A Sufficient Condition for Local Controllability

Transcription

1 A Sufficient Condition for Local Controllability of Nonlinear Systems Along Closed Orbits Kwanghee Nam and Aristotle Arapostathis Abstract We present a computable sufficient condition to determine local controllability of a differential nonlinear control system along a reference closed orbit. Comparisons with already existing results on local controllability along a reference trajectory are made. Assistant Professor in the Department of Electrical Engineering at POSTECH, Pohang, Republic of Korea. Associate Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin. The research of this author was supported in part by the Texas Advanced Research Program (Advanced Technology Program) under Grants No and No , and in part by the Air Force Office of Scientific Research under Grant AFOSR

2 I. Introduction We consider affine nonlinear systems of the form ẋ = f(x)+g(x)u, (1) where x =[x 1,x 2,,x n ] T are the local coordinates on a smooth n-dimensional manifold M, the control u :IR + IR is a scalar piecewise smooth function and f(x), g(x) are the local coordinate representations of smooth vector fields globally defined on M. We denote by e tf (x 0 ) the solution trajectory of the dynamical system ẋ = f(x) which passes through x 0 at t =0. In this communication we present a sufficient condition (Theorem 3) for local controllability of (1) along a reference closed orbit. We also show how this condition extends to systems with multiple inputs (Theorem 4) and to more general nonlinear systems (Theorem 5). The following definitions apply to an arbitrary nonlinear control system defined on a manifold M [5]: Let U be an open connected subset of M and T a nonnegative real number. A point x T is U accessible from x 0 M at time T if there exists a bounded measurable control u(t), generating a trajectory φ t (x 0 ), such that φ t (x 0 ) U for t [0,T] and φ T (x 0 )=x T. The set of all x T which are U accessible from x 0 at time T, is denoted by A(x 0,T,U). Let A(x 0, U) = T 0 A(x 0,T,U). A control system is said to be controllable on U if A(x 0, U) =U, for every x 0 U and locally controllable at x 0 if the system is controllable on some neighborhood of x 0. We call such a neighborhood a controllable neighborhood. An important result regarding the controllability properties of system (1) is stated in the following theorem [6], [8]. Theorem 1. Consider the system (1). Suppose that the Lie algebra L{f,g} generated by f and g has dimension n for all x U, where U is an open neighborhood of x 0 in M. Then, the set A(x 0, U) has a nonempty interior. We also state here a well-known sufficient condition for local controllability at an equilibrium point [7]. Theorem 2. The system (1) is locally controllable at an equilibrium point x e of f if [ b, Ab,,A n 1 b ] has full rank, where b := g(x e ) and A := ( / x ) f(x e ). 2

3 Controllability properties along a reference trajectory have been investigated by Hermes in [3] and [4]. Let φ t (x 0 ) denote a trajectory of a control system with φ 0 (x 0 )=x 0. Hermes defines the system to be locally controllable along φ at time t 0 if all points in some open neighborhood of φ t (x 0 ) can be reached at time t by solutions initiating from x 0. He developed sufficient conditions to determine local controllability along a reference trajectory in [3] and strengthened these results in [4]. As we are going to see in the next section, these conditions are too strong when specialized to a reference trajectory which is a closed orbit. In [2] Bianchini and Stefani utilize graded approximations of vector fields to study the problem of local controllability along a trajectory. This approach allows them to generalize and unify several results in the stationary and nonstationary case. II. Local controllability along a closed orbit Consider (1) and let γ be a closed orbit of the flow e tf with period τ. Let µ>0 be a small number such that τ (n 1)µ >0. With r =[r 1,,r n ] T IR n,we define a map Ψ µ :IR n M M by Ψ µ (r; x) :=e µ(f+r 1g) e µ(f+r n 1g) e (τ (n 1)µ+r n)f (x). (2) Note that, by construction, there exists a neighborhood V of IR n such that every point in Ψ µ (V; x) may be reached from x by a trajectory of (1) with piecewise constant control. We need the following technical lemma: Lemma 1. For each x M, rank DΨ µ (0; x) =rank [ f, g, ad f g, ad 2 f g,... ] (x). (3) Proof: Let V k µ (x) := ε e (k 1)µf e µ(f+εg) e kµf (x), k =1, 2,.... (4) ε=0 3

