EG4321/EG7040. Nonlinear Control. Dr. Matt Turner

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1 EG4321/EG7040 Nonlinear Control Dr. Matt Turner

2 EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner

3 EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt Turner

4 Motivation - Control of a hydraulic Actuator Control Objectives Design Controller to Control position (σ) of load.....by manipulating voltage input (u) Note: System is approximately linear [output] [control]

5 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

6 Motivation - Control of a hydraulic Actuator Adaptive (self-tuning) Control Specify model (desired) behaviour Let an (nonlinear) adaptive algorithm tune the controller Results: 20 Plant/Model state evolution 60 Control signal evolution Position [cm] Control signal Time [sec] Time [sec] Adaptive controller does a pretty good job!

7 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 Mach 6.7 Complex control system

8 Motivation - Nonlinear Flight Control Operational scenarios 1. Rocket powered flight from low altitude/speed to high altitude/speed 2. Re-entry from thin upper atmosphere to denser lower atmosphere 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

9 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 reaction jets 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

10 Motivation - Nonlinear Flight Control Flight 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

11 Timetable Lectures Thursday PHYS LTA Friday BENL LT Seminars Test (EG7040 only) Tuesdays ATT LT3 6th Feb 20th Feb 6th March Friday ENG LT1 9th March

12 Aims of Lecture 1. To motivate the need to examine nonlinear systems and use nonlinear control techniques 2. To provide an overview of the course, the teaching and assessment methods and changes made in response to student feedback

13 Classical Control in a nutshell Typical control configuration d r e K(s) u G(s) y Controller Plant 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

14 Classical Control for Nonlinear Plants What if plant is Nonlinear? d r e u K(s) G(u, d) y Controller Plant 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

15 Nonlinear Control for Linear Plants? d r e K(y,r) u G(s) y Controller Plant A nonlinear controller may give better performance... Example: network congestion control K { ( ) ẋ c (t) = kx c (t T r ) 1 f1(y(t)) f 2(x c(t)) u(t) = x c (t) f 1 (.),f 2 (.) nonlinear...nonlinear controllers need to be treated with caution.

16 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 ordinary differential 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

17 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

18 Background and Pre-requisites Good mathematical ability highly desirable Not necessary to have studied Robust control

19 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 a week reading notes, attempting example questions etc. Directed reading - nonlinear control is an M -level course. Independent investigation required.

20 Assessment Nonlinear Control comprises two modules: EG4321 MEng course 10 credit module Assessment:... 2 hour exam EG7040 MSc course 15 credit module Assessment:... 2 hour exam (2/3)... Test (1/3) Exam: 4 questions Choose 3

21 Books Nonlinear Control Nonlinear Systems, H. Khalil. The classic nonlinear control textbook. 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 some subtleties covered. Background Modern Control Systems, K. Ogata. Standard classical control textbook. Comprehensive. Gives detailed state-space coverage and extensive discussion on classical control techniques Modelling and Analysis of Dynamic Systems Close, Frederick and Newell. Good, easy-to-read background on constructing simple state-space models. Do not consult: Feedback systems: input output properties, Desoer and Vidyasagar. Brilliant book but approaches problems from an inputoutput, rather than Lyapunov, perspective

22 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)

23 Important Background - state-space representations The course deals extensively with systems described in the so-called state-space form ẋ(t) = f(x(t), u(t)) y(t) = g(x(t), u(t)) differential equation algebraic equation x = x 1 x 2. x n u = u 1 u 2. u m y = y 1 y 2. y p x state vector u input vector y output vector

24 Important Background - linearity u G y System is linear if it satifies Homogeneity. Given y = G(u), then αy = G(αu) Superposition. If y 1 = G(u 1 ) y 2 = G(u 2 ) then y 1 +y 2 = G(u 1 +u 2 )

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics