DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 2: Controllability of nonlinear systems Nonlinear Dynamical Control Systems, Chapter 3 See www.math.rug.nl/ arjan (under teaching) for info on course schedule and homework sets. Take-Home Exam I on homepage on March 16.
DISC Systems and Control Theory of Nonlinear Systems 2 Recall: Kinematic model of the unicycle ẋ 1 = u 1 cosx 3 ẋ 2 = u 1 sinx 3 ẋ 3 = u 2 written as a system with two input vector fields and zero drift vector field cosx 3 ẋ = sin x 3 u 1 + u 2 1 The Lie bracket of the two input vector fields is given as sinx 3 sinx 3 cosx 3 = cosx 3 1
DISC Systems and Control Theory of Nonlinear Systems 3 which is a vector field that is independent from the two input vector fields. Claim: This new independent direction guarantees controllability of the unicycle system. Interpretation of the Lie bracket: Proposition 1 Let X, Y be two vector fields such that [X, Y ] = Then the solution flows of the vector fields are commuting. In fact, we may find local coordinates x 1,...,x n such that X = x 1, Y = x 2 Thus, the Lie bracket [X, Y ] characterizes the amount of non-commutativity of the vector fields X, Y.
DISC Systems and Control Theory of Nonlinear Systems 4 In fact, let the control strategy u = col(u 1, u 2 ) be defined by (1, ), t [, ε), ε > (, 1), t [ε, 2ε) u(t) = ( 1, ), t [2ε, 3ε) (, 1), t [3ε, 4ε), Then the motion of the system is described by x(4ε) = x + ε 2 [g 1, g 2 ](x ) + O(ε 3 ). which indicates controllability, since [g 1, g 2 ] is everywhere independent from g 1, g 2. This formula holds in general. This is enough for systems with two inputs and three state variables, but what can we do if the dimension of the state is > 3?
DISC Systems and Control Theory of Nonlinear Systems 5 Answer: consider higher-order Lie brackets. y r θ ϕ x r Example 2 Consider the cart with fixed rear axis x 1 cos(ϕ + θ) d x 2 sin(ϕ + θ) = u dt ϕ sinθ 1 + θ 1 with u 1 the driving input, and u 2 the steering input. u 2
DISC Systems and Control Theory of Nonlinear Systems 6 Define g 1 (x) = cos(x 3 + x 4 ) sin(x 3 + x 4 ), g sin(x 4 ) 2 (x) = } {{ } Drive. 1 }{{} Steer
DISC Systems and Control Theory of Nonlinear Systems 7 Compute [Steer, Drive] = g 1 x g 2 g 2 x g 1 sin(x 3 + x 4 ) sin(x 3 + x 4 ) cos(x 3 + x 4 ) cos(x 3 + x 4 ) = cos(x 4 ) sin(x 3 + x 4 ) cos(x 3 + x 4 ) = =: Wriggle. cos(x 4 ) 1
DISC Systems and Control Theory of Nonlinear Systems 8 Another independent direction is obtained by the third-order Lie bracket sin(x 3 ) cos(x 3 ) [Wriggle, Drive] = =: Slide. This shows that you can manoeuver your car into any parking lot by applying controls corresponding to the Slide direction, i.e., by applying the control sequence {Wriggle, Drive, Wriggle, Drive}.
DISC Systems and Control Theory of Nonlinear Systems 9 What to do with the drift vector field? The system ẋ = f(x) + g(x)u can be considered as a special case of ẋ = g 1 (x)u 1 + g 2 (x)u 2, with u 1 = 1. This means that care has to be taken with respect to brackets involving f: [f, g], [g, [f, g]], [f, [f, g]],...
DISC Systems and Control Theory of Nonlinear Systems 1 Example 3 Consider the system on R 2 ẋ 1 = x 2 2 ẋ 2 = u. Compute the Lie brackets of the vector fields f(x) = x2 2, g(x) = 1, yielding [f, g](x) = 2x 2, [[f, g], g](x) = 2. Clearly, we have obtained two independent directions. However, since x 2 2, the x 1 -coordinate is always non-decreasing. Hence, the system is not really controllable.
DISC Systems and Control Theory of Nonlinear Systems 11 A weaker form of controllability: local accessibility Let V be a neighborhood of x, then R V (x, t 1 ) denotes the reachable set from x at time t 1, following the trajectories which remain in the neighborhood V of x for t t 1, i.e., all points x 1 for which there exists an input u( ) such that the evolution of the system for x() = x satisfies x(t) V, t t 1, and x(t 1 ) = x 1. Furthermore, let R V t 1 (x ) = R V (x, τ). τ t 1 Definition 4 (Local accessibility) A system is said to be locally accessible from x if R V t 1 (x ) contains a non-empty open subset of X for all non-empty neighborhoods V of x and all t 1 >. If the latter holds for all x X then the system is called locally accessible.
