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The regulator problem for a class of linear systems with constrained control
Systems & Control Letters 10 (1988) 357-363 North-Holland
357
The regulator problem for a class of linear systems with constrained control Abdellah BENZAOUIA Laboratoire d'Automatique et d'Analyse des Syst~mes du CNRS, 7 Av du C. Roche, 31077 Toulouse, France and Dep. de Physique, Facuit$ des Sciences, Universit~ Cadi Ayyad, Marrakech, Morocco
Christian BURGAT Laboratoire d'Automatique et d'Analyse des Syst~mes du CNRS, 7 Av du C. Roche, 31077 Toulouse, France
Received 25 November 1987 Revised 18 January 1988
the use of a feedback law u s = Fx k divides the state space into two regions: (a) a region of linear behaviour -~c .;efi~ :d by
= {x
},
R"lus- Fxk
where the linear model in the closed loop, x s + ~ ( A + B F ) x s , is valid; (b) a region R ' \ ~ where the system is subjected to saturations and the resulting model is nonlinear and described by the following equation: x k + ] = A x k + B sat(Fx s )
Abstract: This paper deals with the regulator problem for a class of linear discrete-time systems with nonsymmetrical con~xained controls. For such a class, a nonsymmetrical set of positive invariance and global attractivity exists. This allows a globally asymptoticallystable regulator to be obtained, despite the existence of the constrained controls.
with
sat(Fxk),=
{
(q,),
if ( F X k ) , > (q~),
(Fxk) ,
if x k ~ . ~ c
-(q2),
if (F.~k) , < --(q2),
Keywords: Linear discrete-time systems, Positively invariant
sets, Attractivity, Global asymptotic stability.
1. Introduction The: optimal control theory allows one to solve the regulator problem for linear systems with constrained control. However, in most industrial applicati, oas, optimal control so!ution has not been successful on account of its expensive implementation in terms of memory capacity and computational time, despite major hardware development [1,7,8]. The difficulty ;,n implementing such an optimal control solution justifies the development of different approaches. Fo: ~inear systems with constrained control described by the following equations: Xk + ? --- A x k + Bu k , x k~R',
us~CR
12--- { u ~ R m l - q ~
q;
m, m < ~ n ,
(1)
<~u,
i=l,...,m},
(2)
for i = 1, 2 , . . . , n. A successfully imp:emented approach was presented by Gutman [9] and is described below. In this approach, and in order to guarantee the linear behaviour of the regulator, we must find a positively invariant and asymptotically stable domain ~s such that Ds _cDo. Further, if all possible initial values of the system belong to this domain, i.e., ~0 _c Ds c,_De, then the problem is locally solved. The regulator design proposed by Gutman [9] consists of generating an ellipsoidal stability domain . ~ for continuous-time and discrete-time linear systems; the set 1~ being polyhedral. Since domain .~'~ is of a polyhedral nature, Chegan~as [4] presented an algorithm for computing the regulator ensuring that the stability domain is also polyhedrai by application of the properties of nonqu~dratic Lyapunov functions [5]. This idea was then developed by Bitsoris [3] wo gave the conditions to be satisfied for domain ~c itself to be a positively invariant domain, in the case of symmetrical constrained control. These results
0t67-6911/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
A. Benzaouia, C Burgat / Regulator problem for linear systems
358
were later generalized to the nonsymmetrical case by Benzaouia [2b]. From the specifity of such an approach, it clearly follows that only a local solution to the regulator problem with constrained controls can be obtained. Obviously, we obtain, as most, ~c as domain of positive invariance a n.d asy_m__pt_oficstability. Otherwise, it is well known that for some real regulators with constrained controls, when p ( A ) < 1, the global asymptotic stability property can be obtained. The objective of this paper is to use the properties of invariance and attractivity of some domains, with respect to motions of a class of linear discrete-time systems with nonsymmetrical constrained control, to provide an approach which slightly differs from the aforementioned one. For this class of systems, saturated controls are allowed and global asymptotic stability is guaranteed. This paper is organized as follows: Section 2 deals with notations and definitions. Problem presentation is given in Section 3. In Section 4, we present some preliminary results. The main results are dealt with in Section 5.
2. Definitions and notations In this section, we give all definitions and notations needed in the sequel of this paper. If A denotes a matrix of R ~×~ and x, y are vectors of R ~ then A + (resp. A - ) is the matrix whose components are given by --
up(0,
(resp. a~ = sup(0, - a q ) ) , i, j = 1, 2 , . . . , n ; components
x + (resp. x - ) is the vector of
x,+ = sup(0, x,)
(resp. x [ = sup(0, - x , ) ) ,
i = 1, 2,..., n. For two vectors of Rn, x < y (resp. x < y) if xi
Definition 2.L A subset ~ of R ~ is said to be positively invadant with respect to motions of system (1) with (2), if for every x 0 ~ ~ and every q/k, where
q4 - { u0,
}
is an arbitrary sequence of admissible controls ui~12, i = O , . . . , k - 1 ,
X(Xo, q4, for every k ~ N. Definition 2.2. With respect to motions of system (1) with (2), a set ~ c R n is said to be: (i) contractive, if, for every x k ~ i)(~k~), ~'k >I 1, there exists hi,+ 1 < ~ such that X(Xk, q/k, k ) ~ a ( h k . , ~ ) , for every sequence of admissible controls q/s; (if) attractive for a nonempty set oq'c R n if, for every x o ~ ~rk,~, there exists k 0 ¢ N such that
X(Xo, q/,,, k)E , for every k >I k o and every sequence of admissible controls q/k; (iii) globally attractive if if'= R". Theorem 2.3 [6]. For a matrix A ~ Z - { A Rnx" I a , > 0 for each i a n d a q < 0 whenever i ~ j } , the following statements are equivalent: (1) The real part of each eigenvalue of A is positive. (2) All principal minors of A are positive. (3) All leading principal minors of A are positive. (4) There exists a vector x > 0 such that A x > O. (5) A - 1 exists and .4 - 1 >i O. (See [6] for other equivalent statements.) Definition _2,4. A matrix A ¢ Z is called M-matrix if it satisfies any one of the conditions of Theorem 2.3.
