Outline Linear Matrix Inequalities in Control Carsten Scherer and Siep Weiland 7th Elgersburg School on Mathematical Systems heory Class 3 1 Single-Objective Synthesis Setup State-Feedback Output-Feedback Illustrations 2 Multi-Objective Synthesis Setup rouble and Remedy Mixed synthesis LMI Regions 3 Auxiliary Results Dualization Elimination Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 1 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 2 / 63 Outline 1 Single-Objective Synthesis Setup State-Feedback Output-Feedback Illustrations 2 Multi-Objective Synthesis Setup rouble and Remedy Mixed synthesis LMI Regions 3 Auxiliary Results Dualization Elimination System Interconnection We have seen several analysis specifications stability/performance All formulated in terms of matrix inequalities ry to achieve them by designing a controller F j _jst r ` ]Bt Bjj Generalized plant framework Weights are supposed to be included in system interconnection Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 3 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 4 / 63
Example: Mixed sensitivity design In tracking problems, the shaping of the sensitivity function reference to tracking error is required subject to the constraint that the control effort reference to control does not peak too much and rolls off at high frequency In view of these rough specs, consider Example: Mixed sensitivity design Choose a low-pass weighting function w 1 and a constant or high-pass weighting w 2 Define W 1 = w 1 I and W 2 = w 2 I and consider the following interconnection with weighted performance channels e 1 e 2 W 1 W 2 reference error control K G d y u K G -1-1 which indicates the relevant performance channels hen design a controller K which stabilizes the interconnection and minimizes the H norm of w z = cole 1, e 2 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 5 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 6 / 63 Example: Mixed sensitivity design Suppose closed-loop H norm can be suppressed below γ With S = I GK 1, this means that [ ] w1 iωsiω σ max w 2 iωriω and hence R = KS = KI GK 1 γ for all ω [, ] w 1 iωsiω γ, w 2 iωriω γ for all ω [, ] Example: Mixed sensitivity design In summary: With a stabilizing controller K we try to achieve a small or minimal H -norm for the open-loop interconnection d y u -1 e 1 e 2 W 1 W 2 which is written in the so-called generalized plant format: G and hence Siω γ w 1 iω, Riω γ w 2 iω for all ω [, ] e P d y u Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 7 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 8 / 63
System Descriptions and Problem Signals: Disturbance w, controlled output z, control u, measurement y Open-loop system and controller described as ẋ A B 1 B x z = C 1 D 1 E w ẋc Ac B and = c xc u C y C F u c D c y Controlled closed-loop system described with calligraphic matrices: ξ A B ξ = z w Problem Formulation Find controller that renders A Hurwitz internal stability and achieves desired performance specification for controlled system Dependence on Controller Parameters For the case of General output-feedback: A BD A B c C BC c B 1 BD c F = B c C A c B c F C 1 ED c C EC c D 1 ED c F Special case C = I, F = : Dynamic or static state-feedback: A BD A B c BC c B 1 = B c A c A BDc B or 1 C C 1 ED c EC c D 1 ED c D 1 1 Other information structures or configurations full-information feedback, estimation problems Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 9 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 1 / 63 Design of Quadratic Performance Controllers Qp S Given Π p = p with R p, find controller that renders A S p R p Hurwitz and the quadratic performance specification satisfied Guaranteed for controlled system iff exists X with I X I X, A B X A B I Q p S p I Sp R p I I X, X A X B X A X B I I X, I I I I Q p S p Sp R p I Q p S p I I Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 11 / 63 Recall various ways of writing LMI A X X A X B B X I = X A X B I I = I X A X B I I Π p = I I I X A X B Q p S p I = Sp R p I I Q p S p I I X A X B Sp R p Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 12 / 63
