Stabilization and Passivity-Based Control

DISC Systems and Control Theory of Nonlinear Systems, 2010 1 Stabilization and Passivity-Based Control Lecture 8 Nonlinear Dynamical Control Systems, Chapter 10, plus handout from R. Sepulchre, Constructive Nonlinear Control (pp.25 70, pp.229 239) available at www.montefiore.ulg.ac.be/ sepulch

DISC Systems and Control Theory of Nonlinear Systems, 2010 2 Standard definitions on the stability of an autonomous system ẋ = f(x) where x = (x 1,...,x n ) are local coordinates for X. Let x 0 be an equilibrium point, i.e. f(x 0 ) = 0. The equilibrium point x 0 is stable if for any neighborhood V of x 0 there exists a neighborhood Ṽ of x 0 such that if x Ṽ, then the solution x(t, 0, x) belongs to V for all t 0. The equilibrium x 0 is unstable if it is not stable. The equilibrium x 0 is (locally) asymptotically stable if x 0 is stable and there exists a neighborhood V 0 of x 0 such that all solutions x(t, 0, x) with x V 0, converge to x 0 as t. The equilibrium x 0 is globally asymptotically stable if V 0 = X

DISC Systems and Control Theory of Nonlinear Systems, 2010 3 In Lyapunov s first method the local stability of x 0 is related to the stability of the linearization around the equilibrium point x = A x, with A = f x (x 0). Theorem 1 (First method of Lyapunov) The equilibrium x 0 is asymptotically stable if the matrix A is asymptotically stable, i.e., the matrix A has all its eigenvalues in the open left half plane. The equilibrium point x 0 is unstable if at least one of the eigenvalues of the matrix A has a positive real part.

DISC Systems and Control Theory of Nonlinear Systems, 2010 4 Consider the control system ẋ = f(x, u), where x = (x 1,...,x n ) are local coordinates for a smooth manifold X, u = (u 1,...,u m ) U R m, the input space, and f(, u) a smooth vector field for each u U. We assume U to be an open part of R m and that f depends smoothly on the controls u. Let (x 0, u 0 ) an equilibrium point: f(x 0, u 0 ) = 0. (N.B.: usually u 0 = 0.)

DISC Systems and Control Theory of Nonlinear Systems, 2010 5 Problem 2 Under which conditions does there exist a smooth strict static state feedback u = α(x), α : X U, with α(x 0 ) = u 0, such that the closed loop system ẋ = f(x, α(x)) has x 0 as an asymptotically stable equilibrium?

DISC Systems and Control Theory of Nonlinear Systems, 2010 6 Consider the linearization of the system around the point (x 0, u 0 ) where A = f x (x 0, u 0 ), x = A x + Bū, B = f u (x 0, u 0 ). Define R as the reachable subspace of the linearized system [ ] R = im B AB A n 1 B Clearly R is invariant under A, i.e., AR R, so after a linear change of coordinates d = A 11 A 12 + B 1 ū dt x 2 0 A 22 x 2 0 where the vectors ( x 1, 0) T correspond to vectors in R.

DISC Systems and Control Theory of Nonlinear Systems, 2010 7 Theorem 3 The feedback stabilization problem admits a local solution around x 0 if all eigenvalues of the matrix A 22 are in C, the open left half plane of C. Moreover if one of the eigenvalues of A 22 has a positive real part, then there does not exist a solution to the local feedback stabilization problem.

DISC Systems and Control Theory of Nonlinear Systems, 2010 8 Consider the linearized dynamics around (x 0, u 0 ) and assume all eigenvalues of A 22 belong to C. Then by linear control theory there is a linear state feedback ū = F x which asymptotically stabilizes the linearized system. (We may actually take ū = F 1 x 1.) Then the affine feedback u = u 0 + F(x x 0 ) for the nonlinear system yields the closed loop system ẋ = f(x, u 0 + F(x x 0 )), of which the linearization around x 0 equals x = (A + BF) x. By construction this linear dynamics is asymptotically stable and so by Lyapunov s first method x 0 is an asymptotically stable equilibrium point.

