Lecture 8. Applications

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Lecture 8. Applications Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering May 8, 205 / 3

Logistics hw7 due this Wed, May 20 do an easy problem or CYOA hw8 (design problem) will be the last homework hw6 solutions posted online reading: lmibook Ch 7 2 / 3

Bandpass filter C 2 w(t) C R R 2 z(t) Transfer function from w to z (zero initial conditions, no clipping): ( ) sc 2 R 2 R 2 C s H wz (s) = = sc R (R C s)(r 2 C 2 s) 3 / 3

Baseline design bandpass filter: R = 0K, R 2 = 20K, C = 0nF, C 2 = 5nF 0 0 Hwz(jω) 0-0 -2 300 0 2 0 3 0 4 0 5 0 6 ω Hwz(jω) 250 200 50 00 50 0 2 0 3 0 4 0 5 0 6 ω 4 / 3

State space put system in (minimal) state space form ẋ = Ax B w w, z = C z x, x(0) = 0, where the matrices depend on component values, [ ] [ ] 0 0 A = R R 2C C 2 R C, B w =, C z = [ 0 ]. R 2C 2 R C 2 A is Hurwitz, (A,B w ) controllable, (A,C z ) observable observability Gramian: A T W obs W obs ACz T C z = 0 [ ] 0 W obs = 2(R C R 2 C 2 ) 0 R C R 2 C 2 controllability Gramian: W contr A T AW contr B w Bw T = 0 [ ] R 2 C R2 C 0 W contr = 2(R C R 2 C 2 ) 0 R C 2 5 / 3

Input output norms H 2 -norm: H zw 2 2 = Tr(BwW T obs B w ) ] T 0 = Tr[ R C 2 2(R C R 2 C 2 ) = Tr(C z W contr Cz T ) = Tr [ 0 ] R 2 C 2(R C R 2 C 2 ) R 2 C = 2C C 2 R 2 2C2 2 R R 2 = 0000 = H zw 2 = 00 [ 0 0 R C R 2 C 2 ][ 0 R C 2 ] [ ] R2 C 0 [0 ] T 0 R C 2 6 / 3

Input output norms H -norm: solve the SDP with specific component values: minimize γ subject to [ P 0, ] A T P PACz T C z PB w BwP T 0 γi γ = H wz 2 = C z (si A) B w 2 =.00 can also read H wz directly from Bode magnitude plot H wz = supσ max (H wz (jω)) =.00 ω 7 / 3

Peak gain question. is it possible for op-amp to clip even though H wz =? z 2 z H wz = sup =, H wz pk-gain = sup.4 w 0 w 2 w 0 w.5 z(t) (Volts) 0.5 0-0.5-0 0.2 0.4 0.6 0.8.2.4.6.8 2 0-3 t (sec) answer. yes (depending on supply limits, scale & apply signal above) 8 / 3

Component variations Suppose each component can vary by ±0%. Is stability guaranteed? polytopic LDI. model the state space matrices [ A Bw ] conv {[ A (B w ) ],..., [ AL (B w ) L ]} autonomous stability certificate. find a joint quadratic Lyapunov function V(x) = x T Px, P 0, with A T i P PA i 0, i =,...,L, where the A i are formed by all combinations of parameter variations, i.e., [ ] 0 A = [ A 2 =. R R 2C C 2 R C R 2C 2 0 0.9R R 2C C 2 0.9R C R 2C 2 ] 9 / 3

Guaranteed bounds on output peak for initial condition Suppose E = {x x T Px } is an invariant ellipsoid containing the initial condition x(0), then z(t) T z(t) supx T Cz T C z x x E = λ max (P /2 C T z C z P /2 ) δ, provided there is a matrix P = P T satisfying [ P 0, x(0) T P (Cz ) Px(0), T i (C z ) i δi ] 0, A T i P PA i 0. for input output properties, select x(0) according to (B w ) i. 0 / 3

State estimation observer design. given plant P driven by w, find an observer gain L to minimize some norm of the w-to-e transfer function, H ew. w P y z e ŷ y ŷ E ẑ plant model. ẋ = Ax B w w y = C y x D yw w z = C z x estimator model. ˆx = Aˆx L(ŷ y) ŷ = C yˆx ẑ = C zˆx, e = ẑ z / 3

Luenberger-style architecture plant model. ẋ = Ax B w w y = C y x D yw w z = C z x estimator model. ˆx = Aˆx L(ŷ y) ŷ = C yˆx ẑ = C zˆx, e = ẑ z estimator attempts to replicate plant dynamics without w exogenous input w (e.g., noise) is not directly accessible to estimator C y determines which parts of the state can be measured C z controls performance index (e.g., makes units of x comparable), H ew = C z (ˆx x) example. C z = I, D yw = I with given by H 2 -norm is a Kalman filter 2 / 3

Closed loop system Substitute ŷ = Cˆx and y = Cx D yw w to obtain ˆx = Aˆx LC y (ˆx x)ld yw w ẋ = Ax B w w, and write as an augmented linear system, [ ] [ ][ ] ˆx A LCy ˆx = ˆx ẋ 0 ALC y ˆx x }{{} A cl e = [ 0 C z ] }{{} C cl [ ˆx ˆx x ]. [ ] LD yw w B w LD yw }{{} B cl We aim to minimize the norm of the w-to-e transfer function, H ew = C cl (si A cl ) B cl. 3 / 3

