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On the controllability of switching linear systems
Automatica 41 (2005) 663 – 668 www.elsevier.com/locate/automatica
Brief paper
On the controllability of switching linear systems夡 Mikhail I. Krastanova , Vladimir M. Veliova, b,∗ a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.Bonchev str. 8, 1113 Sofia, Bulgaria b Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstr. 8, A-1040 Vienna, Austria
Received 3 November 2003; received in revised form 13 August 2004; accepted 25 October 2004 Available online 22 December 2004
Abstract This note presents a necessary and sufficient condition for small time controllability of a linear switching system (that is, a collection of linear time-invariant control systems, where a trajectory is any concatenation of trajectories of the individual systems). This result extends the controllability condition recently obtained for unconstrained linear switching systems to the case of control which is constrained in a cone. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Controllability; Switching control systems; Control constraints
such that for almost every t ∈ [tk−1 , tk ) the following relation holds true:
1. Introduction We consider a family of linear control systems (called a switching system) x˙ ∈ Ai x + Ui ,
i = 1, . . . , N,
(1)
where x ∈ Rn , Ai are (n × n)-matrices, Ui ⊂ Rn are closed convex cones in Rn . By definition, a trajectory of the switching system (1) on [0, T ] is any absolutely continuous function x : [0, T ] → Rn for which there exist a finite number of positive integers i1 , i2 , . . . , ip ∈ {1, . . . , N}, points 0 = t0 < t1 . . . < tp = T and integrable functions uk : [tk−1 , tk ) → Uik , k =1, . . . , p, 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. Colaneri under the direction of Editor R. Tempo. This research was partially supported by the Austrian National Bank under contract 9414 and by the Ministry of Science and Higher Education—National Fund for Science Research under Contract MM-1104/01. ∗ Corresponding author. Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstr. 8, A-1040 Vienna, Austria. E-mail addresses: [email protected] (M.I. Krastanov), [email protected] (V.M. Veliov).
0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.10.017
x(t) ˙ = Aik x(t) + uik (t). The investigation of switching control systems is motivated by numerous applications in power systems, electronics, economics, mechanics, etc., as well as for control techniques based on switching between different controllers (see, e.g., Williams & Hoft, 1991; Narendra & Balakrishnan, 1997; Zhivoglyadov & Middleton, 2003). Utilization of switching control systems gives the possibility for intelligent control whereby the controller can operate in multiple environments. Control constraints arise in such systems mainly due to non-negativity requirements for some of the controls. We refer to the recent papers by Li, Wen, and Soh (2001) and Sun, Ge, and Lee (2002) for an extended bibliography. Our concern in the present note is the controllability issue for (1). Denote by A(x, T ) the attainable set of system (1) at time T, starting from the point x ∈ Rn , i.e. A(x, t) = {y ∈ Rn : there exists a trajectory x(·) of (1) such that x(0) = x, x(t) = y}. Definition 1. The switching system (1) is called small-time controllable (STC) if A(0, T ) = Rn for every T > 0.
664
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
Remark 2. We mention that due to the linearity of the systems in (1) and due to the cone constraints for the control, A(0, T ) is a closed convex cone. Therefore STC is equivalent to small-time local controllability (STLC), in the definition of which the requirement “A(0, T ) = Rn ” is replaced with “A(0, T ) contains the origin in its interior”. Remark 3. In the unconstrained case, that is, if Ui are subspaces, STC is equivalent to the same property, but with “for every T > 0” replaced with “for some T > 0”. In the case of control constraints this equivalence between STC and controllability on an arbitrary time horizon is no longer true. In the unconstrained control case (that is, if the cones Ui are all subspaces), a Kalman-type necessary controllability condition for switching linear systems is obtained in Sun and Zheng (2001), where it is also proved that the condition is sufficient for n = 3 and N = 2. In Xie, Zheng, and Wang (2002) the result is extended for n = 3 and an arbitrary N, and in Sun et al. (2002)—for arbitrary n and N. In the present note we extend the controllability condition from Sun et al. (2002) to the case of control constraints, that is, for cones Ui which are not necessarily subspaces. We mention that sufficient and necessary controllability conditions for linear systems with controls restricted to a cone have been obtained in Korobov (1980) (for the case of an arbitrary time horizon), Bianchini (1983), Sussmann (1987b), and Veliov (1988). The proof of the sufficiency of the condition for STC of control constrained linear switching systems presented here utilizes the differential geometric approach in a form similar to that developed in Veliov and Krastanov (1986) for piecewise linear systems and in Veliov (1988) for the case of control constrained linear system. Lemmas 5–8 used in the proof are suitable for analysis of the controllability also for nonlinear switching systems.
ci t di ,
i=1
where di are positive rational numbers and ci > 0, i =1, . . . , , is called a positive rational polynomial. Further, we use the notation o(t) to indicate any family of analytic vector fields o(t, x) parameterized by t > 0, continuous in (t, x) and such that for some > 1 the ratio o(t, x)/t is bounded when t tends to zero, uniformly with respect to x ∈ Bn . By A0 we denote the set of all families of analytic vector fields a(t, x) parameterized by t > 0, continuous in (t, x) and such that for some C > 0 and > 0 the following estimate holds: a(t, x) Ct x, whenever t ∈ [0, 1] and x ∈ Bn . Following some of the ideas proposed in Hermes (1978), Veliov (1988) and Veliov and Krastanov (1986), we define the following set E + of analytic vector fields. Definition 4. E + is the set of all analytic vector fields Z for which there exist a(t) ∈ A0 , o(t), and a positive rational polynomial p(t) such that Exp(tZ + a(t) + o(t))(x) ∈ A(x, p(t)). Lemma 5. Let Z1 , . . . , Zk ∈ E + and 0 ∈ int co {Zi (0): i = 1, . . . , k},
(2)
where “int” and “co” mean the interior and the convex hull, respectively. Then the switching control system (1) is STLC. The following lemmas provide constructions of elements of the set E + . Lemma 6. The set E + is a convex cone. Lemma 7. The constant vector fields ui , i = 1, . . . , N, belong to E + for every ui ∈ Ui .
