EECS 460 Discussion

Winter 2017

Introduction

Jonas Kersulis, GSI

kersulis@umich.edu

PhD student of Dr. Ian Hiskens
Power and Energy major
Controls minor

Week 1 | 2017-01-09/10

Control system representations
Laplace transform
Partial fractions

Motivation

Describe systems using mathematics
Work in frequency domain:

Common operations become easier
Closer to intuitive block form
Transfer function: input/output relationship is algebraic

Laplace transform: tool for going from time domain to frequency domain

$\mathcal{L}\{f(t)\} = F(s) = \int_{0^-}^\infty f(t)e^{-st}dt$

Ex. if $f(t) = e^{-at}u_s(t)$, $F(s) = \frac{1}{s+a}$

Step	Tool
1. Express system using diff. eq.	Physics
2. Take $\mathcal{L}\{\cdot\}$	Transform table
3. Analyze	Algebra
4. Take $\mathcal{L}^{-1}\{\cdot\}$ to obtain time signal	Partial fractions, transform table

Problems

Given $F(s) = \frac{2}{(s+1)(s+2)}$, find $f(t)$.
Given $F(s) = \frac{1}{s^2 + 6s + 9}$, what is $f(t)$?
Given a system with input $u(t)$, output $y(t)$, and dynamics given by $$\frac{d^3y(t)}{dt} + 12\frac{d^2y(t)}{dt} + 32\frac{dy(t)}{dt} = 32u(t),$$
a. Find the transfer function.
b. Solve for the output $y(t)$ given the input $u(t) = u_s(t)$.
Assume zero initial conditions.

Week 2 | 2017-01-17

Control system representations for more complicated systems
State-space <--> transfer function
Block diagram reduction

Motivation

Often we have several states and multiple inputs or outputs.
We also frequently encounter complex systems best expressed with block diagrams.

We still want to be able to find transfer functions.

Tools:

State-space representation
Block diagram reduction

From lecture

Problem 1

Part A: find a state-space representation of the transfer function:

Part B: find a state-space representation of the transfer function:

Problem 2

Go the other way: find the transfer function for the following state-space representation.

$$ \begin{align} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} &= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -a_3 & -a_2 & -a_1 \end{bmatrix}\begin{bmatrix} x_1 \\x_2 \\x_3 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}u \\ y &= \begin{bmatrix} b_2 & b_1 & b_0 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{align} $$

Problem 3

Reduce the block diagram and write its transfer function.

Week 3 | 2017-01-23

Signal flow graphs and Mason's gain formula
Compare to block diagram reduction for finding transfer functions
Linearization

Motivation

Signal flow and Mason's gain formula

Block diagram reduction is messy, and when we want multiple transfer functions, we must reduce multiple times. Gross.
A signal flow diagram encodes the same system information.
Mason's gain formula allows us to read off reusable values and combine them to find multiple transfer functions efficiently. No more system rewriting!

Linearization

Linearization allows us to apply our powerful linear analysis tools to nonlinear systems locally. This is often enough to classify stability of equilibria, and even to design controllers.

Problem 1

Find the transfer function $\frac{E(s)}{U(s)}$.

Problem 2

Part A

Use block diagram reduction to find the transfer function $\frac{Y(s)}{U(s)}$.

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Part B

Find the same transfer function using Mason's gain formula.

Problem 3

Consider the following nonlinear circuit:

Voltage across the nonlinear resistor is related to current as follows: \begin{align*} i(t) &= 2e^{0.1V_R(t)}. \end{align*} Find the differential equation modeling the circuit, then linearize any nonlinear terms. Assume a nominal input of $V_N(t) = 0$.

Week 4 | 2017-01-30/31

BIBO stability
Linearization
Steady-state error

Motivation

Stability is a key feature of control systems

Assess in frequency domain -- no need to try various inputs or simulate time domain behavior

Another linearization example to drive home the process. Make sure you're comfortable with linearization.
Steady-state error: how well will our control systems track various inputs?

Problem 1

Determine stability of the following closed-loop systems. If given, $M(s)$ is the closed-loop transfer function.

