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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 futureoutputs.
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

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Is the state-variable description of a filter H z unique?

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Does the state-variable description fully describe the signal flow graph?

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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 M th-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; theyprovide a more detailed description of a signal flow graph than does the transfer function (although not a full description); and they suggesta large class of alternative implementations. They are even more useful in control theory, which is largely based on state descriptionsof systems.

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Source:  OpenStax, Digital signal processing: a user's guide. OpenStax CNX. Aug 29, 2006 Download for free at http://cnx.org/content/col10372/1.2
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