Analysis of Discrete Time Linear Switched Systems: A Variational Approach

Transcription

1 Analysis of Discrete Time Linear Switched Systems: A Variational Approach 1 Tal Monovich and Michael Margaliot Abstract A powerful approach for analyzing the stability of continuous time switched systems is based on using tools from optimal control theory to characterize the most unstable switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific most unstable switching law. More generally, this so called variational approach was successfully applied to derive nice reachability type results for both linear and nonlinear continuous time switched systems. Motivated by this, we develop in this paper an analogous approach for discrete time linear switched systems. We derive and prove a necessary condition for optimality of the most unstable switching law. This yields a type of discrete time maximum principle (MP). We demonstrate using an example that this MP is in fact weaker than its continuous time counterpart. To overcome this, we introduce the auxiliary system of a discrete time linear switched system, and show that regularity properties of time optimal controls for the auxiliary system imply nice reachability results for the original discrete time linear switched system. Using this approach, we derive several new Lie algebraic conditions guaranteeing nice reachability results. These results, and their proofs, turn out to be quite different from their continuous time counterparts. Keywords: Stability analysis; variational approach; nice reachability; Lie algebraic conditions. Corresponding author: Dr. Michael Margaliot, School of Elec. Eng. Systems, Tel Aviv University, Israel Tel: ; Homepage: michaelm michaelm@eng.tau.ac.il

2 2 I. INTRODUCTION Let N 0 = {0, 1, 2,... } denote the set of natural numbers. Consider the discrete time (DT) linear switched system x(k + 1) = A σ(k) x(k), (1) x(0) = x 0, where x : N 0 R n, and σ : N 0 {0, 1,..., m 1} is the switching law. This models a system that can switch between the m linear subsystems: x(k + 1) = A i x(k), i {0,..., m 1}, with the switching law determining which system is active at each time step. For example, for m = 2 and the switching law 0, k {0,..., k 1 1}, 1, k {k 1,..., k 1 + l 1 1}, σ(k) = 0, k {k 1 + l 1,..., k 1 + l 1 + k 2 1},. the corresponding solution of (1) is of the form... A k 2 0 Al 1 1 A k 1 0 x 0. (2) Switched systems and in particular their stability analysis are recently attracting considerable interest (see, e.g. [27, 45, 11, 23, 44, 15]). Definition 1 We say that the linear switched system (1) is globally uniformly asymptotically stable (GUAS) if lim k x(k) = 0 for any x 0 R n and any switching law σ. When n = 1, A i = a i is a scalar and determining whether the system is GUAS or not is trivial. For example, when n = 1 and m = 2, (2) becomes P P ki li a0 a 1 x 0,

3 3 so (1) is GUAS if and only if a i < 1 for any i. When n > 1, it is clear that a necessary condition for GUAS is that every A i is an asymptotically stable matrix (i.e. its spectral radius is smaller than one), yet this is not a sufficient condition and establishing GUAS is a notoriously difficult problem. The GUAS property is equivalent to ρ({a 0, A 1,..., A m 1 }) < 1, where ρ(σ) denotes the joint spectral radius of the set of matrices Σ. Several hardness results show that computing (or even approximating) the joint spectral radius is extremely hard (see e.g. [23, Chapter 2]). For example, the problem of determining whether ρ({a 0, A 1 }) 1, with A 0, A 1 rational matrices, is Turing undecidable [9]. Other related problems, such as deciding whether a set of matrices is mortal or not, are also known to be extremely hard (see e.g. [10]). In fact, even the case n = m = 2 seems to be highly non trivial. For example, Blondel et al. [8] proved that there exist α, β R such that the switched system (1) with A 0 = α 1 1 and A 1 = β satisfies the following property: for any periodic switching law x(k) 0, yet the switched system is not GUAS (see also [24, 25]). In this paper, we develop a new approach to the analysis of DT linear switched systems which is motivated by the success of the so called variational approach in the analysis of continuous time (CT) switched systems. This allows us to derive new nice reachability type results (see, e.g. [49, 17]) for (1) which are based on Lie algebraic conditions. To explain the notion of nice reachability in our context, let e i m rewrite (1) as the bilinear control system x(k + 1) = (A 0 + denote the ith column of the m m identity matrix, and m 1 i=1 u i (k)b i )x(k), u U, x(0) = x 0, (3) where B i = A i A 0, and U = {0, e 1 m 1, e2 m 1,..., em 1 }. Note that for u(k) = 0 [u(k) = e i m 1], (3) becomes x(k + 1) = A 0 x(k) [x(k + 1) = A i x(k)], so (3) and (1) are equivalent. For a control u U, let x(n; u, x 0 ) denote the solution of (3) corresponding to u at time N.

4 4 The reachable set of (3) at time N corresponding to a subset of controls W U is R(N; W, x 0 ) = {x(n; u, x 0 ) : u W}, i.e. the set of points that can be reached at time N using controls from W. We say that (3) satisfies a nice reachability property if there exists a set of regular controls W U such that for any N N 0 and any x 0 R n, R(N; U, x 0 ) = R(N; W, x 0 ). Roughly speaking, this implies that anything that can be done using a control u U can also be done using a regular control w W. For example, a famous nice reachability result for continuous time linear systems is the bang bang principle stating that any point that can be reached using a measurable control can also be reached using a bang bang control (see e.g. [50]) (see also [2] for a kind of a bang bang principle for DT linear control systems). Nice reachability results have many theoretical and practical implications. Indeed, any point to point control problem (e.g., motion planning, finding optimal controls) over the set U can be reduced to the problem of finding a suitable control from the set of nice controls W. This may assist in the design of efficient numerical algorithms for solving optimal control and motion planning problems [26, 51, 13]. In our context, we are primarily interested in the case where W is the set of all controls satisfying a bound on the number of switches, where the bound is uniform over any final time N. This type of nice reachability results, combined with a necessary condition for GUAS, immediately implies GUAS. The next example demonstrates a trivial nice reachability property and its application to stability analysis. We denote the Lie commutator of the matrices A 0 and A 1 by [A 0, A 1 ] = A 1 A 0 A 0 A 1. We say that u has a switching point at time k 1 if u(k) u(k 1). Example 1 Suppose that m = 2. Note that in this case U is the set of scalar controls taking values in {0, 1}. Assume that [A 0, A 1 ] = 0. In this case, A 0 and A 1 commute, so (2) becomes A P P ki li 0 A 1 x 0. This implies that for any N N 0 and any x 0 R n R(N; U, x 0 ) = R(N; U 1, x 0 ), (4) where U 1 U denotes the set of controls with no more than one switching point.

