The Almost Disturbance Decoupling Problem with ... - Semantic Scholar
Revised for Automatica, April 25, 1997
The Almost Disturbance Decoupling Problem with Internal Stability for Linear Systems Subject to Input Saturation | State Feedback Case Zongli Lin
Ali Saberi
Andrew R. Teel1
Dept. of Applied Mathematics School of Electrical Engr. Dept. of Electrical Engr. and Statistics and Computer Science University of Minnesota SUNY at Stony Brook Washington State University 4-174 EE/CS Bldg, 200 Union St. SE Stony Brook, NY 11794-3600 Pullman, WA 99164-2752 Minneapolis, MN 55455
Abstract
We consider the almost disturbance decoupling problem (ADDP) and/or almost Dbounded disturbance decoupling problem ((ADDP)D ) 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)D ) 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)D , (ADDP/SGAS)D or ADDP/GAS where semi-global or global asymptotic stability is required. Key Words :
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Linear systems subject to input saturation, almost disturbance decoupling.
Research supported in part by the NSF under grant ECS-9309523.
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1. Introduction The problem of disturbance decoupling or almost disturbance decoupling has a vast history behind it, 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 or 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 nd an output feedback control law such that in the closed-loop system the disturbances are quenched, say in the Lp-gain sense, up to any prespeci ed degree of accuracy while maintaining internal stability. In the linear system setting, the above problem is labeled by Willems ([?,?]) 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 as ADDPS (the almost disturbance decoupling problem with internal stability). The ADDPMS for linear systems is now very well understood (see for example, [?,?]). The notion of ADDPMS was also extended to some nonlinear systems having a certain strong relative degree ([?]). In this paper, we make an attempt to extend this notion to linear systems subject to input saturation. u-
d
?j - Saturation
- Linear System
-y
Figure 1.1: Linear system subject to input saturation The recent renewed interest in the linear system 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 linear systems subject to input saturation. Most of these results, however, pertain only to the issues related to global and semi-global internal stabilization of such a class of systems. In regard to global stabilization, it was shown in [?] 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 controls1. Moreover, it was also shown (see [?,?]) 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 [?] and completed in [?,?]. The notion of semi-global stabilization of linear systems subject to input saturation was rst introduced in [?] (and in [?] for the discrete-time counterpart). The semi-global framework for stabilization requires feedback laws that yield a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes an a priori 1 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.
2 given (arbitrarily large) bounded set. In [?] and [?], it was shown that, both for discretetime 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 H1 optimal control theory, alternative ARE-based approaches for designing semi-globally stabilizing low-gain feedback laws were later proposed independently in [?] and [?]. In a recent paper [?] (see also [?]) 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-global 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 ([?]). As a natural development over the internal stabilization, Liu, Chitour and Sontag in [?] recently studied the input/output stabilizability of linear systems subject to input saturation. Their results show that under the assumption that all the poles of the open-loop system are in the closed left-half plane with those on the jw-axis simple and the system in the absence of input saturation is stabilizable and detectable, the linear system subject to input saturation is simultaneously nite-gain Lp-stabilizable and globally asymptotically stabilizable. Using the low-and-high gain design technique proposed in [?,?,?], we showed in [?] that, under the mild condition that the external input is uniformly bounded, the simultaneous Lp-stabilization and internal stabilization is always achievable 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 when global or semi-global asymptotic stabilization is required. In this paper, we further exploit the above mentioned low-and-high gain design technique to study the almost disturbance decoupling problem with internal stability for linear systems subject to input saturation and input-additive disturbance. Our main results show that (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 we need to solve the (ADDP/SGAS)D (almost D-bounded disturbance decoupling problem with semi-global asymptotic stability), (ADDP/GAS)D (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 [?] where under the same conditions the bounded input Lp-stabilization and the internal stabilization were achieved simultaneously, the new results of this paper show that the Lp-gain of the resulting nite gain Lp-stable closed-loop system can be made arbitrarily small. This paper is organized as follows. In Section 2, we de ne the problems at hand. The solutions to these problems are given in Section 3. Finally, we draw a brief concluding remark in Section 4.
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2. Preliminaries and Problem Statement Notation: For x 2 IRn , kxk denotes the Euclidean norm of x. For anyR p 2 [1; 1), we label as Lnp the set of all measurable functions x() : [0; R1) ! IRn such that 01 kx(t)kpdt < 1, and the Lp-norm of x 2 Lnp is de ned as kxkLp = ( 01 kx(t)kpdt)1=p. The label Ln1 denotes the set of all measurable functions x() : [0; 1) ! IRn such that ess supt2[0;1)kx(t)k < 1, and the L1 -norm of x 2 Ln1 is de ned as kxkL1 = ess supt2[0;1)kx(t)k. For any p 2 [1; 1] and any D > 0, Lnp(D) denotes the set of all x 2 Lnp such that kxkL1 D. For a general nonlinear system, : x_ = f (x; d); x 2 IRn; d 2 IRm (2.1) we make the following de nitions.
De nition 2.1. (Lp-Stability) For any p 2 [1; 1], the system is Lp-stable if given any d 2 Lp and x(0) = 0, x 2 Lnp. De nition 2.2. (D-Bounded Input Lp-Stability) For any p 2 [1; 1] and any D > 0, the system is D-bounded input Lp-stable if given any d 2 Lmp (D), and x(0) = 0, x 2 Lnp. De nition 2.3. (Finite Gain Lp-Stability) For any p 2 [1; 1], the system is nite gain Lp-stable if it is Lp-stable and in addition there exists a p > 0 such that kxkLp pkdkLp ; 8d 2 Lmp Further, the smallest such p is called the Lp-gain of the system .