Part A

$$M(s) = \frac{3}{s(s+1)(s+2)}$$

Part B

where $G_1(s) = \frac{7}{2s+1}$.

Part C

where $G_2(s) = \frac{7K}{2s+1}$ (answer should be in terms of $K$).

Problem 2

Consider the system modeled by the following differential equation: \begin{align*} \frac{d^2\theta}{dt^2} + K\frac{d\theta}{dt} + \cos \theta &= u(t) \end{align*} The applied input is $u(t)$, the output is $\theta$, and we are interested in changes around the nominal input value $u_N = 0$.

Part A

Linearize the system to obtain a small-signal state-space model.

Part B

Find the transfer function of the linearized state-space model.

Part C

Comment on the stability of $G(s)$ from Part B.

Part D

Now comment on the stability of the closed-loop system below, where $G(s)$ is the transfer function computed in Part B.

Problem 3

Consider a unity-feedback system with forward path transfer function $G(s) = \frac{1}{s(s+2)}$ Find the steady-state error for step, ramp, and parabolic inputs.

Week 5 | 2017-02-06/07

Second-order systems (GK section 5-6)

Prediction of qualitative responses
Designing to meet specs

Previously looked at steady-state response ($t\to\infty$).
Now looking at transient response ($t$ near 0).

Second-order system

\begin{align} \frac{Y(s)}{R(s)} &= \frac{\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2} \end{align}

Step response: $$ y(t) = 1 - \frac{e^{-\zeta\omega_nt}}{\sqrt{1 - \zeta^2}}\sin\left(\omega_n\sqrt{1-\zeta^2}t + \arccos\zeta\right),~t\geq 0 $$

Poles: \begin{align} s_1,s_2 &= -\zeta\omega_n\pm j\omega_n\sqrt{1 - \zeta^2} = -\alpha \pm j\omega \end{align}

Poles: \begin{align} s_1,s_2 &= -\alpha \pm j\omega \\ \alpha &= \zeta\omega_n \\ \omega &= \omega_n\sqrt{1-\zeta^2} \end{align}

$\alpha$ is the damping factor
$\zeta$ is the damping ratio
$\omega_n$ is the undamped natural frequency
$\omega$ is the damped frequency

Overshoot and peak time

From GK Section 5-6-3, percent maximum overshoot is $$ 100e^{-\pi\zeta/\sqrt{1-\zeta^2}} $$

Peak time is $$ t_\text{max} = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}} $$

Addition of poles and zeros

Read GK Section 5-9! Also good to play with systems in MATLAB.

Problems

Problem 1

Consider the following mechanical system:

where $x(t)$ is displacement (and also output), $f(t)$ is the applied input with nominal $f_N = 10N$, and the nonlinear spring applies a force $f_s(t) = 2x_s^2(t)$.

Part A

Find the linearized state-space representation of the system. Assume that you are concerned with small changes around the equilibrium point.

Part B

Find $\bar{G}(s) = \frac{\Delta Y(s)}{\Delta U(s)}$, the transfer function of the linearized system.

Part C

Comment on the stability of the linearized system.

Part D

Qualitatively predict the step response of this system.

Part E

Now suppose we have a design requirement: overshoot of the step response cannot exceed 5\%. Determine values of $D$ that satisfy this requirement (if any).

Part F

While satisfying the constraint in part (e), is it possible to also achieve a peak time $T_p \leq 2$ seconds? If so, find the values of $D$ that meet both requirements.

Part G

Given these constraints, what type of step response will we see? Revise your predicted qualitative step response.

Problem 2

Suppose $D = 4.2$ in the Problem 1 system, and the transfer function is $$ \bar{G} = \frac{\Delta Y(s)}{\Delta U(s)} = \frac{1}{s^2 + Ds + 4\sqrt{5}} $$ (the constant gain is gone now).

Part A

Suppose a zero is added at $s=-\frac{1}{a}$, so the transfer function becomes $\bar{G}(1+as)$. Qualitatively describe the change in step response we should see. Validate this using MATLAB.