3. Problem presentation In this paper, we are concerned with the study of linear systems given by (1) where x k ~ R n is the state, u k ~ 12 c R % m <~n, is the constrained control. The set of all admissible controls 12 is given by (2), where ql and q2 are given vectors in R '~ +.
A. Benzaouia,C. Burgat / Regulatorproblemfor linearsystems
Matrices A and B are constant matrices of the appropriate size with A satisfying the following assumption: g' = (12~ - A) is an M-matrix,
(3)
where
For this class, there exists a positively invariant and attractive domain -~s with respect to motions of system (1). In other words, for every initial state x 0 ~ R n and every sequence of admissible controls ~k, there exists k0 ~ N such that X(Xo, k, ~ k ) E~R
The set ~ is a nonsymmetrical polyhedral set as is generally the case in practical situations. In the unconstrained case, the regulator problem for system (1) consists of applying a feedback law as
ut, = Fxl,,
F ~ R n'xn,
(5)
Generally, matrix F is chosen such that: (i) if matrix A is stable, p(A + BF) < p(A),
p(A + BF) < 1.
for every k>_.k o. In order to guarantee the asymptotic stability of this class (1), (2) with assumption (3), it suffices to apply a feedback law given by (5) with a matrix F satisfying the two following conditions:
p(A + s t ) < p(A),
00)
&c&.
(11)
The aim of this paper is to design, such a regulator.
4. Preliminary results (6a)
(ii) if matrix A is unstable, (6b)
In this section, we present the properties of positive invariance and attractivity of a nonsymmetrical polyhedral domain with respect to system (1), (2) with assumption (3).
By substituting (5) into (1), system (1) becomes = ( a + B r ) x , = aoX,. In the constrained case, model (7) is not always valid and the asymptotic stability of system (1) is not guaranteed even when condition (6) holds. One. may note that model (7) remains valid only in the region ~c defined by
~=
359
(8)
It is clear that for xk ~ De, one may get xk+~ ~ De. Hence, the necessity to find a stability region or a region of invariance D, with respect tO motions of system (7) such that D, c:_.@¢.
Proof. (i) Condition (3) implies that there exists q= [q~ qt~]t>o such that (12n- A)q > O, which leads to the two inequalitie.~
{x~Rni-q~,<(Fx),<~q~; q~, q~,~R+, i= l , . . . , m } .
I
Lemma 4.1. (i) If 12~- .4 is an M-matrix then p(A) < 1. (ii) If A>~O and p(A)
(9)
When -@o- Do relation (9) ensures that, for every x0 ~ -@o, uk ~ f~ for every k ~ N, and model (7) remains valid. Further, condition (6) guarantees the asymptotic stability of system (7) into the region ~ . Thus, this approach guarantees only local asymptotic stability. However, there exists a class of systems for which global asymptotic stability can be obtained. This class is characterized by property (3).
(ln-A+)ql--A-q2>0 and (I n - A+ )qz - A - q 1 > O. The addition of the latter inequalities yields
(In- IA I)(ql ÷ q~) >0, with I A I = A + + .4-. Then, In - I A I is an M-matrix [6]. This implies that p(A) < 1 [5]. (ii) If A >I0 and p ( A ) < 1 then l n - A is an M-matrix [5]. From Theorem 2.3, there exists q > 0 such that (I n - A)q > 0. Since A - = 0, it follows that
,2n [ In-A0ln°]A
360
A. .'~enzaouia, C Burgat / Regulator problem for linear systems
and (12. - A)~ > 0 for ~ = [qt q,],. Then, 12. - d is an M-matrix. [] We now define the nonsymmetrical polyhedral set . ~ given by ~a--'-- { x ~ n n [ - - p 2 ~ < x ~ < P l ,
P~, P2 ~ Int a + },
(12)
which deafly is a n0nsymmetrical polyhedral set in ~" such that the origin 0 ~ - ~ . We recall now two results given by Benzaouia and Burgat [2a] concerning this domain.
Lemma 4A. If domain ~ is positively invariant with respect to motwns of system (1) with (2), then, for every scalar 8 >i 1 (resp. 8 > 1), domain 8~a ( resp. Int 8 ~ ) possesses the same property. Proof. Since domain ~a is positively invariant, condition (13) holds and is equivalent to (15)
/tp + Bq ~ p.
For every 8 >1 1 (resp. 8 > 1), 8-1/]q
(resp..4p + 8-'/~q < .4p + Bq).
