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ELE7029: Optimal Digital Control System Lecture IV Discrete-time Control Design via LMI Optimization Goals of this lecture · LMIs for Stability, H2 and H∞ performances · Stabilizing state feedback control · H2 state feedback control · H∞ state feedback control · H2 output feedback control · H∞ output feedback control S.-H. Lee ([email protected]) Elec. Engr., Hanyang University Spring 2022 Outline 1. Fundamental LMIs 2. Stabilizing State Feedback Control 3. H2 State Feedback Control 4. H∞ State Feedback Control 5. H2 Output Feedback Control 6. H∞ Output Feedback Control S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 1 Fundamental LMIs Consider a linear discrete-time system Tzw : x(k+ 1) = Φx(k) + Γw(k) z(k) = Cx(k) +Dw(k) Let us describe LMI conditions for stability, H2 and H∞ performances for the system. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 2 ■ Stability: The system x(k+ 1) = Φx(k) is stable iff i) X > 0 s.t. ΦXΦT X < 0 (4.1a) i) X > 0 s.t. X ΦX XΦT X < 0 (4.1b) ii) X > 0 and G s.t. X ΦG GTΦT X (G+GT) < 0 (4.1c) iii) X > 0 and G and W s.t. X ΦW WTΦT WT +ΦX W + XΦT X (G+GT) < 0 (4.1d) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 3 Note: Introducing a slack variable G ∈ Rnx×nx P ΦG GTΦT G+GT P > 0 · Suff: G= GT = P recovers the LMI (4.1b) · Nec: Find that G+GT P> 0 and (P G)TP 1(P G) = P G GT +GTP 1G≥ 0 assures GTP 1G≥ G+GT P Therefore P ΦG GTΦT GTP 1G > 0 Congruence transformation by T= diag(I,G 1P) recovers the standard form. TT P ΦG GTΦT GTP 1G T= P ΦP PΦT P > 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 4 Good to Note: H2 gain with D= 0 Recall ΦPΦT P+ Γ Γ T = 0 → tr(CPCT) = ∥Tzw∥22 ΦPΦT P+ Γ Γ T < 0 → tr(CPCT)> ∥Tzw∥22 P ΦP Γ( ) P 0 ( ) ( ) I > 0 → tr(CPCT) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 5 Find tr(CPCT)< μ CPCT W CP ( ) P > 0, tr(W)< μ S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 6 ■ H2 Performance (Discrete-time): Theorem 4.1 (H2 performance in discrete-time) For a system Tzw with D= 0, the following statements, involving matrix variables X, G, W, Z are equivalent i) ∥Tzw∥2 < ν ii) X > 0 and Z such that X ΦX Γ X 0 ( ) ( ) I > 0, Z CX( ) X > 0, tr(Z)< ν2 (4.2a) iii) X > 0, G and Z such that X ΦG Γ( ) G+GT X 0 ( ) ( ) I > 0, Z CG( ) G+GT X > 0, tr(Z)< ν2 (4.2b) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 7 iv) X > 0, G, W, and Z s.t. X ΦW WTΦT WT +ΦX Γ( ) X (G+GT) 0 ( ) ( ) I < 0, Z CG ( ) G+GT X > 0, tr(Z)< ν2 (4.2c) Exercise: Verify their equivalence given by (4.2). S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 8 ■ Discrete-time Bounded Real Lemma: For the system Tzw, the followings are equivalent: i) ∥Tzw∥∞ = ∥C(zI Φ) 1Γ +D∥∞ < γ ii) ΦTXΦ X + CTC (ΦTXΓ + CTD)(Γ TXΓ +DTD γ2I) 1(ΦTXΓ + CTD)T < 0 with nonsingular Γ TXΓ +DTD γ2I iii) Φ Γ C D T X 0 0 I Φ Γ C D X 0 0 γ2I < 0 iv) Φ Γ γ 1C γ 1D T X 0 0 I Φ Γ γ 1C γ 1D X 0 0 I < 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 9 Verification: Find x(k) w(k) T Φ Γ C D T X 0 0 I Φ Γ C D X 0 0 γ2I x(k) w(k) < 0 leads to xT(k+ 1)Xx(k+ 1) xT(k)Xx(k) + zT(k)z(k) γ2wT(k)w(k)< 0 Let V(k) = xT(k)Xx(k). Then V(k+ 1) V(k) + zT(k)z(k) γ2wT(k)w(k)< 0 → zT(k)z(k)< γ2wT(k)w(k) → ∥Tzw∥∞ < γ S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 10 Find Φ Γ C D T X 0 0 I Φ Γ C D X 0 0 γ2I < 0 ΦTXΦ X + CTC ΦTXΓ + CTD Γ TXΦ+DTC Γ TXΓ +DTD γ2I < 0 ΦTXΦ X + CTC (ΦTXΓ + CTD)(Γ TXΓ +DTD γ2I) 1(ΦTXΓ + CTD)T < 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 11 Also, find x(k) w(k) T Φ Γ γ 1C γ 1D T X 0 0 I Φ Γ γ 1C γ 1D X 0 0 I x(k) w(k) < 0 leads to xT(k+ 1)Xx(k+ 1) xT(k)Xx(k) + γ 2zT(k)z(k) wT(k)w(k)< 0 Let V(k) = xT(k)Xx(k). Then V(k+ 1) V(k) + γ 2zT(k)z(k) wT(k)w(k)< 0 zT(k)z(k)< γ2wT(k)w(k) → ∥Tzw∥2∞ < γ2 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 12 Using P> 0, GTPG P′ < 0 P′ GTP PG P < 0 P PG GTP P′ < 0, for (iii) Φ Γ C D T X 0 0 I Φ Γ C D X 0 0 γ2I < 0, we find its equivalent form X 0 XΦ XΓ0 I C D ΦTX CT X 0 Γ TX DT 0 γ2I < 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 13 Applying permutations leads to X XΦ XΓ 0ΦTX X 0 CT Γ TX 0 γ2I DT 0 C D I A congruence transformation by T= diag(γX 1,γX 1,γ 1,γ) leads to γ 2X 1 Φ(γ2X 1) Γ 0 γ2X 1ΦT γ2X 1 0 γ2X 1CT Γ T 0 I DT 0 C(γ2X 1) D γ2I < 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 14 Applying the change of variables X := γ2X 1 leads to X ΦX Γ 0XΦT X 0 XCT Γ T 0 I DT 0 CX D μI < 0 (4.3) where μ= γ2. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 15 Exercise: Alternatively, let us consider GTPG P< 0 → P GTP PG P < 0 → P PG GTP P < 0 and Φ Γ γ 1C γ 1D T X 0 0 I Φ Γ γ 1C γ 1D X 0 0 I < 0 Again permutations followed by congruence transformations and the change of variables exhibit the LMI X ΦX Γ 0XΦT X 0 XCT Γ T 0 I DT 0 CX D μI < 0 where μ= γ2. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 16 ■ H∞ Performance: Tzw : x(k+ 1) = Φx(k) + Γw(k) z(k) = Cx(k) +Dw(k) Lemma 4.1 (H∞ performance in disscrete-time) The inequality ∥Tzw∥2∞ < μ holds if and only if there exists P> 0 such that P ΦP Γ 0( ) P 0 PCT( ) ( ) I DT ( ) ( ) ( ) μI > 0 (4.4a) is feasible. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 17 With a slack variable G P ΦG Γ 0( ) G+GT P 0 GTCT( ) ( ) I DT ( ) ( ) ( ) μI > 0 (4.4b) Exercise: Verify (4.4b). S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 18 Good to Note: The standard Problem K G, a generalized system w ∈W, a set of disturbances z, a controlled variable Goal: Given W, synthesize K (feedback control) s.t. zTz is as small as possible and Tzw is stable. w S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 19 In the state space with G : x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k) z(k) = C1x(k) +D11w(k) +D12u(k) y(k) = C2x(k) +D21w(k) +D22u(k) (4.5) Applying state feedback control u(k) = Kx(k) leads to a clp Tzw : x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k) z(k) = (C1 +D12K)x(k) +D11w(k) (4.6) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 20 Stabilizing State Feedback Control Let us consider a linear discrete-time system x(k+ 1) = Φx(k) + Γu(k) Applying state feedback control u(k) = Kx(k) leads to x(k+ 1) = (Φ+ Γ2K)x(k) The closed-loop system is asymptotically stable iff there exists a P ∈ Snx such that P (Φ+ Γ2K)P ( ) P > 0 (4.7) is feasible. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 21 Applying the change of variables Z = KP leads to an LMI P ΦP+ Γ2Z ( ) P > 0 (4.8) With P> 0 and Z, the stabilizing control gain K = ZP 1 (4.9) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 22 With a slack variable G ∈ Rnx×nx P (Φ+ Γ2K)G ( ) G+GT P > 0 Applying the change of variables Z = KG leads to an LMI P ΦG+ Γ2Z ( ) G+GT P > 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 23 The closed loop system is asymptotically stable iff there exists a P ∈ Snx , Z ∈ Rnu×nx , G ∈ Rnx×nx such that P ΦG+ Γ2Z ( ) G+GT P > 0 (4.10) With P> 0, Z, and G, the stabilizing control gain K = ZG 1 (4.11) Exercise: Explain why P ∈ Snx satisfying the LMIs is positive definite. Also, explain why G ∈ Rnx×nx satisfying the LMI becomes nonsingular such that computation of stat feedback control gain is possible. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 24 H2 State Feedback Control Consider a linear discrete-time system P : x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k) z(k) = C1x(k) +D11w(k) +D12u(k) y(k) = x(k) (4.12) Applying state feedback control u(k) = Kx(k) leads to a clp Tzw : x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k) z(k) = (C1 +D12K)x(k) +D11w(k) (4.13) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 25 Assuming D11 = 0, ∥Tzw∥22 < μ holds iff there exists P ∈ Snx , W ∈ Snz such that tr(W)< μ, W (C1 +D12K)P ( ) P > 0 (4.14a) P (Φ+ Γ2K)P Γ1( ) P 0 ( ) ( ) I > 0 (4.14b) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 26 Applying the change of variables Z = KP ∈ Rnu×nx leads to tr(W)< μ, W C1P+D12Z ( ) P > 0 (4.15a) P ΦP+ Γ2Z Γ1( ) P 0 ( ) ( ) I > 0 (4.15b) With P> 0, W, and Z, the H2 state feedback control gain K = ZP 1. (4.16) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 27 Using the dilated LMI condition (4.2b) Assuming D= 0, ∥Tzw∥22 < μ holds iff there exists G ∈ Rnx×nx and P ∈ Snx , W ∈ Snz such that tr(W)< μ, W (C1 +D12K)G ( ) G+GT P > 0 (4.17a) P (Φ+ Γ2K)G Γ1( ) G+GT P 0 ( ) ( ) I > 0 (4.17b) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 28 Applying the change of variables Z = KG ∈ Rnz×nx leads to tr(W)< μ, W C1G+D12Z ( ) G+GT P > 0 (4.18a) P ΦG+ Γ2Z Γ1( ) G+GT P 0 ( ) ( ) I > 0 (4.18b) With P> 0, W, Z, and G, the H2 state feedback control gain K = ZG 1. (4.19) Exercise: Find the H2 state feedback control gain, using the dilated cond. (4.2c). S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 29 H∞ State Feedback Control Consider a linear discrete-time system P : x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k) z(k) = C1x(k) +D11w(k) +D12u(k) y(k) = x(k) (4.20) Applying state feedback control u(k) = Kx(k) leads to a clp Tzw : x(k+ 1) = (Φ+ Γ2K)x(k) + Γ1w(k) z(k) = (C1 +D12K)x(k) +D11w(k) (4.21) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 30 ∥Tzw∥2∞ < μ holds, iff there exists P ∈ Snx such that P (Φ+ Γ2K)P Γ1 0( ) P 0 P(C1 +D12K)T( ) ( ) I DT11 ( ) ( ) ( ) μI > 0 (4.22) is feasible. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 31 Applying the change of variables Z = KP ∈ Rnu×nx leads to P ΦP+ Γ2Z Γ1 0( ) P 0 PCT1 + ZTDT12( ) ( ) I DT11 ( ) ( ) ( ) μI > 0 (4.23) With P> 0 and Z, the H∞ state feedback control gain K = ZP 1 (4.24) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 32 Using the dilated cond. (4.4b) ∥Tzw∥2∞
< μ holds, iff there exist P ∈ Snx and G ∈ Rnx×nx such that P (Φ+ Γ2K)G Γ1 0( ) G+GT P 0 GT(C1 +D12K)T( ) ( ) I DT11 ( ) ( ) ( ) μI > 0 (4.25) is feasible. S.-H.

Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 33 Applying the change of variables Z = KG ∈ Rnu×nx leads to P ΦG+ Γ2Z Γ1 0( ) G+GT P 0 GTCT1 + ZTDT12( ) ( ) I DT11 ( ) ( ) ( ) μI > 0 (4.26) With P> 0, Z, and G, the H∞ state feedback control gain K = ZG 1 (4.27) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 34 H2 Output Feedback Control Dynamic output feedback problem: A generalized system P : x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k) z(k) = C1x(k) +D11w(k) +D12u(k) y(k) = C2x(k) +D21w(k) Output feedback control u(k) = ΦK ΓK CK DK : x (k+ 1) = ΦK x (k) + ΓKy(k) u(k) = CK x (k) +DKy(k) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 35 Closed-loop Tzw : x(k+ 1)x (k+ 1) z(k) = Φcl Γcl Ccl Dcl x(k)x (k) w(k) where Φcl = Φ+ Γ2DKC2 Γ2CK ΓKC2 ΦK , Γcl = Γ1 + Γ2DKD21 ΓKD21 , Ccl = C1 +D12DKC2 D12CK , Dcl = D11 +D12DKD21 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 36 The LMIs for the closed-loop H2 performance tr(W)< μ, W CclP ( ) P > 0, P ΦclP Γcl( ) P 0 ( ) ( ) I > 0 With a slackvariable G tr(W)< μ, W CclG ( ) G+GT P > 0, P ΦclG Γcl( ) G+GT P 0 ( ) ( ) I > 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 37 A congruence transformation by T= diag(T,T, I) TT P ΦclG Γcl( ) G+GT P 0 ( ) ( ) I T> 0 TTPT TTΦclGT TTΓcl( ) TT(G+GT P)T 0 ( ) ( ) I > 0 (4.28) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 38 Define K = ΦK ΓK CK DK , G= X 1,2 U 2,2 , G 1 = YT ˉ1,2 VT ˉ2,2 and T = I YT 0 VT Here, and ˉ denote arbitrary matrices with compatible dimensions. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 39 Want to have GG 1 = X 1,2 U 2,2 YT ˉ1,2 VT ˉ2,2 = XYT+ 1,2VT X ˉ1,2+ 1,2 ˉ2,2 UYT+ 2,2VT U ˉ1,2+ 2,2 ˉ2,2 = I 0 0 I with 1,2V T = 0, XYT = I, UYT+ 2,2VT = 0 such that GT = X I U 0 We can find ˉ1,2, ˉ2,2, 1,2, and 2,2 satisfying the above condition. However, the matrices are not to be literally used in our formulation. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 40 Introduce the changes of variables: Q F L R = V YΓ2 0 I ΦK ΓK CK DK U 0 C2X I + Y 0 Φ X 0 (4.29) Po J JT H = TTPT (4.30) S= YX + VU (4.31) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 41 Find Q F L R = VΦKU + YΓ2CKU + VΓKC2X + YΓ2DKC2X + YΦX VΓK + YΓ2DK CKU +DKC2X DK Thus Q= VΦKU + YΓ2CKU + VΓKC2X + YΓ2DKC2X + YΦX F = VΓK + YΓ2DK L= CKU +DKC2X R= DK S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 42 Each block in the LMI (4.28) TTΦclGT = I 0 Y V Φ+ ΓDKC2 Γ2CK ΓKC2 ΦK GT = ΦX + Γ2DKC2X + Γ2CKU Φ+ Γ2DKC2YΦX + YΓ2DKC2X + VΓKC2X + YΓ2CKU + VΦKU YΦ+ YΓ2DKC2 + VΓKC2 = ΦX + Γ2L Φ+ Γ2DKC2 Q YΦ+ FC2 Exercise: Verify the above formulation for TTΦclGT S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 43 TTΓcl = I 0 Y V Γ1 + Γ2DKD21 ΓKD21 = Γ1 + Γ2DKD21 YΓ1 + YΓ2DKD21 + VΓKD21 = Γ1 + Γ2DKD21 YΓ1 + FD21 CclGT = C1 +D12DKC2 D12CK X I U 0 = C1X +D12DKC2X +D12CKU C1 +D12DKC2 = C1X +D12L C1 +D12DKC2 Dcl = D11 +D12DKD21 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 44 Also, find TT(G+GT P)T = X + XT I+ ST I+ S Y + YT Po J JT H Exercise: Verify the above formulation for TT(G+GT P)T. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 45 Thus, in the LMI (4.28) (1, 1) = TTPT = Po J JT H (1,2) = TTΦclGT = ΦX + Γ2L Φ+ ΓRC2 Q YΦ+ FC2 (1, 3) = TTΓcl = Γ1 + Γ2RD21 YΓ1 + FD21 (2, 2) = TT(G+GT P)T = X + XT I+ ST I+ S Y + YT Po J JT H S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 46 A congruence transformation by T= diag(I,T) TT W CclG ( ) G+GT P T< 0 W CclGT ( ) T(G+GT P)TT < 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 47 Find CclGT = C1X +D12L C1 +D12DKC2 TT(G+GT P)T = X + XT Po I+ ST J I+ S JT Y + YT H S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 48 Substituting each block leads to the LMIs for H2 output feedback control synthesis: tr(W)< μ, W C1X +D12L C1 +D12RC2( ) X + XT Po I+ ST J ( ) ( ) Y + YT H > 0 (4.32a) Po J ΦX + Γ2L Φ+ Γ2RC2 Γ1 + Γ2RD21 ( ) H Q YΦ+ FC2 YΓ1 + FD21 ( ) ( ) X + XT Po I+ ST J 0 ( ) ( ) ( ) Y + YT H 0 ( ) ( ) ( ) ( ) I > 0 (4.32b) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 49 Utilizing the solutions S, X, Y, Q F L R , one can find K: Find U and V from (4.31). Compute from (4.29) K = V YΓ2 0 I Q F L R Y 0 Φ X 0 U 0 C2X I (4.33) with V YΓ2 0 I = V YΓ2 0 I T V YΓ2 0 I 1 V YΓ2 0 I T U 0 C2X I = U 0 C2X I T U 0 C2X I U 0 C2X I T 1 Then, we have ΦK, ΓK, CK, and DK. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 50 Discrete-time H2 output feedback control: with D22 = 0 Generalized system x(k+ 1)z(k) y(k) = Φ Γ1 Γ2C1 D11 D12 C2 D21 D22 x(k)w(k) u(k) Output feedback control u(k) = ΦK ΓK CK DK : x (k+ 1) = ΦK x (k) + ΓKy(k) u(k) = CK x (k) +DKy(k) Closed-loop system Gcl(z) = Φcl Γcl Ccl Dcl = Ccl (zI Φcl) 1 Bcl +Dcl S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 51 with Φcl = Φ+ Γ2DKD 1C2 Γ2(I+DKD 1D22)Ck BKD 1C2 ΦK + ΓKD 1D22CK , Γcl = Γ1 + Γ2DKD 1D21 BKD 1D21 , Ccl = C1 +D12DKD 1C2 D12(I+DKD 1D22)CK , Dcl = D11 +D12DKD 1D21 where D = I D22DK S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 52 Now solve for Φ ∈ Rnx×nx , Γ ∈ Rnx×ny , C ∈ Rnu×nx , D ∈ Ruu×ny , X1 > 0, Y1 > 0, Z > 0, and ν ∈ R>0 that minimize ν subject to the following LMIs: X1 I X1Φ+ Γ C2 Φ X1Γ1 + Γ D21 ( ) Y1 Φ+ Γ2D C2 ΦY1 + Γ2C Γ1 + Γ2D D21 ( ) ( ) X1 I 0 ( ) ( ) ( ) Y1 0 ( ) ( ) ( ) ( ) I > 0 Z C1 +D12D C2 C1YT1 +D12C ( ) X1 I ( ) ( ) Y1 > 0 D11 +D12D D21 = 0 X1 I ( ) Y1 > 0 trZ < ν S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 53 The controller is recovered by DK = (I+ DˉD22) 1Dˉ CK = (I DKD22)Cˉ ΓK = Γˉ (I DKD22) ΦK = Φˉ ΓK(I D22DK) 1D22CK where Φˉ Γˉ Cˉ Dˉ = X2 X1Γ2 0 I 1 Φ Γ C D X1ΦY1 0 0 0 YT2 0 C2Y1 I 1 and the matrices X2 and Y2 chosen to satisfy X2Y T 2 = I X1Y1. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 54 Note: If D22 = 0, then Φk = Φˉ, ΓK = Γˉ , CK = Cˉ, and DK = Dˉ. If D11 = 0, D12 = 0, and D21 = 0, then it is open simplest to choose D = 0 in order to satisfy the equality constraint D11 +D12D D21 = 0. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 55 H∞ Output Feedback Control Discrete-time H∞ output feedback control: A discrete-time linear generalized system x(k+ 1) = Φx(k) + Γ1w(k) + Γ2u(k) z(k) = C1x(k) +D11w(k) +D12u(k) y(k) = C2x(k) +D21w(k) Output feedback control u(k) = ΦK ΓK CK DK : x (k+ 1) = ΦK x (k) + ΓKy(k) u(k) = CK x (k) +DKy(k) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 56 Then the closed-loop system Gcl(z) = Φcl Γcl Ccl Dcl = Ccl (zI Φcl) 1 Bcl +Dcl with Φcl = Φ+ Γ2DKC2 Γ2CK ΓKC2 ΦK , Γcl = Γ1 + Γ2DKD21 ΓKD21 , Ccl = C1 +D12DKC2 D12CK , Dcl = D11 +D12DKD21 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV - 57 The LMI for the closed-loop H∞ performance: P ΦclP Γcl 0( ) P 0 PCTcl( ) ( ) I DTcl ( ) ( ) ( ) μI > 0 (4.34) With a slack variable G P ΦclG Γcl 0( ) G+GT P 0 GTCTcl( ) ( ) I