9

648

Control Systems Analysis

in State Space

9–1 INTRODUCTION*

A modern complex system may have many inputs and many outputs, and these may be

interrelated in a complicated manner. To analyze such a system, it is essential to reduce

the complexity of the mathematical expressions, as well as to resort to computers for most

of the tedious computations necessary in the analysis. The state-space approach to system

analysis is best suited from this viewpoint.

While conventional control theory is based on the input–output relationship, or trans-

fer function, modern control theory is based on the description of system equations in

terms of nfirst-order differential equations, which may be combined into a first-order

vector-matrix differential equation. The use of vector-matrix notation greatly simplifies

the mathematical representation of systems of equations. The increase in the number of

state variables, the number of inputs, or the number of outputs does not increase the

complexity of the equations. In fact, the analysis of complicated multiple-input, multiple-

output systems can be carried out by procedures that are only slightly more compli-

cated than those required for the analysis of systems of first-order scalar differential

equations.

This chapter and the next deal with the state-space analysis and design of control sys-

tems. Basic materials of state-space analysis, including the state-space representation of

* It is noted that in this book an asterisk used as a superscript of a matrix, such asA*, implies that it is a con-

jugate transposeof matrixA. The conjugate transpose is the conjugate of the transpose of a matrix. For a real

matrix (a matrix whose elements are all real), the conjugate transposeA* is the same as the transposeAT.

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