DISC Systems and Control Theory of Nonlinear Systems 12 Definition 5 (Accessibility algebra) Consider the system ẋ = f(x) + g 1 (x)u 1 + + g m (x)u m The accessibility algebra C are the linear combinations of repeated Lie brackets of the form [X k, [X k 1, [, [X 2, X 1 ] ]]], k = 1, 2,..., where X i, is a vector field in the set {f, g 1,...,g m }. This linear space is a Lie algebra under the Lie bracket. Definition 6 The accessibility distribution C is the distribution generated by the accessibility algebra C: C(x) = span{x(x) X vector field in C}, x X
DISC Systems and Control Theory of Nonlinear Systems 13 Intermezzo: Distributions on manifolds A distribution D on a manifold X is specified by a subspace D(x) T x X for all x X. Let X 1, X 2,...,X k be vector fields on X. Then D(x) = span(x 1 (x), X 2 (x),...,x k (x)). defines a distribution. A distribution D is called involutive if, whenever f, g D, also [f, g] D. The distribution D is called constant-dimensional whenever the dimension of D(x) is constant.
DISC Systems and Control Theory of Nonlinear Systems 14 Example 7 Let X = R 3 and D = span(f 1, f 2 ), where f 1 (x) = 2x 2 1, f 2(x) = 1. x 2 Since f 1 and f 2 are linearly independent, we have that dim(d(x)) = 2, for all x. Furthermore, we have [f 1, f 2 ](x) = f 2 x (x)f 1(x) f 1 x (x)f 2(x) = 1.
DISC Systems and Control Theory of Nonlinear Systems 15 [f 1, f 2 ] D if and only if rank(f 1 (x), f 2 (x), [f 1, f 2 ](x)) = 2, for all x. However, rank(f 1 (x), f 2 (x), [f 1, f 2 ](x)) = rank for all x. Hence, D is not involutive. 2x 2 1 1 x 2 1 = 3,
DISC Systems and Control Theory of Nonlinear Systems 16 Let D be a nonsingular distribution on X, generated by the independent vector fields f 1,...,f r. Then D is said to be integrable if for each x X, there exists a neighborhood N of x and n r real-valued independent functions h 1 (x),...,h n r (x) defined on N, such that h 1 (x),...,h n r (x) satisfy the partial differential equations for all indices i = 1,...,r, j = 1,...,n r. Frobenius theorem h j x (x)f i(x) =, (1) A constant-dimensional distribution is integrable if and only if it is involutive. The necessity of involutivity for complete integrability is easily seen. Indeed, suppose that (1) is satisfied. This is the same as L fi h j =
DISC Systems and Control Theory of Nonlinear Systems 17 It follows that L [fi,f k ]h j = L fi L fk h j L fk L fi h j = Since the functions h 1 (x),...,h n r (x) are independent, this implies that the Lie brackets [f i, f k ] are (pointwise) linear combinations of the vector fields f 1,...,f r, and are thus contained in the distribution D. A geometric description of Frobenius theorem is as follows. Let the independent functions h 1 (x),...,h n r (x) satisfy (1). Then their level sets, i.e., all sets of the form {x h 1 (x) = c 1,...,h n r (x) = c n r } for arbitrary constants c 1,...,c n r, are well-defined r-dimensional submanifolds of X, to which all the vector fields f 1,...,f r are tangent, and, as a consequence, also all their Lie brackets are tangent.
DISC Systems and Control Theory of Nonlinear Systems 18 Example 8 Consider the following set of partial differential equations = x 1 φ x 1 + x 2 φ x 2 + x 3 φ x 3 = φ x 3 Define the vector fields f 1 (x) = x 1 x 2, f 2(x) =. x 3 1 It is checked that D := span(f 1, f 2 ) has constant dimension = 2 on the set X = {x R 3 x 2 1 + x 2 2 } (that is, R 3 excluding the x 3 -axis), and is involutive. Thus, by Frobenius theorem, D is integrable.
DISC Systems and Control Theory of Nonlinear Systems 19 Consequently, for each x X, there exists a neighborhood N of x and a real-valued function φ(x) with dφ(x) that satisfies the given set of partial differential equations. In fact, φ(x) = lnx 1 lnx 2 is a (global) solution. Note that the solution is not unique. In particular, φ(x) = tan 1 x 2 x 1 is also a global solution.
DISC Systems and Control Theory of Nonlinear Systems 2 By construction, the accessibility distribution C is involutive. Theorem 9 (Local accessibility) A sufficient condition for the system to be locally accessible from x X is dimc(x) = n (2) If this holds for all x X then the system is locally accessible. Conversely, if the system is locally accessible then (2) holds for all x in an open and dense subset of X. We call (2) the accessibility rank condition at x.
DISC Systems and Control Theory of Nonlinear Systems 21 Key idea of the proof Consider the system ẋ = f(x) + g 1 (x)u 1 +...g m (x)u m and its generated system vector fields F = {X u 1,...,u m such that X(x) = f(x) + g 1 (x)u 1 +...g m (x)u m } Then for every k n there exists a submanifold N k around x of dimension k given as N k = {x x = X t k k Xt k 1 k 1 Xt 1 1 (x ), σ i < t i < τ i } with X i F. Indeed, suppose for a certain k < n we cannot construct N k+1. This means that all system vector fields X F are tangent to N k, and hence all vector fields f, g 1,...,g m. This also means that all Lie brackets of these vector fields are tangent to N k, and thus dim C(x) k, which is a contradiction.
