## 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 ωc) between the above two frequency ranges, and below the crossover frequency ωc 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 < ωc).

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 ωc [12]. For instance, your 0 dB crossover ωc must be greater than the magnitude of any plant right-half-plane poles and less than the magnitude of any right-half-plane zeros.

`$\underset{\mathrm{Re}\left({p}_{i}\right)>0}{\mathrm{max}}|{p}_{i}|<{\omega }_{c}<\underset{\mathrm{Re}\left({z}_{i}\right)>0}{\mathrm{min}}|{z}_{i}|.$`

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.