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State-Variable Representation of Discrete-Time Systems

Module by: Douglas L. Jones

Summary: 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.

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State and the State-Variable Representation

Definition 1: State
the minimum additional information at time nn, 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.

fact:

Any LTI (linear, time-invariant) system of finite order MM can be represented by a state-variable description xn+1=Axn+Bun x n 1 A x n B u n yn=Cxn+Dun y n C x n D u n where xx is an Mx1 x M 1 "state vector," un u n is the input at time nn, yn y n is the output at time nn; AA is an MxM x M M matrix, BB is an Mx1 x M 1 vector, CC is a 1xM x 1 M vector, and DD is a 1x1 x 1 1 scalar.

One can always obtain a state-variable description of a signal flow graph.

Example 1: 3rd-Order IIR

yn=- a 1 yn−1− a 2 yn−2− a 3 yn−3+ b 0 xn+ b 1 xn−1+ b 2 xn−2+ b 3 xn−3 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

Figure 1
Figure 1 (fig1State-Space.png)

x 1 n+1 x 2 n+1 x 3 n+1=010001- a 3 - a 2 - a 1 x 1 n x 2 n x 3 n+001un 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 yn=- a 3 b 0 - a 2 b 0 - a 1 b 0 x 1 n x 2 n x 3 n+ b 0 un 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

Exercise 1

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

Exercise 2

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: qn=Txn q n T x n , where TT is a nonsingular MxM x M M matrix, and qn q n is the new state vector. We wish the overall system to remain the same. Note that xn=T-1qn x n T q n , and thus xn+1=Axn+Bun⇒T-1qn=AT-1qn+Bun⇒qn=TAT-1qn+TBun ⇒ 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 yn=Cxn+Dun⇒yn=CT-1qn+Dun ⇒ 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. qn= A ^ qn+ B ^ un q n A ^ q n B ^ u n yn= C ^ qn+ D ^ un y n C ^ q n D ^ u n A ^ =TAT-1 A ^ T A T , B ^ =TB B ^ T B , C ^ =CT-1 C ^ C T , D ^ =D 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 zz transform of the state equations Zxn+1=ZAxn+Bun Z x n 1 Z A x n B u n Zyn=ZCxn+Dun Z y n Z C x n D u n ⇓ ⇓ zXz=AXz+BUz z X z A X z B U z

Note:

Xz X z is a vector of scalar zz-transforms XzT= X 1 z X 2 z… X z X 1 z X 2 z …
Yz=CXn+DUn Y z C X n D U n zI−AXz=BUz⇒Xz=zI−A-1BUz ⇒ z I A X z B U z X z z I A B U z so
Yz=CzI−A-1BUz+DUz=C-zI-1B+DUz Y z C z I A B U z D U z C z I B D U z (1)
and thus Hz=CzI−A-1B+D H z C z I A B D Note that since zI−A-1=±det ( z I - A ) red TdetzI−A z I A ± ( z I - A ) red z I A , this transfer function is an MMth-order rational fraction in zz. The denominator polynomial is Dz=detzI−A D z z I A . A discrete-time state system is thus stable if the MM roots of detzI−A z I A (i.e., the poles of the digital filter) are all inside the unit circle.

Consider the transformed state system with A ^ =TAT-1 A ^ T A T , B ^ =TB B ^ T B , C ^ =CT-1 C ^ C T , D ^ =D D ^ D :

Hz= C ^ zI− A ^ -1 B ^ + D ^ =CT-1zI−TAT-1-1TB+D=CT-1TzI−AT-1-1TB+D=CT-1T-1-1zI−A-1T-1TB+D=CzI−A-1B+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 (2)
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.

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