5 5 Now suppose that, in addition, A 0 and A 1 are asymptotically stable matrices. As the final time N goes to infinity, at least one of k i, l i goes to infinity, so A P P ki li 0 A 1 0, and hence (1) is GUAS. For the case of continuous time (CT) switched systems, a powerful approach for stability analysis is based on optimal control techniques. This so called variational approach was pioneered by E. S. Pyatnitsky [39, 40] in the context of the celebrated absolute stability problem. The variational approach seeks to characterize the most destabilizing switching law σ. If the solution corresponding to σ converges to the origin, the switched system is GUAS. This reduces the problem of analyzing stability for any switching law to analyzing stability for the particular switching law σ. Furthermore, the variational approach can be used to derive nice reachability results which imply GUAS. These are based on analyzing time optimal controls of the so called auxiliary system. In this paper, we extend the variational approach to the case of DT linear switched systems. We begin by deriving a maximum principle (MP) for the problem characterizing the most destabilizing switching law. We provide a simple and self contained proof that requires no convexity assumptions. A similar MP has already been derived by several authors [38, 37, 3]. However, for DT switched systems, applying this MP did not yield powerful results as in the CT case. We provide a possible explanation of this by demonstrating that the discrete time MP is weaker than its CT counterpart. To overcome this, we introduce the analog of the auxiliary system for our DT case. We show that regularity of time optimal controls of the auxiliary system immediately implies nice reachability results for the DT switched system. Using this approach we derive several new Lie algebraic type conditions guaranteeing nice reachability properties for (1), and describe their implications for stability analysis. Some related work includes the pioneering paper of Gurvits [18] that provides a nice reachability type result for CT switched systems with a second order nilpotent Lie algebra, and the paper by Liberzon et al. [28] relating solvability of the Lie algebra to GUAS of the switched system. Kozyakin [24, 25] provides a counter example to the famous finiteness conjecture that includes an interesting and deep analysis of the most unstable switching law. Another related direction is the study of optimal control problems for DT linear switched systems using dynamic programming [52].

6 6 The rest of this paper is organized as follows. Section II briefly reviews the analysis of CT switched systems using the variational approach. We then turn to develop an analog of the variational approach for the case of DT linear switched systems. Section III states a maximum principle (MP) for the most destabilizing control of (3). Section IV introduces the auxiliary system of (1) and shows how it can be used to derive nice reachability results for the original switched system (1). Section V describes several new nice reachability results. These are based on Lie algebraic conditions that imply regularity of time optimal controls for the auxiliary system. These results, and their proofs, turn out to be quite different from those known in the CT case. The final section concludes and describes some possible directions for further research. II. THE VARIATIONAL APPROACH The results in this paper are based on an extension of the variational approach, which proved to be quite successful in the analysis of CT switched systems, to the DT case. We thus begin by a short review of the variational approach in the CT setting. For the sake of simplicity, we consider the case of switching between two linear subsystems. More details can be found in the survey paper [29]. Consider the CT linear switched system: ẋ(t) = A σ(t) x(t), x(0) = x 0, (5) where x : R + R n is the state vector, σ : R + {0, 1} is a piecewise constant function referred to as the switching signal, and A 0, A 1 R n n. There are many variations on the admissible switching laws. For example, some authors restrict the allowed laws to piecewise constant functions with a finite number of switches on any finite time interval (to avoid the possibility of Zeno type behavior). For an in depth treatment of this topic, see [21]. We say that (5) is globally uniformly asymptotically stable (GUAS) if lim t x(t) = 0 for any x 0 R n and any switching law σ. A powerful approach for addressing the GUAS problem is based on variational principles. This approach seeks to characterize the most unstable switching law (MUSL). If the solution

7 7 of (5) corresponding to the MUSL converges to the origin, the system is GUAS. This reduces the problem of analyzing all possible switching laws to the analysis of the system under a specific switching law, the MUSL. The variational approach consists of three steps. The first step is to embed the switched system (5) in the more general bilinear control system [14]: ẋ(t) = Ax(t) + u(t)bx(t), u V, x(0) = x 0, (6) where A = A 0, B = A 1 A 0, and V is the set of measurable functions taking values in [0, 1]. Remark 1 Note that for u 0 [u 1], (6) becomes ẋ = A 0 x [ẋ = A 1 x]. Thus, the set of solutions of (5) is contained in the set of solutions of (6). For u V and T 0, let x(t ; u, x 0 ) denote the solution at time T of (6) corresponding to u. We say that (6) is globally asymptotically stable (GAS) if for any x 0 R n and any control u V: lim x(t; u, x 0) = 0. t It follows from Remark 1 that GAS of (6) immediately implies GUAS of (5). It is possible to prove, using the fact that the reachable set for bang bang controls is a dense subset of the reachable set for measurable controls [46, 6], that GUAS of (5) implies GAS of (6). The second step in the variational approach is to define the most unstable control of (6). Fixing some initial condition x 0 0, and a final time T > 0, let J(T ; u, x 0 ) = x(t ; u, x 0 ) 2, and consider the following optimal control problem. Problem 1 Find a control u V that maximizes J(T ; u, x 0 ). It follows from the definition of the set of admissible controls V that Problem 1 admits a solution [16]. Intuitively, u pushes the trajectory at time T as far as possible from the origin. It is possible to show that (6) is GAS if and only if lim J(T ; T u ( ; T, x 0 ), x 0 ) = 0, for all x 0 R n. (7)

8 8 Under certain mild technical conditions, it is in fact possible to analyze GAS by considering the limit in (7) for a single (and arbitrary) initial condition x 0 0 (see, e.g., [4]). The third step of the variational approach is to characterize u using tools from optimal control theory. These include the Pontrayagin maximum principle (PMP) and, in some cases, the Hamilton-Jacobi-Bellman equation [34, 22]. The variational approach allows the application of sophisticated and powerful tools, such as first and higher orders maximum principles [1, 12, 42] to stability analysis. Some of the results can be generalized to nonlinear control systems and nonlinear switched systems. Indeed, this approach was used to derive the most general stability results currently available for: (1) linear switched systems of order n = 2 [40, 34] and n = 3 [4, 41, 36]; (2) homogeneous switched systems of order n = 2 [22]; and (3) nonlinear switched systems with a nilpotent Lie algebra [43] (see also [31]). We note that the variational approach was also used to study other properties of CT switched systems (see, e.g., [33]). The variational approach was used to derive not only stability results, but more general nicereachability type results for both linear and nonlinear CT switched systems [35, 43, 32] (see also [30] for some related considerations). We now briefly explain this issue. For the sake of simplicity, we focus on the case of a linear switched system. A. The auxiliary system and nice reachability results We refer to the term Ax [ubx] in (6) as the drift term [control term]. It is useful to transform (6) into an equivalent control system that does not include a drift term, i.e. to cancel out the effect of the term Ax. To do so, define y(t) = exp( At)x(t). This yields ẏ(t) = u(t)s(t)y(t), (8) y(0) = x 0, where S(t) = exp( At)B exp(at). This system is usually called the auxiliary system [19, 20] or the pullback system [7]. Note that the auxiliary system does not include a drift term, yet the price paid for this is that (8) is a time varying control system.

9 9 The auxiliary system exposes the crucial role of the commutation relations between A and B in determining the behavior of the control system (6) [48]. Indeed, expanding S(t) as a Taylor series about t = 0 yields S(t) = B + [A, B] t 1! + [A, [A, B]]t2 + [A, [A, [A, B]]]t3 2! 3! +... (9) Sussmann [47] showed how the auxiliary system (8) can be used to derive nice reachability results for (6). To explain this, fix arbitrary T > 0 and u V. Denote p = x(t ; u, x 0 ). We then say that u steers (6) from x 0 to p in time T. Let p = exp( AT )p, so that u steers the auxiliary system from y(0) = x 0 to y(t ; u, x 0 ) = p. Proposition 1 [47] Let v V be a control that steers the auxiliary system from y(0) = x 0 to p in minimal time, i.e. y(t ; v, x 0 ) = p for some T T. Define a control w V by v(t), t [0, T ], w(t) = (10) 0, t (T, T ]. Then x(t ; w, x 0 ) = p. In other words, w steers (6) from x 0 to p in time T, just like u does. Proof. It follows from the definition of w and (8) that y(t ; w, x 0 ) = y(t ; w, x 0 ) = y(t ; v, x 0 ) = p, so x(t ; w, x 0 ) = exp(at )y(t ; w, x 0 ) = exp(at )p = p. Since u V is arbitrary, we conclude that the reachable set R(T ; V, x 0 ) of (6) at time T is equal to the reachable set spanned by controls that are a concatenation of a time optimal control for the auxiliary system and the zero control. If any time optimal control for the auxiliary system is regular in some sense (e.g., bang bang with no more than k switches for any final time T ), then this implies nice reachability results for our original bilinear control system (6). The next example demonstrates this. Example 2 Suppose that [A 0, A 1 ] = 0. Then (8) and (9) yield ẏ = uby. (11) Using the PMP it is straightforward to prove that if v is a time optimal control for (11) then either v 0 or v 1. Hence, the control w in (10) is a bang bang control, with no more than a

10 10 single switch. Letting BB 1 V denote the set of such controls, we conclude that the reachable set of (6) satisfies R(T ; V, x 0 ) = R(T ; BB 1, x 0 ), for any T 0 and any x 0 R n. The results in this paper are based on extending some of these ideas to the case of discrete time linear switched systems. We begin by deriving a maximum principle for the most destabilizing control of (3). III. MAXIMUM PRINCIPLE From here on we consider the DT bilinear control system (3). Recall that U denotes the set of controls taking values in {0, e 1 m 1,..., em 1 m 1}. Fix an arbitrary N N 0. For any u U, let x(n; u, x 0 ) denote the solution of (3) corresponding to u at time N. Let J(N; u, x 0 ) = x(n; u, x 0 ) 2. Consider the following optimal control problem. Problem 2 Find a control u U that maximizes J(N; u, x 0 ). Intuitively, u is a most destabilizing control for (3), as it pushes the solution at time N as far as possible from the origin. Since the set of admissible controls is finite, it is obvious that Problem 2 admits a solution. We refer to any control that maximizes J as an optimal control. The next result provides a necessary condition for a control to be optimal. We use the notation A for the transpose of the matrix A. Theorem 1 (Maximum Principle) Let u be an optimal control for Problem 2, and let x denote the corresponding trajectory. Define the adjoint p : {1, 2,..., N} R n by p(s) = (A 0 + m 1 i=1 u i (s)b i) p(s + 1), p(n) = x (N). (12) Define also m functions r i : {0, 1,..., N 1} R, i = 1,..., m 1, by r i (s) = p (s + 1)B i x (s). (13)

11 11 Then for any s {0, 1,..., N 1}, 0, if r u i (s) < 0 for all i, (s) = e j m 1, if r j(s) > 0 and r j (s) > r i (s) for any i j. (14) The proof, given in the Appendix, is similar in spirit to the proof of the MP for the CT system (6) and Problem 1 given in [29]. In particular, it introduces a perturbed control ũ that is different from u at a single time step, and then estimates the difference x (N) x(n), where x is the trajectory corresponding to ũ. It is possible to state Theorem 1 in the standard Hamiltonian form. To do so, define H : R n R n R m 1 R by H(x, λ, v) = λ (A+ m 1 i=1 v ib)x. Then (14) and (12) can be written as u (s) = arg max v U H(x (s), p(s + 1), v), and p(s) = H x (x (s), p(s + 1), u (s)), respectively. Note that H (s) = H(x (s), p(s + 1), u (s)) is constant. Indeed, H (s) = H(x (s), p(s + 1), u (s)) = p (s + 1)(A + = p (s)x (s) = p (s)(a + m 1 i=1 m 1 i=1 u i (s)b i )x (s) u i (s 1)B i)x (s 1) = H(x (s 1), p(s), u (s 1)) = H (s 1). The next example demonstrates an application of Theorem 1. Example 3 Consider the case where m = 2, A 0 = I, A 1 = 2I, and x 0 0. In this case, it is clear that the only optimal control is u 1. We now show that the MP can indeed be used to

