i need help to find impulse response of system h(t) at time t

Task: For the case E = I, the impulse response of a system is defifined as h(t) = CeAtB + Dδ(t), t ≥ 0,

and represents the system output y(t) when the input u(t) = δ(t).

Use the command expm() to compute the matrix exponential and then plot the impulse response

for an equally spaced time grid t = linspace(0,500,500). (Hint: You only need to compute the

matrix exponential once.) Explain why the impulse response h(t) → 0 as t → ∞.

I have considered the matrix values as below:

A=[-3 -5 -1;1 0 0;0 1 0];

B=[1; 0; 0],

C=[0 0 1]

D=1;

E=eye(3);

Answer 1

[Sam Chak]

Hi @Rushit Bhavsar

I used the impulse() method. but you can try using the expm() method. Look up the example and verify with the impulse response below.

A = [-3 -5 -1; 1 0 0; 0 1 0];
B = [1; 0; 0];
C = [0 0 1];
D = 1;
sys = ss(A, B, C, D)

sys =
 
  A = 
       x1  x2  x3
   x1  -3  -5  -1
   x2   1   0   0
   x3   0   1   0
 
  B = 
       u1
   x1   1
   x2   0
   x3   0
 
  C = 
       x1  x2  x3
   y1   0   0   1
 
  D = 
       u1
   y1   1
 
Continuous-time state-space model.

impulse(sys)

tFinal = 500;
impulse(sys, tFinal)

help expm

 EXPM  Matrix exponential.
    EXPM(A) is the matrix exponential of A and is computed using
    a scaling and squaring algorithm with a Pade approximation.
 
    Although it is not computed this way, if A has a full set
    of eigenvectors V with corresponding eigenvalues D then
    [V,D] = EIG(A) and EXPM(A) = V*diag(exp(diag(D)))/V.
 
    EXP(A) computes the exponential of A element-by-element.
 
    See also LOGM, SQRTM, FUNM.

    Documentation for expm
       doc expm

    Other uses of expm

       gpuArray/expm    sym/expm    symbolic/expm