DISC Systems and Control Theory of Nonlinear Systems, 2010 9 Next suppose that at least one of the eigenvalues of the matrix A 22 has a positive real part. Let u = α(x) be an arbitrary smooth feedback with α(x 0 ) = u 0. Linearizing the dynamics around x 0 yields x = (A + B α x (x 0)) x, which still has the same unstable eigenvalue of the matrix A 22, and thus x 0 is an unstable equilibrium point of the closed loop system.

DISC Systems and Control Theory of Nonlinear Systems, 2010 10 One step further by using center manifold theory Suppose the set of eigenvalues of A, σ(a), can be written as the disjoint union σ(a) = σ σ 0, where the eigenvalues in σ lie in the open left half plane and those in σ 0 lie on the imaginary axis. Let l be the number of eigenvalues (counted with their multiplicity) contained in σ, so that there are n l eigenvalues (counted with their multiplicity) in σ 0. Then there exists a linear coordinate transformation T such that TAT 1 = A0 0 0 A where the (n l, n l)-matrix A 0 and the (l, l)-matrix A have as eigenvalues σ(a 0 ) = σ 0, respectively σ(a ) = σ.

DISC Systems and Control Theory of Nonlinear Systems, 2010 11 In the transformed coordinates z = Tx x 0 the nonlinear system takes the form ż 1 = A 0 z 1 + f 0 (z 1, z 2 ) ż 2 = A z 2 + f (z 1, z 2 ) where f 0 (0, 0) = 0, f (0, 0) = 0, df 0 (0, 0) = 0, df (0, 0) = 0.

DISC Systems and Control Theory of Nonlinear Systems, 2010 12 Theorem 4 (Center Manifold Theorem) For each k = 2, 3,... there exists a δ k > 0 and a C k -mapping φ : {z 1 R n l z 1 < δ k } R l with φ(0) = 0 and dφ(0) = 0, such that the submanifold (the center manifold) z 2 = φ(z 1 ), z 1 < δ k, is invariant under the nonlinear dynamics. Remark 5 (i) In general the nonlinear dynamics does not possess a unique center manifold, but may have an infinite number of such invariant manifolds. (ii) The smooth dynamics has a C k center manifold for each (finite) k = 2, 3,.... However the size of the center manifold (δ k depends on k and may shrink with increasing k. Even in case the system is analytic, there does not necessarily exist an analytic center manifold.

DISC Systems and Control Theory of Nonlinear Systems, 2010 13 Theorem 6 The dynamics on the center manifold are given as ż 1 = A 0 z 1 + f 0 (z 1, φ(z 1 )). If z 1 = 0 is asymptotically stable, stable, or unstable, respectively, for this center manifold dynamics then (z 1, z 2 ) = 0 is asymptotically stable, stable or unstable for the full-order system. The linearized system resulting from applying the linear feedback u = u 0 + F 1 (x 1 x 1 0) + F 2 (x 2 x 2 0) A 11 + B 1 F 1 A 12 + B 1 F 2 0 A 22 Although the matrix F 2 does not affect the eigenvalues, it does affect the orientation of the imaginary eigenspace, and thus the dynamics on the center manifold.

DISC Systems and Control Theory of Nonlinear Systems, 2010 14 Second or direct method of Lyapunov A smooth function L defined on some neighborhood V of x 0 is positive definite if L(x 0 ) = 0 and L(x) > 0 for all x x 0. A set W in M is an invariant set if for all x W the solutions x(t, 0, x) belong to W for all t. Theorem 7 (Second method of Lyapunov) Consider the dynamics ẋ = f(x) around the equilibrium point x 0. Let L be a positive definite function on some neighborhood V 0 of x 0. Then x 0 is stable if L f L(x) 0, x V 0 The function L is called a Lyapunov function.

