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Semi-Global Output Regulation for Linear Systems Subject to Input Saturation | A Low-and-High Gain Design Zongli Lin

Ravi Mantri

Ali Saberi

Dept. of Applied Mathematics School of Electrical Engineering School of Electrical Engineering and Statistics and Computer Science and Computer Science SUNY at Stony Brook Washington State University Washington State University Stony Brook, NY 11794-3600 Pullman, WA 99164-2752 Pullman, WA 99164-2752

August 6, 1994

Abstract

The problem of semi-global output regulation for linear systems subject to input saturation was rst solved in [8] by regulators which are based on low-gain feedback. In this paper, we show that new regulators based on the low-and-high gain design methodology as developed in [6] (see also [5]) and [9] also solve the same problem. These new regulators, inheriting the attributes of the low-and-high gain design methodology, fully utilize the available control capacity to achieve better closed-loop performance.

Submitted to Control { Theory and Advanced Technology

1

1. Introduction In the last few years there has been a surge of interest in the study of linear systems subject to input saturation due to a wide recognition of the inherent constraints on the control input. Although most of the results in this study pertain to the problem of global stabilization (see, for example, [2], [10], [11], [13], [12]) and semi-global stabilization (see, for example, [3], [4], [5], [6], [15], [7]), some attempts have also been made in the solution of output regulation problems. Roughly speaking, this problem is one of controlling a linear system subject to input saturation in order to have its output track (or reject) a family of reference (or disturbance) signals generated by some external system, usually called exosystem. The control laws that solve the output regulation problems are referred to as regulators. A global output regulation problem results if tracking and disturbance rejection is required to occur for all initial conditions of the closed-loop system, while a semi-global output regulation problem results if tracking and disturbance rejection is only required to occur when the initial conditions of the closed-loop system are inside an a priori given (arbitrarily large) bounded set of the state space. The global output regulation problem and semi-global output regulation problem for linear systems subject to input saturation were rst studied in [14] and [8] respectively. Although the global output regulation is appealing by de nition, the semi-global stabilization, as shown in [8], is achievable for a much larger class of systems and allows for linear feedbacks. This paper represents a continued eort of [8] on the study of semi-global output regulation problem for linear systems subject to input saturation. In [8] a set of solvability conditions are given and linear feedback laws, which solve the semi-global output regulation problem are constructed. These feedback laws are constructed in such a way that for any a priori given set of initial conditions the control signals will not saturate after a nite time, which, for large sets of initial conditions, entails very low feedback gains. As a result, because of linearity, whenever the state is close to the origin, the control input will be far away from its maximum allowable value and thus the closed-loop system will be operating far from its full capacity. To avoid such a situation from happening, in this paper, we construct new feedback laws, which also solve the semi-global output regulation problem. These new feedback laws are constructed based on the low-and-high gain design methodology recently developed in [6] (see also [5]) and [9] for the solution of semi-global stabilization problem. These new feedback laws, inheriting the attributes of the low-and-high gain design methodology, fully utilize the available control capacity to achieve better performance. Besides the improvement on the closed-loop performance, we also slightly relax the requirements on the saturation characteristics made in [8]. The remainder of this paper is organized as follows. In Section 2 we recall the problem formulation and the solvability conditions from [8]. The new low-and-high gain state feedback output regulator is given in Section 3, while its error feedback counterpart is given in Section 4. In section 5, we show that the generalized semi-global output regulation problems we formulated earlier in [8], which allow an external driving signal to the exosystem, can also be solved by regulators based on low-and-high gain feedback. Finally, we draw a brief conclusion in Section 6. We will mostly use standard notation in this paper. C 0 denotes the set of continuous functions. For a vector q = (q1; q2; : : :qk )0 we de ne jqj1 := maxi jqij. On the other hand,

2 for a vector-valued function w and T  0 we de ne kwk1 := supt jw(t)j1; kwk1;T := suptT jw(t)j1. For a continuous function V : IRn ! IR+ , a level set LV (c) is de ned as LV (c) = fx 2 IRn : V (x)  cg. Finally k
k denotes the standard Euclidean norm.

2. Preliminaries and Problem Statement We consider the regulator problem when the plant inputs are subject to saturation. More speci cally, we consider a multivariable system with inputs that are subject to saturation together with an exosystem that generates disturbance and reference signals as described by the following system 8 x_ = Ax + B (u) + Pw > > > > < w_ = Sw > > > :

(2:1)

e = Cx + Qw where x 2 IRn, w 2 IRs , u 2 IRm, e 2 IRp, and  is a bounded, globally Lipschitz vector-valued function satisfying (s) = [1(s1); 1(s2);


; 1(sm)]0 (2:2) with 8 > < = si if jsi j  1 (2:3) 1(s) >  ?1 if si < ?1 :  1 if si > 1

Remark 2.1. Unlike in the semi-global stabilization problem (see e.g., [9]), each component

of the saturation function  in the semi-global output regulation problem needs to be exactly known at a neighborhood of the origin for possible disturbance rejection. Without loss of generality, we have assumed that this neighborhood has a radius of 1 and, to ensure the existence of linear feedback regulators, we have also assumed that i is linear in this neighborhood. Following [1] for linear systems in the absence of input saturation, the semi-global linear feedback output regulation problem and semi-global linear observer based error feedback output regulation problem for linear systems subject to input saturation were rst formulated in [8] as follows.

