course schedule - ethz.ch

ETHZ - Control Systems II - Jacopo Tani

Course Schedule

1

Date # Lecture Recitals* Labs23.02 1 Introduction / Control Systems Recap. 12.03 2 Discrete Time / Implementation Challenges 29.03 3 MIMO introduction 3

16.03 4 Robustness Introduction: SISO vs MIMO 423.03 5 Robustness: Analysis 5

NO CLASS (Easter - March 30) NONO CLASS (Easter - April 6) NO

Lab 113.04 6 State Feedback 620.04 7 MIMO control I: LQR 727.04 8 MIMO control II: LQG 8

Lab 24.05 9 MIMO control III: Hinf I NO (May 1st)11.05 10 Example of modern applications: Decentralized control 918.05 11 MIMO control IV: Hinf II 10

Lab 325.05 12 Elements of Nonlinear control: Lyapunov 111.06 13 Exam Review / Catching up 12

* Recitals on Tuesdays after the lecture date

ETHZ - Control Systems II - Jacopo Tani

Staff

2

Lecturer:Dr. Jacopo Tani

Lead Assistant:Claudio Ruch

Teaching Assistants:

Alexandre Didier adidier@student.ethz.chAmir Hadzic hadzica@student.ethz.chBen Weber webeben@student.ethz.chDzenan Lapandic ldzenan@student.ethz.chEugenio Chisari chisarie@student.ethz.chGianmarco Bernasconi gibernas@student.ethz.chGioele Zardini gzardini@student.ethz.chJulien Kindle jkindle@student.ethz.chLaura Guerrini laurague@student.ethz.chLukas Spannagl spalukas@student.ethz.chPhilip Arm parm@student.ethz.chPhilipp Bänninger baephili@student.ethz.ch

ETHZ - Control Systems II - Jacopo Tani

Rooms

3

Lecture: ML D 28 and ML E 12 (live video stream)

Recitals: CHN E 42, CAB G 59, CHN F 46, IFW A 32.1, IFW A 36, LFW C 5

Office Hours:

TAs: in the labsInstructor: ML K 37.3

Labs: ML F 44.2 and ML J 44.4

ETHZ - Control Systems II - Jacopo Tani

Main References

4

Textbook of CS I MIMO systemsAdditional reference for

Robustness

• Lecture slides and notes will be posted on the course web site.

ETHZ - Control Systems II - Jacopo Tani

Grading

5

• Final Written Examination: 100%

• Optional Hardware Exercises: up to +0.5 on final grade

ETHZ - Control Systems II - Jacopo Tani

Questions on Piazza

6

• Use Piazza for content related questions (as in Control Systems I)

Introduction to Control Systems II

ETHZ - Control Systems II - Jacopo Tani

Course level main Intended Learning Outcomes

8

In CS I you learned the fundamentals of controls:

- The importance of system modeling and different approaches- Concepts of system, state, input and output- Properties of LTI SISO systems as stability, controllability, observability- Controller objectives (stability, performance, robustness)- Tools for analysis: Root locus, Bode diagrams, the Nyquist criterion- Tools for controller synthesis: Loop shaping, PID control- Effect of nuisances such as time delays and RHP zeros- The idea of robustness to model uncertainty

In CS II you will extend your knowledge to encompass:

- Analysis and controller synthesis for Multi-input multi-output (MIMO) LTI systems- State space theory- Robustness (robust stability, robust performance)

ETHZ - Control Systems II - Jacopo Tani

A big picture

9

Plant model Control Objective Implementation Controller

No model Stability Analogue (Cont. Time) Loop Shaping(e.g., lead / lag)

SISO LTI Performance Digital (Discrete Time) PID

MIMO LTI Robust Stability (RS) State feedback

LTV (SISO/MIMO) Robust Performance (RP) LQR

Nonlinear LQG

Hinf

MPC

Lyapunov

Learning

Adaptive

(many more)

Italic: treated in CS IBold: will be treated in CS II

ETHZ - Control Systems II - Jacopo Tani

Today’s learning modules

1. Control objectives

• Because we want to get back in the mood and recap on the terminology

2. Sensitivity functions

• Because they are a useful tool for analysis and controller synthesis

3. Fundamental limitations to control

• Because it is not always possible to control a system or achieve arbitrarily good performance