DISC Systems and Control Theory of Nonlinear Systems, 2010 15 Furthermore, x 0 is asymptotically stable if L f L(x) < 0, x x 0 or more generally if the largest invariant set contained in the set W = {x V 0 L f L(x) = 0} equals {x 0 }; i.e. the only solution x(t, 0, x) starting in x W which remains in W for all t 0, coincides with x 0. (This called LaSalle s Invariance principle.)

DISC Systems and Control Theory of Nonlinear Systems, 2010 16 Consider the system ẋ = f(x) + m g i (x)u i i=1 with f(x 0 ) = 0. Suppose there exists a Lyapunov function L for the dynamics with u 0 L f L(x) 0, x V 0. so x 0 is already stable for the system with u = 0.

DISC Systems and Control Theory of Nonlinear Systems, 2010 17 Consider the smooth feedback u = α(x) defined as α i (x) = L gi L(x), i = 1,...,m, x V 0 yielding the closed loop behavior ẋ = f(x) + m g i (x)α i (x). i=1 satisfying L f L(x) + m L αi g i L(x) = L f L(x) i=1 m (L gi L(x)) 2 0, i=1

DISC Systems and Control Theory of Nonlinear Systems, 2010 18 In order to study the asymptotic stability of x 0 we introduce the set W = {x V 0 L f L(x) m i=1 (L g i L(x)) 2 = 0} = {x V 0 L f L(x) = 0, L gi L(x) = 0, i m}. Let W 0 be the largest invariant subset of W under the dynamics. Observe that any trajectory x α (t, 0, x) in W 0 is a trajectory of the original dynamics; this because the feedback u = α(x) is identically zero for each point in W.

DISC Systems and Control Theory of Nonlinear Systems, 2010 19 Consider the distribution D(x) = span{f(x), ad k fg i (x), i m, k 0}, x V 0. Lemma 8 Suppose there exists a Lyapunov-function L for ẋ = f(x) on a neighborhood V 0 of the equilibrium point x 0. Suppose that dimd(x 0 ) = n, which implies that on some neighborhood Ṽ0 V 0 of x 0 dimd(x) = n. Then x 0 is asymptotically stable for the closed loop system. The same holds if dimd(x) = n, for all x V 0 \ {x 0 }

DISC Systems and Control Theory of Nonlinear Systems, 2010 20 Example 9 Consider the equations for the angular velocities of a rigid body with one external torque I ω = S(ω)Iω + bu with ω = (ω 1, ω 2, ω 3 ), 0 ω 3 ω 2 S(ω) = ω 3 0 ω 1 I = ω 2 ω 1 0 I 1 0 0 0 I 2 0 0 0 I 3 I 3 > I 2 > I 1 > 0 denote the principal moments of inertia. Let I 23 = (I 2 I 3 )/I 1, I 31 = (I 3 I 1 )/I 2, I 12 = (I 1 I 2 )/I 3.

DISC Systems and Control Theory of Nonlinear Systems, 2010 21 Then the dynamics may be written as ω 1 ω 2 ω 3 = I 23 ω 2 ω 3 + c 1 u = I 31 ω 3 ω 1 + c 2 u = I 12 ω 1 ω 2 + c 3 u with c = (c 1, c 2, c 3 ) T = I 1 b. An obvious choice for a Lyapunov function for the drift vector field is the kinetic energy of the rigid body, i.e. L(ω) = 1 2 (I 1ω1 2 + I 2 ω2 2 + I 3 ω3) 2. L is a smooth positive definite function having a unique minimum at ω = 0. Computing L f L yields L f L(ω) = I 1 I 23 ω 1 ω 2 ω 3 + I 2 I 31 ω 1 ω 2 ω 3 + I 3 I 12 ω 1 ω 2 ω 3 = 0, which shows that ω = 0 is a stable equilibrium point when u = 0.

