Stability and control of time- delay systemspeople.cs.kuleuven.be/~wim.michiels/courses/socn/Lecture1.pdf · Representation by functional differential equations The initial value

Wim Michiels

Department of Computer Science

K.U.Leuven

Leuven, Belgium

Silviu-Iulian Niculescu

Laboratoire des Systèmes et Signaux

Supélec

Gif-sur-Yvette, France

SOCN Course Leuven

November – December 2011

Stability and control of time-

delay systems

Overview of the course

Lecture 1 motivating examples, basis concepts, eigenvalue problems associated with linear time-delay systems,

spectral properties

Lecture 2 computation of characteristic roots, robustness and performance specifications, computation of H-2

and H-infinity norms

Lecture 3 computation of stability regions in general parameter spaces, stability regions in delay parameter

spaces

Lecture 4 control design I: fundamental limitations induced by delays, design of fixed-order controllers using

eigenvalue optimization

Lecture 5 delay differential equations of neutral type / delay differential algebraic equations: qualitative properties

of solutions, delay sensitivity, design of fixed-order controllers

Lecture 6 control design II: prediction based controllers, using delays as controller parameters

Aim

• the student understands fundamental properties of systems subjected to

time-delays (what becomes different when including a delay?);

• he/she has an overview of methods, techniques, and software for the analysis and

control design -mainly frequency domain based and grounded in numerical

linear algebra and optimization);

• he/she has a view on applications in various domains and on the applicability of

the methods.

Course material

- book “Stability and stabilization of time-delay systems”, SIAM, 2007

- copy of slides (distributed at the beginning of each lecture)

Home-works

- optional (3 instead of 2 ECTS credits if evaluation positive)

- 3 homeworks (distributed after Lectures 2, 3, and 4)

- solutions must be sent to the lecturers by e-mail, one week after the distribution

Contacting lecturers

- Wim Michiels ([email protected])

- Silviu-Iulian Niculescu ([email protected])

Course website

slides, homeworks, supplementary material,…

) http://people.cs.kuleuven.be/wim.michiels/courses/socn/

Overview Lecture I

Examples and applications

Basic properties of time-delay systems

Representation by functional differential equations

The initial value problem

Properties of solutions

Reformulation in a standard, first order equation

Spectrum of linear time-delay systems

Two eigenvalue problems related to linear time-delay systems

Qualitative properties of the spectrum

Continuity propertties of the spectrum

Examples and

applications

• networks

- biology (e.g. interactions between neurons)

- car following models

- time-based spacing of airplanes

- distributed and cooperative control, sensor networks

- congestion control in communication networks

- biochemical networks

• mechanical engineering

- haptic interfaces and motion synchronization

- machine tool vibrations (cutting and milling machines)

• parallel computing (load balancing)

• biological systems

- population dynamics

- cell/virus dynamics

• laser physics (lasers with optical feedback)

Motivating examples

Fluid flow model for a congested router in TCP/AQM

controlled network

pTC

tQtR

QCtR

tWtN

QCtR

tWtN

tQ

tRtptRtR

tRtWtW

tRtW

)()(

0,0,)(

)()(max

0)(

)()(

)(

))(())((

))(()(

2

1

)(

1)(

Hollot et al., IEEE TAC 2002

Model of collision-avoidance type:

W: window-size

Q: queue length

N: number of TCP sessions

R: round-trip-time

C: link capacity

p: probability of packet mark

Tp: propagation delay

AQM is a feedback control problem:

Sender Receiver Bottleneck

router

link c

rtt R

queue Q

acknowledgement

packet marking

• networks







- haptic interfaces



• population dynamics

• cell dynamics, virus dynamics


Motivating examples

Successive passages of teeth

delay

Rotation of each tooth

periodic coefficients

Delay inversely proportional to

speed

Goal: increasing efficiency while avoiding undesired oscillations

(chatter)

Model:

Rotating milling machines

• networks















Motivating examples

Delays appear as intrinsic components of the system, or in approximatations

of (mostly PDE) models describing propragation and wave phenomena

Heating system

Linear system of dimension 6,

5 delays

temperature to be controlled setpoint

Lab. Tomas Vyhlidal, CTU Prague

Model

Control law (PI+ state feedback)

• networks















Motivating examples

Haptic Interfaces and Motion Synchronization


Smith predictor: transforming interconnections

“delay-out-of-the-loop”



• networks















Motivating examples

Evolution of anti-cancer T cells (T) Evolution of cancer cells (C)

Immune Dynamics in Leukemia Models

Interactions between T/C cells

Immune Dynamics in Leukemia Models

Post-transplantation dynamics of the immune response to chronic

myelogenous leukemia:

where:

T: anti-cancer cell population,

C: cancer cell population (functions of time t).

