On discrete-time linear systems over cones

Systems & Control Letters 6 (1985) 271-275 North-Holland

October 1985

Paolo d’ALESSANDR0, Manuela DALLA MORA, Elena DE SANTIS Istituto di Elettrotecnica, Facolth di Ingegwria. Uttioersit6 di

L’Aquila, Monrcluco. 6 7100 L’Aquila, Italy

The theory of controlled and conditioned invariant cones extends the results for the linear case given in [l]. The application given in Theorem 6 pertaining to the possibility of confining a trajectory to a convex cone also extends the result given in [l] for the case of linear subspaces.

Received 3 November 1984 Revised 24 June 1985

Abstract: The subject of this paper is the theory of linear discrete-lime systems with input constrained to belong to a closed convex cone. After discussing the generality of this class

of systems, a number of invariant cones of states are inlro- duced. In the last section these latter are shown to allow the extension of some classical results of the theory of linear

unconstrained systems.

There is limited but interesting literature on constrained systems. Without any attempt to com- pleteness, references [2-71 give a significant sam- ple.

Of course the rather extended literature on positive systems deals with the special case of linear constraints (see e.g. [S]).

For the concepts of convex analysis the main reference is the classical book by Rockafellar [9]. See also [lo].

Keywords: Linear systems, Constrained inputs, Invariant cones.

1. Introduction 2. Homogenization of linear systems with linearly constrained inputs

This paper deals with the theory of linear discrete-time systems with input values constrained to belong to a convex cone. In this setting it is possible to extend some major results of linear theory.

Section 2 is devoted to assessing the generality of the theory. In fact it is shown that, thanks to the concept of homogenization of a system, the case of inputs bounded to a convex set can be reduced to the case of inputs bounded to a convex cone.

In Section 3 the mathematical tools are devel- oped. A number of invariant cones are introduced, that enjoy the property of being invariant with

Consider a linear system S described by the recursive equations

x(c+l)=Ax(t)+Bu(t), v(I+l)=Qx(t+l), Vr x(t)ER”,u(t)ERP,y(t)ER’,

with any initial time iE R (but for simplicity it is convenient to assume i = 0 unless otherwise stated), initial state x(i) = X, t >, i. Consider a closed convex set A4 in R”, and assume that the dehomogenization of the homogenization (see e.g. [lo]) of M is M itself. Let now the inputs be constrained by

respect to the dynamical matrix of a system. Finally in Section 4 some system-theoretic re-

u(t) EM, t=o, 1,. . . .

sults exploiting the machinery of invariant cones Note that our hypothesis includes the case of are stated. linear constraints in which th,e set M is a .poly-

Here the spirit is that of giving some significant hedron [7]. examples. Further results of linear theory may be Consider in the space Rp+” the set D = M extended to systems over cones. x {Z} and then the homogenization 9(D) qf this

set in Rp+“+‘. The projection of this space onto l This work was supported by C.N.R. under grant No. CT

RP+l is denoted by P,,, and then the projection 83.01052.93. of this space onto R”+’ is denoted by P,,+,.

0167-6911/85/%3.30 b 1985, Elsevier Science Publishers B.V. (North-Holland) 271

On discrete-time linear svstems over cones * d

Volume 6, Number 4 SYSTEMS & CONTROL LETTERS October 1985

Next with reference to the augmented state space R”+’ and input value space RF+’ consider the recursive equations

This system with initial conditions and inputs specified as above will be called the homogenized system. Note that in the homogenized system the input is constrained to belong to a closed convex cone.

Finally the dehomogenization of the homogenized system is introduced. The solution of the original system can be retrieved by projecting on R” a solution of the homogenized system with component x”+’ identically 1.

In this way it is shown that without great restriction of generality, it can be assumed that the constraint on the input requires its values to belong to a closed convex cone.

3. Some invariant cones

Before initiating our treatment it might be use- ful to avoid ambiguities to make a few stipulations OF terminology.

A convex cone is a nonvoid subset P of a linear space such that aP c P for any real a > 0 and P + P c P. The polar of a set P is the set

For the sake of simplicity a convex cone is simply called a cone in most instances.

Let C be a convex cone in R’, B be a linear operator on RP to R” and A a linear operator on R”. Consider the minimal convex and the minimal closed convex cones which are invariant with respect to A and contain the convex cone BC, to be denoted by R(A, BC) and G(A, BC) respec-

tively. These are well defined and unique because R” is a (closed) convex cone containing BC, and moreover, the intersection of any family of (closed) convex cones invariant with respect to A is a cone of the same sort invariant with respect to A. Therefore

R( A, BC) = n( C: C convex cone invariant with respect to A, C 3 BC},

G( A, BC) =fl (C: C closed convex cone invariant with respect to A, C 13 BC).