12 12 conclude this. Since A 0 = B 1 = I, x (s + 1) = (1 + u (s))x (s), x(0) = x 0, p(s) = (1 + u (s))p(s + 1), p(n) = x (N), r 1 (s) = p (s + 1)x (s). Thus, r 1 (N 1) = p (N)x (N 1) = (x (N)) x (N)/(1 + u (N 1)) > 0. By (14), this implies that u (N 1) = 1. Hence, x (N 1) = x (N)/2, p(n 1) = 2p(N) = 2x (N), and r 1 (N 2) = p (N 1)x (N 2) = 2(x (N)) x (N 1)/(1 + u (N 2)) = (x (N)) x (N)/(1 + u (N 2)) > 0. By (14), this implies that u (N 2) = 1, and proceeding in this fashion we conclude that u 1. MPs similar to the one described in Theorem 1 have been derived already in the 1970s [38, 37, 3] (see also [5]). This was motivated by an attempt to extend the variational approach, that was originally developed to analyze the CT absolute stability problem, to its DT analog. However, in the DT case this approach proved to be less successful than for CT systems. We believe that this is because the discrete time MP is inherently weaker than its CT counterpart. The next example demonstrates this. Example 4 Consider the case where m = 2, [A 0, A 1 ] = 0. In this case, we already know that the reachable set satisfies the nice reachability property (4). Furthermore, the continuous time MP yields that in this case ṙ(t) = 0, where r(t) is the switching function analogous

13 13 to r 1 (s + 1) defined in (13). It is natural to expect that in this case the discrete time MP will yield r 1 (s+1) r 1 (s) = 0. We will show that this is not the case. Denote D(s) = A 0 +u 1 (s)b 1, and assume that D(s) is non singular for any s. Then r 1 (s + 1) = p (s + 2)B 1 x (s + 1) = p (s + 1)D 1 (s + 1)B 1 D(s)x (s), (15) so r 1 (s + 1) r 1 (s) = p (s + 1) ( D 1 (s + 1)B 1 D(s) B 1 ) x (s). Since [A 0, A 1 ] = 0, B 1 D(s) = D(s)B 1, so r 1 (s + 1) r 1 (s) = p (s + 1) ( D 1 (s + 1)D(s)B 1 B 1 ) x (s). However, this does not imply that r 1 (s + 1) r 1 (s) = 0, and in fact it does not seem possible to use the MP to derive more explicit information on u even in this simple case. To overcome this difficulty, we turn to develop an auxiliary system for (3). IV. THE AUXILIARY SYSTEM AND NICE REACHABILITY Consider the DT bilinear control system (3). From here on we pose the following. Assumption 1 The matrix A 0 is non singular. It is useful to transform (3) into an equivalent control system that is drift free, i.e. to cancel out the effect of the term A 0 x(k). To do so, define a new state vector by: Using (3), (16), and the definition of B i yields y(k + 1) = y(k) + y(k) = A k 0 x(k). (16) m 1 i=1 u i (k)(q i (k) I)y(k), y(0) = x(0), (17)

14 14 where Q i (k) = A (k+1) 0 A i A k 0. Note that for u(k) = 0 [u(k) = e i m 1 ], (17) becomes y(k + 1) = y(k) [y(k + 1) = Q i(k)y(k)]. We refer to (17) as the auxiliary system corresponding to (3). It is interesting to compare the properties of the CT auxiliary system (8) with its DT analog (17). For the sake of simplicity, we do this for the case m = 2 for which (17) becomes y(k + 1) y(k) = u 1 (k)(q 1 (k) I)y(k). (18) The left hand side here is the DT analog of ẏ, whereas the right hand side includes only a control term. The price paid for achieving this drift free form is that the auxiliary system is time varying. Second, the auxiliary system (18) reveals the important part of the Lie commutators of A 0 and A 1 in determining the behavior of the control system. To demonstrate this, let with C R n n. Then a calculation yields Z(k) = A k 0 CAk 0, (19) Z(k + 1) Z(k) = A (k+1) 0 [A 0, C]A k 0. (20) This implies that if we let M(k) = A 0 Q(k) = A k 0 A 1 A k 0, then M(k+1) M(k) = A (k+1) 0 [A 0, A 1 ]A k 0. So the first order differences of M(k) depend on [A 0, A 1 ]. Denoting N(k) = A 0 (M(k + 1) M(k)) = A k 0 [A 0, A 1 ]A k 0, yields N(k + 1) N(k) = A (k+1) 0 [A 0, [A 0, A 1 ]]A k 0. Note that M(k +1) M(k) may be viewed as a corrected version of the first order differences of Q, whereas N(k + 1) N(k) is the corrected version of the second order differences of Q. The dependence of higher order (corrected) differences on higher order Lie brackets is somewhat similar to the expansion of S(t) in (9). Third, we now show that regularity of time optimal controls of the auxiliary system (17) implies nice reachability results for the DT bilinear control system (3).

15 15 A. Nice reachability Fix arbitrary u U, p R n, and N N 0. Let q = x(n; u, p), i.e. u steers the DT system (3) from x(0) = p to x(n) = q. The next result shows that there always exists a control w that does the same, where w is a concatenation of a time optimal control for the auxiliary system (17) and the zero control. Proposition 2 Fix arbitrary u U, N N 0, and p R n. Let q = x(n; u, p). Then there exists an integer N N and a control w in the form u (k), k {0, 1,..., N 1}, w(k) = 0, k {N, N + 1,..., N 1}, with u a time optimal control for the auxiliary system (17), such that x(n; w, p) = q. Proof. By (16), y(n; u, p) = A N 0 x(n; u, p) = A N 0 q. Let u be the control that steers the auxiliary system from y(0) = p to A N 0 q in minimal time. Then y(n ; u, p) = A N 0 q for some N N. Consider the control w defined in (21). By (17), y(n; w, p) = y(n ; w, p) = A N 0 q, (21) so (16) yields x(n; w, p) = A N 0 y(n; w, p) = q, and this completes the proof. Proposition 2 forms the basis of all the nice reachability results derived in this paper. The basic idea is to show that certain Lie algebraic conditions imply that time optimal controls u for (17) are regular in some sense, and then use the fact that any control u U can be replaced by a control w in the form (21). The next result demonstrates this. Claim 1 Consider the bilinear control system (3). Suppose that [A i, A j ] = 0 for all i, j. Then for any N > 0 and any x 0 R n the reachable set of (3) satisfies R(N; U, x 0 ) = R(N; U m 1, x 0 ), where U m 1 U is the set of controls with no more than m 1 switching points.

