Unforced System Responses. Perturbations in Initial Conditions

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# Unforced System Responses. Perturbations in Initial Conditions

The eigenvalues of the A matrix in the state space model provide information about stability and the relative speed of response. The eigenvectors provide information about the directional dependence of the speed of response. The MATLAB EIG routine is used for eigenvalue/eigenvector calcuations. Recall that a positive eigenvalue is unstable, while a negative eigenvalue is stable. A large magnitude eigenvalue is ``faster'' than a small magnitude eigenvalue.

The eigenvalues for this system are (see appendix)

(slow) and (fast)

The eigenvectors are

The second eigenvalue and eigenvector tell us that a perturbation in the initial condition of the second state variable will have a fast response. The first eigenvalue and eigenvector tell us that a perturbation in the direction of the

eigenvector will have a slow response.

##### Slow Response

Consider an initial perturbation in the slow direction. Let the perturbation be of magnitude 5

The physical state variables (tank and jacket temperature) are

The responses for a perturbation in the slow direction are shown in Figure 6.

##### Fast Response

Consider an initial perturbation in the fast direction. Let the pertubation be

The physical state variables (tank and jacket temperature) are

The responses for a perturbation in the fast direction are shown in Figure 7 (note the time scale change from Figure 6).

In Figures 6 and 7 we have shown the effect of initial condition ``direction''. This can be illustrated more completely by viewing the phase-plane behavior.

##### Phase-Plane Behavior

The phase-plane behavior is shown in Figure 8, where the Tank Temperature is plotted on the x-axis and the Jacket Temperature is plotted on the y-axis. Notice that initial conditions in the

direction respond much more slowly than initial conditions in the

direction.

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B. Wayne Bequette, bequeb@rpi.edu