EG4321/EG7040

Nonlinear Control

Dr. Matt Turner

EG4321/EG7040

[An introduction to]Nonlinear Control

Dr. Matt Turner

EG4321/EG7040

[An introduction to]Nonlinear [System Analysis]

and Control

Dr. Matt Turner

Motivation - Control of a hydraulic Actuator

Control Objectives

Design Controller to

◮ Control position (σ) of load... [output]

◮ ..by manipulating voltage input (u) [control]

Note:

◮ System is approximately linear

Motivation - Control of a hydraulic Actuator

Controller Design

System is linear so could use many design methods

◮ Classical Control - PID, lead-lag etc

◮ State-space based - pole placement, LQR etc

Motivation - Control of a hydraulic Actuator

Adaptive (self-tuning) Control

◮ Specify model (desired) behaviour

◮ Let an (nonlinear) adaptive algorithm tune the controller

Results:

Time [sec]0 5 10 15

Pos

ition

[cm

]

0

2

4

6

8

10

12

14

16

18

20Plant/Model state evolution

Time [sec]0 5 10 15

Con

trol

sig

nal

-60

-40

-20

0

20

40

60Control signal evolution

Adaptive controller does a pretty good job!

Motivation - Nonlinear Flight Control

NASA/USAF X15 Experimental Rocket Powered Aircraft

◮ Rocket powered high speed (Mach 6.7 top speed) aircraft

◮ High altitude (30 km +, some flights technically space flight)

◮ Wide flight envelope: 0km-100km altitude, Mach 0.8 - Mach 6.7

◮ Complex control system

Motivation - Nonlinear Flight Control

Operational scenarios

1. Rocket powered flight from“low” altitude/speed to highaltitude/speed

2. “Re-entry” from thin upperatmosphere to denser loweratmosphere

3. Glide landing

Two sets of control effectors

◮ Conventional (lower atmosphere)

◮ Rocket thrusters (upper atmosphere)

◮ ....and a blend of the two....quite difficult to control

Motivation - Nonlinear Flight Control

Three X15 aircraft built

1. X15-1 conventional (linear) automatic control system

2. X15-2 conventional (linear) automatic control system

3. X15-3 MH-96 Adaptive Flight Control System

The [nonlinear] Adaptive Flight Control System was the most advanced:

◮ Blended (automatically) conventional control surfaces and reactionjets

◮ Control gains updated automatically

◮ In principle able to adapt to flight condition

Surely the adaptive system was the best.....?

Motivation - Nonlinear Flight Control

Flight 191 - Disaster

◮ Limit cycle oscillation

◮ Break-up of aircraft

◮ Death of pilot

Investigation

◮ Adaptive control system implicated in contributing to crash

◮ Nonlinear stability analysis inadequate/absent?

Message: nonlinear control methods need appropriate supporting analysis

Timetable

Lectures

09.00-10.00 Thursday PHYS LTA

14.00-15.00 Friday BENL LT

Seminars

17.00-18.00 Tuesdays ATT LT36th Feb20th Feb6th March

Test (EG7040 only)

17.00-18.00 Friday ENG LT19th March

Aims of Lecture

1. To motivate the need to examine nonlinear systems and usenonlinear control techniques

2. To provide an overview of the course, the teaching and assessmentmethods and changes made in response to student feedback

Classical Control in a nutshell

Typical control configuration

yu

re

d

Controller Plant

K(s) G(s)

Objective: Given G(s), design linear controller K (s) such that

1. Closed-loop system is stable

2. System is insensitive to disturbances (at appropriate frequencies)

3. Error is small (at appropriate frequencies)

4. System has sufficient stability margins

Implicit assumption: Plant is linear, or approximately linear

Classical Control for Nonlinear Plants

What if plant is Nonlinear?

yu

re

d

Controller Plant

K(s) G(u, d)

Approximate nonlinear plant with linearised version:

G(., .) → G(s)

But:

1. Linear model only approximates nonlinear plant locally

2. Linearisation may be difficult

3. Linearisation may not preserve salient features

◮ Linearisation may not yield a “good” linear controller

◮ A nonlinear controller may be more suitable for a nonlinear system

Nonlinear Control for Linear Plants?

yu

re

d

Controller Plant

G(s)K(y , r)

◮ A nonlinear controller may give better performance...