2. Preliminaries In this section we develop some technical tools for controllability analysis that are useful also for nonlinear systems. The lemmas below use only the semi-group property of the reachable set A(x, t): A(A(x, t), ) ⊂ A(x, t + ),
∀x ∈ Rn , ∀t, 0.
Below Bn := {y ∈ Rn : y 1} is the unit ball. Given an analytic vector field Z on Rn , we denote by Exp(tZ)x0 the value of the solution of the equation x( ˙ ) = tZ(x()),
Sussmann (1987a). A function of the type
x(0) = x0 ,
at time = 1. Below we shall use also the notion Exp(Zt )(x0 ) for an arbitrary family of analytic vector fields {Zt : t ∈ R}, depending continuously on t, as defined in
Lemma 8. Let ±Z ∈ E + and let Y (x) = Ai x for some i ∈ {1, . . . , N}. Then ±[Z, Y ] ∈ E + , where the Lie bracket [Z, Y ] is defined as [Z, Y ] :=
jY jZ Z− Y. jx jx
(3)
The proofs of the above lemmas are given in Appendix. 3. A necessary and sufficient controllability condition Below Rec(V ) stands for the maximal subspace contained in the convex cone V ⊂ Rn , Lin(Z) is the minimal linear subspace of Rn that contains the set Z, and Inv(Z) is the minimal linear subspace of Rn that contains the set Z and is invariant with respect to all matrices Ai , i = 1, . . . , N.
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
This means that there exist i 0, and u¯ i ∈ U¯ i with u¯ i =1, i = 1, . . . , N, such that
Introduce successively the subspaces N L1 = Inv Rec co , Ui
L2 = Inv Rec co
i=1 N
N
Ui ∪ L 1
Lk+1 = Inv Rec co
N
,
i = 1.
(8)
i=1
The definition of (Ui ) implies existence of elements li ∈ Lm , i = 1, . . . , N, such that
Ui ∪ L k
N
i u¯ i = 0,
i=1
i=1
............
665
ui := u¯ i + li ∈ Ui ,
,
Since u¯ i = (ui ) and u¯ i = 1, it follows that ui ∈ Ui \Lm . Taking into account (8), we obtain that
i=1
............ Theorem 9. The switching control system (1) is STC if and only if Lm = Rn
i = 1, . . . , N.
(4)
for some m n. Remark 10. If the sets Ui are subspaces, then Lm = L1 for every m. In this case the claim of the theorem reduces to the main result in Sun et al. (2002). Proof of Theorem 9. Sufficiency: According to Remark 2, it is enough to prove that (1) is STLC. According to Lemmas 6 and 7 we obtain that co( N i=1 Ui ) is a subset of E + . Then applying Lemma 8, we obtain successively that every subspace Li , i = 1, . . . , m, is a subset of E + . Condition (4) implies existence of a finite number of elements w1 , . . . , ws of Lm such that co{w1 , . . . , ws } contains a neighborhood of the origin in Rn . Hence inclusion (2) is fulfilled for Zi (x) = wi , i = 1, . . . , s. Applying Lemma 5, we complete the proof of the sufficiency.
N
i ui =
i=1
N
i (u¯ i + li ) =
i=1
N
i li ∈ Lm .
(9)
i=1
Then for some index j for which j > 0 we obtain the existence of an element l ∈ Lm such that 1 −uj = i ui − l . j i=j
Since uj ∈ Uj \Lm , we obtain that uj ∈ Lm+1 \Lm , which contradicts (5). Hence, N 0∈ / co (U¯ i ∩ S) . (10) i=1
Then we can find a nonvanishing vector h ∈ L¯ and a positive real > 0 such that for every u¯ i ∈ U¯ i ∩ S, i = 1, . . . , N, the following relation holds: h, u¯ i .
(11)
Necessity: If (4) is not fulfilled, then there exists m < n such that
The last relation and the linearity of the considered control systems (see Sussmann, 1978, Theorem 1) imply that
Lm = Lm+1 = Rn .
