Question 26.1: Consider a stable 2 × 2 MIMO system having a nominal model g...
Consider a stable 2 × 2 MIMO system having a nominal model given by
G_{o}=\frac{1}{(s+1)^{2}(s+2)}\begin{bmatrix} 2(s+1) & -1 \\ (s+1)^{2} & (s+1)(s+2) \end{bmatrix} (26.2.10)
Choose a suitable matrix Q(s) to control this plant,using the affine parameterization, in such a way that the MIMO control loop is able to track references of bandwidths less than, or equal to, 2[rad/s] and 4[rad/s], in channels 1 and 2, respectively.
We will attempt a decoupled design, i.e. to obtain a complementary sensitivity matrix given by
T_{0}(s)= diag(T_{11}(s), T_{22}(s)) (26.2.11)
where T_{11}(s) and T_{22}(s) will be chosen of bandwidths 2[rad/s] and 4[rad/s], in channels 1 and 2, respectively.
Then, Q(s) must ideally satisfy
Q(s)=[G_{o}(s)]^{-1}T_{o}(s) (26.2.12)
We also note that matrix G_{o}(s) is stable and with zeros strictly inside the LHP. Then the only difficulty to obtain an inverse of G_{o}(s) lies in the need to achieve aproper (biproper) Q. This may look simple to achieve, since it would suffice to add a large number of fast, stables poles to T_{11}(s) and T_{22}(s). However, usually having the relative degrees larger than strictly necessary adds unwanted lag at high frequencies. We may also be interested to obtain a biproper Q(s) to implement anti wind-up architectures (see later in the chapter). Here is where interactors play a useful role. We choose the structure (26.2.8),
Q(s)=[A_{L}(s)]^{-1}\xi _{L}(s)D_{Q}(s) (26.2.8)
from where we see that T_{o}(s) = D_{Q}(s). Hence, the relative degrees of T_{11}(s) and T_{22}(s) will be chosen equal to the degrees of the first and second column of the left interactor for G_{o}(s) respectively. We then follow section §25.4.1 to compute the left interactor ξ_{L}(s). This leads to
\xi _{L}(s)= diag((s+\alpha )^{2}, (s+\alpha )); α ∈ \mathbb{R}^{+} (26.2.13)
Then (26.2.12) can also be written as
Q(s)=[\xi _{L}(s)G_{o}(s)]^{-1} \xi _{L}(s)T_{o}(s) (26.2.14)
Hence, Q(s) is proper if and only if T_{o}(s) is chosen so as to make ξ_{L}(s)T_{o}(s) proper.
Thus, possible choices for T_{11}(s) and T_{22}(s) are
T_{11}(s)=\frac{4}{s^{2}+3s+4}; and T_{22}(s)=\frac{4(s+4)}{s^{2}+6s+16} (26.2.15)
The reader is invited to check that these transfer functions have the required bandwidths.
To obtain the final expression for Q(s), we next need to compute [G_{o}(s)]^{-1}, which is given by
[G_{o}(s)]^{-1}=\frac{s+2}{2s+5}\begin{bmatrix} (s+1)(s+2) & 1 \\ -(s+1)^{2} & 2(s+1) \end{bmatrix} (26.2.16)
from where we finally obtain
Q(s)=[G_{o}(s)]^{-1}T_{o}(s)=\frac{s+2}{2s+5}\begin{bmatrix} \frac{ 4(s+1)(s+2)}{s^{2}+3s+4} & \frac{4(s+4)}{s^{2}+6s+16} \\ \frac{-4(s+1)^{2}}{s^{2}+3s+4} & \frac{8(s+4)(s+1)}{s^{2}+6s+16} \end{bmatrix} (26.2.17)
The above design can be tested using SIMULINK file mimo4.mdl.