33 Discrete-Time Compensation Technique for Real-Time Hybrid Simulation 353 GFF z 1 D A z 1 BC.z 1/ H z 1 (33.9) H z 1 is the FIR and defined as below H z 1 Dh 1 Ch2z 1 C Chnz nC1 (33.10) where h1, h2, : : : hn are n unknown parameters that are determined by an optimization scheme. Different number of parameters (n) can be chosen to design this compensator. The delay compensation performance up to a desired frequency (cut-off frequency fc) range will be guaranteed by the Optimization Schemes explained in the next section. Frequency range from 0 Hz to the cut-off frequency is called passband and the frequency range beyond the cut-off frequency is called stopband. Compensated system transfer function can be written as TFc z 1 DGFF z 1 TF z 1 DB 1 z 1 H z 1 (33.11) 33.3 Optimization Schemes Three optimization schemes, namely least square (LS), weighted least square (WLS), and minimax (MM), are designed to find unknown parameters h1, h2, : : : hn in the FIR. The objective functions of these three optimization schemes are to minimize the magnitude of the compensated system transfer function, TFc z 1 , defined in Eq. (33.11). For WLS scheme, the objective function is defined as min( X i wi:kTFc.i/ 1k 2) ; wi 1 for i 2passband wi D1 for i 2stopband (33.12) where i indicates discretized points in frequency domain (e.g., Fourier transform point); TFc(i) is the value of the compensated system transfer function (Eq. 33.11) at the ithpoint; wi represents the weight assigned to the correspondingith point. Awi larger than 1 indicates higher importance is given to the corresponding error normTFc.i/ 1; jj jj represents modulus of complex number. In this study, wi D1000 is chosen for the frequency points located in the passband, whereas wi D1 for the points located in the stopband. The LS scheme has the same objective function as the WLS scheme does, except the weights (wi) for all the frequency points are assigned as equal to 1. For MM scheme, the objective function is given as min max i 2 stopbandk TFc.i/k (33.13) For all the above schemes, same constraints are applied during the optimization process: 1. Steady-state gain should equal one TFc .z D1/ D1 (33.14) 2. Magnitude of the compensated system transfer function in passband range should be one sXi .TFc.i/ 1/ 2 D0 (33.15) 3. Phase of compensated system transfer function in pass-band range should be zero sXi fargŒTFc.i/ g 2 D0 (33.16) Where arg calculates the arguments of a complex number.
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