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Feedback control of linear discrete-time systems under state and
control constraints
In this paper the problem of stabilizing linear discrete-time systems under state and
control linear constraints is studied. Based on the concept of positive invariance,
existence conditions of linear state feedback control laws respecting both the
constraints are established. These conditions are then translated into an algorithm
of linear programming.
1. Introduction
Most industrial systems must operate within fixed bounds and are subject to strict
control limitations. The determination of closed-loop controls for such systems by
state or output feedback often reduces to solving an associated unconstrained
problem and then modifying the solution by superimposition of state and control
limitations. The global stability of these control schemes is usually not guaranteed.
Another approach that is more rigorous, consists of explicitly introducing the
constraints in the lagrangian formulation of an optimal control problem (Mouradi
1979, Franckena and Sivan 1979, Gauthier and Bornard 1983). However, its
implementation isnot simple, because as an open-loop scheme it implies considerableoff-line computation and as a closed-loop scheme it is represented by a non-linear
controller.
The concept of invariance (or positive invariance), which is related to the notion of
Lyapunov functions, is a convenient tool both f or guaranteeing stability and
respecting the constraints. In the general case of constrained controllers for linear
systems, Gutman and Hagander (1985) used quadratic Lyapunov functions to
determine non-linear feedback controllers. However, for linear systems with linear
constraints on state and control variables, non-quadratic Lyapunov functions must be
used in order to generate the biggest positively invariant set included in the domain of
constraints. Such Lyapunov functions have already been applied for improving linearconstrained contr ollers of linear systems characterized by a stable non-negative
dynamic matrix (Chegan
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2. Problem statement
Throughout the paper, capital letters generally denote real matrices, lower case
letters denote column vectors or scalars, R n denotes the euclidean n-space and R n xm
the set of real n x m matrices. For a real matrix A = (aij), IAI denotes the matrix IAI =
(Ia i j !) ' For vectors x =[XI Xz ... Xn]T and Ixi =[ixil IXzl Ixnl]T. Finally:lL
denotes a unity matrix.We consider discrete-time linear system described by the difference equation
where X E R n, UE R m, A ER n XI', B E R n xm and k belongs to the set of non-negative
integers T= {a, 1,2, ... }.
The control vector u( k) is subject to constraints
where P = [PI pz ... P m ]T with P i> 0, i = 1,2, ... , m.
There is also given a bounded set of initial states Xodefined by the inequalities
whereGERqXnwithq~n,rankG=nandw=[wl Wz ... w q]Tw ith w i> O,i=
1,2, ..., q. These inequalities can also be considered as state constraints.
The problem to be studied is the determination of a linear state feedback control
law
that satisfy constraints (2) are transferred asymptotically to the origin while the
control vector u(k) does not violate the constraints (1). We call this problem the linear
constrained regulation problem (LCRP).
If the equilibrium x = of the open-loop system
is stable in the sense of Lyapunov or asymptotically stable, then the above problem
admits the trivial solution u(k) = 0. If, on the contrary, the open-loop system is
unstable, then the LCRP may not possess any solution. Therefore, we shall say that
constraints (1) and (2) are compatible with respect to system (S) if the LCRP has at
least one solution.
3. Existence conditions of linear constrained controllers
Let us associate to each linear state feedback control law u(k) = Fx(k) withF ERm x n the set
R(F, p ) = {x ER": - P~ Fx ~ p}
It is clear that the polyhedral set R(F, p ) isthe region of initial states of the closed-loopsystem (4) at which the linear state feedback control u(k) = Fx(k) does not initially
violate the constraints (1).
According to the above notation the set of initial states defined in (2) is expressed
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It is obvious that the control law u= Fx is a solution of the LCRP ifand only iftheresulting closed-loop system (4) is asymptotically stable and every trajectory x(k; xo)
emanating from the region R(G, w) does not leave the region R(F, p ) for any instant
k ET. This condition can also be expressed as follows (Bitsoris 1988 b).
