Engg. tutorials

Engg. tutorials

5.17 For a system $A=\begin{pmatrix}-3 & 5 \\ -1 & -5 \end{pmatrix}$, $B=\begin{pmatrix}0 \\ 5 \end{pmatrix}$, $C=\begin{pmatrix}1 & 0 \end{pmatrix}$,
$D=0$, design a state feedback plus integral control, so that the system characteristics equation will be, $s^3+12s^2+50s+100=0$.
Identify the gains for state feedback and gain for integral control.

5.18 Let us consider a system, $A=\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 &1 \\ 0 & 0 & 0 \end{pmatrix}$, $B=\begin{pmatrix}1 & 0 \\ 1 & 0 \\ 1 & 0 \end{pmatrix}$, $C=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0\end{pmatrix}$

a) Comment on system stability.

b) Comment on observability and controllability.

c) Let us assume that all the states are available for feedback. Find out the feedback gain to place the poles at -1, -2, -5 position.

d) Now let us assume that states are available only from the measurements. Construct an observer with the poles placed according to your wish. Take any initial condition of the truth state and plot the truth states. Take any initial condition of observer states (not equal to initial truth state) and plot the estimated states and error. You may use MATLAB to plot . You may take unity step input if required.

Courtesy: Davison, IEEE Auto. Control, Aug 1975, pp 516

a) Comment on system stability.

b) Comment on observability and controllability.

c) Let us assume that all the states are available for feedback. Find out the feedback gain to place the poles at -1, -2, -5 position.

d) Now let us assume that states are available only from the measurements. Construct an observer with the poles placed according to your wish. Take any initial condition of the truth state and plot the truth states. Take any initial condition of observer states (not equal to initial truth state) and plot the estimated states and error. You may use MATLAB to plot . You may take unity step input if required.

Courtesy: Davison, IEEE Auto. Control, Aug 1975, pp 516

5.19 Model of an Inverted pendulum (after linearlization) is obtained
as $A=\begin{pmatrix}0 & 1 & 0 & 0\\ 20.601 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\\ -0.4905 & 0 & 0 &0\end{pmatrix}$;
$B=\begin{pmatrix}0 \\-1 \\0 \\ 0.5 \end{pmatrix}$; $C=\begin{pmatrix}0 & 0 & 1 & 0 \end{pmatrix}$;
(recall inverted pendulum problem in earlier exercise).
To obtain a reasonable speed and damping in the response of the designed system (for example, the settling time of approximately $4 - 5$ sec and the maximum overshoot of $15\% - 16\%$ in the step response of the cart), let us choose the desired closed-loop poles at $-1\pm \sqrt3j $, -5, -5, -5. Determine the feedback gains and integral gain. Using MATLAB plot the states of the system with time, without feedback gain and with feedback gain.

Fig 5.6 State feedback control with integrator

5.20 Write a short article (around 4 pages) on rank of a matrix and it's physical interpretation.
You discussion must include column rank as well as row rank.

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