Exercise 6

We will first discuss handin 4 and left-over problems from exercise 5.

Problem 1

Use the appropriate Riccati equation to prove the Kalman filter identity

$LaTeX: R_2 + C_2(sI - A)^{-1}R_1(s_I - A^\top)^{-1} C_2^\top = [I_p + C_2(s_i - A)^{-1}L]R_2[I_p + C_2(-sI - A^\top)^{-1}L]^\top$ $R_2 + C_2(sI - A)^{-1}R_1(s_I - A^\top)^{-1} C_2^\top = [I_p + C_2(s_i - A)^{-1}L]R_2[I_p + C_2(-sI - A^\top)^{-1}L]^\top$

Use duality to deduce the return difference formula

$LaTeX: Q_2 + B^\top(-sI - A^\top)^{-1}Q_1(sI - A)^{-1}B = [I_m + K(-sI - A^\top)^{-1}B]^\top Q_2[I_m + K(sI - A)^{-1}B]$ $Q_2 + B^\top(-sI - A^\top)^{-1}Q_1(sI - A)^{-1}B = [I_m + K(-sI - A^\top)^{-1}B]^\top Q_2[I_m + K(sI - A)^{-1}B]$

Problem 2

Consider the Doyle-Stein LTR example from the LQG lecture

$LaTeX: G(s)=\frac{s+2}{(s+1)(s+3)}$ $G(s)=\frac{s+2}{(s+1)(s+3)}$

See the slides, or their articles, for more details.

Evaluate the $LaTeX: \mathcal H_2$ $\mathcal H_2$ -norm for the system from $v$ to $z$ where $LaTeX: z^\top z = x^\top Q_1 x + u^\top Q_2 u$ $z^\top z = x^\top Q_1 x + u^\top Q_2 u$ and the maximum sensitivity $M_S$ of the closed-loop system for $q = 0$
Plot the $LaTeX: \mathcal H_2$ $\mathcal H_2$ -norm versus $M_S$ for varying values of $q$ . Is much $LaTeX: \mathcal H_2$ $\mathcal H_2$ -optimality lost to obtain robustness?

Problem 3

Consider the Rosenbrock example from the "Going MIMO" lecture

$LaTeX: P(s) = \begin{bmatrix} \frac{1}{s + 1} & \frac{2}{s+3} \\ \frac{1}{s+1} & \frac{1}{s+1} \end{bmatrix}$ $P(s) = \begin{bmatrix} \frac{1}{s + 1} & \frac{2}{s+3} \\ \frac{1}{s+1} & \frac{1}{s+1} \end{bmatrix}$ ,

which has a multivariable zero in LaTeX: s = 1 $s = 1$ . Design a controller using LQG and try to achieve a gain crossover frequency $LaTeX: \omega_{gc} > 0.5 \textrm{rad/s}$ $\omega_{gc} > 0.5 \textrm{rad/s}$ and reasonable robustness.

Problem 4

The file quadtank.m Download quadtank.m contains a linear model of a symmetric quadtank. The outputs are the levels of the two lower tanks. The parameter setting $LaTeX: \gamma = 0.3$ $\gamma = 0.3$ corresponds to a non-minimum phase system that is difficult to control.

Make an LQG design with reasonable performance that has integral action using either:

Explicit integration augmenting the system with integrator states $LaTeX: \dot x_9 = y - 3 \in \mathbb R^2$ $\dot x_9 = y - 3 \in \mathbb R^2$ (these two states are directly measured, and should not be estimated by the Kalman filter)
Augmenting the system with a constant input disturbance model, i.e. $LaTeX: \dot x = Ax + Bu + Bd$ $\dot x = Ax + Bu + Bd$ where $LaTeX: \dot d = 0$ $\dot d = 0$ and $LaTeX: u = -K\hat x - \hat d$ $u = -K\hat x - \hat d$

In both cases, plot the gain (singular values vs frequency) of the resulting controller and the GOF. Also verify that the step responses of the closed loop system look reasonable.

Problem 5

Prove the formulas mentioned in the Robust Control lecture:

$LaTeX: \text{gain margin} \geq \frac{1 + b_{P, K}}{1 - b_{P, K}}$ $\text{gain margin} \geq \frac{1 + b_{P, K}}{1 - b_{P, K}}$

$LaTeX: \text{phase margin} \geq 2 \arcsin(b_{P, K})$ $\text{phase margin} \geq 2 \arcsin(b_{P, K})$

Problem 6

Find a rational controller LaTeX: C(s) $C(s)$ that stabilizes both $LaTeX: P(s) = \frac{1}{s}$ $P(s) = \frac{1}{s}$ and $LaTeX: P(s) = -\frac{1}{s}$ $P(s) = -\frac{1}{s}$ or prove that it is impossible.

