09.Design limitations of feedback control

Proof The first inequality follows directly from Corollary 9.17 after adding the simplifying hypotheses. The second inequality follows from straightforward manipulation of the inequality

√π−Θp(ωb)

2 Θp(ωb) ≤ Tmax.

The upshot of the corollary is that if one wishes to make the lower bound on the H -form

√ ∞

of TL smaller than 2, then the closed-loop bandwidth will exceed the location of the real, unstable pole p.

Often in applications one wishes to impose the conditions (9.5) and (9.6) together, with

ω2 > ω1, and noting that (9.6) implies that |SL(iω)| < 1 for |ω| > ω2. The picture is

2

shown in Figure 9.6: one wishes to design the sensitivity function so that it remains below

|SL(iω)|

1

Figure 9.6 Combined sensitivity and complementary sensitivity function restrictions

the shaded area. Thus one will want to minimise the sensitivity function over a range of frequencies, as well as attenuate the complementary sensitivity function for high frequencies. Assuming that z C+ is a zero for RL, computations like those used to prove Corollary 9.16 then give

The implications of this lower bound are examined in a simple case in Exercise E9.4.

9.3 The robust performance problem

So-called “H∞ design” has received increasing attention in the recent control literature. The basic methodology has its basis in the frequency response ideas outlined in Section 8.5.2 and Section 9.2. The idea is that, motivated by Proposition 8.24, one wishes to minimise kSLk∞. However, as we saw in Section 9.2, this is not an entirely straightforward task. Indeed, certain plant features—unstable poles and/or nonminimum phase zeros—can make this task attain a subtle nature. In this section we look at some ways of getting around these

di culties to specify realistic performance objectives. The approach makes contact with the topic of robust stability of Section 7.3, and enables us to state a control design problem, the so-called “robust performance problem.” The resulting problem, as we shall see, takes the form of a minimisation problem, and its solution is the subject of Chapter 15.

We shall continue in this section to use the unity gain feedback loop of Figure 6.25 that we reproduce in Figure 9.7.

Figure 9.7 Unity gain feedback loop for robust performance problem

9.3.1 Performance objectives in terms of sensitivity and transfer functions If you ever thought that it was possible to arbitrarily assign performance specifications to a system, it is hoped that the content of Sections 9.1 and 9.2 have made you think di erently. But perhaps by now one is frightened into thinking that there is simply no way to specify performance objectives that are achievable. This is not true, of course. But what is true is that one needs to be somewhat cagey about how these specifications are stated.

Let us begin by reinforcing our results about the di culty facing us by trying something na¨ıve. According to Proposition 8.24, if we wish to minimise the energy of the error, we should minimise the H∞-norm of SL. Thus, we could try to specify > 0 and then set out to design a controller with the property that kSLk∞ ≤ . This objective may be impossible

setting, we cannot possibly specify a H∞-norm bound for SL which is less than 1.

2.Based upon Proposition 8.22 we have two situations that provide limitations on possible sensitivity reduction.

(a)If RP and RC have no poles in C+ and if either have a zero in C+, then RL is analytic in C+ and has a zero z C+. From Proposition 8.22 we have SL(z) = 1. Thus it follows from the Maximum Modulus Principle (Theorem D.9) that |SL(iω)| ≥ 1 for some ω R.

(b)This is much like the previous situation, except we reflect things about the imag-

inary axis. Thus, if RP and RC have no poles in C− and if either have a zero in C−, then RL is analytic in C− and has a zero z C−. Arguing as above, we see that |SL(iω)| ≥ 1 for some ω R.

3.If the relative degree of RL = RC RP exceeds 1 and if RL is strictly proper, then, from Theorem 9.7 we know that kSLk∞ > 1. Furthermore, this same result tells us that this e ect is exacerbated by RL having unstable poles.

4.If RL is nonminimum phase and if we minimise |SL| over a certain range of frequencies, then the result will be an increase in kSLk∞.

5.The previous point is reiterated in the various corollaries to the Poisson integral formu-

lae given in Theorem 9.13. Here the location of nonminimum phase zeros is explicitly seen in the estimate for the H∞-norm for the sensitivity function.

The above, apart from providing a neat summary of some of the material in Section 9.2, indicates that the objective of minimising |SL(iω)| over the entire frequency range is perhaps not a good objective. What’s more, from a practical point of view, it is an excessively stringent condition. Indeed, remember why we want to minimise the sensitivity function: to reduce the L2-norm of the error to a given input signal. However, in a given application, one will often have some knowledge about the nature of the input signals one will have to deal with. This knowledge may come in the form of, or possibly be converted into the form of, frequency response information. Let us give some examples of how such a situation may arise.

9.20 Examples 1. Suppose that we are interested in tracking sinusoids with frequencies in a certain range with minimal error. Thus we take a class of nominal signals to be sinusoids of the form

{r(t) = a sin ωt | 0 ≤ a ≤ 1} .

Now we bias a subset of these inputs as follows:

Lref = {|Wp(iω)a| sin ωt | 0 ≤ a ≤ 1} ,

where Wp R(s) is a function for which the magnitude of restriction to the imaginary axis captures the behaviour we wish be giving more weight to certain frequencies. With this set of inputs, the maximum error is

sup kek∞ = sup |Wp(iω)SL(iω)|

Note here that the measure of performance we use is the L∞-norm of the error. In the following two examples, di erent measures will be used.