Theorem 4.2 [2a]. The set ~ defined by (12) is positively invariant with respect to motions of system (1), .~:ubject to the constraints (2), if and only if
From (15), we should have .4p + 8-1Bq ~
(12 . - 2 ) p ~/~q,
g'(Sp ) >I Bq (resp. g'(Sp) >/~q)
(13)
(see notation of (4)). El
where' the matrix .4 is given by (4),
q:
p:[g
and
B-
qt]',
This lemma allows us to derive the contractivity property of domain D,.
.]
B+ "
Let us define the set ,~' of all p ~ R 2, which satisfy condH:,on (13), .o~° : {p ~ R2"1(12, - .4)/, >I Bq }.
(14)
Property 4.3 [2a]. There exists a smallest positively invariant polyhedral domain D m, in the sense of inclusion, with respect to motions of system (1) with (2) and assumption (3). This domain is defined by
"~m'-pN+..~'~:--{X~R"I--P=,<~x<~Pm2}, with pm= [ p,~l ptm2lt: ( 1 2 ~ - , 4 ) - ' B q .
Note that domain ~m alwaysexists from assumption (3). Indeed, if matrix q, is an M-matrix, then matrix ~,-1 exists and is nonnegative [6]. We now give a result concerning the properties of contraetivity and attractbAty of domains ~a with respect to system (1) with (2), according to Definition 2.2. The following lemma generalizes Theorem 4.2.
Property 4.5. (i) Every positively invari,~nt set Da is contractive with respect to motions of system (1)-(3). (ii) Every positively invariant set Da, satisfying Da D Dm, is globally attractive with respect to mob,ions of system (1)-(3). Proof. Let x k ~ R ' \ D a ; it is clear that there exists ~k > 1 such that x k ~ a(~kD~) for p ¢ Int R~. where, by Lemma 4.4, domain lnt ~k.~a is positively invafiant with respect to motions of system
(1). Note that from the definition of domain ~a, Xk ~ k l , ~ if and only if (16)
-~'kP2 <~Xk <~~kPl"
By decomposing A into the form A + - A - and premultiply/ng (16) by A + and - A - respectively, we obtain - k k A + p 2 <~A+xk <~kkA+pl
and - ~ k A - p l <~ - A - x k <~~kA-p2.
Adding these two inequalities yields + A-e
)
<
+
(17)
A. Benzaouia, C. Burgut / Regulator problem for finear systems
For every admissible control satisfying - q 2 ~< Uk ~
+ B - q 2 ).
(18)
361
where ~a is a given positively invoriant set such that ~a D ~m, i.e., "~a- 8~m and 8 > 1, then, system (1)-(3) with (5) i:~ ~ymptotically stable for every x o E R n.
It follows from the addition of (17) and (18) that --?~k(A+P2 + A - p l ) -- ( B +q2 + B-q1) <~xk+ , < ?~k(a +p, + A-P2) + ( B +ql + B-q2)"
Since, by Lemma 4.4, h kAP + / ] q < hk P, we have --hkp 2 < xk+ 1 <
Proof. Since domain -@a= &@m with 8 > 1 is positively invaxiant and attractive relatively to motions of system (1) with (2) and (3) (Property 4.5), there exists k o ~ N such that
h i p l,
that is Xk+t ~ Int ~kDa. Then, if Xk ~ Da, W~. ~hould have two possibilities: (a) xk+ t ~ D~; in firs ~ s e we have the attracti~ty property. (b) xk+ l ~ , ; then there exists hk+1 > I, such that Xk+l ¢ 8 ( h k + , ~ a ) and taking into account the fact that xk, ~ ~ I n t ~ a , it follows that hi,+ ~ < ?~. That is the contractivity property. It follows that every positively invariant set ~ , is contractive, in particular domain ~m itself. The contractivity property of ~m implies the attracfivity property of every positively invariant set ~ , ~m, i.e., ~ , ffi B~m, ~ > Io Consequently, since ~a and ~ are by definition compact sets, there exists k o such that
for every k >t k o, every Xo ~ R n, and every sequence of admissible controls ~k. Further, since condition (20) implies that for every k I> k o, model ('?) remains valid and taking into account condition (19), system (7) is asymptotically stable, rl It is well known that the theory of pole assignment allows one to compute matrices F satisfying condition (19) under the controllability assumption of (A, B). Further, for a given matrix F generating domain -@c,there always exists a scalar 0 < a < 1, such that aF satisfies condition (20). Neither the existence nor the computation of a matrix F satisfying concurrently conditions (19) and (20) are available in the literature. In order to design such regulator, we therefore give the following theorem.
X(Xo; ko) Then the attractivity of "@a relatively to any initial
value x o (global attractivity) holds, provided that x o ¢ R n \ ~ a and p~, p2 ~ lnt R~. n We can now apply these results to the design of a regulator which satisfies conditions (10) and (11). This is dealt with in the following section.
Theorem 5.2. There exists a scalar a ~ ]0,1] such that matrix F = aFt, guarantees the global asymptotic stability of system (1)-(3) with (5), where Fo -
-
-' B'eA,
P is the unique symmetric positive definite matrix solution to the equation ?~-2AtpA - p -- _ Q,
5. Application to the regular problem
with ~ a chosen parameter such that p(A) < ~ ~<1, for a given symmetric positive definite matrix Q.
In this section, we give our main results which enable us to design the required regulator, taking into account the existence of a nonsymmetrical polyhedral domain of positive invariance and global attractivity.