De nition 2.1. (Semi-Global Linear State Feedback Output Regulation Problem) Consider the system (2.1) and a compact set W0  IRs. The semi-global linear state

feedback output regulation problem is de ned as follows. For any a priori given (arbitrarily large) bounded set X0  IRn , nd, if possible, a linear static feedback law u = Fx + Gw such that 1. The equilibrium x = 0 of x_ = Ax + B(Fx) (2:4) is locally exponentially stable with X0 contained in its basin of attraction;

3 2. For all x(0) 2 X0 and w(0) 2 W0, the solution of the closed-loop system satis es lim e(t) = 0:

t!1

(2:5)

De nition 2.2. (Semi-Global Linear Observer Based Error Feedback Output Regs ulation Problem) Consider the system (2.1) and a compact set W0  IR . The semi-global linear observer based error feedback output regulation problem is de ned as follows. For any a priori given (arbitrarily large) bounded sets X0  IRn and Z0  IRn+s , nd, if possible, an error feedback law of the form 8 _           > > < x^_ = A P x^ + B  (u) + LA (e ? [ C Q ] x^ ) w^ 0 S w^ 0 LS w^ (2:6) > > : u = F x^ + Gw^ such that 1. The equilibrium (x; x^; w^) = (0; 0; 0) of 8 > > < x_ = Ax + B (F x^ + Gw^ ))   _        (2:7) > > : x^_ = A P x^ + B  (u) + LA ([ C Q ] x ? x^ ) ?w^ LS 0 0 S w^ w^ is locally exponential stable with X0  Z0 contained in its basin of attraction. 2. For all (x(0); x^(0); w^(0)) 2 X0  Z0 and w(0) 2 W0, the solution of the closed-loop system satis es lim e(t) = 0: (2:8) t!1

The solvability conditions for these problems were also established in [8] and are recalled as follows.

Theorem 2.1. Consider the system (2.1) and the given compact set W0  IRs. The semiglobal linear state feedback regulator problem is solvable if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane; 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P > : C + Q = 0

(2:9)

(b) There exist a  > 0 and a T  0 such that k?wk1;T  1 ?  for all w with w(0) 2 W0.

4

Theorem 2.2. Consider the system (2.1) and the given compact set W0  IRs. The semi-

global error feedback regulator problem is solvable if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. Moreover, the pair    [ C Q ] ; A0 PS is detectable; 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations :

8 > < S = A + B ? + P > : C + Q = 0

(2:10)

(b) There exists a  > 0 and a T  0 such that k?wk1;T  1 ?  for all w with w(0) 2 W0. The low-gain based regulators that solve the above problems were also explicitly constructed in [8]. As explained in the introduction, these low-gain based designs result in under-utilization of the available control capacity. The goal of this paper is to provide a new design methodology based on the low-and-high gain design technique as developed in [6] (see also [5]) and [9], which will lead to full utilization of the control capacity and hence better closed-loop performance. We will also show that this new design methodology is also applicable to the so-called generalized output regulation problems we formulated earlier in [8].

3. Low-and-High Gain State Feedback Regulator Design In this section we construct a family of low-and-high gain based linear state feedback laws, parameterized in " and , and then show that such a family of low-and-high gain state feedback laws solves the semi-global output regulation problem. Signi cant improvement on the closed-loop performance over the earlier design ([8]) is then shown by an example. The low-and-high gain based state feedback regulator design is carried out in the following two steps. Step 1 { Solution of an Algebraic Riccati Equation : We start by choosing a continuous function H : (0; 1] ! IRnn such that H (") is positive de nite for each " 2 (0; 1] and lim H (") = 0

"!0

(3:1)

While a simple choice of H (") is H (") = "I , the choice of H (") does signi cantly aect the closed-loop performance. We leave the issue of judicious choice of H (") for future investigation.

5 Next, we form the following algebraic Riccati equation (ARE),

A0X + XA ? XBB 0X + H (") = 0

(3:2)

We have the following lemma regarding the ARE (3.2). This lemma was proved in [8].

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 X (") > 0 which solves the ARE (3.2). Moreover, lim X (") = 0 (3:3) "!0 Step 2 { Composition of Linear Low-and-High State Feedback Laws:

u = ?( + 1)B 0X (")x + [( + 1)B 0X (") + ?]w;   0

(3:4)

We note that here when  = 0 the low-and-high gain state feedback laws as given in (3.4) reduce to the low-gain based linear state feedback laws as given in [8] and " is referred to as a low-gain parameter. Moreover, as shown next, for any value of , the family of state feedback laws (3.4) also solve the semi-global linear state feedback output regulation problem and for this reason  is referred to as a high-gain parameter. We then have this following result.

