Post on 06-Jul-2018
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)