Robust Stability Robust stability against time-invariant and time-varying uncertainties Parameter dependent Lyapunov functions Semi-infinite LMI problems From nominal to robust performance 1/24
Time-Invariant Parametric Uncertainty Consider linear time-invariant system ẋ(t) = A(δ)x(t) where A(.) is a continuous function of the parameter vector δ = (δ 1,..., δ p ) which is only known to be contained in uncertainty set δ R p. Robust stability analysis Is system asymptotically stable for all possible parameters δ δ? Example: Mass- or load-variation of controlled mechanical system. 2/24
Example Academic example with rational parameter-dependence: 1 2δ 1 2 ẋ = δ 2 2 1 x. 3 1 The parameters δ 1, δ 2, δ 3 are bounded as δ 3 10 δ 1 +1 δ 1 [ 0.5, 1], δ 2 [ 2, 1], δ 3 [ 0.5, 2]. Hence δ is actually a polytope (box) with eight generators: δ = [ 0.5, 1] [ 2, 1] [ 0.5, 2] = δ 1 = co δ 2 : δ 1 { 0.5, 1}, δ 2 { 2, 1}, δ 3 { 0.5, 2}. δ 3 3/24
Relation to Optimization 1 Spectral abscissa of square matrix A: ρ a (A) = max (λ + λ). λ λ(a) 2 Obvious fact: A(δ) is Hurwitz for all δ δ if and only if Two main sources for trouble: ρ a (A(δ)) < 0 for all δ δ. ρ a (A(δ)) is far from convex/concave in the variable δ. Inequality has to hold at infinitely many points. Hence typically computational approaches fail: Cannot find global maximum of ρ a (A(δ)) over δ. Even if δ is a polytope, not sufficient to check the generators. Even more trouble if δ is no polytope. 4/24
Quadratic Stability The uncertain system ẋ = A(δ)x with δ δ is defined to be quadratically stable if there exists X 0 with A(δ) T X + XA(δ) 0 for all δ δ. Why name? V (x) = x T Xx is quadratic Lyapunov function. Why relevant? Implies that A(δ) is Hurwitz for all δ δ. How to check? Easy if A(δ) is affine in δ and δ = co{δ 1,..., δ N } is a polytope with moderate number of generators: Verify whether X 0, A(δ k ) T X + XA(δ k ) 0, k = 1,..., N is feasible. 5/24
Example If A(δ) is not affine in δ, a parameter transformation often helps! In example introduce δ 4 = δ 3 10 + 12. Test quadratic stability of δ 1 + 1 1 2δ 1 2 δ 2 2 1, (δ 1, δ 2, δ 4 ) δ = [ 0.5, 1] [ 2, 1] [ 9, 8]. 3 1 δ 4 12 LMI-Toolbox: System quadratically stable for (δ 1, δ 2, δ 4 ) rδ with largest possible factor r 0.45. Quadratically stable for shrunk set 0.45 δ. Not for rδ with r > 0.45. The critical factor is called quadratic stability margin. 6/24
Time-Varying Parametric Uncertainties Now assume that the parameters δ(t) vary with time, and that they are known to satisfy δ(t) δ for all t. Check stability of ẋ(t) = A(δ(t))x(t), δ(t) δ. The uncertain system with time-varying parametric uncertainties is exponentially stable if there exists X 0 with A(δ) T X + XA(δ) 0 for all δ δ. Proof will be given for a more general result in full detail. Therefore quadratic stability does in fact imply robust stability for arbitrary fast time-varying parametric uncertainty. If bounds on velocity of parameters are known, this test is conservative. 7/24
Rate-Bounded Parametric Uncertainties Let us hence assume that the parameter-curves δ(.) are continuously differentiable and are only known to satisfy δ(t) δ and δ(t) v for all time instances. Here δ R p and v R p are given compact sets (e.g. polytopes). Robust stability analysis Verify whether the linear time-varying system ẋ(t) = A(δ(t))x(t) is exponentially stable for all parameter-curves δ(.) that satisfy the above described bounds on value and velocity. Search for suitable Lyapunov function. 8/24
Main Stability Result Theorem Suppose X(δ) is continuously differentiable on δ and satisfies p X(δ) 0, k X(δ)v k + A(δ) T X(δ) + X(δ)A(δ) 0 k=1 for all δ δ and v v. Then there exist constants K > 0, a > 0 such that all trajectories of the uncertain time-varying system satisfy x(t) Ke a(t t0) x(t 0 ) for all t t 0. Covers many tests in literature. Study the proof to derive variants! Condition is in general sufficient only! Is also necessary in case that v = {0}: Time-invariant uncertainty. 9/24