De nition 2.4. (D-Bounded Input Finite Gain Lp-Stability) For any p 2 [1; 1]and any D > 0, the system is D-bounded input nite gain Lp-stable if it is D-bounded input Lp-stable and in addition there exists a D;p > 0 such that kxkLp D;p kdkLp ; 8d 2 Lmp (D) Further, the smallest such D;p is called the D-bounded input Lp-gain of the system . We now consider a linear system subject to input saturation and input-additive disturbance d, d : x_ = Ax + B(u + d); x 2 IRn; u 2 IRm ; d 2 IRm (2.2) where (A; B ) is stabilizable, and : IRm ! IRm is a saturation function de ned as follows.
De nition 2.5. A function : IRm ! IRm is called a saturation function if 1. (u) is decentralized, i.e., 2
1 (s1) 3 6 (s ) 7 6 7 (s) = 66 2 .. 2 77 4 . 5 m (sm)
(2.3)
4 2. i is locally Lipschitz; 3. si(s) > 0 whenever s 6= 0; 4. minflims!0+ is(s) ; lims!0? is(s) g > 0; 5. lim infjsj!1 ji(s)j > 0.
Remark 2.1. Graphically, the assumption on is that each i is in the rst and third
quadrants and there exist a > 0 and a k > 0 such that the nonlinearity lies in the region between the vertical axis and the graph (s; ksat(s)), where sat(s) = sgn(s) minfjsj; g. i.e., s[i ( s) ? ksat(s)] 0; 8 1 (2.4) Moreover, for simplicity but without loss of generality, throughout this paper, we will assume that k = 1.
Remark 2.2. Since each i is locally Lipschitz, given a > 0, there exists a continuous, nondecreasing function L : IR+ ! IR+ such that, for each i = 1 to m, ji(si + di) ? i (si)j L(jdij)jdij; 8si 2 fs 2 IR : jsj g and 8di 2 IR (2.5) We next pose the following three problems.
De nition 2.6. Almost D-bounded disturbance decoupling problem with local asymptotic stability via linear static state feedback ((ADDP/LAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/LAS)D for the system d is de ned as follows. For any a priori given (arbitrarily small) > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable; 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D)
De nition 2.7. Almost D-bounded disturbance decoupling problem with semiglobal asymptotic stability via linear static state feedback ((ADDP/SGAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/SGAS)D for the system d is de ned as follows. For any a priori given (arbitrarily large) bounded set W IRn and any a priori given (arbitrarily small) number > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable with W contained in its domain of attraction;
5 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D) De nition 2.8. Almost disturbance decoupling problem with global asymptotic stability via linear static state feedback (ADDP/GAS) For any p 2 [1; 1], the
ADDP/SGAS for the system d is de ned as follows. For any a priori given (arbitrarily small) number > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; 2. The closed-loop system is nite gain Lp-stable and its Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp De nition 2.9. Almost D-bounded disturbance decoupling problem with global asymptotic stability via linear static state feedback ((ADDP/GAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/GAS)D for the system d is de ned as follows. For any a priori given (arbitrarily small) > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D)
3. Main Results In these section, we rst present our main results in three theorems and then give proofs for them. Theorem 3.1. (ADDP/LAS)D Consider the system d and given any D > 0. Then the (ADDP/LAS)D is solvable for any p 2 (1; 1]. Theorem 3.2. (ADDP/SGAS)D Consider the system d and given any D > 0. Assume that all the eigenvalue of A are in the closed left-half plane. Then the (ADDP/SGAS)D is solvable for any p 2 (1; 1]. Theorem 3.3. (ADDP/GAS)D and ADDP/GAS Consider the system d and any D > 0. Assume that all the eigenvalues of A are in the open left-half plane. Then the (ADDP/GAS)D is solvable for any p 2 (1; 1]. Moreover, if is globally Lipschitz, then ADDP/GAS is solvable for any p 2 (1; 1].
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Remark 3.1. The solvability of (ADDP/LAS)D , (ADDP/SGAS)D , (ADDP/GAS)D and ADDP/GAS for p = 1 remains unknown.
?? : We prove this theorem by rst explicitly constructing a family of state feedback laws u = ?F ()x, parameterized in , and then showing that, for any given > 0, there exists a such that, for each , each p 2 (1; 1], and each D > 0 the
Proof of Theorem
closed-loop system is D-bounded input nite gain Lp-stable with its D-bounded input nite gain Lp-gain from d to x less than or equal to , 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 of u = ?(1 + )B 0 Px; 0 (3.1) where P is the unique positive de nite solution to the following algebraic Riccati equation, A0P + PA ? 2PBB 0P + Q = 0 (3.2) and where Q is any positive de nite matrix. With this family of state feedback laws, the closed-loop system takes the form of x_ = Ax + B(?(1 + )B 0Px + d) (3.3) We now pick a Lyapunov function V = x0Px (3.4) Let c be such that x 2 LV (c) implies that kB 0Pxk , where LV (c) is a level set de ned as LV (c) = fx 2 IRn : V (x) cg. We rst consider part 1 of the (ADDP/LAS)D 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 that for all x 2 LV (c), V_ = ?x0 Qx + 2x0 PB [(?(1 + )B 0 Px) + B 0Px] = ?x0 Qx ? 2
m X i=1
vi[i ((1 + )vi) ? sat(vi )]
?x0 Qx where we have de ned v 2 IRm by v = ?B 0 Px.
