13.Advanced synthesis,including PID synthesis

22/10/2004 13.2 Synthesis using pole placement 513

to be strictly proper, then we must allow the closed-loop characteristic polynomial to have degree 2n + k. While we do not produce the explicit formula for the coe cients in NC and

One may also wish to specify that the numerator of RC have roots at some specified locations. The following result tells us when this can be done, and the degree of the closedloop polynomial necessary to guarantee the required behaviour. We omit the details of the proof, as these go much like the proofs of Theorem 13.2 and Proposition 13.10, except that there are more complications.

13.15 Proposition Consider the interconnection of Figure 13.8 and suppose that (NP , DP ) if the c.f.r. for RP . For F R[s] a monic polynomial of degree k, the following statements hold:

(i) if deg(NP ) ≤ n − 1 and if P R[s] is monic and degree 2n + k − 1, then there exists a controller RC R(s) with c.f.r. (NC, DC) with the following properties:

(a)RC is proper;

(b)NC has F as a factor;

(c)deg(DC) = n + k − 1;

(d) the closed-loop characteristic polynomial of the interconnection, DCDP + NP NC, is exactly P ;

(ii)if deg(NP ) ≤ n and if P R[s] is monic and degree 2n + k, then there exists a controller RC R(s) with c.f.r. (NC, DC) with the following properties:

(a)RC is strictly proper;

(b)NC has F as a factor;

(c)deg(DC) = n + k;

(d) the closed-loop characteristic polynomial of the interconnection, DCDP + NP NC, is exactly P .

Note that if the GCD of NC and DP is F , then F is guaranteed to appear as a factor in the closed-loop characteristic polynomial.

13.2.3 Achievable poles using PID control We begin by noting that in this section, as in the rest of this chapter, we use a PID controller that renders proper the derivative term in the controller. Thus we take a controller transfer function of the form

The advantage of this from our point of view in this section is contained in the following result.

13.16 Lemma Let R R[s] be any rational function of the form

R(s) = a2s2 + a1s + a0 . s2 + b1s

then the closed-loop polynomial of the interconnection of Figure 13.8 is P2;

a2 = −τ22 − dτ2τ12τ22 + τ1τ2(−1 + aτ2) + τ12(−1 + aτ2 − bτ22) + τ(τ1 + τ2 − aτ1τ2 + cτ12τ22) a1 = −τ1 − τ2 + aτ1τ2 − cτ12τ22 − dτ2τ1τ2(τ1 + τ2) + τ(1 − bτ1τ2 + dτ12τ22 + cτ1τ2(τ1 + τ2))

a0 = −(d(τ − τ1)τ1(τ − τ2)τ2) b2 = −(τ − τ1)(τ − τ2)

b1 = τ1 + τ2 − aτ1τ2 − cτ2τ1τ2 + dτ3τ1τ2 + τ(−1 + bτ1τ2),

then the closed-loop polynomial of the interconnection of Figure 13.8 is P2.

13.18 Remarks 1. Using Lemma 13.16 one can turn the controllers of Proposition 13.17 into

Let us illustrate a PID design using Proposition 13.17.

13.19 Example (Example 12.14 cont’d) So that we may contrast our design with that of the ad hoc PID design of Section 12.2.3, we take RP = s12 . Let us design a controller with poles at {−5, −5, −2 ± 2i}. Thus we require the characteristic polynomial

P = s4 + 14s3 + 73s2 + 180s + 200.

By contrast, the closed-loop characteristic polynomial of Example 12.14 is

s3 + 134 s2 + 3910 s + 1320 ,

which has roots of approximately {−0.197, −1.53 ± 0.984i}. The lower degree of the characteristic polynomial is Example 12.14 is a consequence of the derivative term not being proper as in (13.5). In any event, an application of Proposition 13.17, or more conveniently

of Proposition 13.10, gives

RC(s) = 73s2 + 180s + 200. s(s + 14)

turn our attention to a richer specification of the closed-loop transfer function to allow the determination of not only the closed-loop characteristic polynomial, but also additional features of the closed-loop transfer function. To do this we must consider a more complicated interconnection, and we consider the interconnection of Figure 13.14. There are now two

Rfb(s)

Figure 13.14 A two controller feedback loop

controllers we may specify, and we call Rfb the feedback controller and R the feedforward controller. It certainly makes sense that having two controllers makes it possible to do more than was possible in Section 13.2. The objective of this section is to quantify how much can be done with the richer configuration, and to detail exactly how to do what is possible.

13.3.1 Implementable transfer functions First let us be clear about what we are after.

13.20 Definition Let RP R(s) be a proper plant. R R(s) is implementable for RP if there exists proper feedback and feedforward controllers Rfb, R R(s) so that

(i)R is the closed-loop transfer function for the interconnection of Figure 13.14,

(ii)the interconnection is IBIBO stable, and

(iii)the only zeros of R in C+ are those of RP , including multiplicities.

We denote by I (RP ) the collection of implementable closed-loop transfer functions for the plant RP .

The next result tells us that it is possible to achieve an implementable transfer function with an interconnection of the form Figure 13.14.