Theorem 3.1. Consider the system (2.1) and the given compact set W0  IRs . The family

of linear state feedback laws as given in (3.4) solves the semi-global linear state feedback output regulation problem if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane; 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P (3:5) > : C + Q = 0

(b) There exist a  > 0 and a T  0 such that k?wk1;T  1 ?  for all w with w(0) 2 W0. More speci cally, for any a priori given (arbitrarily large) bounded set X0  IRn, there exists an " > 0 such that for each " 2 (0; "] and for each   0, i ) The equilibrium x = 0 of

x_ = Ax + B(?(1 + )B 0X (")x) is locally exponentially stable with X0 contained in its basin of attraction.

(3:6)

6 ii ) For all x(0) 2 X0 and w(0) 2 W0, the solutions of the closed-loop system satis es lim e(t) = 0

t!1

(3:7)

Proof : The fact that there exists an "1 2 (0; 1] such that for each " 2 (0; "1] and for each   0, the equilibrium x = 0 of (3.6) is locally exponentially stable with X0 contained in its basin of attraction has been established in [9]. We next show that there exists an "2 2 (0; 1] such that for each " 2 (0; "2] and for each   0, item ii) of the theorem holds. To this end, let us introduce an invertible, triangular coordinate change  = x ? w. Using condition 2(a), we have

_ = A + B [(u) ? ?w] (3:8) With the family of state feedback laws given by (3.4), the closed-loop system can be written as _ = A + B [(?w ? ( + 1)B 0X (")) ? ?w] (3:9) By Condition 2(b), k?wk1;T < 1 ? . Moreover, for any x(0) 2 X0 and any w(0) 2 W0, (T ) belongs to a bounded set, say UT , independent of " since X0 and W0 are both bounded and (T ) is determined by a linear dierential equation with bounded inputs (
) and ?w. We next pick a Lyapunov function V () = 0 X (") and let c > 0 be such that

c

sup

2UT ;"2(0;1]

0 X (")

(3:10) (3:11)

Such a c exists since lim"!0 X (") = 0 by Lemma 3.1 and UT is bounded. Let "2 2 (0; 1] be such that  2 LV (c) implies that jB 0X (")j1  . The existence of such an "2 is again due to the fact that lim"!0 X (") = 0. The evaluation of V_ , t  T , inside the set LV (c), using (3.2), now shows that V_ = ?0 (H (")+ X (")BB 0X ("))  +20 X (")B [(?w ? ( +1)B 0X (")) ? ?w + B 0X (")]  ?0H (") ? 2v0[(( + 1)v + ) ? v ? ] m X (3.12) = ?0H (") ? 2 vi[1(( + 1)vi + i) ? vi ? i] i=1

where we have denoted v := ?B 0X (") and  := ?w with their ith component denoted by vi and i respectively. Noting that sign(1(s)) = sign(s) and for " 2 (0; "1], jvi + ij  1, we observe that,

j( + 1)vi + ij  1 =) vi[1(( + 1)vi + i) ? vi ? i] = vi2  0 ( + 1)vi + i > 1 =) vi > 0 =) vi[1(( + 1)vi + i) ? vi ? i]  0 ( + 1)vi + i < ?1 =) vi < 0 =) vi[1(( + 1)vi + i) ? vi ? i]  0

Hence, we conclude that

V_  ?0 H (")

(3:13)

7 which shows that any trajectory of (3.9) starting at t = 0 from f = x ? w : x 2 X0; w 2 W0g remain inside the set LV (c) and approaches the equilibrium  = 0 as t ! 1, which implies that e(t) = C(t) ! 0 as t ! 1. Finally, setting " = minf"1; "2g, we conclude our proof of Theorem 3.1. We now demonstrate the improvement on the closed-loop performance as the high gain parameter  increases by an example.

Example 3.1: Consider the following system:

2 ?1 0 3 2 1 03 21 03 1 1 0 777 x + 666 0 1 777 (u) + 666 0 1 777 w 4 0 05 ?2 ?1:5 ?0:5 5 4 0 0 5 0 0 0 0 6 4:5 ?0:5  (3:14) 1 w 0 1:5 ?0:5  x +  ?0:5 0  w 2 3 1:5 ?0:5 0 ?1 n p o with w(0) 2 W0 where W0 = w : kwk < 2=4 It is straightforward to show that, the solvability conditions for the semi-global linear state feedback output regulation problem are satis ed. More speci cally, the matrices, 2 1 ?1 3 6 :5 0 777 ; ? =  1:25 0:25  (3:15)  = 664 ?1:025 1:75 5 ?0:25 ?1:25 3:25 ?0:75 solve the linear matrix equations (2). Also,  = 0:4697, since k?wk1  0:5303 for all w(0) 2 W0. Let the set X0 be given by X0 = fx : kxk  14; x 2 IR4g. We choose 2 2 " 0 0 03 7 6 2 (3:16) H (") = 10?4  664 00 "0 0" 00 775 0 0 0 " Then, following the proof of Theorem 3.1, a choice of " is 1:5  10?4 . For " = ", the feedback law (3.4) is given by   10?3 1:1918  10?2 3:3628  10?3 ?2:9767  10?3  x u = ?( + 1) 61::4203 1918  10?2 2:2623  10?2 6:1139  10?3 ?5:6505  10?3   ?2:9799  10?3 3:3755  10?3   1:25 0:25  + ( + 1) ?4:2588  10?3 4:6980  10?3 + ?0:25 ?1:25 w;   0 2 8 1 > 6 > > x_ = 66 1 > 4 ?1 > > <  50 > > w_ = ?1 > >  > > :e = 3 2

For the initial conditions x0 = (7; 7; 7; 7), w0 = (0:1; 0:1), Figures 3.1 and 3.2 show the control action and the closed-loop performance for low gain feedback ( = 0) and low-and-high gain feedback ( = 1000) respectively. The simulation results illustrate that the low-and-high gain feedback regulators signi cantly outperform the low gain feedback ones as introduced in [8].

