optimal control for enignieering , Study notes of Optimization Techniques in Engineering

Download optimal control for enignieering and more Study notes Optimization Techniques in Engineering in PDF only on Docsity!

Chapter 1

Introduction

Introduction

• Optimization

• Optimal control

• Calculus of variations

Classical (conventional) control

theory

• (^) Single input and single output (SISO)
• (^) Based on Laplace transform
• (^) System representation in block diagram form

• The input u ( t ) to the plant is determined by the error e ( t ) and the compensator.
• (^) All the variables are not readily available for feedback. In most cases only one output variable is available for feedback.

Modern Control configurations

• (^) The input u ( t ) is determined by the controller (consisting of error detector and compensator) driven by system states x ( t ) and reference signal r ( t ).
• (^) All or most of the state variables are available for control.
• (^) It depends on well-established matrix theory, which is amenable for large scale computer simulation.

Components of a Modern Control

System

• System dynamics ( modeling ) in terms of differential or difference equations based on the Lagrangian function
• Performance to find out mainly stability of the system by Lyapunov
• System synthesis (design) by applying control theory such as optimal control

Optimization classification

• (^) Static Optimization is concerned with controlling a

plant under steady state conditions.

• (^) The system variables are not changing with respect

to time.

• (^) The plant is then described by algebraic equations.
• (^) Techniques used are ordinary calculus, Lagrange

multipliers, linear and nonlinear programming.

1.3 Optimal Control

• (^) To determine control signals that will cause a process

(plant)

• (^) To satisfy some physical constraints and at the same

time extremize (maximize or minimize) a chosen

performance criterion (performance index or cost

function)

The formulation of optimal control

problem

• (^) A mathematical description (or model) of the process

to be controlled (generally in state variable form).

• (^) A specification of the performance index.
• (^) A statement of boundary conditions and the physical

constraints on the states and/or controls.

1.3.1 Plant

1.3.2 Performance Index

Classical control design

• (^) Linear, time-invariant, single-input, single output

(SISO) systems

• (^) Typical performance criteria are system time response

to step or ramp input characterized by rise time,

settling time, peak overshoot, and steady state

accuracy;

• (^) And the frequency response of the system

characterized by gain and phase margins, and

bandwidth.