Outline. Linear Matrix Inequalities in Control. Outline. Time-Invariant Parametric Uncertainty. Robust stability analysis

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1 Outline Linear Matrix nequalities in Control Carsten Scherer and Siep Weiland 7th Elgersburg School on Mathematical Systems Theory Class 4 1 Robust Stability against arametric Uncertainties Quadratic stability Rate-bounded uncertainties Robust performance 2 Multipliers Linear fractional representations Relaxations of robust linear matrix inequalities 3 Robust Controller Synthesis Setup Robust synthesis inequalities Summary Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 1 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 2 / 54 Outline Time-nvariant arametric Uncertainty 1 Robust Stability against arametric Uncertainties Quadratic stability Rate-bounded uncertainties Robust performance 2 Multipliers Linear fractional representations Relaxations of robust linear matrix inequalities 3 Robust Controller Synthesis Setup Robust synthesis inequalities Summary Consider the linear time-invariant (LT system ẋ(t = A(δx(t here A( is a continuous function of a (constant parameter vector δ = ( δ 1... δ p hich is knon to be contained in an uncertainty set δ R p. Robust stability analysis s this system asymptotically stable for all δ δ? Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 3 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 4 / 54

2 Example: Load variation in a mechanical system Example: Academic Differential eq. (kernel model: Mẍ Bẋ Kx = 0 State-space model: (ẋ ( = ẏ 0 1 M 1 K M 1 B here x is position and y = ẋ is velocity. arameter load changes on the mass matrix M M ( x y b k x m load variation in a MSD system Academic example ith rational parameter-dependence 1 2δ 1 2 ẋ = δ δ 3 10 δ 1 1 here the parameters δ 1, δ 2, δ 3 are bounded as x δ 1 [ 0.5, 1], δ 2 [ 2, 1], δ 3 [ 0.5, 2]. Hence δ is actually a polytope (box ith eight generators: δ = [ 0.5, 1] [ 2, 1] [ 0.5, 2] = δ 1 = conv δ 2 : δ 1 { 0.5, 1}, δ 2 { 2, 1}, δ 3 { 0.5, 2}. δ 3 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 5 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 6 / 54 Relation to optimiation Quadratic Stability A(δ is Hurit for all δ δ if and only if ρ(a(δ := max re λ(a(δ < 0 for all δ δ Definition The uncertain system ẋ = A(δx ith δ δ is said to be quadratically stable if there exists X 0 ith To main sources for trouble: spectral abscissa ρ(a(δ is not convex/concave in δ. inequality has to hold at infinitely many points. Consequences: Computational approaches fail: Cannot find global maximum of ρ(a(δ over δ. Even if δ is a polytope, not sufficient to check its generators. Even more trouble if δ is not a polytope. A (δx XA(δ 0 for all δ δ. Why is the name? V (x = x Xx is a quadratic Lyapunov function. Why this is relevant? mplies that A(δ is Hurit for all δ δ. Ho to check? Easy, if A(δ is affine in δ and δ = conv{δ 1,..., δ N } is a polytope ith moderate number of generators: Verify hether is feasible. X 0, A (δ k X XA(δ k 0, k = 1,..., N Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 7 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 8 / 54

