Farid Golnaraghi

Simon Fraser University

Vancouver, Canada

ISBN-13: 978-1259643835

ISBN-10: 1259643832

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INTRODUCTION• In Chapter 3 we presented the concept and definition of state

variables and state equations for linear continuous-data.

• In Chapter 4 we used block diagram and signal-flow-graph (SFG) methods to obtain the transfer function of linear systems.

• In this chapter, the SFG concept is extended to the modeling of the state equations, and the result is the state diagram.

• In contrast to the transfer-function approach to the analysis and design of linear control systems, the state-variable method is regarded as modern.

• The basic characteristic of the state-variable formulation is that linear and nonlinear systems, time-invariant and time-varying systems, and single-variable and multivariable systems can all be modeled in a unified manner.

• Transfer functions, on the other hand, are defined only for linear time-invariant systems.

• The objective of this chapter is to introduce the basic methods of state variables and state equations to provide a working knowledge of the subject.

• Specifically, the closed-form solutions of linear time-invariant state equations are presented.

• Various transformations that may be used to facilitate the analysis and design of linear control systems in the state-variable domain are introduced.

• The relationship between the conventional transfer-function approach and the state-variable approach is established.

• Controllability and Observability of linear systems are defined and their applications investigated.

• Some state-space controller design problems appear in the end.

• At the end of the chapter, we also present MATLAB tools to solve most state-space problems.

8-2 BLOCK DIAGRAMS, TRANSFER FUNCTIONS, AND STATE DIAGRAMS

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8-3 SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS: STATE EQUATIONS

8-4 VECTOR-MATRIX REPRESENTATION OF STATE EQUATIONS

8-5 STATE-TRANSITION MATRIX

8-6 STATE-TRANSITION EQUATION

The state-transition equation is defined as the solution of the linear homogeneous state equation.

8-7 RELATIONSHIP BETWEEN STATE EQUATIONS AND HIGH-ORDER

DIFFERENTIAL EQUATIONS

As shown in Chapter 2, the state variables of an nth-order differential equation in Eq. (2-97)

are intuitively defined, as shown in Eq. (2-105). The results are the n state equations in Eq.

(2-106).

8-8 RELATIONSHIP BETWEEN STATE EQUATIONS AND TRANSFER FUNCTIONS

8-9 CHARACTERISTIC EQUATIONS, EIGENVALUES, AND EIGENVECTORS

8-10 SIMILARITY TRANSFORMATION

8-10-1 Invariance Properties of the Similarity Transformations

8-10-4 Controllability Canonical Form (CCF)

8-10-5 Observability Canonical Form (OCF)

8-10-6 Diagonal Canonical Form (DCF)

8-10-7 Jordan Canonical Form (JCF)

8-11

DECOMPOSITIONS OF

TRANSFER

FUNCTIONS

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8-11-1 Direct Decomposition

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8-11-2 Direct Decomposition to CCF

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10-2-3 State Diagram

A state diagram is constructed following all the rules of the SFG using the Laplace-transformed

state equations.

Important uses of the state diagram:

1. A state diagram can be constructed directly from the system’s

differential equation.This allows the determination of the state variables and the

state equations.

2. A state diagram can be constructed from the system’s transfer

function. This step is defined as the decomposition of transfer functions (Section 10-10).

3. The state diagram can be used to program the system on an analog

computer or for simulation on a digital computer.

4. The state-transition equation in the Laplace transform domain may be

obtained from the state diagram by using the SFG gain formula.

5. The transfer functions of a system can be determined from the state

diagram.

6. The state equations and the output equations can be determined from

the state diagram.

10-2-4 From Differential Equations to State Diagrams

10-2-6 From State Diagrams to State and Output Equations

10-2-5 From State Diagrams to Transfer Functions