(3.5)
This shows that the equilibrium x = 0 of the closed-loop system (??) with d 0 is locally asymptotically stable for all 0. It remains to show that there exists a > 0 such that, for all , the closed-loop system (??) is also D-bounded input nite gain Lp-stable with its D-bounded Lp-gain from d to x less than or equal to . To this end, we evaluate the derivative of this V along the trajectories of the system (??) yielding that for all x 2 LV (c), V_ = ?x0 Qx + 2x0 PB [(?(1 + )B 0Px + d) + B 0 Px] = ?x0 Qx ? 2
m X i=1
vi[i ((1 + )vi + di) ? sat(vi)]
(3.6)
7 Recalling Remarks ?? and ??, we have that, jvij jdij =) ?vi [i ((1 + )vi + di) ? sat(vi)] 0 and
jvij jdij =) ?vi[i ((1 + )vi + di?)i (vi) + i (vi) ? sat(vi)] jdij jvij1? ji ((1 + )vi + di) ? i(vi )j 2 kB 0 P k1? L(2D)kxk1? jdij1+
where 2 (0; 1) is such that 1 + p. Hence, we conclude that V_ ? 1 V + 1 1 V 1?2 kdk1+ (3.7) for some positive constants 1 and 1 independent from . Choose 1+ ! 1 D 1 1 = 1 c 1+2 then, LV (c) is an invariant set for all 1. 1+ Letting W = V 2 , it follows from (??) that 1? 1? 2 V_ = 1 +2 W 1+ W_ ? 1 W 1+ + 1 1 W 1+ kdk1+ (3.8) which, in turn, shows that, for x 6= 0, (3.9) W_ ? 2 W + 1 2 kdk1+ for some positive constants 2 and 2 independent from . As will be seen shortly, we work with W because its behavior can be compared to that of a stable linear system through standard comparison theorems [?]. A similar reasoning was also used in [?, pp. 286-289]. Recall that d 2 Lmp for p 2 (1; 1]. This means that, if p 2 (1; 1), Z 1
1
kd(t)kpdt p
kdkLmp = 0 0 such that, for each p 2 (1; 1], each D > 0 and each ("), " 2 (0; "], the closed-loop system is D-bounded input nite gain Lp-stable with its D-bounded input Lp-gain from d to x less than or equal to , and, in the absence of the disturbance d, its equilibrium x = 0 is locally asymptotically stable with W contained in its domain of attraction. This family of state feedback laws takes the form of u = ?(1 + )B 0 P (")x; " > 0; 0 (3.18) where P (") is the unique positive de nite solution to the following algebraic Riccati equation, A0P (") + P (")A ? 2P (")BB 0P (") + "Q = 0 (3.19) where Q is any positive de nite matrix. Before moving on, let us recall a property on P (") from [?] in a lemma. For completeness, we also include the proof of this lemma.
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Lemma 3.1. Assume (A; B ) is stabilizable and A has all its eigenvalues in the closed left-
half plane. Then, for all " > 0 there exists a unique matrix P (") > 0 which solves the ARE (??). Moreover, lim P (") = 0 (3.20) "!0
Proof of Lemma ?? : The existence of a unique positive de nite solution P (") for all " > 0
has been established in [?]. The same paper established that for " = 0 there is a unique solution P (0) = 0 for which A ? BB 0 P (0) has all eigenvalues in the closed left-half plane. Continuity of the solution of the algebraic Riccati equation for " = 0 (in other words that P (") ! P (0) = 0 as " ! 0) has been established in [?]. 2 With the family of state feedback laws (??), the closed-loop system takes the form of x_ = Ax + B(?(1 + )B 0P (")x + d) (3.21) We now pick the Lyapunov function V = x0P (")x (3.22) and let c > 0 be such that c sup x0 P (")x x2W ;"2(0;1]
Such a c exists since lim"!0 P (") = 0 by Lemma ?? and W is bounded. Let " 2 (0; 1] be such that, for each " 2 (0; "], x 2 LV (c) implies that kB 0P (")xk , where the level set LV (c) is de ned as LV (c) = fx 2 IRn : V (x) cg. The existence of such an " is again due to the fact that lim"!0 P (") = 0. The evaluation of the derivative of V along the trajectories of the closed-loop system in the absence of d, using Remark ??, shows that for all x 2 LV (c), V_ = ?"x0 Qx + 2x0 P (")B [(?(1 + )B 0 P (")x) + B 0 P (")x] = ?"x0 Qx ? 2
m X i=1
vi[i ((1 + )vi ) ? sat(vi)]
?"x0 Qx (3.23) where we have de ned v 2 IRm by v = ?B 0 P (")x. (??) shows that, for all " 2 (0; "] and 0 the equilibrium x = 0 of the closed-loop system (??) with d 0 is locally asymptotically stable with W LV (c) contained in its
domain of attraction. It remains to show that for each " 2 (0; "], there exists a (") > 0 such that, for all ("), " 2 (0; "], the closed-loop system (??) is D-bounded input nite gain Lp-stable with its D-bounded input Lp-gain from d to x less than or equal to . This can be shown in a similar way as we did in the proof of Theorem ??. Proof of Theorem ?? : Again, we prove this theorem by rst explicitly constructing a family of state feedback laws u = ?F ()x, parameterized in , and then showing that, for any given
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> 0, there exists a such that, for all > and for each p 2 (1; 1]and each D > 0, the closed-loop system is D-bounded input nite gain Lp -stable ( nite gain Lp stable if is globally Lipschitz) with its D-bounded input Lp-gain (Lp gain if 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 of u = ?B 0 Px; 0 (3.24) where P is the unique positive de nite solution of the following Lyapunov equation, A0P + PA + Q = 0 (3.25) and where Q is any positive de nite matrix. With this family of state feedback laws, the closed-loop system takes the form of x_ = Ax + B(?B 0 Px + d) (3.26) We now pick a Lyapunov function V = x0Px (3.27) The evaluation of this Lyapunov function along the trajectories of the closed-loop system in the absence of the disturbance d, using Condition 3 of De nition ??, shows that V_ = ?x0 Qx + 2x0 PB [(?B 0Px)] = ?x0 Qx ? 2
m X
vii (vi)
i=1
?x0 Qx (3.28) where we have de ned v 2 IRm by v = ?B 0 Px. (??) shows that the equilibrium x = 0 of the closed-loop system (??) with d 0 is
globally asymptotically stable. It remains to show that there exists a > 0 such that, for all , the closed-loop system (??) is also D-bounded input nite gain Lp-stable ( nite gain Lp stable if is globally Lipschitz) with its D-bounded Lp-gain (Lp gain if is globally Lipschitz) from d to x less than or equal to . To this end, we evaluate the derivative of this V along the trajectories of the system (??) yielding that, V_ = ?x0 Qx + 2x0PB(?B 0 Px + d) m
X = ?x0 Qx ? 2 v (v
i=1
i i
Recalling Remarks ?? and ??, we have that, jvij jdij =) ?vi i (vi + di) 0
i + di )
(3.29)
11 and
jvi j jdij =) j d i j 1? ?vi i(vi + di) = ?vi [i (vi + di) ? i (0)] jvij [i (vi + di) ? i (0)] 8 2 0 1? 1? 1+ if is globally Lipschitz > < kB P k kxk jdi j >: 2 0 1? 1? 1+ if kdk D kB P k L(2D)kxk jdij where 2 (0; 1) is such that 1 + p, and is a Lipschitz condition for the function .
