FIGURE 46.1 State variable compensator employing full-state feedback in series with a full-state observer.

 

A key question that arises in the design of state variable compensators is whether or not all the poles of the closed-loop system can be arbitrarily placed in the complex plane. Recall that the poles of the closed-loop system are equivalent to the eigenvalues of the system matrix in state variable format. As we shall see, if the system is controllable and observable, then we can accomplish the design objective of placing the poles precisely at the desired locations to meet the performance specifications.

Full-state feedback design commonly relies on pole-placement techniques. It is important to note that a system must be completely controllable and completely observable to allow the

flexibility to place all the closed-loop system poles arbitrarily.

 

Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. The system must be considered controllable in order to implement this method.

 

If the closed-loop input-output transfer function can be represented by a state space equation, see State space (controls),

then the poles of the system are the roots of the characteristic equation given by

 

Full state feedback is utilized by commanding the input vector . Consider an input proportional (in the matrix sense) to the state vector,

.

Substituting into the state space equations above,

The roots of the FSF system are given by the characteristic equation, . Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.

Example 46.1

Consider a control system given by the following state space equations

The uncontrolled system has closed-loop poles at s = − 1 and s = − 2. Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at

s = − 1 and s = − 5. The desired characteristic equation is then s ² + 6 s + 5 = 0.

Following the procedure given above, , and the FSF controlled system characteristic equation is

.

Upon setting this characteristic equation equal to the desired characteristic equation, we find

.

Therefore, setting forces the closed-loop poles to the desired locations, affecting the response as desired.

 

Determining the gain matrix K is the objective of the full-state feedback design procedure.

The beauty of the state variable design process is that the problem naturally separates into a full-state feedback component and an observer design component.

These two design procedures can occur independently, and in fact, the separation principle provides the proof that this approach is optimal. We will show later that the stability of the closed-loop system is guaranteed if the full-state feedback control law stabilizes the system (under the assumption of access to the complete state) and the observer is stable (the tracking error is asymptotically stable).

Observer design will be discussed in Lecture 4.7.

 

Example 46.2

Let’s consider a 3^rd order control system which is described by the following differential equation

We may choose the following state variables

Then we may describe this system in matrix form

and

If the state variable feedback matrix is

and

then the closed-loop system is

The state feedback matrix is

and the characteristic equation is

(46.1)

If we seek a rapid response with a low overshoot, we choose a desired characteristic equation such as (see Equation 5.18 and Table 5.2)

We choose = 0.8 for minimal overshoot and to meet the settling time requirement.

If we want a settling time (with a 2% criterion) equal to 1 second, then

If we choose = 6, the desired characteristic equation is

(46.2)

Comparing Equations (46.1) and (46.2) yields the three equations

Therefore, we require that = 9.4, = 79.1, and =170.8. The step response has no overshoot and a settling time of 1 second, as desired.

 

Example 46.3

Let’s consider a control system with full-state feedback control (output variable ).

This system is using a DC motor which is controlled by an exciting circuit.

The transfer function of this DC motor is as follows

where = the complex transfer coefficient of the DC motor.

We suppose that =1 and =5.

As you may see from Fig.46.2 the control system has 3 feedback loops: on position, motor speed, and excitation current.

Suppose, that the coefficient of feedback loop on position is equal to -1 (look at Fig.46.3).

Fig.46.2 A control system with full-state feedback control (output variable ).