How to add intgeral action to Full state feedback control with two inputs two ouputs system

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Emad Alhani on 15 Feb 2021

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Edited: Pavl M. on 5 Dec 2024 at 8:41

TF.m

How to add integral action to Full state feedback control with two inputs and two outputs system. The system is fourth-order.

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Answer 1

Pavl M. on 4 Dec 2024 at 10:24

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https://in.mathworks.com/matlabcentral/answers/745982-how-to-add-intgeral-action-to-full-state-feedback-control-with-two-inputs-two-ouputs-system#answer_1553413

Edited: Pavl M. on 5 Dec 2024 at 8:41

Open in MATLAB Online

Subm1A.m

Good and valued: Solution ver. 1,

clc

clear all

close all

%% consider a 2 input, 2 output system,

nstates = 4;

ninputs = 2;

noutputs = 2;

A1=[0 0 -178 178;0 0 590 -500;-256 256 0 0; 0 -1 0 0.0000666];

B1=[8 -5;48 -63;-0.01 2.5; 0 0];

C1=[1 0 0 0;0 0 0 1];

D1=[0 0; 0 0];

Ts = 10e-3; % Sampling time in s

% Aa=[0 0 -178 178;0 0 590 -500;-256 256 0 0; 0 -1 0 0.0000666];

% Ba=[8 -5;48 -63;-0.01 2.5; 0 0];

% Ca=[1 0 0 0;0 0 0 1];

% Da = [0 0; 0 0];

% % you can use 'rss' and 'ssdata' to get random A, B, C, D matrices

% sys = rss(2,2,2) % to generate a random 2x2 system

% [A,B,C,D] = ssdata(sys); % to provide A, B, C, D matrices for this system

[num1 den1] = ss2tf(A1,B1,C1,D1,1) % iu = 1

num1 = 2×5

1.0e+06 * 0 0.0000 0.0000 -3.4081 0.0004 0 0 -0.0000 0.0000 3.3956

den1 = 1×5

1.0e+06 * 0.0000 -0.0000 -0.1971 0.0000 -4.1011

[num2 den2] = ss2tf(A1,B1,C1,D1,2); % iu = 2

%First input to first output only:

sys1 = tf(num1(1,:),den1) %contin

sys1 = 8 s^3 + 1.779 s^2 - 3.408e06 s + 386.3 ------------------------------------------------------ s^4 - 6.66e-05 s^3 - 1.971e05 s^2 + 13.09 s - 4.101e06 Continuous-time transfer function.

%Second input to second output only:

sys2 = tf(num2(2,:),den2); %contin

%First input to second output only:

sys3 = tf(num1(2,:),den1); %continuous

%Second input to first output only:

sys4 = tf(num2(1,:),den2); %continuous

[A, B, C, D] = tf2ss(num1(1,:),den1)

A = 4×4

1.0e+06 * 0.0000 0.1971 -0.0000 4.1011 0.0000 0 0 0 0 0.0000 0 0 0 0 0.0000 0

B = 4×1

1 0 0 0

C = 1×4

1.0e+06 * 0.0000 0.0000 -3.4081 0.0004

D = 0

asys = ss(A,B,C,D)

asys = A = x1 x2 x3 x4 x1 6.66e-05 1.971e+05 -13.09 4.101e+06 x2 1 0 0 0 x3 0 1 0 0 x4 0 0 1 0 B = u1 x1 1 x2 0 x3 0 x4 0 C = x1 x2 x3 x4 y1 8 1.779 -3.408e+06 386.3 D = u1 y1 0 Continuous-time state-space model.

opt2 = c2dOptions('Method','tustin','ThiranOrder',4,'DelayModeling','state');

%dsys=c2d(asys,Ts,opt2)

% observer model

%poles_obsv = exp(Ts*[-700 -310 -100 -90]);

poles_obsv = [-700 -310 -100 -90];

L=place(asys.a',asys.c',poles_obsv);

L=L';

Ah = asys.A;

bh = [asys.B L];

%cTh = eye(nstates);

%dh = [0 0;0 0;0 0;0 0];

%asysh=ss(Ah,bh,cTh,dh);

%figure

%step(asysh)

%add a state variable for the integral of the output error. This has the effect of adding an integral term to our controller which is known to reduce steady-state error.

