# Robust Design Using Loop-Shaping

• Richard Alden Hyde
Part of the Advances in Industrial Control book series (AIC)

## Abstract

Loop-shaping is well accepted and widely used for scalar feedback design. For a scalar system it is a straightforward task to convert closed-loop disturbance rejection requirements into loop-shape requirements. However, when the approach is generalised for multivariable systems this task is somewhat more complicated. The problem arises in that different loops may have very different gains which in turn means that the directionality of references, disturbances and plant uncertainty become very important when specifying and evaluating stability and performance. An effect of different loop gains is that a compensator used to give good properties at one break point in the loop may give poor properties at another. To see this, consider Figure 4.1 where a compensator K is to be designed by loop-shaping to give good rejection of disturbances d 1 and d 2. Consider shaping GK i.e. we break the loop at point B. If the condition number of G, defined as
$$k(G)\,\underline{\underline \Delta } \,\frac{{\overline \sigma \left( G \right)}}{{\underline \sigma \left( G \right)}}$$
(4.1)
is large with $$\underline \sigma \left( G \right)\, < < \,1\,and\,\overline \sigma \left( G \right)\, > > \,1$$ over the frequency range for which disturbances are to be rejected, then designing K to increase gain in the directions for which the gain of G is small so that $$\underline \sigma \left( {GK} \right)\, > > \,1$$ for this frequency range will give good rejection of d 2 disturbances. If we now consider breaking the loop at point A, it can be seen that the rejection of d 1 is dictated by the gain of KG. If K has been designed only to increase gain in certain directions, KG may have low gain in certain directions i.e. $$\overline \sigma \left( {KG} \right)\, < < \,1$$ over the frequency range for which disturbances are to be rejected. Hence rejection of disturbances at A may be much poorer than rejection of disturbances at B. In the case that $$k\left( G \right)\, \simeq \,1$$ and K is chosen such that $$k\left( K \right)\, \simeq \,1,\,then\,k\left( {GK} \right)\, \simeq \,k\left( {KG} \right)\, \simeq \,1$$, and properties at one loop breaking point will reflect those at another.

### Keywords

Torque Fractional Distillation Lution Hyde Rounded