Hence, we conclude that (3.30) V_ ? 1 V + 1 1 V 1?2 kdk1+ for some positive constants 1 and 1 independent from . Noting that (??) is identical to (??) in the proof of Theorem ??, the rest of the proof follows the same way as in the proof of Theorem ??. Finally, we conclude this section with a remark regarding Theorem ??.
Remark 3.2. It was shown in [?] (see also [?]) 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 = ?B 0 x achieves global asymptotic stabilization in the absence of the disturbance d and and nite gain Lp-stabilization for any > 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)D ) as ! 1. The following example shows that, in general, this is not the case.
Example 3.1. Consider the following linear system subject to input saturation,
0 1 x_ = ?1 0 x + 11 sat1(u + d) (3.31) where jdj 1=5. The open loop system has two poles at j . Pick the family of state feedback laws as u = ?B 0 x = ? [ 1 1 ] x; > 1 (3.32) Assuming that the saturation element is nonexistent in the closed-loop system, we calculate the impulse response from d to u as p p2 ? 1) p p ( + 2 ?1)t (? + 2 ? 1) ?(? 2 ?1)t ? ( + ? p2 ? 1 e h(t) = ? p2 ? 1 e It can be shown that Z 1 jh(t)jdt 4
0
12 which, in turn, shows that ku + dkL1 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 1 s + 1 H (s) = s2 + 2s + 1 s ? 1 Hence H (0) = ?11 which shows that for a constant disturbance d, jdj 1=5, the steady state of the state will remain a constant of the same magnitude.
4. Concluding Remarks We have considered the almost disturbance decoupling problem (ADDP) and/or almost Dbounded disturbance decoupling problem (ADDP)D 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 open loop system are, and the locations of these poles play a role only in the solution of (ADDP/SGAS)D , (ADDP/GAS)D or ADDP/GAS where semi-global or global asymptotic stability is required.
References [1] A.T. Fuller, \In-the-large stability of relay and saturating control systems with linear controller", Int. J. Control, Vol. 10, No. 4, pp. 457-480, 1969. [2] Z. Lin and A. Saberi, \Semi-global exponential stabilization of linear systems subject to `input saturation' via linear feedbacks", Systems & Control Letters, vol. 21, no. 3, pp. 225-239, 1993. [3] Z. Lin and A. Saberi, \Semi-global exponential stabilization of linear discrete-time systems subject to `input saturation' via linear feedbacks", Systems & Control Letters, vol 24, pp. 125-132, 1995. [4] Z. Lin, A.A. Stoorvogel and A. Saberi, \Output regulation for linear systems subject to input saturation," Automatica, to appear. [5] Z. Lin and A. Saberi, \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," Proceedings of the 32nd CDC, San Antonio, pp. 1820-1821, December 1993. [6] Z. Lin and A. Saberi, \A Semi-global low-and-high gain design technique for linear systems with input saturation { stabilization and disturbance rejection," Special Issue on Control of Systems with Saturating Actuator of International Journal of Robust and Nonlinear Control, to appear.
