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[STATE-SPACE](#page-14-0) ANALYSIS

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STATE-SPACE ANALYSIS

In Sec. 1.10, basic notions of state variables were introduced. In this chapter, we shall discuss state variables in more depth.

Most of this book deals with an external (input–output) description of systems. As noted in Ch. 1, such a description may be inadequate in some cases, and we need a systematic way of finding a system’s internal description. State-space analysis of systems meets this need. In this method, we first select a set of key variables, called the state variables, in the system. Every possible signal or variable in the system at any instant t can be expressed in terms of the state variables and the input(s) at that instant t. If we know all the state variables as a function of t, we can determine every possible signal or variable in the system at any instant with a relatively simple relationship. The system description in this method consists of two parts:

    1. A set of equations relating the state variables to the inputs (the state equation).
    1. A set of equations relating outputs to the state variables and the inputs (the output equation).

The analysis procedure, therefore, consists of solving the state equation first, and then solving the output equation. The state-space description is capable of determining every possible system variable (or output) from knowledge of the input and the initial state (conditions) of the system. For this reason, it is an internal description of the system.

By its nature, state variable analysis is eminently suited for multiple-input, multiple-output (MIMO) systems. A single-input, single output (SISO) system is a special case of MIMO systems. In addition, the state-space techniques are useful for several other reasons, mentioned in Sec. 1.10, and repeated here.

    1. The state equations of a system provide a mathematical model of great generality that can describe not just linear systems, but also nonlinear systems; not just time-invariant systems, but also time-varying parameter systems; not just SISO systems, but also MIMO systems. Indeed, state equations are ideally suited for analysis, synthesis, and optimization of MIMO systems.
    1. Compact matrix notation along with powerful techniques of linear algebra greatly facilitates complex manipulations. Without such features, many important results of

modern system theory would have been difficult to obtain. State equations can yield a great deal of information about a system even when they are not solved explicitly.

    1. State equations lend themselves readily to digital computer simulation of complex systems of high order, with or without nonlinearities, and with multiple inputs and outputs.
    1. For second-order systems (N = 2), a graphical method called phase-plane analysis can be used on state equations, whether they are linear or nonlinear.

10.1 MATHEMATICAL PRELIMINARIES

This chapter requires some understanding of matrix algebra. Section B.6 introduces basic concepts of matrix algebra, but misses a few needed mathematical concepts, which we present next.

10.1-1 Derivatives and Integrals of a Matrix

Elements of a matrix need not be constants; they may be functions of a variable. For example, if

A=[eβˆ’2tsin⁑teteβˆ’t+eβˆ’2t]\mathbf{A} = \begin{bmatrix} e^{-2t} & \sin t \\ e^t & e^{-t} + e^{-2t} \end{bmatrix}

(10.1)

then the matrix elements are functions of t. Here, it is helpful to denote A by A(t). Next, we define the derivative and integral of A(t).

The derivative of a matrix A(t) (with respect to t) is defined as a matrix whose ijth element is the derivative (with respect to t) of the ijth element of the matrix A. Thus, if

A(t)=[aij(t)]mΓ—n\mathbf{A}(t) = [a_{ij}(t)]_{m \times n}

then

ddt[A(t)]=[ddtaij(t)]mΓ—norAΛ™(t)=[aΛ™ij(t)]mΓ—n\frac{d}{dt}[\mathbf{A}(t)] = \left[\frac{d}{dt}a_{ij}(t)\right]_{m \times n} \quad \text{or} \quad \dot{\mathbf{A}}(t) = [\dot{a}_{ij}(t)]_{m \times n}

Thus, the derivative of the matrix in Eq. (10.1) is given by

AΛ™(t)=[βˆ’2eβˆ’2tcos⁑tetβˆ’eβˆ’tβˆ’2eβˆ’2t]\dot{\mathbf{A}}(t) = \begin{bmatrix} -2e^{-2t} & \cos t \\ e^t & -e^{-t} - 2e^{-2t} \end{bmatrix}

Similarly, we define the integral of A(t) (with respect to t) as a matrix whose ijth element is the integral (with respect to t) of the ijth element of the matrix A:

∫A(t)dt=(∫aij(t)dt)mΓ—n\int \mathbf{A}(t) dt = \left( \int a_{ij}(t) dt \right)_{m \times n}

Thus, for the matrix A in Eq. (10.1), we have

∫A(t)dt=[∫eβˆ’2tdt∫sin⁑dt∫etdt∫(eβˆ’t+2eβˆ’2t)dt]\int \mathbf{A}(t) dt = \begin{bmatrix} \int e^{-2t} dt & \int \sin dt \\ \int e^t dt & \int (e^{-t} + 2e^{-2t}) dt \end{bmatrix}