Entering Matrices

Transcription

1 MATLAB Basic

2 Entering Matrices

3 Transpose

4 Subscripts

5 The Colon Operator

6

7 Operators

8 Generating Matrices

9 Element-by-element operation

10 Creating a Plot

11

12 Multiple Data Sets in One Graph

13

14 Line Styles and Colors

15

16 Flow Control: for

17 Control System Toolbox

18 Linear Model Representations

19 DC Motor

20

21 Converting Model

22 Transfer Function

23 Transfer Function

24 MIMO Transfer Functions

25 Interconnecting Linear Models

26 Interconnecting Linear Models

27 Feedback Interconnection

28 Frequency and Time Response

29 Time Response

30 MIMO Model Responses

31 Step Response

32 Right-Click Menus

33 Data Markers

34 lsim >> g=tf([1],[1 1 1]) Transfer function: s^2 + s + 1 >> t=0:0.1:10; >> u=ones(size(t)); >> lsim(g,u,t)

35 bode >> g=tf([1],[1 1 1]) Transfer function: s^2 + s + 1 >> bode(g)

36 bode >> w=logspace(-1,1,100); >> bode(g,w)

37 bode >> w=logspace(-1,1,100); >> g=tf([1],[1 1 1]) Transfer function: s^2 + s + 1 >> [mag,phase]=bode(g,w); >> for i=1:100 db(i)=20*log10(mag(:,:,i)); end >> semilogx(w,db) >> grid

38 Example 2.3

39 Example 2.3

40 % Example 2.3. Computes the principle gains. clear % Define the state model matrices. A=[ ]; B=[ ]; C=[ ]; D=zeros(2); % Define the frequencies for computing the principle gains. w=logspace(-2,2); % Define an identity matrix with the right dimensions. I=eye(size(A)); % Compute the principle gains. for k=1:50, Hjw=C*inv(j*w(k)*I-A)*B; Pgains(:,k)=svd(Hjw); end

41 % Plot the results. set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') figure(1) clf semilogx(w,20*log10(pgains)) xlabel('frequency (rad/sec)') ylabel('gain (db)') grid % % Alternative code uses the sigma command: figure(2) clf sigma(a,b,c,d)

42 Observer Feedback State Feedback

43 Example 2.9

44 Example 2.9

45 Example 2.9. Computes the state feedback gain and simulates. clear % Define the state model matrices. A=[ ]; B=[0 1]; C=eye(2); D=[0 0]'; % Define the desired pole locations. p=[-2+2*j -2-2*j]; % Find the state feedback gain matrix. K=place(A,B,p) % Define the initial condition. X0=[1 0]'; % Generate the intial condition solution. [y,x,t]=initial(a-b*k,b,c,d,x0);

46 % Plot the results. set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') figure(1) clf plot(t,y(:,1),'r-',t,y(:,2),'r--') grid xlabel('time (sec)') ylabel('amplitude') legend('angle','angle rate')

47 Observers

48

49 x ˆˆ( t) = ( A GC) x() t + Gm() t + ( B GD) u() t = Axˆˆ( t) GCx() t + Gm() t + Bu() t GDu() t = Axˆˆ( t) + Bu() t + G m() t Cx() t Du() t ( )

50 Example xt ˆˆˆ () xt () ut () = + + mt () [ 1 0 ] xt ()

51

52 % Example Generate the observer and simulate. clear % Define the state model matrices. A=[ ]; B=[0 1]; C=[1 0]; D=0; % Define the desired observer pole locations. p=[-8-8]; % Generate the observer. G=acker(A',C',p)' F=A-G*C H=B-G*D % Set the plant initial condition. X0=[1-2]';

53 % Generate the plant initial condition response. t=0:.01:5; [y,x]=initial(a,b,c,d,x0,t); % Set the observer initial condition. Xhat0=[0 0]'; % Simulate the observer. [yh,xh]=lsim(f,g,c,d,y,t,xhat0); % Plot the results. set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') figure(1) clf plot(t,x(:,1),'r-',t,x(:,2),'r--',t,xh(:,1),'r-.',t,xh(:,2),'r:') grid xlabel('time (sec)') ylabel('amplitude') legend('angle','angle rate','est. angle','est. angle rate')

54 Deterministic Separation Principle

55 Example 2.12

56 Example 2.12

57 % Example Generate the observer feedback controller and simulate. clear % Define the state model matrices. A=[ ]; B=[0 1]; C=[1 0]; D=0; % Define the state feedback pole locations. p=[-2+2*j -2-2*j]; % Generate the state feedback gain matrix. K=place(A,B,p);

58 % Define the desired observer pole locations. po=[-8-8]; % Generate the observer. G=acker(A',C',po)'; F=A-G*C; H=B-G*D; % Form the closed loop state model: Acl=[ A -B*K G*C F-B*K]; Bcl=[ ]'; Ccl=eye(4); Dcl=[ ]'; % Set the plant initial condition. X0=[ ]'; % Simulate the closed loop system. t=0:0.01:3; [y,x]=initial(acl,bcl,ccl,dcl,x0,t);

