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

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1 Outline Linear Matrix Inequalities and Robust Control Carsten Scherer and Siep Weiland 7th Elgersburg School on Mathematical Sstems Theor Class 4 version March 6, Robust Stabilit Against Uncertainties Quadratic stabilit Rate-bounded uncertainties Robust performance 2 Robust Controller Snthesis Robust State-Feedback Snthesis Robust Estimation Summar 3 Linear Parametricall-Varing Controller Snthesis Class 4 version March 6, / Class 4 version March 6, / Outline Time-Invariant Parametric Uncertaint 1 Robust Stabilit Against Uncertainties Quadratic stabilit Rate-bounded uncertainties Robust performance 2 Robust Controller Snthesis Robust State-Feedback Snthesis Robust Estimation Summar 3 Linear Parametricall-Varing Controller Snthesis Consider the linear time-invariant LTI sstem ẋt Aδxt where A is a continuous function of a constant parameter vector δ δ 1... δ p which is known to be contained in an commonl closed uncertaint set δ R p. Robust stabilit analsis Is this sstem asmptoticall stable for all δ δ? Class 4 version March 6, / Class 4 version March 6, /

2 Example: Load variation in a mechanical sstem Example: Academic Differential eq. kernel model: Mẍ + Bẋ + Kx 0 State-space model: ẋ v 0 I M 1 K M 1 B where x is position and v ẋ is velocit. Uncertain parameter: load changes in the mass matrix M M x v b k x m load variation in an MSD sstem Academic example with rational parameter-dependence 1 2δ 1 2 ẋ δ δ 3 10 δ 1 +1 where 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 actuall a poltope 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 Class 4 version March 6, / Class 4 version March 6, / Quadratic Stabilit Example: Academic cont d Definition The uncertain sstem ẋ Aδx with δ δ is said to be quadraticall stable if there exists X 0 with A δx + XAδ 0 for all δ δ. Wh is the name? V x x Xx is a quadratic Lapunov function. Wh this is relevant? Implies that Aδ is Hurwitz for all δ δ. How to check? Eas, if Aδ is affine in δ and δ co{δ 1,..., δ N } is a poltope with moderate number of generators: Verif whether If Aδ is not affine in δ, a parameter transformation often helps! Regarding the previous example, introduce δ 4 δ Test δ quadratic stabilit 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? LMI-Toolbox: Sstem quadraticall stable for δ 1, δ 2, δ 4 rδ with largest possible factor r is feasible. X 0, A δ k X + XAδ k 0, k 1,..., N Quadraticall stable for a deflated set 0.49 δ. Not for rδ with r > This critical factor is called the quadratic stabilit margin. Class 4 version March 6, / Class 4 version March 6, /

3 Example - implementation LMI toolbox LMI-toolbox commands: quadstab, pss and pvec Define sstem matrices >> S0 ltiss[ ; ; ]; >> S1 ltiss[ ; ; ], zeros3; >> S2 ltiss[ ; ; ], zeros3; >> S4 ltiss[ ; ; ], zeros3; Define parameter ranges >> pv pvec box,[-0.5 1; -2 1; ]; Define affine parameter dependent sstem >> pdss pss pv, [S0,S1,S2,S4] ; Find quadratic stabilit margin >> [tau,x] quadstabpdss, [1 1 1e8]; Class 4 version March 6, / Time-Varing Dnamic Uncertainties Now, assume that the parameters δt var with time, and that the are known to satisf δt δ for all t. Check stabilit of ẋt Aδtxt, δ : R δ. Theorem The uncertain sstem with time-varing uncertainties is exponentiall stable if there exists X 0 with A δx + XAδ 0 for all δ δ. The proof will be given for a more general result in full detail. Quadratic stabilit does in fact impl robust stabilit for arbitrar fast time-varing uncertaint. If bounds on the rate of variations of the parameters are known, this test is conservative. Class 4 version March 6, / Rate-Bounded Uncertainties Let us hence assume that the trajectories δ. are continuousl differentiable and are onl known to satisf δt δ and δt v for all time instances. Here δ R p and v R p are given compact sets e.g., poltopes. Robust stabilit analsis Verif whether the linear time-varing sstem ẋt Aδtxt is exponentiall stable for all trajectories δ. that satisf the above described bounds on value and variation. Ke idea: Search for a suitable quadratic Lapunov function. Class 4 version March 6, / Main Stabilit Result Theorem Suppose Xδ is continuousl differentiable w.r.t. δ and satisfies Xδ 0, k Xδv k + A δxδ + XδAδ 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-varing sstem satisf xt Ke at t 0 xt 0 for all t t 0. Covers man tests in literature. Stud the proof to derive variants! In general, this condition is onl sufficient! It is necessar in case v {0}: Time-invariant static uncertaint. Note that, in the Theorem, the dependenc structure is not restricted! Class 4 version March 6, /

