**Subject :**Advance Control System

## Pole Placement for Plants in Phase-Variable Form

**Pole Placement for Plants in Phase-Variable Form**

To apply pole-placement methodology to plants represented in phase-variable form,

We take the following steps:

1. Represent the plant in phase-variable form.

2. Feedback each phase variable to the input of the plant through a gain, ki.

3. Find the characteristic equation for the closed-loop system represented in Step 2.

4. Decide upon all closed-loop pole locations and determine an equivalent characteristic equation.

5. Equate like coefficients of the characteristic equations from Steps 3 and 4 and solve for ki.

Following these steps, the phase-variable representation of the plant is given by Eq. (1), with

The k1’s are the phase variables' feedback gains. Using Eq. (2a) with Eqs. (3) ()and (6), the system matrix, A - BK, for the closed-loop system is

Since Eq. (7) is in phase-variable form, the characteristic equation of the closedloop system can be written by inspection as

(8)

Notice the relationship between Eqs. (4) and (8). For plants represented in phase-variable form, we can write by inspection the closed-loop characteristic equation from the open-loop characteristic equation by adding the appropriate ki to each coefficient. Now assume that the desired characteristic equation for proper pole placement is Now that we have found the denominator of the closed-loop transfer function, let us find the numerator. For systems represented in phase-variable form, we learned that the numerator polynomial is formed from the coefficients of the output coupling matrix, C. Since Figures 2(A) and (b) are both in phase-variable form and have the same output coupling matrix, we conclude that the numerators of their transfer functions are the same.

(9)

Where the di’s are the desired coefficients. Equating Eq. (8) and (9), we obtain