%model for integral action

Ai = [[asys.A zeros(nstates,1)];-asys.C 0];

bi = [asys.B;0];

br = [zeros(nstates,1);1];

ci = [asys.C 0];

di = [0];

asysi=ss(Ai,bi,ci,di);

%Other augmentation scheme:

% Plant augmentation

Aaug=[asys.A zeros(nstates,1); zeros(1,nstates+1)];

Aaug(nstates+1,nstates)=1;

Baug=[asys.B;0];

Caug=eye(nstates+1);

Daug=zeros(nstates+1,1);

Plantcs=ss(Aaug,Baug,Caug,Daug);

% feedback controller

p1 = -800 + 800i;

p2 = -800 - 800i;

p3 = -400 - 400i;

%p1 = -110;

%p2 = -310;

%p3 = -500;

p4 = -400 + 400i;

p5 = -90;

%poles_k = exp(Ts*[p1 p2 p3 p4 p5]);

poles_k = [p1 p2 p3 p4 p5];

K = place(asysi.a,asysi.b,poles_k)

K = 1×5

1.0e+17 * 0.0000 -0.0000 -0.0000 2.9661 -0.0000

Ko = place(asys.A,asys.B,[p1 p2 p3 p4])

Ko = 1×4

1.0e+11 * 0.0000 0.0000 0.0154 4.0960

s4 = size(asys.A,1);

Z = [zeros([1,s4]) 1];

N = (1\([asys.A,asys.B;asys.C,asys.D]))*Z';

Nx = N(1:s4);

Nu = N(s4+1);

Nbar=Nu + Ko*Nx

Nbar = 2.4000e+03

%observer design:

At = [ asys.A-asys.B*Ko asys.B*Ko

zeros(size(asys.A)) asys.A-L*asys.C ];

Bt = [ asys.B*Nbar

zeros(size(asys.B)) ];

Ct = [ asys.C zeros(size(asys.C)) ];

%If you'd like to eliminate steady state error as much use Nbar:

% compute Nbar

%poles_k_d = exp(Ts.*poles_k)

%K_d = place(dsysi.a,dsysi.b,poles_k_d)

Ki=K(nstates+1);

K=K(1:nstates);

s4 = size(asys.A,1);

Z = [zeros([1,s4]) 1];

N = (1\([asys.A,asys.B;asys.C,asys.D]))*Z';

Nx = N(1:s4);

Nu = N(s4+1);

Nbarq=Nu + K*Nx

Nbarq = 2.5043e+03

%Well performing (adjusted) system:

syso = ss(At,Bt,Ct,0);

isstable(syso)

ans = logical

1

syso2 = minreal(syso)

4 states removed. syso2 = A = x1 x2 x3 x4 x1 -2400 -2.88e+06 -1.536e+09 -4.096e+11 x2 1 0 0 0 x3 0 1 0 0 x4 0 0 1 0 B = u1 x1 2400 x2 0 x3 0 x4 0 C = x1 x2 x3 x4 y1 8 1.779 -3.408e+06 386.3 D = u1 y1 0 Continuous-time state-space model.

%isminphase(syso)

f = 3;

t = 0:Ts:f;

figure

step(syso2,[0 f])

title('Continuous With observer flat(cold) start')

sysod = c2d(syso2,Ts,opt2)

sysod = A = x1 x2 x3 x4 x1 -0.9962 -390.2 -6.724e+04 -7.685e+06 x2 1.876e-05 -0.9512 -336.2 -3.842e+04 x3 9.381e-08 0.0002439 -0.6811 -192.1 x4 4.69e-10 1.22e-06 0.001595 0.0394 B = u1 x1 0.04503 x2 0.0002251 x3 1.126e-06 x4 5.629e-09 C = x1 x2 x3 x4 y1 -0.1448 -1977 -8.128e+05 2.966e+08 D = u1 y1 -1.738 Sample time: 0.01 seconds Discrete-time state-space model.

