Modern Control System Theory and Design, 2nd Edition

2.22.  STATE-VARIABLE DIAGRAM

The state-variable diagram provides a physical picture that is useful in understanding the state-variable concept. In addition, the differential equations relating the state variables are easily obtained by inspection of the diagram. A state-variable diagram consists of integrators, summing devices, and amplifiers. Outputs from the integrators denote the state variables. It should be noted that the state-variable diagram is the same as an analog computer simulation diagram [15].

Example 1. As an example of determining the state-variable diagram, consider a system whose transfer function is given by

Dividing numerator and denominator by s³, to obtain the result in terms of integrators:

Defining the error node of the control system to be E(s),

Eq. (2.223) may be rewritten as follows:

From Eq. (2.225) and the relation

the state-variable diagram can easily be obtained as indicated in Figure 2.30*. The state variables are indicated in the diagrams as x₁(t), ...