chapter 9 (state space techniques)

STATE-SPACE REPRESENTATION

CHAPTER 9

Objective• Students should be able to:

- Find a mathematical model, called a state-space representation, for a linear, time-invariant system

- Convert between transfer function and state-space models

5.1 THE GENERAL STATE-SPACE REPRESENTATION

• A system is represented in state space by the following equations:

As an example, for a linear, invariant, 2nd order with single input, v(t) the state equation could be the following:

x1 and x2 are the state variables.

If there is a single input, the output equation could take on the following term:

5.2 CONVERTING A TRANSFER FUNCTION TO STATE SPACE

• One advantage of the state-space representation is that it can be used for the simulation of physical systems on the digital system.

• Thus, if we want to simulate a system that is represented by a transfer function, we must first convert the transfer function representation to state space.

And differentiating both side yields

SUMMARY

Solutions:

Separate the system into two cascaded blocks, as shown in Figure 5.4 (b). The first block contains the denominator, and the second block contains the numerator.Find the state equations for the block containing the denominator. The state equation is same with Ex 5.3 except that the system’s input matrix is 1. Hence the state equation is :

We can produce an equivalent block diagram as shown in Figure 5.2 (c).