16 16 Proof. Since A 0 and A i commute, Q i (k) = A (k+1) 0 A i A k 0 = A 1 0 A i, and the auxiliary system becomes the time invariant system y(k + 1) = y(k) + m 1 i=1 u i (k)(a 1 0 A i I)y(k). (22) Let u be a time optimal control for this system. Suppose that u (s) = 0 for some time s {0, 1,..., N 1}. If u (k) = 0 for all k {s,..., N 1}, then y (N ) = y (s), so clearly u is not time optimal. Thus, there exists a minimal time k {s+1,..., N 1} such that u (k) = v, with v {e 1 m 1,..., em 1}. Define a new control ũ U by u (j), j {0,..., s 1}, ũ(j) = v, j = s, u (j + k s), j {s + 1,..., N 1}. Then the corresponding trajectory of (22) satisfies ỹ(n k+s) = y (N ), and so u is not time optimal. We conclude that any time optimal control satisfies u (s) 0 for any s. Intuitively speaking, this may be explained as follows. Since u (s) = 0 implies that y (s + 1) = y (s), using a zero control may be justified only if we are waiting for a better value Q i (l) for some time l > s. However, when every Q i is time invariant, there is no point in waiting. Consider the case m = 2. In this case, we conclude that the only time optimal control is u (k) 1 for all k. Proposition 2 now implies that we may replace any control u in (3) with a control w that contains up to a single switch. Thus, Claim 1 holds when m = 2. Consider now the case m = 3. Fix some arbitrary control u U. By Proposition 2, we can replace u with a control w satisfying w(k) = u (k), k {0,..., N 1}, with u time optimal control for the auxiliary system y(k + 1) = y(k) + (u 1 (k)(a 1 0 A 1 I) + u 2 (k)(a 1 0 A 2 I))y(k), and w(k) = 0 for k {N,..., N 1}. Consider the bilinear control system for this control w, i.e. x(k + 1) = (A 0 + w 1 (k)b 1 + w 2 (k)b 2 )x(k), (23) a

17 17 where B i = A i A 0. Since u (k) 0, u (k) {e 1 2, e 2 2} for any k {0,..., N 1}. Hence, on this time interval, (23) is equivalent to x(k + 1) = (A 1 + v(k)(a 2 A 1 ))x(k), (24) with v(k) = 0 if u (k) = e 1 2, and v(k) = 1 if u (k) = e 2 2. Note that [A 1, A 2 A 1 ] = [A 1, A 2 ] = 0. Since we already proved Claim 1 for the case m = 2, we know that we can replace v with a control θ that includes no more than a single switch such that x(n ; v, x 0 ) = x(n ; θ, x 0 ). This implies that we may replace w with a control that includes no more than two switches. Thus, Claim 1 holds for m = 3. Proceeding in this fashion, we conclude that Claim 1 holds for any m. Remark 2 The result in Claim 1 may seem obvious. However, note that we were not able to derive this result using the discrete time MP even for the particular case m = 2. Thus the derivation above demonstrates how analyzing time optimal controls for the auxiliary system can be used to circumvent the weakness of the discrete time MP. In the next section, we derive several new Lie algebraic conditions guaranteeing that time optimal controls for the auxiliary system (17) are regular in some sense. By Proposition 2, these results immediately yield nice reachability type results for the DT bilinear control system (3). V. REGULARITY OF TIME OPTIMAL CONTROLS We will make frequent use of the following well known and easy to demonstrate property. Proposition 3 (Principle of Optimality.) Suppose that {u (N 1),..., u (1), u (0)} is a time optimal control for (17) steering the system from y (0) = x(0) to y (N ) = q. Fix an arbitrary s {1,..., N 1}, and let p = y (s). Then the subsequence {u (N 1),..., u (s+ 1), u (s)} steers the auxiliary system from p at time s to q at time N in minimal time. In other words, any subsequence {u (N 1),..., u (s + 1), u (s)} of a time optimal control must also be a time optimal control. Remark 3 An immediate implication of Proposition 3 is that if u is a time optimal control, then u (N 1) 0. Indeed, if u (N 1) = 0, then y (N ) = y (N 1), so the subsequence {u (N 1)} is not time optimal.

18 18 A calculation yields Q i (k + 1) Q i (k) = A (k+2) 0 A i A k+1 0 A (k+1) 0 A i A k 0 = A (k+2) 0 [A 0, A i ]A k 0. (25) We already considered the case [A i, A j ] = 0 for all i, j, implying in particular that Q i (k + 1) Q i (k) = 0, and derived a nice reachability result for this case. It is natural to consider next the case where Q i (k + 1) Q i (k) is a constant, but not necessarily zero, matrix. A. The case Q i (k + 1) Q i (k) = const Proposition 4 Suppose that for any i {1,..., m 1} and any k 1, [[A 0, A i ], A k 0] = 0. (26) Then Q i (k) = kp i + Q i (0), (27) with P i = A 2 0 [A 0, A i ]. If, furthermore, P i P j = 0 and P i Q j (0) = 0, for all i, j, (28) then any time optimal control u for the auxiliary system satisfies the following property. If there exists an index r {0,..., N 1} such that u (r) = 0, then u (r) = u (r 1) = = u (0) = 0. (29) Proof. Condition (26) implies that the matrices [A 0, A i ] and A k 0 commute, so (25) yields Q i (k + 1) Q i (k) = P i. This proves (27). To prove (29), suppose that u (r) = 0. If r = 0 then (29) holds vacuously. So assume that r 1 and, seeking a contradiction, that (29) does not hold. Combining this with Remark 3 implies that there exists k 0 such that {u (N 1), u (N 2),..., u (k)} = {e l N 1 m 1,..., e l k+j+1 m 1, 0,..., 0, e l k m 1 },