59 % Plot the results. set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') figure(1) clf plot(t,x(:,1),'r-',t,x(:,2),'r--',t,x(:,3),'r-.',t,x(:,4),'r:') grid xlabel('time (sec)') ylabel('amplitude') legend('angle','angle rate','est. angle','est. angle rate')

60 Mu-Analysis and Synthesis Toolbox

61 SYSTEM Matrices

62

63 VARYING Matrices

64

65

66 Plotting VARYING Matrices

67 Frequency Domain Functions

68

69

70 Example 4.5

71 % Example 4.5. System 2-norm computation clear % Define the Plant. A=[ ]; Bu=[0 1]'; Bw=[ ]; Cm=[1 0]; Cy=[ ]; Dmw=[0 1]; Dyu=[0 1]'; Dyw=zeros(2);

72 % Define the Controller. Ac=[ ]; Bcm=[ ]'; Cc=[ ]; Dcm=0; % Generate the closed loop state model. Acl=[A+Bu*Dcm*Cm Bu*Cc Bcm*Cm Ac ]; Bcl=[Bw+Bu*Dcm*Dmw Bcm*Dmw ]; Ccl=[Cy+Dyu*Dcm*Cm Dyu*Cc]; Dcl=Dyw+Dyu*Dcm*Dmw;

73 % Compute the system 2-norm. Lc=lyap(Acl,Bcl*Bcl'); J=sqrt(trace(Ccl*Lc*Ccl')) % Alternative computation of the system 2-norm. Lo=lyap(Acl',Ccl'*Ccl); J=sqrt(trace(Bcl'*Lo*Bcl)) % A function for computing the system 2-norm is provided in the Matlab % mu-syn toolbox. Gcl=pck(Acl,Bcl,Ccl,Dcl); J=h2norm(Gcl) % MATLAB2010b Gcl=ss(Acl,Bcl,Ccl,Dcl); J=norm(Gcl,2)

74 Example 4.6

75 % Example 4.6. Infinity-norm computation. clear % Define the closed loop system: p1=4; p2=0.1; k1=10; k2=0.01; k3=10; k4=150; a=0.4; b=4; Acl=[-p1 0 -k1*k4 k1 k2 -p k4*(b-a) -b ]; Bcl=[ k1*k4 0 -k1*k4 0 k2*k k4*(a-b) 0 k4*(b-a)]; Ccl=[ k4 1]; Dcl=[1 0 0 k4 0 -k4];

76 % Compute the principal gains of the closed loop system. w=logspace(-2,3,200); Gcl=pck(Acl,Bcl,Ccl,Dcl); fr=frsp(gcl,w); pgains=vsvd(fr); % Plot the principle gains. figure(1) set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') clf vplot('liv,lm',pgains) xlabel('frequency (rad/sec)') ylabel('magnitude') axis([ ]) grid

77 % The infinity-norm is the maximum over frequency of the maximum % principal gain. [nr,nc]=size(pgains); Hinf=max(pgains(1:nr-1,1)) % An alternative algorithm for computing upper and lower bounds on % the infinity-norm is given in the Mu-Synthesis and Analysis Toolbox. temp=hinfnorm(gcl); % The infinity-norm is given as the average of the bounds. Hinf=(temp(1)+temp(2))/2 %MATLAB2010b Gcl=ss(Acl,Bcl,Ccl,Dcl); J=norm(Gcl,inf)

78 Hinf = Hinf =

79 % Example 4.6. Infinity-norm computation. MATLAB2010 clear % Define the closed loop system: p1=4; p2=0.1; k1=10; k2=0.01; k3=10; k4=150; a=0.4; b=4; Acl=[-p1 0 -k1*k4 k1 k2 -p k4*(b-a) -b ]; Bcl=[ k1*k4 0 -k1*k4 0 k2*k k4*(a-b) 0 k4*(b-a)]; Ccl=[ k4 1]; Dcl=[1 0 0 k4 0 -k4];

80 % Compute the principal gains of the closed loop system. w=logspace(-2,3,200); Gcl=ss(Acl,Bcl,Ccl,Dcl); fr=frd(gcl,w); pgains=svd(fr); % Plot the principle gains. figure(1) set(0,'defaultaxesfontname','times') set(0,'defaultaxesfontsize',16) set(0,'defaulttextfontname','times') clf P = bodeoptions; P.MagUnits = 'abs'; P.MagScale = 'log'; P.PhaseVisible = 'off'; P.XLim = {[ ]}; P.YLim = {[ ]}; P.Grid = 'on'; bode(pgains(1),'c',pgains(2),'r',p)

81 % The infinity-norm is the maximum over frequency of the maximum % principal gain. [response, frequency]=frdata(pgains(1)); Hinf=max(response) % MATLAB2010 Gcl=ss(Acl,Bcl,Ccl,Dcl); Hinf=norm(Gcl,inf)

82 Hinf = Hinf = 10 3 Bode Diagram Magnitude (abs) Frequency (rad/sec)