Fine-Tuning the Target Loop Shape to Meet Design Goals

If your first attempt at loopsyn design does not achieve everything you wanted, you will need to readjust your target desired loop shape Gd. Here are some basic design tradeoffs to consider:

• Stability Robustness. Your target loop Gd should have low gain (as small as possible) at high frequencies where typically your plant model is so poor that its phase angle is completely inaccurate, with errors approaching ±180° or more.

• Performance. Your Gd loop should have high gain (as great as possible) at frequencies where your model is good, in order to ensure good control accuracy and good disturbance attenuation.

• Crossover and Roll-Off. Your desired loop shape Gd should have its 0 dB crossover frequency (denoted ω_c) between the above two frequency ranges, and below the crossover frequency ω_c it should roll off with a negative slope of between –20 and –40 dB/decade, which helps to keep phase lag to less than –180° inside the control loop bandwidth (0 < ω < ω_c).

Other considerations that might affect your choice of Gd are the right-half-plane poles and zeros of the plant G, which impose fundamental limits on your 0 dB crossover frequency ω_c [12]. For instance, your 0 dB crossover ω_c must be greater than the magnitude of any plant right-half-plane poles and less than the magnitude of any right-half-plane zeros.

$$\underset{\mathrm{Re}\left(p_i\right)>0}{\max }\left|p_i\right|<{\omega }_c<\underset{\mathrm{Re}\left(z_i\right)>0}{\min }\left|z_i\right|.$$

If you do not take care to choose a target loop shape Gd that conforms to these fundamental constraints, then loopsyn will still compute the optimal loop-shaping controller K for your Gd, but you should expect that the optimal loop L = G*K will have a poor fit to the target loop shape Gd, and consequently it might be impossible to meet your performance goals.