19 19 where j denotes the number of consecutive zeros in this sequence. The corresponding solution of (17) satisfies y (N ) = Q ln 1 (N 1)... Q lk+j+1 (k + j + 1) I... I Q lk (k)y (k), with the matrix I repeated j times. Using (27) yields y (N ) = ((N 1)P ln 1 + Q ln 1 (0))... ((k + j + 1)P lk+j+1 + Q lk+j+1 (0)) (kp lk + Q lk (0))y (k), (30) Applying (28) shows that many terms in this product vanish and we are left with y (N ) = (Q ln 1 (0)... Q lk+j+1 (0)Q lk (0) + Q ln 1 (0)... Q lk+j+1 (0)kP lk )y (k). (31) Consider the control {u(n j 1),..., u(k + 1), u(k)} = {e l N 1 m 1,..., el k+j+1 m 1, el k m 1 }. (32) The resulting trajectory satisfies y(n k j) = Q ln 1 (N j 1)... Q lk+j+1 (k + 1)Q lk (k)y (k) = ((N j 1)P ln 1 + Q ln 1 (0))... ((k + 1)P lk+j+1 + Q lk+1 (0)) (kp lk + Q lk (0))y (k). Applying (28) shows that this is equal to the term on the right hand side of (31), i.e. y(n k j) = y (N ). In other words, the control (32) steers the auxiliary system from y (k) at time k to y (N ) in N k j time steps. The control u does the same in N k time steps. Proposition 3 implies that u is not time optimal. This contradiction completes the proof Example 5 Consider the case n = 3, m = 3, A 0 = 0 1 0, A 1 = 0 4 8, and

20 5 0 3 A 2 = It is easy to verify that (26) and (28) hold. For small values of N it is straightforward to determine time optimal controls by explicitly calculating the reachable set for any final time N (recall that the set of admissible controls is finite). Using such a calculation shows for example that for y(0) = (1, 2, 3) the time optimal controls of length 3 are: 20 {u (2), u (1), u (0)} ={e 1 2, 0, 0}, {e 2 2, 0, 0}, {e 1 2, e 1 2, 0}, {e 1 2, e 2 2, 0}, {e 2 2, e 1 2, 0}, {e 2 2, e 2 2, 0}, {e 1 2, e 1 2, e 1 2}, {e 1 2, e 1 2, e 2 2}, {e 1 2, e 2 2, e 1 2}, {e 1 2, e 2 2, e 2 2}, {e 2 2, e1 2, e1 2 }, {e2 2, e1 2, e2 2 }, {e2 2, e2 2, e1 2 }, {e2 2, e2 2, e2 2 }. This agrees, of course, with Proposition 4. Specializing Proposition 4 to the case m = 2 yields the following. Corollary 1 Consider the auxiliary control system (17) with m = 2. Suppose that for any k 1, [[A 0, A 1 ], A k 0 ] = 0. (33) Then Q 1 (k) = kp 1 + Q 1 (0), (34) with P 1 = A 2 0 [A 0, A 1 ]. If, furthermore, P 2 1 = 0 and P 1Q 1 (0) = 0, (35) then any time optimal control u for the auxiliary system has no more than a single switch. Proof. Seeking a contradiction, assume that u admits more than one switch. Then by Remark 3, there exists k 0 such that {u (N 1), u (N 2),..., u (k)} = {1, 1,..., 1, 0,..., 0, 1}. But this is a contradiction to (29).

21 Example 6 Consider the case n = 3, m = 2, A 0 = and A 1 = In this case, k 9k [A 0, A 1 ] = 0 0 0, Ak 0 = 0 1 0, and it is easy to verify that (33) holds. Eq. (34) yields Q 1 (k) = kp 1 + Q 1 (0), with P 1 = A 2 0 [A 0, A 1 ] = 0 0 0, and Q 1(0) = A 1 0 A 1 = It is straightforward to show that (35) also holds. We conclude that any time optimal control contains up to a single switch. Combining this with Remark 3 implies that either {u (N 1),..., u (0)} = {1,..., 1, 0,..., 0} or {u (N 1),..., u (0)} = {1,..., 1}. A calculation shows that for y(0) = (1, 2, 3) there are five time optimal controls of length 5: {u (4),..., u (0)} = {1, 0, 0, 0, 0}, or {1, 1, 0, 0, 0}, or {1, 1, 1, 0, 0}, or {1, 1, 1, 1, 0}, or {1, 1, 1, 1, 1, 1}. 21 This agrees, of course, with Corollary 1. Combining Corollary 1 with Proposition 2 yields the following. Corollary 2 Suppose that the conditions of Corollary 1 hold. Then for any N > 0 and any x 0 the reachable set of (3) satisfies R(N; U, x 0 ) = R(N; U 2, x 0 ), (36) where U 2 U is the set of controls with no more than two switches. If the matrices A i, i = 0, 1, are asymptotically stable, then the DT switched system (1) is GUAS. We now turn to consider the case where Q i (k + 2) Q i (k) = 0.

22 22 B. The case Q i (k + 2) Q i (k) = 0 Proposition 5 If then for any i, [A 2 0, A i ] = 0, i {1,..., m 1}, (37) Q i (2k) = Q i (0) and Q i (2k + 1) = Q i (1) for any k, (38) and any time optimal control u cannot include two consecutive zeros. Proof. A calculation yields Q i (k + 2) Q i (k) = A (k+3) 0 A i A k+2 0 A (k+1) 0 A i A k 0 = A (k+3) 0 [A 2 0, A i]a k 0, so (37) implies that Q i (k + 2) = Q i (k), and this proves (38). To prove the second part of the proposition, assume that u is a time optimal control satisfying u (s) = u (s + 1) = 0 for some s {0,..., N 2}. Let q = y (N ). Using (38), it is straightforward to verify that the control u (k), k {0,..., s 1}, v(k) = u (k + 2), k {s,..., N 3}, steers the auxiliary system from y(0) = x(0) to y(n 2) = q. This implies that u is not time optimal Example 7 Consider the case n = 3, m = 3, A 0 = 0 5 8, A 1 = and A 2 = It is straightforward to verify that (37) holds. We conclude that any time optimal control cannot include two consecutive zeroes. For y(0) = (1, 2, 3) the only time