Static State-Feedback Synthesis Analysis Inequalities A BDc X, X X A BD c X B 1 B1 X Π p I C 1 ED c D 1 I C 1 ED c D 1 Perform congruence transformation with X 1 and diagx 1, I: X 1 X 1 A BD, c A BD c X 1 B 1 B1 I I C 1 ED c X 1 Π D p 1 C 1 ED c X 1 D 1 Define the new variables Y = X 1 and M := D c X 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 13 / 63 Static State-Feedback Synthesis Leads to synthesis inequalities for static state-feedback design: AY BM Y, AY BM B 1 B1 Π p I C 1 Y EM D 1 I C 1 Y EM D 1 Since R p this is easily turned into a genuine LMI as seen below Synthesis Procedure Check whether the synthesis inequalities have solution Y, M If no we are sure that quadratic performance cannot be achieved If yes then D c = MY 1 achieves quadratic performance Note that X = Y 1 proves quadratic performance for controlled system! Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 14 / 63 Example: Floating Platform Y Example: Floating Platform With actuator dynamics we use the following interconnection structure: F sea level sea floor Minimize drift Y resulting from lateral force F Controller should act on low-frequency component of force only Suppress resonance of M φ moment, angle wrt vertical axis Keep thruster actuation bounded Specifications Keep Y t below 25cm and φt below 3 o hruster actuation ut should stay below 3 Push resonance peak of M φ down below 15 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 15 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 16 / 63
Example: State-Feedback Design H state-feedback design with LMI s for F M Closed-loop poles and time-domain specifications: Ȳ 1 φ 5 u Example: State-Feedback Design Frequency domain-domain characteristics: F to Y,phi,u M to Y,phi,u 1 1 1 5 2 1 2 1 1 1 1 1 1 1 5 1 2 1 1 2 1 8 6 4 2 2 5 1 15 2 1 2 2 1 5 1 2 1 1 2 1 2 1 1 1 1 1 1 1 2 2 5 1 15 2 5 1 15 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 17 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 18 / 63 Output-Feedback: the Big picture Output-Feedback: the Big picture Quadratic performance synthesis problem: Find X and A c, B c, C c, D c st I I I X, I Q p S p I X A X B I X A X B Sp R p Essential idea is to construct a nonlinear mapping X, A c, B c, C c, D c v := X, Y, K, L, M, N and clever Y defining the congruence transformations where Xv := Av Bv Cv Dv := Y I I X AY BM A BNC B 1 BNF K AX LC XB 1 LF C 1 Y EM C 1 EN 1 ENF are all affine functions of the new decision variables v = X, Y, K, L, M, N his is the main reasoning! he mapping X, A c, B c, C c, D c v := X, Y, K, L, M, N Y X Y Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 19 / 63 is a convexification transformation and turns synthesis problem affine in variables v up to a point Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 2 / 63
And now the details Block ransformations Recall that A is partitioned Partition accordingly: X U X = U, X 1 Y V = V Why? Short computation reveals Y Y I X Y = I X, Observe that XY UV = I used later and that Y I I Y := V, Z := satisfy Y X = Z X U Controller transformation ransform controller parameters as K L XAY U XB Ac B = c V M N I CY I C c D c Y X AY Y X B = CY D AY BM A BNC B 1 BNF = K XA LC XB 1 LF C 1 Y EM C 1 EN 1 ENF he last block appeared after the congruence transformation on slide 19 We obtain affine dependence on X, Y and K, L, M, N! Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 21 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 22 / 63 Congruence ransformation For necessity: Can assume wlog that Y has full column rank For sufficiency: We will make sure that Y is square and non-singular ransform X, I I X A X B I Q p S p I Sp R p by congruence with Y and diagy, I into Y X Y, I Q p S p I Sp R p I I X A X B I I Y X AY Y X B CY D Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 23 / 63 Synthesis Inequalities Formulas on slide 22 lead to inequalities in X, Y, K, L, M, N Synthesis inequalities Y I, I X I I I I Q p S p I I AY BM A BNC B 1 BNF I K XA LC XB 1 LF Sp R p C 1 Y EM C 1 EN 1 ENF he second inequality is a quadratic matrix inequality since R p it can be turned into a genuine LMI Schur Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 24 / 63
esting Feasibility is LMI Problem Constant V, S, and affine Qv, Uv, W v in variables v Linearization Lemma esting existence of v with Uv and is LMI problem V W v Proof Second inequality reads as Qv S S Uv 1 V W v V QvV V SW v W v S V W v Uv 1 W v Hence both inequalities are equivalent to the following single LMI: V QvV V SW v W v S V W v W v Uv Output-Feedback: Controller Construction Synthesis Procedure Solve synthesis inequalities to compute X, Y and K, L, M, N Determine non-singular U, V with U V = I XY Analysis inequalities on slide 11 or 19 are satisfied for Ac B c C c D c 1 Y V I X = I X U 1 U XB K XAY L = I M N V CY I A c has the same size as A Construct full order controller Freedom in choice of U, V : Controller state-coordinate change ypical choice: U = X, V = X 1 Y or U = Y 1 X, V = Y 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 25 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 26 / 63 General Procedure Variables