,) - sat, (vi)].
(11)
Recalling Remarks 2.1 and 2.2, we have bv,I 2 IW 3 -~,[a~((1 + ph + 4 - sat, (v,)l 50, lbvil 5 Id,13 - vibdU + phi + 4) - u,,(v,)+ ai(s) - satA(vi)1
Theorem
3.1: (ADDP/LAS)o. Consider the system Zd, Then, given any D > 0, the (ADDP/LAS), is solvable for
bii(U + phi + 4) - u,(v,)l
ml.
Theorem 3.2: (ADDPISGAS),. Consider the system Zd, and assume that all the eigenvalue of A are in the closed left half-plane. Then, given any D >O, the (ADDP/SGAS)n is solvable for any p E (1, a]. 3.3: (ADDPICAS),
system
to the
and where Q is any positive definite matrix. With this family of state feedback laws, the closed-loop system takes the form
11~11~~ 5 17l141~p Vd E L,“(D).
Theorem
(6)
solution
,=I
as follows. For any a priori given (arbitrarily small) n >O, find, if possible, a linear state feedback law u = -Fx such that
the
p _z 0,
where P is the unique positive-definite algebraic Riccati equation
Vd E L;.
Definition 2.9: Almost D-bounded disturbance decoupling problem with global asymptotic stability via linear static state feedback ((ADDP/GAS)o). For any given p E [l, m] and any D > 0, the (ADDPIGAS), for the system Z”, is defined
any p E (1,
of (ADDP/LAS),, and ADDP/GAS for
We now pick a Lyapunov function
Definition 28: Almost disturbance decoupling problem with global asymptotic stability via linear static state feedback (ADDPICAS). For any p E [l, m], the ADDP/SGAS for
lI~ll~.~5 r) Ildll~,
621
Xd, any
and ADDPICAS.
any D >O.
Assume
that
Consider all the
+
IIB’PII’-vL(2D)
/(x/I’--B(dil’+B,
where 0 E (0,l) is such that 1 + 195 p. Hence we conclude that (12)
622
Brief Papers
for some positive Choose
constants
(Y, and ,f$ independent
(
I,; :
from p
This completes
our proof.
Proof ojf Theorem 3.2. We prove this theorem by first explicitly constructing a family of state feedback laws (I : ~F(F, p)x, parameterized in F and p, and then showing that. for any a priori given (arbitrarily large) bounded set W and any a priori given (arbitrarily small) number n > 0, there exists an P* >O and for each F E (0, ~$1, there exists a P*(F) >O such that, for each p t (1, m], each D >O and each 61‘p*(c). E E (0. F*], the closed-loop system is D-bounded input finite-gain L,,-stable. with its D-bounded input L,, gain from rl to x less than or equal to n, and, in the absence of the distrubance d. its equilibrium x = 0 is locally asymptotically stable. with -ti. contained in its domain of attraction. This family of state feedback laws takes the form
,,‘(:rl_l .:_. )“Y
Then L, (c) is an invariant set for all p > p:‘. Letting W y V (’ “)“, it follows from (12) that
which in turn shows that. for x f 0.
I, = For some poaitivc constants cy? and /!I: independent of p. As will be seen shortlv. WC work with W because its behavior can be compared with that of a stable linear system through standard comparison theorems (Walter, 1970). A similar reasoning was also used in Vidyasagar (1993. pp. 2866289). Recall that d E L;:’ for p F (I. x]. This means that. if p E (1. x).
1I’ Il4II;:’ or. if ,’
SC1,
(15)
x
ll(/ll,
7
css,,“up
0
i/l/ii ‘. x
(I +p)B’P(c)x.
where P(c) is the unique algebraic Riccati equation A’P(F)
+
e>o.
positive-definite
P(F)A -. 2f(r)BB’f(~)
pzo.