3 Example: Robust State-Feedback Synthesis Uncertain (affine control system ẋ = A(δx B(δu, δ δ. Find F such that ẋ = [A(δ B(δF ]x is quadratically stable on δ. Solved if there exist some X such that X 0 and [A(δ B(δF ] X X[A(δ B(δF ] 0 for all δ δ. Standard trafo: Y = X 1 and K = F X 1. Equivalent to finding K and Y ith Y 0 and A(δY Y A (δ B(δK (B(δK 0. Affine A(δ, B(δ, finitely generated δ: LM! Very unfortunate fact: Does not ork for output-feedback control! Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 9 / 54 Example: Academic cont d f A(δ is not affine in δ, a parameter transformation often helps! Regarding the previous example, introduce δ 4 = δ Test δ 1 1 quadratic stability of 1 2δ 1 2 δ 2 2 1, (δ 1, δ 2, δ 4 δ = [ 0.5, 1] [ 2, 1] [ 17 2, 17 2 ]. 3 1 δ What s the price? LM-Toolbox: System quadratically stable for (δ 1, δ 2, δ 4 rδ ith largest possible factor r Quadratically stable for a deflated set 0.49 δ. Not for rδ ith r > This critical factor is called quadratic stability margin. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 10 / 54 Example - implementation LM toolbox LM-toolbox routines: quadstab, psys and pvec Define system matrices >> S0 = ltisys([ ; ; ]; >> S1 = ltisys([ ; ; ], eros(3; >> S2 = ltisys([ ; ; ], eros(3; >> S4 = ltisys([ ; ; ], eros(3; Define parameter ranges >> pv = pvec( box,[-0.5 1; -2 1; ]; Define affine parameter dependent system >> pdsys = psys( pv, [S0,S1,S2,S4] ; Find quadratic stability margin >> [tau,x] = quadstab(pdsys, [1 1 1e8]; Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 11 / 54 Time-Varying arametric Uncertainties No, assume that the parameters δ(t vary ith time, and that they are knon to satisfy δ(t δ for all t. Check stability of ẋ(t = A(δ(tx(t, δ : R δ. Theorem The uncertain system ith time-varying parametric uncertainties is exponentially stable if there exists X 0 ith A (δx XA(δ 0 for all δ δ. The proof ill be given for a more general result in full detail. Quadratic stability does in fact imply robust stability for arbitrary fast time-varying parametric uncertainty. f bounds on velocity of parameters are knon, this test is conservative. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 12 / 54

4 Rate-Bounded arametric Uncertainties Let us hence assume that the parameter trajectories δ(. are continuously differentiable and are only knon 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 hether the linear time-varying system ẋ(t = A(δ(tx(t is exponentially stable for all parameter trajectories δ(. that satisfy the above described bounds on value and variation. Key idea: Search for a suitable (quadratic Lyapunov function. Main Stability Result Theorem Suppose X(δ is continuously differentiable.r.t. δ and satisfies X(δ 0, k X(δv k A (δx(δ X(δA(δ 0 for all δ δ and v v. Then, there exist constants K > 0, a > 0 such that all state trajectories of the uncertain time-varying system satisfy x(t Ke a(t t 0 x(t 0 for all t t 0. Covers many tests in literature. Study the proof to derive variants! n general, this condition is only sufficient! t is necessary in case v = {0}: Time-invariant uncertainty. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 13 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 14 / 54 roof Continuity/compactness exist α, β, γ > 0 such that for all δ δ, v v: α X(δ β, k X(δv k A (δx(δ X(δA(δ γ. Suppose that δ(t is an admissible parameter trajectory and let x(t denote a compatible state-trajectory of the system. Here is the crucial point: [ ] d dt x (tx(δ(tx(t = x (t }{{} k X(δ(t δ k (t x(t ξ(t [ ] x (t A(δ (tx(δ(t X(δ(tA(δ(t x(t. Since δ(t δ and δ(t v e can hence conclude α x(t 2 2 ξ(t β x(t 2 2, d dt ξ(t γ x(t 2 2. roof The conclusion of the proof is routine: We infer x(t α ξ(t, ξ(t β x(t 2 2, ξ(t γ β ξ(t. The latter inequality leads to ξ(t ξ(t 0 e γ β (t t 0 With the former inequalities e infer for all t t 0. x(t 2 2 β α e γ β (t t 0 x(t for all t t 0. Choose K = β/α and a = γ/(2β to complete proof. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 15 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 16 / 54

