1.3 State-variable representation of discrete-time systems

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.

One can always obtain a state-variable description of a 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: $q(n)=Tx(n)$ , 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 $x(n)=T^{(-1)}q(n)$ , and thus $$(x(n+1)=Ax(n)+Bu(n), T^{(-1)}q(n)=AT^{(-1)}q(n)+Bu(n), q(n)=TAT^{(-1)}q(n)+TBu(n))$$ $$(y(n)=Cx(n)+Du(n), y(n)=CT^{(-1)}q(n)+Du(n))$$ 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. $$q(n)=\hat{A}q(n)+\hat{B}u(n)$$ $$y(n)=\hat{C}q(n)+\hat{D}u(n)$$ $\hat{A}=TAT^{(-1)}$ , $\hat{B}=TB$ , $\hat{C}=CT^{(-1)}$ , $\hat{D}=D$

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 $$Z(x(n+1))=Z(Ax(n)+Bu(n))$$ $$Z(y(n))=Z(Cx(n)+Du(n))$$ $$$$ $$zX(z)=AX(z)+BU(z)$$

Consider the transformed state system with $\hat{A}=TAT^{(-1)}$ , $\hat{B}=TB$ , $\hat{C}=CT^{(-1)}$ , $\hat{D}=D$ :

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.