(23)
solution
to the
+ FQ = 0,
(24)
where Q is any positive-detinitic matrix. Before moving on. let us recall a property on P(E) from Lin et ul. (1996) in a lemma. For completeness, we also include the proof of this lemma. Lrrnma 3.1. Assume that (A, H) is stabilizable and that A has all its eigenvalues in the closed left half-plane. Then, for all F j 0. there exists a unique matrix P(F) > 0 that solves the ARE (24). Moreover.
116)
hm P(F)
= 0.
(25)
* 4,
It is then [I t (1. x).
clear
that
11~111’ ‘I’ F L,,.,,
, ,,,. Furthermore.
11 f’rrwf ctf f~cwmu 3.1. The cxistencc of a unique positivedefinite solution P(F) for all F >O has been established in Willems (1971). The same paper established that for F = 0 there is a unique solution P(0) = 0 for which A - BB’P(0) has all eigcnvalues in the closed left half-plane. Continuity of the solution of the algebraic Riccati equation for e =O (in other words that P(F) + P(0) = 0 as F+O) has been established in Trentelman (1987). n
LJsing standard comparison thcorcms on (14). find that W t I-,, ,, ,,), and furthermore that
WC easily
With the family of state feedback closed-loop system takes the form .\: = n.u + Btr( mm(1 + P)B’P(F)X WC now pick the Lyapunov
,’
Finally.
a
,;(,,N;;
/i.Yll.-_tulW”’ for some have
positive
constant
0
the
(26)
(27)
(1%
and let ( bt) be such that
(20)
Such a c exists. since lim, +(,P(F) = 0 by Lemma 3.1 and ‘N‘is bounded. Let F* t (0. I] be such that, for each F E (0, F*]. .t F Lv(c’) implies that l~B’P(f),r;~ 51, where the level set f.,,(c) is defined as L, (c) = {I t KY’: V(x) 5 c}. The existence of such an F* is again due to the fact that lim,_,,, P(F) = 0. ‘The evaluation of the derivative of V along the trajectories of the closed-loop system in the absence of d. using Remark 2. I. shows that. for all .\- F L, (c).
1
that U’ x~V”
recalling
H
+ d).
(23)
function
v =.r’P(e).<
g
laws
“I“, WC‘have It”
I independent
of 11. Hence
WC
which shows that
=. for some poaitivc constant (\, independent of I’. This shows the finite-gain stability of the closed-loop system. Choosing ,’
we have
max{p;,(~)““‘\.
F.V
Qr .
(28)
whcrc we have defined u t W”’ hy LI= pB’P(f)x. The relation (2X) shows that, for all I‘ FI (0, F*] and p 20, the equilibrium x = 0 of the closed-loop system (26) with d - 0 is locallv asymptotically stable. with ‘Urc L,(c) contained in its domam of attraction. It remains to show that. for each F t (0, F*], them exists a /I.~(?.) ,t) such that, for all p ?p*(e), F c (0. F*], the closed-loop system (26) is D-bounded input finite-gain I.,,-stable. with its D-hounded input L,, gain from rl to x less
623
Brief Papers than or equal to TJ.This can be shown in a similar way as in 0 the proof of Theorem 3.1.
Proof of Theorem 3.3. Again, we prove this theorem by first explicitly constructing a family of state feedback laws u = -F(p)x, parameterized in p, and then showing that, for any given 77>O, there exists a p* such that, for all p >p* and for each p E (1, m] and each D >O, the closed-loop system is D-bounded input finite-gain &-stable (finite-gain Lp stable if (Tis globally Lipschitz), with its D-bounded input L,, gain (L, gain if g is globally Lipschitz) from d to x less that or equal to ?, and, in the absence if the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable. This family of state feedback laws takes the form ll = -pB’Px, p 2 0, (29) where P is the unique Lyapunov equation
positive-definite
solution
of the
A’P+PA+Q=O
(30)
and Q is any positive-definite matrix. With this family of state feedback laws, the closed-loop system takes the form 1= Ax + Bv( -pB’Px
+ d).
(31)
We now pick a Lyapunov function v = X’PX.