DISC Systems and Control Theory of Nonlinear Systems, 2010 22 Define the smooth feedback u = L c L(ω) = (c 1 I 1 ω 1 + c 2 I 2 ω 2 + c 3 I 3 ω 3 ). Assume that c 1 c 2 c 3 0. (So the control axis is not perpendicular to any of the principal axes of the rigid body.) Then the above feedback asymptotically stabilizes ω = 0.

DISC Systems and Control Theory of Nonlinear Systems, 2010 23 Passive systems theory A system with outputs ẋ = f(x) + g(x)u y = h(x) is called passive if there exists a function S(x) with S(x) 0 for all x if and only if d dt S yt u In physical situations S is often the stored energy in the system, while y T u is the supplied power. Formally, S(x) is called the storage function and s(y, u) = y T u the supply rate.

DISC Systems and Control Theory of Nonlinear Systems, 2010 24 d dt S yt u is equivalent to for all x and u, or equivalently T S x (x)[f(x) + g(x)u] ht (x)u L f S(x) = T S x (x)f(x) 0 h(x) = g T (x) S x (x) Hence, passivity can be regarded as an extension of the notion of Lyapunov function.

DISC Systems and Control Theory of Nonlinear Systems, 2010 25 Furthermore, the above feedback u = α(x) can be seen to be equal to leading to u = g T (x) S (x) = y x d dt S yt u = y 2 and asymptotic stability of a minimum of S is guaranteed if the largest invariant subset contained in the set where the passive output y is equal to zero is equal to this minimum.

DISC Systems and Control Theory of Nonlinear Systems, 2010 26 Two fundamental properties of passive systems 1) In general a passive system has uniform relative degree 1. Indeed, since y j = L gj S the input-output decoupling matrix is generally given as [L gi L gj S] i,j=1,,p which has full rank if the Hessian matrix of the storage function S has full rank. 2) The zero-dynamics of a passive system is stable. Indeed, by substituting y = 0 in d dt S yt u it follows that the zero-dynamics satisfies d dt S 0

DISC Systems and Control Theory of Nonlinear Systems, 2010 27 Both properties are if and only if conditions for passifiability. Rewrite the system satisfying both conditions as ż = q(z, ξ) ξ = a(z, ξ) + b(z, ξ)u y = ξ with b(z, ξ) full rank. Furthermore, factorize q(z, ξ) = q(z, 0) + p(z, ξ)ξ Since q(z, 0) is asymptotically stable there exists a Lyapunov function W(z) for ż = q(z, 0). Then S(z, ξ) := W(z) + 1 2 ξt ξ is a Lyapunov function for the system after applying the feedback u = b 1 (z, ξ)( a(z, ξ) L p(z,ξ) W + v)

DISC Systems and Control Theory of Nonlinear Systems, 2010 28 Passivity interpretation of backstepping procedure for stabilization Suppose we want to stabilize the system ẋ 1 = f 1 (x 1, x 2 ) ẋ 2 = f 2 (x 1, x 2, x 3 ) ẋ 3 = f 3 (x 1, x 2, x 3, u) Construct a passive output y = x 3 α 2 (x 1, x 2 ) such that the zero-dynamics ẋ 1 = f 1 (x 1, x 2 ) ẋ 2 = f 2 (x 1, x 2, α 2 (x 1, x 2 )) is asymptotically stable.

DISC Systems and Control Theory of Nonlinear Systems, 2010 29 How to find α(x 1, x 2 )? This is a stabilization problem for a reduced-order dynamics. Can be solved by constructing a passive output y = x 2 α 1 (x 1 ) such that the zero dynamics ẋ 1 = f 1 (x 1, α 1 (x 1 )) is asymptotically stable. This procedure can be generalized to arbitrary systems that are feedback linearizable.

