JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 73, No. 2, MAY 1992
Controllability Properties of Constrained Linear Systems *
M. FASHORO, 2 O. HAJEK, 3 AND K. LOPARO 4
Communicated by G. Leitmann
Abstract. tn this paper, we present results on constrained controllabil- ity for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1o
We prove that controllability of an arbitrary point p in R" is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties of G, the set of all nonnegative times at which p is controllable, and of C, the set of all controllable points. We characterize the set C for the given system and provide additional spectrum-dependent structure.
We show that, for the given linear system, several notions of constrained controllability of the point p are the same, and thus the set C is open. We also provide a necessary condition for small-time (differential or local) constrained controllability of p,
Key Words. Linear systems, constrained controls, reachable sets, complete constrained controllability, local constrained controllability, isochronous controllability.
1. Introduction
In a c o m p a n i o n paper (Ref. 1), we investigated the geometr ic properties o f reachable sets to a given point p in R" for l inear t ime-invariant cont inuous systems with const ra ined controls. We showed that, under certain simple
~This work was supported in part by NSF Grant ECS-86-09586. 2Department of Electrical Engineering, Vanderbitt University, Nashville, Tennessee. 3Department of Mathematics and Statistics, Systems Engineering Department, Case Western Reserve University, Cleveland, Ohio.
4Systems Engineering Department, Case Western Reserve University, Cleveland, Ohio.
conditions, the reachable set ~ ( p ) to p is a proper subset of the reachable set ~ ( q ) to another point q in R n. We also provided conditions for the equality of two reachable sets.
Among other properties of ~ (p) shown in Ref. 1, we presented results on the geometric shape of ~ ( p ) in terms of the spectrum of the state coefficient matrix of the linear constrained control system [the earliest results on the geometry of ~ ( p ) were developed in Ref. 2]. We also showed that
(p) is a proper subset of R n when the spectrum of the system lies in both halves of the complex plane.
This paper presents results on the controllability properties of the constrained linear control system using the geometric properties of ~ ( p ) . In Section 2, we describe several notions for constrained controllability to and from a point p in R": complete constrained controllability, constrained controllability at bounded times, isochronous constrained controllability, and local (small-time or differential) constrained controllability.
In Section 3, we show that complete constrained controllability to p for a given linear system with control constraint set U, at a nonnegative time s, is equivalent to the origin being in the interior of the reachable set to p, for the same linear system and at the same time, with a new control constraint set equal to U -A p . We also prove that this result is equivalent to the reachable set to point p at time t - 0 being contained in the reachable set to p at time t + s, s -> 0. Furthermore, we describe properties of the set of all times t -- 0 at which the given linear system is completely constrained controllable to p and the set of all points p for which the system is constrained controllable. In particular, we characterize the set of all controllable points for the given linear system.
We show that, for linear systems, (complete) constrained controllability coincides with isochronous controllability. An extension of this result proves that a necessary condition for ~ ( p ) to be open is the constrained controlla- bility of point p. Finally, we provide a necessary condition for local con- strained controllability of the system to point p.
Examples to illustrate the results are presented in Section 4.
2. Definitions
Consider the linear system on R n
Yc=Ax-u, u~ U, (1)
with A c R "2, u: [0, T]-~ U a measurable vector-valued function, and U a compact subset of R" containing the origin.
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2.1. Reachable Set. For the system (1), the reachable set to point p at a nonnegative time t, denoted by ~ , ( p ) , is the set of all states from which a given point p can be reached at time t. As shown in Ref. 1,
2.2. Attainable Set. ag,(p), the attainable set from point p, at nonnega- rive time t, is the set of all states which can be reached from p with admissible control functions. Hence,
f, sdt(p)=exp(-At)p- jo exp(As)Uds=exp(At)(p-g~,), (5)
with
~ , = ~,(0) = e x p ( - A s ) Uds. (6)
a4(p), the attainable set from point p, is the union
ae(p) = U ae,(p) (7)
We will use the notation zg in lieu of J ( 0 ) and the notation ~2 instead of ~ ( 0 ) in this presentation.
