Adaptive Control

Stochastic Adaptive Control (02421)

www.imm.dtu.dk/courses/02421

Niels Kjølstad Poulsen

Build. 303B, room 048 Section for Dynamical Systems

Dept. of Applied Mathematics and Computer Science The Technical University of Denmark

Email: nkpo@dtu.dk phone: +45 4525 3356 mobile: +45 2890 3797

2019-04-29 20:09

Adaptive systems (L22-24)

L23

Wikipedia:Adaptive behavior is a type of behavior that is used to adjust to another type of behavior or situation.

Here: device, algorithm or method with the ability ajust itself (or its behavior) to the actual system.

Prediction, Filtering and smoothing Detection, isolation and fault estimation

Control

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Adaptive Control

w y

e

u System and disturbances

Model of Objectives Constraints

System Controller

Design

Model

y e u

ID Uncertainty

Knowledge Objective

System

Self Tuning Controller (STC)

w

Controller System

Design

e u

y ID

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Adaptive Control - the line of arguments

PID: Process information aroundwc. Robust. Not necessarily optimal (rarely optimal wrt.

disturbances).

Stochastic Control: requires a precise model (also of the disturbances).

w y

e

u System and disturbances

Model of

Objectives Constraints

System Controller

Design

System Identification: Parameter estimation. Validation.

Model

y e

u

ID Uncertainty

Knowledge Objective

System

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Adaptive Control

Adaptive Control: Parameter changes (time variations, nonlinear system).

w

Controller System

Design

e u

y ID

w y e

u System and disturbances

Model of

Objectives Constraints

System Controller

Design

-3 -2 -1 0 1 2 3 4 5

0 2 4 6 8 10 12 14 16 18 20

Optimal

Suboptimal

→hhn et al.

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Gain Scheduling (GS), Linear Parameter Variation (LPV)

Controller System

Schedule

Measured point of operation

w u

e

y

Examples: Tank system. Wind turbine. Ship. Aircraft.

To be considered as an open loop adaptive controller or a methods for dealing with nonlinear systems.

System Controller

Adaption Model

w

u y

error ym

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The Basic Self Tuner

w

Controller System

Design

e u

y ID

Explicit STC

Implicit Cautious

CE based STC

Suboptimal dual controllers Optimal dual controllers

MRAC STC Gain Schedule

Dual

Gradient optimization

Stability optimization

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Recap: Polynomials and vectors

Let us consider the result of a polynomium operating on a signal S(q⁻¹)yt = s0yt+s1y_t−1+ ...+sny_t−n

= γ^Tϑ where

γ^T =

yt y_t−1 ... y_t−n ϑ^T=

s0 s1 ... sn

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The Basic Self Tuner (Explicit version)

System:

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et

ID:

yt=ϕ^⊤_tθ+et

θˆt=arg M in

t

X

i=0

ε²_i pem or plr

Control:MV.

ut=arg M inEn y_t+k² o

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et

ID:

θ^⊤ = (... ai, ... bi, ... ci...)

ϕ^⊤_t = (... −y_t−i, ... u_t−i, ... e_t−i...)

θˆt = θˆ_t−1+ P¯tψt

1 +ψ_t^⊤P¯tψt

εt εt=yt−ϕ^⊤_t θˆt

Pt = P¯t− P¯tψψ^⊤_t P¯t

1 +ψ^⊤_tP¯tψt

P¯t=f unk(Pt−1, εt,ϕˆt, ψt) Forgetting ψt= 1

Cˆ_t−1

ˆ ϕt

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Control: The Basic Self Tuner - Explicit version

J=E n

y_t+k² o

Minimum variance control

Cˆ= ˆAG+q^−kS yt+k= 1

Cˆ

BGuˆ t+Syt

+Get+k

Rut=−Syt R= ˆBG

Controller:

Rut+Syt= 0

γt = (ut, u_t−1, ... yt, y_t−1, ...)^⊤ ϑ = (r0, , r1, ... s0, s1, ...)^⊤ ut=arg Sol

γ_t^Tϑ= 0

System:

yt−0.9y_t−1= 1u_t−2+et NB: an ARX system Model:

yt+ ˆa y_t−1= ˆb u_t−2+εt

Estimation:

εt=yt+ ˆa_t−1y_t−1−ˆb_t−1u_t−2 ϕ^T_t =

−y_t−1 u_t−2

θ^T= a b

θˆt = θˆ_t−1+ P¯tϕt

1 +ϕ^⊤_t P¯tϕt

εt

Pt = P¯t− P¯tϕtϕ^⊤_t P¯t

1 +ϕ^⊤_tP¯tϕt

P¯t=f unk(Pt−1, εt,ϕˆt, ψt)

Je=

t

X

i=1

ε²_i ≃tσ² εt=et for correct parameters

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Example

Design:

1 = (1 + ˆatq⁻¹)(1 +gq⁻¹) +q⁻²s G= 1−ˆatq⁻¹ S= ˆa²_t Controller:

ˆbt(1−aˆtq⁻¹)ut=−ˆa²_tyt

or

ut= ˆatu_t−1−ˆa²_t ˆbt

yt

Jc=

t

X

i=0

y_i²≃E n

y_t² o

t≃1.81σ²t In closed loop (for correct parameters)

yt= (1 + 0.9q⁻¹)et

0 20 40 60 80 100 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

System parameters

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Example

0 20 40 60 80 100

-1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5

Control parameters

0 20 40 60 80 100 50

100 150 200

Accumulated control and est. losses

Acc. Loss

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Example

20 40 60 80 100 0

50 100 150 200 Loss

Acc. Loss

20 40 60 80 100 50

100 150 200 250 Loss

Acc. Loss

20 40 60 80 100 50

100 150 200 250 Loss

Acc. Loss

20 40 60 80 100 0

200 400 600 800 1000 Loss

Acc. Loss

J=E n

y_t+k² o

A(q⁻¹)yt=B(q⁻¹)u_t−k+C(q⁻¹)et

Cˆ= ˆAG+q^−kS yt+k= 1

Cˆ

BGuˆ t+Syt

+Get+k

Rut=−Syt R= ˆBG

Controller:

Rut+Syt= 0

γt = (ut, u_t−1, ... yt, y_t−1, ...)^⊤ ϑ = (r0, , r1, ... s0, s1, ...)^⊤ ut=arg Sol

γ_t^Tϑ= 0

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The Basic Self Tuner - Explicit simple version

n y_t+1²

o

A(q⁻¹)yt=B(q⁻¹)ut−1+et

k= 1 C(q⁻¹) = 1

1 = ˆA+q⁻¹S G(q⁻¹) = 1 S(q⁻¹) =q(1−A) = ˜ˆ A yt+1=Buˆ t+Syt

+et+1

Rut=−Syt R= ˆB

Controller:

Rut+Syt= 0

γt = (ut, u_t−1, ... yt, y_t−1, ...)^⊤ ϑ = (r0, , r1, ... s0, s1, ...)^⊤ ut=arg Sol

γ_t^Tϑ= 0

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The Basic Self Tuner II (Implicit version)

A(q⁻¹)yt=B(q⁻¹)ut−1+et C= 1 k= 1

MV: J=En

y²_t+1o

C=AG+q^−kS y_t+k= 1

C

BGut+Syt

+Ge_t+k Rut=−Syt R=BG

1 =A∗1 +q⁻¹S G= 1 S=q(1−A) =−a1−a2q⁻¹− ... −anq¹⁻ⁿ yt+1= [But+Syt]+et+1 But=−Syt Notice the simple design

yt+1= [Syt+Rut]+et+1=ϕ^⊤_t+1θ+et+1

ϕ^⊤_t+1=

yt y_t−1 ... ut ...

θ^T =

s0 s1 ... r0 ...