13.21 Theorem Let RP be a proper plant with c.f.r. (NP , DP ), and let R R(s) have c.f.r. (N, D). The following conditions are equivalent.

multiplicities;

(iii)the following three conditions hold:

(a)D is Hurwitz;

(b)deg(D) − deg(N) ≥ deg(DP ) − deg(NP );

(c)the roots of N in C+ are exactly the roots of N in C+, including multiplicities.

Proof (i) = (ii) If R is implementable, the interconnection of Figure 13.14 is IBIBO stable. In particular, the closed-loop transfer function is IBIBO stable, and since this is by hypothesis R, we have R RH+∞. The transfer function from the plant input uˆ to the reference rˆ should also be BIBO stable. This means that

this implies that deg(DNP ) ≥ deg(NDP ). Since the degree of the product of polynomials is the sum of the degrees, it follows that

deg(D) + deg(NP ) ≥ deg(N) + deg(DP ),

C+ must also be roots of N, and vice versa. Thus (iii c) holds.

the GCD of N and NP and write N = F N and NP = F NP . Then define P1 = DNP . Note

that by (iii a) and (iii c), P1 is Hurwitz. Now let P2 be an arbitrary Hurwitz polynomial having the property that deg(P1P2) = 2n − 1, where n = deg(DP ). By Corollary 13.5 we may find polynomials D and Nfb so that

the interconnection of Figure 13.14 is IBIBO stable. The relevant transfer functions that must be checked as belonging to RH+∞ are

Rfb

T7 = 1 + RP R Rfb .

NEED T3 and T7

To see that all of the transfer functions are have no poles in C+ first note that they can all be written as rational functions with denominator

˜

P2 is Hurwitz by design, P1 is Hurwitz by (iii a) and (iii c), and N is Hurwitz by (iii c). Thus

all transfer functions T1 and T7 are analytic in C+.

Let us now give the numerator for each of the transfer functions T1 through T7 when put over the denominator (13.8), and use this to ascertain that these transfer functions are all

T5 = NP D .

P1P2

We have deg(DP ) = n and deg(P1P2) = 2n − 1. Since D satisfies (13.7), deg(D ) ≤ n − 1. This shows that T5 is proper.

Finally we check that the closed-loop transfer function is R. Indeed, the closed-loop transfer function is T2 and we have

˜

N NNP

R = D = ˜ =

DNP

using (13.10).

Thus we have shown that the interconnection of Figure 13.14 is IBIBO stable with transfer

class of interconnections more general than that of Figure 13.14. Indeed, if (S, G) is any interconnected SISO linear system with the property that every forward path from the reference rˆ to the output yˆ passes through the plant, then one readily sees that the

proof that (i) implies (ii) in Theorem 13.21 below). Thus we have the following statement:

If (S, G) is an interconnected SISO linear system with the property that every forward path from the input to the output passes through the plant, then the

interconnection is IBIBO stable with transfer function R R(s) only if R, R

RP

RH+∞.

(This is Exercise E6.14.) This indicates that the conditions R, R RH+ are the weak-

RP ∞

est one can impose on a closed-loop transfer function if it is to be realisable by some “reasonable” interconnection. The additional hypothesis that R have no nonminimum phase zeros other than those of RP is reasonable: the nonminimum phase zeros of RP must appear in R, and we would not want any more of these than necessary, given the discussions of Chapters 8 and 9 concerning the e ects of nonminimum phase zeros.

2.As with our results of Section 13.2, Theorem 13.21 is constructive.

3.Note that the interconnection of Figure 13.14 is the same as that of Figure 10.9. Indeed, there is a relationship between the controllers constructed in Theorem 13.21 and the combination of the observer combined with static state feedback in Theorem 10.48. The procedure of Theorem 13.21 is a bit more flexible in that the order of the closed-loop characteristic polynomial is not necessarily 2n.

Let us first illustrate via an example that implementability is indeed di erent that stabilisation of the closed-loop system.

We wish to determine how the choice of the parameter θ a ects the properties of the closedloop system. For this purpose, let us suppose that we decide that we wish to place all poles of the closed-loop system in some region Cdes in the complex plane. Note that we can always do this, since by, for example, Theorem 10.27 or Theorem 13.2, we may construct controllers that achieve any closed-loop characteristic polynomial, provided it has su ciently high degree (twice the order of the plant for a strictly proper controller, and one less than this for a proper controller). Thus we are permitted to talk about the set of all proper controllers for which the closed-loop poles lie in Cdes, and let us denote this set of controllers by Spr(RP , Cdes). The following result describes these controllers using the Youla parameterisation.

13.24 Proposition Let RP be a proper plant and Cdes C be the set of desirable closed-loop pole locations. Also let (P1, P2) be a coprime fractional representative for RP with (ρ1, ρ2) a coprime factorisation of (P1, P2). Then

The result is perhaps surprisingly “obvious,” at least in statement, and clearly provides a useful tool for controller design.

13.5Summary

1.Ziegler-Nichols tuning is available for doing PID control design.