Part B

Suppose instead a pole is added at $s=-1/\tau$, so the transfer function becomes $\frac{\bar{G}(s)}{\tau s + 1}$. Again describe the change in step response you expect, and verify with MATLAB.

Week 6 | 2017-02-13/14

Stability via Routh-Hurwitz
Stability via Kharitanov's Theorem
Oscillation

Routh-Hurwitz

GK 2-13 on page 78, lecture 14, handout 7
Stability properties without root finding
Just a table and computations similar to 2x2 determinants

Kharitanov's Theorem

Lecture 15
Stability properties for family of devices
Robustness to coefficient variation
Four Routh tables provides strong conclusion for infinitely many systems

Problems

Problem 1

Part A

Assess the stability of a system with the following transfer function:

$$ G(s) = \frac{s+1}{s^2 + 4s + 1} $$

Part B

Now suppose we connect the Part A system in series with a gain $K$, and introduce unity feedback.

Use Routh-Hurwitz to give a range of $K$ for which the closed-loop system is stable.

Part C

Suppose $K=-4$. Comment on the stability of the system.

Part D

Suppose we don't know $K$ exactly, but we know it will lie somewhere in the range $[-2,5]$. Use Kharitanov's Theorem to show that the system will not be stable for every $K$ in this range.

Problem 2

Suppose you have eight similar devices in your system with slightly different transfer functions. You know the eight characteristic polynomials for these devices:

\begin{align*} s^3 + 18s^2 + 2s + 2 & \qquad s^3 + 18s^2 + 3s + 6 \\ s^3 + 18s^2 + 2s + 3 &\qquad s^3 + 16s^2 + 2s + 4 \\ s^3 + 18s^2 + 1s + 8 &\qquad s^3 + 20s^2 + 3s + 4 \\ s^3 + 16s^2 + 3s + 5 &\qquad s^3 + 18s^2 + 2s + 3 \end{align*}

You also know that the overall system will be stable if and only if all eight components are stable. Will the system be stable?

Problem 3

Consider the following transfer function of a feedback control system. Assume $K>0$.

$$ G(s) = \frac{K(s+3)}{s^4 + 7s^3 + 14s^2 + (8+K)s + 3K} $$

Part A

Determine the range of $K$ so that the system is asymptotically stable.

Part B

Determine the value of $K$ that renders the system marginally stable, and determine the frequency of oscillation, if applicable.

Week 7 | 2017-02-20/21

Double office hour: Tuesday 2-4 PM at EECS 2420

Root locus criteria
Review of signal flow diagrams, FVT, and second-order prototype response
Your questions

Study hint: be proficient at things we've done several times.

Differential equations to transfer functions to state-space (and back)
Linearization

Problems

Problem 1

Consider the block diagram below, where $K$ is a tunable gain. Is there a value of $K$ for which $s_1 = -4 + j2$ is a pole of the closed-loop system? If so, what is that value of $K$?

Problem 2

Consider the block diagram shown below.

Part A

Find the transfer function $\frac{C(s)}{R(s)}$. You can use any method you like.

Part B

Assume that the signal labeled $A(s)$ is known. Find $C(s)$.

Part C

Assume $G(s) = K_f>0$.

Give conditions on $K_f$ and $K$ such that the overall closed-loop system is stable.
When $R(s)$ is a unit step function, what is the steady-state value of $C(s)$ (i.e. $\lim_{t\to\infty}c(t)$)? Explain the qualitative effect of $K_f$ on this steady-state value.
Explain why the extra branch with $G(s) = K_f$ might have been put into the system (hint: consider the qualitative effect of $K_f$ on the overall transfer function).

Part D

Now assume $G(s) = 0$ and $K>0$.

Provide conditions for $K$ that yield underdamped, critically damped, and overdamped responses.
For what value of $K$ will the real parts of the poles be -0.25? Find the corresponding imaginary parts of the poles.
What will the rise time and 5% settling time be for the value of $K$ you computed above?
Use your computations to sketch the step response.