θˆ: yt=ϕ^⊤_tθˆ_t−1+εt; Je=1 t

t

X

i=0

ε²_i ut: ϕ^⊤_t+1ˆθt= 0 Jc=E

n y²_t+1

o

A(q⁻¹)yt=q^−kB(q⁻¹)ut+et

C= 1 k≥1

MV: J=E

n y²_t+ko

BGut=−Syt

1 =AG+q^−kS

y_t+k= [Syt+BGut]+Get+k

y_t+k = [Syt+Rut]+Get+k

= ϕ^⊤_t+kθ+Get+k

θˆ: yt=ϕ^⊤_tθˆ_t−1+εt; Je=1 t

t

X

i=0

ε²_i

ut: ϕ^⊤_t+kθˆt= 0 Jc=E n

y²_t+k o

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The Basic Self Tuner IV

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et

MV: J=En

y²_t+ko

BGut=−Syt

C=AG+q^−kS y_t+k= 1

C[Syt+BGut]+Get+k

RLS: yt+k = [Syt+Rut] +Get+k Notice the missing C

= ϕ^⊤_t+kθ+Ge_t+k θˆ: yt=ϕ^⊤_tθˆt−1+εt; Je= 1

N

t

X

i=0

ε²_i

ut: ϕ^⊤_t+kθˆt= 0 Jc=E n

y²_t+k o

Advantage:

Design simple, RLS (even ifC6= 1), Je≃Jc

Disadvantage:

More parameters (k >>1),

Not all strategies can be transformed into an implicitte strategy.

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Convergence analysis

Isθ0a possible convergence point.

S : yt= 1

C ϕ^⊤_tθ0+Get

M : yt= ϕ^⊤_tθˆ+εt

En ϕtεt

o

= 0

εt = yt−ϕ^⊤_tθˆ= 1

C ϕ^⊤_tθ0+Get−ϕ^⊤_t θˆ

= Get + 1−C

C ϕ^⊤_t θ0 for θˆ=θ0

Synergy Control:Jc=En

y²_t+ko

=En ε²_t+ko Estimation:Je= 1

N Pε²_i

Fixation

Controller:

ut=−S Ryt

Model:

yt+k= [Rut+Syt] +Get+k

1 2 <r0

b0

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Stochastic Adaptive Control (02421)

www.imm.dtu.dk/courses/02421

Niels Kjølstad Poulsen

Build. 303B, room 048 Section for Dynamical Systems

Dept. of Applied Mathematics and Computer Science The Technical University of Denmark

Email: nkpo@dtu.dk phone: +45 4525 3356 mobile: +45 2890 3797

2019-04-29 20:09

Adaptive Control II (L23)

Explicit STC

Implicit Cautious

CE based STC

Suboptimal dual controllers Optimal dual controllers

MRAC STC Gain Schedule

Dual

Gradient optimization

Stability optimization

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Certainty Equivalence principle

State space control

xt+1=Axt+But+vt

yt=Cxt+et

Complete state information ut=−Lxt

Incomplete state information ut=−Lˆxt

CE valid

Adaptive control

A(q⁻¹)y=q^−kB(q⁻¹)ut+C(q⁻¹)et

MV for known system C=AG+q^−kS BGut=−Syt

Adaptive MV Cˆ= ˆAG+q^−kS BGuˆ t=−Syt

CE convinient

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Explicit STC (CE based)

w

Controller System

Design

e u

y ID

Minimalvariance, MV0

PZ, GSP GMV (MVi) GPC (or MPC) LQG (SS or Xreg) Deadbeat, PID ao.

Kalmanfilter/observer Polplacement controller LQG

Robust

System (assumed known):

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et+d Cost:

J=En

(yt+k−wt)²o

Controller:

BGut=Cwt−Syt−Gd Design:

C=AG+q^−kS

G(0) = 1 ord(G) =k−1 ord(S) =max(na−1, nc−k)

wt

−Gd

ut C

et d

yt A−1

−S 1

C BG q−k B

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Main loop

measinit; % Initilialise the measurement system for it=1:nstp,

w=wt(it);

[y,t]=meas; % Measure output

u=...

act(u); % Actuate control

end

Rut=Qwt−Syt−δ

x^r_t+1 = A^rx^r_t+B^r wt

yt

ut = C^rx^R_t +D^r wt

yt

+u0

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Cannonical realization

%--- [A,B,k,C,s2]=sysinit(dets); % Determine linear model (ie. get system)

%--- [Q,R,S,G]=dsnmv0(A,B,k,C);

%--- [Ar,Br,Cr,Dr]=qrs2ss(Q,R,S);

nr=length(Ar); Xr=zeros(nr,1);

%--- measinit; % Initilialise the measurement system

for it=1:nstp,

w=wt(it); wf=wft(it);