h, y 0
(5)
¯ T ), for every y ∈ A(0,
(12)
Let L¯ be the orthogonal complement of Lm to Rn , and let : ¯ Since Lm is R n → L¯ be the orthogonal projection onto L. invariant with respect to all of the matrices Ai , i =1, . . . , N, we can define correctly the linear operators A¯ i on L¯ by the relations
where T > 0 is sufficiently small. So, system (6) is not STLC. On the other hand, if x(·) satisfies the relation
A¯ i (x) = (Ai x),
on some interval [t1 , t2 ], then x(t) ¯ = (x(t)) satisfies the relation
x ∈ R n , i = 1, . . . , N,
and the control systems ˙¯ ∈ A¯ i x(t) ¯ + U¯ i , x(t)
˙¯ ∈ A¯ i x(t) x(t) ¯ + U¯ i (6)
¯ T) where U¯ i = (Ui ), i = 1, . . . , N. We denote by A(0, the reachable set of the switching system (6) at time T > 0 starting from the origin. ¯ Let us assume that Let S be the unit sphere in L. N 0 ∈ co (U¯ i ∩ S) . (7) i=1
x(t) ˙ ∈ Ai x(t) + Ui
(13)
on the same interval. This implies that if x(·) is a trajectory of the control system (1), then x(t) ¯ := (x(t)) is a trajectory ¯ T ), and of (6) defined on [t1 , t2 ]. Hence (A(0, T )) ⊆ A(0, since system (6) is not STLC, we obtain that system (1) is also not STLC. This completes the proof of the theorem. Remark 11. One can easily verify that condition (4) is necessary and sufficient for STC also in the case of the relaxed
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M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
definition of a trajectory of the switching system, where a trajectory is any solution of the differential inclusion x˙ ∈ co {Ai x + Ui : i = 1, . . . , N}. Example 12. To illustrate the proposed approach we consider the following simple switching system , determined by the linear control systems 1 and 2 : ( 1 )
x˙ = u, y˙ = px + v, z˙ = x + v,
(2 )
x˙ = −x + w, y˙ = −y + w, v 0, z˙ = −z, w 0,
where p is a parameter. According to Bianchini (1983) or Veliov (1988), each of the systems 1 and 2 is not STC. We have L1 = {(, p, ) : ∈ R, ∈ R}, hence, dim L1 = 2. Moreover, Rec(co(U1 ∪ U2 ∪ L1 )) = Rec(co((0, 1, 1), (1, 1, 0), L1 )). Since = (0, −1, p) is an orthogonal vector to L1 , it holds that L2 = L1 if and only if , (0, 1, 1). , (1, 1, 0) < 0, which holds if and only if (−1 + p).(−1) < 0, i.e. p > 1. Thus, according to Theorem 9, the switching system is STC if and only if p > 1. Notice that the construction of the spaces Li and Lemmas 6–8 give qualitative information of how to reach a given state from x = 0. To reach the points from L1 it is enough to move only along a trajectory of 1 resulting from v =0 and a piecewise constant u with a single jump point. To reach a point from L2 \L1 one could apply first to 1 appropriate controls u and v as above, then either a constant control w to 2 , or a constant control v to 1 , depending on which side of L1 is the given target point from. Given the end time and the target, corresponding switching times and control values can easily be found explicitly (cf. the proof of Lemma 5).
4. Conclusions The necessary and sufficient condition for small time controllability of linear switching systems proved in this paper extends that from Sun et al. (2002) to the case of control which is constrained in a cone. The proof is based on a suitable differential-geometric technique which can also be applied for analysis of the controllability for nonlinear switching systems. The proposed approach is constructive. The proofs of Lemmas 5–8 show how to determine the switching law and how to design the control input for the purpose of controllability. The verification of the condition, however, involves determination of the recessive subspace of a convex hull, which goes beyond the pure linear-algebraic considerations typical for the case of unconstrained control.
Appendix Our proofs are based on the formula of Campbell–Baker– Hausdorff (C–B–H): if X and Y are analytic vector fields on Rn , then Exp(t1 X) ◦ Exp(t2 Y )(x)
t1 t2 = Exp t1 X + t2 Y + [X, Y ] 2
t1 t22 t12 t2 + [Y, [Y, X]] + [X, [X, Y ]] + · · · (x), (14) 12 12
where ◦ means superposition, and the infinite sum on the right-hand side is convergent for sufficiently small |t1 | and |t2 |. Proof of Lemma 5. According to Definition 4 there exist T > 0, positive rational polynomials pi , families of vector fields ai (t) ∈ A0 , and oi (t), i = 1, . . . , k, such that for every x ∈ Bn and for every t ∈ [0, T ] Exp(tZ i + ai (t) + oi (t))(x) ∈ A(x, pi (t)),
(15)
and the following estimates hold: ai (t, x) Ci t i x and oi (t, x) ci t i , with i > 1, i = 1, . . . , k. Then R(t1 , . . . , tk , x) := Exp(tZ 1 + a1 (t) + o1 (t)) ◦ · · · ◦ Exp(tZ k + ak (t) + ok (t))(x) k ∈ A x, pi (t) . (16) i=1
Let tˆ ∈ Tk := {(t1 , . . . , tk )|ti ∈ [0, T ]}. Applying the formula of C–B–H we obtain that there exist > 0, > 1, C > 0 and c > 0 such that R(tˆ, x) = R(t1 , . . . , tk , x) k := Exp ti Zi + a(t1 , . . . , tk ) i=1
+o(t1 , . . . , tk )) (x), where for each s 0 the following inequalities hold true: max
tˆ∈ Tk , ∈Bn
a(st 1 , . . . , st k , x) Cs x,
whenever xBn , and max
tˆ∈ Tk , x∈Bn
o(st 1 , . . . , st k , x) cs .