Proof. From assumption (3) and Lemma 4.1, we have p ( A ) < 1. Let us consider the new linear discrete-time system given by yk+, = ~yk + Buk
Theorem 5.1. I f there exists a matrix F such that p(A+BF)
(19)
(20)
(21)
with A'= h-~A. It is obvious that p(A) < 1; then, for every symmetric positive definite matrix Q, there exists a unique symmetric positive definite
362
A ne.-.:a,uta, C. Burgat / Regulator problem for linear systems
matrix P solution to the Lyapunov equation [10]
A'PA - P =
-Q.
(22)
Let us apply a feedback law u k = GYk to system (21), with G = a G o . Consider the candidate Lyapunov function j,(y)=ytpy. The rate of inorease of this function with respect to system (21) is g~ven by the following equation:
Ao(y) =f(A'PA -P)y + O(y),
(23)
where O(y) is giver, by
Remark 5.3, For ?~- p(A), we have p(A) = 1. In this critical case, the solution of equation (22) is not unique for a given symmetric positive senddefinite matrix Q, but all the solutions are not positive definite [11]. The dynamics of the system is conditioned by the choice of the parameter h. In particular, ?~can be chosen such that X = p(A) + e, e being a given positive scalar (as small as we like). In this case, we are able to design a regulator which guarantees the global asymptotic stability for the system without decrease of the dynamics.
O ( y ) = f [ a:( BGo)tP( BGo) + 2a( BGo)'PA~ y. The result of Theorem 5.2 leads to the following algorithm which ensures the computation of the required regulator.
By substituting Go = _ ( BtpB ) - 1Btp~
into the expression of function O(y), one would have
Algorithm 5.4. Step 1. Choose ~ such that P(A) < ~ ~< 1. Step 2. Give a symmetric positive definite matrix
O ( y ) = a( a - 2) y'[ ( BGo)tP( BGo)] y;
Q and find the unique symmetric positive definite matrix P solution to the equation
then, for 0 ~
~- 2AtPA _ p ffi - Q.
p(A+BG)
(24)
Let us now choose
. . .
Fo = ~,Go = - ( BtPB )- ' BtpA,
p(X-'A + ~-'B(aFo) ) < 1.
(25)
Since h is positive, relation (25) is equivalent to < X for every a ¢ [0, 2].
(26)
Further, domain ~c is given, in tlfis case, by
~c = { x ~ R"l - q2 < aFox < q, ; q,, q2 e R'~ } = i~ ~ R,, I _ q2cx <~Fox ~
Fo - - ( B t P B ) - 1 B t P A , with P computed in Step 2. Step 4. Find a 0 ¢ ]0,1] and 8 > 1 such that 8 ~ m c ~ c. Step 5. The regulator F ffi % F o guarantees the global asymptotic stability of system (1)-(3) with
(s).
since A ffi h-1.4. In this case, the spectral radius relation (24) becomes
p(A + aaFo)
Step 3. Compute
"
It is obvious that there always exists a ~ ]0,1] and 8 > 1 such that &@mc.@~. Then, by virtue of Theorem 5.1, the global asymptotic stability is guaranteed for system (1)-(3) with (5). []
Conclusion In this paper, a class of discrete-time linear systems with constrained control is selected so as to admit a positively invariant set of global attractivity. The latter is used to guarantee the global asymptotic stability of this class despite the constraints on the control. An algorithm has been given to compute the regulator ensuring through a feedback law, the global asymptotic stability of system (1)-(3). Tiffs class contains the class of systems with nonnegative stable matrix A. References [1] M. Athans ,and P.L Falb, Optimal Control (McGraw-Hill, New York, 1966).
A. Benzaouia, C. Burgut / Regulator problem for linear systems
[2] A. Benzaouia and C. Burgat, (a) Existence of positively invariant domains for discrete-time linear systems with constrained control, Rapport Interne LAAS, No 86354, Submitted to J. Optim. Theory AppL (1987). (b) On the regulator problem for linear discrete-time systems with nonsymmetrical constrained control, Submitred to Internat. J. Control (1987). [3] G. Bitsoris, Positively invariant polyhedral sets of discrete-time linear systems, Internat. J. Control (1988), to appear. [4] J. Chegancas, Sur ie concept d'invariance positive appliqu6 a i'~tude de la commande constrainte des syst~mes dynamiques. Thesis No 85325, LAAS, Toulouse (1985). [5] J. Chegancas and C. Burgat, Polyhedral cones associated to M-matrices and stability of time-varying discrete-time systems, J. Math. Anal. Appl. 118 (1) (1986) 88-96. [6] M. Fielder and V. Ptak, On matrices with non positive off
363
diagonal elements and positive principal minors, Czech Math. J. 12 (1962) 382-400. [7] J.F. Frankena and R. Sivan, A new linear optimal control law for linear systems, lnternat. J. Control. 30 (1) (1979) 159-178. [8] J.P. Gauthier and G Bornard, Commande multivariable en pr~.sence de constraintes du type in6galit6, RAIRO Automatique 17 (13) (1983) 205-222. [9] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems, IEEE Trans. Auto. mat. Control 30 (1) (1985) 22-33. [10] R.E. Kalman and J.E. Bertram, Control systems analysis and design via the second method of Lyapunov. Trans. ASME Set D J. Basic Engug. 82 (1960) 394-400. [11] J.P. LasaHe, The Stability of Dynamical ,Systems, SIAM Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, PA, 1976).