Theorem 1. The cones R( A, BC) and G(A, BC) are given by

R(A, BC)=%(U{A’BC:i=0,1,2 ,... )). G(A, BC)= R(A, BC)-

= %‘o- {lJ{ A’BC: i = 0, 1, 2,. . . })

(where%‘o( .) (%o-( .)) denotes tire (closed) conuex conical extension ).

Proof. That %(U{ A’BC }) is an invariant cone that contains BC is obvious by direct verification. On the other hand, any convex cone invariant with respect to A that contains BC, must also contain the cones ABC, A’BC,.. ., so that the proof of the first statement is finished.

Next note that if a convex cone D is invariant with respect to an operator A, so is its closure. In fact AD C D inplies AD C D-. On the other hand if x E D- there exists a sequence (x,,} in D that converges to x and {Ax,,} is in D. But A is continuous, thus {Ax,,} converges to Ax which is in D-. This implies R(A, BC)-2 G( A, BC). Conversely by definition G( A, BC) I R(A, BC) but since the cone on the left-hand side is closed it is also true that G( A, BC) 3 R( A, BC)-.

Note that

%(“(AiBC:i=O, l,...,k))= ;: A’BC. r=O

Thus it might be written symbolically

%(U{ A’BC}) = E A’BC. i-0

Note also that even when finite sums are closed infinite sums might well be not closed.

A most important (both from practical and theoretical standpoint) special case of closed cones is that of closed polyhedral cones. For such cones

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Volume 6. Number 4 SYSTEMS & CONTROL LETTERS October 1985

both finite sums and images under linear operators are still closed polyhedral cones.

The next theorem establishes a duality with Theorem 1.

Theorem 2. The polars of the cones R( A, BC) and G( A, BC) ure given by

R(A,BC)P=G(A, BC)P

=n{A*-‘B*-‘CP:i=O, l,...).

This cone is the (unique) maximal closed conuex cone that is contained in the cone (BC)* and is invariant with respect to A*.

(Here inverses are understood in the sense 01 inverse image operators.)

Proof. It suffices to demonstrate the second equality. First compute the polar of the cone BC:

(BC)‘={y:(y, Bz)<O,VZEC}

={y:(B*y, i)<O,b’z~C)

={y:B*y~CP)=fl*-‘Cp.

Then apply the formula

(uq)p=np). Next if a cone D is invariant with respect to A,

i.e. AD c D, then it follows by taking polars that

A*D” c D”

so that DP is invariant with respect to A*. Finally suppose that a closed cone F is con-

tained in (BC)P and is invariant with respect to A*. Then FP contains BC and is invariant with respect to A and therefore FP I) G(A, BC). But this implies FPP = F c G(A, BC)P. This com- pletes the proof.

Further invariant cones are introduced in Defi- nitions 1 and 2 below.

Definition 1. A convex cone C is called a controlled invariant cone with respect to a convex cone D under the operator A if

ACcC+D.

The cone C is called for short (A, D)-controlled invariant cone.

Lemma 1. If C, and C, are (A, D)-controlled inuariant cones, then so is C, + Cz.

Proof. Let C, and C, be any two such cones then

AC, c C, + D, AC,cC,+D

imply

A(C,+C,)cC,+Cz+D.

Definition 2. A convex cone C is called a conditioned invariant cone under A with respect to a convex cone D if

A(CnD)CC.

The cone C is called for short (A, D)-conditioned invariant cone.

Lemma 2. The intersection of two conditioned invariant cones is a conditioned inoariant cone.

Proof. Let C, and Cz any two such cones, then

A(C, nD)cC,, A(C2nD)cC2

imply

A((C,nD)n(C,nD))=A((C,nC,)nD),

A(C,nD)nA(CznD)CC,nCz.

Lemma 3. The polar of a’)? (A, Dj-contra/led invariant cone is an (A*, DP)-conditioned invariant cone. Conversely if C is an (A, D)-conditioned invariant cone, and the cones C, D and CP + Dp are closed, then CP is an (A*, DP)-controlled invariant cone.

Proof. Let C be a controlled invariant cone under A with respect to D, so that

ACcCi-D.

This implies

(AC)P~(C+D)P, A*-‘CP 2 (C+ D)“,

A*( C + D)P c CP. A*(Cpn Dp)c CP,

so that the first statement is proved. Conversely if

A(CnD)CC,

then since C, D and C’P + DP are closed,

A(CPP n Dpp) c C,

A(CP + Dp)’ c C,

CPcA*-‘(CP+DP), A*CP c CP + DP.

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Volume 6, Number 4 SYSTEMS & CONTROL LETTERS October 1985

Note again that no hypothesis is required in this lemma for the case of closed polyhedral cones.

It is now convenient to introduce and study some further important invariant cones.

Let X and D be convex cones. Denote by m(A, D, X) the least (A, D)-condi-

tioned invariant cone that contains X; by mc( A, D, X) the least closed (A, D)-conditioned invariant cone that contains X; and by M(A, D, X) the greatest (A, D)-controlled invariant cone that is contained in X.