2.3. Constrained Controllability. There are several reasonable candi- dates for the concept of controllability of a point p.
One notion is that p belongs to the interior of ~2(p); that is, all points x in some neighborhood of p can be steered to p, but at possibly various times, depending on the location of the point x. When the times are specified, for example
p c int {._j ~ , ( p ) ,
this is controllability within a bounded time interval [0, T]. Another example is
p e i n t U { ~ , ( p ) : I t-T]<e}, for all e > 0 .
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This is controllability at the approximate time T. It will be shown and this is a major result of the paper, that these concepts are all equivalent to isochronous controllability of p; that is,
p e in t 9~t(p), for some t > 0 fixed,
at least for linear control systems; the question remains open in the nonlinear case.
A related, but different concept, is the following: p is locally (differential or small-time) controllable if
p~ in t (_J{~(p ) :O<- t<~} , for all small E>0 .
A classical result for the case p = 0 is that, if the control constraint set U is compact and symmetric about the origin, then the origin is controllable if and only if it is locally controllable.
The set C of controllable points for the system (1) is the collection of all points p which are constrained controllable at some time t---0.
3. Controllability Properties
3.1. Constrained Controllability. The following lemma presents a condition for constrained controllability of a given point p in R" for the system (1).
Lemma 3.1. For the given system (1), p e int ~s (p), for some s -> 0, if and only if the origin is contained in the interior of the reachable set at time s --- 0 for the auxiliary control system
£ = A x - v , v s U - A p . (8)
Proof. The condition p e int ~s (P) defines the notion of constrained controllability to point p at time s -> 0; see Ref. 1, Section 3. Let V denote the unit ball, centered at the origin in R ~. Then, p ~ int ~ ( 0 ) implies that there exists ~ > 0 such that
Thus,
p + ~V c ~ ( p ) = exp( -As )p + 9~s.
~V c ( exp ( -As ) - l )p + Yts
= { ( d / d w ) ( e x p ( - A w ) p ) + e x p ( - A w ) U } dw,
~ V c e x p ( - A w ) ( U - A p ) dw.
(9a)
(9b)
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Hence, the origin is contained in the interior of the reachable set at time s for the system (8). Since all the steps are reversible, this completes the proof. U]
From Lemma 3.1, a necessary and sufficient condition for controllability to point p at time s-> 0 is that
fo 0~in t exp(-Aw)(U-Ap) dw. (t0)
Here, U - A p denotes { u - A p : u E U}, a translation of the elements of U by the vector -Ap. The lemma relates the controllability of the system (1) to point p to the controllability of the system (8) to the origin. When the origin is a relative boundary point of U-Ap, Brammer's theorem (Ref. 3, Theorem 8.3.1, p. 149), developed for linear systems with constrained controls, provides conditions for controllability to the origin.
The next lemma establishes the equivalence between the results in Lemma 3.2 of Ref. 1 and the previous lemma.
Lemma 3.2. Consider the system (1) with U convex and compact. Let s >--0 be fixed. Then, the following statements are equivalent:
(i) ~ ( p ) c i n t Y2,+,(p), for all t->0; (ii) 0eintlo exp(-Aw)(U-Ap) dw; (iii) p is controllable at time s; i.e., p ~ int ~s(P).
Proof. The equivalence of (i) and (ii) is proved in Lemma 3.2 of Ref. t. From Lemma 3.1, (ii) and (iii) are equivalent. []
Next, we develop the properties of G, the set of all times t >-0 for which p is controllable.
Lemma 3.3. Let G -- {t: p ~ int ~f(p)}. Then, the set G of times t-> 0 at which the point p is controllable is an open additive subset of R += {t: t-->O}.
Proof. From Lemma 3.1, if p ~ int ~ t (P) for some t > 0, there exists > 0 such that
p + 6 V = ~ , (p) = exp( -At )p + Y~t,
where V is the unit ball centered about the origin in R". Thus,
6Vc (exp(-At) - I )p+~t , for some 6 >0 .