5 Extreme Cases Remarks arameters are time-invariant: v = {0}. We need to find a X(δ satisfying X(δ 0, A (δx(δ X(δA(δ 0 for all δ δ. arameters vary arbitrarily fast: We need to find a parameter-independent X satisfying X 0, A (δx XA(δ 0 for all δ δ. This is identical to the quadratic stability test! We can apply the subsequently suggested numerical techniques in both cases! We have derived general results based on Lyapunov functions hich still depend quadratically on the state (hich is restrictive, but hich allo for non-linear (smooth dependence on the uncertain parameters. ure algebraic test hich does not involve system- nor parameter-trajectories. Not easy to apply: Have to find a function satisfying a partial differential LM. Have to make sure that inequality holds for all δ δ, v v. Allos to easily derive special cases hich are or can be implemented ith LM solvers. (Affine dependence is just around the bend We ill only consider a couple of examples. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 17 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 18 / 54 Example: Affine Dependence - Affine Lyapunov Matrix Suppose A(δ depends affinely on the parameters: A(δ = A 0 δ 1 A 1 δ p A p. arameter- 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 an affine parameter dependent X(δ: X(δ = X 0 δ 1 X 1 δ p X p and hence k X(δ = X k. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 19 / 54 Example: Affine Dependence - Affine Lyapunov Matrix With δ 0 = 1 observe that k X(δv k A (δx(δ X(δA(δ = = X k v k ν=0 µ=0 δ ν δ µ (A ν X µ X µ A ν. s affine in X 1,..., X p and v 1,..., v p but quadratic in δ 1,..., δ p. Relaxation nclude an additional constraint: A ν X ν X ν A ν 0. mplies that it suffices to guarantee the required inequality at generators. Why? artially convex function on the box! Extra condition renders test stronger and numerically verifiable. Still sufficient for robust stability! Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 20 / 54

6 Example: Affine Dependence - Affine Lyapunov Matrix Robust exponential stability guaranteed if There exist X 0,..., X p ith A ν X ν X ν A ν 0, ν = 1,..., p, and X k δ k 0 k=0 X k v k for all δ δ g and v v g and δ 0 = 1. ν=0 µ=0 δ ν δ µ (A ν X µ X µ A ν 0 This test is implemented in the LM/Robust control toolbox For rate-bounded uncertainties, it is often much less conservative than the quadratic stability test. Useful to understand the proof and derive your on generaliation(s. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 21 / 54 Robust Quadratic erformance Given an uncertain system described as ẋ(t = A(δ(tx(t B(δ(t(t (t = C(δ(tx(t D(δ(t(t ith continuously differentiable parameter trajectories δ(. that satisfy δ(t δ and δ(t v (δ, v R p compact. Robust quadratic performance property Existence of an ɛ > 0 such that, for all solution trajectories ((t, (t, δ(t ith x(0 = 0, it holds that 0 ( (t (t L 2 -gain, passivity,... ( Qp S p S p R p } {{ } p ( (t (t dt ɛ 2 2. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 22 / 54 Sufficient Condition for Robust Quadratic erformance Theorem Assume that R p 0. Suppose that there exists a continuously differentiable symmetric-valued X(δ such that X(δ 0 and k X(δv k A (δx(δx(δa(δ S p X(δB(δ R p B (δx(δ 0 ( ( ( 0 Qp S p 0 0 C(δ D(δ C(δ D(δ for all δ δ, v v. Then, the uncertain system satisfies the robust quadratic performance property. Numerical search for X(δ: like for stability! Easily extended to other LM performance specifications! Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 23 / 54 Sketch of the roof Exponential stability: Left-upper block is k X(δv k A (δx(δ X(δA(δ C (δr p C(δ 0. }{{} 0 Hence, e can apply our general result on robust exponential stability. erformance: Adding ɛ (small ɛ > 0 to the right-loer block (compactness. Left- & right-multiply the inequality ith col(x(t, (t: ( ( d (t (t dt x (tx(δ(tx(t (t p ɛ (t(t 0. (t ntegrate over [0, T ] and use x(0 = 0 to obtain T ( ( x (t (t (T X(δ(T x(t (t p (t 0 dt ɛ T 0 (t(tdt. Since X(δ(T 0, x (T X(δ(T x(t can be dropped and the limit T is taken to obtain the required quadratic performance inequality. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 24 / 54