4 Proof Continuit/compactness exist α, β, γ > 0 such that for all δ δ, v v: αi Xδ βi, k Xδv k + A δxδ + XδAδ γi. Suppose that δt is an admissible parameter trajector and let xt denote a compatible state-trajector of the sstem. Here is the crucial point: [ ] d dt x txδtxt x t }{{} k Xδt δ k t xt+ ξt [ ] + x t A δtxδt + XδtAδt xt. Since δt δ and δt v we can hence conclude α xt 2 2 ξt β xt 2 2, d dt ξt γ xt 2 2. Proof The conclusion of the proof follows b recognizing that xt α ξt, ξt β xt 2 2, ξt γ β ξt. The latter inequalit leads to ξt ξt 0 e γ β t t 0 With the former inequalities we infer for all t t 0. xt 2 2 β α e γ β t t 0 xt for all t t 0. Choose K β/α and a γ/2β to complete proof. Class 4 version March 6, / Class 4 version March 6, / Extreme Cases Remarks Parameters are time-invariant: v {0}. We need to find a Xδ satisfing Xδ 0, A δxδ + XδAδ 0 for all δ δ. Parameters var arbitraril fast: We need to find a parameter-independent X satisfing X 0, A δx + XAδ 0 for all δ δ. This is identical to the quadratic stabilit test! Can appl the subsequentl suggested numerical techniques in both cases! We have derived general results based on Lapunov functions which still depend quadraticall on the state which is restrictive, but which allow for non-linear smooth dependence on the uncertain parameters. Pure algebraic test which does not involve state nor δt trajectories. Not eas to appl: Have to find a function satisfing a partial differential LMI. Have to make sure that inequalit holds for all δ δ, v v. Allows to easil derive special cases which are or can be implemented with LMI solvers. Affine dependence is just around the bend We will onl consider a couple of examples. Class 4 version March 6, / Class 4 version March 6, /

5 Example: Affine Dependence - Affine Lapunov Matrix Suppose Aδ depends affinel on the parameters: Aδ A 0 + δ 1 A δ p A p. Parameter- and rate-constraints are intervals box structure: δ {δ R p : δ k [δ k, δ k ] }, v {v R p : v k [v k, v k ] } Identical to 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 Xδ: Xδ X 0 + δ 1 X δ p X p and hence k Xδ X k. Example: Affine Dependence - Affine Lapunov Matrix With δ 0 1 observe that k Xδv k + A δxδ + XδAδ X k v k + ν0 µ0 δ ν δ µ A ν X µ + X µ A ν. Is affine in X 1,..., X p and v 1,..., v p but quadratic in δ 1,..., δ p. Relaxation Include an additional constraint: A ν X ν + X ν A ν 0. Implies that it suffices to guarantee the required inequalit at the generators. Wh? Partiall convex function over the box! The extra condition renders the test conservative, but numericall verifiable. Still sufficient for robust stabilit! Class 4 version March 6, / Class 4 version March 6, / Example: Affine Dependence - Affine Lapunov Matrix Robust exponential stabilit guaranteed if There exist X 0,..., X p with A ν X ν + X ν A ν 0, ν 1,..., p, and X k δ k 0 k0 X k v k + for all δ δ g and v v g and δ 0 1. ν0 µ0 δ ν δ µ A ν X µ + X µ A ν 0 This test is implemented in the LMI toolbox within RCT. For rate-bounded uncertainties, it is often much less conservative than the quadratic stabilit test. Useful to understand the proof and derive our own generalizations. Robust Quadratic Performance Given an uncertain sstem described as ẋt Aδtxt + Bδtwt zt Cδtxt + Dδtwt with continuousl differentiable parameter trajectories δ. that satisf δt δ and δt v δ, v R p compact. Robust quadratic performance propert Robust stabilit and existence of an ɛ > 0 such that, for all solution trajectories wt, zt, δt with x0 0, it holds that 0 wt zt Qp S p S p R p } {{ } P p wt zt dt ɛ w 2 2. Class 4 version March 6, / L 2 -gain, passivit,... Class 4 version March 6, /

