21.6: Robust Disturbance Rejection (SISO)

As shown earlier, the disturbance rejection requirement could be converted to a robust stability problem with two blocks of uncertainty, as in Figure 21.2, where �1 and �2 are SISO stable systems. Hence 6� is the set of 2 � 2 diagonal complex matrices (which result from evaluating � at each frequency).

Now, since this is a two-block problem, it should be possible to �nd � by in�mizing �max(D;1MD). We have D � diag(d1� d2), so that

 

with the \pure" robust stability requirement performance requirement on the lower right. Setting � � d2�d1 and �xing !, and taking the de�nition of A(�) from (21.5), we have

 

Now, for nominal performance, we require that

 

For robust stability, we need

 

For robust performance, the necessary and su�cient condition is

 

A bit of algebra yields

 

from which we have

 

This minimum occurs at

 

which is not equal to 1 in general, so that sup! � � kMk1. In other words, � is a less conservative measure than k�k1 in this case.

Once again, there is a graphical interpretation of the SISO robust disturbance rejection problem, in terms of the Nyquist criterion. From (21.12), we have

 

Letting L(j!) represent the nominal loop gain P0K(j!), this can be rewritten as:

 

Graphically, we can represent this at each frequency ! as a circle centered at ;1 of radius jW2j, and a second circle centered at L(j!) of radius jW1L(j!)j. Robust performance will be achieved as long as the two circles never intersect.

Loop-shaping Revisited

Loop-shaping is a well-established method of control design that concentrates on the frequency-domain characteristics of the open-loop transfer function L � P0K. Based primarily on design experience, there are certain characteristics of the loop transfer function that translate into desirable control performance. Other open-loop characteristics are known by experience to result in undesirable or unpredictable behavior. This method di�ers from �-synthesis and H1 methods, which concentrate on optimizing the characteristics of the closed-loop transfer function. Since, presumably, a controller with good behavior designed by loop-shaping should be similar in some way to a controller designed by more recent methods, it is of interest to look for parallels in the heuristic rules of loop-shaping and the more methodical methods of �-synthesis and H1.

Identifying the sensitivity and complementary sensitivity functions from (21.14), we can write the RP requirement as

 

Model uncertainty typically increases with frequency, so it is important that the complementary sensi- tivity function decreases with increasing frequency. For disturbance rejection, which is typically most critical over a low frequency range, we require that S(j!) remain small. The weighting functions W1 and W2 are designed to re�ect this, and so might take on the form of Figure 21.5. Normally, at low

 

frequency, L(j!) �� 1 and at high frequency, L(j!) �� 1. Now,

 

so that at low frequency, T0 � 1 and S0 � 1�L. Thus we can approximate the RP requirement at the low end as:

 

At high frequency, the approximation is T0 � L and S0 � 1, which leads to:

 

These constraints are summarized in Figure 21.6, which also notes another design rule, which is that the 0 dB crossing should occur at a slope no more negative than -40 dB per decade. If W1 and W2 do not overlap signi�cantly in frequency, then the upper and lower bounds reduce to jW2j and 1�jW1j, respectively.

Example \(\PageIndex{21.2}\) (Loop Shaping)

Assume P0 is minimum phase stable with relative degree 1. Designing a controller by shaping the loop gain L � P0K is not a�ected by P0� just the relative degree is needed.

 

Suppose the multiplicative uncertainty is described by

 

i.e., the multiplicative perturbations of the plant are upper bounded by W1(j!) at each frequency.

The objective is to track sinusoidal signals at the reference input in the frequency range [0� 1] rad�s. We would like to make the tracking error small� however, we do not know yet by how much. Let W2(j!) have the following frequency response

Note that this may not correspond to a stable W2(s)� however, this does not a�ect the resulting loop shape. We are going to exhibit the design by trial and error. Let

 

At high frequency, ! � 20,

 

If we pick c � 1, then the largest value for b such that the above is satis�ed is b � 20. Hence

 

At low frequency, ! � 1, j

 

Since jL(j!)j is decreasing and jW1(j!)j is increasing in the range [0� 1], the largest a can be solved for:

 

which implies that a � 13:15. Checking the RP condition

 

which implies RP is achieved and the tracking error is smaller than 1�13:15 in the range [0� 1]. If a better performance is desired, a possibly more complicated L needs to be used.

The discussion in this chapter has focused on perturbations that are arbitrary dynamic systems. This alowed us to think of any class of structured perurbations as sets of arbitrary (structured) matrices at each frequency point. These matrices correspond to evaluating the dynamic system at a given frequency.

In practical applications, some perturbations may be static and not dynamic. These arise in problems with real parameter uncertainties. We can still proceed as before and transform such problems to the general M-� diagram. In this case, � will have a combination of both static and dynamic perturbations. � for such a class can be de�ned as before, and it will provide a necessary and su�cient condition for robust stability.

The main issue here is computing a good upper bound for �. Of course, we can always embed this class of perturbations in a larger class containing dynamic perturbations and use D-scaling to obtain an upper bound. This, however, gives conservative conditions. Computing non-conservative upper bounds of � for such perturbations remains an active area of research.