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
These estimates imply that a(t1 , . . . , tk , 0) = 0, and R(t1 , . . . , tk , 0) =
k
ti Zi (0) + (t1 , . . . , tk )
(17)
i=1
with ˆ max (st 1 , . . . , st k ) cs
ˆ
tˆ∈ Tk
ˆ > 1. for some cˆ > 0 and some On the other hand, relation (2) implies the existence of a neighborhood of the origin such that
⊆ co {Z1 (0), . . . , Zk (0)}. The last inclusion, (16), and (17), imply the claim of the lemma by virtue of Lemma 4 in Sussmann (1978). Proof of Lemma 6. It is straightforward to prove Lemma 6 by applying the C–B–H formula. Proof of Lemma 7. Let ui ∈ Ui , and t > 0. We set P (x) = Ai x and Q(x) = ui . Then Exp(tP + tQ)(x) ∈ A(x, t). Taking into account that Ai x Ai x we obtain that tP ∈ A0 , and hence ui ∈ E + according to Definition 4. Proof of Lemma 8. According to Definition 4 there exist positive numbers ± and ± > 1, positive rational polynomials p± , two families of vector fields a ± (t) ∈ A0 and o± (t), such that Exp(±tZ + a ± (t) + o± (t))(x) ∈ A(x, p ± (t)),
(18)
where for x ∈ Bn and for every sufficiently small t > 0 ± a ± (t, x) C ± t x,
and ±
o± (t, x) c± t . Let us choose a rational number b ∈ (0, 1) satisfying the inequality: b min{+ , − } > 1.
(19)
By setting t := b and substituting in (18), we obtain that S(x, ) := Exp(b Z + a + (b ) + o+ (b )) ◦ Exp(1−b Y ) ◦ Exp(−b Z + a − (b ) + o− (b ))(x) ∈ A(x, p+ (b ) + 1−b + p − (b )).
(20)
667
We apply the C–B–H formula two times and obtain that ˆ ) ˆ ) + o( S(x, ) = Exp b Z + 1−b Y + [Z, Y ] + a( 2 ◦ Exp(−b Z + a − (b ) + o− (b ))(x) = Exp([Z, Y ] + 1−b Y + a ∗ () + o∗ ())(x). (21) In the above equalities a( ˆ ) is the sum of a + (b ) and all Lie brackets appearing in the C–B–H formula that involve a + (b ) and Y. Notice that 1−b Y ∈ A0 (since Y (0) = 0 and 1 − b > 0) and that [a1 (), a2 ()] ∈ A0 for every elements a1 () and a2 () of A0 . Thus a( ˆ ) ∈ A0 . By o( ˆ ) we have denoted the sum of all remainder Lie brackets appearing in the C–B–H formula, i.e. o( ˆ ) denotes the sum of o+ (b ) and all Lie brackets such that each of them contains at last once o+ (b ), or at last once Z and Y. So, o( ˆ )/˜ + + ˜ := min{b , 1 + b , 1 + b, 2 − b}. is bounded, where Similarly, a ∗ () ∈ A0 and o∗ ()/ is bounded, where = min{b± , 1 + b± , 1 + b, 2 − b}. Then from the definition of E + and (19) we conclude that [Z, Y ] ∈ E + . Changing the places of Z and −Z, we obtain also −[Z, Y ] ∈ E + . References Bianchini, R. M. (1983). Instant controllability of linear autonomous systems. Journal of Optimization Theory and Application, 39, 237–250. Hermes, H. (1978). Lie algebras of vector fields and local approximation of attainable sets. SIAM Journal on Control and Optimization, 16, 715–727. Korobov, V. (1980). A geometrical criterion for local controllability of dynamic systems with restrictions on controls. Differential Equations, 15, 1136–1142. Li, Z. G., Wen, C. Y., & Soh, Y. C. (2001). Switched controllers and their applications in bilinear systems. Automatica, 37(3), 477–481. Narendra, K. S., & Balakrishnan, J. (1997). Adaptive control using multiple models. IEEE Transactions on Automatic Control, 42, 171–187. Sun, Z., Ge, S., & Lee, T. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38, 775–786. Sun, Z., & Zheng, D. (2001). On reachability and stabilization of switched linear control systems. IEEE Transactions on Automatic Control, 46, 291–295. Sussmann, H. (1978). A sufficient condition for local controllability. SIAM Journal on Control and Optimization, 16, 790–802. Sussmann, H. (1987a). A general theorem on local controllability. SIAM Journal on Control and Optimization, 25, 158–194. Sussmann, H. (1987b). Small-time local controllability and continuity of the optimal time function for linear systems. Journal of Optimization Theory and Application, 53, 281–296. Veliov, V. M. (1988). On the controllability of control constrained linear systems. Mathematica Balkanica New Series, 2(2/3), 147–155. Veliov, V. M., & Krastanov, M. I. (1986). Controllability of piecewise linear systems. Systems & Control Letters, 7, 335–341. Williams, S. M., & Hoft, R. G. (1991). Adaptive frequency domain control of ppm switched power line conditioneer. IEEE Transactions on Power Electron, 6, 665–670. Xie, G., Zheng, D., & Wang, L. (2002). Controllability of switched linear systems. IEEE Transactions on Automatic Control, 47(8), 1401–1405. Zhivoglyadov, P., & Middleton, R. (2003). Networked control design for linear systems. Automatica, 39(4), 743–750.
668
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668 Mikhail Krastanov received the M.Sc. and Ph.D. degrees in mathematics from the University of Sofia, Bulgaria, in 1983 and 1989, respectively. He is currently an associate professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. His areas of interests are nonsmooth and set-valued analysis, geometrical methods in control theory, mathematical economics.
Vladimir Veliov received the M.Sc. and Ph.D. degrees in mathematics from the University of Sofia, Bulgaria, in 1978 and 1982, respectively. He is currently an associate professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, and a research fellow in the Vienna University of Technology. His areas of interest are non-smooth and set-valued analysis, optimal control theory, structured control systems, mathematical economics.