10

Control objectives

Credits

Intended Learning Outcomes Prerequisites

Control objectives

• Recall terminology• Enunciate controller

objectives• Refresh intuition on Stability,

Performance and Robustness

• Control Systems I

• Jacopo Tani - ETHZ - Feb. 2018

ETHZ - Control Systems II - Jacopo Tani

Feedback control loop with signals

13

-1

+ + +

y

r e u

d n

yvF C P

The controller: designed The process / environment: given

⌘Feedforward Controller Feedback Controller Plant

Disturbance Noise

Measured Output

Real Output

DisturbedInputInput

Negative Feedback

Reference TrackingError

η

ETHZ - Control Systems II - Jacopo Tani

Controller objectives

14

1. Nominal stability 2. Nominal performance3. Robust stability (RS)4. Robust performance (RP)

A. Analysis: given a controller, how can we check if the above is satisfied?

B. Synthesis: given a plant, how to design a controller that satisfies the above?

To achieve them, we require tools for:

ETHZ - Control Systems II - Jacopo Tani

Stability

15

• Lyapunov stability: A system is called Lyapunov stable if, for any bounded initial condition, and zero input, the state remains bounded, i.e.,

• A system is called asymptotically stable if, for any bounded initial condition, and zero input, the state converges to zero, i.e.,

• A system is called unstable if not stable

kx0k < ✏ and u = 0 ) kx(t)k < �, 8t � 0

• Bounded-Input, Bounded-Output stability: A system is called BIBO-stable if, for any bounded input, the output remains bounded, i.e.,

kx0k < ✏ and u = 0 ) limt!+1

kx(t)k = 0.

ku(t)k < ✏, 8t � 0, and x0 = 0 ) ky(t)k < �, 8t � 0.

ETHZ - Control Systems II - Jacopo Tani

Performance

16

• Regulation problem: determine an input such that the system maintains a reference value despite disturbances.

• Tracking / servo problem: find the input that allows the system output to closely follow a time varying reference signal.

Main challenges are disturbance and noise attenuation

ETHZ - Control Systems II - Jacopo Tani

Robustness

17

A control system is robust when it is insensitive to model uncertainties (bounded variation in the model parameters)

All models are wrong, but some are usefulTypical model mismatch cause:

- Aging: plant model parameters change with time / use;- System identification: poorly modeled high frequency behaviors, open loop unstable

plants.

• Robust stability (RS): The system is stable for all perturbed plants about the nominal model up to the worst-case model uncertainty.

• Robust performance (RP): The system satisfies the performance specifications for all perturbed plants about the nominal model up to the worst-case model uncertainty.

ETHZ - Control Systems II - Jacopo Tani

Tools for Analysis so far?

18

How do we check for:

1. Nominal stability:

- Root Locus: track the closed loop poles while varying the feedback gain.- Bode Diagram: analyse the frequency response of a system: magnitude and phase.- Nyquist plot and criterion: polar plot representation of the frequency response.

2. Nominal performance:

- Time domain and frequency domain specifications for disturbance rejection, noise attenuation, and transient performances.

3. Robustness

- Margins (gain, phase, stability)

ETHZ - Control Systems II - Jacopo Tani 19

Looked at loop shaping (e.g., lead / lag) in the frequency domain, and PID control in the time domain

• Both address nominal stability and performance

• Robustness: PID control, loop shaping

Tools for Synthesis so far?

Loop shaping techniques give good insight, let us recap those ideas

Sensitivity Functions

Credits

Intended Learning Outcomes Prerequisites

Loop transfer functions, or, "The gang of six” (four)

• Transfer functions• Sensitivity functions• Frequency domain

specifications

• Jacopo Tani - ETHZ - Feb.

ETHZ - Control Systems II - Jacopo Tani

Why transfer functions?

22

Analysis made easier: Many control problems are easier to visualize and address in the frequency domain than the time domain

Why?

- Key idea of propagation of sinusoidal signals in feedback loop leads to important results (e.g., Nyquist criterion). Evaluated by .

- Math is easier (e.g., multiplication for systems in series instead of convolutions)

Synthesis made easier: By understanding closed loop system properties from (open) loop transfer function, it is easier to evaluate the effect of the controller

Prof. E. Frazzoli, CS I, ETHZ, 2017C(s)P (s) vs.

C(s)P (s)

1 + C(s)P (s)