and Blocks Rewrite analysis inequalities in terms of blocks X, X A, X B, C, D Find formal congruence transformation involving Y to transform into inequalities in terms of blocks Y X Y, Y X AY, Y X B, CY, D State-Feedback: v = Y, M and Av Bv AY BM B1 Xv = Y, = Cv Dv C 1 Y EM D 1 Obtain synthesis inequalities by substitution Y Y X Y Xv, [X A]Y Y [X B] Av Bv Cv Dv with affine Av, Bv, Cv, Dv in new variables v Controller construction independent of particular analysis inequalities! Construction leads to controller of same order as plant Works both in continuous-time and discrete-time in identical fashion Output-Feedback: v = X, Y, K, L, M, N and Y I Xv = I X AY BM A BNC B Av Bv 1 BNF = K XA LC XB Cv Dv 1 LF C 1 Y EM C 1 EN 1 ENF Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 27 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 28 / 63
Illustration for H Control Problem H -Synthesis here exists a controller that renders A Hurwitz and achieves Example: Floating Platform With actuator dynamics we use the following interconnection structure: CsI A 1 B D < γ if and only if there exists v with I I Av Bv Cv Dv Xv I γi I 1 γ I I I Av Bv Cv Dv Linearization Lemma: Can directly compute minimal achievable γ since 1 γ I = γi 1 I I Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 29 / 63 Specifications Keep Y t below 25cm and φt below 3 o hruster actuation ut should stay below 3 Push resonance peak of M φ down below 15 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 3 / 63 Example: Output-Feedback Design H design with LMI s for F M Ȳ 1 φ 5 u Closed-loop poles and time-domain specifications: Example: Output-Feedback Design Frequency domain-domain characteristics: F to Y,phi,u M to Y,phi,u 1 1 1 5 2 1 2 1 1 1 1 1 1 5 1 5 1 2 1 1 2 1 8 6 4 2 2 5 1 15 2 1 1 2 2 1 5 1 2 1 1 2 1 2 1 1 1 1 1 1 1 2 2 5 1 15 2 5 1 15 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 31 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 32 / 63
Illustration for positive real design Positive-real Synthesis here exists a controller that renders A Hurwitz and achieves iω iω if and only if there exists v with Xv for all ω R { } Av Av Bv Cv Bv Cv Dv Dv his is an LMI in v = X, Y, K, L, M, N Apply synthesis procedure on slide 26 to find controller A c, B c, C c, D c together with Lyapunov function V ξ = ξ X ξ Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 33 / 63 Illustration for H 2 -Synthesis Recall: A Hurwitz and CsI A 1 B D 2 < γ iff exists P with D =, AP PA γ 1 BB, tracecpc < γ iff Schur exist X and Z with tracez < γ and A D =, X X A X B X C B, X γi C Z Apply formal congruence trafo with diagy, I H 2 -Synthesis Exists a controller which renders A Hurwitz and closed-loop H 2 -norm smaller than γ iff exist v and Z with tracez < γ and Av Dv =, Av Bv Xv Cv Bv, γi Cv Z Allowed: Affine equality constraints & auxiliary variables Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 34 / 63 Remarks No hypotheses on system required ake precautions to render controller construction well-conditioned Can directly optimize affine functional of involved variables For example it is possible to directly compute inf CsI A stable A 1 B D k, k = 2, Warning: Optimal controllers do in general not exist If γ approaches optimum for H or H 2 problem, the poles of closed-loop system move to imaginary axis and/or the controller parameters blow up Stay away from optimality! Remove fast poles by residualization! An interesting and important variation: Estimation Problems Interconnection for estimation: e Estimator z y System Derivation of convexifying parameter transformation as exercise Find estimator which minimizes H -norm of w e worst case error as small as possible Find estimator which minimizes H 2 -norm of w e optimally reduce asymptotic variance against white noise w Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 35 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 36 / 63
Outline Multi-Objective Control 1 Single-Objective Synthesis Setup State-Feedback Output-Feedback Illustrations 2 Multi-Objective Synthesis Setup rouble and Remedy Mixed synthesis LMI Regions 3 Auxiliary Results Dualization Elimination Design controller which achieves multiple objectives on different channels of closed-loop system: Loop shaping: w1 z 1 < γ 1 Disturbance attenuation: w2 z 2 2 < γ 2 No loss of generality F[ @ F[ E j[ E _jst F[ 4 r ]Bt Bjj Relevant channels are w k z k, k = 1,, q on diagonal ` j[ @ j[ 4 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 37 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 38 / 63 Multi-Channel System Description Open-loop system and controller: ẋ A B 1 B q B x z 1 C 1 D 1 D 1q E 1 w 1 z q C q D q1 D q E q w q y C F 1 F q u, Controlled closed-loop system: ξ A B 1 B q z 1 = C 1 D 1 D 1q z q C q D q1 D q ẋc Ac B = c xc u C c D c y ξ w 1 w q Multi-Objective H 2 /H Control Find controller such that A is Hurwitz and C 1 si A 1 B 1 D 1 < γ 1, C 2 si A 1 B 2 D 2 2 < γ 2 Related analysis inequalities: A X 1 X 1 A X 1 B 1 C 1 B1 X 1 γ 1 I D1 C 1 D 1 γ 1 I A X 2 X 2 A X 2 B 2 X B2 X, 2 C 2, tracez < γ 2 γ 2 I C 2 Z 2, D 2 = In general need X 1 X 2 Untractable in state-space Relaxation: Introduce extra constraint X 1 = X 2 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 39 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 4 / 63