Proposition 1
The control law u= Fx with FE Rm x n is a solution to the LCRP if and only if
(a) the eigenvalues Ai> i= 1,2, ... ,n , of the matrix A+ BF are in the open disklA d
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Proposition 2 (Bitsoris 1988 a)
The polyhedral set R( G, w) is a positively invariant set of system (S) if and only if
there exists a matrix H ERq x q such that
(IHI- :ll.)w :::; 0
GA-HG=O
By a direct application of this result to the closed-loop system described by (4) we
establish the following.
Proposition 3
If F ER '" x n and there exists a matrix HE R q x q such that
(IHI-:ll.)w:::;O
GA + GBF= HG
(iii) the eigenvalues Ai = 1,2, ...,n of the matrix A +B F are in the open disk IAi l
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Now, setting y= Gx in (10) and (11) we conclude that if
IGxl~w
or, equivalently ifx E R ( G, w) then
IF(GTG) -I GT
Gxl = IFxl ~ p
It must be noted that in the case where G ER " x n, inequality (9) becomes IFG-11
w ~ p and is,in addition, anecessary condition for R(G, w) c:; : R(F, p ) (Bitsoris 1988 b).
Now, by combining the results stated in Pr opositions 3 and 4 we conclude that if
FE R m x n and there exists an asymptotically stable matrix HE R q x q satisfying (6), (7)
and (9), then with the control law u= Fx all the states Xo ER(G, w) are transferredasymptotically to the origin while the control and state vectors satisfy inequalities (I)
and (2) respectively.
4. Design by means of linear programming
A straightforward application of the preceding result to the design of constrained
linear controllers seems to be a v ery difficult problem. However, by an appropriate
transformation of conditions (6), (7) and (9), the determination of a solution to the
LCRP can be reduced to a linear programming problem.
Observe that (7) is satisfied with
Now, by introducing matrix DE Rm
xq such that
D = F( GTG) - 1GT
relation (12) can be written as
H=GA(GTG)-IGT +GBD
Insertion of (13) and (14) into (9) and (6), respectively, yields
IDlw~p
IGA(GTG)-IGT +GBDlw~w
(15)
( 16)
These conditions on matrix D guarantee the existence of a linear control law
u= Fx = DGx such that R(G, w) is a positively invariant set of the resulting closed-loop system
and R(G, w) c :; : R(F, p) . However, conditions (15) and (16) do not imply the asympto-
tic stability of system (17). The asymptotic stability of (17) is guaranteed ifinequality
(16) is strictly satisfied. For this reason, inequality (16) is replaced by the inequality
IGA(GT G) -I GT + GBDlw ~ GW (18)
where G is a real number such that 0~ G
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Proposition 5
The matrix inequality
where Y = (Y iJ, Y ER P x r, r:J .E R' and fJ ER P with r:J .j ; ; : :'0 and f J i ; ;: :,0, is equivalent to the
set of equations
where e s = [ e 1S ez s e rsJT denotes one of the distinct vectors e s E R' with
components equal to +1or -1.
Proof
Assume that Y = (Y ij), Y E R Px r satisfies inequalities (19). Then
r r
r
IY ijejs r:J . j ~ I!Y ijllejs l r:J . j = IIY ij ! r:J . j ~ fJ ij~l j~l j~ l
for all i= 1,2, ... ,p and s= 1,2, ... ,2r, because r: J .j;;::' O.Therefore, (19) implies (20).
Conversely, if Y = (Y ij), Y ER P xr satisfies (20) then for ejS = sign (Y ij) we obtain
r r
I!Y ij! r:J . j= IY ijejs r:J . j ~ fJ i ' i= 1,2,... , pj~ 1 j~ 1
The application of this result to the determination of a solution to the LCRP isstraightforward. The system of inequalities (15) and (18) can be equivalently replaced
by a system of linear inequalities with unknown variables, the elements d ij of matrix D
and the positive variable B. If the set of solutions to these inequalities is non-empty,
then such a solution can be obtained by minimizing any linear function of the
unknown variables d ij and B.
