Consider the feedback control loop of Figure 3.14, where the open-loop system,L, is a proper transfer function of the complex variablez.
b - i- -b
6
FIGURE 3.14. Discrete-time feedback control system.
As in (3.8), the sensitivity and complementary sensitivity functions of the loop in Figure 3.14 are defined as
S(z) = 1
1+L(z) and T(z) = L(z)
1+L(z), (3.43) respectively. In common with the continuous-time case, the sensitivity and complementary sensitivity functions for the discrete case satisfy interpo- lations constraints imposed by zeros and poles of the open-loop system.
Indeed, ifLis free of unstable pole-zero cancelations,11thenShas zeros at the unstable open-loop plant poles andThas zeros at the nonminimum phase open-loop plant zeros; i.e., Lemma 3.1.3 holds with +replaced by the region outside the open unit disk, which we denote by c.
11Recall from Chapter 2 that, for discrete-time systems, a transfer function is nonminimum phase if it has zeros outside the open unit disk, , and it is unstable if it has poles outside . Thus, is free of unstable pole-zero cancelations if there are no cancelations of zeros and poles outside between the plant and controller whose cascade connection form .
76 3. SISO Control For the purpose of deriving Poisson integral constraints, we need to extract fromSandT all the zeros outside theclosed12unit disk — denoted by c. Letqi,i=1, . . . , nq, be the zeros ofLin c, andpi,i=1, . . . , np, be the poles ofLin c, all counted with multiplicities. It follows from our previous discussion that theqi’s are the zeros ofT in c, and thepi’s are
the zeros ofSin c. We introduce thediscrete Blaschke productsof the zeros ofSandT, given by
BS(z) =
np
Y
i=1
pi
|pi| z−pi
1−piz , and BT(z) =
nq
Y
i=1
qi
|qi|
z−qi
1−qiz . (3.44) We will defineBS(z) = 1ifL is stable, andBT(z) = 1 ifLis minimum phase. Similar to their continuous-time counterpart, the discrete Blaschke products are “all-pass” functions, since their magnitude is constant and equal to one on the unit circle|z| = 1. Using (3.44), and extracting the zeros at infinity in a similar fashion as was done in (2.7) in Chapter 2, we can factor the open-loop transfer function,L, as
L(z) =B−1S (z)BT(z)L(z)˜ z−δ, (3.45) where ˜Lis a stable, minimum-phase transfer function, having relative de- gree zero. It is then easy to see thatSandTin (3.43) can be factored as
S(z) =S(z)B˜ S(z),
T(z) =T˜(z)BT(z)z−δ , (3.46) where ˜Sand ˜Thave no zeros in the region outside the closed unit disk (in- cluding infinity). Note that here the zeros at infinity ofT(which are branch points of logT, see §A.9.2 in Appendix A) have to be factored explicitly since they will contribute to the value of the Poisson integral of logT on the unit circle. This is because the point at infinity is an interior point of the region “encircled” by the unit circle, c, and thus it should be treated as any other finite singularity of logT in c. Note that this is not the case for continuous-time systems, i.e., the singularities at infinity of logT aris- ing from a strictly properLdo not add to the value of the Poisson integral of logTon the imaginary axis (see Corollary A.6.3 in Appendix A, and the proof of Theorem 3.3.2).
Assuming stability of the closed-loop system, and since, as just seen, ˜S and ˜T in (3.46) have no zeros in c, then the functions log ˜Sand log ˜T are analytic in c. The real parts of these functions are harmonic in c, and
12Recall that, in the continuous-time case, only the zeros and poles of the plant in the ORHP impose integral constraints. Those on the imaginary axis do not contribute to the value of the integrals (see Lemma A.6.2 and Corollary A.6.3 in Appendix A).
3.4 Discrete Systems 77 they thus satisfy the conditions of Corollary A.6.5 in Appendix A. We then obtain the following results.
Theorem 3.4.1 (Poisson Integral forS). Let Sbe the discrete sensitivity function defined by (3.43). Assume that the open-loop systemL can be factored as in (3.45) and letq = rqejθq be a zero ofL in c. Then, if the closed-loop system is stable,
Zπ
−π
log|S(ejθ)| r2q−1
1−2rqcos(θ−θq) +r2qdθ=2πlog
B−1S (q)
. (3.47) Proof. Under the assumptions of the theorem,Scan be factored as in (3.46), where ˜Sis a stable, minimum-phase transfer function, having relative de- gree zero. It follows that the function log|S|˜ is harmonic in c. Using
Corollary A.6.5 in Appendix A withu = log|S|˜ ands0 = q, and noting that|S(e˜ jθ)| = |S(ejθ)| (since
BS(ejθ)
= 1,∀θ) and that ˜S(q) = B−1S (q)
(sinceS(q) =1), yields the desired result.
The corresponding result forT is as follows.
Theorem 3.4.2 (Poisson Integral forT). LetT be the complementary sen- sitivity function defined by (3.43). Assume that the open-loop systemL can be factored as in (3.45) and letp=rpejθpbe a pole ofLin c. Then, if
the closed-loop system is stable, Zπ
−π
log|T(ejθ)| r2p−1
1−2rpcos(θ−θp) +r2pdθ=2πlog
B−1T (p)
+2πδlog|p|. (3.48) Proof. The proof uses Corollary A.6.5 in Appendix A with u = log|T˜|, where ˜T is given in (3.46), ands0=p. Then (3.48) follows by similar argu- ments to those used in the proof of Theorem 3.4.1.
As in the continuous-time case, two technical comments are in order.
We first note that Theorems 3.4.1 and 3.4.2 still hold whenL has poles or zeros on the unit circle. Second, if the zeros and/or poles ofLoutside the unit disk have multiplicities greater than one, then additional integral constraints on the derivatives of log ˜Sand of log ˜T can be derived.
Also similar to the Poisson integral constraints for continuous-time sys- tems, the relations (3.47) and (3.48) represent a balance between weighted areas of sensitivity or complementary sensitivity attenuation and ampli- fication. This is because: (i) forz0 = r0ejθ0 with r0 > 1, the weighting function (or Poisson kernel for the unit disk)
Wz0(θ), r20−1
1−2r0cos(θ−θ0) +r20 (3.49)
78 3. SISO Control is positive for allθ, and (ii) the RHSs of both equations are nonnegative since, for a discrete Blaschke productB, we have that
B−1(z)
≥ 1 for
|z| > 1. These facts imply that the weighted area of sensitivity increase must be, at least, as large as the weighted area of sensitivity reduction.
This situation is aggravated if the open-loop system is both unstable and nonminimum phase, since in this case both
B−1S (q) and
B−1T (p) on the RHSs of (3.47) and (3.48) are strictly greater than one. Moreover, the closer the unstable zeros and poles are to each other, the larger the RHSs of both equations become.
Note also the similarities between Theorem 3.4.2 and its counterpart for continuous-time systems, Theorem 3.3.2. In particular, the term due to the relative degree of the plant in the discrete case in (3.48) is analogous to that due to a pure time delay in (3.30). Hence, the constraints imposed onT by open loop time delays and unstable poles are also present in the discrete case, and again, the “more unstable” the pole, and the larger the time delay, the worse these constraints will be.
In the following section we will discuss some of the design trade-offs implied by Theorems 3.4.1 and 3.4.2.