Symmetric Root Locus. LQR Design

Transcription

1 Leture 5 Symmetri Root Lous LQR Design State Estimation Seletion of 'otimal' oles for SISO ole laement design: SRL LQR design examle Predition and urrent estimators L5:1

2 L5:2 Otimal ole laement for SISO systems For a single-inut system, we might selet one artiular outut (y = Cx to onstrain in a ost funtion: r = + u x = Ax + B u y J = ( 2 2 ρ y ( t + u ( t dt Cx y = Plant transfer funtion: x K Y ( s 1 N ( s = C ( s I A B =, say U ( s D ( s It an be shown that: J will be minimised by the ontrol law u = Kx the eigenvalues of the losed-loo system are the left half-lane roots of the 2n-degree olynomial equation α ( α ( s = D ( s D ( s + ρ N ( s N ( s = s Ref: Kailath, Linear Systems, 198

3 L5:3 Otimal ole laement for SISO systems α ( α ( s = D ( s D ( s + ρ N ( s N ( s = s Hene we an see the effets of the outut-weighting fator ρ on the losed-loo oles by lotting a root lous for N ( s N ( s 1 + ρ = D ( s D ( s This symmetri root lous (SRL will be a 18º (º lous if the oen-loo ole-zero exess is even (odd The roots of α (s = are guaranteed stable Hene if branhes of the 18º SRL lie on the imaginary axis, lot the º lous instead The SRL thus rovides a basis for seifying CLPs in a SISO ole-laement design

4 The SRL is lotted for the equation Examle: u SRL for disrete systems M y Oen-loo TF: D ( z.134( = U ( z ( z b z + 1 N D ( ( ρ 1 2 m d z z.7( z + ( z.91 N D ( ( z z =.82( z + ± j.39 Comliant struture: M = 1, m =.1 b =.36, =.91 States: x = [ d d y y ] T Samle eriod: T =.4 s 7.74 L5:4 See srit flex_srl.m and funtion srl.m

5 Disrete SRL for omliant struture Symmetri root lous Real Axis L5:5 Imag Axis

6 s A xi g a Im Use rlofind to selet trial CLPs L5:6» [rho,l]=rlofind(gol rho = l = i i i i i i i i Real Axis» dl=l(find(abs(l<1 dl = i i i i

7 SRL ole-laement design (see flex_srl.m G = ss(a, B, C, D,'InutName','fore u','oututname','dis d'; set(g, 'StateName', {'d', 'ddot', 'y', 'ydot'} T =.4; % samle eriod Gd = 2d(G, T; % disrete ss model L5:7 % Plot symmetri root lous and get desired CL oles [rho, dl] = srl(gd; % MCG-written funtion % State-feedba ontrol gains Phi = Gd.a; Gam = Gd.b; K = aer(phi, Gam, dl; % Closed-loo regulator system Gdu = [Gd; 1]; % get d and u as oututs Gdux = augstate(gdu; % augment lant oututs with states Gl = feedba(gdux, K, [1], [3:6]; % feed-ba 4 states to u Gl = Gl(1:2,1; % suress state oututs Gl.oututn={'dis d'; 'trl u'}; % Initial ondition resonse x = [1 ]'; % d( = 1 initial(gl, x

8 SRL ole-laement design Initial ondition resonse 1 d Time (se. L5:8 Amlitude To: dis d u.5 To: trl u

9 LQR design for omliant struture (see flex_lqr.m % Disrete model T =.4; Gd = 2d(G, T; Phi = Gd.a; Gam = Gd.b; % Otimal ontrol: trial and error R = 1; % Salar inut % Weight dislaements but not veloities % Get trial diagonal elements for Q Q11 = inut('enter ontroller Q11: '; Q22 = inut('enter ontroller Q22: '; Q = diag([q11 Q22 ]; K = dlqr(phi, Gam, Q, R L5:9

10 LQR design, Q11 = 1, Q22 = 1, R = 1 d y u Time (se. L5:1 Amlitude To: dis y To: dis d To: fore u

11 LQR with redition estimator u( m x( Γ + q 1 C y( L5:11 x y Control law: u ( = K x ˆ( Estimator: x ˆ( Plant: ( ( = Φ = Cx ( 1 x = Φ + ( x ˆ( L + Γ 1 u ( 1 + Γ Plant [ ( ˆ( 1 ] y C x K u ( Feedba omensator u( x ˆ ( 1 Φ Γ q 1 Φ C + y ˆ ( Estimator + L

12 Full-order redition estimator Plant: Estimator: x ( + 1 = Φ x ( + Γ u ( y ( = Cx ( L5:12 [ y ( C ˆ( 1 ] x ˆ ( + 1 = Φ x ˆ( 1 + Γ u ( + L x Estimation error: ~ x ( ˆ( 1 ( = x x ~ ~ Estimation error dynamis: x ( + 1 = [ Φ L C ] ( x Charateristi olynomial: α ( z = det [ Φ + C ] L zi L n oeffiients ontaining n unnowns Desired harateristi olynomial: α e ( z = ( z β ( z β 2 ( z β 1 n n hosen oeffiients For MIMO systems, Matlab s lae funtion uses the extra DoF to give a robust solution