[y,t]=meas;

u=Cr*Xr+Dr*[wf;y]+u0;

act(u); % Actuate control Xr=Ar*Xr+Br*[wf;y];

end

Rut=Qwt−Syt−δ Rut−Qwt+Syt+δ= 0

ϕ^T

t =

ut, u_t−1, ...−wt,−w_t−1, ... yt, y_t−1, ...1

ϕ=ϕr

θ^T =

r0, r1, ... q0, q1, ... s0, s1, ... δ

ut=−ϕ˜^T

tθ˜ r0

θ˜=θ(2 :end);

ϕ^T

t−1=

u_t−1, u_t−2, ... −w_t−1,−w_t−2, ... y_t−1, y_t−2, ...1

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Direct realization

%--- [A,B,k,C,d,s2]=sysinit(dets); % Determine linear model (ie. get system)

%--- [Q,R,S,G]=dsnmv0(A,B,k,C);

%--- nr=length([R Q S])+1; fir=zeros(nr,1); thr=[R Q S G(1)*d]’;

pil=1+[0 length(R) length([R Q])];

%--- measinit; % Initilialise the measurement system

for it=1:nstp, w=wt(it);

[y,t]=meas; % Measure output

% Ru=Qw-Sy-Gd

fir(2:end)=fir(1:end-1);

fir(pil)=[0 -w y];

u=-fir’*thr/thr(1);

fir(1)=u;

act(u); % Actuate control end

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et+d

ID:

θ^⊤ = (... ai, ... bi, ... ci... d)

ϕ^⊤_t = (... −y_t−i, ... u_t−i, ... e_t−i...1)

θˆt = θˆ_t−1+ P¯tψt

1 +ψ_t^⊤P¯tψt

εt εt=yt−ϕ^⊤_t θˆt

Pt = P¯t− P¯tψψ^⊤_t P¯t

1 +ψ^⊤_tP¯tψt

P¯t=f unk(Pt−1, εt,ϕˆt, ψt) Forgetting ψt= 1

Cˆ_t−1

ˆ ϕt

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Explicit MV

controller

%--- [A,B,k,C,d,s2]=sysinit(dets); % Determine linear model (ie. get system)

%--- na=length(A)-1; mb=length(B); nc=length(C)-1;

th=[A(2:end) B C(2:end) d]’; th0=th;

th=th*0;

pil=[na mb];

pil=[0 cumsum(pil)]+1;

fi=zeros(size(th));

p0=10000;

P=eye(size(th,1))*p0;

%--- [Q,R,S,G]=dsnmv0(A,B,k,C); % just for the structure

%--- nr=length([R Q S]); fir=zeros(nr,1); fir=[fir; 1]; thr=[R Q S G(1)*d]’;

pilr=1+[0 length(R) length([R Q])];

%---

Main loop

measinit; % Initilialise the measurement system for it=1:nstp,

w=wt(it);

[y,t]=meas; % Measure output

% ID block res=y-fi’*th;

K=P*fi/(1+fi’*P*fi);

P=P-K*fi’*P;

th=th+K*res;

% Design block

A=[1 th(1:na)’]; B=th(pil(2):pil(3)-1)’;

C=[1 th(pil(3):end)’]; d=th(end);

[Q,R,S,G]=dsnmv0(A,B,k,C);

thr=[R Q S G(1)*d]’;

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Explicit MV

controller

% Ru=Qw-Sy Controller

fir(2:end)=fir(1:end-1);

fir(pilr)=[0 -w y];

u=-fir’*thr/thr(1);

fir(1)=u;

fi(2:end)=fi(1:end-1);

fi(pil(1:2))=[-y u]’;

act(u); % Actuate control end

System:

Ayt=q^−kBut+Cet+d Cost:

J=En

(Amy_t+k−Bmwt)²o

Controller:

BGut=BmCwt−Syt−Gd Design:

AmC=AG+q^−kS

G(0) = 1 ord(G) =k−1 ord(S) =max(na−1, nc+nam−k)

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Explicit PZ controller

%--- [A,B,k,C,d,s2]=sysinit(dets); % Determine linear model (ie. get system)

%--- na=length(A)-1; mb=length(B); nc=length(C)-1;

th=[A(2:end) B C(2:end)d]’; th0=th;

th=th*0;

pil=[na mb];

pil=[0 cumsum(pil)]+1;

fi=zeros(size(th));

p0=10000;

P=eye(size(th,1))*p0;

%--- Am=[1 -0.6];

Bm=sum(Am);

[Q,R,S,G]=dsnpz(A,B,k,C,Am,Bm); % just for the structure

%--- nr=length([R Q S]); fir=zeros(nr,1); fir=[fir; 1]; thr=[R Q S G(1)*d]’;

pilr=1+[0 length(R) length([R Q])];

%---

Main loop

measinit; % Initilialise the measurement system for it=1:nstp,

w=wt(it);

[y,t]=meas; % Measure output

% ID block res=y-fi’*th;

K=P*fi/(1+fi’*P*fi);

P=P-K*fi’*P;

th=th+K*res;

% Design block

A=[1 th(1:na)’]; B=th(pil(2):pil(3)-1)’;

C=[1 th(pil(3):end)’]; d=th(end);

[Q,R,S,G]=dsnpz(A,B,k,C,Am,Bm);

thr=[R Q S G(1)*d]’;

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Explicit PZ-control

% Ru=Qw-Sy Controller

fir(2:end)=fir(1:end-1);

fir(pilr)=[0 -w y];

u=-fir’*thr/thr(1);

fir(1)=u;

fi(2:end)=fi(1:end-1);

fi(pil(1:2))=[-y u]’;

act(u); % Actuate control end

0 10 20 30 40 50 60 70 80 90 -1

-0.5 0 0.5 1

Y, W

t in sec

yt wt

0 10 20 30 40 50 60 70 80 90

-1 -0.5 0 0.5 1

U

t in sec ut

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Explicit PZ-control

0 10 20 30 40 50 60 70 80 90

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

θ

t in sec

θ

0 10 20 30 40 50 60 70 80 90 -0.4

-0.2 0 0.2 0.4

Y, W

t in sec

yt wt

0 10 20 30 40 50 60 70 80 90

-0.4 -0.2 0 0.2 0.4

U

t in sec

ut

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Explicit PZ-control

0 10 20 30 40 50 60 70 80 90 100

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Je

t in sec Accumulated losses

0 10 20 30 40 50 60 70 80 90 100

0 0.5 1 1.5 2

Ju

t in sec

0 10 20 30 40 50 60 70 80 90 -2

-1.5 -1 -0.5 0 0.5 1 1.5 2

θ

t in sec

θ

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Implicit Adaptive Control

w

Controller System

Design

e u

y ID

Implicit Adaptive Control (STC)

The model is rephrased in terms of the control parameters, which are estimated. The adaptation mechanism (estimation procedure) is working on the control parameters directly.

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Implicit MV

J=En

(yt+k−wt)²o

A(q⁻¹)yt=q^−kB(q⁻¹)ut+C(q⁻¹)et+d

C=AG+q^−kS R=BG δ=G(1)d ξt+k = yt+k−wt

= 1

C[Syt+Rut−Cwt+δ] +Get+k

= 1

Cϕ^⊤_t+kθ+ ¯et+k

Model: ξt = yt−w_t−k

= ϕ^⊤_tθˆ_t−1+εt NoC

Control: ut = argSol

ϕ^⊤_t+kθˆt= 0.

RLS-algorithm

Computer burden (design)

Active (estimation) and passive (control) have similar cost function.

Has to knowk.

The min phase problem.

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Implicit MV

J=E n

(yt+k−wt)²o

1 Measureyt.

2 Createξt=yt−w_t−k.

3 Createϕt= (y_t−k, ... , u_t−k, ... ,−w_t−k,1)^Tandϕt+k. 4 Update the estimates:

ǫt = ξt−ϕtθˆ_t−1 (1)

P_t⁻¹ = P_t−⁻¹₁+ϕtϕ^T_t (2)

θˆt = θˆ_t−1+Ptϕtǫt (3)

5 Determineutsuch that:

ϕ^T_t+kθˆt= 0 (4)

6 Actuate the control.

J=E n

(A^m(q⁻¹)yt+k−B^m(q⁻¹)wt)²o

A^mC=AG+q^−kS R=BG Q=B^mC δ=Gd ξt+k = A^myt+k−B^mwt

= 1

C[Syt+Rut−Qwt+δ] +Ge_t+k

= 1

Cϕ^⊤_t+kθ+ ¯e_t+k

Model: ξt = A^myt−B^mw_t−k

= ϕ^⊤_tθˆ_t−1+εt

Control: ut = argSol

ϕ^⊤_t+kθˆt= 0.