Since it is very important not only to stabilize the system but also to increase the
rate of convergence to the equilibrium, we can choose as the objective function of the
linear programming problem, the function
J(D, B) = BIndeed, if B satisfy inequality (18), then by virtue of (14)
!H lw ~BW
v(x ) = max {!(G X );l}t Wi
is positive-definite. Therefore v(x) can be considered as the distance of x from the
origin. (It can easily be proved that v(x)
represents the distance of x from the origin, inthe space R" with the distance
{
V ( X ) + v(y)d(x, y) =
o
ifx#y
if x = Y
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Now, taking into account (7), we get
v(x(k + 1))= m~x f(GX(~~ 1))il}
= m~x {I(G (A + ~ :)X (k) );i}
= m~x fHG:;k))il}
~ m~x {(IHII~~(k)l)i}~ w(x(k))
because from (22) it follows that
Therefore, minimization of 8increases the rate of convergence of the state variable
x(k) to the origin.
If the optimal solution D*, 8* of the linear programming problem is such that
8*
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A~ [ _ ~ : ~~Jare Al = 1+jO-4 and .1 .2 = 1 - jO 4.Now, setting
B~[~ lG~ [-:5 ~ lw~[la p~7and taking into account that det G # - 0, conditions (15) and (18) become
IDlw":;p
IGA(GTG)-lGT + GBDlw= IGAG-I + GBDlw,,:;ew
51d11+ 1OId21 ,, : ; 7
510'87 +2 dll + 1010'58+ 2 d21 ,, : ; 5e
51-0'305 +2 dll + 1011'13+ 2d21, , : ; lO e
(26 a)
( 2 6 b )
(26 c)
Thus, the LCRP for system (23) under constraints (24) and (25) is reduced to the
determination of dl, d2 and e which minimize the objective function
under constraints (26).
Transformation of inequalities (26) to a system of linear inequalities and
application of a standard algorithm of linear programming gave the optimal values
u~ [-0435-04825{-:5 ~J
[::JWith this control, the resulting closed-loop system becomes x(k + 1)= (A
+ BD*G)x(k) where
[
08A+BD*G=
-0,11125
05 ]
-0,635
The eigenvalues of matrix A + BD*G are Al = 0'76, .1.2 = 059 and it is worth noticingthat the optimal value ofparameter e is a good upper bound of max ( 1.1.11 , 1.1 .21 ). Due tothe optimality of control law (26), the intersection of the boundary of set R ( G, w ) and
the boundary of set R(F, p) is not empty. This is shown in the following Figure.
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5. Conclusion
The Linear Constrained R egulation Problem has been analysed by determining
positively invariant sets associated to non-quadratic Lyapunov functions. Existence
conditions oflinear state feedback control laws have been obtained and translated into
an efficient algorithm based on linear programming. The control laws obtained by this
approach not only transfer to the origin all the initial states belonging to a polyhedral
subset of the state space but also optimize the convergence rate, while respecting
control constraints.
Although in the formulation of the LCRP no constraints on the state vector have
been imposed, the proposed algorithm also provides a solution to the case where the
state vector must satisfy linear inequalities.
REFERENCES
BITSORIS,G., 1986, Sur I'existence des ensembles invariants polyhedraux des systemes lineaires,
Technical Report 86015 (L.A.A.S.-CN.R.S, Toulouse, France); 1988 a, Int. J. Control,
47, 1713; 1988 b, J. Large-scale Systems, to be published.
BITSORIS,G., and BURGAT,C, 1977, Int. J. Control, 25, 413.
CHEGANc;AS,J., and BURGAT, C, 1985, Actes du Congres Automatique d'AFCET, Toulouse,
France, 193.
FRANKENA,J. F., and SIVAN, R., 1979, Int. J. Control, 30, 159.
GANTMACHER,F. R., 1960, The Theory of Matrices (New Yor k: Chelsea).
GAUTHIER, 1. P., and BORNARD, G., 1983, Rev. Autom. Inf. Res. Oper., 17, 205.GUTMAN,P.O., and HAGANDER, P., 1985, IEEE Trans. autom. Control, 30, 22.MOURADI,M., 1979, Rev. Autom. Inf. Res. Oper., 13, 127.
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