8 600 400 200 0 -200 0

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0.2 0.1 0 -0.1 -0.2 -0.3 0

0.4 0.2 0 -0.2 -0.4 -0.6 0

Figure 3.1: " = 1:5  10?4 ,  = 0. a) e1; b) e2; c) 1(u1); d) 1(u2).

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1 0.5 0 -0.5 -1 0

1 0.5 0 -0.5 -1 0

Figure 3.2: " = 1:5  10?4 ,  = 1000. a) e1; b) e2; c) 1(u1); d) 1(u2).

10

4. Low-and-High Gain Error Feedback Regulator Design As in the previous section, in this section we rst construct a family of low-and-high gain error feedback laws, parameterized in a low-gain parameter ", a high-gain parameter  and an observer parameter `, and then show that such a family of low-and-high gain error feedback laws solves the semi-global output regulation problem. Signi cant improvement on the closed-loop performance over the earlier design ([8]) is again shown by an example. The family of low-and-high gain error feedback laws we construct is linear observer based and is composed by implementing the low-and-high gain state feedback laws developed in the previous section with the state of a fast observer. More speci cally, it takes the following form 8 x^_ = Ax^ + B (u) + P w^ + L (`)C (x ? x^) + L (`)Q(w ? w^ ) A A > > > > < w^_ = S w^ + LS (`)C (x ? x^) + LS (`)Q(w ? w^) > > > : 0 0

(4:1)

u = ?(1 + )B X (")^x + (( + 1)B X (") + ?)w^ where LA (`) and LS (`) are constructed in the following three steps. Step 1 : Assuming that the pair    [ C Q ] ; A0 PS is observable, we choose a nonsingular state transformation TS and a nonsingular output transformation TO such that 3 2 b11 Ip1?1 b21 0


bk 1 0 66 b12 0 b22 0


bk 2 0 777 66 0 b23 Ip2?1


bk3 0 77 A P  66 b13 ? 1 0 b24 0


bk 4 0 777 TS 0 S TS = 66 b14. . ... 7 . . . . 66 .. .. .. .. .. .. 7 64 b 0 b22k?1 0


bk2k?1 Ipk?1 75 12k?1 b12k 0 b22k 0


bk2k 0 21 0 0


0 0 3 1p1 ?1 0 66 0 0 1 01p2?1


0 0 777 6 ... 77 ... ... ... . . . ... TO?1 [ C Q ] TS = 66 ... 64 0 0 0 0


1 01pk ?1 75 0 0 0 0


0 0 where Ipi?1 is an identity matrix of dimension pi ?1pi ?1, and the integers pi , i = 1; 2;


; k are the so-called observability indices. Such a transformation exists and is usually called Brunovski transformation. Step 2 : For each i = 1 to k , nd an li1 2 IRpi?11 and a scalar li2 such that the following matrix is Hurwitz,  l I  i1 pi?1 A~1 = ? ?l 0 i2

11 Step 3 : Compose the observer gain as follows 2 b + S (`)l b21


bk 1 66 11b12 +p1`?p11 l12 11 b22


bk 2 66 b b + S ( ` ) l


b 13 23 p2 ?1 21 k3 66 p 2l b b + `


b 14 23 k4 L(`) = TS 66 ... ... ... 22 ... 66 64 b12k?1 b22k?1


bk2k?1 + Spk ?1(`)lk1

b12k

where for an integer r,

b22k






bk2k + `pk lk2

3

0 0 777 0 77 0 777 TO?1 0 777 05 0

2` 0


0 3 66 0 `2


0 77 Sr (`) = 64 ... ... . . . ... 75

0 0


`r Finally, partition the matrix L(`) to obtain LA (`) 2 IRnp and LS (`) 2 IRsp as follows  L (`)  A LS (`) = L(`) We then have the following result.

Theorem 4.1. Consider the system (2.1) and the given compact set W0  IRs . Then the

family of low-and-high gain error feedback laws (4.1) solves the semi-global linear observer based error feedback regulation problem if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. Moreover, the pair   A P  [C Q]; 0 S is observable; 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations :

8 > < S = A + B ? + P > : C + Q = 0

(4:2)

(b) There exists a  > 0 and a T  0 such that k?wk1;T  1 ?  for all w with w(0) 2 W0. More speci cally, for any a priori given (arbitrarily large) sets X0  IRn and Z0  IRn+s , there exists an " 2 (0; 1], and for each " 2 (0; "] and each   0, there exists an `("; ) > 0 such that for each " 2 (0; "], each   0 and each `  `("; ),