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Implicit PZ-regulator

J=En

(A^m(q⁻¹)yt+k−B^m(q⁻¹)wt)²o

1 Measureyt.

2 Createξt=Am(q⁻¹)yt−Bm(q⁻¹)w_t−k.

3 Createϕt= (y_t−k, ... , u_t−k, ... ,−w_t−k,1)^Togϕ^T_t+k. 4 Update the estimates:

ǫt=ξt−ϕtθˆ_t−1 (5)

P_t⁻¹=P_t−1⁻¹ +ϕtϕ^T_t (6)

θˆt= ˆθ_t−1+Ptϕtǫt (7)

5 Determineutsuch that:ϕ^T_t+kθˆt= 0.

%--- [A,B,k,C,d,s2]=sysinit(dets); % Determine linear model (ie. get system)

%--- Am=[1 -0.6];

Bm=sum(Am);

[Ax,Bx,Cx,Dx]=armax2ss(1,Bm,k,Am);

nx=length(Ax); Xm=zeros(nx,1);

%--- [Q,R,S,G]=dsnpz(A,B,k,C,Am,Bm);

nr=length([R Q S d]); fi=zeros(nr,1);

th=[R Q S G(1)*d]’; th0=th; % th=th*0;

pil=1+[0 length(R) length([R Q])];

p0=10000;

P=eye(nr)*p0;

th(pil(2))=Q(1); % First coefficient in Q=C*Bm is known P(pil(2),pil(2))=0;

%th(pil(1))=R(1); % Fixed b0

%P(pil(1),pil(1))=0;

%---

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Implicit PZ-regulator

measinit; % Initilialise the measurement system for it=1:nstp,

w=wt(it);

[y,t]=meas; % Measure output

xi=Cx*Xm+Dx*[-w;y];

% ID block res=xi-fi’*th;

K=P*fi/(1+fi’*P*fi);

P=P-K*fi’*P;

th=th+K*res; % th’

fi(2:end)=fi(1:end-1);

fi(pil)=[0 -w y]’;

u=-fi’*th/th(1);

fi(pil(1))=u;

act(u); % Actuate control Xm=Ax*Xm+Bx*[-w;y];

end

Deterministic:

0 10 20 30 40 50 60 70 80 90

-1 -0.5 0 0.5 1

Y, W

t in sec

yt wt

0 10 20 30 40 50 60 70 80 90

-1 -0.5 0 0.5 1

U

t in sec ut

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Implicit PZ-regulator

0 10 20 30 40 50 60 70 80 90

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

θ

t in sec

θ

J=En By

Ay

y_t+k−Bw

Aw

wt

2

+ρ Bu

Au

ut

2

o

A^mC=ByAG+q^−kS

R=AuBG+αBuC δ=Gd

ξt+k = y˜t+k−w˜t+α˜ut

= 1

C[Sˇyt+Rˇut−Qwˇt+δ] +Get+k

= 1

Cϕ^⊤_t+kθ+ ¯et+k

ˇ y= 1

Ay

y wˇ= ˜w=Bw

Aw

w uˇ= 1 Au

u

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Implicit GMV-regulator

Model: ξt = ˜yt−w˜_t−k+α˜u_t−k

= ϕ^⊤_tθˆ_t−1+εt

Control: ut = argSol

ϕ^⊤_t+kθˆt= 0.

J=EnBy

Ay

yt+k−Bw

Aw

wt

2

+ρ Bu

Au

ut

2

o

1 Measureyt.

2 Createξt=y˜_t−w˜_t−k+α˜u_t−k h

α=_b^ρ

0

i .

3 Createϕt= (ˇy_t−k, ...,uˇ_t−k, ...,−wˇ_t−k, ...,1)^Togϕt+k. 4 Update the estimates:

ǫt=ξt−ϕtθˆ_t−1 (8)

P_t⁻¹=P_t−⁻¹₁+ϕtϕ^T_t (9)

θˆt= ˆθ_t−1+Ptϕtǫt (10)

5 Determineuˇtsuch that:ϕ^T_t+kθˆt= 0.

6 Determineut=Auuˇt.

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