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That is,
- ( e x p ( - A t ) - I)p ~ int ~ , . (11)
Thus, there is a ball B, centered at ( - e x p ( - A t ) + I)p, such that B c i n t ~t. Since the boundaries of the reachable sets move continuously with the time, there exists e > 0 such that
B c i n t ~s, Is-tl<~. Hence, G is an open set.
Let ~t denote the reachable set to the origin at time t ~> 0 for the auxiliary system
:~=Ax-v, v~ U-Ap.
By Lemma 3.1, if p is controllable at times t and s, then 0~ in t ~ , and 0 ~ int ~ ; in particular,
0 e int ~ , and 0c e x p ( - A t ) ~ s .
By the addition theorem,
0~ int ~ , + e x p ( - A t ) ~ s c i n t (~ , + e x p ( - A t ) ~ s )
= int ~,+s. (12)
Again, by Lemma 3.1, p is controllable at time t+s; i.e.,
t, s ~ G implies that t + s ~ G.
This completes the proof of Lemma 3.3. []
A consequence of this lemma is the following result.
Lemma 3.4. Given a controllable point p of the system (1), then the set G of times t --> 0 at which p is controllable contains some right unbounded interval (0, co) in R +.
Proof. By assumption, the set G of times at which p is controllable is not empty. By the preceding lemma, G is open. Thus, G contains some interval (a, b) with 0 < a < b <oo. Since G is also additive, it must contain all intervals (/ca, kb), for k = 1, 2, 3 , . . . . These intervals begin overlapping for large k, indeed for all k > a / (b -a ) . Thus, G contains an unbounded interval of the form (0, ~) . [i]
The geometric properties of the set C of all constrained controllable points for the system (1) are developed next.
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Theorem 3.1. Given that (the origin of) the system (1) is constrained controllable, with U compact and convex, then the set C of controllable points is nonvoid, convex, and open. I f U is symmetric (about the origin), then so is C.
Proof. To show the convexity of C, let the system be controllable to points p, q~ C. Then, from Lemma 3.2, there exists numbers Op(Oq) such that the system (1) is controllable to p(q) at all times t >-- 0p(0q). Thus, there exists a common 0 - m a x ( 0 p , 0q) such that the system is controllable to both p and q for all times t > 0. From Lemma 3.1, there exists 6 > 0 with
6V c ( e x p ( - A 0 - I)p + ff~o,
6 V c ( e x p ( - A 0 ) - I )q + ~o.
Let A s [0, 1]. Then,
6V c A ( ( e x p ( - a 0 ) - I)p + ~o)+ (1 - A)((exp(-A0) - I )q + ~o),
or equivalently,
3 V c ( e x p ( - a o ) - I ) ( A p + ( 1 - A ) q ) + ~ o. (13)
Since ~o is convex, Ap + ( 1 - A)q is controllable at time 0 and C is convex. Let p e C be controllable at time s > 0. Then, fi'om Lemma 3.1,
0 ~ in t ( (exp( -As) - I)p + ~s) = ( exp ( -As ) - I)p + int ~ ,
Equivalently,
- ( e x p ( - A s ) - I )p s int ~ c i n t ~.
By continuity, there exists e > 0 such that, for all q, t q -p l < ~,
- ( e x p ( - A s ) - I)q E int ~s c i n t ~.
Hence, C is open. The hypothesis that the system (1) is controllable to the origin ensures
that C is nonempty, since at least it contains the origin. To show symmetry, if p is controllable at time t - -0 , then
0 e i n t e x p ( - A w ) ( U - A p ) dw.
Thus, - p is controllable at time t-> 0 if
0E int e x p ( - A w ) ( U + A p ) dw.
If U is symmetric (about the origin),
u(t) e U - A p implies - u ( t ) ~ U - A ( - p ) ,
and C is symmetric.
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To completely characterize the set C for the system (1), the following mathematical notions are useful. Define
Thus,
/~ = {M c R": for any x, y 6 M, there is an admissible trajectory from x to y}.
M r ~ ( x ) , for all x ~ M .