Brief paper
On the controllability of switching linear systems夡 Mikhail I. Krastanova , Vladimir M. Veliova, b,∗ a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.Bonchev str. 8, 1113 Sofia, Bulgaria b Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstr. 8, A-1040 Vienna, Austria
Received 3 November 2003; received in revised form 13 August 2004; accepted 25 October 2004 Available online 22 December 2004
Abstract This note presents a necessary and sufficient condition for small time controllability of a linear switching system (that is, a collection of linear time-invariant control systems, where a trajectory is any concatenation of trajectories of the individual systems). This result extends the controllability condition recently obtained for unconstrained linear switching systems to the case of control which is constrained in a cone. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Controllability; Switching control systems; Control constraints
such that for almost every t ∈ [tk−1 , tk ) the following relation holds true:
1. Introduction We consider a family of linear control systems (called a switching system) x˙ ∈ Ai x + Ui ,
i = 1, . . . , N,
(1)
where x ∈ Rn , Ai are (n × n)-matrices, Ui ⊂ Rn are closed convex cones in Rn . By definition, a trajectory of the switching system (1) on [0, T ] is any absolutely continuous function x : [0, T ] → Rn for which there exist a finite number of positive integers i1 , i2 , . . . , ip ∈ {1, . . . , N}, points 0 = t0 < t1 . . . < tp = T and integrable functions uk : [tk−1 , tk ) → Uik , k =1, . . . , p, 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. Colaneri under the direction of Editor R. Tempo. This research was partially supported by the Austrian National Bank under contract 9414 and by the Ministry of Science and Higher Education—National Fund for Science Research under Contract MM-1104/01. ∗ Corresponding author. Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstr. 8, A-1040 Vienna, Austria. E-mail addresses: [email protected] (M.I. Krastanov), [email protected] (V.M. Veliov).
0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.10.017
x(t) ˙ = Aik x(t) + uik (t). The investigation of switching control systems is motivated by numerous applications in power systems, electronics, economics, mechanics, etc., as well as for control techniques based on switching between different controllers (see, e.g., Williams & Hoft, 1991; Narendra & Balakrishnan, 1997; Zhivoglyadov & Middleton, 2003). Utilization of switching control systems gives the possibility for intelligent control whereby the controller can operate in multiple environments. Control constraints arise in such systems mainly due to non-negativity requirements for some of the controls. We refer to the recent papers by Li, Wen, and Soh (2001) and Sun, Ge, and Lee (2002) for an extended bibliography. Our concern in the present note is the controllability issue for (1). Denote by A(x, T ) the attainable set of system (1) at time T, starting from the point x ∈ Rn , i.e. A(x, t) = {y ∈ Rn : there exists a trajectory x(·) of (1) such that x(0) = x, x(t) = y}. Definition 1. The switching system (1) is called small-time controllable (STC) if A(0, T ) = Rn for every T > 0.
664
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
Remark 2. We mention that due to the linearity of the systems in (1) and due to the cone constraints for the control, A(0, T ) is a closed convex cone. Therefore STC is equivalent to small-time local controllability (STLC), in the definition of which the requirement “A(0, T ) = Rn ” is replaced with “A(0, T ) contains the origin in its interior”. Remark 3. In the unconstrained case, that is, if Ui are subspaces, STC is equivalent to the same property, but with “for every T > 0” replaced with “for some T > 0”. In the case of control constraints this equivalence between STC and controllability on an arbitrary time horizon is no longer true. In the unconstrained control case (that is, if the cones Ui are all subspaces), a Kalman-type necessary controllability condition for switching linear systems is obtained in Sun and Zheng (2001), where it is also proved that the condition is sufficient for n = 3 and N = 2. In Xie, Zheng, and Wang (2002) the result is extended for n = 3 and an arbitrary N, and in Sun et al. (2002)—for arbitrary n and N. In the present note we extend the controllability condition from Sun et al. (2002) to the case of control constraints, that is, for cones Ui which are not necessarily subspaces. We mention that sufficient and necessary controllability conditions for linear systems with controls restricted to a cone have been obtained in Korobov (1980) (for the case of an arbitrary time horizon), Bianchini (1983), Sussmann (1987b), and Veliov (1988). The proof of the sufficiency of the condition for STC of control constrained linear switching systems presented here utilizes the differential geometric approach in a form similar to that developed in Veliov and Krastanov (1986) for piecewise linear systems and in Veliov (1988) for the case of control constrained linear system. Lemmas 5–8 used in the proof are suitable for analysis of the controllability also for nonlinear switching systems.
ci t di ,
i=1
where di are positive rational numbers and ci > 0, i =1, . . . , , is called a positive rational polynomial. Further, we use the notation o(t) to indicate any family of analytic vector fields o(t, x) parameterized by t > 0, continuous in (t, x) and such that for some > 1 the ratio o(t, x)/t is bounded when t tends to zero, uniformly with respect to x ∈ Bn . By A0 we denote the set of all families of analytic vector fields a(t, x) parameterized by t > 0, continuous in (t, x) and such that for some C > 0 and > 0 the following estimate holds: a(t, x) Ct x, whenever t ∈ [0, 1] and x ∈ Bn . Following some of the ideas proposed in Hermes (1978), Veliov (1988) and Veliov and Krastanov (1986), we define the following set E + of analytic vector fields. Definition 4. E + is the set of all analytic vector fields Z for which there exist a(t) ∈ A0 , o(t), and a positive rational polynomial p(t) such that Exp(tZ + a(t) + o(t))(x) ∈ A(x, p(t)). Lemma 5. Let Z1 , . . . , Zk ∈ E + and 0 ∈ int co {Zi (0): i = 1, . . . , k},
(2)
where “int” and “co” mean the interior and the convex hull, respectively. Then the switching control system (1) is STLC. The following lemmas provide constructions of elements of the set E + . Lemma 6. The set E + is a convex cone. Lemma 7. The constant vector fields ui , i = 1, . . . , N, belong to E + for every ui ∈ Ui .
