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17. Linear State Space Models

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We have seen that there are many alternative model formats that can be used for linear dynamic systems. In simple SISO problems, any representation is probably as good as any other. However, as we move to more complex problems (especially multivariable problems) it is desirable to use special model formats. One of the most flexible and useful structures is the state space model. As we saw in Chapter 3, this model takes the form of a coupled set of first order differential (or difference) equations. This model format is particularly useful with regard to numerical computations.

State space models were briefly introduced in Chapter 3. Here we will examine linear state space models in a little more depth for the SISO case. Note, however, that many of the ideas will directly carry over to the multivariable case presented later. In particular, we will study

  • similarity transformations and equivalent state representations
  • state space model properties
    • controllability, reachability and stabililizability
    • observability, reconstructability and detectability
  • special (canonical) model formats

The key tools used in studying linear state space methods are linear algebra and vector space methods. The reader is thus encouraged to briefly review these concepts as a prelude to reading this chapter.

Summary
  • State variables are system internal variables, upon which a full model for the system behavior can be built. The state variables can be ordered in a state vector.
     
  • Given a linear system, the choice of state variables is not unique. However,
    • the minimal dimension of the state vector is a system invariant,
    • there exists a nonsingular matrix which defines a similarity transformation between any two state vectors, and
    • any designed system output can be expressed as a linear combination of the states variables and the inputs.
       
  • For linear, time invariant systems the state space model is expressed in the following equations:
     
    \begin{align}\intertext{continuous time systems} \dot{x}(t)&=\ensuremath{\math... ...h{\mathbf{C}} _\delta x[k]+\ensuremath{\mathbf{D}} _\delta u[k]\\ \end{align}
     
  • Stability and natural response characteristics of the system can be studied from the eigenvalues of the matrix $\ensuremath{\mathbf{A}} $ ( $\ensuremath{\mathbf{A}} _q$, $\ensuremath{\mathbf{A}} _\delta$).
     
  • State space models facilitate the study of certain system properties which are paramount in the solution control design problem. These properties relate to the following questions
    • By proper choice of the input u, can we steer the system state to a desired state (point value)? controllability
    • If some states are or uncontrollable, will these states generate a time decaying component? (stabilizability)
    • If one knows the input, u(t) for $t\geq t_0$, can we infer the state at time t=t0 by measuring the system output, y(t) for $t\geq t_0$? (observability)
    • If some of the states are unobservable, do these states generate a time decaying signal? (detectability)
       
  • Controllability tells us about the feasibility to control a plant.
     
  • Observability tells us about whether it is possible to know what is happening in a given system by observing its outputs.
     
  • The above system properties are system invariants. However, changes in the number of inputs, in their injection points, in the number of measurements and in the choice of variables to be measured may yield different properties.
  • A transfer function can always be derived from a state space model.
     
  • A state space model can be built from a transfer function model. However, only the completely controllable and observable part of the system is described in that state space model. Thus the transfer function model might be only a partial description of the system.
     
  • The properties of individual systems do not necessarily translate unmodified to composed systems. In particular, given two systems completely reachable, observable, controllable and reconstructible, their cascade connection:
    • is not completely observable if a pole of the first system coincides with a zero of the second system (pole-zero cancellation),
    • is not detectable if the pole-zero cancellation affects an unstable pole,
    • is not completely controllable if a zero of the first system coincides with a pole of the second system (zero-pole cancellation), and
    • is not stabilizable if the zero-pole cancellation affects a NMP zero
       
  • this chapter provides a foundation for the design criteria which states that one should never attempt to cancel unstable poles and zeros.