DTcl ( ) ( ) ( ) μI > 0 (4.35) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 58 A congruence transformation by T= diag(T,T, I, I) TT P ΦclG Γcl 0( ) G+GT P 0 GTCTcl( ) ( ) I DTcl ( ) ( ) ( ) μI T> 0 T TPT TTΦclGT T TΓcl 0 ( ) TT(G+GT P)T 0 TTGTCTcl ( ) ( ) I DTcl ( ) ( ) ( ) μI > 0 (4.36) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 59 Thus, in the LMI (4.36) (1, 1) = TTPT = Po J JT H (1,2) = TTΦclGT = ΦX + Γ2L Φ+ ΓRC2 Q YΦ+ FC2 (1, 3) = TTΓcl = Γ1 + Γ2RD21 YΓ1 + FD21 (2, 2) = TT(G+GT P)T = X + XT I+ ST I+ S Y + YT Po J JT H (2,4) = TTGTCTcl = XTCT1 + L TDT12 CT1 + C T 2R TDT12 (3, 4) = DTcl = DT11 +D T 21R TDT12 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 60 Substituting each expression leads to an LMI for H∞ output feedback control synthesis: Po J ΦX + Γ2L Φ+ Γ2RC2 Γ1 + Γ2RD21 0 ( ) H Q YΦ+ FC2 YΓ1 + FD21 0 ( ) ( ) X + XT Po I+ ST J 0 XTCT1 + LTDT21 ( ) ( ) ( ) Y + YT H 0 CT1 + CT2RTDT21 ( ) ( ) ( ) ( ) I DT11 +DT21RTDT11 ( ) ( ) ( ) ( ) ( ) μI > 0 (4.37) S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 61 Utilizing the solutions S, X, Y, Q F L R , one can compute K: Find S, X, Y → U, VT Compute K = V YΓ2 0 I Q F L R Y 0 Φ X 0 U 0 C2X I (4.38) with V YΓ2 0 I = V YΓ2 0 I T V YΓ2 0 I 1 V YΓ2 0 I T U 0 C2X I = U 0 C2X I T U 0 C2X I U 0 C2X I T 1 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 62 Discrete-time H∞ output feedback control: with D22 = 0 Generalized system x(k+ 1)z(k) y(k) = Φ Γ1 Γ2C1 D11 D12 C2 D21 D22 x(k)w(k) u(k) Output feedback control u(k) = ΦK ΓK CK DK : x (k+ 1) = ΦK x (k) + ΓKy(k) u(k) = CK x (k) +DKy(k) Closed-loop system Gcl(z) = Φcl Γcl Ccl Dcl = Ccl (zI Φcl) 1 Bcl +Dcl S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 63 with Φcl = Φ+ Γ2DKD 1C2 Γ2(I+DKD 1D22)Ck BKD 1C2 ΦK + ΓKD 1D22CK , Γcl = Γ1 + Γ2DKD 1D21 BKD 1D21 , Ccl = C1 +D12DKD 1C2 D12(I+DKD 1D22)CK , Dcl = D11 +D12DKD 1D21 where D = I D22DK S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 64 Solve for Φ ∈ Rnx×nx , Γ ∈ Rnx×ny , C ∈ Rnu×nx , D ∈ Rnu×ny , X1 > 0, Y1 > 0, and γ ∈ R>0 that minimizes γ subject to the following LMIs: X1 I X1Φ+ Γ C2 Φ X1Γ1 + Γ D21 0 ( ) Y1 Φ+ Γ2D C2 ΦY1 + Γ2C Γ1 + Γ2D D21 0 ( ) ( ) X1 I 0 CT1 + CT2DT12 ( ) ( ) ( ) Y1 0 Y1CT1 + C TDT12 ( ) ( ) ( ) ( ) γI DT11 +DT21D TDT12 ( ) ( ) ( ) ( ) ( ) γI > 0 X1 I ( ) Y1 > 0 S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 65 The controller is recovered by DK = (I+ DˉD22) 1Dˉ CK = (I DKD22)Cˉ ΓK = Γˉ (I DKD22) ΦK = Φˉ ΓK(I D22DK) 1D22CK where Φˉ Γˉ Cˉ Dˉ = X2 X1Γ2 0 I 1 Φ Γ C D X1ΦY1 0 0 0 YT2 0 C2Y1 I 1 and the matrices X2 and Y2 chosen to satisfy X2Y T 2 = I X1Y1. If D22 = 0, then Φk = Φˉ, ΓK = Γˉ , CK = Cˉ, and DK = Dˉ. S.-H. Lee : ODC / Design via LMI Optim. Elec. Engr., Hanyang University IV – 66