2. Preliminaries In this section we develop some technical tools for controllability analysis that are useful also for nonlinear systems. The lemmas below use only the semi-group property of the reachable set A(x, t): A(A(x, t), ) ⊂ A(x, t + ),
∀x ∈ Rn , ∀t, 0.
Below Bn := {y ∈ Rn : y 1} is the unit ball. Given an analytic vector field Z on Rn , we denote by Exp(tZ)x0 the value of the solution of the equation x( ˙ ) = tZ(x()),
Sussmann (1987a). A function of the type
x(0) = x0 ,
at time = 1. Below we shall use also the notion Exp(Zt )(x0 ) for an arbitrary family of analytic vector fields {Zt : t ∈ R}, depending continuously on t, as defined in
Lemma 8. Let ±Z ∈ E + and let Y (x) = Ai x for some i ∈ {1, . . . , N}. Then ±[Z, Y ] ∈ E + , where the Lie bracket [Z, Y ] is defined as [Z, Y ] :=
jY jZ Z− Y. jx jx
(3)
The proofs of the above lemmas are given in Appendix. 3. A necessary and sufficient controllability condition Below Rec(V ) stands for the maximal subspace contained in the convex cone V ⊂ Rn , Lin(Z) is the minimal linear subspace of Rn that contains the set Z, and Inv(Z) is the minimal linear subspace of Rn that contains the set Z and is invariant with respect to all matrices Ai , i = 1, . . . , N.
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
This means that there exist i 0, and u¯ i ∈ U¯ i with u¯ i =1, i = 1, . . . , N, such that
Introduce successively the subspaces N L1 = Inv Rec co , Ui
L2 = Inv Rec co
i=1 N
N
Ui ∪ L 1
Lk+1 = Inv Rec co
N
,
i = 1.
(8)
i=1
The definition of (Ui ) implies existence of elements li ∈ Lm , i = 1, . . . , N, such that
Ui ∪ L k
N
i u¯ i = 0,
i=1
i=1
............
665
ui := u¯ i + li ∈ Ui ,
,
Since u¯ i = (ui ) and u¯ i = 1, it follows that ui ∈ Ui \Lm . Taking into account (8), we obtain that
i=1
............ Theorem 9. The switching control system (1) is STC if and only if Lm = Rn
i = 1, . . . , N.
(4)
for some m n. Remark 10. If the sets Ui are subspaces, then Lm = L1 for every m. In this case the claim of the theorem reduces to the main result in Sun et al. (2002). Proof of Theorem 9. Sufficiency: According to Remark 2, it is enough to prove that (1) is STLC. According to Lemmas 6 and 7 we obtain that co( N i=1 Ui ) is a subset of E + . Then applying Lemma 8, we obtain successively that every subspace Li , i = 1, . . . , m, is a subset of E + . Condition (4) implies existence of a finite number of elements w1 , . . . , ws of Lm such that co{w1 , . . . , ws } contains a neighborhood of the origin in Rn . Hence inclusion (2) is fulfilled for Zi (x) = wi , i = 1, . . . , s. Applying Lemma 5, we complete the proof of the sufficiency.
N
i ui =
i=1
N
i (u¯ i + li ) =
i=1
N
i li ∈ Lm .
(9)
i=1
Then for some index j for which j > 0 we obtain the existence of an element l ∈ Lm such that 1 −uj = i ui − l . j i=j
Since uj ∈ Uj \Lm , we obtain that uj ∈ Lm+1 \Lm , which contradicts (5). Hence, N 0∈ / co (U¯ i ∩ S) . (10) i=1
Then we can find a nonvanishing vector h ∈ L¯ and a positive real > 0 such that for every u¯ i ∈ U¯ i ∩ S, i = 1, . . . , N, the following relation holds: h, u¯ i .
(11)
Necessity: If (4) is not fulfilled, then there exists m < n such that
The last relation and the linearity of the considered control systems (see Sussmann, 1978, Theorem 1) imply that
Lm = Lm+1 = Rn .