Example: network congestion control

K ∼

{

xc(t) = kxc(t − Tr )(

1− f1(y(t))f2(xc (t))

)

u(t) = xc(t)f1(.), f2(.) nonlinear

.....Nonlinear controllers need to be treated with caution.

What this module is about

◮ The limitations of linear design/analysis techniques

◮ An introduction to a subset of nonlinear analysis techniques

◮ An introduction to a subset of nonlinear synthesis techniques

The sorts of themes covered in this module will be the following:

◮ The characteristics of nonlinear systems described by ordinarydifferential equations (ODEs)

◮ Asymptotic (stability) properties of nonlinear systems◮ The richness of this behaviour◮ The difficulties in assessing asymptotic behaviour

◮ “Weakly” nonlinear systems◮ How aspects of linear systems theory can help us

◮ Controller design methods for nonlinear systems

Overview of syllabus

◮ Brief revision of state-space concepts

◮ Introduction to nonlinear systems◮ Representation; Distinguishing features◮ Phase portraits (qualitative analysis)

◮ Lyapunov analysis of nonlinear systems◮ Fundamentals of Lyapunov’s 2nd method◮ Circle/Popov Criterion for interconnected systems◮ Introduction to passivity

◮ Controller design◮ Nonlinear dynamic inversion (feedback linearisation)◮ Adaptive control

Background and Pre-requisites

◮ Good mathematical ability highly desirable

◮ Not necessary to have studied Robust control

Teaching

◮ Lectures

Lecture A: Theory (mainly slides)Lecture B: Examples class (board work)

Attendence of lectures highly recommended

◮ Private study - Very important. Aim to spend a couple of hours aweek reading notes, attempting example questions etc.

◮ Directed reading - nonlinear control is an “M”-level course.Independent investigation required.

Assessment

“Nonlinear Control” comprises two modules:

EG4321 MEng course10 credit moduleAssessment: . . . . . . 2 hour exam

EG7040 MSc course15 credit moduleAssessment: . . . . . . 2 hour exam (2/3)

. . . . . . Test (1/3)

Exam: 4 questionsChoose 3

Books

Nonlinear Control

◮ “Nonlinear Systems”, H. Khalil. The classic nonlinear controltextbook. Well-written and comprehensive. Quite technical.

◮ “Nonlinear Control Systems”. H.J. Marquez. Similar style to Khalil.A little easier to read perhaps. A little less comprehensive but somesubtleties covered.

Background

◮ “Modern Control Systems”, K. Ogata. Standard classical controltextbook. Comprehensive. Gives detailed state-space coverage andextensive discussion on classical control techniques

◮ “Modelling and Analysis of Dynamic Systems” Close, Frederick andNewell. Good, easy-to-read background on constructing simplestate-space models.

Do not consult:“Feedback systems: input output properties”, Desoer andVidyasagar. Brilliant book but approaches problems from an input-output, rather than Lyapunov, perspective

Student Feedback

The 2016 students liked the following aspects of the module

◮ Good lecturer

◮ Handouts/slides

The 2016 students had the following complaints about the module

◮ No Panopto

◮ Challenging

This year

◮ Seminars integrated into course

◮ Textual summary of each part of course(as requested)

Important Background - state-space representations

The course deals extensively with systems described in the so-calledstate-space form

x(t) = f (x(t), u(t)) differential equation

y(t) = g(x(t), u(t)) algebraic equation

x =

x1x2...xn

u =

u1u2...um

y =

y1y2...yp

x state vectoru input vectory output vector

Important Background - linearity

Gyu

System is linear if it satifies

◮ Homogeneity. Given y = G(u), then

αy = G(αu)

◮ Superposition. If

y1 = G(u1) y2 = G(u2)

theny1 + y2 = G(u1 + u2)