s = j!<latexit sha1_base64="vCaYnm4syFLJmMebebat9fLnT2Q=">AAAB8nicbVA9SwNBEJ2LXzF+xVjaLAbBKtzZRAshYGMjRPBMJHeEvc1esmZ379jdE8IR8FfYWKjY+mvs/DduPgpNfDDweG+GmXlRypk2rvvtFFZW19Y3ipulre2d3b3yfuVOJ5ki1CcJT1Q7wppyJqlvmOG0nSqKRcRpKxpeTvzWI1WaJfLWjFIaCtyXLGYEGysFGl2gBxQkgvZxt1x1a+4UaJl4c1JtVO6vnwCg2S1/Bb2EZIJKQzjWuuO5qQlzrAwjnI5LQaZpiskQ92nHUokF1WE+vXmMjq3SQ3GibEmDpurviRwLrUcisp0Cm4Fe9Cbif14nM/FZmDOZZoZKMlsUZxyZBE0CQD2mKDF8ZAkmitlbERlghYmxMZVsCN7iy8vEP62d19wbG0YdZijCIRzBCXhQhwZcQRN8IJDCM7zCm5M5L8678zFrLTjzmQP4A+fzBxgikg0=</latexit><latexit sha1_base64="lvL6PiJ7GeRBJpcETSOcltyPsks=">AAAB8nicbVA9SwNBEN2LH4nxK8bSZjEIVuHOJloIARsbIYJnIskR5jZ7yZrdvWN3TwhHfoC92Fio2Ppr7Pw3bj4KTXww8Hhvhpl5YcKZNq777eRWVtfW84WN4ubW9s5uaa98q+NUEeqTmMeqFYKmnEnqG2Y4bSWKggg5bYbDi4nffKBKs1jemFFCAwF9ySJGwFipo/E5vsedWNA+dEsVt+pOgZeJNyeVevnu6in/CI1u6avTi0kqqDSEg9Ztz01MkIEyjHA6LnZSTRMgQ+jTtqUSBNVBNr15jI+s0sNRrGxJg6fq74kMhNYjEdpOAWagF72J+J/XTk10GmRMJqmhkswWRSnHJsaTAHCPKUoMH1kCRDF7KyYDUECMjaloQ/AWX14m/kn1rOpe2zBqaIYCOkCH6Bh5qIbq6BI1kI8IStAzekVvTuq8OO/Ox6w158xn9tEfOJ8/c6iTEg==</latexit><latexit sha1_base64="lvL6PiJ7GeRBJpcETSOcltyPsks=">AAAB8nicbVA9SwNBEN2LH4nxK8bSZjEIVuHOJloIARsbIYJnIskR5jZ7yZrdvWN3TwhHfoC92Fio2Ppr7Pw3bj4KTXww8Hhvhpl5YcKZNq777eRWVtfW84WN4ubW9s5uaa98q+NUEeqTmMeqFYKmnEnqG2Y4bSWKggg5bYbDi4nffKBKs1jemFFCAwF9ySJGwFipo/E5vsedWNA+dEsVt+pOgZeJNyeVevnu6in/CI1u6avTi0kqqDSEg9Ztz01MkIEyjHA6LnZSTRMgQ+jTtqUSBNVBNr15jI+s0sNRrGxJg6fq74kMhNYjEdpOAWagF72J+J/XTk10GmRMJqmhkswWRSnHJsaTAHCPKUoMH1kCRDF7KyYDUECMjaloQ/AWX14m/kn1rOpe2zBqaIYCOkCH6Bh5qIbq6BI1kI8IStAzekVvTuq8OO/Ox6w158xn9tEfOJ8/c6iTEg==</latexit>

ETHZ - Control Systems II - Jacopo Tani

Poles and Zeros

23

x(t) = Ax(t) +Bu(t)