23 23 optimal controls of length 3 are {u (2), u (1), u (0)} ={e 1 2, e 1 2, 0}, {e 1 2, e 2 2, 0}, {e 2 2, e 1 2, 0}, {e 2 2, e 2 2, 0}, {e 1 2, 0, e1 2 }, {e1 2, 0, e2 2 }, {e2 2, 0, e1 2 }, {e2 2, 0, e2 2 }, {e 1 2, e1 2, e1 2 }, {e1 2, e1 2, e2 2 }, {e1 2, e2 2, e1 2 }, {e1 2, e2 2, e2 2 }. This agrees, of course, with Proposition 5. If we pose a stronger condition, then it is possible to say more on the structure of time optimal controls. Recall that the possible control values are 0, e 1 m 1,..., e m 1 m 1. We use 0 to denote a value that is not 0, i.e. 0 can be any of the values in the set {e 1 m 1,..., e m 1 m 1}. We use 0 to denote a pair of control values where the first one is not 0. Proposition 6 Suppose that for any i, j {1,..., m 1}, [A 2 0, A i] = 0, (39) and A 0 A j A i A i A j A 0 = 0. (40) Then any time optimal control has the form { 0, 0,..., 0, 0}, {u (N 1),..., u (0)} = { 0, 0,..., 0 }, N is odd, N is even. (41) Proof. We already know that the first condition in (39) implies that Q i (2k) = Q i (0) and Q i (2k + 1) = Q i (1) for any k, i, and that any time optimal control cannot include two consecutive zeros. A calculation yields [Q i (0), Q j (1)] = [A 1 0 A i, A 2 0 A j A 0 ] = A 2 0 A ja i A 1 0 A ia 2 0 A ja 0 = A 3 0 (A 0A j A i A 2 0 A ia 2 0 A ja 0 ) = A 3 0 (A 0 A j A i A i A j A 0 ),

24 24 where the last equation follows from (39). Hence, (40) implies that Q i (0) and Q j (1) commute. Consider the case N odd. Let {u (N 1),..., u (0)} be a time optimal control. Seeking a contradiction, suppose that u (0) = 0. Let k + 1 denote the number of zeros in u. The corresponding trajectory is of the form y (N ) = Q jr (1)... Q j1 (1)Q il (0)... Q i1 (0)I k Iy(0). (42) where we used the fact that Q i (0) and Q j (1) commute. Here, Q i1 (0) corresponds to u (l) = e i 1 m 1 with l even, and Q j1 (1) corresponds to u (q) = e j 1 m 1 with q odd. For example, for N = 5 and {u (4),..., u (0)} = {e 2 m 1, 0, e 3 m 1, e 1 m 1, 0}, (43) y (5) = Q 2 (0)IQ 3 (0)Q 1 (1)Iy(0) = Q 1 (1)Q 2 (0)Q 3 (0)I 1 Iy(0). Note that the values l, k, r satisfy k l + r = N, l (N 1)/2, r (N 1)/2. (44) Consider a control {u(n 2),..., u(0)} defined by u(q) = u (q) for q odd, and u(l) = u (l+2) for l even. In other words, u is obtained from u by keeping the odd entries unchanged, and shifting every even entry two places to the right. For example, for u given in (43), {u(3),..., u(0)} = {0, e 2 m 1, e1 m 1, e3 m 1 }. It is straightforward to verify that u steers the auxiliary system to the same final location as in (42) at time N 1, i.e. y(n 1; u) = y (N ). This implies that u is not time optimal. Thus, when N is odd any time optimal control satisfies u (0) 0. To prove the remainder of the proposition for the case N odd, we use induction. For N = 1, any optimal control is of the form u (0) = 0. For N = 3, the candidates for the optimal control {u (2), u (1), u (0)} are in one of the forms: { 0, 0, 0}, { 0, 0, 0}, { 0, 0, 0}, and { 0, 0, 0}. The first of these is ruled out by Proposition 5, and the third is ruled out since we already showed that u (0) 0. We conclude that the only possible forms are {u (2), u (1), u (0)} = { 0, 0, 0} or { 0, 0, 0},

25 25 so (41) holds for N = 3. Assume that the proposition holds for N = 1, 3,..., 2l 1. Consider the case N = 2l + 1 and an optimal control {u (2l),..., u (2), u (1), u (0)}. We know that u (2l) 0 and u (0) 0. By the principle of optimality and the induction hypothesis, {u (2l),..., u (2)} contains only sequences of the form { 0 }, and u (2) 0. Thus, {u (2l),..., u (2), u (1), u (0)} = { 0,..., 0, 0, u (1), 0}. Clearly, this implies that (41) holds for N = 2l + 1. This completes the proof for the case N odd. Consider the case where N is even. The proposition holds for N = 2, as the only possible form for time optimal controls is {u (1), u (0)} = { 0 }. For N = 4, the candidates for a time optimal control {u (3), u (2), u (1), u (0)} are: { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, and { 0, 0, 0, 0}. The first, the second and the fifth of these are ruled out by Proposition 5. The sixth is ruled out because we can steer the system to the same final location using the control {u(2), u(1), u(0)} = {u (2), u (3), u (0)} (recall that Q i (0) and Q j (1) commute). We conclude that the only possible time optimal controls are: { 0, 0, 0, 0}, { 0, 0, 0, 0}, { 0, 0, 0, 0}, and { 0, 0, 0, 0}, so (41) holds for N = 4. Assume that the proposition holds for N = 2, 4,..., 2l. Consider the case N = 2l + 2 and a time optimal control {u (2l + 1),..., u (2), u (1), u (0)}. By the principle of optimality and the induction hypothesis, {u (2l + 1),..., u (2)} contains only sequences of the form 0. The possible values for the pair {u (1), u (0)} are obviously {0, 0}, {0, 0}, { 0, 0}, and { 0, 0}.

