State-Variable Representation of Discrete-Time Systems 1.1 2003/07/14 15:34:31 GMT-5 2004/12/27 12:48:51.204 US/Central Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Kyle Clarkson kclarks@rice.edu discrete-time systems state state space state variable representation state variable transformation state variables State-variable, or state-space, representations provide a general description of all linear, time-invariant (LTI) systems that is useful both for their analysis and for generating alternate forms with more convenient implementation or with less sensitivity to quantization.

State and the State-Variable Representation State the minimum additional information at time n, which, along with all current and future input values, is necessary to compute all future outputs. Essentially, the state of a system is the information held in the delay registers in a filter structure or signal flow graph. Any LTI (linear, time-invariant) system of finite order M can be represented by a state-variable description x n 1 A x n B u n y n C x n D u n where x is an x M 1 "state vector," u n is the input at time n, y n is the output at time n; A is an x M M matrix, B is an x M 1 vector, C is a x 1 M vector, and D is a x 1 1 scalar. One can always obtain a state-variable description of a signal flow graph. 3rd-Order IIR y n a 1 y n 1 a 2 y n 2 a 3 y n 3 b 0 x n b 1 x n 1 b 2 x n 2 b 3 x n 3

x 1 n 1 x 2 n 1 x 3 n 1 0 1 0 0 0 1 a 3 a 2 a 1 x 1 n x 2 n x 3 n 0 0 1 u n y n a 3 b 0 a 2 b 0 a 1 b 0 x 1 n x 2 n x 3 n b 0 u n

Is the state-variable description of a filter H z unique? Does the state-variable description fully describe the signal flow graph?

State-Variable Transformation Suppose we wish to define a new set of state variables, related to the old set by a linear transformation: q n T x n , where T is a nonsingular x M M matrix, and q n is the new state vector. We wish the overall system to remain the same. Note that x n T q n , and thus ⇒ x n 1 A x n B u n T q n A T q n B u n q n T A T q n T B u n ⇒ y n C x n D u n y n C T q n D u n This defines a new state system with an input-output behavior identical to the old system, but with different internal memory contents (states) and state matrices. q n A ^ q n B ^ u n y n C ^ q n D ^ u n A ^ T A T , B ^ T B , C ^ C T , D ^ D These transformations can be used to generate a wide variety of alternative stuctures or implementations of a filter.

Transfer Function and the State-Variable Description Taking the z transform of the state equations Z x n 1 Z A x n B u n Z y n Z C x n D u n ⇓ z X z A X z B U z X z is a vector of scalar z-transforms X z X 1 z X 2 z … Y z C X n D U n ⇒ z I A X z B U z X z z I A B U z so Y z C z I A B U z D U z C z I B D U z and thus H z C z I A B D Note that since z I A ± ( z I - A ) red z I A , this transfer function is an Mth-order rational fraction in z. The denominator polynomial is D z z I A . A discrete-time state system is thus stable if the M roots of z I A (i.e., the poles of the digital filter) are all inside the unit circle. Consider the transformed state system with A ^ T A T , B ^ T B , C ^ C T , D ^ D : H z C ^ z I A ^ B ^ D ^ C T z I T A T T B D C T T z I A T T B D C T T z I A T T B D C z I A B D This proves that state-variable transformation doesn't change the transfer function of the underlying system. However, it can provide alternate forms that are less sensitive to coefficient quantization or easier to analyze, understand, or implement. State-variable descriptions of systems are useful because they provide a fairly general tool for analyzing all systems; they provide a more detailed description of a signal flow graph than does the transfer function (although not a full description); and they suggest a large class of alternative implementations. They are even more useful in control theory, which is largely based on state descriptions of systems.