The following properties hold:
(i) I f N ~ M e Ix, then N ~/~. (ii) I f {Mi} is a monotone sequence of sets in Ix, then U~ Mi e Ix. (iii) I f M, N ~ Ix and M c~ N = O, then M u N ~ Ix. (iv) I f Mi is open and has a nonvoid interior for all i, then there
is a disjoint decomposi t ion of R" such that LJ~ M~ = R" with maximal M~ e Ix.
To show proper ty (iii), choose a ~ M c~ N. I f x ~ M and y E M r~ N, then either y ~ M, which means that there is a trajectory from x to y, or y ~ N. When y e N, there are trajectories from x to a and from a to y. Thus, there is also a trajectory from x to y.
For property (iv), certainly U~M~ c R"; we need to show that R " c Us M~. From the hypothesis, consider the case when M~ = {x}. Then, R" c
U,M;.
Lemma 3.5. Consider a maximal M ~/z which satisfies properties (i) to (iv) above. Then, for the system (1), the set C of all controllable points, satisfies M n C = int M.
Proof. Consider a point x ~ R ~. I f x c M c T C , then x ~ C and C is open. Also, x ~ M and M ~/~. Thus, there exists a ball of radius ~ about x, say S~(x), such that S~(x)~ Ix. Since M E Ix and x ~ M c7 S~(x), we have M n S~(x)6 Ix. But M is maximal; thus, S~(x )c M (that is, M is open) and x ~ int M. This proves that M n C c i n t M. Conversely, let x ~ int M; since M c ~ ( x ) , by definition, we have x e int ~ ( x ) . That is, x 6 C. Thus, int M c M n C. []
Lemma 3.6. Given {Ms} which satisfy properties (i)-(iv), there exists precisely one maximal M ~ IX such that C c M
Proof. From Lemma 3.5,
C = R" n C = ( U M~) c~ C = U int Mi.
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Thus, C is the union of disjoint open sets Mr. From Theorem 3.1, C is convex, which implies that C is connected. Thus, there is only one, nonveid, term in the decomposit ion. This term is
C = i n t M o c Mo. (14) []
Corollary 3.1. The set C c Yt n sO.
Proof. 0 ~ C c M and M c ~ (x), for all x ~ M. tn particular, M c (0) = ~ and C c ~ . Since C is invariant with respect to a change in time
orientation, C c sO. Thus, C c ~ c7 M. []
From this corollary, C is the interior of the union of closed trajectories through the origin. Hence, the following result ensues.
Corollary 3.2. Consider a point p ~ R n. For the system (t) , if p ~ C, then the union of closed trajectories through p has a nonempty interior.
Proof. We proceed by contradiction. Assume that p ~ C, but that the union of closed trajectories through p has an empty interior. This im- plies that there exists an open set which includes p that has the point-to- point steering property. Hence, p ~ C, which is a contradiction of the hypothesis. .~
We will now completely characterize the set C of controllable points.
Theorem 3.2. Consider the system (t) . The set C of all controllable points for the system is such that
C = ~ n s / . (15)
Proof. From Corollary 3.1, C c ~ ~ s~. It remains to show that ~ s~ c C. Consider a point p ~ Y2 n sO; that is, p ~ Y~ and p c ~ . From Lemmas 3.4 and 3.5 in Ref. 1, when ~ ( p ) = ~ , ~ ( p ) is an open convex set such that 0 e ~ ( p ) . Consider a point q e C such that q ~ ~ n sO. Since 0 ~ Y~(p) and p s ~/, there exists a trajectory from q to p and a trajectory from p to q. This implies that p e (7.
Alternatively, p ~ .~¢ n ~ implies that there exists a ball of radius 6 about p, say S~(p), such that S~(p)c ~ n sg. Since Y2 and s~ are open sets, this implies that p ~ C. D
A consequence of this theorem is the next result, relating points in C and the equivalence of reachable sets.
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Corollary 3.3. For the system (1), p s C if and only if ~ ( p ) = 8 .