13 [7] Z. Lin, A. Saberi and A.R. Teel, \Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation { state feedback case," Systems & Control Letters, to appear. [8] W. Liu, Y. Chitour and E. Sontag, \Remarks on nite gain stabilizability of linear systems subject to input saturation," Proceedings of the 32nd CDC, San Antonio, pp. 1808-1813, December 1993. [9] H.K. Ozcetin, A. Saberi and P. Sannuti, \Design for H1 almost disturbance decoupling problem with internal stability via state or measurement feedback { singular perturbation approach," International Journal of Control, Vol. 55, No. 4, pp. 901944, 1993. [10] A. Saberi, Z. Lin and A.R. Teel, \Control of linear systems subject to input saturation," submitted to IEEE Trans. Auto. Contr., 1994. [11] A. Saberi and P. Sannuti, \Global stabilization with almost disturbance decoupling of a class of uncertain nonlinear systems," International Journal of Control, Vol. 45, 1655-1704, 1988. [12] E.D. Sontag and H.J. Sussmann, \Nonlinear output feedback design for linear systems with saturating controls," Proc. 29th IEEE Conf. Decision and Control, pp. 3414-3416, 1990. [13] E.D. Sontag and Y. Yang, \Global stabilization of linear systems with bounded feedback," Report 91-09, Rutgers Center for Systems and Control, July 1991. [14] H.J. Sussmann, E.D. Sontag and Y. Yang, \A general result on the stabilization of linear systems using bounded controls", IEEE Transactions on Automatic Control, Vol. 39, No. 12, pp. 2411-2425, 1994. [15] H.J. Sussmann and Y. Yang, \On the stabilizability of multiple integrators by means of bounded feedback controls", Proc. 30th CDC, Brighton, U.K., pp. 70-72, 1991. [16] A.R. Teel, \Global stabilization and restricted tracking for multiple integrators with bounded controls", Systems & Control Letters, Vol. 18, No.3, pp. 165-171, 1992. [17] A.R. Teel, \Semi-global stabilization of linear controllable systems with input nonlinearities," IEEE Transaction on Automatic Control, Vol. 40, No. 1, pp. 96-100, 1995. [18] H.L. Trentelman, \Families of linear-quadratic problems: continuity properties", IEEE Trans. Aut. Control, Vol. 32, pp. 323-329, 1987. [19] M. Vidyasagar, Nonlinear Systems Analysis, 2nd edition, Englewood Cli s, NJ: Princeton-Hall, 1993. [20] W. Walter, Di erential and Integral Inequalities, Berlin: Springer-Verlag, 1970. [21] S. Weiland and J.C. Willems, \Almost disturbance decoupling," IEEE Trans. Aut. Control, Vol. 34, pp. 277-286, 1989. [22] J.C. Willems, \Least squares stationary optimal control and the algebraic Riccati equation", IEEE Trans. Aut. Control, Vol. 16, pp. 621-634, 1971. [23] J.C. Willems, \Almost invariant subspaces: an approach to high gain feedback design. Part I: Almost controlled invariant subspaces," IEEE Trans. Auto. Contr. Vol. 26, pp. 235-252, 1981.
14 [24] J.C. Willems, \Almost invariant subspaces: an approach to high gain feedback design. Part II: Almost conditionally invariant subspaces," IEEE Trans. Auto. Contr. Vol. 27, pp. 1071-1085, 1982.
The Almost Disturbance Decoupling Problem with Internal Stability for Linear Systems Subject to Input Saturation | State Feedback Case Zongli Lin
Ali Saberi
Andrew R. Teel1
Dept. of Applied Mathematics School of Electrical Engr. Dept. of Electrical Engr. and Statistics and Computer Science University of Minnesota SUNY at Stony Brook Washington State University 4-174 EE/CS Bldg, 200 Union St. SE Stony Brook, NY 11794-3600 Pullman, WA 99164-2752 Minneapolis, MN 55455
Abstract
We consider the almost disturbance decoupling problem (ADDP) and/or almost Dbounded disturbance decoupling problem ((ADDP)D ) 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)D ) 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)D , (ADDP/SGAS)D or ADDP/GAS where semi-global or global asymptotic stability is required. Key Words :
1
Linear systems subject to input saturation, almost disturbance decoupling.
Research supported in part by the NSF under grant ECS-9309523.
1
1. Introduction The problem of disturbance decoupling or almost disturbance decoupling has a vast history behind it, 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 or 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 nd an output feedback control law such that in the closed-loop system the disturbances are quenched, say in the Lp-gain sense, up to any prespeci ed degree of accuracy while maintaining internal stability. In the linear system setting, the above problem is labeled by Willems ([?,?]) 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 as ADDPS (the almost disturbance decoupling problem with internal stability). The ADDPMS for linear systems is now very well understood (see for example, [?,?]). The notion of ADDPMS was also extended to some nonlinear systems having a certain strong relative degree ([?]). In this paper, we make an attempt to extend this notion to linear systems subject to input saturation. u-
d
?j - Saturation
- Linear System
-y
Figure 1.1: Linear system subject to input saturation The recent renewed interest in the linear system 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 linear systems subject to input saturation. Most of these results, however, pertain only to the issues related to global and semi-global internal stabilization of such a class of systems. In regard to global stabilization, it was shown in [?] 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 controls1. Moreover, it was also shown (see [?,?]) 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 [?] and completed in [?,?]. The notion of semi-global stabilization of linear systems subject to input saturation was rst introduced in [?] (and in [?] for the discrete-time counterpart). The semi-global framework for stabilization requires feedback laws that yield a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes an a priori 1 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.
2 given (arbitrarily large) bounded set. In [?] and [?], it was shown that, both for discretetime 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 H1 optimal control theory, alternative ARE-based approaches for designing semi-globally stabilizing low-gain feedback laws were later proposed independently in [?] and [?]. In a recent paper [?] (see also [?]) 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-global 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 ([?]). As a natural development over the internal stabilization, Liu, Chitour and Sontag in [?] recently studied the input/output stabilizability of linear systems subject to input saturation. Their results show that under the assumption that all the poles of the open-loop system are in the closed left-half plane with those on the jw-axis simple and the system in the absence of input saturation is stabilizable and detectable, the linear system subject to input saturation is simultaneously nite-gain Lp-stabilizable and globally asymptotically stabilizable. Using the low-and-high gain design technique proposed in [?,?,?], we showed in [?] that, under the mild condition that the external input is uniformly bounded, the simultaneous Lp-stabilization and internal stabilization is always achievable 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 when global or semi-global asymptotic stabilization is required. In this paper, we further exploit the above mentioned low-and-high gain design technique to study the almost disturbance decoupling problem with internal stability for linear systems subject to input saturation and input-additive disturbance. Our main results show that (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 we need to solve the (ADDP/SGAS)D (almost D-bounded disturbance decoupling problem with semi-global asymptotic stability), (ADDP/GAS)D (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 [?] where under the same conditions the bounded input Lp-stabilization and the internal stabilization were achieved simultaneously, the new results of this paper show that the Lp-gain of the resulting nite gain Lp-stable closed-loop system can be made arbitrarily small. This paper is organized as follows. In Section 2, we de ne the problems at hand. The solutions to these problems are given in Section 3. Finally, we draw a brief concluding remark in Section 4.