DISC Systems and Control Theory of Nonlinear Systems 22 If there is no drift vector field then we obtain real controllability: Theorem 1 Consider ẋ = g 1 (x)u 1 + g 2 (x)u 2 +... + g m (x)u m If dimc(x) = n for all x X then the system is controllable. Consider the map (t 1,...,t n ) X t n n X t n 1 n 1 Xt 1 1 (x ), σ i < t i < τ i having image N n, which is an n-dimensional open part of X. Now let s 1,...,s n be such that σ i < s i < τ i. Then the map (t 1,...,t n ) ( X 1 ) s 1 ( X 2 ) s 2 ( X n ) s n X t n n X t n 1 n 1 Xt 1 1 (x ), σ i < t i < τ i has an image which is an open neighborhood of x. Thus the reachable set R(x ) from x contains an open neighborhood of x.
DISC Systems and Control Theory of Nonlinear Systems 23 Suppose now that the reachable set is smaller than X. Then take any point on the boundary of the reachable set R(x ). Then the set of points reachable from this point is again open. Contradiction. Thus the unicycle and the cart are indeed controllable. Note that the actual construction of the input functions which steers the system from x to x 1 has not been addressed.
DISC Systems and Control Theory of Nonlinear Systems 24 Sometimes local accessibility is heavily depending on the flow of the drift vector field; consider for example the system ẋ 1 = 1 ẋ 2 = u This system is locally accessible, but of course very far from controllability. In order to improve the situation we look at a stronger form of accessibility: local strong accessibility A system is locally strongly accessible from x if for any neighborhood V of x the set R V (x, t 1 ) contains a non-empty set for any t 1 > sufficiently small. If the latter holds for all x X then the system is called locally strongly accessible. (The example given above is not locally strongly accessible.)
DISC Systems and Control Theory of Nonlinear Systems 25 Define C as the smallest algebra which contains g 1,...,g m and satisfies [f, w] C for all w C. Define the corresponding involutive distribution C (x) := span{x(x) X vector field in C }. We refer to C and C as the strong accessibility algebra and the strong accessibility distribution, respectively.
DISC Systems and Control Theory of Nonlinear Systems 26 Notice that the strong accessibility algebra C does not contain the drift vector field f). Theorem 11 (Strong accessibility) A sufficient condition for the system to be locally strongly accessible from x is dimc (x) = n Furthermore, the system is locally strongly accessible if this holds for all x. Conversely, if the system is locally strongly accessible then it holds for all x in an open and dense subset of X. The system given before: ẋ 1 = x 2 2 ẋ 2 = u. is not only locally accessible, but also locally strongly accessible, since g(x) and [[f, g], g](x) are everywhere independent.
DISC Systems and Control Theory of Nonlinear Systems 27 Let us apply the theory developed above to a linear system ẋ = Ax + m b i u i, x R n, i=1 where b 1,...,b m are the columns of the input matrix B. Clearly, the Lie brackets of the constant input vector fields given by the input vectors b 1,...,b m are all zero, i.e., [b i, b j ] =, for all i, j = 1,...,m. Furthermore, the Lie bracket of the linear drift vector field Ax with an input vector field b i yields the constant vector field [Ax, b i ] = Ab i. The Lie brackets of Ab i with Ab j or b j are again all zero, while [Ax, Ab i ] = A 2 b i.
DISC Systems and Control Theory of Nonlinear Systems 28 Hence we conclude that C is spanned by all constant vector fields b i, Ab i, A 2 b i,..., i m, together with the linear drift vector field Ax, i.e., C = {Ax, b i, Ab i, A 2 b i...,a n 1 b i, i = 1,...,m}. while C = columns of (B, AB, A 2 B...,A n 1 B) We see that for linear systems the rank condition for strong accessibility coincides with the Kalman rank condition for controllability. Hence, if we would not have known anything special about linear systems, then at least a linear system which satisfies the Kalman rank condition is locally strongly accessible.
DISC Systems and Control Theory of Nonlinear Systems 29 Example 12 (Actuated rotating rigid body) Consider ω 1 A 1 ω 2 ω 3 α 1 ω 2 = A 2 ω 3 ω 1 + u 1 + α 2 u 2 ω 3 A 3 ω 1 ω 2 with α 1, α 2. Here the constants A 1, A 2, A 3 are determined by the moments of inertia a 1, a 2, a 3. Compute [g 1, f](ω) = α 1 A 2 ω 3 [g 2, f](ω) = α 1 A 3 ω 2 α 2 A 1 ω 3 α 2 A 3 ω 1
DISC Systems and Control Theory of Nonlinear Systems 3 On the other hand [g 2, [g 1, f]] = α 1 α 2 A 3 Thus the system is locally strongly accessible if A 3 which is equivalent to a 1 a 2. In fact, this is the if and only if condition. Indeed, if A 3 = then ω 3 =, showing that the system is not locally strongly accessible. Remark 13 Due to the specific properties of the drift vector field, i.c. Poisson stability, it can be shown that the system is in fact controllable if and only if the two first moments of inertia a 1 and a 2 are different.)