Noting that (35) is identical to (12) in the proof of Theorem 3.1, the rest of the proof follows the same way as in 0 the proof of Theorem 3.1. Finally, we conclude this section with a remark regarding Theorem 3.3. Remark 3.2. It was shown in Liu er al. (1993) (see also Lin er al., 1995) that when the open-loop system has only simple jw poles and the matrix A is, without loss of generality,
skew-symmetric, the state feedback u = -pB’x achieves global asymptotic stabilization in the absence of the disturbance d and finite-gain Lp stabilization for any p > 0. One might naturally wonder if the same class of state feedback laws would achieve almost D-bounded disturbance decoupling with global asymptotic stability ((ADDP/GAS),) as p + 00. The following example shows that, in general, this is not the case. Example 3.1. Consider the following linear system subject to
input saturation: i=[_y
where (dl~ 4. The open-loop system has two poles at *j. Pick the family of state feedback laws as
(32)
The evaluation of this Laypunov function along the trajectories of the closed-loop system in the absence of the disturbance d, using Condition 3 of Definition 2.5, shows that
i]x+[:]sat,(u+d)
U = -pB’x = -p[l
11x, p > 1
(37)
Assuming that the saturation element is nonexistent in the closed-loop system, we calculated the impulse response from d to u as
ri = -x’Qx + Zx’PB[cr(-pB’Px)]
qq = _ P(P+ G=Q(p+q?)t
$77
= -x’Qx - 2 2 u;u,(pu;) i=l 5
_P(-P+m)
-x'Qx,
where we have defined v E KY”by v = -B’Px. The relation (33) shows that the equilibrium x = 0 of the closed-loop system (31) with d = 0 is globally asymptotically stable. It remains to show that there exists a p* > 0 such that, for all p 2 p*, the closed-loop system (31) is also D-bounded input finite-gain &-stable (finite-gain L, stable if (+ is globally Lipschitz), with its D-bounded L, gain (L, gain if (T is globally Lipschitz) from d to x less than or equal to 7. To this end, we evaluate the derivative of this V along the trajectories of the system (31), yielding li = -x’Qx + 2x’PBu( -pB’Px
e-(p-*)t
k$Fi
(33) It can be shown that
Ih(
dt 5 4,
which in turn shows that 11~+ dllL, 5 1, and hence the closed-loop system will operate linearly even in the presence of the saturation element. For the linear closed-loop system, the transfer function from d to x is given by H(s) =
+ d)
L1
s+l sz + 2ps + 1 s - 1
l
Hence = -x’Qx
- 2 2 u,u,(pv, + d,).
(34)
,=I
Recalling Remarks 2.1 and 2.2, we have IPU,/2 Id,13 -v,(Pu, IPU,I5 Id,13 -w,(/w = -4~,b,
which shows that, for a constant disturbance d, IdI5 4, the steady state of the state will remain a constant of the same magnitude.
+ d,) 50,
+ 4)
+ 4) - o,(O)]
~~~~l~-~[u,(pq +d,) - a,(O)]
I;
5 {
IIB'Pll'-e Ilx~\'-eld,I'+e
if 17is globally Lipschitz,
jj IIB’PII’-B L(2D)
Ilxll’~~eld,l’+e
where f3 E (0, 1) is such that 1 + f?sp, condition for the function (T. Hence, we conclude that
if lldll 5 D,
and 6 is a Lipschitz
4. Concluding remarks We have considered the almost disturbance decoupling problem (ADDP) and/or almost D-bounded disturbance decoupling problem (ADDP), with various internal stability for linear systems subject to input saturation and input-additive disturbance via linear static state feedback. Our main results show that the almost D-bounded disturbance decoupling problem with local asymptotic stability is always solvable as long as the system in the absence of input saturation is stabilizable, no matter where the poles of the oper-loop system are, and the locations of these poles play a role only in the solution of (ADDP/SGAS),, (ADDP/GAS), of ADDP/GAS where semi-global or global asymptotic stability is required. Acknowledgement-The
for some positive constants (Y, and PI independent
of p.
(A.R.T.) was supported ECS-9309523.
research of one of the authors in part by the NSF under Grant
624
Brief Papers
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