12 i ) The equilibrium (x; x^; w^) = (0; 0; 0) of 8 0 0 > < x_ = Ax + B (?( + 1)B X (")^x + (( + 1)B X (") + ?)w^ )) _          (4:3) > : x^_ = A P x^ + B  (u) + LA (`) ([ C Q ] x ? x^ ) w^ 0 S w^ 0 LS (`) ?w^ is locally exponentially stable with X0  Z0 contained in its basin of attraction. ii ) For all (x(0); x^(0); w^ (0)) 2 X0  Z0 and w(0) 2 W0, the solution of the closed-loop system satis es lim e(t) = 0: (4:4) t!1

Proof : With the family of feedback laws (4.1), the closed-loop system can be written as, 8 x_ = Ax + B (?w^ ? (1 + )B 0 X (")(^x ? w^ )) + Pw > > > > < x^_ = Ax^ + B(?w^ ? (1 + )B 0X (")(^x ? w^)) + P w^ + LA(`)C (x ? x^) + LA(`)Q(w ? w^ ) > > > :_

w^ = S w^ + LS (`)C (x ? x^) + LS (`)Q(w ? w^)

We then adopt the invertible change of state variable, 8  = x ? w > > > < x~ = x ? x^ > > > :

w~ = w ? w^ and rewrite the closed loop system (4.5) as 8_  = A + B(?(1+ )B 0X (") +?w ? ?w~ +(1+ )B 0X (")~x? (1+ )B 0X (")w~) > > > +(A ? S + P )w > > < > x~_ = (A ? LA(`)C )~x + (P ? LA (`)Q)w~ > > > : w~_ = ?L (`)C x~ + (S ? L (`)Q)w~ S

(4:6)

(4:7)

S

A further state transformation of the form   x~ ~ w~ = TS S (`)~z takes (4.7) into the following form, 8_ > <  = A + B [ (?(1+ )B 0X (") +( + 1)MTS S~(`)~z + ?w) ? ?w] > : z~_ = `A~z~

(4:5)

(4:8)

13 where

and

" 0 B X (")

#

0 M= ? 0 ?( +1 + B 0X (")) 3 2 Sp1 (`)=`p1 +p 0


0 77 6 0 Sp2 (`)=`p2 +p


0 ~S(`) = 666 77 ... ... ... ... 5 4 p + p k 0 0


Spk (`)=` 03 0 77 ... 75 0


A~k

2 A~ 0


1 66 0 A~2


A~ = 64 .. .. . . . . .

0

where p = maxfp1; p2; :::; pkg. We are now in a position to show items i) and ii) of the theorem. To show that item i) of the theorem holds, we note that (4.3) is equal to (4.5) for w = 0. Hence (4.8) reduces to 8_ > <  = A + B (?(1+ )B 0X (") + ( + 1)MTS S~(`)~z ) (4:9) >_ : ~ z~ = `Az~ For any x(0) 2 X0, (T + 1) belongs to a bounded set, say UT , independent of " since X0 is bounded and (T + 1) is determined by a linear dierential equation with bounded input (
). Since A~ is Hurwitz, let P~ be the unique positive de nite solution to the Lyapunov equation A~0P~ + P~ A~ = ?I (4:10) and de ne a Lyapunov function

V (; z~) = 0X (") + z~0P~ z~

(4:11)

c1 

(4:12)

We next let c1 > 0 be such that sup

2UT ;kz~k1;"2(0;1]

V (; z~)

Such a c1 exists since lim"!0 X (") = 0 by Lemma 3.1 and UT is a bounded set. Let "1 2 (0; 1] be such that  2 LV (c1) implies that jB 0X (")j1  . The existence of such an "1 is again due to the fact that lim"!0 X (") = 0. Consider the second equation of (4.9), it is straightforward to verify that there exists an  `1 > 1 such that for all `  `1 and for all (^x(0); w^(0)) 2 Z0,

kz~(T + 1)k  1

(4:13)

14 On the other hand, the evaluation of V_ , t  T + 1, inside the set LV (c1), using (3.2) and (4.10) and following the argument used in proving (3.13), shows that V_ = ?0(H (") + X (")BB 0X (")) +20X (")B [(?(1 + )B 0X (") + ( + 1)MTS S~(`)~z ) + B 0X (")] ? `z~0z~  ?0H (") ? `z~0z~ ? 0 X (")BB 0X (") +20X (")B [(?(1 + )B 0X (") + ( + 1)MTS S~(`)~z ) ? (?(1 + )B 0X ("))]  ?0H (") ? `kz~k2 + 2 ( + 1)kMTS S~(`)kk0X (")B kkz~k ? k0X (")B k2 (4.14) where is the Lipschitz constant of the function . It is now clear that, for each " 2 (0; "1] and each   0, there is an `2("; ) > 0 such that, for `  `2("; ), (; z~) 2 LV (c) =) V_  ?0H (") ? 1 `z~0z~ (4:15) 2 which, in turn, shows that, for any a priori given set X0 and Z0, there exists an "1 > 0 such that for each " 2 (0; "1] and any   0, there exists an `2("; )  `1, such that for each " 2 (0; "1],   0, `  `2("; ), the equilibrium of (4.3) is locally exponentially stable with X0  Z0 contained in its basin of attraction. Now, in order to show that item ii) of the theorem holds, consider (4.8). Again, for any x(0) 2 X0 and any w(0) 2 W0, (T + 1) belongs to a bounded set, say UwT , independent of " since X0 is bounded and (T + 1) is determined by a linear dierential equation with bounded inputs (
) and ?w. For the same Lyapunov function as given by (4.11), we choose c2 > 0 such that

c2 

sup

2UwT ;kz~k1;"2(0;1]

