lAd < 1;. (b) there exists a positively invariant sett Q £; W of the resulting closed-loop system (4) such that. There are many ways to use this result for obtaining a ...
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 is not simple, because as an open-loop scheme it implies considerable off-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 for 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 linear constrained controllers of linear systems characterized by a stable non-negative dynamic matrix (Chegan 0, i = 1,2, ... , m. There is also given a bounded set of initial states Xo defined by the inequalities
whereGERqXnwithq~n,rankG=nandw=[wl Wz ... wq]Twithwi>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 F E Rm x n the set R(F, p)
law u(k)
= Fx(k) with
= {x E R": - P ~ Fx ~ p}
It is clear that the polyhedral set R(F, p) is the region of initial states of the closed-loop system (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
It is obvious that the control law u = Fx is a solution of the LCRP ifand only if the resulting 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 E T. This condition can also be expressed as follows (Bitsoris 1988 b). Proposition 1 The control
law u
=
(a) the eigenvalues lAd < 1;
Fx with FE Rm
Ai>i
xn
is a solution
to the LCRP if and only if
= 1,2, ... , n, of the matrix A + BF are in the open disk
(b) there exists a positively system (4) such that
invariant
sett
Q £; W of the resulting
There are many ways to use this result for obtaining a solution (Bitsoris 1986). An approach can be based upon the following preceding proposition. Corollary
closed-loop
on to the LCRP corollary of the
1
If F is a real m x n matrix such that (a) the eigenvaluesA;,
i
= 1, 2, ... , n of the matrix A + BF are in the open disk IAI< 1,
(b) R( G, w) is a positively invariant
set of the closed-loop
system (4); and
(c) R(G, w) £; R(F, p) then the control law u = Fx is a solution to the LCRP and the state vectors satisfy constraints (2) for all initial states Xo E R( G, w) and k E T. It is clear that for applying this result to the derivation of a solution to the LCRP, one must first establish conditions guaranteeing that R( G, w) is a positively invariant set of the resulting closed-loop system (4). It is well known that an asymptotically stable system possesses positively invariant sets having a quadratic boundary. Such positively invariant sets are generated by quadratic Lyapunov functions of the type v(x) = xT Px where P E Rn xn is a symmetric positive-definite matrix. By using Lyapunov-like functions we can also generate positively invariant polyhedral sets R(G, w). It can be easily seen that the polyhedral set R(G, w) can be equivalently defined by the expression
v *( x )
LI} =t::. max {I(GX -1 ~i~q
Wi
(Gx); denoting the ith component of the vector Gx. The following proposition provides a necessary and sufficient condition for v*(x) defined by (5) to be a Lyapunov function, and, accordingly, for R(G, w) to be a positively invariant set of system (S).
tA Xo E Q
non-empty subset Q of R" is said to be a positively implies x(k; xo) E Q for all k E T.
invariant
set of (4) if and only if
Proposition
2
(Bitsoris
1988 a)
The polyhedral set R( G, w) is a positively invariant there exists a matrix H E Rq x q such that
(IHI-
set of system (S) if and only if
:ll.)w:::; 0
GA-HG=O By a direct application establish the following. Proposition If FER'"
of this result to the closed-loop
system described by (4) we
3 x n
and there exists a matrix HE Rq
x
q such that
(IHI-:ll.)w:::;O GA (iii) the eigenvalues
Ai
+ GBF=
HG
= 1,2, ... , n of the matrix A + BF are in the open disk IAil