Problem 7

Calculate the nu-gap $LaTeX: \delta_\nu$ $\delta_\nu$ for varying parameters LaTeX: a $a$ between the processes

$LaTeX: G_1 = \frac{1}{s + a}$ $G_1 = \frac{1}{s + a}$ and $LaTeX: G_2(s) = \frac{1}{s-a}$ $G_2(s) = \frac{1}{s-a}$
$LaTeX: G_1 = \frac{1}{s + a}$ $G_1 = \frac{1}{s + a}$ and $LaTeX: G_2(s) = \frac{1}{a-s}$ $G_2(s) = \frac{1}{a-s}$
$LaTeX: G_1 = \frac{1}{s + 1}$ $G_1 = \frac{1}{s + 1}$ and $LaTeX: G_2(s) = \frac{a}{s + a}$ $G_2(s) = \frac{a}{s + a}$
$LaTeX: G_1 = \frac{1}{s + 1}$ $G_1 = \frac{1}{s + 1}$ and $LaTeX: G_2(s) = \frac{1}{(s + 1)^2}$ $G_2(s) = \frac{1}{(s + 1)^2}$
$LaTeX: G_1(s) = \frac{1}{s - 1}$ $G_1(s) = \frac{1}{s - 1}$ and $LaTeX: G_2(s) = \frac{1}{(s - 1)^2}$ $G_2(s) = \frac{1}{(s - 1)^2}$
$LaTeX: G_1(s) = \frac{1}{s - 1}$ $G_1(s) = \frac{1}{s - 1}$ and $LaTeX: \frac{1}{(s-1)(s+1)}$ $\frac{1}{(s-1)(s+1)}$

Hint: Use the matlab command gapmetric.

Problem 8

The solution to the $LaTeX: \mathcal H_\infty$ $\mathcal H_\infty$ problem presented at the lecture (implemented in the matlab-routine hinfsyn) solves the so-called sub-optimal problem: Given $LaTeX: \gamma$ $\gamma$ , determine if a controller exists giving a closed loop with

$LaTeX: \|T_{zw}\|_\infty < \gamma$ $\|T_{zw}\|_\infty < \gamma$ .

The optimal level, $LaTeX: \gamma_\star$ $\gamma_\star$ can then be found by decreasing $LaTeX: \gamma$ $\gamma$ until no solution exists. To study the optimal $LaTeX: \mathcal H_\infty$ $\mathcal H_\infty$ controller, consider the system given in the matlab-file goldenratioex.m. Download goldenratioex.m.

The system describes the feedforward optimization problem

$LaTeX: \begin{aligned} \dot x_1 & = -x_1 + u \\ \dot x_2 & = -x_2 + x_1 + w \\ z & = \begin{bmatrix} x_2 / \rho \\ u \end{bmatrix} \\ y & = w \end{aligned}$ $\begin{aligned} \dot x_1 & = -x_1 + u \\ \dot x_2 & = -x_2 + x_1 + w \\ z & = \begin{bmatrix} x_2 / \rho \\ u \end{bmatrix} \\ y & = w \end{aligned}$

For the case $LaTeX: \rho = 1$ $\rho = 1$ , use hinfsyn to find the optimal controller $u = K(s) w$ when $LaTeX: \gamma = \gamma_\star$ $\gamma = \gamma_\star$ . Compare with the analytical solution $LaTeX: K(s) = \dfrac{(s + 1)}{k(s + 1) + 1}$ $K(s) = \dfrac{(s + 1)}{k(s + 1) + 1}$ with $k = 1.3953$ . Hint: the optimal value is $LaTeX: \gamma_\star \approx 0.7167$ $\gamma_\star \approx 0.7167$
Do the same for $LaTeX: \rho = \sqrt{2}$ $\rho = \sqrt{2}$ . What order will the controller given by hinfsyn be now? Hint: The optimal value is $LaTeX: \gamma_\star = \frac{1}{\sqrt{3}}$ $\gamma_\star = \frac{1}{\sqrt{3}}$ .

For the interested, more details can be found in an article by Bernhardsson and Hagander from 1990.

Problem 9

Use mixsyn to do control of the motor

$LaTeX: G(s) = \frac{20}{s(s + 1)}$ $G(s) = \frac{20}{s(s + 1)}$

achieving $LaTeX: |\frac{1}{\omega}S(j\omega)| \leq k_1$ $|\frac{1}{\omega}S(j\omega)| \leq k_1$ and $LaTeX: |C(j_\omega)S(j\omega)|_\infty < k_2$ $|C(j_\omega)S(j\omega)|_\infty < k_2$ . Plot the region in the LaTeX: (k_1, k_2) $(k_1, k_2)$ -plane that you were able to achieve. Hint: you can use the file motorex6.m Download motorex6.m to get started.

Problem 10

Use the file aircglover.m Download aircglover.m to do Glover-MacFarlane design with loopsyn on the aircraft example. in the design, the 3 PI controllers had the same parameters. Redo the design and try to reduce the control peak due to a change of height reference (first input, i.e., the first column in figure 6) while maintaining a settling time of 1 second, good damping and mainly diagonal (noninteracting) response.