Introduction re Stochastic self- Tuners self-tuning regulator 1.Introduct ion EStimation 2. Minimum variance co nt ro Controller 3. Est imat ion of noise models parameters 4. Stochast ic self-tuners Controller Process 5. Feedforward cont rol 6. P redict ive co nt rol 7. Conclusio ns nt eresting res ults Here An e amp Process dynamics y(t+1)+ay(t)=bu(t)+e(t+1)+ce(t) If parameters are know n the co nt rol law is Mini mum Variance and Moving Average Control u(t)=-y(t) b,3(t . Motivat io n The output then beco mes yt=et . The general case Notice Out put is white noise Innovations represent ation of c< 1 C K.J. Ast ro m and B Wittenmark
Stochastic Self-Tuners 1. Introduction 2. Minimum variance control 3. Estimation of noise models 4. Stochastic Self-tuners 5. Feedforward control 6. Predictive control 7. Conclusions Introduction Same idea as before Process parameters Controller design Estimation Controller Process Controller parameters Reference Input Output Specification Self-tuning regulator But now use Design based on stochastic control theory Some very interesting results Here is where it started Minimum Variance and Moving Average Control Motivation An Example The General Case An Example Process dynamics y(t + 1) + ay(t) = bu(t) + e(t + 1) + ce(t) If parameters are known the control law is u(t) = y(t) = c a b ; y(t) The output then becomes y(t) = e(t) Notice Output is white noise Prediction very simple with the model Innovations representation Importance of jcj < 1 c K. J. Åström and B. Wittenmark 1
The C-poly nomial The model xam C(a=z+ (+)=A1(q (t) Spect ral density Distur bances e=dc(e c(e-in 1(q e() But A2(q) C(z)C(z-)=(z+2)(z-+2) 4(z+0.5)(z-1+0.5) y(+)=m(t)+v(t) B1 q C u(t)+ The disturbance We can write t his as y(t)=e(1)+2e(t-1) A(g)y(t)=b(gu(t)+ C(ae(t) with Ee2= l can thus be represent ed as The st and ard model y(t)=∈(t)+0.5e(t-1) ith ee Prediction The general case . Model c st a ble Process model c(a) y(k+m)=A(g) e(k+m) (q-1) (k+m) A(gy(h)= b(gu(k)+c(g)e(k) deg a-deg b=d, deg C=n F(a e(h+ m)+9 4(q-1) (k+m) C stable SIso. Innovat io n model Predict Desig n criteria: Mini mize G under the condition that the closed loo p Predict ion error stable May ass ume any causal no nlinear con- (k+m|k)=F(q-1)e(k+m) Optimal predictor dynamics C(a) C K. J. Ast ro m and B. Wittenmark
The Model Process dynamics x(t) = B1(q) A1(q) u(t) Disturbances v(t) = C1(q) A2(q) e(t) Output y(t) = x(t) + v(t) = B1(q) A1(q) u(t) + C1(q) A2(q) e(t) We can write this as A(q)y(t) = B(q)u(t) + C(q)e(t) The standard model!!! The C-polynomial Example C(z) = z + 2 Spectral density (e i!h) = 1 2 C(e i!h)C(ei!h) But C(z)C(z1 )=(z + 2)(z1 + 2) = 4(z + 0:5)(z1 + 0:5) The disturbance y(t) = e(1) + 2e(t 1) with Ee2 = 1 can thus be represented as y(t) = (t)+0:5(t 1) with E2 = 4 The General Case Process model A(q)y(k) = B(q)u(k) + C(q)e(k) deg A deg B = d, deg C = n C stable SISO, Innovation model Design criteria: Minimize E(y 2 + u2 ) under the condition that the closed loop system is stable May assume any causal nonlinear controller Prediction Model C stable y(k + m) = C(q) A(q) e(k + m) = C (q1 ) A (q1 ) e(k + m) = F (q1 )e(k + m) + qm G (q1 ) A (q1 ) e(k + m) Predictor y^(k + mjk) = G (q1 ) C (q1 ) y(k) = qG(q) C(q) y(k) Prediction error y~(k + mjk) = F (q1 )e(k + m) Optimal predictor dynamics C(q) c K. J. Åström and B. Wittenmark 2
Mi nim um var ance cont rol Minim um vari ance control ▲ Output y(t) System wit h st able inve rse B(a C( y(t+dolt A(g-1)9“a(k)+Cg2 B"(q A(-)e(k) Pre dict the out put C Input u(t) y(k+d) A'(g-e(+d)+B(9 F(a e(h+d) u(t)=? 4(-e)+B2(g-1u(k) (q-1) Com pute old innov at io ns A Choose do = d and u(k) such t hat e(8)=y(8)-g-分 u(h) y(k+dk)=0! Minim um vari ance control contd Pole Placem ent Interpretat: on Process =F以(b+d)+c(6)+-9() A"y bu+c'e a(k) is fu nct ion of y (k), y(k-1),..and Closed loop system u(k-1),u(k-2 (A·BF+q-B"G)y=BF"C Ey2(k+)=E(F e(k+d) H ence B Fx R=B F (k) We h (1+f2+…+ bt ained fo u(k) B*(g-1)p*(0-y(k)=-Gg flay(e) whe re deg F*=n-l. he nce (zF(2+2-G=2-C(a Minim um variance cont roller K.J. Ast rom and b Witte nm
Minimum variance control = Prediction Output y(t) t Input u(t) u( t)=? y ˆ (t + d0 t)=? t + d0 t Choose d0 = d and u(k) such that y^(k + djk)=0! Minimum Variance Control System with stable inverse y(k) = B(q) A(q) u(k) + C(q) A(q) e(k) = B (q1 ) A (q1 ) qdu(k) + C (q1 ) A (q1 ) e(k) Predict the output y(k + d) = C (q1 ) A (q1 ) e(k + d) + B (q1 ) A (q1 ) u(k) = F (q1 )e(k + d) + G (q1 ) A (q1 ) e(k) + B (q1 ) A (q1 ) u(k) Compute old innovations e(k) = A C y(k) qd B C u(k) Minimum Variance Control Cont'd y(k + d) = F e(k + d) + G C y(k) qd BG AC u(k) + B A u(k) = F e(k + d) + G C y(k) + BF C u(k) u(k) is function of y(k); y(k 1);::: and u(k 1); u(k 2);::: . Then Ey 2 (k + d)=EF e(k + d)2 + E G C y(k) + BF C u(k)2 It follows that Ey2 (k + d) 1 + f 2 1 + + f 2 d12 Equality is obtained for u(k) = G (q1 ) B (q1 )F (q1 ) y(k) = G(q) B(q)F (q) y(k) Minimum variance controller Pole Placement Interpretation Controller u = G BF y Process Ay = qdBu + Ce Closed loop system (ABF + qdBG )y = BF Ce Hence R = BF S = G We have C (z1 ) A (z1 ) = F (z1 ) + zm G (z1 ) A (z1 ) We can also write this as (A (z1 )F (z1 ) + zmG (z1 ) = C (z1 ) where deg F = n 1. Hence A(z)F (z) + znmG = zn1C(z) c K. J. Åström and B. Wittenmark 3
Adapt:ve Cont rol Process dy nam ics Minim umn vari ance self-tuners y(t+1)+ay(t)=bu(t)+e(t+1)+ce(t) ple in principle Est im ate param ete rs in the mod How to estim ate? Ay= Bu+ce l y(t+1)=6y(t)+u(t) An Ecam ple The le res est im ate is · Surprise d! · A sim ple case 6(1)=25y((k+1)-u(8) (t)=-6(t)y(t) How to Estim ate Noise models An Exam ple An exam ple y(t+1)+ay()=bu()+e(t+1)+ce(t) y(t+1)+ay(t)=bu(t)+e(t+1)+ce(t 0.9. and c=-03Mi nIm um variance cont roller u(t) A reg ression model estim ates a(0)=c0)=0 and b(0)=1 y(t+1)=-ay(t)+bu()+ce(t)+e(t+1 le do not know e(t) but we can appro cimate with E(t) He nce =(a,b,c) (t)=(-y(t),u(t),∈(t) (+)=y(t)-92(t-1)(t-1) (t)=6(t-1)+K(t)e(t) a=A+(t-1)P(t-1)y(t-1) Self-tu hing co nt rol K(+)=P(t-1)p(t-1)=a Minim um variance co P(t)=(I-K(t)P(t-1)y2(t-1)P(t-1) K.J. Ast rom and b Witte nm ark
