Contents

% Skills Assessment - Chap 10

P10.1 Inverted pendulum

clear variables
g=9.8;
disp('Sampling time:'), Ts=.1
A=[0 1; g 0]; B=[0; 1]; C=[1 0]; D=0;
IP=ss(A,B,C,D);
disp('Discrete-time model:')
IPz=c2d(IP,Ts);
Ad=IPz.a, Bd=IPz.b,
f1=@(x,u) Ad*x+Bd*u;
u=rand(1,10); x=rand(2,1);
for i=1:length(u), x(:,i+1)=f1(x(:,i),u(i)); end
y=C*x;
disp('Random input:'), disp(u)
disp('Model output:'), disp(y)

Sampling time:
Ts =
    0.1000
Discrete-time model:
Ad =
    1.0494    0.1016
    0.9961    1.0494
Bd =
    0.0050
    0.1016
Random input:
  Columns 1 through 7
    0.5578    0.3134    0.1662    0.6225    0.9879    0.1704    0.2578
  Columns 8 through 10
    0.3968    0.0740    0.6841
Model output:
  Columns 1 through 7
    0.4024    0.5250    0.7038    0.9546    1.3037    1.7898    2.4585
  Columns 8 through 11
    3.3722    4.6225    6.3318    8.6706

P10.2 Airplane dynamics

A=[-1 1;-7 -.5]; B=[-.1;-5];
C=eye(2); D=[0;0];
AP=ss(A,B,C,D);
disp('Sampling time:'), Ts=.01
APz=c2d(AP,Ts);
APz.y={'AOA','q'};
disp('Discrete-time model:')
Ad=APz.a, Bd=APz.b,
disp('Transfer functions:')
tf(APz)
x=Bd;
f2=@(x,u) Ad*x;
for i=1:length(u), x(:,i+1)=f1(x(:,i),u(i)); end
disp('Model outputs:')
disp('AOA:')
disp(x(1,:))
disp('Pitch rate:')
disp(x(2,:))

Sampling time:
Ts =
    0.0100
Discrete-time model:
Ad =
    0.9897    0.0099
   -0.0695    0.9947
Bd =
   -0.0012
   -0.0498
Transfer functions:

ans =
 
  From input to output...
         -0.001244 z + 0.0007424
   AOA:  -----------------------
         z^2 - 1.984 z + 0.9851
 
        -0.04983 z + 0.04941
   q:  ----------------------
       z^2 - 1.984 z + 0.9851
 
Sample time: 0.01 seconds
Discrete-time transfer function.

Model outputs:
AOA:
  Columns 1 through 7
   -0.0012   -0.0036   -0.0018    0.0021    0.0103    0.0276    0.0534
  Columns 8 through 11
    0.0867    0.1318    0.1924    0.2757
Pitch rate:
  Columns 1 through 7
   -0.0498    0.0032    0.0316    0.0483    0.1160    0.2324    0.2887
  Columns 8 through 11
    0.3823    0.5279    0.6928    0.9882

P10.3 DC motor

R=1; L=.01; J=.01; b=.1; k=.02;
A=[R/L -k/L; k/J -b/J]; B=[1/L; 0];
C=[0 1]; D=[0];
DC=ss(A,B,C,D);
Ts=.01;
DCz=c2d(DC,Ts);
disp('Discrete-time model:')
Ad=DCz.a, Bd=DCz.b,
disp('Pole placement controller:')
syms k1 k2 real
Acl=Ad-Bd*[k1 k2];
p=charpoly(Acl);
k=solve(p(2)+2*exp(-1), p(3)-exp(-2),k1,k2);
k=double([k.k1 k.k2])
DC1=DCz;
DC1.a=DCz.a-DCz.b*k;
disp('Deadbeat controller design:')
k=solve(p(2), p(3),k1,k2);
k=double([k.k1 k.k2])
DC2=DCz;
DC2.a=DCz.a-DCz.b*k;
disp('Step responses:')
c=evalfr(DC1,1)/evalfr(DC2,1);
step(DC1, DC2)
legend('pole placement controller','deadbeat controller')

Discrete-time model:
Ad =
    2.7179   -0.0330
    0.0330    0.9046
Bd =
    1.7182
    0.0139
Pole placement controller:
k =
    1.6057    9.1701
Deadbeat controller design:
k =
    1.8968   26.0751
Step responses:

P10.4 Flexible beam

wn=1e2; zi=.05;
A=[0 1; -wn^2 -2*zi*wn]; B=[0; wn]; C=[wn 0];
FB=ss(A,B,C,0);
FBz=c2d(FB,.001)
disp('Controller design:')
k=place(FBz.a,FBz.b,exp(.25*[-1+j -1-j]))
FBz.a=FBz.a-FBz.b*k;
disp('Step response:')
step(FBz)

FBz =
 
  a = 
              x1         x2
   x1      0.995  0.0009934
   x2     -9.934     0.9851
 
  b = 
              u1
   x1  4.979e-05
   x2    0.09934
 
  c = 
        x1   x2
   y1  100    0
 
  d = 
       u1
   y1   0
 
Sample time: 0.001 seconds
Discrete-time state-space model.

Controller design:
k =
  879.2029    4.3001
Step response:

P10.5 Airplane dynamics

A=[-1 1;-7 -.5]; B=[-.1;-5];
C=eye(2); D=[0;0];
AP=ss(A,B,C,D);
AP.y={'AOA';'PR'};
Ts=.05;
APz=c2d(AP,Ts);
Ad=APz.a; Bd=APz.b;
disp('Pole placement controller:')
k=place(Ad,Bd,exp(Ts*[-5+j,-5-j]))
H1=tf({k(1),k(2)},1);
T1=feedback(APz,H1);
disp('Deadbeat controller:')
syms k1 k2 real
Acl=Ad-Bd*[k1 k2];
p=charpoly(Acl);
k=solve(p(2), p(3),k1,k2);
k=double([k.k1 k.k2])
H2=tf({k(1),k(2)},1);
T2=feedback(APz,H2);
disp('Step responses:')
c=evalfr(T1(2),1)/evalfr(T2(2),1);
step(T1(1), T2(1))
title('AOA')
legend('pole placement controller','deadbeat controller')
figure, step(T1(2), T2(2))
title('Pitch rate')
legend('pole placement controller','deadbeat controller')

Pole placement controller:
k =
   -1.5336   -1.3747
Deadbeat controller:
k =
  -77.1633   -4.3400
Step responses:

P10.6 Human postural dynamics

G=tf([10],[1 0 -10]);
G=ss(G);
Ts=.1;
disp('Discrete-time model:')
Gz=c2d(G,Ts);
[Ad,Bd]=ssdata(Gz)
disp('State feedback controller:')
%K=place(G.a,G.b,[-5+j -5-j]);
K1=place(Ad,Bd,exp([-4+3*j -4-3*j]))
disp('Deadbeat controller:')
syms k1 k2 real
Acl=Ad-Bd*[k1 k2];
p=charpoly(Acl);
k=solve(p(2), p(3),k1,k2);
K2=double([k.k1 k.k2])
G1=Gz; G1.a=Gz.a-Gz.b*K1;
G2=Gz; G2.a=Gz.a-Gz.b*K2;
step(G1,G2)
legend('pole placement controller','deadbeat controller')

Discrete-time model:
Ad =
    1.0504    0.2542
    0.4067    1.0504
Bd =
    0.2034
    0.0403
State feedback controller:
K1 =
    7.7127   14.1001
Deadbeat controller:
K2 =
    7.6244   13.6464

Published with MATLAB® R2015b