Simulating the step response

The step input is widely used to characterize the transient response of a system. MATLAB’s step function conveniently simulates the step response of an LTI system model.

[ys_a,t_a] = step(sys);
disp([t_a(1:6),ys_a(1:6,1:4)]) % print a little

         0         0         0    1.0000         0
    0.0002    0.0018    0.0056    0.9082    0.0573
    0.0005    0.0071    0.0106    0.8245    0.1093
    0.0007    0.0155    0.0152    0.7482    0.1565
    0.0010    0.0267    0.0194    0.6786    0.1993
    0.0012    0.0405    0.0232    0.6151    0.2381

The vector t_a contains values of time and array ys_a contains a vector of time-series values for each output. If one would like the output for a step input \(k u_s(t)\) (scaled unit step \(u_s(t)\)), by the principle of superposition for linear systems, one can scale the output by \(k\). The outputs are plotted in figure 4.14.

It is also interesting to inspect the power flow and energy associated with each element. Since we have simulated both the across and the through variable for each element, we can compute the instantaneous power by simply taking the product of them at each time step. Moreover, we can cumulatively compute the energy contribution of that power for each element. For energy storage elements, this is the change in energy stored or supplied; for energy dissipative elements, this is the change in energy dissipated; for source elements, this is the energy supplied or absorbed. The results are plotted in figure 4.15.

P = NaN*ones(size(ys_a,1),size(ys_a,2)/2);
E = NaN*ones(size(P));
j = 0;
for i = 1:2:size(ys_a,2)
    j = j+1;
    P(:,j) = ys_a(:,i).*ys_a(:,i+1);
    E(:,j) = cumtrapz(t_a,P(:,j));
end
disp('power:');
disp(P(1:6,1:4)) % print a little
disp('energy change:')
disp(E(1:6,1:4)) % print a little

power:
         0         0         0         0
    0.0000    0.0520    0.0000    0.0052
    0.0001    0.0901    0.0000    0.0191
    0.0002    0.1171    0.0000    0.0392
    0.0005    0.1352    0.0000    0.0635
    0.0009    0.1465    0.0000    0.0907
energy change:
   1.0e-03 *
         0         0         0         0
    0.0000    0.0064    0.0000    0.0006
    0.0000    0.0239    0.0000    0.0036
    0.0001    0.0494    0.0000    0.0108
    0.0001    0.0805    0.0000    0.0235
    0.0003    0.1152    0.0000    0.0425