Proof Continuity & compactness... exist α, β, γ > 0 such that for all δ δ, v v: p αi X(δ) βi, k X(δ)v k + A(δ) T X(δ) + X(δ)A(δ) γi. k=1 Suppose that δ(t) is any admissible parameter-curve and let x(t) denote a corresponding state-trajectory of the system. Here is the crucial point: [ p ] d dt x(t)t X(δ(t))x(t) = x(t) T k X(δ(t)) δ k (t) x(t)+ k=1 + x(t) T [ A(δ(t)) T X(δ(t)) + X(δ(t))A(δ(t)) ] x(t). Since δ(t) δ and δ(t) v we can hence conclude α x(t) 2 x(t) T X(δ(t))x(t) β x(t) 2, d dt x(t)t X(δ(t))x(t) γ x(t) 2. 10/24
Proof The proof is now finished as the one for LTI systems given earlier. Define ξ(t) := x(t) T X(δ(t))x(t) to infer that x(t) 2 1 α ξ(t), ξ(t) β x(t) 2, ξ(t) γ β ξ(t). The latter inequality leads to ξ(t) ξ(t 0 ) e γ β (t t 0) for all t t 0. With the former inequalities we infer x(t) 2 β α e γ β (t t 0) x(t 0 ) 2 for all t t 0. Can choose K = β/α and a = γ/(2β). 11/24
Extreme Cases Parameters are time-invariant: v = {0}. Have to find X(δ) satisfying X(δ) 0, A(δ) T X(δ) + X(δ)A(δ) 0 for all δ δ. Parameters vary arbitrarily fast: Have to find parameter-independent X satisfying X 0, A(δ) T X + XA(δ) 0 for all δ δ. Is identical to quadratic stability test! Can apply subsequently suggested numerical techniques in both cases! 12/24
Specializations Have derived general results based on Lyapunov functions which still depend quadratically on the state (which is restrictive) but which allow for non-linear (smooth) dependence on the uncertain parameters. Is pure algebraic test and does not involve system- or parameter-trajectory. Not easy to apply: Have to find function satisfying partial differential LMI Have to make sure that inequality holds for all δ δ, v v. Allows to easily derive specializations which are or can be implemented with LMI solvers. We just consider a couple of examples. 13/24
Example: Affine System - Affine Lyapunov Matrix Suppose A(δ) depends affinely on parameters: A(δ) = A 0 + δ 1 A 1 + + δ p A p. Parameter- and rate-constraints are boxes: δ = {δ R p : δ k [δ k, δ k ] }, v = {v R p : v k [v k, v k ] } These are the convex hulls of δ g = {δ R p : δ k {δ k, δ k } }, v g = {v R p : v k {v k, v k } } Search for affine parameter dependent X(δ): X(δ) = X 0 + δ 1 X 1 + + δ p X p and hence k X(δ) = X k. 14/24
Example: Affine System - Affine Lyapunov Matrix With δ 0 = 1 observe that p k X(δ)v k + A(δ) T X(δ) + X(δ)A(δ) = k=1 p p p = X k v k + δ ν δ µ (A ν X µ + X µ A ν ). k=1 ν=0 µ=0 Is affine in X 1,..., X p and v 1,..., v p but quadratic in δ 1,..., δ p. Relaxation: Include additional constraint A T ν X ν + Xν T A ν 0. Implies that it suffices to guarantee required inequality at generators. Why? Partially convex function on box! Extra condition renders test stronger. Still sufficient for RS! 15/24
Partially Convex Function on Box Suppose that S R n. The Hermitian-valued function F : S H m is said to be partially convex if the one-variable mapping t F (x 1,..., x l 1, t, x l+1,..., x n ) defined on {t R : (x 1,..., x l 1, t, x l+1,..., x n ) S} is convex for all x S and l = 1,..., n. Suppose S = [a 1, b 1 ] [a n, b n ] and that F : S H m is partially convex. Then F (x) 0 for all x S if and only if F (x) 0 for all x {a 1, b 1 } {a n, b n }. Proof is simple exercise. Various variants suggested in literature. 16/24
Example: Affine System - Affine Lyapunov Matrix Robust exponential stability guaranteed if There exist X 0,..., X p with A T ν X ν + X ν A ν 0, ν = 1,..., p, and p p p p X k δ k 0, X k v k + δ ν δ µ (A T ν X µ + X µ A ν ) 0 k=0 k=1 ν=0 µ=0 for all δ δ g and v v g and δ 0 = 1. This test is implemented in LMI-Toolbox. For rate-bounded uncertainties often much less conservative than quadratic stability test. Understand the arguments in proof to derive your own variants. 17/24