3
2. Preliminaries and Problem Statement Notation: For x 2 IRn , kxk denotes the Euclidean norm of x. For anyR p 2 [1; 1), we label as Lnp the set of all measurable functions x() : [0; R1) ! IRn such that 01 kx(t)kpdt < 1, and the Lp-norm of x 2 Lnp is de ned as kxkLp = ( 01 kx(t)kpdt)1=p. The label Ln1 denotes the set of all measurable functions x() : [0; 1) ! IRn such that ess supt2[0;1)kx(t)k < 1, and the L1 -norm of x 2 Ln1 is de ned as kxkL1 = ess supt2[0;1)kx(t)k. For any p 2 [1; 1] and any D > 0, Lnp(D) denotes the set of all x 2 Lnp such that kxkL1 D. For a general nonlinear system, : x_ = f (x; d); x 2 IRn; d 2 IRm (2.1) we make the following de nitions.
De nition 2.1. (Lp-Stability) For any p 2 [1; 1], the system is Lp-stable if given any d 2 Lp and x(0) = 0, x 2 Lnp. De nition 2.2. (D-Bounded Input Lp-Stability) For any p 2 [1; 1] and any D > 0, the system is D-bounded input Lp-stable if given any d 2 Lmp (D), and x(0) = 0, x 2 Lnp. De nition 2.3. (Finite Gain Lp-Stability) For any p 2 [1; 1], the system is nite gain Lp-stable if it is Lp-stable and in addition there exists a p > 0 such that kxkLp pkdkLp ; 8d 2 Lmp Further, the smallest such p is called the Lp-gain of the system .
De nition 2.4. (D-Bounded Input Finite Gain Lp-Stability) For any p 2 [1; 1]and any D > 0, the system is D-bounded input nite gain Lp-stable if it is D-bounded input Lp-stable and in addition there exists a D;p > 0 such that kxkLp D;p kdkLp ; 8d 2 Lmp (D) Further, the smallest such D;p is called the D-bounded input Lp-gain of the system . We now consider a linear system subject to input saturation and input-additive disturbance d, d : x_ = Ax + B(u + d); x 2 IRn; u 2 IRm ; d 2 IRm (2.2) where (A; B ) is stabilizable, and : IRm ! IRm is a saturation function de ned as follows.
De nition 2.5. A function : IRm ! IRm is called a saturation function if 1. (u) is decentralized, i.e., 2
1 (s1) 3 6 (s ) 7 6 7 (s) = 66 2 .. 2 77 4 . 5 m (sm)
(2.3)
4 2. i is locally Lipschitz; 3. si(s) > 0 whenever s 6= 0; 4. minflims!0+ is(s) ; lims!0? is(s) g > 0; 5. lim infjsj!1 ji(s)j > 0.
Remark 2.1. Graphically, the assumption on is that each i is in the rst and third
quadrants and there exist a > 0 and a k > 0 such that the nonlinearity lies in the region between the vertical axis and the graph (s; ksat(s)), where sat(s) = sgn(s) minfjsj; g. i.e., s[i ( s) ? ksat(s)] 0; 8 1 (2.4) Moreover, for simplicity but without loss of generality, throughout this paper, we will assume that k = 1.
Remark 2.2. Since each i is locally Lipschitz, given a > 0, there exists a continuous, nondecreasing function L : IR+ ! IR+ such that, for each i = 1 to m, ji(si + di) ? i (si)j L(jdij)jdij; 8si 2 fs 2 IR : jsj g and 8di 2 IR (2.5) We next pose the following three problems.
De nition 2.6. Almost D-bounded disturbance decoupling problem with local asymptotic stability via linear static state feedback ((ADDP/LAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/LAS)D for the system d is de ned as follows. For any a priori given (arbitrarily small) > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable; 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D)
De nition 2.7. Almost D-bounded disturbance decoupling problem with semiglobal asymptotic stability via linear static state feedback ((ADDP/SGAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/SGAS)D for the system d is de ned as follows. For any a priori given (arbitrarily large) bounded set W IRn and any a priori given (arbitrarily small) number > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is locally asymptotically stable with W contained in its domain of attraction;
5 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D) De nition 2.8. Almost disturbance decoupling problem with global asymptotic stability via linear static state feedback (ADDP/GAS) For any p 2 [1; 1], the
ADDP/SGAS for the system d is de ned as follows. For any a priori given (arbitrarily small) number > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; 2. The closed-loop system is nite gain Lp-stable and its Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp De nition 2.9. Almost D-bounded disturbance decoupling problem with global asymptotic stability via linear static state feedback ((ADDP/GAS)D) For any given p 2 [1; 1]and any D > 0, the (ADDP/GAS)D for the system d is de ned as follows. For any a priori given (arbitrarily small) > 0, nd, if possible, a linear state feedback law u = ?Fx such that 1. In the absence of the disturbance d, the equilibrium x = 0 of the closed-loop system is globally asymptotically stable; 2. The closed-loop system is D-bounded input nite gain Lp-stable and its D-bounded input Lp-gain from d to x is less than or equal to , i.e.,
kxkLp kdkLp ; 8d 2 Lmp (D)
3. Main Results In these section, we rst present our main results in three theorems and then give proofs for them. Theorem 3.1. (ADDP/LAS)D Consider the system d and given any D > 0. Then the (ADDP/LAS)D is solvable for any p 2 (1; 1]. Theorem 3.2. (ADDP/SGAS)D Consider the system d and given any D > 0. Assume that all the eigenvalue of A are in the closed left-half plane. Then the (ADDP/SGAS)D is solvable for any p 2 (1; 1]. Theorem 3.3. (ADDP/GAS)D and ADDP/GAS Consider the system d and any D > 0. Assume that all the eigenvalues of A are in the open left-half plane. Then the (ADDP/GAS)D is solvable for any p 2 (1; 1]. Moreover, if is globally Lipschitz, then ADDP/GAS is solvable for any p 2 (1; 1].
