Systems with Delays

Analysis, Control, and Computations

A. V. Kim, A. V. Ivanov

• Publisher: Wiley-Scrivener
• Year: 2015
• Print ISBN: 9781119117582
• eBook ISBN: 9781119117735
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

The main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays. Referred to as "systems with delays" in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations. Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems

At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician. there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems.

Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.

Chapter 1

Linear time-delay systems

1.1 Introduction

1.1.1 Linear systems with delays

In this book we consider methods of analysis, control and computer simulation of linear systems with delays

(1.1)

where A(t), A_τ(t) are n × n matrices with piece-wise continuous elements, G(t, s) is n × n matrix with piece-wise continuous elements on R × [−τ, 0], u is a given n-dimensional vector-function, τ(t) : R → [−τ, 0] is a continuous function, τ is a positive constant.

Much attention will be paid to the special class of linear time-invariant systems

(1.2)

where A, A_τ are n × n constant matrices, G(s) is n × n matrix with piece-wise continuous elements on [−τ, 0], τ is a positive constant¹.

Usually we will consider u as the vector of control parameters. There are two possible variants:

1) u = u(t) is the function of time t;

2) u depend on the current and previous state of the system, for example,

(1.3)

Consider some models of control systems with delays.

1.1.2 Wind tunnel model

A linearized model of the high-speed closed-air unit wind tunnel is [134, 135]

(1.4)

with

.

The state variable x₁, x₂, x₃ represent deviations from a chosen operating point (equilibrium point) of the following quantities: x₁ = Mach number, x₂ = actuator position guide vane angle in a driving fan, x₃ = actuator rate. The delay represents the time of the transport between the fan and the test section.

The system can be written in matrix form

(1.5)

where

1.1.3 Combustion stability in liquid propellant rocket motors

A linearized version of the feed system and combustion chamber equations, assuming nonsteady flow, is given by²

(1.6)

Here

ϕ(t) = fractional variation of pressure in the combustion chamber,

t is the unit of time normalized with gas residence time,

θ_g, in steady operation,

= value of time lag in steady operation,

= pressure in combustion chamber in steady operation,

= const for some number γ,

μ(t) = fractional variation of injection and burning rate,

ψ(t) = relative variation of p₁,

p₁ = instantaneous pressure at that place in the feeding line where the capacitance representing the elasticity is located,

ξ = fractional length for the constant pressure supply,

J = inertial parameter of the line,

P = pressure drop parameter,

μ₁(t) = fractional variation of instantaneous mass flow upstream of the capacitance,

Δp = injector pressure drop in steady operation,

p₀ = regulated gas pressure for constant pressure supply,

E = ...

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• Preface
• Chapter 1: Linear Time-Delay Systems
• Chapter 2: Stability Theory
• Chapter 3: Linear Quadratic Control
• Chapter 4: Numerical Methods
• Chapter 5: Appendix
• Bibliography
• Index