2.One can also think of specifying the character of reference signals by providing bounds on the H2-norm of the signal. Thus we could think of nominal signals as being those of the form

{rnom : [0, ∞) → R | krˆnomk2 ≤ 1} ,

and these are shaped to a set of possible reference signals

Lref = {r : [0, ∞) → R | rˆ = Wprnom, krˆnomk2 ≤ 1} ,

where Wp is a specified transfer function that shapes the energy spectrum of the signal to a desired shape. Simple manipulation then shows that

where we have used part (i) of Theorem 5.21. Again, we have arrived at a specification of the form of kWpSLk∞ < .

3. The above argument can be carried out for nominal reference signals

{rnom : [0, ∞) → R | pow(ˆrnom) ≤ 1} .

Similar arguments, now asking that the power spectrum of the error be minimised, lead in the same way (using part (ix) of Theorem 5.21) to a condition of the form kWpSLk∞ < .

4.Suppose that one knows from past experience that a controller will perform well if the frequency response of the sensitivity function lies below that of a given transfer function. Thus one would have a specification like

Note that all specifications like kWpSLk∞ < are by simple scaling transfered to a condition of the form kWpSLk∞ < 1. This is the usual form for these conditions to take, in practice.

The above examples present, in sort of general terms, possible natural ways in which one can arrive at performance criterion of the form kWpSLk∞ < 1, for some Wp R(s). In making such specifications, one in only interested in the magnitude of Wp on the imaginary axis. Therefore one may as well suppose that Wp has no poles or zeros in C+.1 In this book, we shall alway deal with specifications that come in the form kWpSLk∞ < 1. Note also that

for performance specifications in the case when the loop gain RL is the product of RC with RP . There is nothing in principle stopping us from using these other transfer functions; our choice is motivated by a bald-faced demand for simplicity. Below we shall formulate criterion that use our performance conditions of this section, and these results are complicated if one uses the any of the other transfer functions in place of the sensitivity function.

There is a readily made graphical interpretation of the condition kWpSLk∞ < 1, mirroring the pictures in Figures 7.21 and 7.25. To see how this goes, note the following:

kWpSLk∞ < 1

|Wp(iω)| < |−1 − RL(iω)| , ω R.

1If Wp has, say, a zero z C+, then we can write Wp = (s − z)kW (s) where W (z) 6= 0. One can then

axis. Doing this for all poles and zeros, we see that we can produce a function with no poles or zeros in C+ that has the same magnitude on iR.

Now note that |−1 − RL(iω)| is the distance away from the point −1+i0 of the point RL(iω) on the Nyquist contour for RL. Thus the interpretation we make is that the Nyquist contour at frequency ω remain outside the closed disk centred at −1 + i0 of radius |Wp(iω)|. This is depicted in Figure 9.8. Note that we could just as well have described the con-

|Wp(iω)|

−1 + i0

RL(iω)

Figure 9.8 Interpretation of weighted performance condition

dition kWpSLk∞ < 1 just as we did in Figures 7.21 and 7.25 by saying that the circle of radius |Wp(iω)| and centre RL(iω) does not contain the point −1 + i0. However, the interpretation we have given has greater utility, at least as we shall use it. Also, note that one can make similar interpretations using any of the other performance conditions

|WpTL|∞ , |WpRC SL|∞ , |WpRP SL|∞ < 1.

9.3.2 Nominal and robust performance Now that we have a method for producing frequency domain performance specifications that we can hope are manageable, let us formulate some problems based upon combining this strategy with the uncertainty models of Section 4.5 and the notions of robust stability of Section 7.3. The situation is roughly this: we have a set of plants P and a performance weighting function Wp that gives a performance specification kWpSLk∞ < 1. We wish to examine questions dealing with stabilising all plants in P while also meeting the performance criterion.

The following definition makes precise the forms of stability possible in the framework just described.

and P+(RP , Wu) and for RP P×(RP , Wu) or RP P+(RP , Wu) denote by SL(RP ) and

TL(RP ) the corresponding sensitivity and closed-loop transfer functions, respectively. Let Wp R(s) have no poles or zeros in C+.

¯ ¯

(i) RC provides nominal performance for P×(RP , Wu) (resp. P+(RP , Wu)) relative to Wp if

¯¯

(a)RC provides robust stability for P×(RP , Wu) (resp. P+(RP , Wu)) and

¯

(b)WpSL ∞ < 1.

Thus by nominal performance we mean that the controller provides robust stability, and meets the performance criterion for the nominal plant. However, robust performance requires that we have robust stability and that the performance criterion is met for all plants in the uncertainty set. Thus nominal performance consists of the two conditions

To state a theorem on robust performance we need some notation. For R1, R2 R(s) we define a R-valued function of s by

s 7→R|1(s)| + |R2(s)| .

We denote this function by |R1| + |R2|, making a slight, but convenient, abuse of notation. Although this function is not actually in the class of functions for which we defined the H∞-norm, we may still denote

This notation and the calculations of the preceding paragraph are useful in stating and proving the following theorem.

22/10/2004 9.3 The robust performance problem 387

TL(RP ) the corresponding sensitivity and closed-loop transfer functions, respectively. Let