6 Sufficient Condition for Robust Quadratic Performance Theorem Assume that R p 0. Suppose that there exists a continuousl differentiable smmetric-valued Xδ such that Xδ 0 and k Xδv k +A δxδ+xδaδ S p XδBδ R p + B δxδ 0 0 I Qp S + p 0 I 0 Cδ Dδ Cδ Dδ for all δ δ, v v. Then, the uncertain sstem satisfies the robust quadratic performance propert. Numerical search for Xδ: like for stabilit! Can be easil extended to other LMI performance specifications! Class 4 version March 6, / Sketch of the Proof Exponential stabilit: Left-upper block is k Xδv k + A δxδ + XδAδ + C δr p Cδ 0. }{{} 0 Hence, we can appl our general result on robust exponential stabilit. Performance: Adding ɛi small ɛ > 0 to the right-lower block compactness. Left- & right-multipl the inequalit with colxt, wt: d wt wt dt x txδtxt + P zt p + ɛw twt 0. zt Integrate over [0, T ] and use x0 0 to obtain T x wt wt T XδT xt + P zt p zt 0 dt ɛ T 0 w twtdt. Since XδT 0, x T XδT xt can be dropped and the limit T is taken to obtain the required quadratic performance inequalit. Class 4 version March 6, / Outline Configuration for Robust Controller Snthesis 1 Robust Stabilit Against Uncertainties Quadratic stabilit Rate-bounded uncertainties Robust performance Design a controller guaranteeing: robust stabilit robustl a desired performance specification on w z z Sstemδt Controller w u 2 Robust Controller Snthesis Robust State-Feedback Snthesis Robust Estimation Summar 3 Linear Parametricall-Varing Controller Snthesis Consider the following approach: Sstem matrices depend affinel on parameter δ Parameter varies in poltope: δ co{δ 1,..., δ N } Emplo parameter-independent storage function Goal: Robust stabilit and quadratic performance for controlled sstem. Class 4 version March 6, / Class 4 version March 6, /

7 Sstem Descriptions Uncontrolled uncertain sstem: Controller: Controlled uncertain sstem: ẋ Aδtx + B 1 δtw + Bδtu z C 1 δtx + D 1 δtw + Eδtu Cδtx + F δtw ẋ c A c x c + B c u C c x c + D c ξ Aδtξ + Bδtw z Cδtξ + Dδtw We consider static state-feedback snthesis and estimator snthesis. Class 4 version March 6, / Static State-Feedback Snthesis Find X 0 and D c such that for δ co{δ 1,..., δ N }: Aδ + BδDc X + X Aδ + BδD c X B 1 δ B 1 δ + X 0 0 I + P p 0 C 1 δ + EδD c D 1 δ Variable change Y X 1 and M : D c X 1 as earlier: Find Y 0 and M such that for δ co{δ 1,..., δ N }: AδY + BδM + AδY + BδM B 1 δ B 1 δ 0 + P p + 0 I C 1 δy + EδM D 1 δ 0 This is an LMI problem! Does not work for output-feedback snthesis! Class 4 version March 6, / Robust Estimation Uncertain sstem: ẋ z Aδt C 1 δt Cδt B 1 δt D 1 δt x w F δt Suppose Aδ is quadraticall stable on parameter poltope. Robust estimator: ẋc ẑ Ac B c C c D c xc Robust Estimation: Is Output-Feedback Snthesis Uncertain sstem with estimator output as control input: ẋ Aδt B 1 δt 0 x z ẑ C 1 δt D 1 δt I w Cδt F δt 0 u Robust estimator viewed as controller: ẋc Ac B c u C c D c xc Design problem: For given γ > 0 check existence of an estimator such that A c is Hurwitz and z ẑ 2 sup 0< w 2 < w 2 < γ Design problem: For given γ > 0 check existence of an estimator such that A c is Hurwitz and z ẑ 2 sup 0< w 2 < w 2 < γ Class 4 version March 6, / Class 4 version March 6, /