Proof. From Theorem 3.2, p ~ C is equivalent to p ~ ~ n M. From Lemma 3.4 of Ref. 1, p ~ ~ n M if and only if ~ = ~ ( p ) . []
The theorem can be used to obtain the geometric shape of C when the spectrum of the system matrix A is known. See Ref. 1 for geometric results related to the spectrum of A.
Corollary 3.4. For the system (1), when Re(spA) < 0, C = M, a proper subset o f R". For Re(spA) = O, C = R". I f Re(spA) > 0, C = ~ , also a proper subset o f R".
Note that Re(spA) is the real part o f the spectrum of A.
Proof. For Re(spA) < 0 and Re(spA) = 0, ~ ~ M = M; see Ref. 1, Lemma 4.5 and Corollary 4.3. When Re(spA) > 0, Ref. 1, Lemma 4.2, shows that ~t n M = ~ . []
Corollary 3.5. For the system (1), given any point p ~ R" such that p is controllable, (that is, p ~ ~ n M), then ~ n M = ~ ( p ) n M(p) = C.
Proof. From Lemma 3.4 of Ref. 1, if p ~ ~ n M, ~ = ~ (p) and M = M(p). Hence,
~ n M = ~ ( p ) n M(p). []
We next show that, in linear systems, constrained controllability is equivalent to isochronous constrained controllability.
Theorem 3.3. Consider the system (1) with 0 ~ U. Assume a weak form of controllability of the origin,
int U St is nonempty. t_>0
Then, whenever a point p satisfies
p ~ int U ~ ( p ) = int ~ ( p ) , (16) t ~ O
necessarily p c i n t ~ s ( P ) for some s -> 0.
Proof. This consists of two steps. In the first step, f rom 0 ~ U, it follows that the reachable sets (to the origin) gt, increase monotonically with the time t. Then, we have the reduction of a reachable set to a countable union,
U ~, = ~3 ~k. (17) t-->O k = l
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Here, we have assumed that the set U,_>o 5~t has a nonvoid interior. From Baire's theorem, it follows that
int 5~k # Z , for some k = 1 , 2 , . . . , (18)
where O is the null set. By monotonicity, then also
int ~ , # O, for all t>-k. (19)
In the second step, it is convenient to introduce the set
/~t = ~ t (P ) - P = ( exp ( -At ) - l )p + ~t. (20)
The advantage is that this set satisfies an addition property,
tz~+~ =tx t+exp( -At ) t% t, s>-O. (21)
This property is easily obtained from (20); see Ref. 1. The change from ~t(P) to lz, is effected by the mapping x~-~x-p for all points xE ~t(p). This is an affine isomorphism. In particular, p ~ ~ I (P) if and only if 0 c/~,, and p ~ int Y~,(p) if and only if 06 int/z,; see Ref. 1, Lemma 3.2. Mso,
U ~ , ( p ) = p + U / z l . (22) t > O 1 > 0
Now, consider the set
O = {O: 0 ~/~o}.
From (20), the set is closed under addition. From (16), it contains some time ~'>0. Hence, it also contains all times ~-, 2¢,3z, . . . . In conjunction with (19), we conclude that there exists a time 0 = kz such that
0 ~/~o, int/z0 # ~ .
Now, fix 0 and also a point q ~ int tzo, Then,
eq = (1 - e )0+ eq c int/~o, (23)
for all 0-~ e -< 1. The assumption
p ~ int ~_) ~t(P) t > O
yields that there is a ball B about 0, with
B c U ~z,, 1_>0
Choose e small enough so that
E(-exp(-AO)q) ~ B.
Thus,
e(-exp(AO)q)ctxr, for some t > 0 ,
0 = Eq +exp(-AO)(e(-exp(AO). q))
c (int tXo) + exp(-AO)tz, ~ int lxo+t.
(24a)
(24b)
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This shows that, with s = 0 + t, we have 0 ~ int/zs; that is, p ~ int ~ ( p ) . This theorem gives a necessary condition for the reachable set of ~ ( p ) to be open. []
Corollary 3.6. Consider the system (1) with 0~ U and a weak form of controllability of the origin; that is,
int U ~t is nonvoid. t ~ 0
I f p ~ int U,~0 ~ , (P ) , then the set ~ ( p ) = ~.J,~_o ~ , (P ) is open.
