Question 7.3: plant has a nominal model given by Go(s) = 1/s−1. The contro...
A plant has a nominal model given by G_{o}(s) = \frac{1}{s-1}. The control aim is to track a sinusoidal reference of 2[rad/s] of unknown amplitude and phase. Design a controller which stabilizes the plant and which provides zero steady state control error.
Polynomial pole assignment will be used. We first consider the order of the polynomial A_{cl}(s) to be specified. The requirement of zero steady error for the given reference implies that
S_{o}(\pm j2) = 0 \Longleftrightarrow G_{o}(\pm j2)C(\pm j2) = \infty (7.2.31)
The constraint in equation (7.2.31) can be satisfied for the given plant if and only if the controller C(s) has poles at \pm j2 . Thus
C(s) = \frac{P(s)}{L(s)} = \frac{P(s)}{(s^{2}+4)L_{ad}(s)} (7.2.32)
This means that A_{cl}(s) can be arbitrarily specified if its degree is , at least, equal to 3. We choose A_{cl}(s)= (s^{2}+ 4s + 9)(s + 10). This leads to L_{ad}(s) = 1 and P(s) = p_{2}s^{2} + p_{1}s + p_{0} The pole assignment equation is then
(s − 1)( s^{2} + 4 )+ p_{2}s^{2} + p_{1}s + p_{0}= (s^{2} + 4s + 9)(s + 10) (7.2.33)
On expanding the polynomial products and equating coefficients we finally obtain
P(s) = 15 s^{2} + 45s + 94 \Longrightarrow C(s) = \frac{15 s^{2}+ 45s+ 94}{s^{2}+ 4} (7.2.34)
For a more numerically demanding problem , the reader may wish to use the MATLAB program paq.m in the accompanying diskette.