DISC Systems and Control Theory of Nonlinear Systems, 2010 30 Dual interpretation Stabilize ẋ 1 = f 1 (x 1, x 2 ) ẋ 2 = f 2 (x 1, x 2, x 3 ) ẋ 3 = f 3 (x 1, x 2, x 3, u) by working top-down : first consider the system ẋ 1 = f 1 (x 1, v) and construct a stabilizing feedback v = β 1 (x 1 ). Define a new coordinate z 2 = x 2 β(x 1 )

DISC Systems and Control Theory of Nonlinear Systems, 2010 31 and rewrite the system as ẋ 1 = f 1 (x 1, β(x 1 ) + z 2 ) ż 2 = f 2 (x 1, z 2, x 3 ) ẋ 3 = f 3 (x 1, z 2, x 3, u) and stabilize the first two lines by a feedback x 3 = β 2 (x 1, z 2 ), etc.

DISC Systems and Control Theory of Nonlinear Systems, 2010 32 Passivity is a compositional property Consider k passive subsystems with input-output pairs u i, y i, i = 1,, k and storage functions S i (x i ), interconnected in such a way that y1 T u 1 + y2 T u 2 + + yk T u k = y T u Then the overall system is again passive with storage function S 1 (x 1 ) + S 2 (x 2 ) + + S k (x k )

DISC Systems and Control Theory of Nonlinear Systems, 2010 33 From a network modeling point of view, passive systems arise as port-hamiltonian systems: Port-based network modeling leads to a representation of a physical system as a graph, where each edge is decorated with a (vector) pair of flow variables f R m, and effort variables e R m, i.e., a bond graph R 1 IC : f = 0 H 1 fh1 e H1 0 1 T H 2 H 3 0 f R 2 e R2 R 2 Figure 1: Port-based network modeling

DISC Systems and Control Theory of Nonlinear Systems, 2010 34 and each vertex corresponds to one of the following ideal elements: Energy-storing elements H: ẋ = f H e H = H x (x), H(x 1,, x m ) R energy Power-dissipating elements R: R(f R, e R ) = 0, e T Rf R 0 Power-conserving elements: transformers T, gyrators GY, ideal constraints IC. 0- and 1-junctions: e 1 = e 2 = = e k, f 1 + f 2 + + f k = 0 f 1 = f 2 = = f k, e 1 + e 2 + + e k = 0

DISC Systems and Control Theory of Nonlinear Systems, 2010 35 0- and 1-junctions are the basic conservation laws of the system, and are also power-conserving: e 1 f 1 + e 2 f 2 + + e k f k = 0 Transformers, gyrators are energy-routing devices, and may correspond to exchange between different types of energy. All power-conserving elements have the following properties in common. They are described by linear equations: Ff + Ee = 0, f, e R l whose solutions f, e satisfy e T f = e 1 f 1 + e 2 f 2 + + e l f l = 0, [ ] rank F E = l All power-conserving elements taken together define a Dirac structure.

DISC Systems and Control Theory of Nonlinear Systems, 2010 36 A geometric definition of port-based network models Take all power-conserving elements (T, G, IC, 0- and 1-junctions) together in a single power-conserving interconnection structure: R 1 D IC : f = 0 H 1 f H1 e H1 0 1 T H 2 H 3 0 f R 2 e R2 R 2 Figure 2: Power-conserving interconnection structure

DISC Systems and Control Theory of Nonlinear Systems, 2010 37 Input-state-output port-hamiltonian systems: Particular case is a Dirac structure D(x) T x X T x X F F given as the graph of the skew-symmetric map f x = J(x) g(x) e x e P g T (x) 0 f P leading (f x = ẋ, e x = H x (x)) to a port-hamiltonian system as before ẋ = J(x) H x (x) + g(x)f P, x X, f P R m, e P = g T (x) H x (x), e P R m

DISC Systems and Control Theory of Nonlinear Systems, 2010 38 Power-dissipation is included by terminating some of the ports by static resistive elements f R = F(e R ), where e T RF(e R ) 0, for alle R. d dt H et Pf P This leads, e.g. for linear damping, to input-state-output port-hamiltonian systems in the form ẋ = [J(x) R(x)] H x (x) + g(x)f P e P = g T (x) H x (x) where J(x) = J T (x), R(x) = R T (x) 0 are the interconnection and damping matrices, respectively.