Delay Description

four distinct delays: and ;

relevant values approximately

Lossless Propagation: from hyperbolic PDEs to DDAEs

nonlinear characteristic of the tunnel diode.

Non linear circuit with LC line and tunnel diode


Cauchy problem defined on the domain:

“propagation triangle”.

Propagation infinite angle Propagation triangle


denote:

• networks















Motivating examples

Biochemical Network

(Monotone) Cyclic System with delay

subject to “negative feedback”

given by:

or

leading to:

where:

Basic properties of

time-delay systems

Representation by a functional differential equation

: Banach space of continuous function over [-¿, 0], equipped

with the maximum norm,

t t-¿

Functional Differential Equation of retarded type

functional

Linear FDE of retarded type

F: bounded variation in [-¿, 0]

F(0)=0

) unifying theory available

Linear FDE

F: bounded variation in [-¿, 0]

F(0)=0

examples:

pointwise

(discrete) delay

distributed

(continuous) delay

F

µ

-¿

-A0-A1

-A0

0

+A1

+A0


Delay differential equation

linear

initial data required = function segment

x

-¿ 0 t

Ordinary differential equation

linear

x

0 t


recall:

Theorem holding under mild conditions of f:

t

x

-¿ 0 t-¿ t

→ infinite-dimensional system

Properties of solutions

“Method of steps (single delay)”

Divide the solution in “couplets”:

General property: solutions become smoother as time evolves !

“Small solutions”

Backward continuation of solutions is in general not possible !

Asymptotic growth rate of solutions and stability

t

x

-¿ 0 t-¿ t

-¿ 0 µ

t=0

t=t1

t1

x

-¿ 0 t1-¿ t

Reformulation in a standard, first order form

Abstract ordinary differential equation over the function space X

-¿ 0 µ

t=0

t=t1

t1

x

-¿ 0 t1-¿ t

a time-delay system is a distributed parameter system with a special

structure: “spatial” and “temporal” variable are coupled

Related PDE formulation

Abstract ordinary differential equation

Functional differential equation

Delay systems are distributed parameter systems /infinite-dimensional

scalar examples

oscillatory solutions

chaotic attractor

Analysis: complex behavior

Controller synthesis:

any control design problem involving the determinination of a finite number of

controller parameters is a reduced order controller design problem

! inherent limitations

! control design almost exclusively ends up in

an optimization problem

! two main type of representations lead two

two mainstream control design techniques

The spectrum of linear time-

delay systems

Two eigenvalue problems associated with linear time-

delay systems

Finite-dimensional

nonlinear eigenvalue problem

Infinite-dimensional

linear eigenvalue problem

Functional differential equation Abstract ODE

substitution of exponential solution

Extension to general linear system

Outlook to Lecture 2:

Computing characteristic roots via a two-step approach

1. discretize linear-infinite-dimensional operator;

compute eigenvalues of resulting matrix

2. correct the individual characteristic root approximations

using the nonlinear equation (2)

(2)

Qualitative properties of the spectrum

Real axis

Continuity properties of the spectrum

“Shower example”

The following situations may occur.

• no smoothing of solutions

• infinitely many characteristic roots in a right half plane

• exponential and asymptotic stability not equivalent for linear systems

• multiple delays: stability may be sensitive to infinitesimal delay

perturbations

The presented results do NOT carry over time-delay systems of

neutral type (Lecture 5)

or, more generally,

Conclusions of Lecture 1

• Examples

• Properties of time-delay systems of retarded type

(solutions, representation, spectral properties)

Note: generalization of Lyapunov’s second method

Delay differential equation of retarded type Ordinary differential equation

sufficient condition for exponential stability of null solution

converse theorems exist, quadratic function(al) works in linear case

But: V is a functional ! gap between sufficient and

necessary conditions when restricting

to subclass characterized by a finite

number of parameters

system candidate Lyapunov functional

exact stability regions:

a¿

k¿

bounded by red curves

Example:

Note: time-stepping (numerical integration of solutions)

Main difference with time-steppers for ordinary differential equations

• memory / dependence of past data, often interpolation needed

• taking into account discontinuities in derivatives of solution

Overview of software available at

http://twr.cs.kuleuven.be/research/software/delay/software.shtml

Documents

Stability and control of time- delay systemspeople.cs.kuleuven.be/~wim.michiels/courses/socn/Lecture1.pdf · Representation by functional differential equations The initial value