h, y 0
(5)
¯ T ), for every y ∈ A(0,
(12)
Let L¯ be the orthogonal complement of Lm to Rn , and let : ¯ Since Lm is R n → L¯ be the orthogonal projection onto L. invariant with respect to all of the matrices Ai , i =1, . . . , N, we can define correctly the linear operators A¯ i on L¯ by the relations
where T > 0 is sufficiently small. So, system (6) is not STLC. On the other hand, if x(·) satisfies the relation
A¯ i (x) = (Ai x),
on some interval [t1 , t2 ], then x(t) ¯ = (x(t)) satisfies the relation
x ∈ R n , i = 1, . . . , N,
and the control systems ˙¯ ∈ A¯ i x(t) ¯ + U¯ i , x(t)
˙¯ ∈ A¯ i x(t) x(t) ¯ + U¯ i (6)
¯ T) where U¯ i = (Ui ), i = 1, . . . , N. We denote by A(0, the reachable set of the switching system (6) at time T > 0 starting from the origin. ¯ Let us assume that Let S be the unit sphere in L. N 0 ∈ co (U¯ i ∩ S) . (7) i=1
x(t) ˙ ∈ Ai x(t) + Ui
(13)
on the same interval. This implies that if x(·) is a trajectory of the control system (1), then x(t) ¯ := (x(t)) is a trajectory ¯ T ), and of (6) defined on [t1 , t2 ]. Hence (A(0, T )) ⊆ A(0, since system (6) is not STLC, we obtain that system (1) is also not STLC. This completes the proof of the theorem. Remark 11. One can easily verify that condition (4) is necessary and sufficient for STC also in the case of the relaxed
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M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
definition of a trajectory of the switching system, where a trajectory is any solution of the differential inclusion x˙ ∈ co {Ai x + Ui : i = 1, . . . , N}. Example 12. To illustrate the proposed approach we consider the following simple switching system , determined by the linear control systems 1 and 2 : ( 1 )
x˙ = u, y˙ = px + v, z˙ = x + v,
(2 )
x˙ = −x + w, y˙ = −y + w, v 0, z˙ = −z, w 0,
where p is a parameter. According to Bianchini (1983) or Veliov (1988), each of the systems 1 and 2 is not STC. We have L1 = {(, p, ) : ∈ R, ∈ R}, hence, dim L1 = 2. Moreover, Rec(co(U1 ∪ U2 ∪ L1 )) = Rec(co((0, 1, 1), (1, 1, 0), L1 )). Since = (0, −1, p) is an orthogonal vector to L1 , it holds that L2 = L1 if and only if , (0, 1, 1). , (1, 1, 0) < 0, which holds if and only if (−1 + p).(−1) < 0, i.e. p > 1. Thus, according to Theorem 9, the switching system is STC if and only if p > 1. Notice that the construction of the spaces Li and Lemmas 6–8 give qualitative information of how to reach a given state from x = 0. To reach the points from L1 it is enough to move only along a trajectory of 1 resulting from v =0 and a piecewise constant u with a single jump point. To reach a point from L2 \L1 one could apply first to 1 appropriate controls u and v as above, then either a constant control w to 2 , or a constant control v to 1 , depending on which side of L1 is the given target point from. Given the end time and the target, corresponding switching times and control values can easily be found explicitly (cf. the proof of Lemma 5).
4. Conclusions The necessary and sufficient condition for small time controllability of linear switching systems proved in this paper extends that from Sun et al. (2002) to the case of control which is constrained in a cone. The proof is based on a suitable differential-geometric technique which can also be applied for analysis of the controllability for nonlinear switching systems. The proposed approach is constructive. The proofs of Lemmas 5–8 show how to determine the switching law and how to design the control input for the purpose of controllability. The verification of the condition, however, involves determination of the recessive subspace of a convex hull, which goes beyond the pure linear-algebraic considerations typical for the case of unconstrained control.
Appendix Our proofs are based on the formula of Campbell–Baker– Hausdorff (C–B–H): if X and Y are analytic vector fields on Rn , then Exp(t1 X) ◦ Exp(t2 Y )(x)
t1 t2 = Exp t1 X + t2 Y + [X, Y ] 2
t1 t22 t12 t2 + [Y, [Y, X]] + [X, [X, Y ]] + · · · (x), (14) 12 12
where ◦ means superposition, and the infinite sum on the right-hand side is convergent for sufficiently small |t1 | and |t2 |. Proof of Lemma 5. According to Definition 4 there exist T > 0, positive rational polynomials pi , families of vector fields ai (t) ∈ A0 , and oi (t), i = 1, . . . , k, such that for every x ∈ Bn and for every t ∈ [0, T ] Exp(tZ i + ai (t) + oi (t))(x) ∈ A(x, pi (t)),
(15)
and the following estimates hold: ai (t, x) Ci t i x and oi (t, x) ci t i , with i > 1, i = 1, . . . , k. Then R(t1 , . . . , tk , x) := Exp(tZ 1 + a1 (t) + o1 (t)) ◦ · · · ◦ Exp(tZ k + ak (t) + ok (t))(x) k ∈ A x, pi (t) . (16) i=1
Let tˆ ∈ Tk := {(t1 , . . . , tk )|ti ∈ [0, T ]}. Applying the formula of C–B–H we obtain that there exist > 0, > 1, C > 0 and c > 0 such that R(tˆ, x) = R(t1 , . . . , tk , x) k := Exp ti Zi + a(t1 , . . . , tk ) i=1
+o(t1 , . . . , tk )) (x), where for each s 0 the following inequalities hold true: max
tˆ∈ Tk , ∈Bn
a(st 1 , . . . , st k , x) Cs x,
whenever xBn , and max
tˆ∈ Tk , x∈Bn
o(st 1 , . . . , st k , x) cs .