8 Modified Transformation Interconnection is ẋ Aδt 0 B 1 δt ξ B c Cδt A c B c F δt z C 1 δt D c Cδt C c D 1 δt D c F δt Proceed as for output-feedback snthesis with factorization Y T X Z where Y T I Y 1 V Y 1 0, Z I 0 X U x ξ. w With the suggested factorization consider Y T X AY Y T X B C c Y D Y 1 0 A 0 I I Y 1 0 B1 X U B c C A c V T Y 1 0 X U B c F I I C1 D c C C c V T Y 1 D 0 1 D c F. Modified Transformation Y 1 A 0 I I Y 1 B 1 XA + UB c C UA c V T Y 1 0 XB 1 + UB c F C1 D c C C c V T Y 1 C 1 D c C D 1 D c F Y 1 A Y 1 A Y 1 B 1 XA + UB c C + UA c V T Y 1 XA + UB c C XB 1 + UB c F C1 D c C C c V T Y 1 C 1 D c C D 1 D c F ZA ZA ZB 1 XA + LC + K XA + LC XB 1 + LF C1 NC M C 1 NC D 1 NF for Z : Y 1 and K L M N UAc V T Y 1 UB c C c V T Y 1 D c The latter involves no sstem parameters!. Class 4 version March 6, / Class 4 version March 6, / Robust Estimator Snthesis Summar The snthesis inequalities with quadratic performance spec read as Z Z 0 and Z X I 0 0 I I 0 0 I Q p 0 0 S p 0 0 I I ZAδ ZAδ ZB 1 δ 0 0 I XAδ+LCδ+K XAδ+LCδ XB 1 δ+lf δ 0 0 Sp T 0 0 R p C 1 δ NCδ M C 1 δ NCδ D 1 δ NF δ Find X, Z and K, L, M, N such these hold for δ co{δ 1,..., δ N }. Choose invertible U, V with U V I XZ 1. Invert transformation on previous slide to find A c, B c, C c, D c. Robustness Analsis and Snthesis Techniques to search for Lapunov functions if the state-space model of the sstem depends affinel on the uncertain parameters. Did not reveal role of linear fractional representations and multipliers if the sstem depends rationall on uncertain parameters. Did not sketch extension to nonlinearities / non-linear uncertainties. Robust Controller Snthesis Robust state-feedback controller design is convex! Robust estimator design is convex! Output feedback controller design non-convex! Multipliers allow for heuristic algorithms. Man open problems! If parameters are measurable on-line: LPV design... Class 4 version March 6, / Class 4 version March 6, /

9 Outline 1 Robust Stabilit Against Uncertainties Quadratic stabilit Rate-bounded uncertainties Robust performance 2 Robust Controller Snthesis Robust State-Feedback Snthesis Robust Estimation Summar 3 Linear Parametricall-Varing Controller Snthesis Configuration for LPV Snthesis Open-loop sstem: ẋ z Controller: ẋc u Controlled Sstem: ξ z Aδt B 1 δt Bδt C 1 δt D 1 δt Eδt Cδt F δt 0 Ac δt B c δt C c δt D c δt Aδt Bδt Cδt Dδt xc ξ w x w u Class 4 version March 6, / Parameter trajectories satisf δt δ co{δ 1,..., δ N }. Class 4 version March 6, / Analsis If there exists a X 0 such that 0 X 0 0 [ ] X Q p S p 0 0 Sp R p I 0 Aδ Bδ 0 I Cδ Dδ 0 for all δ δ one has achieved robust quadratic performance for the controlled sstem. Reduces to LMIs in generators δ 1,..., δ N if Aδ Bδ is affine in δ. Cδ Dδ How can we guarantee that? What is the resulting design procedure? Hpotheses on the Sstem Description Hpothesis on the open-loop sstem: Aδ B 1 δ B C 1 δ D 1 δ E is affine in δ. C F 0 Observe independence of B, E and C, F from δ can be guaranteed via simple filters. Hpothesis on the controller: Ac δ B c δ C c δ D c δ is affine in δ. Aδ Bδ Then is affine in δ. Cδ Dδ Class 4 version March 6, / Class 4 version March 6, /

