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168 CHAPTER 4. DEMONSTRATION EXAMPLES
4.2.5 D-K Iteration
We will now perform one D-K iteration to generate the controller Kmu. Significant
robustness and performance improvement is achieved with only one iteration.
Transfer functions are fit to the D-scales from the previous robust performance µ test.
Here we preselect an order of 2 for each D-scale. This has been found to give a
satisfactory result.
[Dsys,Dinvsys] = musynfit(D,blk,nmeas,ncon,sens,[],2)
Now a new weighted interconnection is formed by pre- and post-multiplying by the
D-scale approximations. A second H
∞
design is performed to get Kmu.
Pd = Dsys*P*Dinvsys
glimits = [0;20]
Kmu = hinfsyn(Pd,nmeas,ncon,glimits,{tol=0.1,epr=1e-10,epp=1e-4})
Test bounds: 0.0000 < gamma <= 20.0000
gamma Hx
eig X eig Hy eig Y eig nrho xy p/f
20.000 2.9e-02 3.5e-10 1.2e-01 -1.2e-15 0.0001 p
10.000 2.9e-02 3.3e-10 1.2e-01 -3.8e-16 0.0006 p
5.000 2.9e-02 3.7e-10 1.2e-01 -3.7e-16 0.0022 p
2.500 2.9e-02 3.1e-10 1.2e-01 -3.7e-16 0.0090 p
1.250 2.9e-02 3.6e-10 1.2e-01 -1.8e-15 0.0364 p
0.625 2.8e-02 3.4e-10 1.2e-01 -2.1e-16 0.1518 p
0.312 2.8e-02 3.8e-10 1.2e-01 -2.2e-29 0.7425 p
0.156 2.1e-02 -4.3e+06 1.5e-01 -2.2e-17 18.9961 f
0.234 2.6e-02 3.6e-10 1.3e-01 -2.3e-16 1.7666 f
0.273 2.7e-02 3.8e-10 1.2e-01 -9.8e-17 1.0735 f
0.293 2.7e-02 3.9e-10 1.2e-01 -1.9e-16 0.8828 p
Gamma value achieved: 0.2930
Note that this value of γ is significantly lower than even the µ value from the Ghinf
closed loop system. Again, both the controller and closed loop system are stable.
A frequency response of Kmu is calculated and plotted.