7 Outline Rational arameter Dependence 1 Robust Stability against arametric Uncertainties Quadratic stability Rate-bounded uncertainties Robust performance 2 Multipliers Linear fractional representations Relaxations of robust linear matrix inequalities 3 Robust Controller Synthesis Setup Robust synthesis inequalities Summary Consider the uncertain system ( 1 2δ1 ẋ = F (δx ith F (δ = 1 δ δ 2 here δ 1 r, δ 2 r and r > 0. Ho can e handle rational parameter dependence? ẋ = F (δx can be represented as } ẋ = Ax B = (δ = Cx D ith (δ depending linearly on the parameters δ. (δ ẋ = Ax B = Cx D Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 25 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 26 / 54 Derivation Rerite ξ = F (δx as ( ( 1 0 ξ1 = 0 1 δ 1 ξ 2 or as ( ξ1 ξ 2 or as ( ξ1 ξ 2 ( ( 1 2δ1 x1 1 ( 4 3δ 2 (1 δ 1 x 2 ( ( ( ( x1 1 1 = 1 4 3δ 2 x 2 1 = δ 1 1, 1 = ξ 2, 2 = δ 1 2, 2 = x δ 2 ( ( ( ( x = 1 4 x ( = δ 1 1, 1 = ξ 2, 2 = δ 1 2, 2 = x 2, 3 = δ 2 3, 3 = x 2 2. Derivation Hence ξ = F (δx can be ritten as ξ = ( ( x = 0 1 x 0 0 0, = Therefore e can choose ( A B = C D , (δ = δ δ δ 2 δ δ δ 2.. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 27 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 28 / 54

8 Single parameter derivation Suppose that G(s = A B(s D 1 C realies the transfer function G(s = F (1/s. Then ( 1 1 F (δ = A B δ D C = A Bδ( Dδ 1 C is represented as ẋ F (δ x = ẋ δ ẋ = Ax B = Cx D x Linear Fractional Representation (LFR Theorem Suppose F (δ 1,..., δ p is a matrix-valued function that is rational in δ = (δ 1,..., δ p and non-singular at ero. Then one can construct matrices A, B, C, D such that ( F (δ = A B (δ ( D (δ 1 A B C =: (δ C D here (δ = ( δ1 d δ p dp and dk is identity of sie d k. f F (δ depends affinely on δ, one can choose D = 0. The operation is often called loer linear fractional transformation or star-product. Our example illustrates ho to prove this result in general. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 29 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 30 / 54 Linear Fractional Representation (LFR The LFR is called ell-posed on the set δ R p if D (δ is non-singular for all δ δ. Simple and poerful algebraic operations preserve LFR structure: E.g. sums, products and LFR s of LFR s are LFR s. There exists a nice toolbox for orking ith LFR s developed by J.F. Magni, ONERA-CERT, Toulouse. The Robust Control Toolbox (RCT supports construction of LFR s. Here is a simple recipe of ho to derive LFR s and ho to ork ith them, even if the uncertainties are not parametric. Vie uncertainties as systems processing signals and just employ the usual techniques for manipulating system interconnections. Nonconservative Robust Stability Test Let ẋ = F (δx be represented as } ẋ = Ax B = (δ = Cx D ith (δ being linear in δ δ. Define the transfer matrix T (s = D C(s A 1 B. Theorem (δ ẋ = Ax B = Cx D The LFR is ell-posed and ẋ = F (δx is Hurit for all δ δ iff det( T (iω (δ 0 for all ω R { }, δ δ. Testing robust stability is reduced to a robust non-singularity condition for the matrices T (iω, ω R { }, and the set (δ. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 31 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 32 / 54