10 From Analsis to Snthesis LPV Snthesis Inequalities We achieve robust quadratic performance for the controlled sstem if there exists X 0 such that for all k 1,..., N: [ ] 0 X 0 0 X Q p S p 0 0 Sp R p I 0 Aδ k Bδ k 0 I Cδ k Dδ k 0. How to proceed? Appl our general convexifing transformation X, A c δ k, B c δ k, C c δ k, D c δ k X, Y, K k, L k, M k, N k. Snthesis inequalities for k 1,..., N Y I 0 I X I 0 0 I I 0 0 I Q p 0 0 S p 0 0 I I Aδ k Y +BM k Aδ k +BN k C B 1 δ k +BN k F 0 0 I K k XAδ k +L k C XB 1 δ k +L k F 0 0 Sp T 0 0 R p C 1 δ k Y +EM k C 1 δ k +EN k C D 1 δ k +EN k F Class 4 version March 6, / Class 4 version March 6, / LPV Controller Construction Comments Solve inequalities for X, Y, K k, L k, M k, N k, k 1,..., N. Construct X and the extreme-point controllers Ac,k B c,k, k 1,..., N C c,k D c,k as in the standard snthesis procedure. Let δ δ, represented as δ N λ k δ k with λ k 0, Then, the analsis inequalities are satisfied with Ac B c C c D c N Ac,k λ k C c,k B c,k D c,k N λ k 1.. For controller simulation and implementation one has to proceed as follows: At time t, find convex combination coefficients in δt N λ k tδ k and use N to define the dnamics of the LPV controller. Ac,k B λ k t c,k C c,k D c,k Determination of λ k t requires the solution of an LP. In order to ensure uniqueness e.g. to assure continuit in time one could enforce, in addition, that, e.g., N λ2 k t is minimized. This simplest procedure for designing LPV controllers is implemented in the Robust Control Toolbox in Matlab. Class 4 version March 6, / Class 4 version March 6, /

11 High-Performance Aircraft Sstem Main Idea u n z q u: Control input α: Measurable parameter n z : Tracked output Rewrite as linear parameter-varing sstem α Kδ 1 [ an δ 2 2 +b n δ 2 +c n 2 δ 1 /3 α+d n u ] +q q δ 1 2 [ a m δ 2 2 +b m δ 2 c m 7 8δ 1 /3 α+d m u ] n z δ 1 2 [ a n δ 2 2 +b n δ 2 +c n 2 δ 1 /3 α+d n u ] Nonlinear sstem description with aerodnamic effects: α KM [ a n α 2 +b n α+c n 2 M/3 α+d n u ] +q q M 2[ a m α 2 +b m α c m 7 8M/3 α+d m u ] n z M 2[ a n α 2 +b n α+c n 2 M/3 α+d n u ] with bounds 2 δ 1 t 4 and 20 δ 2 t 20. Design good controller scheduled with δ 1 t, δ 2 t Is good controller for nonlinear sstem Class 4 version March 6, / Class 4 version March 6, / Interconnection Structure Application to Aircraft Model Snthesis with Convex Hull Relaxation Mt decreases in 5 seconds from 4 to 2. Mt decreases in 5 seconds from 4 to Normal Normal acceleration acceleration Reference Reference Response Response Normal acceleration Model-Matching Let controlled sstem approximatel act like ideal model w ideal Time Class 4 version March 6, / Class 4 version March 6, / 42/50

12 That s all folks!! Class 4 version March 6, 2015 /