V (; z~)

(4:16)

Such a c2 exists since lim"!0 X (") = 0 by Lemma 3.1 and UwT is a bounded set. Let "2 2 (0; 1] be such that  2 LV (c1) implies that jB 0X (")j1  . The existence of such an "2 is again due to the fact that lim"!0 X (") = 0. Now, the evaluation of V_ , t  T + 1, inside the set LV (c2), using the same technique as above, yields, V_ = ?0(H (") + X (")BB 0X (")) ? `z~0z~ +20 X (")B [(?(1 + )B 0X (") + ( + 1)MTS S~(`)~z + ?w) ? ?w + B 0X (")]  ?0H (") ? v0v ? 2v0[(( + 1)v +  + ) ? v ? ] ? z~0z~  ?0H (") ? v0v ? 2v0[(( + 1)v +  + ) ? v ? ] ? 2z~0z~ m m m X X X (4.17)  ?0H (") ? vi2 ? 2 vi[1(( + 1)vi + i + i) ? vi ? i] ? i2 i=1

i=1

i=1

where we have denoted v := ?B 0X ("),  := ( + 1)MTS S~(`)~z , and  := ?w with their ith components denoted by vi, i and i respectively. We have also chosen `3("; ) > 1 such that for each " 2 (0; "2],   0, `  `3("; ), k( + 1)MTS S~(`)k < , which together with (4.13) ensures that ji + ij < 1.

15 Noting that sign(1(s)) = sign(s), and for " 2 (0; "2], `  `3("; ), jvi + ij  1 and ji + ij < 1, we observe that, j( + 1)vi + i + ij  1 =) vi[1(( + 1)vi + i + i) ? vi ? i] = vi2 + vii ( + 1)vi + i + i > 1 =) vi > 0 =) vi[1(( + 1)vi + i + i) ? vi ? i]  0 ( + 1)vi + i + i < ?1 =) vi < 0 =) vi[1(( + 1)vi + i + i) ? vi ? i]  0 Now, without loss of generality, let's assume that j( + 1)vi + i + ij  1; i = 1 : : : m for some 0  m  m. Hence we conclude that, m X V_  ?0H (") ? (vi + i)2

 ?0H (")

i=1

(4.18) Thus, we have shown that, for all (x(0); x^(0); w^(0)) 2 X0  Z0, there exists an "2 > 0, such that for each " 2 (0; "2] and each   0, there exists an `3 ("; ) such that for each " 2 (0; "2],   0, `  `3 ("; ), the solutions of the closed-loop system (4.7) satisfy lim e(t) = 0 (4:19) t!1 Finally, taking " = minf"1; "2g and `("; ) = maxf`2 ("; ); `3("; )g, we complete our proof of Theorem 4.1. Example 4.1 : We consider the same plant and the exosystem as in Example 3.1. However, this time, the state x and w are not available for feedback, which force us to use error feedback regulators. Let the sets W0 and X0 be same as those in Example 3.1. Let the set Z0, be given by Z0 = fz : kzk  1; z 2 IR6g. Following the proof of Theorem 4.1, a suitable choice of " is 1:5  10?4 . Placing all eigenvalues of A~ at ?4, a suitable choice of ` for   1200 is 5. For " = ",   1200 and ` = `, the feedback laws (4.1) are given by 8_ > > x^ = Ax^ + B (u) + P w^ + LA (5)C (x ? x^) + LA (5)Q(w ? w^ ) > > > > w^_ = S w^ + LS (5)C (x ? x^) + LS (5)Q(w ? w^) > <  6:4203  10?3 1:1918  10?2 3:3628  10?3 ?2:9767  10?3  x^ > u = ? (  + 1) > > 1:1918  10?2 2:2623  10?2 6:1139  10?3 ?5:6505  10?3 > >    ?3 ?3   > > : + ( + 1) ?2:9799  10?3 3:3755  10?3 + 1:25 0:25 w^

?4:2588  10

4:6980  10

?0:25 ?1:25

(4:20) where   0 and 2 ?6:6148  103 ?8:8425  102 3 6  103 ?5:3047  103 777 ; LS (5) =  ?3:5526  103 9:4147  102  LA (5) = 664 ?22::8146 4153  104 ?6:6806  103 5 2:0833  103 ?4:0093  103 2:4947  104 ?4:7511  104 For the initial conditions x0 = (7; 7; 7; 7), w0 = (0:1; 0:1), x^0 = (0; 0; 0; 0), w^0 = (0; 0), Figures 4.1 and 4.2 show the control action and the closed-loop performance for low gain feedback ( = 0,` = 5) and low-and-high gain feedback ( = 1200,` = 5) respectively. The simulation results illustrate that the low-and-high gain feedback regulators signi cantly outperform the low gain feedback ones as introduced in [8].