y(t) = Cx(t) +Du(t)<latexit sha1_base64="blXoCCpLm79uNizFkSdDGHUo+mE=">AAACHXicbZDLSgMxFIYz9VbrrerSTbAIFaFMRakuhGpduKxgbaEdSibNtKGZC8kZcRj6JG58FTcuVFy4Ed/GtDOCth4IfPz/OZyc3w4EV2CaX0Zmbn5hcSm7nFtZXVvfyG9u3So/lJQ1qC982bKJYoJ7rAEcBGsFkhHXFqxpD2tjv3nHpOK+dwNRwCyX9D3ucEpAS938cafnA74vwj4+w+cJHOALHI6h08lFiVP7cS4nTjdfMEvmpPAslFMooLTq3fyH3kNDl3lABVGqXTYDsGIigVPBRrlOqFhA6JD0WVujR1ymrHhy3gjvaaWHHV/q5wGeqL8nYuIqFbm27nQJDNS0Nxb/89ohOCdWzL0gBObRZJETCgw+HmeFe1wyCiLSQKjk+q+YDogkFHSiOR1CefrkWWgclk5L5vVRoVpJ08iiHbSLiqiMKqiKrlAdNRBFD+gJvaBX49F4Nt6M96Q1Y6Qz2+hPGZ/fUqab7Q==</latexit><latexit sha1_base64="blXoCCpLm79uNizFkSdDGHUo+mE=">AAACHXicbZDLSgMxFIYz9VbrrerSTbAIFaFMRakuhGpduKxgbaEdSibNtKGZC8kZcRj6JG58FTcuVFy4Ed/GtDOCth4IfPz/OZyc3w4EV2CaX0Zmbn5hcSm7nFtZXVvfyG9u3So/lJQ1qC982bKJYoJ7rAEcBGsFkhHXFqxpD2tjv3nHpOK+dwNRwCyX9D3ucEpAS938cafnA74vwj4+w+cJHOALHI6h08lFiVP7cS4nTjdfMEvmpPAslFMooLTq3fyH3kNDl3lABVGqXTYDsGIigVPBRrlOqFhA6JD0WVujR1ymrHhy3gjvaaWHHV/q5wGeqL8nYuIqFbm27nQJDNS0Nxb/89ohOCdWzL0gBObRZJETCgw+HmeFe1wyCiLSQKjk+q+YDogkFHSiOR1CefrkWWgclk5L5vVRoVpJ08iiHbSLiqiMKqiKrlAdNRBFD+gJvaBX49F4Nt6M96Q1Y6Qz2+hPGZ/fUqab7Q==</latexit><latexit sha1_base64="blXoCCpLm79uNizFkSdDGHUo+mE=">AAACHXicbZDLSgMxFIYz9VbrrerSTbAIFaFMRakuhGpduKxgbaEdSibNtKGZC8kZcRj6JG58FTcuVFy4Ed/GtDOCth4IfPz/OZyc3w4EV2CaX0Zmbn5hcSm7nFtZXVvfyG9u3So/lJQ1qC982bKJYoJ7rAEcBGsFkhHXFqxpD2tjv3nHpOK+dwNRwCyX9D3ucEpAS938cafnA74vwj4+w+cJHOALHI6h08lFiVP7cS4nTjdfMEvmpPAslFMooLTq3fyH3kNDl3lABVGqXTYDsGIigVPBRrlOqFhA6JD0WVujR1ymrHhy3gjvaaWHHV/q5wGeqL8nYuIqFbm27nQJDNS0Nxb/89ohOCdWzL0gBObRZJETCgw+HmeFe1wyCiLSQKjk+q+YDogkFHSiOR1CefrkWWgclk5L5vVRoVpJ08iiHbSLiqiMKqiKrlAdNRBFD+gJvaBX49F4Nt6M96Q1Y6Qz2+hPGZ/fUqab7Q==</latexit>

• Given a LTI SISO system described by the following state space representation:

• The output is given by:

y(t) = CeAtx(0) + C

Z t

0eA(t�⌧)Bu(⌧)d⌧ +Du(t)

<latexit sha1_base64="kvhbdodDE0LgWPpAxBD+DflTJCk=">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</latexit><latexit sha1_base64="qXMeiu9rX4l56P+e2QSKCvtqth0=">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</latexit><latexit sha1_base64="qXMeiu9rX4l56P+e2QSKCvtqth0=">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</latexit>

• In general, the transfer function between input and output is a rational function:

• The zeros of the system are the:

P (s) = C(sI �A)�1B +D =bn�1sn�1 + bn�2sn�2 + · · ·+ b0

sn + an�1sn�1 + · · ·+ a0+ d

<latexit sha1_base64="VVNwWe/ZJYCww2z3fWGajyvmSXM=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">AAACXHicbVHLattAFB2pcZvYdes2kE0JDAkFB2MjGUqSRSGPLpqdC3Ee2I4ZjUb2kNFIzFwVjNBPZpds8h3Z9UpOoHV6YdA55547j6MgVdKC59077pu12tt36xv1xvvmh4+tT58vbJIZLoY8UYm5CpgVSmoxBAlKXKVGsDhQ4jK4PS37l7+FsTLR57BIxSRmMy0jyRmgNG3BoG336Hd62rZn3eO9m7zrFye0Q3+gNo4M43kwzTWK9qb6dGjF+0veRz4OE7C01L0iL9UCCVsZejExNHXCaWvX63lV0dfAfwa7R99OPPF0vT2Ytu5wA57FQgNXzNqR76UwyZkByZUo6uPMipTxWzYTI4SaxcJO8iqdgn5FJaRRYnBpoJX690TOYmsXcYDOmMHcrvZK8X+9UQbRwSSXOs1AaL48KMoUhYSWUdNQGsFBLRAwbiTelfI5w0wBf0gdQ/BXn/waDPu9w573C8PYJ8taJ1/IDmkTn+yTI/KTDMiQcPLgEGfDqTuPbs1tuM2l1XWeZzbJP+Vu/QHubrFW</latexit><latexit sha1_base64="C39OhB+IczRcjLNINXH29e9lt8M=">AAAB2HicbZDNSgMxFIXv1L86Vq1rN8EiuCpTN+pOcOOygmML7VAymTttaCYzJHeEMvQFXLhRfDB3vo3pz0KtBwIf5yTk3hMXSloKgi+vtrW9s7tX3/cPGv7h0XGz8WTz0ggMRa5y04+5RSU1hiRJYb8wyLNYYS+e3i3y3jMaK3P9SLMCo4yPtUyl4OSs7qjZCtrBUmwTOmtowVqj5ucwyUWZoSahuLWDTlBQVHFDUiic+8PSYsHFlI9x4FDzDG1ULcecs3PnJCzNjTua2NL9+aLimbWzLHY3M04T+zdbmP9lg5LS66iSuigJtVh9lJaKUc4WO7NEGhSkZg64MNLNysSEGy7INeO7Djp/N96E8LJ90w4eAqjDKZzBBXTgCm7hHroQgoAEXuDNm3iv3vuqqpq37uwEfsn7+Aap5IoM</latexit><latexit sha1_base64="P10G41lk8/3QNO4Kt9Q75+f/hSg=">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</latexit><latexit sha1_base64="P10G41lk8/3QNO4Kt9Q75+f/hSg=">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</latexit><latexit sha1_base64="Iv2aiLdFnGtvDCm6G7wSKLpjjYE=">AAACXHicbVHLSsNAFJ3EVx9Wq4IbN4NFaCmWpBt1IVTrQncVrBbaWibTiQ5OJmHmRighP+lON/6Kk7SCVi8MOefcc+dx4kWCa3Ccd8teWV1b3ygUS+XNytZ2dWf3XoexoqxPQxGqgUc0E1yyPnAQbBApRgJPsAfvpZv1H16Z0jyUdzCL2DggT5L7nBIw0qQKvbpu4HPcreub44vGY3Lsppe4ia+MNvIVoYk3SaQR9WP+aeKct+e8bfhoGoLGme6kSaamhpCloW8TMabmdFKtOS0nL/wXuAtQQ4vqTapvZgMaB0wCFUTroetEME6IAk4FS0ujWLOI0BfyxIYGShIwPU7ydFJ8ZJQp9kNllgScqz8nEhJoPQs84wwIPOvlXib+1xvG4J+OEy6jGJik84P8WGAIcRY1nnLFKIiZAYQqbu6K6TMxmYL5ISUTgrv85L+g326dtZxbp9Y5WaRRQAfoENWRi05QB12jHuojij4sZBWtkvVpr9lluzK32tZiZg/9Knv/C0Vjrps=</latexit><latexit sha1_base64="Iv2aiLdFnGtvDCm6G7wSKLpjjYE=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="C39OhB+IczRcjLNINXH29e9lt8M=">AAAB2HicbZDNSgMxFIXv1L86Vq1rN8EiuCpTN+pOcOOygmML7VAymTttaCYzJHeEMvQFXLhRfDB3vo3pz0KtBwIf5yTk3hMXSloKgi+vtrW9s7tX3/cPGv7h0XGz8WTz0ggMRa5y04+5RSU1hiRJYb8wyLNYYS+e3i3y3jMaK3P9SLMCo4yPtUyl4OSs7qjZCtrBUmwTOmtowVqj5ucwyUWZoSahuLWDTlBQVHFDUiic+8PSYsHFlI9x4FDzDG1ULcecs3PnJCzNjTua2NL9+aLimbWzLHY3M04T+zdbmP9lg5LS66iSuigJtVh9lJaKUc4WO7NEGhSkZg64MNLNysSEGy7INeO7Djp/N96E8LJ90w4eAqjDKZzBBXTgCm7hHroQgoAEXuDNm3iv3vuqqpq37uwEfsn7+Aap5IoM</latexit><latexit sha1_base64="P10G41lk8/3QNO4Kt9Q75+f/hSg=">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</latexit><latexit sha1_base64="P10G41lk8/3QNO4Kt9Q75+f/hSg=">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</latexit><latexit sha1_base64="Iv2aiLdFnGtvDCm6G7wSKLpjjYE=">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</latexit><latexit sha1_base64="Iv2aiLdFnGtvDCm6G7wSKLpjjYE=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">AAACXHicbVHLattAFB2pcZvYdes2kE0JDAkFB2MjGUqSRSGPLpqdC3Ee2I4ZjUb2kNFIzFwVjNBPZpds8h3Z9UpOoHV6YdA55547j6MgVdKC59077pu12tt36xv1xvvmh4+tT58vbJIZLoY8UYm5CpgVSmoxBAlKXKVGsDhQ4jK4PS37l7+FsTLR57BIxSRmMy0jyRmgNG3BoG336Hd62rZn3eO9m7zrFye0Q3+gNo4M43kwzTWK9qb6dGjF+0veRz4OE7C01L0iL9UCCVsZejExNHXCaWvX63lV0dfAfwa7R99OPPF0vT2Ytu5wA57FQgNXzNqR76UwyZkByZUo6uPMipTxWzYTI4SaxcJO8iqdgn5FJaRRYnBpoJX690TOYmsXcYDOmMHcrvZK8X+9UQbRwSSXOs1AaL48KMoUhYSWUdNQGsFBLRAwbiTelfI5w0wBf0gdQ/BXn/waDPu9w573C8PYJ8taJ1/IDmkTn+yTI/KTDMiQcPLgEGfDqTuPbs1tuM2l1XWeZzbJP+Vu/QHubrFW</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit><latexit sha1_base64="zPIgoOPRZJVmRtKd6R215tNR4Y0=">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</latexit>