M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668
These estimates imply that a(t1 , . . . , tk , 0) = 0, and R(t1 , . . . , tk , 0) =
k
ti Zi (0) + (t1 , . . . , tk )
(17)
i=1
with ˆ max (st 1 , . . . , st k ) cs
ˆ
tˆ∈ Tk
ˆ > 1. for some cˆ > 0 and some On the other hand, relation (2) implies the existence of a neighborhood of the origin such that
⊆ co {Z1 (0), . . . , Zk (0)}. The last inclusion, (16), and (17), imply the claim of the lemma by virtue of Lemma 4 in Sussmann (1978). Proof of Lemma 6. It is straightforward to prove Lemma 6 by applying the C–B–H formula. Proof of Lemma 7. Let ui ∈ Ui , and t > 0. We set P (x) = Ai x and Q(x) = ui . Then Exp(tP + tQ)(x) ∈ A(x, t). Taking into account that Ai x Ai x we obtain that tP ∈ A0 , and hence ui ∈ E + according to Definition 4. Proof of Lemma 8. According to Definition 4 there exist positive numbers ± and ± > 1, positive rational polynomials p± , two families of vector fields a ± (t) ∈ A0 and o± (t), such that Exp(±tZ + a ± (t) + o± (t))(x) ∈ A(x, p ± (t)),
(18)
where for x ∈ Bn and for every sufficiently small t > 0 ± a ± (t, x) C ± t x,
and ±
o± (t, x) c± t . Let us choose a rational number b ∈ (0, 1) satisfying the inequality: b min{+ , − } > 1.
(19)
By setting t := b and substituting in (18), we obtain that S(x, ) := Exp(b Z + a + (b ) + o+ (b )) ◦ Exp(1−b Y ) ◦ Exp(−b Z + a − (b ) + o− (b ))(x) ∈ A(x, p+ (b ) + 1−b + p − (b )).
(20)
667
We apply the C–B–H formula two times and obtain that ˆ ) ˆ ) + o( S(x, ) = Exp b Z + 1−b Y + [Z, Y ] + a( 2 ◦ Exp(−b Z + a − (b ) + o− (b ))(x) = Exp([Z, Y ] + 1−b Y + a ∗ () + o∗ ())(x). (21) In the above equalities a( ˆ ) is the sum of a + (b ) and all Lie brackets appearing in the C–B–H formula that involve a + (b ) and Y. Notice that 1−b Y ∈ A0 (since Y (0) = 0 and 1 − b > 0) and that [a1 (), a2 ()] ∈ A0 for every elements a1 () and a2 () of A0 . Thus a( ˆ ) ∈ A0 . By o( ˆ ) we have denoted the sum of all remainder Lie brackets appearing in the C–B–H formula, i.e. o( ˆ ) denotes the sum of o+ (b ) and all Lie brackets such that each of them contains at last once o+ (b ), or at last once Z and Y. So, o( ˆ )/˜ + + ˜ := min{b , 1 + b , 1 + b, 2 − b}. is bounded, where Similarly, a ∗ () ∈ A0 and o∗ ()/ is bounded, where = min{b± , 1 + b± , 1 + b, 2 − b}. Then from the definition of E + and (19) we conclude that [Z, Y ] ∈ E + . Changing the places of Z and −Z, we obtain also −[Z, Y ] ∈ E + . References Bianchini, R. M. (1983). Instant controllability of linear autonomous systems. Journal of Optimization Theory and Application, 39, 237–250. Hermes, H. (1978). Lie algebras of vector fields and local approximation of attainable sets. SIAM Journal on Control and Optimization, 16, 715–727. Korobov, V. (1980). A geometrical criterion for local controllability of dynamic systems with restrictions on controls. Differential Equations, 15, 1136–1142. Li, Z. G., Wen, C. Y., & Soh, Y. C. (2001). Switched controllers and their applications in bilinear systems. Automatica, 37(3), 477–481. Narendra, K. S., & Balakrishnan, J. (1997). Adaptive control using multiple models. IEEE Transactions on Automatic Control, 42, 171–187. Sun, Z., Ge, S., & Lee, T. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38, 775–786. Sun, Z., & Zheng, D. (2001). On reachability and stabilization of switched linear control systems. IEEE Transactions on Automatic Control, 46, 291–295. Sussmann, H. (1978). A sufficient condition for local controllability. SIAM Journal on Control and Optimization, 16, 790–802. Sussmann, H. (1987a). A general theorem on local controllability. SIAM Journal on Control and Optimization, 25, 158–194. Sussmann, H. (1987b). Small-time local controllability and continuity of the optimal time function for linear systems. Journal of Optimization Theory and Application, 53, 281–296. Veliov, V. M. (1988). On the controllability of control constrained linear systems. Mathematica Balkanica New Series, 2(2/3), 147–155. Veliov, V. M., & Krastanov, M. I. (1986). Controllability of piecewise linear systems. Systems & Control Letters, 7, 335–341. Williams, S. M., & Hoft, R. G. (1991). Adaptive frequency domain control of ppm switched power line conditioneer. IEEE Transactions on Power Electron, 6, 665–670. Xie, G., Zheng, D., & Wang, L. (2002). Controllability of switched linear systems. IEEE Transactions on Automatic Control, 47(8), 1401–1405. Zhivoglyadov, P., & Middleton, R. (2003). Networked control design for linear systems. Automatica, 39(4), 743–750.
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M.I. Krastanov, V.M. Veliov / Automatica 41 (2005) 663 – 668 Mikhail Krastanov received the M.Sc. and Ph.D. degrees in mathematics from the University of Sofia, Bulgaria, in 1983 and 1989, respectively. He is currently an associate professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. His areas of interests are nonsmooth and set-valued analysis, geometrical methods in control theory, mathematical economics.
Vladimir Veliov received the M.Sc. and Ph.D. degrees in mathematics from the University of Sofia, Bulgaria, in 1978 and 1982, respectively. He is currently an associate professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, and a research fellow in the Vienna University of Technology. His areas of interest are non-smooth and set-valued analysis, optimal control theory, structured control systems, mathematical economics.