357
The regulator problem for a class of linear systems with constrained control Abdellah BENZAOUIA Laboratoire d'Automatique et d'Analyse des Syst~mes du CNRS, 7 Av du C. Roche, 31077 Toulouse, France and Dep. de Physique, Facuit$ des Sciences, Universit~ Cadi Ayyad, Marrakech, Morocco
Christian BURGAT Laboratoire d'Automatique et d'Analyse des Syst~mes du CNRS, 7 Av du C. Roche, 31077 Toulouse, France
Received 25 November 1987 Revised 18 January 1988
the use of a feedback law u s = Fx k divides the state space into two regions: (a) a region of linear behaviour -~c .;efi~ :d by
= {x
},
R"lus- Fxk
where the linear model in the closed loop, x s + ~ ( A + B F ) x s , is valid; (b) a region R ' \ ~ where the system is subjected to saturations and the resulting model is nonlinear and described by the following equation: x k + ] = A x k + B sat(Fx s )
Abstract: This paper deals with the regulator problem for a class of linear discrete-time systems with nonsymmetrical con~xained controls. For such a class, a nonsymmetrical set of positive invariance and global attractivity exists. This allows a globally asymptoticallystable regulator to be obtained, despite the existence of the constrained controls.
with
sat(Fxk),=
{
(q,),
if ( F X k ) , > (q~),
(Fxk) ,
if x k ~ . ~ c
-(q2),
if (F.~k) , < --(q2),
Keywords: Linear discrete-time systems, Positively invariant
sets, Attractivity, Global asymptotic stability.
1. Introduction The: optimal control theory allows one to solve the regulator problem for linear systems with constrained control. However, in most industrial applicati, oas, optimal control so!ution has not been successful on account of its expensive implementation in terms of memory capacity and computational time, despite major hardware development [1,7,8]. The difficulty ;,n implementing such an optimal control solution justifies the development of different approaches. Fo: ~inear systems with constrained control described by the following equations: Xk + ? --- A x k + Bu k , x k~R',
us~CR
12--- { u ~ R m l - q ~
q;
m, m < ~ n ,
(1)
<~u,
i=l,...,m},
(2)
for i = 1, 2 , . . . , n. A successfully imp:emented approach was presented by Gutman [9] and is described below. In this approach, and in order to guarantee the linear behaviour of the regulator, we must find a positively invariant and asymptotically stable domain ~s such that Ds _cDo. Further, if all possible initial values of the system belong to this domain, i.e., ~0 _c Ds c,_De, then the problem is locally solved. The regulator design proposed by Gutman [9] consists of generating an ellipsoidal stability domain . ~ for continuous-time and discrete-time linear systems; the set 1~ being polyhedral. Since domain .~'~ is of a polyhedral nature, Chegan~as [4] presented an algorithm for computing the regulator ensuring that the stability domain is also polyhedrai by application of the properties of nonqu~dratic Lyapunov functions [5]. This idea was then developed by Bitsoris [3] wo gave the conditions to be satisfied for domain ~c itself to be a positively invariant domain, in the case of symmetrical constrained control. These results
0t67-6911/88/$3.50 © 1988, ElsevierScience Publishers B.V. (North-Holland)
A. Benzaouia, C Burgat / Regulator problem for linear systems
358
were later generalized to the nonsymmetrical case by Benzaouia [2b]. From the specifity of such an approach, it clearly follows that only a local solution to the regulator problem with constrained controls can be obtained. Obviously, we obtain, as most, ~c as domain of positive invariance a n.d asy_m__pt_oficstability. Otherwise, it is well known that for some real regulators with constrained controls, when p ( A ) < 1, the global asymptotic stability property can be obtained. The objective of this paper is to use the properties of invariance and attractivity of some domains, with respect to motions of a class of linear discrete-time systems with nonsymmetrical constrained control, to provide an approach which slightly differs from the aforementioned one. For this class of systems, saturated controls are allowed and global asymptotic stability is guaranteed. This paper is organized as follows: Section 2 deals with notations and definitions. Problem presentation is given in Section 3. In Section 4, we present some preliminary results. The main results are dealt with in Section 5.
2. Definitions and notations In this section, we give all definitions and notations needed in the sequel of this paper. If A denotes a matrix of R ~×~ and x, y are vectors of R ~ then A + (resp. A - ) is the matrix whose components are given by --
up(0,
(resp. a~ = sup(0, - a q ) ) , i, j = 1, 2 , . . . , n ; components
x + (resp. x - ) is the vector of
x,+ = sup(0, x,)
(resp. x [ = sup(0, - x , ) ) ,
i = 1, 2,..., n. For two vectors of Rn, x < y (resp. x < y) if xi
Definition 2.L A subset ~ of R ~ is said to be positively invadant with respect to motions of system (1) with (2), if for every x 0 ~ ~ and every q/k, where
q4 - { u0,
}
is an arbitrary sequence of admissible controls ui~12, i = O , . . . , k - 1 ,
X(Xo, q4, for every k ~ N. Definition 2.2. With respect to motions of system (1) with (2), a set ~ c R n is said to be: (i) contractive, if, for every x k ~ i)(~k~), ~'k >I 1, there exists hi,+ 1 < ~ such that X(Xk, q/k, k ) ~ a ( h k . , ~ ) , for every sequence of admissible controls q/s; (if) attractive for a nonempty set oq'c R n if, for every x o ~ ~rk,~, there exists k 0 ¢ N such that
X(Xo, q/,,, k)E , for every k >I k o and every sequence of admissible controls q/k; (iii) globally attractive if if'= R". Theorem 2.3 [6]. For a matrix A ~ Z - { A Rnx" I a , > 0 for each i a n d a q < 0 whenever i ~ j } , the following statements are equivalent: (1) The real part of each eigenvalue of A is positive. (2) All principal minors of A are positive. (3) All leading principal minors of A are positive. (4) There exists a vector x > 0 such that A x > O. (5) A - 1 exists and .4 - 1 >i O. (See [6] for other equivalent statements.) Definition _2,4. A matrix A ¢ Z is called M-matrix if it satisfies any one of the conditions of Theorem 2.3.