13 Examle: Regulation of omliant struture with redition estimator u M b m Comliant struture: M = 1, m =.1 b =.36, =.91 States: x = [ d d y y ] T L5:13 y d Samle eriod: T =.4 s Sensed oututs: y = [d y] T A revious ontroller design K = [ ] to lae oles at z =.9 ± j.5,.8 ± j.4 (orresonding to [ζ, ω n ] = [.88,.29], [.23, 1.19] Try laing estimator oles 5-times faster: dls = 1/T*log(dlz; % ontroller s-oles des = 5*dls; dez = ex(t*des; % estimator z-oles

14 L5:14 Predition estimator design Disretised lant model: T =.4; Gd = 2d(G,T; Phi = Gd.a; C = Gd.; Estimator gains: L = lae(phi',c',dez L = Regulator: H = reg(gd, K, L Feedba: Gl = feedba(gd, H, +1 Che resonse to initial ondition d( = 1 Exeriment with estimator ole seletion, et. (Matlab srit flex_redit.m

15 Resonse to initial ondition d = d( M u y Time (se. L5:15 Amlitude To: dis d.4.2 y( To: dis y u( To: fore u b m d

16 Atual states omared with redition estimates esonse of states and reditive estimates to x 1 ( = x 1 =d x 1 hat 5 x 2 =ddot x 2 hat Time (s 5 1 Time (s.4.2 x 3 =y x 3 hat 1.5 x 4 =ydot x 4 hat Time (s 5 1 Time (s L5:16

17 Current estimators L5:17 Using the revious redition estimator, the ontrol at instant is based on sensor data u to the revious samling instant: u ( = K x ˆ( 1 We now that delays (lateny in a feedba loo are de-stabilising Hene, if we were able to quily udate the state estimate at instant, based on the urrent measurement, it would be worthwhile: [ y ( C ˆ( 1 ] x ˆ ( = x ˆ( 1 + L x This measurement udate is erformed on the model redition (a time udate: x ˆ ( 1 = Φ x ˆ( Γ u ( We need to organise the alulations so that the measurement udate an be erformed raidly 1

18 LQR with urrent estimator u( m x( Γ + q 1 C Plant: x ( + 1 = Φ x ( + Γ u ( y ( = Cx ( Plant Φ Control law: u ( = K x ˆ( u( Γ + Estimator: Time udate: Φ x ˆ ( + 1 = Φ x ˆ( + Γ u ( x ˆ ( Measurement udate: K + L [ y ( C ˆ( 1 ] x ˆ ( = x ˆ( 1 + L x q 1 x ˆ ( 1 C L5:18 y( y ˆ ( +

19 Comarison between redition and urrent estimates For the urrent estimator we have x ˆ ( + 1 = Φ x ˆ( + Γ u ( and [ y ( C ˆ( 1 ] x ˆ ( = x ˆ( 1 + L x Hene x ˆ ( + 1 = Φ x ˆ( 1 + Γ u ( + Φ L y ( C x ˆ( [ 1 ] The redition estimation error is ~ x ( + 1 = x ˆ( + 1 x ( + 1 = Φ x ˆ( = Φ x 1 + Γ u ( + Φ L ( Γ [ Φ Φ L C ] ~ x ( u ( [ Cx ( Cx ˆ( 1 ] L5:19

20 Comarison between redition and urrent estimates... L5:2 For the urrent estimator we have the redition error ~ x ( + 1 = x ˆ( + 1 x ( + 1 = Φ Φ L C ~ x ( [ ] Similarly, we an show that the urrent estimate error is ~ x ( + 1 = x ˆ( x ( + 1 = Φ L C Φ ~ x ( [ ] The two error equations have the same eigenvalues (both are oututs from the same dynami system hene an use either as a basis for alulating L For the redition estimator: ~ x ( + 1 = L ~ [ Φ C ] ( x Hene for urrent estimator we an use: L = lae(phi, (C*Phi, dez

21 Imlementation of urrent estimator in Matlab Given x ˆ ( + 1 = Φ x ˆ( + Γ u ( and [ y ( C ˆ( 1 ] x ˆ ( = x ˆ( 1 + L x we an show that the state equations for the estimator, with inut y( and outut u(, are x ˆ( + 1 u ( = K I = [ Φ Γ K ][ I L C ] x ˆ( 1 + [ Φ Γ ] [ L C ] x ˆ( 1 KL y ( K L y ( Srit flex_urrent omares resonses of urrent and redition estimators L5:21

22 1.5 Initial Condition Results Time (se. L5:22 Amlitude To: dis d.4.2 To: dis y urrent estimator redition estimator To: fore u