9 Multipliers Recall quadratic stability test for ẋ = F (δx: Find X 0 such that ( ( ( 0 X F 0 for all δ δ R p. (δ X 0 F (δ Full Block S-rocedure LFR is ell-posed and (QS holds if there exists a multiplier ith ( (δ that also satisfies ( ( 0 0 X A B X 0 ( (δ 0 for all δ δ ( ( ( A B C D C D f δ is compact, then the statement holds ith iff. (QS (OS (DE Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 33 / 54 Sketch of proof of if ( ( ( (δ (DE implies 0. Due to (OS, and D D D hence also D (δ are nonsingular. This implies ell-posedness. Left- and right-multiply (DE ith ( (δ( D (δ 1 and C to infer ith simple computation: ( (δ( D (δ 1 C ( ( ( 0 X F (δ X 0 F (δ ( ( [( D (δ 1 C] (δ (δ ( D (δ 1 C 0. }{{} 0 due to (OS This obviously implies (QS. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 34 / 54 nterpretation: Separation by Dissipation Numerical mplementation? Uncertainty anti-dissipative Nominal System ( Nominal system dissipative T ( AT X XA XB B T X 0 0 Separation ith same Multiplier 0 C D T 0 0 C D Need to find a nicely parameteried set of to satisfy (OS. Example: Small-Gain Suppose that δ 1 1,..., δ p 1 for all δ δ. Then, e can choose = ( 0 0 roof. Since the components of δ and the magnitude bound by one, e infer (δ 1 (spectral norm and hence (δ (δ and hence ( (δ ( 0 0. ( (δ 0. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 35 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 36 / 54

10 Numerical mplementation? Much less conservatism ith scalings from µ-theory. Example: Diagonal Scalings Suppose that δ 1 1,..., δ p 1 for all δ δ. We can choose = ( diag(q1,..., Q p diag(s 1,..., S p diag(s 1,..., S p diag(q 1,..., Q p ith arbitrary Q 1,..., Q p 0 and ske-symmetric S 1,..., S p. roof. t is simple to check that ( (δ ( (δ Since Q j 0 and 1 δ 2 j = diag(q j (1 δj 2 δ j (S j Sj = diag(q j (1 δj 2. 0 e infer (OS. Numerical mplementation? Example: Full Block Multipliers Suppose that δ = conv{δ 1,..., δ N }. We can choose satisfying ( 0 ( 0 0, ( (δ j ( (δ j Set of multipliers described by finitely many LM constraints. roof. The first condition implies that δ ( (δ ( (δ 0, j = 1,..., N. is concave. Therefore (OS is valid iff positivity holds at the generators δ 1,..., δ N. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 37 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 38 / 54 Example Determine largest r such that the non-singularity test is successful. Largest r Small Gain Diagonal Multipliers ith S=0 Diagonal Multipliers Frequency Observation: The larger the class of multipliers, the better the test! Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 39 / 54 Comments Many sets of multipliers have been suggested in the literature. The larger the set, the better the test. The smaller the set, the better the computational efficiency. With affine Q 0 (v, R 0 (v, S 0 (v in decision variables v, the same technique applies to finding v hich robustly satisfies ( ( ( Q0 (v S 0 (v F (δ S0 (v R 0, R 0(v F (δ 0 (v 0. Examples: Discrete-time stability, eigenvalue-location in LM region. LFR s form the basis for advanced robustness analysis We have reduced robust stability test to a non-singularity test Multiplier relaxation schemes have been developed to verify robust stability Extends to stable structured dynamic uncertainties that satisfy (iω for all ω R { }. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 40 / 54

11 Example: Nonlinearity in the Loop A system ith linear fractional representation ẋ = Ax B, = Cx D, = (. roof Exists ɛ > 0 ith ( A X XA XB B X 0 ( 0 C D ( 0 C D ( ɛx is exponentially stable if there exist X 0 and a multiplier ith ( 0 A B such that ( 0 X X 0 ( ( ( 0 A B ( ( ( 0 C D 0 for all. ( 0 C D 0 Choose an arbitrary system trajectory. Then left-multiply this inequality ith col(x(t, (t and right-multiply ith col(x(t, (t: ( d (t dt x (txx(t (t ( (t (t ɛx (txx(t 0. Note that (t = ((t and the second inequality imply ( (t (t ( (t (t = ( ((t (t Hence d dt x (txx(t ɛx (txx(t 0. Done! ( ((t (t 0. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 41 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 42 / 54 Example: Multipliers for Saturation Nonlinearity Outline As long as min max have ( ( a(b ( 0. This translates into ( ( ( ( 2 a b ( 0 a b 2ab }{{} =b =a Test ith fixed multiplier guarantees stability for all trajectories for hich min (t max. With a = 0, e can infer global stability. For robust performance, choose a multiplier family q and search for the real scalar q 0. min max 1 Robust Stability against arametric Uncertainties Quadratic stability Rate-bounded uncertainties Robust performance 2 Multipliers Linear fractional representations Relaxations of robust linear matrix inequalities 3 Robust Controller Synthesis Setup Robust synthesis inequalities Summary Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 43 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 44 / 54