3. Problem presentation In this paper, we are concerned with the study of linear systems given by (1) where x k ~ R n is the state, u k ~ 12 c R % m <~n, is the constrained control. The set of all admissible controls 12 is given by (2), where ql and q2 are given vectors in R '~ +.
A. Benzaouia,C. Burgat / Regulatorproblemfor linearsystems
Matrices A and B are constant matrices of the appropriate size with A satisfying the following assumption: g' = (12~ - A) is an M-matrix,
(3)
where
For this class, there exists a positively invariant and attractive domain -~s with respect to motions of system (1). In other words, for every initial state x 0 ~ R n and every sequence of admissible controls ~k, there exists k0 ~ N such that X(Xo, k, ~ k ) E~R
The set ~ is a nonsymmetrical polyhedral set as is generally the case in practical situations. In the unconstrained case, the regulator problem for system (1) consists of applying a feedback law as
ut, = Fxl,,
F ~ R n'xn,
(5)
Generally, matrix F is chosen such that: (i) if matrix A is stable, p(A + BF) < p(A),
p(A + BF) < 1.
for every k>_.k o. In order to guarantee the asymptotic stability of this class (1), (2) with assumption (3), it suffices to apply a feedback law given by (5) with a matrix F satisfying the two following conditions:
p(A + s t ) < p(A),
00)
&c&.
(11)
The aim of this paper is to design, such a regulator.
4. Preliminary results (6a)
(ii) if matrix A is unstable, (6b)
In this section, we present the properties of positive invariance and attractivity of a nonsymmetrical polyhedral domain with respect to system (1), (2) with assumption (3).
By substituting (5) into (1), system (1) becomes = ( a + B r ) x , = aoX,. In the constrained case, model (7) is not always valid and the asymptotic stability of system (1) is not guaranteed even when condition (6) holds. One. may note that model (7) remains valid only in the region ~c defined by
~=
359
(8)
It is clear that for xk ~ De, one may get xk+~ ~ De. Hence, the necessity to find a stability region or a region of invariance D, with respect tO motions of system (7) such that D, c:_.@¢.
Proof. (i) Condition (3) implies that there exists q= [q~ qt~]t>o such that (12n- A)q > O, which leads to the two inequalitie.~
{x~Rni-q~,<(Fx),<~q~; q~, q~,~R+, i= l , . . . , m } .
I
Lemma 4.1. (i) If 12~- .4 is an M-matrix then p(A) < 1. (ii) If A>~O and p(A)
(9)
When -@o- Do relation (9) ensures that, for every x0 ~ -@o, uk ~ f~ for every k ~ N, and model (7) remains valid. Further, condition (6) guarantees the asymptotic stability of system (7) into the region ~ . Thus, this approach guarantees only local asymptotic stability. However, there exists a class of systems for which global asymptotic stability can be obtained. This class is characterized by property (3).
(ln-A+)ql--A-q2>0 and (I n - A+ )qz - A - q 1 > O. The addition of the latter inequalities yields
(In- IA I)(ql ÷ q~) >0, with I A I = A + + .4-. Then, In - I A I is an M-matrix [6]. This implies that p(A) < 1 [5]. (ii) If A >I0 and p ( A ) < 1 then l n - A is an M-matrix [5]. From Theorem 2.3, there exists q > 0 such that (I n - A)q > 0. Since A - = 0, it follows that
,2n [ In-A0ln°]A
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A. .'~enzaouia, C Burgat / Regulator problem for linear systems
and (12. - A)~ > 0 for ~ = [qt q,],. Then, 12. - d is an M-matrix. [] We now define the nonsymmetrical polyhedral set . ~ given by ~a--'-- { x ~ n n [ - - p 2 ~ < x ~ < P l ,
P~, P2 ~ Int a + },
(12)
which deafly is a n0nsymmetrical polyhedral set in ~" such that the origin 0 ~ - ~ . We recall now two results given by Benzaouia and Burgat [2a] concerning this domain.
Lemma 4A. If domain ~ is positively invariant with respect to motwns of system (1) with (2), then, for every scalar 8 >i 1 (resp. 8 > 1), domain 8~a ( resp. Int 8 ~ ) possesses the same property. Proof. Since domain ~a is positively invariant, condition (13) holds and is equivalent to (15)
/tp + Bq ~ p.
For every 8 >1 1 (resp. 8 > 1), 8-1/]q
(resp..4p + 8-'/~q < .4p + Bq).