Proof. Let x c ~ ( p ) . Then, from Theorem 3.3,
q:= x - p ~ l._J tzt, t ~ O
so that q~/zt , for some t---0. Hence,
q = q + e x p ( - A t ) . 0 ~ / z t + exp(-At) ( in t /xs)
c in t /z ,+s c i n t U / z , , t_>0
and thus x E int ~ ( p ) .
(25)
[]
3.2. Local Constrained Controllability. The next result gives a necessary condition for local constrained controllability of points p in R ", given the system (1) and the control constraint U.
Lemma 3.7. If the system (1) is local constrained controllable to point p in R n, then Ap ~ U.
Proof. From the definition of local (differential or small-time) con- strained controllability, if the system is local constrained controllable to point p, then for some positive time sequence s = sk ~ 0, sk ~ 0, we have
~t(P) c ~,+s(P) = e x p [ - A ( t + s)] .p + ~,+s,
that is,
Thus,
~ t (P) = e x p ( - A t ) -p + ~t c e x p [ - A ( t + s)] .p + ~,+s.
e x p ( - A t ) . p + ~ t c e x p [ - A ( t + s ) ] . p + ~ , + e x p ( - A t ) ~ s . (26)
From the assumptions on U, ~ , is nonvoid, convex, and compact. It can be eliminated from both sides of the inclusion; see Ref. 4, Proposition 1, p. 108. Thus,
e x p ( - A t ) - p c e x p [ - A ( t + s)]- p + e x p ( - A t ) ~ .
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Premultiplying both sides by exp(At), we have
p e e x p ( - A s ) .p + Ns, (27a)
0 ~ ( exp ( -As ) - I)p + ~,. (27b)
Hence, for s ¢ 0,
0 e ( exp ( -As ) - I)p + ~,)/s. (28)
For s = sk, on taking limits as sk --> 0, we have
0 ~ -Ap + lira ( ~ / s ) . (29)
But the limit of ~, / s as s->0 is U; see Ref. 4, Theorem on p. 129. Thus, OE-Ap+ U and Ape U. []
This lemma shows that the set of points p for local constrained con- trollability is small relative to the set of constrained controllable points.
4. Examples
Example 4.1. Local Constrained Controllability. Consider the system
Ix~:] = [~ ~] [ x l ] - [ i ] u , lul--< 1, (30a)
xl (final) = Pi, i = 1, 2, P = P2 "
For this system,
[2p~] (31a) Ap= L p2 2'
[ul] ) U i s t h e s e t : ] u ~ l ~ l , i = l , 2 , U2
U - Ap is the set { I ul ] : Ul 6 [ - 1 - 2pl , l - 2pt], U2
u2~ [ -1 -P2 , 1 -P2]} •
Pl If the system is local constrained controllable to p = [ p2 ], then
P6{IPl]:Pl=O'5p2andp26(-I '0'I"O)}
(31b)
(31c)
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At the interval endpoints,
[0"51 P = ± 1.0 "
Brammer's theorem (Ref. conditions are:
(i)
3) checks for local controllability. Brammer's
The unconstrained system is completely controllable since the rank of[~ ~] i s2 .
(ii) The eigenvectors of the system are [ko~] and [°2], with ki ~ 0, i = -0.5 1, 2. Also, A T = A. For p = [ -1.o],
: 0--< ui ----- 2, i = 1,2 . U2
= [1.o], At p 0.5
: -2~u i<- -0 , i = 1 , 2 . U2
0.5 For either endpoint p = ± [ 1.o], the eigenvectors of the system are not exterior normals to U - Ap at the origin. Hence, the endpoints are locally controllable and the necessary condition Ap ~ U holds.