12 Configuration for Robust Controller Synthesis System Descriptions Design a controller guaranteeing: robust stability robustly a desired performance specification on p p. Consider the folloing approach: p y System Controller Use robust performance characteriation ith multipliers Try to satisfy the multiplier characteriation ith a suitable controller Just for notational simplicity, concentrate on robust stabiliation. Consider time-varying parametric uncertainty and quadratic stability. u p Uncontrolled LT part: ẋ = Ax B 1 Bu = C 1 x D 1 Eu y = Cx F Controller: ẋ c = A c x c B c y u = C c x c D c y Controlled LT part: ξ = Aξ B = Cξ D Uncertainty: (t = (δ(t(t. y : uncertainty input : uncertainty output u: control input y: measured output (δ(t System Controller u Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 45 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 46 / 54 Robust Stability Analysis nequalities Robust Synthesis nequalities Assume δ(t δ = conv{δ 1,..., δ N } (polytope containing ero. Robust stability guaranteed if exist X and Q, R, S ith ( (δ Q 0, k ( ( Q S (δ k S 0, k = 1,..., N R X 0, X A X B X A X B Q S 0 0. C D 0 0 S R C D Apply standard procedure from analysis to synthesis. Exists controller guaranteeing robust stability if exist v, Q, R, S: Q 0, X(v 0, ( (δ k 0 A(v B(v 0 C(v D(v ( Q S S R ( (δ k Q S 0 0 S R 0, Unfortunately not convex in all variables v and Q, R, S! k = 1,..., N 0 A(v B(v 0 C(v D(v 0. No technique knon ho to convexify in general! Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 47 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 48 / 54

13 Dual Robust Synthesis nequalities There is a controller guaranteeing robust stability, if exist v, Q, R, S: Q 0, ( ( ( Q S (δ k S R (δ k 0, A (v C T (v A (v X(v 0, B (v D (v 0 0 Q S S R k = 1,..., N C (v 0 B (v D (v 0 ( ( 1 Q S Q S Note that multipliers are related as S = R S. R No progress in general. Hoever, it helps for state-feedback synthesis! 0. Static State-Feedback Synthesis - Lucky Case! Recall block substitution: ( A(v B(v C(v D(v Last column does not depend on v... ( AY BM B1 = C 1 Y EM D 1... dual inequalities are affine in all variables robust state-feedback synthesis possible ith LM s!... control la D c = MY 1 Similar results for robust estimation! Very good exercise. p Estimator y System p Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 49 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 50 / 54 Output-Feedback - A Heuristic ath Remarks Scale the uncertainty set as conv{rδ 1,..., rδ N }, r [0, 1]. Solve the problem for the system ẋ A B 1 B = rc 1 rd 1 re y C F 0 Start ith the nominal design for r = 0. ncrease r by iterating beteen the folloing steps: For a fixed controller, maximie r over X and. For a fixed, maximie r over X and the controller. x u. Technique orks for general robust performance tests ith multipliers. Generalies to QC s. State-feedback and robust estimator design lead to LM problems. ntroducing parameter r is only one of a multitude of possibilities. Example: teratively fix any combination of (X, Y, (K, L, M, N, (Q, R, S to render the problem convex in other variables. Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 51 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 52 / 54

14 Summary Robustness Analysis Techniques to search for Lyapunov functions if the system depends affinely on uncertain parameters. Revealed role of linear fractional representations and multipliers if the system depends rationally on uncertain parameters. Sketched extension to non-linear uncertainties. That s all folks!! Robust Controller Synthesis State-feedback controller design is convex! Output feedback controller design non-convex! Suggested ideas behind heuristic algorithms. Many open problems! f parameters are measurable: LV design... Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 53 / 54 Carsten Scherer and Siep Weiland (Elgersburg Linear Matrix nequalities in Control Class 4 54 / 54