• The poles of the system are zeros (!) of the denominator of P(s), i.e., the values of s such that the characteristic polynomial of A, i.e. det(sI-A) = 0.

zi : P (zi) = 0<latexit sha1_base64="4xLTrIizJayN9dZAZvqyhnBrU4Y=">AAAB93icbVC7SgNBFL0bXzE+smphYTMYhNiEjU1UEAI2lhFcE0iWZXYymwyZfTAPIVkC/oeNhYqtv+An2Pk3Th6FJh643MM59zJ3TpByJpXjfFu5ldW19Y38ZmFre2e3aO/t38tEC0JdkvBEtAIsKWcxdRVTnLZSQXEUcNoMBtcTv/lAhWRJfKeGKfUi3ItZyAhWRvLt4shn6BI1yqafXjnIt0tOxZkCLZPqnJTqh/rzEQAavv3V6SZERzRWhGMp21UnVV6GhWKE03GhoyVNMRngHm0bGuOISi+bHj5GJ0bpojARpmKFpurvjQxHUg6jwExGWPXlojcR//PaWoXnXsbiVCsak9lDoeZIJWiSAuoyQYniQ0MwEczcikgfC0yUyapgQqgufnmZuGeVi4pza8KowQx5OIJjKEMValCHG2iACwQ0PMELvFoj69l6s95nozlrvnMAf2B9/ABKL5M/</latexit><latexit sha1_base64="djZ67qJpMHDb01Rn/03Vx2bqGsg=">AAAB93icbVC7SgNBFL3rM8ZHVi0sbAaDEJuwsYkKQsDGMoJrAsmyzE5mkyGzD+YhJEu+wQ+wsVCx9Rf8BDt/xNrJo9DEA5d7OOde5s4JUs6kcpwva2l5ZXVtPbeR39za3inYu3t3MtGCUJckPBHNAEvKWUxdxRSnzVRQHAWcNoL+1dhv3FMhWRLfqkFKvQh3YxYygpWRfLsw9Bm6QPWS6SeXDvLtolN2JkCLpDIjxdqB/nhol77rvv3Z7iRERzRWhGMpWxUnVV6GhWKE01G+rSVNMenjLm0ZGuOISi+bHD5Cx0bpoDARpmKFJurvjQxHUg6iwExGWPXkvDcW//NaWoVnXsbiVCsak+lDoeZIJWicAuowQYniA0MwEczcikgPC0yUySpvQqjMf3mRuKfl87JzY8KowhQ5OIQjKEEFqlCDa6iDCwQ0PMIzvFhD68l6td6mo0vWbGcf/sB6/wFVTpTI</latexit><latexit sha1_base64="djZ67qJpMHDb01Rn/03Vx2bqGsg=">AAAB93icbVC7SgNBFL3rM8ZHVi0sbAaDEJuwsYkKQsDGMoJrAsmyzE5mkyGzD+YhJEu+wQ+wsVCx9Rf8BDt/xNrJo9DEA5d7OOde5s4JUs6kcpwva2l5ZXVtPbeR39za3inYu3t3MtGCUJckPBHNAEvKWUxdxRSnzVRQHAWcNoL+1dhv3FMhWRLfqkFKvQh3YxYygpWRfLsw9Bm6QPWS6SeXDvLtolN2JkCLpDIjxdqB/nhol77rvv3Z7iRERzRWhGMpWxUnVV6GhWKE01G+rSVNMenjLm0ZGuOISi+bHD5Cx0bpoDARpmKFJurvjQxHUg6iwExGWPXkvDcW//NaWoVnXsbiVCsak+lDoeZIJWicAuowQYniA0MwEczcikgPC0yUySpvQqjMf3mRuKfl87JzY8KowhQ5OIQjKEEFqlCDa6iDCwQ0PMIzvFhD68l6td6mo0vWbGcf/sB6/wFVTpTI</latexit>

ETHZ - Control Systems II - Jacopo Tani 24

Which transfer functions are relevant?

• There are three signals that enter the system: (r, d, n)

• These signals drive the dynamics of the other variables

-1

+ + +

y

r e u

d n

yvF C P

The controller: designed The process / environment: given

⌘Feedforward Controller Feedback Controller Plant

Disturbance Noise

Measured Output

Real Output

DisturbedInputInput

Negative Feedback

Reference TrackingError

η

ETHZ - Control Systems II - Jacopo Tani

Loop transfer functions

25

-1y

r e u

d n

yvF C P

⌘

2

66664

y⌘vue

3

77775=

2

666664

PCF1+PC

P1+PC

11+PC

PCF1+PC

P1+PC

�PC1+PC

CF1+PC

P1+PC

�PC1+PC

CF1+PC

�PC1+PC

�C1+PC

F1+PC

�P1+PC

�11+PC

3

777775

2

4rdn

3

5

• A performant controller minimizes the error between reference and plant output:

• Other variables of interest:

✏ = r � ⌘ = (1� PCF

1 + PC)r � P

1 + PCd+

PC

1 + PCn

ETHZ - Control Systems II - Jacopo Tani

The "Gang of Six”

26

S(s) = 11+P (s)C(s)

T (s) = P (s)C(s)1+P (s)C(s)

P (s)S(s) = P (s)1+P (s)C(s)

C(s)S(s) = C(s)1+P (s)C(s)

T (s)F (s) = P (s)C(s)F (s)1+P (s)C(s)

C(s)F (s)S(s) = C(s)F (s)1+P (s)C(s)

• Then:

• And:

• Observation: Many terms are common. Define:

2

66664

y⌘vue

3

77775=

2

66664

TF PS STF PS �TCFS PS �TCFS �T �CSFS �PS �S

3

77775

2

4rdn

3

5

" = (1� T )r � PSd+ Tn

ETHZ - Control Systems II - Jacopo Tani

On Feedforward

=

Example from Prof. Sandipan Mishra, FF and Iterative Learning Control, RPI, 2012

ETHZ - Control Systems II - Jacopo Tani

On Feedforward

• Feedback is error based, compensates for unexpected / unmodeled phenomena (disturbances, noise, model uncertainty)

• Feedforward works well with prior knowledge (of reference, plant, and/or disturbance). The key idea is that of plant inversion

"Perfect control” example: we want to find input such that:

y(s) = P(s)u(s)

y(t) = r(t)

r(s) = P(s)u(s)⇒ u = P−1r

• This is not possible when:- The plant has RHP zeros (unstable inverse)- Time delays (non causal inverse)- More poles than zeros (unrealizable inverse)- Model uncertainty (unknown inverse, cancellation does not occur)

ETHZ - Control Systems II - Jacopo Tani

The gang of four

29

(F = 1)

S(s) =1

1 + P (s)C(s)=

1

1 + L(s)

T (s) =P (s)C(s)

1 + P (s)C(s)=

L(s)

1 + L(s)

P (s)S(s) =P (s)

1 + P (s)C(s)

C(s)S(s) =C(s)

1 + P (s)C(s)

Sensitivity function:

Complementary sensitivity:

Load sensitivity:

Noise sensitivity:

2

66664

y⌘vue

3

77775=

2

66664

T PS ST PS �TCS PS �TCS �T �CSS �PS �S

3

77775

2

4rdn

3

5 " = (1� T )r � PSd+ Tn

How does the disturbance affect the output?

How does noise affect the input?

ETHZ - Control Systems II - Jacopo Tani

Notes on S and T

30

• The sensitivity function measures how much a (relative) variation in the plant model influences the transfer function from the reference to the output:

S(s) + T (s) = (1 + L)�1 + L(1 + L)�1 = 1, 8s

Pyr(s) = T (s) =P (s)C(s)

1 + P (s)C(s)

@TT@PP

=P

T

@T

@P=

P

T

C(1 + PC)� CPC

(1 + PC)2=

P

T

1

P

PC

(1 + PC)

1

(1 + PC)= S

• S(s) and T(s) depend only on L(s), the (open) loop transfer function

• The complementary sensitivity function news its name to:

ETHZ - Control Systems II - Jacopo Tani

Performance specifications

31

• How can we translate the insight provided by the sensitivity functions in control objectives?

"(s) = S(s)r � P (s)S(s)d+ T (s)n

• Good tracking and disturbance attenuation: "small" S(s), or “high" L(s)

• Noise Rejection: "small" T(s), or "small" L(s)

log|L(j!)|

!!c0dBLow frequencyHigh gainGood disturbance att.

High frequencyLow gainGood noise rejection

ETHZ - Control Systems II - Jacopo Tani

References

Feedback Systems, Chapter 11

SISO Design constraints

Credits

Intended Learning Outcomes Prerequisites

SISO Design constraints

• Fundamental limitation to controller design

• Sensitivity functions

• Jacopo Tani - ETHZ - Feb. 2017

ETHZ - Control Systems II - Jacopo Tani

Storyline

We:

• Determined the objectives of a good controller

• Defined the specifications in the frequency domain through the sensitivity functions

• What are the constraints? What stands in our way?