16 600 400 200 0 -200 0

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Figure 4.1: " = 1:5  10?4,  = 0, ` = 5. a) e1; b) e2; c) 1(u1); d) 1(u2).

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Figure 4.2: " = 1:5  10?4 ,  = 1200, ` = 5. a) e1; b) e2; c) 1(u1); d) 1(u2).

18

5. Generalized Semi-Global Output Regulation Problems In an eort to broaden the class of disturbance and reference signals, in [8], we formulated the generalized semi-global linear feedback regulator problem, in which an external driving signal to the exosystem is included. More speci cally, we consider a multivariable system with inputs that are subject to saturation together with an exosystem that generates disturbance and reference signals as described by the following system 8 x_ = Ax + B (u) + Pw > > > > < w_ = Sw + r > > > :

(5:1)

e = Cx + Qw where x 2 IRn, w 2 IRs, u 2 IRm, e 2 IRp, r 2 C 0 is an external signal to the exosystem, and  is a vector-valued saturation function as de ned by in Section 2. The generalized semi-global linear state feedback regulation problem and the generalized semi-global error feedback regulation problem are formulated as follows.

De nition 5.1. (Generalized Semi-Global Linear State Feedback Regulator Problem) Consider the system (5.1), a compact set W0  IRs and a compact set R  C 0. The

generalized semi-global linear state feedback regulator problem is de ned as follows. For any a priori given (arbitrarily large) bounded set X0  IRn , nd, if possible, a linear static feedback law u = Fx + Gw + Hr, such that 1. The equilibrium x = 0 of x_ = Ax + B(Fx) (5:2) is locally exponentially stable with X0 contained in its basin of attraction, 2. For all x(0) 2 X0, w(0) 2 W0 and r 2 R, the solution of the closed-loop system satis es lim e(t) = 0:

t!1

(5:3)

De nition 5.2. (Semi-Global Linear Observer Based Error Feedback Output Regs ulation Problem) Consider the system (5.1), a compact set W0  IR and a compact set R  C 0. The semi-global linear observer based error feedback output regulation problem is de ned as follows. For any a priori given (arbitrarily large) bounded sets X0  IRn and Z0  IRn+s , nd, if possible, an error feedback law of the form 8 _           > < x^_ = A P x^ + B  (u) + LA (e ? [ C Q ] x^ ) w^ LS w^ 0 S w^ 0 (5:4) > > : u = F x^ + Gw^ + Hr such that

19 1. The equilibrium (x; x^; w^) = (0; 0; 0) of 8 > > < x_ = Ax + B (F x^ + Gw^ )) _          (5:5) x ^ A P x ^ B x ? x ^ L > A : _ = + ([ C Q ]  ( u ) + ) w^ 0 S w^ 0 LS ?w^ is locally exponential stable with X0  Z0 contained in its basin of attraction. 2. For all (x(0); x^(0); w^(0)) 2 X0  Z0, w(0) 2 W0, and all r 2 R, the solution of the closed-loop system satis es lim e(t) = 0: (5:6) t!1 The solvability conditions for these problems were also established in [8] and are recalled as follows. Theorem 5.1. Consider the system (5.1) and given compact sets W0  IRs and R  C 0. The generalized semi-global linear state feedback regulator problem is solvable if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P (5:7) > : C + Q = 0 (b) For each r 2 R, there exists a function r~ 2 C 0 such that r = B r~. (c) There exists  > 0 and T > 0 such that k?w + r~k1;T  1 ?  for all w with w(0) 2 W0 and all r 2 R. Theorem 5.2. Consider the system (5.1) and the given compact sets W0  IRs and R  C 0. Let l = n + s. The semi-global linear-observer-like based error feedback regulator problem is solvable if 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. Moreover, the pair h i " A P #! C Q ; 0 S is detectable 2. There exists matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P > : C + Q = 0

(5:8)

20 (b) For each r 2 R, there exists a function r~(t) 2 C 0 such that r(t) = B r~(t) for all t  0. (c) There exists  > 0 such that k?w + r~k1  1 ?  for all w with w(0) 2 W0 and all r 2 R. In this section we show that the low-high-gain feedback based design methodology as developed in the previous section can also be used to construct the regulators to solve the generalized semi-global output regulation problems. We summarize these results in the following two theorems. Theorem 5.3. Consider the system (5.1) and given compact sets W0  IRs and R  C 0. If 1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. 2. There exist matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P (5:9) > : C + Q = 0 (b) For each r 2 R, there exists a function r~ 2 C 0 such that r = B r~. (c) There exists  > 0 and T > 0 such that k?w + r~k1;T  1 ?  for all w with w(0) 2 W0 and all r 2 R. then the following family of low-and-high gain state feedback laws solve the semi-global linear state feedback output regulation problem, u = ?( + 1)B 0X (")x + [( + 1)B 0X (") + ?]w + r~;   0 (5:10) where X (") is as given in Section 3. More speci cally, for any a priori given (arbitrarily large) bounded set X0  IRn , there exists an " > 0 such that for each " 2 (0; "] and for each   0, i ) The equilibrium x = 0 of x_ = Ax + B(?(1 + )B 0X (")x) (5:11) is locally exponentially stable with X0 contained in its basin of attraction. ii ) For all x(0) 2 X0, w(0) 2 W0, and r 2 R the solutions of the closed-loop system satis es lim e(t) = 0 (5:12) t!1

Proof : The proof of this theorem is similar, mutatis mutandis, to that of Theorem 3.1,

except that (3.8) takes the following slightly dierent form _ = A + B ((u) ? ?w ? r~)

(5:13)

21

Theorem 5.4. Consider the system (5.1) and the given compact sets W0  IRs and R  C 0.