Theorem 4.2 [2a]. The set ~ defined by (12) is positively invariant with respect to motions of system (1), .~:ubject to the constraints (2), if and only if
From (15), we should have .4p + 8-1Bq ~
(12 . - 2 ) p ~/~q,
g'(Sp ) >I Bq (resp. g'(Sp) >/~q)
(13)
(see notation of (4)). El
where' the matrix .4 is given by (4),
q:
p:[g
and
B-
qt]',
This lemma allows us to derive the contractivity property of domain D,.
.]
B+ "
Let us define the set ,~' of all p ~ R 2, which satisfy condH:,on (13), .o~° : {p ~ R2"1(12, - .4)/, >I Bq }.
(14)
Property 4.3 [2a]. There exists a smallest positively invariant polyhedral domain D m, in the sense of inclusion, with respect to motions of system (1) with (2) and assumption (3). This domain is defined by
"~m'-pN+..~'~:--{X~R"I--P=,<~x<~Pm2}, with pm= [ p,~l ptm2lt: ( 1 2 ~ - , 4 ) - ' B q .
Note that domain ~m alwaysexists from assumption (3). Indeed, if matrix q, is an M-matrix, then matrix ~,-1 exists and is nonnegative [6]. We now give a result concerning the properties of contraetivity and attractbAty of domains ~a with respect to system (1) with (2), according to Definition 2.2. The following lemma generalizes Theorem 4.2.
Property 4.5. (i) Every positively invari,~nt set Da is contractive with respect to motions of system (1)-(3). (ii) Every positively invariant set Da, satisfying Da D Dm, is globally attractive with respect to mob,ions of system (1)-(3). Proof. Let x k ~ R ' \ D a ; it is clear that there exists ~k > 1 such that x k ~ a(~kD~) for p ¢ Int R~. where, by Lemma 4.4, domain lnt ~k.~a is positively invafiant with respect to motions of system
(1). Note that from the definition of domain ~a, Xk ~ k l , ~ if and only if (16)
-~'kP2 <~Xk <~~kPl"
By decomposing A into the form A + - A - and premultiply/ng (16) by A + and - A - respectively, we obtain - k k A + p 2 <~A+xk <~kkA+pl
and - ~ k A - p l <~ - A - x k <~~kA-p2.
Adding these two inequalities yields + A-e
)
<
+
(17)
A. Benzaouia, C. Burgut / Regulator problem for finear systems
For every admissible control satisfying - q 2 ~< Uk ~
+ B - q 2 ).
(18)
361
where ~a is a given positively invoriant set such that ~a D ~m, i.e., "~a- 8~m and 8 > 1, then, system (1)-(3) with (5) i:~ ~ymptotically stable for every x o E R n.
It follows from the addition of (17) and (18) that --?~k(A+P2 + A - p l ) -- ( B +q2 + B-q1) <~xk+ , < ?~k(a +p, + A-P2) + ( B +ql + B-q2)"
Since, by Lemma 4.4, h kAP + / ] q < hk P, we have --hkp 2 < xk+ 1 <
Proof. Since domain -@a= &@m with 8 > 1 is positively invaxiant and attractive relatively to motions of system (1) with (2) and (3) (Property 4.5), there exists k o ~ N such that
h i p l,
that is Xk+t ~ Int ~kDa. Then, if Xk ~ Da, W~. ~hould have two possibilities: (a) xk+ t ~ D~; in firs ~ s e we have the attracti~ty property. (b) xk+ l ~ , ; then there exists hk+1 > I, such that Xk+l ¢ 8 ( h k + , ~ a ) and taking into account the fact that xk, ~ ~ I n t ~ a , it follows that hi,+ ~ < ?~. That is the contractivity property. It follows that every positively invariant set ~ , is contractive, in particular domain ~m itself. The contractivity property of ~m implies the attracfivity property of every positively invariant set ~ , ~m, i.e., ~ , ffi B~m, ~ > Io Consequently, since ~a and ~ are by definition compact sets, there exists k o such that
for every k >t k o, every Xo ~ R n, and every sequence of admissible controls ~k. Further, since condition (20) implies that for every k I> k o, model ('?) remains valid and taking into account condition (19), system (7) is asymptotically stable, rl It is well known that the theory of pole assignment allows one to compute matrices F satisfying condition (19) under the controllability assumption of (A, B). Further, for a given matrix F generating domain -@c,there always exists a scalar 0 < a < 1, such that aF satisfies condition (20). Neither the existence nor the computation of a matrix F satisfying concurrently conditions (19) and (20) are available in the literature. In order to design such regulator, we therefore give the following theorem.
X(Xo; ko) Then the attractivity of "@a relatively to any initial
value x o (global attractivity) holds, provided that x o ¢ R n \ ~ a and p~, p2 ~ lnt R~. n We can now apply these results to the design of a regulator which satisfies conditions (10) and (11). This is dealt with in the following section.
Theorem 5.2. There exists a scalar a ~ ]0,1] such that matrix F = aFt, guarantees the global asymptotic stability of system (1)-(3) with (5), where Fo -
-
-' B'eA,
P is the unique symmetric positive definite matrix solution to the equation ?~-2AtpA - p -- _ Q,
5. Application to the regular problem
with ~ a chosen parameter such that p(A) < ~ ~<1, for a given symmetric positive definite matrix Q.
In this section, we give our main results which enable us to design the required regulator, taking into account the existence of a nonsymmetrical polyhedral domain of positive invariance and global attractivity.