Decoupling w [ ˆx x [ ] 0 0 ˆx C z e decomposition. the second states are driving the first states, so we do not need the entire dynamics to compute H ew. [ ] [ ][ ] [ ] ˆx A LCy ˆx LD = yw w, e = [ ] [ ] ˆx 0 C ˆx ẋ 0 ALC y ˆx x B w LD z yw ˆx x ( ˆx ẋ) = (ALC y )(ˆx x)(b w LD yw )w, e = C z (ˆx x) H ew = C z (si (ALC y )) (B w LD yw ) 4 / 3

H 2 case The norm H ew 2 is the square root of the optimal value of minimize Tr ( (B w LD yw ) T P(B w LD yw ) ) subject to P 0 (ALC y ) T P P(ALC y )C T z C z 0 convex program in P = P T and W = PL (assuming D yw D T yw = I and B w D T yw = 0) minimize Tr(B T wpb w )Tr(W T P W) subject to P 0 A T P PAC T z C z C T y W T WC y 0 optimal estimator gain is L = (P ) W, and is independent of C z alternate solution. set L = QC T y in the solution Q of the algebraic Riccati equation QA T AQ B w B T w QC T y C y Q = 0. 5 / 3

H case The norm H ew is the square root of the optimal value of minimize γ subject to [ P 0 ] (ALCy ) T P P(ALC y )Cz T C z P(B w LD yw ) (B w LD yw ) T 0 P γi dissipation condition: V = x T Px, P 0, V e T e γw T w 0 convex program in P = P T, γ, and W = PL minimize γ subject to P ( 0 ) A T P PACz T C z Cy T W T WC y BwP T DywW T T PB w WD yw 0 γi optimal estimator gain is L = (P ) W 6 / 3

Example: H 2 vs H estimator nominal plant model (DC motor inspired) 0. 0 0 A nom = 0., B w = 0, C y = [ 0 0 ] 0. estimator parameters C z = I 3, D yw = we will track estimator performance with three noise types w(t) = 0, w(t) =, w(t) = cos(2πt) would like estimator to be robust with respect to (polytopic) model perturbations. 7 / 3

Example: H 2 vs H estimator.8.6 A = A nom, w = 0 H 2 est. H est..4.2 e(t) 2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 8 / 3

Example: H 2 vs H estimator.8.6 A = A nom, w = H 2 est. H est..4.2 e(t) 2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 9 / 3

Example: H 2 vs H estimator.8.6 A = A nom, w = cos(2πt) H 2 est. H est..4 e(t) 2.2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 20 / 3

Example: H 2 vs H estimator 2.5 A = 0.5A nom, w = 0 H 2 est. H est. 2 e(t) 2.5 0.5 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 2 / 3

Example: H 2 vs H estimator 3.5 3 A = 0.5A nom, w = H 2 est. H est. 2.5 e(t) 2 2.5 0.5 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 22 / 3

Example: H 2 vs H estimator 2.5 A = 0.5A nom, w = cos(2πt) H 2 est. H est. 2 e(t) 2.5 0.5 0 0 5 0 5 20 25 30 t L (H 2) = (0.3,0.08,0.02) L (H ) = (0.00,0.00,.00) 23 / 3

Example: robust H 2 vs robust H estimator.8.6 A = 0.5A nom, w = 0 H 2 est. H est..4 e(t) 2.2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 redesigned estimators, polytopic model A conv{0.25a nom,.75a nom } t L (H 2) = (0.3,0.08,0.03) L (H ) = (.59,0.20,.04) 24 / 3

Example: robust H 2 vs robust H estimator.8.6 A = 0.5A nom, w = H 2 est. H est. e(t) 2.4.2 0.8 0.6 0 5 0 5 20 25 30 redesigned estimators, polytopic model A conv{0.25a nom,.75a nom } t L (H 2) = (0.3,0.08,0.03) L (H ) = (.59,0.20,.04) 25 / 3

Example: robust H 2 vs robust H estimator.8.6 A = 0.5A nom, w = cos(2πt) H 2 est. H est..4 e(t) 2.2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 redesigned estimators, polytopic model A conv{0.25a nom,.75a nom } t L (H 2) = (0.3,0.08,0.03) L (H ) = (.59,0.20,.04) 26 / 3

Example: robust H 2 vs robust H estimator.8.6 A = 0.5A nom, w = cos(0πt) H 2 est. H est..4 e(t) 2.2 0.8 0.6 0.4 0.2 0 0 5 0 5 20 25 30 redesigned estimators, polytopic model A conv{0.25a nom,.75a nom } t L (H 2) = (0.3,0.08,0.03) L (H ) = (.59,0.20,.04) 27 / 3

Implementation of Luenberger observer c y ˆx s ˆx a l ˆx 2 s ˆx 2 b d l 2 ˆx = (ALC y )ˆx Ly, ALC y = [ ] a b, L = c d [ l l 2 ] 28 / 3

Analog computing elements gain-inverter. V i R R 2 V o = R2 R V i V o sum junction. R V R 2 V 2 R 3 V 3 R 0 V o = k R 0V k R k V o integrator. C V i R V o = src V i V o 29 / 3

Luenberger observer analog computer y R C ˆx R /a R 2/c R l R 2 l 2 R 3 C R 2 R 2 ˆx 2 R /b R 2/d 4 30 / 3

Notes op-amps and 2 are integrators, 3 and 4 are sum junctions R and C chosen so the time constant is RC = second R, R 2 arbitrary, chosen so all internal signals stay within supply voltage bounds internal signal-to-noise ratios are not too small gain ratios specified by constants a, b, c, d y is external (voltage) signal ˆx and ˆx 2 are negative (voltage) state estimates 3 / 3

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