Introduction to State Vectors

Cuthbert Nyack
The Transfer function approach to the analysis of control systems works for low order linear systems. Although limited in scope it provides a convenient approach because of concepts as poles and zeros, root locus and bode plots. Because of this, it is still used for an introductory analysis of control systems.

The State Vector approach can be used for small and large systems as well as for linear and nonlinear systems. In this approach the system is described by a set of first order differential equations as shown below. The state of the system is described by the column vector x, u is the control input, y is the output vector and A is the system matrix. The system matrix is usually derived from the differential equations describing the dynamics of the system, because the transfer function approach has been used so far, we introduce here the use of state vectors derived from the transfer function. One of the features of this approach is that there can be several ways of deriving the differential equations of the system and this is illustrated below for a third order system.  This approach gives the system matrix as a companion matrix. Below the system transfer function is treated as a product of 3 terms.   Below the transfer function is treated as the sum of 3 terms derived from a partial fraction expansion.   The State vector approach makes other approaches to control system design possible. Two of these methods are Pole placement and Optimal control. The diagram below shows how a system is modified to implement pole placement design. When the different components of the state vector are accessible, then each can be used as the source of feedback signals. For the Companion matrix G2(s) and G3(s) are integrators and for the Product case, they are lag factors.

In the State vector approach, stability is determined from the system matrix A. For the system to be stable all eigenvalues of the system matrix A (ie |sI - A|) must lie in the left half of the s plane.

The Following applets show the transient response of the state vector for second and third order systems as well as how the pole placement approach affects the transient response, bode plot and root locus.