Theorem 3. The cones m(A, D, X), mc( A, D, X) and M( A, D, X) are well defined and unique. Moreover

m(A, D, X)= lim{Y,:i=O,1,2. ..}

where

y,=x, yI+, = Y,+A(x:nD).

Proof. The argument for existence and uniqueness of m( A, D, X) and mc( A, D, X) is by now obvious.

As to M( A, D, X) note that the family ~4 of all (A, D)-controlled invariant cones that are contained in X is nonvoid (because the cone (0) is in &). Consider a chain in such a family. Then the union of the chain is clearly an (A, D)-controlled invariant cone contained in ‘X that contains every member of the chain. Thus by the maximum prin- ciple there exists a maximal element in ~4. Con- sider any two distinct maximal elements. If they do not coincide then their sum is an (A, D)-controlled invariant cone contained in X that prop- erly contains both. This contradiction shows that the maximal element is unique.

Next note that the sequence in the statement is an increasing sequence of convex cones containing X. It is clear that any (A, D)-conditioned cone containing X contains each member of the sequence and hence also the union. It remains to be shown that the union is (A, D)-conditioned. But a point.is in the union if and only if it is in some element of the sequence. Because

A(~flD)c~+,~ lim{q},

the desired conclusion follows.,

4. System-theoretic results

The preceding sections have set up the ingredi- ents that allow to parallel for systems over cones

274’

some results pertaining to systems over linear spaces, which have applications in control prob- lems (see e.g. [ll, 121). The subsequent analysis is confined to the most basic results.

Consider a linear system as in Section 2 and a closed convex cone C. Then the following theorems can be stated:

Theorem 4. The cone R( A, BC) is the reachable set of a linear discrete-time system with the constraint u(t) E C, Qt.

Theorem 5. The cone G( A’, Q’CP) is the set of ail initial states such that the corresponding free solution y, satisfies the constraint y, (t) E C, Qt.

The proof of these two theorems is rather straight-forward and can safely be omitted.

Note that a duality result for the case of continuous-time positive systems is given in [8].

Theorem 6. Given two cones X and C, the trajectory of a linear system with the constraint u(t) E C, Qt, can be controlled to be contained in X if and only if x belongs to M( A, - BC, X).

Proof. Because

AM(A, -BC, X) c M( A, -BC, X) - BC,

in view of the equation

Ax(O) =x(l) - Bu(O),

if x(O)= x E M( A, - BC, X) it is possible to choose u(0) in C so that x(l) also belongs to M(A, - BC, X). Thus, by induction, the thesis follows.

Necessity: Suppose a trajectory 4 (that is the range of a solution x on [0, + co)) is contained in X. Note that for %(e) in view of t,he system’s equation it is true that

A%(z)c k%(s)-BC

and this shows that %(‘a) is an (A, - BC)-controlled invariant cone contained in X and hence %(c)cM(A, -BC, X).

References

[l] G. Basile and G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optinz. The- ory Appl. 3 (5)(1969).

volume 6, Number 4 SYSTEMS % CONTROL LETTERS October 1985

[2] D.H. Jacobson, Extensions of Linear-Quadratic Corrfrof. Optiminrrtiotr and Mafrix Theory (Academic Press, New York, 1977).

[3] D.H. Jacobson, D.H. Martin. M. Pachter and T. Geveci. Extettsions of Linear -Qua&uric Co~~frol Theor>> (Springer, Berlin-New York, 1980).

[4] R. Conti, Notes from lectures held al Mathematical In- stitute U. Dini, Firenze.

[S] R.M. Bianchini Tiberio. Instant comrollability of linear autonomous systems, J. @rim. Theory Appl. 39 (2) (1983).

[a] R.M. Bianchini Tiberio, Local controllability, rest states and cyclic points, SIAM J. Conrroi Optim. 21 (5) (1983).

[7] P. d’Alessandro, M. Dalla Mora and E. De Santis, On consistency of linear linearly constrained discrete time systems. J. Franklin Insf.. to appear.

[8] Y. Ohta, H. Maeda and S. Kodama, Reachability. ob- servability, and realizability of continuous-time positive systems, Siam J. Control Oprim. 22 (2) (March 1984).

[9] R.T. Rockafellar, Convex Ano!vsis (Princeton University Press, Princeton, NJ, 1982).

[lo] J. Steer and C. Whzgall, Cotroexiry a& Op/imizaricn it1 Finite Dimemions I (Springer, Berlin-New York, 1970).

[ll] G. Basile and G. Marro, Luoghi caralteristici dello spazio degli stati relativi al controllo dei sistemi lineari, L’EIer- rroreo~ica 55 (12) (Dec. 1968).

[12] G. Basile. R. Laschi and G. Marro, Luoghi caratteristici delle traiettorie dei sistemi lineari nello spazio delle uscite. L’Elerrrorecnica 56 (5) (May 1969).

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On discrete-time linear systems over cones