6
Remark 3.1. The solvability of (ADDP/LAS)D , (ADDP/SGAS)D , (ADDP/GAS)D and ADDP/GAS for p = 1 remains unknown.
?? : We prove this theorem by rst explicitly constructing a family of state feedback laws u = ?F ()x, parameterized in , and then showing that, for any given > 0, there exists a such that, for each , each p 2 (1; 1], and each D > 0 the
Proof of Theorem
closed-loop system is D-bounded input nite gain Lp-stable with its D-bounded input nite gain Lp-gain from d to x less than or equal to , 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 of u = ?(1 + )B 0 Px; 0 (3.1) where P is the unique positive de nite solution to the following algebraic Riccati equation, A0P + PA ? 2PBB 0P + Q = 0 (3.2) and where Q is any positive de nite matrix. With this family of state feedback laws, the closed-loop system takes the form of x_ = Ax + B(?(1 + )B 0Px + d) (3.3) We now pick a Lyapunov function V = x0Px (3.4) Let c be such that x 2 LV (c) implies that kB 0Pxk , where LV (c) is a level set de ned as LV (c) = fx 2 IRn : V (x) cg. We rst consider part 1 of the (ADDP/LAS)D 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 that for all x 2 LV (c), V_ = ?x0 Qx + 2x0 PB [(?(1 + )B 0 Px) + B 0Px] = ?x0 Qx ? 2
m X i=1
vi[i ((1 + )vi) ? sat(vi )]
?x0 Qx where we have de ned v 2 IRm by v = ?B 0 Px.
(3.5)
This shows that the equilibrium x = 0 of the closed-loop system (??) with d 0 is locally asymptotically stable for all 0. It remains to show that there exists a > 0 such that, for all , the closed-loop system (??) is also D-bounded input nite gain Lp-stable with its D-bounded Lp-gain from d to x less than or equal to . To this end, we evaluate the derivative of this V along the trajectories of the system (??) yielding that for all x 2 LV (c), V_ = ?x0 Qx + 2x0 PB [(?(1 + )B 0Px + d) + B 0 Px] = ?x0 Qx ? 2
m X i=1
vi[i ((1 + )vi + di) ? sat(vi)]
(3.6)
7 Recalling Remarks ?? and ??, we have that, jvij jdij =) ?vi [i ((1 + )vi + di) ? sat(vi)] 0 and
jvij jdij =) ?vi[i ((1 + )vi + di?)i (vi) + i (vi) ? sat(vi)] jdij jvij1? ji ((1 + )vi + di) ? i(vi )j 2 kB 0 P k1? L(2D)kxk1? jdij1+
where 2 (0; 1) is such that 1 + p. Hence, we conclude that V_ ? 1 V + 1 1 V 1?2 kdk1+ (3.7) for some positive constants 1 and 1 independent from . Choose 1+ ! 1 D 1 1 = 1 c 1+2 then, LV (c) is an invariant set for all 1. 1+ Letting W = V 2 , it follows from (??) that 1? 1? 2 V_ = 1 +2 W 1+ W_ ? 1 W 1+ + 1 1 W 1+ kdk1+ (3.8) which, in turn, shows that, for x 6= 0, (3.9) W_ ? 2 W + 1 2 kdk1+ for some positive constants 2 and 2 independent from . As will be seen shortly, we work with W because its behavior can be compared to that of a stable linear system through standard comparison theorems [?]. A similar reasoning was also used in [?, pp. 286-289]. Recall that d 2 Lmp for p 2 (1; 1]. This means that, if p 2 (1; 1), Z 1
1
kd(t)kpdt p
kdkLmp = 0 0 such that, for each p 2 (1; 1], each D > 0 and each ("), " 2 (0; "], the closed-loop system is D-bounded input nite gain Lp-stable with its D-bounded input Lp-gain from d to x less than or equal to , and, in the absence of the disturbance d, its equilibrium x = 0 is locally asymptotically stable with W contained in its domain of attraction. This family of state feedback laws takes the form of u = ?(1 + )B 0 P (")x; " > 0; 0 (3.18) where P (") is the unique positive de nite solution to the following algebraic Riccati equation, A0P (") + P (")A ? 2P (")BB 0P (") + "Q = 0 (3.19) where Q is any positive de nite matrix. Before moving on, let us recall a property on P (") from [?] in a lemma. For completeness, we also include the proof of this lemma.
