Exercise 5

HINT: For problems 1 and 2, use a generalized plant description and the feedback or LFT command in matlab.

Discuss Handin 3

Leftovers from Exercise 4:

Solve Problem 4 and Problem 6

Problem 1

Consider the following midranging architecture based on error feedback where

$P_{1} = \frac{e^{-s}}{s + 1}$, and  $P_2 = \frac{e^{-5s}}{10s + 1}$. The controllers $C_1 = \frac{1 + 2/(3s) + s/2}{s/2 + 1}$ and $C_2 = \frac{1 + 1/(10s) + 3s}{2s + 1}$ have been tuned to give reasonable nice performance in feedback with $P_1$ and $P_2$ respectively.

.

1. Why do we use positive feedback with $C_2$?, i.e. why $u_2 = C_2(u - u_r)$ and not $C_2(u_r - u)$?
2. Simulate the responses to $y$, $u_1$ and $u_2$ for reference step changes to $y_r$ and $u_r$ and input load disturbances
3. Generate bode plots of the closed loop responses from $(y_r, d_1, d_2, n) \mapsto (y, u_1, u_2)$, where $d_1$ and $d_2$ are the corresponding input load disturbances and $n$ is measurement noise acting on $y$. Generate the responses for $k_\text{ff} = 0$ and $k_\text{ff} = -P_2(0) / P_1(0)$.

Problem 2

With $P_1 = \frac{e^{-5s}}{10s + 1}$ and $P_{2} = \frac{e^{-s}}{s + 1}$, consider a Cascade architecture as in the block diagram below:

Design reasonable controllers for the cases

$C_1$	$C_2$
PD	PID
PID	PD
PID	PID

 

1. Discuss which closed-loop transfer functions are interesting, and plot their frequency responses. What's Gang of Four here?
2. How would you implement anti-windup in these three cases. Introduce actuator limitations and simulate. To help you get started you can use the following simulink implementation: get_started_p2.zip Download get_started_p2.zip

Problem 3

Explain why the standard Smith predictor does not work for processes with integration or unstable dynamics

Problem 4

The standard Smith predictor for a process $P(s) = P_0(s)e^{-sL}$ with time delay can be factored into

$C(s) = C_0(s)C_\text{pred}(s), \qquad C_\text{pred}(s) = \frac{1}{1 + \hat P_0(s) C_0(s)(1-e^{-sL})}$ where $C_0$ is the nominal controller for the process $P_0$ without delay and $L$ is the time delay. The transfer function $C_\text{pred}(s)$ is actually a good predictor that also can be used for loop shaping.

1. Explore the Bode plot of the predictor $C_\text{pred}(s)$ for $P_0(s) = \frac{1}{s + 1}, \quad C_0(s) = 2 + \frac{4}{s}$. Demonstrate that $C_\text{pred}(s)$ is indeed a good phase lead compensator
2. The properties of the compensator $C_\text{pred}(s)$ changes qualitatively with $L$, explain the nature of the changes and find the smallest value of $L$ for which the change occurs.
3. Use the insight from B) to discuss fundamental limitations of the Smith predictor $C_\text{pred}$
4. Smith's predictor can be used a as lead compensator without reference to time delays. Explore if it can be used as a lead compensator for the oscillatory system $P(s) = \frac{1}{s^2 + 0.02s + 1}$

Problem 5

In the same set up as Problem 1 construct an integrating LQG controller $\begin{bmatrix}u_1 \\ u_2 \end{bmatrix} = C\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, where $y_1 = y + n_1$ and $y_2 = u_1 + n_2$. Can you tune the controller to get midranging behavior?

Smith Predictor

The control structure is based on the idea to use a model $\hat P_0(s)$to predict the output of a delayed system, $P(s) = P_0(s) e^{-sL}$. The hat is introduced to differentiate between model and reality.

The controller becomes $C(s) = \frac{C_0}{1 + C_0 \hat P_0(1 - e^{-sL})}$. If $\hat P_0 = P_0$, the closed-loop becomes $T = \frac{P_0 C_0}{1 + P_0 C_0} e^{-sL}$.