4 We use the following expansion (see [1], [9]) V k µ (x) = µ i i! ( 1)i+1 {k i (k 1) i }ad i 1 f g(x). (5) Observe that Ψ µ r k (0; x) =V k µ (x), for k =1,...,n 1 and consider k Ṽµ k (x) := Vµ j (x) = j=1 µ i i! ( 1)i+1 k i ad i 1 f g(x), k =1,...,n 1. Since the coefficients of { Ṽµ 1 (x),...,ṽ µ n 1 (x) } in terms of { (µ i /i!)ad i 1 f g(x) } form a Vandermonde matrix, we obtain rank [ Ṽµ 1 (x),, Ṽ µ n 1 (x) ] = rank [ Ψ µ Ψ µ (0; x),, (0; x) ] r 1 r n 1 = rank [ g(x), ad f g(x), ad 2 f g(x),... ]. Finally, since Ψ µ r n (0; x) =f(x), the result follows. Theorem 3. Consider the system (1) and let γ be a closed orbit of the flow e tf with period τ. Suppose that for some p γ rank [ f, g, ad f g, ad 2 f g,... ] (p) =n. (6) Then, given an open neighborhood U of γ, there exists a further neighborhood W of γ, W Usuch that every pair of points of W may be joined by a trajectory of (1) resulting from piecewise constant control action and lying in U. Proof: Lemma 1 and (6) imply that Ψ µ ( ; p) is a local diffeomorphism from a neighborhood V of 0 IR n onto a neigborhood of p M. As noted before, if µ and V are sufficiently small, then for each r V, there corresponds a trajectory of (1) denoted by Φ µ,r (t; p), resulting from piecewise constant control action which steers p at t = 0 to Ψ µ (r; p) att = τ + r n. In addition, due to the continuity of Ψ µ (r; p) and of Φ µ,r (t; p) with respect to µ and r we may choose µ and V sufficiently small such that e tf ( Ψ µ (V; p) ) U, t [0,τ] (7) 4

5 and Φ µ,r (t; p) U, t [0,τ + r n ], r V. (8) Applying the same argument to the time-reversed system, we conclude that there exists a neighborhood N of p such that each point of N may be steered to p via a trajectory of (1), resulting from piecewise constant control action and lying in U, and satisfying e tf ( N ) U, t [0,τ]. (9) Define W 0 := N Ψ µ (V; p) and W := 0 t τ { e tf ( W 0 ) e (τ t)f ( W 0 ) }. (10) Then, W is an open neighborhood of γ, which, by (7) and (9), is contained in U and we claim that it satisfies the conclusions of the Theorem. Indeed, let p 1,p 2 W. Observe that, by (10), either p 2 W 0, or there exists p 2 W 0 such that e t f (p 2)= p 2 for some t [0,τ] and e tf (p 2) Wfor all t [0,t ]. Then, by construction of W, p 1 can be steered to a point p 1 W 0, via the action of the vector field f; p 1 can then be steered to p which in turn can be steered to p 2 all by trajectories which lie in U. Remark: This technique may be used along an arbitrary trajectory as follows: Let p, q 1 and q 2 be points on M such that p = e t 1 (q 1 ) and q 2 = e t 2 (p), with t 1,t 2 > 0. Then, provided that (6) holds, there exists a neighborhood U of p such that, if p is an arbitrary point in U, one may find trajectories of (1), resulting from piecewise control action, which steer q 1 to p and p to q 2. We exhibit, through an example, that the condition in Proposition 2 of [3] is rather strong when specialized to our problem. Consider a vector field f in IR 2 pocessing a closed orbit γ with period τ and let p γ. Construct a vector field g defined in some open neighborhood V of p such that f and g are linearly independent and ad f g = 0. Select an open neighborhood N of p with N V. By the smooth Urysohn Lemma, there exists a smooth function α :IR 2 IR such that α =1on N and α =0onV c. Define g = α g. Clearly, rank [ g, ad f g, ad 2 f g,... ] (x) 1, x γ 5

6 and therefore the condition in Proposition 2 of [3] is not satisfied. On the other hand the hypothesis of Theorem 3 does hold. Note, though, that the conclusion of Theorem 3 does not imply local controllability in the sense of Hermes. Theorem 3 can be extended to systems with multiple inputs, i.e., of the form ẋ = f(x)+ or the more general nonlinear system m g i (x)u i, (11) ẋ = F (x, u), (12) where F (x, u) is a smooth function of u U, with U some open set of IR m containing the origin. The appropriate statements, whose proofs are analogous to that of Theorem 3, are as follows. Theorem 4. Consider the system (11) and let γ be a closed orbit of the flow e tf with period τ. Fork =0, 1,..., define G k := { ad k f g i,,...,m }. Suppose that rank [ f, G 0, G 1, G 2,... ] (p) =n, for some p γ. (13) Then the conclusions of Theorem 3 follow. Theorem 5. Consider the system (12) and define f(x) := F (x, 0), g i (x) := ( / u i )F (x, 0). Suppose that γ is a closed orbit of the flow e tf with period τ. Then, under the hypothesis (13), the conclusions of Theorem 3 follow. As a final remark, it may initially appear that the definition of local controllability in Section I is stronger than the conclusion of Theorem 3. However, upon closer investigation, these two are equivalent. Indeed, one can show that, with U and W as in Theorem 3, there exists a neighborhood V, W V U, such that every pair of points in V may be joined by a trajectory of (1) which lies in V. Hence, V becomes a controllable neighborhood. 6

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