Proof. From assumption (3) and Lemma 4.1, we have p ( A ) < 1. Let us consider the new linear discrete-time system given by yk+, = ~yk + Buk
Theorem 5.1. I f there exists a matrix F such that p(A+BF)
(19)
(20)
(21)
with A'= h-~A. It is obvious that p(A) < 1; then, for every symmetric positive definite matrix Q, there exists a unique symmetric positive definite
362
A ne.-.:a,uta, C. Burgat / Regulator problem for linear systems
matrix P solution to the Lyapunov equation [10]
A'PA - P =
-Q.
(22)
Let us apply a feedback law u k = GYk to system (21), with G = a G o . Consider the candidate Lyapunov function j,(y)=ytpy. The rate of inorease of this function with respect to system (21) is g~ven by the following equation:
Ao(y) =f(A'PA -P)y + O(y),
(23)
where O(y) is giver, by
Remark 5.3, For ?~- p(A), we have p(A) = 1. In this critical case, the solution of equation (22) is not unique for a given symmetric positive senddefinite matrix Q, but all the solutions are not positive definite [11]. The dynamics of the system is conditioned by the choice of the parameter h. In particular, ?~can be chosen such that X = p(A) + e, e being a given positive scalar (as small as we like). In this case, we are able to design a regulator which guarantees the global asymptotic stability for the system without decrease of the dynamics.
O ( y ) = f [ a:( BGo)tP( BGo) + 2a( BGo)'PA~ y. The result of Theorem 5.2 leads to the following algorithm which ensures the computation of the required regulator.
By substituting Go = _ ( BtpB ) - 1Btp~
into the expression of function O(y), one would have
Algorithm 5.4. Step 1. Choose ~ such that P(A) < ~ ~< 1. Step 2. Give a symmetric positive definite matrix
O ( y ) = a( a - 2) y'[ ( BGo)tP( BGo)] y;
Q and find the unique symmetric positive definite matrix P solution to the equation
then, for 0 ~
~- 2AtPA _ p ffi - Q.
p(A+BG)
(24)
Let us now choose
. . .
Fo = ~,Go = - ( BtPB )- ' BtpA,
p(X-'A + ~-'B(aFo) ) < 1.
(25)
Since h is positive, relation (25) is equivalent to < X for every a ¢ [0, 2].
(26)
Further, domain ~c is given, in tlfis case, by
~c = { x ~ R"l - q2 < aFox < q, ; q,, q2 e R'~ } = i~ ~ R,, I _ q2cx <~Fox ~
Fo - - ( B t P B ) - 1 B t P A , with P computed in Step 2. Step 4. Find a 0 ¢ ]0,1] and 8 > 1 such that 8 ~ m c ~ c. Step 5. The regulator F ffi % F o guarantees the global asymptotic stability of system (1)-(3) with
(s).
since A ffi h-1.4. In this case, the spectral radius relation (24) becomes
p(A + aaFo)
Step 3. Compute
"
It is obvious that there always exists a ~ ]0,1] and 8 > 1 such that &@mc.@~. Then, by virtue of Theorem 5.1, the global asymptotic stability is guaranteed for system (1)-(3) with (5). []
Conclusion In this paper, a class of discrete-time linear systems with constrained control is selected so as to admit a positively invariant set of global attractivity. The latter is used to guarantee the global asymptotic stability of this class despite the constraints on the control. An algorithm has been given to compute the regulator ensuring through a feedback law, the global asymptotic stability of system (1)-(3). Tiffs class contains the class of systems with nonnegative stable matrix A. References [1] M. Athans ,and P.L Falb, Optimal Control (McGraw-Hill, New York, 1966).
A. Benzaouia, C. Burgut / Regulator problem for linear systems
[2] A. Benzaouia and C. Burgat, (a) Existence of positively invariant domains for discrete-time linear systems with constrained control, Rapport Interne LAAS, No 86354, Submitted to J. Optim. Theory AppL (1987). (b) On the regulator problem for linear discrete-time systems with nonsymmetrical constrained control, Submitred to Internat. J. Control (1987). [3] G. Bitsoris, Positively invariant polyhedral sets of discrete-time linear systems, Internat. J. Control (1988), to appear. [4] J. Chegancas, Sur ie concept d'invariance positive appliqu6 a i'~tude de la commande constrainte des syst~mes dynamiques. Thesis No 85325, LAAS, Toulouse (1985). [5] J. Chegancas and C. Burgat, Polyhedral cones associated to M-matrices and stability of time-varying discrete-time systems, J. Math. Anal. Appl. 118 (1) (1986) 88-96. [6] M. Fielder and V. Ptak, On matrices with non positive off
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diagonal elements and positive principal minors, Czech Math. J. 12 (1962) 382-400. [7] J.F. Frankena and R. Sivan, A new linear optimal control law for linear systems, lnternat. J. Control. 30 (1) (1979) 159-178. [8] J.P. Gauthier and G Bornard, Commande multivariable en pr~.sence de constraintes du type in6galit6, RAIRO Automatique 17 (13) (1983) 205-222. [9] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems, IEEE Trans. Auto. mat. Control 30 (1) (1985) 22-33. [10] R.E. Kalman and J.E. Bertram, Control systems analysis and design via the second method of Lyapunov. Trans. ASME Set D J. Basic Engug. 82 (1960) 394-400. [11] J.P. LasaHe, The Stability of Dynamical ,Systems, SIAM Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, PA, 1976).