Reference Tracking and Optimal Control

Examine why reference error remains even with a stable controller, and compare tracking performance with control-input cost.

Starting conditions

$A=\begin{bmatrix}0&1\\-2&-3\end{bmatrix}$, $B=\begin{bmatrix}0\\1\end{bmatrix}$, $C=\begin{bmatrix}1&0\end{bmatrix}$, $K=\begin{bmatrix}6&3\end{bmatrix}$, $r=1$, x(0)=0.

Procedure

1. Use $N=1$ and $N=8$ in turn to calculate unit-step responses on the same time axis. Record the responses and steady-state outputs side by side.
2. For $0=(A-BK)x_{ss}+BNr$ under $r=1$, derive $x_{ss}$ in terms of $N$. Use $y_{ss}=Cx_{ss}$ to find $N$ satisfying $y_{ss}=1$, and show your calculations.
3. Compare the directly calculated $N$ with the final values of both responses, and explain what the reference gain changed.

Results to retain

• Both step responses and steady-state values, and the equation used to obtain N.

Starting conditions

$Q=\operatorname{diag}(q,1)$, $q\in\{1,25\}$, and $R=\begin{bmatrix}1\end{bmatrix}$. Use $x(0)=\begin{bmatrix}1\\0\end{bmatrix}$, $r=0$.

Procedure

1. Solve the CARE for each Q and calculate $K=R^{-1}B^\top P$.
2. Plot position and control input on the same time axis.
3. Compare the integral of squared position error and the integral of squared input separately.

Results to retain

• Position and input plots and a comparison table of both integrals.

Starting conditions

Design using nominal B, and change only the actual input gain to $B_{\mathrm{actual}}=gB$ and $g\in\{.8,1,1.2\}$. Use $K=\begin{bmatrix}6&3\end{bmatrix}$, $N=8$, $r=1$.

Procedure

1. Find the poles and final step-response value for each actual model.
2. Check whether N calculated from the nominal model also achieves the exact reference in every actual model.
3. Distinguish the roles of reference-gain adjustment and integral-error feedback.

Results to retain

• Three response curves and a table of final values and errors.

• SciPy · Pole placement
• SciPy · Continuous-time CARE