Example 4.2. Constrained Controllability. Consider the system
[ : : ] = [ - i 11] [ x ; ] - [11] u' [u]-< 1' (32a)
xi (final) = p~, i = 1, 2, P = P2 "
As shown in Ref. 1, from the bang-bang theorem,
~[ e-'(pl cos t - p 2 s i n t)+5o e-S( c°s s - s i n s)uds] Y~'(P) = [.[e-'(pl sin t+p2 cos t)+J'o e-S( sin s + c o s s)udsJ'
measurable u: [0, t] --, {-1, 1},/. (33) l t~[0,~r)
is the same as
[e-~(pl cos t -p2 sin t)+So e-S(cos s - s i n s)(1) ds ~t(P) = L e_~(pl sin t+p2 cos t)+So e-S( sin s +cos s)(1) ds
+I'~ e-S( c°s s - s i n s ) ( -1 ) ds], +S~ e-S(sin s + cos s ) ( -1 ) ds_] or ~ [0, t], t c [0, 7r), (34)
where cr is the time to switch from u = -1 to u = 1 or vice versa. Computer plots of ~ t (P ) for p = [o°], [_o°.5], [°65] are shown in Figs. 1, 2, 3, respectively.
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1
343
× 1
Fig. 1
The points p = [o °] and [_°.5] are cons t ra ined control lable , since
~ , ( p ) = int Y~,+s(p), for all t--- 0, s - 0.
For the example , ~¢ is all o f R ", while Y~ is a p rope r subset o f R ". Thus, C = ~ c~ M = ~t and plots o f M, are shown in Fig. 1. Figure 3 indicates that the given sys tem is cons t ra ined cont ro l lable at p = [°65] for 2.5 < t < z,, since p = [°65] is con ta ined in the inter ior o f ~ , ( p ) for all t such tha t 2.5 < t < ~r.
For p = [o °] and [_o°5], Figs. 1 and 2 show that the system is local const ra ined control lable , since for small t imes t > 0,
~ , ( p ) c i n t ~ ,+~(p) , t>_0, s_>0.
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× 12
3 . 0
1 . 5 I |
J I I I
I - ' 0 . 5
)c 1
Fig. 2
The necessary condition for local controllability Ap ~ U is also satisfied by these two points.
5. Conclusions
In this paper, we have derived controllability properties for linear systems with controls constrained within a compact set U and an arbitrary terminal point p other than the origin of R ".
We showed that a necessary and sufficient condition for constrained controllability to the point p at s--- 0 is the inclusion of the origin of the
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×12
345
-----×~
Fig. 3
system in the interior of the reachable set at time s for the same system with new control constraint U - Ap. We also proved that constrained con- trollability to p at s --- 0 is equivalent to the inclusion of the reachable set to point p, at nonnegative time t -> 0, within the reachable set to p at time t+s>_O.
We characterized properties of G, the set of all times t-> 0 at which the point p is controllable. G was shown to be an open additive subset of R + which contains a right unbounded interval (0, oo) in R +. We showed that C, the set of all constrained controllable points for the system, is nonvoid, convex, and open. In particular, we derived a result which com- pletely characterizes the geometric shape of C in terms of reachable and
346 JOTA: VOL. 73, NO. 2, MAY 1992
attainable sets of the given system and showed that the set depends on the spectrum of the system. We proved that, in linear systems, constrained controllability is equivalent to isochronous constrained controllability. A consequence of this result is the fact that ~ ( p ) is open. In conclusion, we showed that a necessary condition for local (differential or small-time) controllability of the point p for the given system is that Ap ~ U.
References
1. FASHORO, M., H.~OEK, O., and LOPARO, K., Reachability Properties of Con- strained Linear Systems, Journal of Optimization Theory and Applications, Vol. 73, No. 1, pp. 169-195, 1992.
2. FASHORO, M., Reachability and Controllability Properties for Constrained Linear Systems, PhD Thesis, Case Western Reserve University, 1987.
3. BRAMMER, R. F., Controllability of Linear Autonomous Systems with Positive Controllers, SIAM Journal on Control, Vol. 10, No. 2, pp. 339-353, 1972.
4. HAJEK, O., Pursuit Games, Academic Press, New York, New York, 1975.
