8.0: Prelude to State Variable Models

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This chapter discusses algebraic methods to analyze the state variable models of dynamic systems. These models comprise a set of first-order linear differential equations that describe the system behavior using system state variables. A common choice includes the natural variables associated with the energy storage elements present in the system. Alternate variables in equal number can also be selected.

The state equations include a set of coupled first-order ODE’s that describe the behavior of the system. The state equations can be collectively integrated using a matrix exponential, i.e., the state-transition matrix, as an integrating factor. The state-transition matrix contains natural modes of system response and plays a fundamental role in the evolution of system trajectories. The general solution to the homogeneous state equations is a weighted sum of the columns of the state-transition matrix.

Given a state variable model of the system, a transfer function representation can be obtained by applying the Laplace transform. Generally, the degree of the denominator polynomial in the transfer function model equals the number of state variables used to represent it. A pole-zero cancellation in the transfer function would, however, cause some of the response modes to be absent from the input-output system description.

A given transfer function model can be realized into a state variable model in multiple ways. Different choices of state variables accord different structures to the system matrix. Specific realization structures may be preferred for ease of computing the system response or determining its stability characteristics.

The modal realization reveals the natural modes of system response. The controller realization facilitates the controller design using state variable methods. The diagonal representation decouples the system variables into a set of independent first-order ordinary differential equations (ODEs) that can be easily integrated. The state variable vectors for these alternate models are related through bilinear transformations.

This chapter describes the analysis techniques for state variable models of the continuous-time systems. The discrete-time system models are covered later in Chapter 8. The analysis is restricted to the single-input single-output (SISO) systems, that exhibit a rational transfer function, \(G\left(s\right)\). The methods, however, can be generalized to include multi-input multi-output (MIMO) systems.