Mixed-Objective H 2 /H Control Mixed analysis inequalities Find controller such that there exist X and Z with A X X A X B 1 C 1 B1 X γ 1I D1 C 1 D 1 γ 1 I A X X A X B 2 X C B2 X γ, 2, tracez < γ 2I Z 2, D 2 = C 2 Solvability of mixed problem implies stability of A and the desired norm inequalities Can hence conclude in general that Minimal mixed γ 2 Minimal multi-objective γ 2 X 1 = X 2 often implies that there is a gap and the inequality is strict Solution of Mixed H 2 /H Control But X 1 = X 2 implies tractability: Can apply general procedure! Mixed synthesis inequalities Av Av B 1 v C 1 v B 1 v γ 1 I D 1 v C 1 v D 1 v γ 1 I Av Av B 2 v Xv C2 v B 2 v, γ 2 I C 2 v Z tracez < γ 2, D 2 v =, Can be solved by standard algorithms controller construction as usual controller order identical to order of system! Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 41 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 42 / 63 Extensions Example: Floating Platform With actuator dynamics we use the following interconnection structure: For fixed α 1, α 2 and variable γ 1, γ 2, optimize α 1 γ 1 α 2 γ 2 Analyze trade-off between specifications by playing with α 1, α 2 Improve relaxation with tuning parameter α > : X 1 = αx 2 Line-search over α Might reduce conservatism Can include more than two LMI performance on different channels Never forget conservatism Possible to include other type of constraints Important example: Closed-loop poles in convex LMI region Specifications Keep Y t below 25cm and φt below 3 o hruster actuation ut should stay below 3 Push resonance peak of M φ down below 15 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 43 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 44 / 63
Example: Mixed Design F Ȳ H -bound 8: H M 1 φ 2 -minimization: Closed-loop poles and time-domain specifications: F M u Example: Mixed Design Frequency domain-domain characteristics: F to Y,phi,u M to Y,phi,u 1 1 1 5 2 1 1 2 1 1 1 1 1 1 5 1 2 1 1 2 1 8 6 4 2 2 5 1 15 2 1 2 2 1 5 1 2 1 1 2 1 2 1 1 1 1 1 2 2 1 1 5 1 15 2 5 1 15 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 45 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 46 / 63 LMI Regions Definition For a real symmetric 2m 2m matrix P the set of complex numbers L P := { z C : I zi is called the LMI region described by P P I zi } Half-plane: Rez < α z z 2α z L P1 Circle: z < r zz r 2 z L P2 Sector: Rez tanφ < Imz z L P3 where 2α 1 r P 1 =, P 1 2 = 2 sinφ cosφ, P 1 3 = cosφ sinφ Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 47 / 63 Properties of LMI Regions he intersection of finitely many LMI regions is an LMI region Can hence describe quite general subsets of C as LMI regions Proof rivially follows from usual stacking property of LMI s: I Q1 S 1 I zi S1 R, 1 zi if and only if I I zi zi Q 1 S 1 Q 2 S 2 S 1 R 1 S 2 R 2 I Q2 S 2 I zi S2 R 2 zi I I zi zi Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 48 / 63
Recap Generalization Standard Stability Characterization Recall the definition of the Kronecker product A 11 B A 1n B A B = for A C m n, B C k l A m1 B A mn B It is easy to prove following properties for compatible dimensions: 1 A = A = A 1 A B C = A C B C A BC D = AC BD A B = A B and A B 1 = A 1 B 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 49 / 63 heorem All eigenvalues of A R n n are contained in the LMI-region { z C : I zi Q S S R if and only if there exists a K such that I A I K Q K S K S K R I zi I A I } Beautiful generalization of standard stability test! Chilali, Gahinet 96 Proof Along lines of algebraic proof for standard LI stability and by exploiting the properties of the Kronecker product Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 5 / 63 Closed-loop Poles in Convex LMI Regions Eigenvalues of A in LMI-region defined by Q, R, S iff exists X with I X Q X S I X, A I X S X R A I Example: Floating Platform With actuator dynamics we use the following interconnection structure: or equivalently X, X Q X A S S X A A X A R Assumption Suppose that R Factorize as R = U 1 with U LMI s equivalent to Schur and properties of Kronecker product: X Q X A S S X A X A X A X U Formal congruence trafo with diagy I, Y I Done! Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 51 / 63 Specifications Keep Y t below 25cm and φt below 3 o hruster actuation ut should stay below 3 Push resonance peak of M φ down below 15 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 52 / 63