If

1. (A; B ) is stabilizable and A has all eigenvalues in the closed left half plane. Moreover, the pair h i " A P #! C Q ; 0 S

is observable; 2. There exists matrices  and ? such that: (a) They solve the following linear matrix equations : 8 > < S = A + B ? + P > : C + Q = 0

(5:14)

(b) For each r 2 R, there exists a function r~(t) 2 C 0 such that r(t) = B r~(t) for all t  0. (c) There exists  > 0 such that k?w + r~k1  1 ?  for all w with w(0) 2 W0 and all r 2 R. Then, the generalized semi-global linear observer based error feedback output regulation problem can be solved by the following family of low-and-high gain feedbacks, 8 x^_ = Ax^ + B (u) + P w^ + L (`)C (x ? x^) + L (`)Qw ? w^ ) A A > > > > < w^_ = S w^ + LS (`)C (x ? x^) + LS (`)Q(w ? w^) > > > : 0 0

(5:15)

u = ?(1 + )B X (")^x + (( + 1)B X (") + ?)w^ + r~ where X (") is as given in Section 3 and LA(`) and LS (`) are as given in Section 4. More speci cally, for any a priori given (arbitrarily large) sets X0  IRn and Z0  IRn+s , there exists an " 2 (0; 1], and for each " 2 (0; "] and each   0, there exists an `("; ) > 0 such that for each " 2 (0; "], each   0 and each `  `("; ), i ) The equilibrium (x; x^; w^) = (0; 0; 0) of 8 0 0 > > < x_ = Ax + B (?( + 1)B X (")^x + (( + 1)B X (") + ?)w^ )) _          (5:16) > : x^_ = A P x^ + B  (u) + LA (`) ([ C Q ] x ? x^ ) w^ 0 S w^ 0 LS (`) ?w^ is locally exponentially stable with X0  Z0 contained in its basin of attraction. ii ) For all (x(0); x^(0); w^ (0)) 2 X0  Z0 and w(0) 2 W0, the solution of the closed-loop system satis es lim e(t) = 0: (5:17) t!1

22

Proof : The proof of this theorem is similar, mutatis mutandis, to that of Theorem 4.1, except that, in this case, (4.7) takes the following form 8_  = A + B(?(1+ )B 0X (") +?w ? ?w~ +(1+ )B 0X (")~x ? (1+ )B 0X (")w~ + r~) > > > > +(A ? S + P )w ? r > < > x~_ = (A ? LA (`)C )~x + (P ? LA(`)Q)w~ > > > > : w~_ = ?L (`)C x~ + (S ? L (`)Q)w~ S

S

(5:18)

6. Conclusions New regulators based on the low-and-high gain design technique were constructed to solve the semi-global output regulation problems for linear systems subject to input saturation. These regulators, inheriting the attributes of the low-and-high gain design technique, fully utilize the available control capacity to achieve better closed-loop system performance.

References [1] B.A. Francis, \The linear multivariable regulator problem", SIAM J. Contr. & Opt., Vol. 15, No. 3, pp. 486-505, 1977. [2] 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. [3] 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. [4] Z. Lin and A. Saberi, \Semi-global exponential stabilization of linear discrete-time systems subject to `input saturation' via linear feedbacks", to appear in Systems & Control Letters. [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," Proc. 32nd IEEE Conference on Decision and Control, 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," submitted to International Journal of Robust and Nonlinear Control, 1993. [7] Z. Lin, A. Saberi and A. Stoorvogel, \semi-global stabilization of linear discrete-time systems subject to input saturation via linear feedback { an ARE-based approach," submitted to IEEE Transactions on Automatic Control, 1994. [8] Z. Lin, A. Stoorvogel and A. Saberi, \Output regulation for linear systems subject to input saturation," submitted to Automatica, 1993.

23 [9] A. Saberi, Z. Lin and A.R. Teel, \Control of linear systems with saturating actuators," submitted to IEEE Transactions on Automatic Control, 1994. [10] 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. [11] 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. [12] H.J. Sussmann, E.D. Sontag and Y. Yang, \A general result on the stabilization of linear systems using bounded controls", Preprint, 1993. [13] 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. [14] A.R. Teel, Feedback Stabilization: Nonlinear Solutions to Inherently Nonlinear Problems, Ph.D dissertation, College of Engineering, University of California, Berkeley, CA, 1992. [15] A.R. Teel, \Semi-global stabilization of linear controllable systems with input nonlinearities," to appear in IEEE Transaction on Automatic Control, 1993.