Automatic Control III Computer Exercise 1 Solutions part 2

Automatic Control III Computer Exercise 1 Solutions part 2 Andreas Svensson [email protected] November 3, 2013 Exercise 4.1 The order of the system is the number of states. Since the A-matrix A4tank is 4 x 4, the order is 4. The system is found being controllable using, e.g., this Matlab code if rank(ctrb(A4tank,B4tank)) == length(A4tank) display(’Controllable’) end In a similar way, the system can be found being observable. A controllable and observable state space description is a minimal realization.

Exercise 4.2 Since the the state space description is a minimal realization of the system, the poles of the system are the eigenvalues of the A-matrix1 . The eigenvalues, and hence the poles, is found using the command eig to be -0.0161, -0.0111, -0.0435, -0.0333. Since the real parts of all poles are negative, the system is stable.

Exercise 4.3 At the end of Section 2 in the instructions, the approximate relation for for the systems impulse response g ptq (for t " 0 and a system with diagonalizable A-matrix) is derived: log g ptq  p1 t

C

(1)

For plotting the logarithm of g ptq, the follwing matlab code can be used: [Y,T] = impulse(G4tank); subplot(221); plot(T,log(Y(:,1,1))); ... subplot(224); plot(T,log(Y(:,2,2))); For each separate element in the transfer function, the following slopes2 (and, hence, dominating poles) can be found for t " 0: (1,1): (1,2): (2,1): (2,2):

-0.016 -0.016 -0.011 -0.011

These poles were also found in Exercise 4.2.

1 Check: If the state space description is not a minimal realization, what is the relation between the eigenvalues of A and the poles then? 2 An easy way to find the approximate slope is to use the data cursor tool in the plots.

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Exercise 4.4 Using the command tf, the following transfer function is found:

 Gpsq  

0.042 s 0.016

ps

ps

0.00052 0.033 s 0.011

qp

q

0.0010 0.044 s 0.016

qp

0.031 s 0.011

q



(2)

Compare the dominating poles to the findings in Exercise 4.3.

Exercise 4.5 For computing zeros of MIMO-system, the Matlab command tzero is to be used, according to the Matlab documentation! tzero(G4tank) gives the zeros -0.017 and -0.059, and hence is the system minimum phase. Try to find out why tzero(G), where G is the transfer function from Exercise 4.4, does not give the same answer!

Exercise 4.6 -

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