9
Lemma 3.1. Assume (A; B ) is stabilizable and A has all its eigenvalues in the closed left-
half plane. Then, for all " > 0 there exists a unique matrix P (") > 0 which solves the ARE (??). Moreover, lim P (") = 0 (3.20) "!0
Proof of Lemma ?? : The existence of a unique positive de nite solution P (") for all " > 0
has been established in [?]. The same paper established that for " = 0 there is a unique solution P (0) = 0 for which A ? BB 0 P (0) has all eigenvalues in the closed left-half plane. Continuity of the solution of the algebraic Riccati equation for " = 0 (in other words that P (") ! P (0) = 0 as " ! 0) has been established in [?]. 2 With the family of state feedback laws (??), the closed-loop system takes the form of x_ = Ax + B(?(1 + )B 0P (")x + d) (3.21) We now pick the Lyapunov function V = x0P (")x (3.22) and let c > 0 be such that c sup x0 P (")x x2W ;"2(0;1]
Such a c exists since lim"!0 P (") = 0 by Lemma ?? and W is bounded. Let " 2 (0; 1] be such that, for each " 2 (0; "], x 2 LV (c) implies that kB 0P (")xk , where the level set LV (c) is de ned as LV (c) = fx 2 IRn : V (x) cg. The existence of such an " is again due to the fact that lim"!0 P (") = 0. The evaluation of the derivative of V along the trajectories of the closed-loop system in the absence of d, using Remark ??, shows that for all x 2 LV (c), V_ = ?"x0 Qx + 2x0 P (")B [(?(1 + )B 0 P (")x) + B 0 P (")x] = ?"x0 Qx ? 2
m X i=1
vi[i ((1 + )vi ) ? sat(vi)]
?"x0 Qx (3.23) where we have de ned v 2 IRm by v = ?B 0 P (")x. (??) shows that, for all " 2 (0; "] and 0 the equilibrium x = 0 of the closed-loop system (??) with d 0 is locally asymptotically stable with W LV (c) contained in its
domain of attraction. It remains to show that for each " 2 (0; "], there exists a (") > 0 such that, for all ("), " 2 (0; "], the closed-loop system (??) is D-bounded input nite gain Lp-stable with its D-bounded input Lp-gain from d to x less than or equal to . This can be shown in a similar way as we did in the proof of Theorem ??. Proof of Theorem ?? : Again, we prove this theorem by rst explicitly constructing a family of state feedback laws u = ?F ()x, parameterized in , and then showing that, for any given
10
> 0, there exists a such that, for all > and for each p 2 (1; 1]and each D > 0, the closed-loop system is D-bounded input nite gain Lp -stable ( nite gain Lp stable if is globally Lipschitz) with its D-bounded input Lp-gain (Lp gain if 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 of u = ?B 0 Px; 0 (3.24) where P is the unique positive de nite solution of the following Lyapunov equation, A0P + PA + Q = 0 (3.25) and where Q is any positive de nite matrix. With this family of state feedback laws, the closed-loop system takes the form of x_ = Ax + B(?B 0 Px + d) (3.26) We now pick a Lyapunov function V = x0Px (3.27) The evaluation of this Lyapunov function along the trajectories of the closed-loop system in the absence of the disturbance d, using Condition 3 of De nition ??, shows that V_ = ?x0 Qx + 2x0 PB [(?B 0Px)] = ?x0 Qx ? 2
m X
vii (vi)
i=1
?x0 Qx (3.28) where we have de ned v 2 IRm by v = ?B 0 Px. (??) shows that the equilibrium x = 0 of the closed-loop system (??) with d 0 is
globally asymptotically stable. It remains to show that there exists a > 0 such that, for all , the closed-loop system (??) is also D-bounded input nite gain Lp-stable ( nite gain Lp stable if is globally Lipschitz) with its D-bounded Lp-gain (Lp gain if is globally Lipschitz) from d to x less than or equal to . To this end, we evaluate the derivative of this V along the trajectories of the system (??) yielding that, V_ = ?x0 Qx + 2x0PB(?B 0 Px + d) m
X = ?x0 Qx ? 2 v (v
i=1
i i
Recalling Remarks ?? and ??, we have that, jvij jdij =) ?vi i (vi + di) 0
i + di )
(3.29)
11 and
jvi j jdij =) j d i j 1? ?vi i(vi + di) = ?vi [i (vi + di) ? i (0)] jvij [i (vi + di) ? i (0)] 8 2 0 1? 1? 1+ if is globally Lipschitz > < kB P k kxk jdi j >: 2 0 1? 1? 1+ if kdk D kB P k L(2D)kxk jdij where 2 (0; 1) is such that 1 + p, and is a Lipschitz condition for the function .
Hence, we conclude that (3.30) V_ ? 1 V + 1 1 V 1?2 kdk1+ for some positive constants 1 and 1 independent from . Noting that (??) is identical to (??) in the proof of Theorem ??, the rest of the proof follows the same way as in the proof of Theorem ??. Finally, we conclude this section with a remark regarding Theorem ??.
Remark 3.2. It was shown in [?] (see also [?]) 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 = ?B 0 x achieves global asymptotic stabilization in the absence of the disturbance d and and nite gain Lp-stabilization for any > 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)D ) as ! 1. The following example shows that, in general, this is not the case.
Example 3.1. Consider the following linear system subject to input saturation,
0 1 x_ = ?1 0 x + 11 sat1(u + d) (3.31) where jdj 1=5. The open loop system has two poles at j . Pick the family of state feedback laws as u = ?B 0 x = ? [ 1 1 ] x; > 1 (3.32) Assuming that the saturation element is nonexistent in the closed-loop system, we calculate the impulse response from d to u as p p2 ? 1) p p ( + 2 ?1)t (? + 2 ? 1) ?(? 2 ?1)t ? ( + ? p2 ? 1 e h(t) = ? p2 ? 1 e It can be shown that Z 1 jh(t)jdt 4
0
12 which, in turn, shows that ku + dkL1 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 1 s + 1 H (s) = s2 + 2s + 1 s ? 1 Hence H (0) = ?11 which shows that for a constant disturbance d, jdj 1=5, the steady state of the state will remain a constant of the same magnitude.
4. Concluding Remarks We have considered the almost disturbance decoupling problem (ADDP) and/or almost Dbounded disturbance decoupling problem (ADDP)D 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 open loop system are, and the locations of these poles play a role only in the solution of (ADDP/SGAS)D , (ADDP/GAS)D or ADDP/GAS where semi-global or global asymptotic stability is required.
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