26 26 The first of these is ruled out because a time optimal control cannot contain two consecutive zeros. We now show that the second option can also be ruled out. Indeed, {u (1), u (0)} = {0, 0} implies that {u (2l + 1),..., u (0)} = {u (2l + 1),..., u (2), 0, 0}. By the principle of optimality, the subsequence {u (2l + 1),..., u (2), 0} is time optimal, but this contradicts our results for the case where N is odd derived above. We conclude that either {u (1), u (0)} = { 0, 0} or {u (1), u (0)} = { 0, 0}. Combining this with the induction hypothesis implies that {u (2l+1),..., u (2), u (1), u (0)} = { 0,..., 0 }, and this completes the proof. When m = 2, the only possible value for 0 is 1. Hence, specializing Proposition 6 to this case yields the following. Corollary 3 Suppose that m = 2, [A 2 0, A 1] = 0 and [A 2 1, A 0] = 0. (45) Then any time optimal control has the form {1, 1,..., 1, 1}, {u (N 1),..., u (0)} = {1, 1,..., 1 }, N is odd, N is even, where stands for either 1 or Example 8 Consider the case n = 3, m = 2, A 0 = and A 1 = It is easy to verify that (45) holds. For y(0) = (1, 2, 3) and N = 6, a calculation of the reachable sets of the auxiliary system shows that the only time optimal controls {u (5),..., u (0)} are: {1, 0, 1, 0, 1, 0}, {1, 1, 1, 0, 1, 0}, {1, 0, 1, 1, 1, 0}, {1, 1, 1, 1, 1, 0}, {1, 0, 1, 0, 1, 1}, {1, 1, 1, 0, 1, 1}, {1, 0, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1},

27 27 whereas for N = 7, the only time optimal controls {u (6),..., u (0)} are: {1, 0, 1, 0, 1, 0, 1}, {1, 1, 1, 0, 1, 0, 1}, {1, 0, 1, 1, 1, 0, 1}, {1, 1, 1, 1, 1, 0, 1}, {1, 0, 1, 0, 1, 1, 1}, {1, 1, 1, 0, 1, 1, 1}, {1, 0, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1}. This agrees, of course, with Corollary 3. VI. DISCUSSION The variational approach, pioneered by E. S. Pyatnitsky, proved to be a powerful tool in the analysis of continuous time linear and nonlinear switched systems. This approach is based on characterizing the most unstable switching law using optimal control techniques and, in particular, the MP. It was also used to derive more general nice reachability results based on analyzing time optimal controls of the auxiliary system. For DT switched systems the variational approach received far less attention. We believe that this is due to the fact that the DT MP is inherently weaker than its CT counterpart. In this paper, we tried to overcome this limitation by introducing the auxiliary system of a DT bilinear control system, and showing that regularity of time optimal controls for the auxiliary system implies nice reachability of the original DT bilinear control system. We derived several Lie algebraic conditions guaranteeing regularity (in some sense) of time optimal controls of the auxiliary system. These results are demonstrated here using synthetic examples. More work is needed in order to examine their usefulness for analyzing real world examples. We believe that the approach presented here can be further developed in several directions including: (1) establishing other Lie algebraic conditions that imply regularity of time optimal controls; and (2) extending the approach to the analysis of DT nonlinear switched systems. Appendix: Proof of Theorem 1 For 0 a b N, let C (b, a) denote the solution of the matrix difference equation C(k + 1, a) = (A 0 + C(a, a) = I. m 1 i=1 u i (k)b i)c(k, a),

28 28 Then x (b) = C (b, a)x (a), i.e. C (b, a) is the transition matrix from time a to time b corresponding to the optimal control u. Fix arbitrary s {0, 1,..., N 1} and v {0, e 1 m 1,..., em 1 m 1}, and define a control ũ by: v, if j = s, ũ(j) = u (j), otherwise. Note that ũ U, i.e. ũ is an admissible control. Let x denote the trajectory corresponding to ũ. Then (46) yields x(n) = C (N, s + 1)(A 0 + m 1 i=1 v i (s)b i )C (s, 0)x 0. (46) Since x (N) = C (N, s + 1)(A 0 + m 1 i=1 u i (s)b i)c (s, 0)x 0, we have m 1 x (N) x(n) = (u i (s) v i )C (N, s + 1)B i C (s, 0)x 0. (47) By the optimality of u, x (N) 2 x(n) 2. This implies i=1 (x (N)) (x (N) x(n)) 0, and using (47) yields m 1 (u i (s) v i)z (s + 1)B i x (s) 0, (48) i=1 where z (s + 1) = (x (N)) C (N, s + 1). (49) It is straightforward to verify using (49) and (12) that z(s) = p(s) for all s, so (13) and (48) yield m 1 (u i (s) v i)r i (s) 0. (50) i=1 We can now prove (14). Suppose that r i (s) < 0 for all i {1, 2,..., m 1}. We need to show that u (s) = 0. Seeking a contradiction, assume that u (s) = e j m 1 for some j {1,..., m 1}.

29 29 Then for v = 0, (50) yields: r j (s) 0, This contradiction implies that u (s) = 0. Suppose that r j (s) > 0 and r j (s) > r i (s) for any i j. We need to show that u (s) = e j m 1. Seeking a contradiction, assume that u (s) e j m 1. Then either u (s) = 0 or u (s) = e k m 1 for k j. In the first case, taking v = e j m 1 in (50) yields: r j (s) 0. In the second case, substituting v = e j m 1 in (50) yields: r k (s) r j (s) 0. In both cases we obtained a contradiction, so u (s) = e j m 1. ACKNOWLEDGMENTS We are grateful to the anonymous reviewers for their constructive comments. REFERENCES [1] A. A. Agrachev and Y. L. Sachkov. Control Theory From The Geometric Viewpoint, volume 87 of Encyclopedia of Mathematical Sciences. Springer-Verlag, [2] Z. Artstein. Discrete and continuous bang-bang and facial spaces or: Look for the extreme points. SIAM Review, 22: , [3] N. E. Barabanov. The Lyapunov indicator of discrete inclusions: Part II. Automat. Remote Control, 3:24 29, [4] N. E. Barabanov. On the Aizerman problem for third-order nonstationary systems. Diff. Eqns., 29: , [5] N. E. Barabanov. Lyapunov exponent and joint spectral radius: Some known and new results. In Proc. 44th IEEE Conf. on Decision and Control, pages , Seville, Spain, [6] S. C. Bengea and R. A. DeCarlo. Optimal control of switching systems. Automatica, 41:11 27, 2005.

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