1. ME190L Loop Shaping Course Introduction UC Berkeley Copyright , Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

2. Course Structure (Fall 2010)
Class detail
1 hour of lecture/week, Friday, 10:00-11:00 AM.
3113 Etcheverry Hall
BSpace
Slides, Homeworks, Files
Workload
Weekly Homework assignments (1-2 hours/week workload)
Hand in a clean notebook with all problems on Friday, December 10
Work steady, but go back and fix things as your understanding improves
Goal (serious): everybody eventually does every problem correctly
During semester, come to office hours, show me your work, and I’ll assess it
Access to Matlab, Simulink, Control System Toolbox
ME students can get accounts in 2107, 2109; others see me if you need access
1 Final exam (take-home), available Dec 3, due (w/ homework) on Dec 10
Prereq
ME 132, or equivalent
Me:
Andrew Packard, 5116 Etcheverry Hall,
Office hours: M 1:30-2:30, Tu 3:30-4:30, W 10:30-11:30

3. Supplementary Reading Material
SP: Skogestad and Postlewaite, Multivariable Feedback Control: Analysis and Design, 2nd edition, Wiley, 2005.
Chapter 2 (pp , especially pp )
Section 9.4 (pp )
DFT: Doyle, Francis, Tannenbaum, Feedback Control Theory, Macmillan, 1992, out-of-print
available at
Chapter 7, Loopshaping
Chapter 6, Analytic Design constraints
FPE: Franklin & Powell & Emami-Naeini, Feedback Control of Dynamic Systems, Prentice Hall, 5th edition.
popular, general undergrad control textbook, decent reference
MG: McFarlane and Glover, A Loop Shaping Design Procedure using H∞ Synthesis
pp , vol 37, June 1992, IEEE Trans. on Automatic Control.

4. We’ll learn this starting in next lecture
Basic Feedback Loop n d2 u d1 y P C - r
Start with simplest feedback topology
Closed-Loop Transfer functions
Stability
Roots of characteristic equation in open-left-half plane. Alternatively…
Nyquist plot of P(jω)C(jω) must encircle -1 the correct number of times
Disturbance rejection
Goal is that d1 and/or d2 have little effect on y
Noise insensitivity
Goal is that n has little effect on u and y
Adequate robustness margins
Adequate gain/phase/time delay margins,
We’ll learn this starting in next lecture

5. What is Loopshaping?
Control-Design technique: shaping, by choice of C, the magnitude/phase of PC, so that the closed-loop system has desired properties
Stable
Disturbance rejection
Noise insensitivity
Adequate robustness margins
Advantages
PC depends linearly (simple) on C, moreover |PC| = |P| |C|
Some closed-loop properties are very simply related to |PC|
Requires:
Understanding how open-loop gain (|PC|) is related to closed-loop properties
Understanding what closed-loop properties are achievable for a given plant
Techniques (graphical, computer-based, etc) to shape PC
Easy for benign plants, with standard goals; more challenging for others
unstable poles, RHP zeros, flexible modes, etc, unusual objectives
The harder aspects can be partially automated
theory
computation

6. Disturbance rejection- n d2 u d1 y P C r
Disturbance rejection
Transfer functions
Open-Loop (C=0)
Closed-Loop
Improvement Ratio: Closed-Loop/Open-Loop
Usual goals of feedback
Make |S(jω)|<<1 in frequency ranges where d1 and d2 are large
Keep |S(jω)|<2 in all frequency ranges
Question: Can S be made small at all frequencies?
Typical S for PI feedback around simple P

7. - P C Noise Insensitivity Transfer functions Noted2 u d1 y P C r
Noise Insensitivity
Transfer functions
Open-Loop
Closed-Loop
Feedback always introduces sensor noise into the loop
Note
Consequently, at frequencies where |S(jω)|<<1, there will be direct transmission, (Gn→y≈-1) of n to y
Basic Limitation:

8. Disturbance/Noise Tradeoff- n d2 u d1 y P C r
Disturbance/Noise Tradeoff
Basic Limitation: at all frequencies
So… for y to be unaffected by d2 and n, we need
at frequencies where n is large, it must be that
d2 is small
|T| is small
at frequencies where d2 is large, it must be that
n is small
|S| is small S
Sensitivity function T
Complimentary Sensitivity function

9. - P C Conditions on |PC| Transfer functionsy P C r
Conditions on |PC|
Transfer functions
Sensitivity and Complementary Sensitivity
Simple (approximate) inequalities: for small β (relative to 1)
How do closed-loop stability and robustness margins enter?
Need large loop-gain where |S| is to be small relative to 1
Need small loop-gain where |T| is to be small relative to 1

10. - P C Margins/Stability Gain, Phase, Time-Delay margins Stabilityu d1 y P C r
Margins/Stability
Gain, Phase, Time-Delay margins
All measure how close P(jω)C(jω) approaches the point -1 from different, special directions.
TImeDelay margin takes into account frequency, ω, too
So, what is important for these margins to not be too small, is the phase of PC, when |PC|≈1
previous bounds on |PC| were for very large and very small values
So, ensuring adequate margins is not addressed by the previous constraints
Analytic function theory tells us (soon) that and are related, so in loopshaping, margins are accounted for by properly adjusting the slope of |PC| in the frequency range where |PC|≈1
Stability
Nyquist theorem: closed-loop system is stable if and only if the plot of P(jω)C(jω) encircles -1 the “correct number of times”
These are starting to both sound challenging, with regard to shaping PC by choosing C. But, we’ll address them.

11. - P C Also in this course… Theoretical Toold2 u d1 y P C r
Also in this course…
Theoretical Tool
P is given
User specifies a candidate (ie, proposed) controller Cprop
This is chosen (typically) to satisfy the easy (|PC| large, |PC| small) constraints
The issues of closed-loop stability and adequate margins are ignored
A “magic” process determines if there is a controller which
preserves the large-loop gain of PCprop
preserves the small-loop gain of PCprop
achieves closed-loop stability, with modest gain/phase margins
We will learn/derive the theory behind this, as well as use it in a series of examples.