Week 8 | 2017-03-06/07

Exam review
Root locus

Problems

Problem 1

Part A

Sketch the root locus for the system below.

Part B

Verify your root locus sketch with MATLAB.

Problem 2

Part A

Sketch the standard root locus for the system shown below.

Part B

For what (positive) values of $K$ is the closed-loop system stable?

Week 9 | 2017-03-13/14

Lead compensator design
PD controller design
Sensor delay effects

Lead compensator design

Add positive phase to the system, shifting root locus left.

Angle of deficiency

Phase needed to cause a chosen point to satisfy angle criterion.

$$ \phi = -180^\circ - \sum_i \angle(s + z_i) + \sum_j \angle (s + p_j) $$

Bisection method

Many ways to place the pole and zero of a lead compensator. Bisection tends to work well.

PD controller design

Add only a zero: speed response, increase overshoot

$$ G_c(s) = K_c(s+z) $$

Useful heuristic, analogous to bisection:

$$ z = d_R + \frac{d_I}{\tan \phi} $$

For lead compensator and PD controller, try to keep $K$ low. Less amplification typically means cheaper parts.

Sensor delay effects

Actual transfer function:

$$ e^{-sT}, $$

where $T$ is delay in seconds.

All-pass approximation (always has magnitude 1):

$$ e^{-sT} \approx \frac{\left(\frac{2n}{T} - s\right)^n}{\left(\frac{2n}{T} + s\right)^n} $$

Sensor delay can push the system towards instability.

How much should you spend on your sensor?

Problems

Problem 1

Consider the system shown below.

Part A

Sketch the root locus ($K>0$) of the system.

Part B

Design a controller that causes overshoot to be no more than 10%, and 5% settling time to be no more than 2s.

Part C

Now try to meet the spec using a PD controller.

Part D

How is performance affected if there is some sensor delay? Assume a second-order (all-pass) model and a delay of $T=0.2$s.

Week 10 | 2017-03-20/21

Precompensator design
Phase lag compensator design
2-DoF design

Precompensator design

Introduce $K(s)$ immediately after input.

To fix transient, use $K(s)$ to cancel zeros of $G_{cl}(s)$.
To fix steady-state, calculate $e_{ss}$ and determine what $K(0)$ should be to reduce $e_{ss}$ according to spec.
Transfer functions become:

\begin{align*} \frac{Y(s)}{R(s)} &= K(s)\frac{G(s)}{1 + G(s)} \\ \frac{E(s)}{R(s)} &= \frac{1 + G(s) - K(s)G(s)}{1 + G(s)}~, \end{align*}

Phase lag compensation

$$G_\text{lag}(s) = K\frac{s+z}{s+p},~p

Draw root locus and check transient properties; fix with lead compensation if needed.
Check $e_{ss}$. If too large, determine $z/p$ for lag compensator.
Place $z$ and $p$ close to origin.
Adjust $K$ to ensure dominant poles do not move much.

2-DoF design

We care about stability, steady-state error, and transient behavior. Adding a second degree of freedom lets us target multiple objectives.

Problems

Problem 1

Consider the system shown below.

Part A

Roughly sketch the root locus ($K>0$) of the system.

Part B

Stabilize the system and include compensation so that $\zeta = 0.174$ (approx. 57% overshoot).

Part C

With the compensation designed in Part B, calculate the steady-state error to a step input.

Part D

Design a precompensator, $\bar{K}(s)$, to reduce the steady-state error to a step input by a factor of 10. See figure below.

Part E

Repeat Part D using a phase-lag controller. See figure below.

Part F

Verify your results in Parts D and E using MATLAB or Simulink.

Part G

Notice that the overshoot from Part E is greater than 40%. Design a precompensator $\tilde{K}(s)$ to reduce the overshoot by a factor of 10. See figure below. Verify your results with MATLAB or Simulink.

Week 11 | 2017-03-27/28

Lab time!
PID controller design: cascaded PD and PI method

Lab

Homework 10 connected to lab: individual component
Lab assignment asks you to complete a 2-DOF design
Sign up in groups for a lab session, choose one "best" design from your group before coming to lab.
Soon you will levitate a magnetic ball and be the envy of all your Snapchat followers.