isstable(sysod)

ans = logical

1

u = 0.001*ones(size(t));

x0 = [0.01 0 0 0];

figure

lsim(sysod,zeros(size(t)),t,[x0]);

title('Discrete sampled with observer and conditions hot start')

xlabel('Time (sec)')

ylabel('Output')

res_agent = sysod;

figure

w = logspace(2,6,1000);

bode(res_agent,w),grid

figure

rlocus(res_agent)

figure

nyquist(res_agent)

ms = minreal(res_agent);

Gcc = ms;

zms = zero(ms) % closed-loop zeros

zms = 4×1

-1.0000 -1.8838 -0.5310 1.0000

pms = pole(ms)

pms =

-0.7561 + 0.1951i -0.7561 - 0.1951i -0.5385 + 0.3077i -0.5385 - 0.3077i

tsim = 1;

setpointamp = 1; %

setpointapptime = 0.001; %

defaultsetpointpos = 0; %

Conf = RespConfig(Bias=defaultsetpointpos,Amplitude=setpointamp,Delay=setpointapptime)

Conf =

RespConfig with properties: Bias: 0 Amplitude: 1 Delay: 1.0000e-03 InitialState: [] InitialParameter: []

%figure

%step(Gcc,[0 tsim], Conf)

%title('Plant+Controller Closed Loop system step response')

[wngcc,zetagcc,pgcc] = damp(Gcc)

wngcc = 4×1

266.5610 266.5610 289.9608 289.9608

zetagcc = 4×1

0.1792 0.1792 0.0853 0.0853

pgcc =

-0.5385 + 0.3077i -0.5385 - 0.3077i -0.7561 + 0.1951i -0.7561 - 0.1951i

sigma(Gcc)

margin(Gcc)

diskmargin(Gcc)

ans = struct with fields:

GainMargin: [1 1] PhaseMargin: [0 0] DiskMargin: 0 LowerBound: 0 UpperBound: 0 Frequency: NaN WorstPerturbation: [1x1 ss]

stepinfo(Gcc)

ans = struct with fields:

RiseTime: 0 TransientTime: 0.2000 SettlingTime: 0.7500 SettlingMin: -1.8775 SettlingMax: 2.3386 Overshoot: 1.0331e+08 Undershoot: 8.2937e+07 Peak: 2.3386 PeakTime: 0.0200

disp('Whether the system is stable, minimum phase and proper:')

Whether the system is stable, minimum phase and proper:

isstable(sysod)

ans = logical

1

isminphase(tfdata(tf(sysod),'v'))

ans = logical

0

isproper(sysod)

ans = logical

1

Qc= ctrb(A,B)

Qc = 4×4

1.0e+05 * 0.0000 0.0000 1.9711 0.0001 0 0.0000 0.0000 1.9711 0 0 0.0000 0.0000 0 0 0 0.0000

controllab = rank(Qc)

controllab = 4

hasdelay(sysod)

ans = logical

0

%tdc3 = totaldelay(sysod)

%sysndc3 = absorbDelay(sysod)

isallpass(tfdata(tf(sysod),'v'))

ans = logical

0

%Less important:

%sys_cl = ss(Ai-bi*[K Ki],br,ci,di)

%[a,b] = ss2tf(sys_cl.A,sys_cl.B,sys_cl.C,sys_cl.D)

%[A,B,C,D] = tf2ss(Nbar*a,b)

%sys_cl = ss(A,B,C,D)

%figure

%step(sys_cl,t)

%discrete system

%hold on

%sys_cld = c2d(sys_cl,Ts)

%figure

%step(sys_cld)

%title('Discrete sampled System')

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How to add intgeral action to Full state feedback control with two inputs two ouputs system

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How to add intgeral action to Full state feedback control with two inputs two ouputs system

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