ME190L Loop Shaping Course Introduction UC Berkeley Copyright 2007-10, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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, apackard@berkeley.edu Office hours: M 1:30-2:30, Tu 3:30-4:30, W 10:30-11:30

Supplementary Reading Material SP: Skogestad and Postlewaite, Multivariable Feedback Control: Analysis and Design, 2nd edition, Wiley, 2005. Chapter 2 (pp. 15-66, especially pp. 42-54) Section 9.4 (pp. 364-382) DFT: Doyle, Francis, Tannenbaum, Feedback Control Theory, Macmillan, 1992, out-of-print available at http://www.control.utoronto.ca/people/profs/francis/dft.html 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. 759-769, vol 37, June 1992, IEEE Trans. on Automatic Control.

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

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

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

- P C Noise Insensitivity Transfer functions Note d2 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:

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

- P C Conditions on |PC| Transfer functions y 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

- P C Margins/Stability Gain, Phase, Time-Delay margins Stability u 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.

- P C Also in this course… Theoretical Tool d2 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.