PID controller design

$$ G_{PID} = K_P + K_Ds + K_I/s $$

Two useful methods:

Method of Root Contours (see Handout 14)
Cascading PD and PI.
We can view this as a PD controller cascaded with a PI controller:
\begin{align*} G_{PID}(s) &= K_P + K_Ds + K_I/s = (1 + \tilde{K}_Ds)(\tilde{K}_P + \tilde{K}_I/s) \\ \implies K_P &= \tilde{K}_P + \tilde{K}_D\tilde{K}_I \\ K_D &= \tilde{K}_D\tilde{K}_P \\ K_I &= \tilde{K}_I \end{align*}

Design process

PD: "anticipatory" | Section 9-2. Place the zero of the PD controller to increase damping and reduce rise time, settling time, and overshoot.
PI: "low-pass" | Section 9-3. Place the zero of the PI controller close to the origin and far from dominant poles to improve steady-state error and overshoot.

Problems

Problem 1

Consider the third-order attitude control system with forward-path transfer function

$$ G_p(s) = \frac{2.718\times 10^9 }{s(s^2 + 3408.3s + 1,204,000)} = \frac{2.718\times 10^9}{s(s+400.26)(s+3008)} $$

Design a PID controller such that the closed-loop system meets the following specifications:

Steady-state error due to a ramp input $\frac{1}{2}t^2u_s(t)\leq 0.2$.
Maximum overshoot $\leq 5\%$.
Rise time $t_r \leq 0.005s$.
5% settling time $t_s \leq 0.005s$.

Week 12 | 2017-04-03/04

Lab sessions begin this week!
Bode plots
Nyquist plots
Stability margins

Bode plots

Logarithmic plots of transfer function magnitude and phase w.r.t. frequency.
Logarithm converts multiplication and division into addition and subtraction.

Five basic elements

Constant gain: $K$

Magnitude: $20\log_{10}K ~\forall ~\omega$
Phase: $0^\circ~\forall ~\omega$

Poles and zeros at the origin: $(j\omega)^n$

Magnitude: $20n$ dB/decade; 0 dB at $\omega = 1$ ($n$ can be positive or negative)
Phase: $90^\circ\times n~\forall~\omega$ ($n$ can be positive or negative)

Simple zeros and poles: $1+j\omega T$ and $1/(1+j\omega T)$

Magnitude: 0 dB to left of corner frequency $1/T$; $\pm 20$ dB/decade to right (positive for zero, negative for pole)
Phase: $0^\circ$ to $\pm 90^\circ$ over the course of two decades, centered at corner frequency $1/T$ (positive for zero, negative for pole)

4. Quadratic poles and zeros

See GK 2-2-8.

5. Time delay

Only affects phase. See GK 2-2-9.

Nyquist plots

Based on Cauchy's Principle of the Argument
Define Nyquist path containing entire RHP
Map onto $L(j\omega)$ plot
Stability: enclosure of critical point
Stability margins from real-axis crossing

Problems

Problem 1

Part A

Draw the Bode plot for the system $G(s)$ shown below.

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

Part B

Now generate a Bode plot with Matlab and comment on stability margins.

Problem 2

Part A

Consider the system below. Sketch the Nyquist plot.

Part B

Use the Nyquist plot to assess stability.

Part C

Validate your Part B stability conclusion using root locus techniques.

Week 13 | 2017-04-10/11

Lab sessions continuing this week
Towards frequency design
Design information from Bode plots
Time delays

Design with Bode plots

Determine GM, PM, but can also play what-if

Time delays

No approximation needed with frequency domain techniques

Problems

Problem 1

The following Bode diagram of the forward-path transfer function of a unity-feedback control system was obtained experimentally.

Part A

Find the gain and phase margins of the system. Find the gain- and phase- crossover frequencies.

Part B

Repeat Part A if the gain is doubled from its nominal value.

Part C

Repeat Part A if the gain is ten times its nominal value.