Minimum Variance Self-tuners Simple in principle How to estimate? Ay = Bu + C e Cheating!! An Example Surprised!! A simple case A general result Adaptive Control Process dynamics y(t + 1) + ay(t) = bu(t) + e(t + 1) + ce(t) Estimate parameters in the model y(t + 1) = y(t) + u(t) The Least squares estimate is ^ (t) = Pt1 k=0 y(k)y(k + 1) u(k) Pt1 k=0 y2 (k) Control law u(t) = ^ (t)y(t) How to Estimate Noise Models An example y(t + 1) + ay(t) = bu(t) + e(t + 1) + ce(t) A regression model y(t + 1) = ay(t) + bu(t) + ce(t) + e(t + 1) We do not know e(t) but we can approximate it with (t). Hence = (a; b; c) '(t)=(y(t); u(t); (t)) (t) = y(t) ' T (t 1)(t 1) (t) = (t 1) + K(t)(t) = + 'T (t 1)P (t 1)'(t 1)) K(t) = P (t 1)'(t 1)= P (t)=(I K(t)P (t 1)'T (t 1))P (t 1) An Example Consider y(t + 1) + ay(t) = bu(t) + e(t + 1) + ce(t) with a = 0:9; b = 3, and c = 0:3. Minimum variance controller u(t) = 0:2y(t) Initial estimates a^(0) = ^c(0) = 0 and ^ b(0) = 1. 0 100 200 300 400 500 −5 0 5 0 100 200 300 400 500 −2 0 2 Time Time y u 0 100 200 300 400 500 0 200 400 600 Time Self-tuning control Minimum variance control c K. J. Åström and B. Wittenmark 4
An direct selftune Explanat io n ocess Explain the surprising result y(t+1)+ay(t=but+et+1)+cet Parameters a=_0.9.6=3 and c =_0.3 Direct self-tuner based o a(t 2(k) y(t+1)=rout+ Soy (t Cortrd law with fixed ro=1. Contrd law ut=-e(ty(t ro P ty()+u(y() =t∑(6t-6()y() Sdf-turing contrd Vinimumvariance contrd r(1)=mE∑(+1)y()=0 D irect self-tune rs The direct selftune Use the drect sdf-tuner withE " Parameter To=bo is either fixed estimated Estimate parameters in Property 1: If the reg ression vectors are y(t+d=R(g-y t+s(g-)y,(t baunded the dased-loop system has the R"(Q-l)=ro+rig-1 properties y(t+T)y(t=0 T=dd 4(5=9(o-1y ut y(t+Tus=0 T=dd+1,., d+k where k deg R*and l= deg sw (t y(t Prop erty 2: If the ith least squares. Use cortrd law Ag)y=b(gut+ caet R(q-lut=-S(g-y(t and if min(k, l)>n-1 then ice d and samping period are key desig n (t+ T) t=0 =dd If parameters converge we will thus obtain moving average cortrol K.J. AstO m and B wittenmark
An Direct Self-tuner Process y(t + 1) + ay(t) = bu(t) + e(t + 1) + ce(t) Parameters: a = 0:9; b = 3, and c = 0:3 Direct self-tuner based on y(t + 1) = r0u(t) + s0y(t) with xed r0 = 1. Control law u(t) = s^0 r^0 y(t) 0 100 200 300 400 500 0.0 0.5 1.0 1.5 Time s^0=r^0 0 100 200 300 400 500 0 200 400 600 Time Self-tuning control Minimum variance control Explanation Explain the surprising result ^ (t) = Pt1 k=0 y(k)y(k + 1) u(k) Pt1 k=0 y2 (k) Control law u(t) = ^ (t)y(t) Properties 1 t Xt1 k=0 y(k + 1)y(k) = 1 t Xt1 k=0 ^ (t)y 2 (k) + u(k)y(k) = 1 t Xt1 k=0 ^ (t) ^ (k) y 2 (k) r^ y (1) = lim t!1 1 t Xt1 k=0 y(k + 1)y(k)=0 The Direct Self-tuner Estimate parameters in y(t + d) = R (q1 )uf (t) + S (q1 )yf (t) R (q1 ) = r0 + r1q1 + + rkqk S (q1 ) = s0 + s1q1 + + slql uf (t) = Q (q1 ) P (q1 ) u(t) yf (t) = Q (q1 ) P (q1 ) y(t) with least squares. Use control law R (q1 )u(t) = S (q1 )y(t) Notice d and sampling period are key design parameters. Direct Self-tuners Use the direct self-tuner with Q=P = 1. Parameter r0 = b0 is either xed or estimated. Property 1: If the regression vectors are bounded the closed-loop system has the properties y(t + )y(t)=0 = d; d + 1;::: ;d + l y(t + )u(t)=0 = d; d + 1;::: ;d + k where k deg R and l = deg S Property 2: If the process is described by A(q)y = B(q)u(t) + C(q)e(t) and if min(k; l) n 1 then y(t + )y(t)=0 = d; d + 1;::: If parameters converge we will thus obtain moving average control! c K. J. Åström and B. Wittenmark 5