Example: Mixed Pole-Placement Reduce resonance by pushing resonating pole away from axis Closed-loop poles and time-domain specifications: Example: Mixed Pole-Placement Frequency domain-domain characteristics: F to Y,phi,u M to Y,phi,u 1 1 1 5 2 1 1 2 1 1 1 1 1 1 5 1 2 1 1 2 1 8 6 4 2 2 5 1 15 2 1 2 2 1 5 1 2 1 1 2 1 2 1 1 1 1 1 2 2 1 1 5 1 15 2 5 1 15 2 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 53 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 54 / 63 Summary Outline View mixed synthesis as Lyapunov shaping technique: Improve existing controller by adding extra specs on closed-loop system Analyze designed mixed controller with different Lyapunov matrices Exist solutions to genuine multi-objective control problem: Can initialize with mixed controller Can compute upper bounds and lower bounds Expensive Controller order might be large 1 Single-Objective Synthesis Setup State-Feedback Output-Feedback Illustrations 2 Multi-Objective Synthesis Setup rouble and Remedy Mixed synthesis LMI Regions 3 Auxiliary Results Dualization Elimination Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 55 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 56 / 63
A Dualization Lemma Application to Quadratic Performance Synthesis Dualization Lemma Let P be nonsingular hen I P I and I W P I W Consider quadratic performance specification with nonsingular cost matrix Qp S Π p = p Qp Sp, Π 1 Sp R p = p S p satisfying R p, Qp R p are equivalent to I Note that P 1 I W im I and W I P 1 W I equals orthogonal complement of I im W In general: Let P = P be nonsingular with k negative eigenvalues If the subspace S with dimension k is P -negative then S is P -positive Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 57 / 63 Primal Synthesis Inequalities Exists controller that renders A Hurwitz and quadratic performance specification for Π p satisfied iff there exists v with I I I Xv, I Q p S p I Av Bv I Av Bv Cv Dv Sp R p Cv Dv Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 58 / 63 Application to Quadratic Performance Synthesis Elimination of ransformed Controller Parameters Consider quadratic performance specification with cost matrix Qp S Π p = p Qp Sp, Π 1 Sp R p = p S p satisfying R p, Qp R p Dual Synthesis Inequalities Exists controller that renders A Hurwitz and quadratic performance specification for Π p satisfied iff there exists v with Av Cv I Xv, Bv Dv Q p S Av Cv p Bv Dv I I I I S p R p I A crucial general observation Unstructured matrix variables in one LMI can often be eliminated For example let us recall the particular structure Av Bv = Cv Dv AY A B 1 XA XB 1 C 1 Y C 1 D 1 B I E K L M N K L Can eliminate in synthesis inequalities How? M N I C F Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 59 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 6 / 63
Elimination Consider the quadratic matrix inequality in Z I U ZV W P I U ZV W where P is non-singular, its right-lower block is positive semi-definite, and the left-upper block of P 1 is negative semi-definite Elimination Lemma Let U, V be basis matrices of keru, kerv hen QI is solvable iff V I I P V W W & U W P 1 W U I I QI Application to Quadratic Performance Synthesis Consider quadratic performance specification with nonsingular cost matrix Qp S Π p = p Qp Sp, Π 1 Sp R p = p S p satisfying R p, Qp R p Primal Synthesis Inequalities Exists controller that renders A Hurwitz and quadratic performance specification for Π p satisfied iff there exists v with I I I Xv, I Q p S p I Av Bv I Av Bv Cv Dv Sp R p Cv Dv Can explicitly construct solution Z if solvability conditions satisfied Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 61 / 63 Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 62 / 63 QP Synthesis Inequalities after Elimination With basis matrices Φ of ker B E and Ψ of ker C F, the synthesis inequalities after elimination read as Y I, I X I I I Ψ I Q p S p I XA XB 1 I XA XB 1 Ψ, C 1 D 1 Sp R p C 1 D 1 Φ Y A Y C1 B1 D1 I I I Qp Sp I S p Rp Y A Y C1 B1 D1 I I Much fewer variables! Nice system theoretic interpretation! Φ Carsten Scherer and Siep Weiland Elgersburg Linear Matrix Inequalities in Control Class 3 63 / 63