Part D

Find out how much the gain must be changed from its nominal value to obtain a gain margin of 40 dB.

Part E

Find out how much the loop gain must be changed from its nominal value to obtain a phase margin of 45$^\circ$.

Part F

Find the steady-state error of the system if the reference input to the system is a unity-step function.

Part G

The forward path now has a pure time delay of $T_d$ seconds, so that the forward path transfer function is multiplied by $e^{-T_ds}$. Find the gain margin and the phase margin for $T_d = 0.1$ sec (with nominal gain).

Part H

With the gain set at nominal, find the maximum time delay $T_d$ the system can tolerate without going into instability.

Problem 2

Consider the system with forward-path transfer function $$ \frac{100Ke^{-T_ds}}{s(s^2 + 10s + 100)} $$

Part A

When $K=1$, determine the maximum time delay $T_d$ for the closed-loop system to be stable.

Part B

When the time delay is 1s, find the maximum value of $K$ for system stability.

Week 14 | 2017-04-17/18

Frequency domain interpretation of specs
Lead compensator design
Lead/lag compensator design

Frequency domain specs

It's sometimes challenging to translate time domain specs into the frequency domain.
Relationship between $\zeta$ and $PM$

Lead compensator design

$$ G_c(s) = \frac{1+aTs}{1+Ts},\quad a > 1 $$

Obtain Bode plot of loop transfer function. Determine gain and phase margins, $\phi_m$. Solve $a = \frac{1 + \sin\phi_m}{1 - \sin\phi_m}$
Solve for $T$: $\omega_n = \frac{1}{\sqrt{a}T}$. Adjust gain.
Verify design using Bode plot.
Write final controller $G_c(s)$.

Lead/lag compensator design

Discussed during lecture, example included in discussion notes.

Problems

Problem 1

Given the system below, design a lead compensator to yield 20% overshoot and a peak time $T_p$ of 0.1 seconds.

Problem 2

Suppose there is now a 0.01 sec delay in the Problem 1 system. Can we still achieve our design specs of 20% overshoot and $T_p = 0.1$ sec?

Problem 3

Identify the critical gain value for the compensated system.

Problem 4

Verify your Problem 3 answer using root locus.

Problem 5

The system below may be used for positioning of a shaft by a command from a remote location.

Part A

Design a lead compensator so that the closed-loop system is as fast as possible, consistent with the following specifications:

$PM\geq 60^\circ$
Open-loop gain rolled off to at least -20 dB at 60 rad/sec. This is to ensure stability robustness to unmodeled flexing in the belt connecting the load to the motor.

Part B

Compute the steady-state positioning error caused by a unit step torque disturbance.

Part C

If necessary, add additional compensation so that the steady-state positioning error caused by a unit step torque disturbance is $\leq 1$%.

EECS 460 Discussion

Introduction

Jonas Kersulis, GSI

Meta

Office hours in EECS 2420

Discussion format

Week 1 | 2017-01-09/10

Motivation

Problems

Week 2 | 2017-01-17

Motivation

From lecture

From lecture

Problem 1

Problem 2

Problem 3

Week 3 | 2017-01-23

Motivation

Problem 1

Problem 2

Part A

Part B

Problem 3

Week 4 | 2017-01-30/31

Motivation

Problem 1

Part A

Part B

Part C

Problem 2

Part A

Part B

Part C

Part D

Problem 3

Week 5 | 2017-02-06/07

Second-order system

Overshoot and peak time

Addition of poles and zeros

Problems

Problem 1

Part A

Part B

Part C

Part D

Part E

Part F

Part G

Problem 2

Part A

Part B

Week 6 | 2017-02-13/14

Routh-Hurwitz

Kharitanov's Theorem

Problems

Problem 1

Part A

Part B

Part C

Part D

Problem 2

Problem 3

Part A

Part B

Week 7 | 2017-02-20/21

Problems

Problem 1

Problem 2

Part A

Part B

Part C

Part D

Week 8 | 2017-03-06/07

Problems

Problem 1

Part A

Part B

Problem 2

Part A

Part B