Signals and Systems

10.2 STATE MODEL OF A FIRST-ORDER SYSTEM

Let us consider the following difference equation:

 

y(k + 1) + 4y(k) = u(k)     (10.2)

 

where y and u represent the output and input respectively.

Let us start from y (k + 1). We get y (k) using a delay element shown in Fig. 10.2.

 

 

Fig. 10.2

In Fig. 10.3, x₁(k) represents y(k). From Eq. (10.1) we get

 

 

Fig. 10.3

 

y(k + 1) = u(k) − 4y(k) = −4x₁(k) + u(k)     (10.3)

 

Figure 10.3 shows the block diagram of Eq. (10.3).

The state equation and output equation are

 

y(k + 1) = −4x₁(k) + u(k)     (10.4) ...