ETHZ - Control Systems II - Jacopo Tani

(Input-Output) Controllability

• Is it always possible to design a controller for a given plant?

No. Controllability (You can’t always find an input that affects the system how you would like)

• Is there anything we can do when a system is not controllable?

Maybe.

- Choose outputs wisely: such that input have direct, rapid effect on them- Choose inputs wisely: large effect on outputs

But controllability is a property of the plant (not of the controller)

ETHZ - Control Systems II - Jacopo Tani

Recall ideal situation (plant inversion)

Suppose a system is controllable. Why not just plant inversion?

• Are there other fundamental limitations that affect the ability to design performant controllers?

• This is not possible when:- The plant has RHP zeros (unstable inverse)- Time delays (non causal inverse)- More poles than zeros (unrealizable inverse)- Model uncertainty (unknown inverse, cancellation does not occur)

ETHZ - Control Systems II - Jacopo Tani

S(s) +T(s) = 1, 8s

• We already noticed that S, T cannot be arbitrarily determined, at each given frequency, because of the complementarity constraint.

• This consideration led us to shape the open loop function as we did previously.

• But can we make S small on an arbitrary big frequency range? Not always.

ETHZ - Control Systems II - Jacopo Tani

Bode’s integral formula

G. Stein: Respect the Unstable

(a.k.a. First waterbed formula)

• The sensitivity function cannot be shaped arbitrarily either. There is a fundamental relation:

Z 1

0log |S(j!)|d! =

Z 1

0log

1

|1 + L(j!)|d! = ⇡X

pk

• Theorem (Bode’s integral formula) let the loop transfer function of a feedback system satisfy , and have unstable poles . Then, lim

s!1sL(s) = 0 k pk

• Proof: non trivial, see additional resources in references section.

Bad things, something we don’t want. Let’s call it “dirt”.

• This is a conservation law, like in physics. The "principle of conservation of dirt”.

ETHZ - Control Systems II - Jacopo Tani

Principle of conservation of dirt

G. Stein: Respect the Unstable

(a.k.a. Waterbed effect)

• Low sensitivity is desirable across a broad range of frequencies. It implies disturbance rejection and good tracking.

• The meaning of this principle is that “so much (dirt) we remove (at some frequency), that much we need to add (at some other frequency)” [waterbed effect].

ETHZ - Control Systems II - Jacopo Tani

Second waterbed formula (RHP-zero)

• Theorem (Second waterbed formula) Suppose that L(s) has a single real RHP-zero or a complex conjugate pair of zeros , and has RHP-poles, . Let denote the complex conjugate of . Then for closed-loop stability the sensitivity function must satisfy:

z = x± jyz

Np pi pipi

Z 1

0ln |S(j!)| · w(z,!)d! = ⇡

NpY

I=1

����pi + z

pi � z

����

w(z,!) =2z

z2 + !2=

2

z

1

1 + (!z )2

w(z,!) =x

x2 + (y � !)2+

x

x2 + (y + !)2

if the zero is real:

if the zero is complex:

Dirt goes to infinity as pi ! z

• Proof: non trivial, see additional resources in references section.

ETHZ - Control Systems II - Jacopo Tani

Second waterbed formula (RHP-zero)Z 1

0ln |S(j!)| · w(z,!)d! = ⇡

NpY

I=1

����pi + z

pi � z

����

Dirt goes to infinity as pi ! z

• Note:

Unstable poles close to RHP-zeros make a plant very difficult to control.

• Weighting function such that argument of integral is negligible at ! > z

RHP-zero reduces “effective" frequency range where dirt can be distributed. This implies higher S(s) peak hence disturbance amplification.

ETHZ - Control Systems II - Jacopo Tani

Time Delays

• Time delay introduces a linearly increasing (in frequency) phase lag.

Recall Summary of how to deal with delays:

ETHZ - Control Systems II - Jacopo Tani

Today’s Summary

• We looked at the objectives of controller design: stability, performance, and robustness.

• We reviewed the relevance of transfer functions and sensitivity functions.

• We reviewed some fundamental limitations to the design of feedback controllers.

ETHZ - Control Systems II - Jacopo Tani

In the next episode

We will look at practical limitations as opposed to fundamental ones. In particular, related with digital implementation of controllers (discrete time domain SISO LTI systems).

ETHZ - Control Systems II - Jacopo Tani

References

Additional resources:

Gunther Stein Bode Lecture: https://www.youtube.com/watch?v=9Lhu31X94V4

Respect the Unstable paper

Multivariable Feedback Control: Chapter 5

Bode formula’s proof: Feedback Systems, pg. 338

Feedback Systems, Murray,- 9.4 - Chapter 11

Proof of second waterbed formula: Freudenberg and Looze (1985; 1988).