General Recipe to Reduce to Finite Dimensions Restrict the search to a chosen finite-dimensional subspace. For example choose scalar continuously differentiable basis functions b 1 (δ),..., b N (δ) and search for the coefficient matrices X 1,..., X N in the expansion N N X(δ) = X ν b ν (δ) with k X(δ) = X ν k b ν (δ). ν=1 ν=1 Have to guarantee for all δ δ and v v that ( N N p ) X ν b ν (δ) 0, X ν k b ν (δ)v k +[A(δ) T X ν +X ν A(δ)]b ν (δ) 0. ν=1 ν=1 k=1 Is finite dimensional but still semi-infinite LMI problem! 18/24
Remarks If systematically extending the set of basis functions one can improve the sufficient stability conditions. Example: Polynomial basis b (k1,...,k p)(δ) = δ k 1 1 δ kp p, k ν = 0, 1, 2,..., ν = 1,..., p. If δ is star-shaped one can prove: If the partial differential LMI has a continuously differentiable solution then it also has a polynomial solution (of possibly high degree). Polynomial basis is a generic choice with guaranteed success. One can just grid δ and v to arrive at finite system of LMI s. Trouble: Huge LMI system. No guarantees at points not in grid. 19/24
Semi-Infinite LMI-Constraints Nominal stability could be reduced to an LMI feasibility test: Does there exist a solution x R n of some LMI F 0 + x 1 F 1 + + x n F n 0. We have now seen that testing robust stability can be reduced to the following question: Does there exist some x R n with F 0 (δ) + x 1 F 1 (δ) + + x n F n (δ) 0 for all δ δ where F 0 (δ),..., F n (δ) are Hermitian-valued functions of δ δ. Is a generic formulation for the robust counterpart of an LMI feasibility test in which the data matrices are affected by uncertainties. 20/24
From Nominal to Robust Performance Recall how we obtained from nominal stability characterizations the corresponding robust stability tests against time-varying rate-bounded parametric uncertainties. As a major beauty of the dissipation approach, this generalization works without any technical delicacies for performance as well! Only for time-invariant uncertainties the frequency-domain characterizations make sense. For time-varying uncertainties (and time-varying systems) we have to rely on the time-domain interpretations. We provide an illustration for quadratic performance! 21/24
Robust Quadratic Performance Uncertain input-output system described as ẋ(t) = A(δ(t))x(t) + B(δ(t))w(t) z(t) = C(δ(t))x(t) + D(δ(t))w(t). with continuously differential parameter-curves δ(.) that satisfy δ(t) δ and δ(t) v (δ, v R p compact). Robust quadratic performance: Exponential stability and existence of ɛ > 0 such that for x(0) = 0, for all parameter curves and for all trajectories ( ) T ( w(t) 0 z(t) L 2 -gain, passivity,... Q p S p Sp T R p ) ( ) w(t) dt ɛ w 2 2. z(t) 22/24
Characterization of Robust Quadratic Performance Suppose there exists a continuously differentiable Hermitian-valued X(δ) such that X(δ) 0 and p k X(δ)v k +A(δ) T X(δ)+X(δ)A(δ) X(δ)B(δ) k=1 + B(δ) T X(δ) 0 ( ) T ( ) 0 I 0 I + P p 0 C(δ) D(δ) C(δ) D(δ) for all δ δ, v v. Then the uncertain system satisfies the robust quadratic performance specification. Numerical search for X(δ): Same as for stability! Extends to other LMI performance specifications! 23/24
Sketch of Proof Exponential stability: Left-upper block is p k X(δ)v k + A(δ) T X(δ) + X(δ)A(δ) + C(δ) T R p C(δ) 0. }{{} k=1 0 Can hence just apply our general result on robust exponential stability. Performance: Add to right-lower block ɛi for some small ɛ > 0 (compactness). Left- and right-multiply inequality with col(x(t), w(t)) to infer ( ) T ( ) d w(t) w(t) dt x(t)t X(δ(t))x(t) + P p + ɛw(t) T w(t) 0. z(t) z(t) Integrate on [0, T ] and use x(0) = 0 to obtain ( ) T ( ) T x(t ) T w(t) w(t) X(δ(T ))x(t )+ P p 0 z(t) z(t) T dt ɛ w(t) T w(t) dt. 0 Take limit T to arrive at required quadratic performance inequality. 24/24