11.Ad hoc methods I.The root-locus method

This version: 22/10/2004

Chapter 11

Ad hoc methods I: The root-locus method

The root-locus method we study in this section was put forward in the papers of Evans [1948, 1950]. The study is of roots of polynomials when the coe cients depend linearly on a parameter. In control systems, the parameter is typically the gain of a feedback loop, and our interest is in choosing the gain so that the closed-loop system is IBIBO stable. As we saw in Section 10.2.2, with static output feedback of SISO systems, there naturally arises a control problem where one has a polynomial with coe cients linear in a parameter, and stabilisation requires choosing this parameter so that the polynomial is Hurwitz. In Section 11.1.1 below, we discuss some control problems where this scenario arises. The manner of studying such problems in this chapter is to study how the roots move in the complex plane as functions of the parameter. That is to say, we look at the locus of all roots of the polynomial as the parameter varies, hence the name “root-locus.”

In many introductory texts, one will find a laying out of a “design method” using rootlocus methods. We do not devote significant e ort to this for the reason that, according to Horowitz [1963], “It appears, therefore, that the root locus approach to the sensitivity problem is justified only in systems where there are few dominant poles and zeros.” These systems are quite well understood in any case (see Section 13.2.3).

Contents

11.1 The root-locus problem, and its rˆole in control

The aim in this section is to provide some situations in control where a certain type of problem involving a certain type of polynomial arises. Once this has been nailed down, we can talk in generality about this problem, and some of its broad properties. We reserve for Section 11.2 a more or less complete discussion of the properties of such polynomials.

11.1.1 A collection of problems in control Our first task is to indicate that there are a collection of control problems which can be reduced to a problem of a certain type.

11.1 Examples 1. To get things rolling, let us consider the unity gain feedback loop of Figure 11.1 where the loop gain is known up to a multiplicative constant K, and let us

Figure 11.1 Root-locus from a negative feedback configuration

suppose that we ask that K be nonnegative. The transfer function from rˆ(s) to yˆ(s) is,

as usual,

yˆ(s) = KRL(s) . rˆ(s) 1 + KRL(s)

If RL has the c.f.r. (NL, DL), then the characteristic polynomial for the interconnection is DL + KNL. If RL is strictly proper, as is quite often the case, then we have deg(NL) < deg(DL). The design objective is to determine whether there is a constant K0 ≥ 0 so that the characteristic polynomial DL + K0NL is Hurwitz. One may even want to choose K so that certain performance objectives are met. This is discussed in Section 11.3.

2.As we saw in Section 6.4.2, the closed-loop characteristic polynomial for static output feedback of a SISO linear system Σ = (A, b, ct, 01) is given by

PA(s) + F ctadj(sIn − A)b

where F is the output feedback constant. Since ctadj(sIn −A)b is a polynomial of degree at most n − 1, this will have the form of P1 + F P2 where deg(P2) < deg(P2).

3.It is possible that the variable constant K may not appear in as simple a manner as indicated in Figure 11.1. However, in some such cases, one can still reduce the characteristic polynomial to one of the desired form. Suppose that we have a plant with

RP (s) = ms1 2 and we wish to stabilise it in a unity gain feedback loop with a PID con-

troller RC (s) = K(1 + TDs + 1 s). Let us suppose that, for whatever reason, we are

TI

interested only in changing the reset time TI , and not the gain K. Furthermore, we restrict our interest to TI > 0. Thus we are not immediately in the situation illustrated in Figure 11.1. Nevertheless, we proceed. The closed-loop characteristic polynomial is

s3 + KTDs2 + Ks + K .

TI

Now note that we may write this closed-loop characteristic polynomial as P1 + αP2 if we take

Now, as TI runs from 0 to ∞, α runs from ∞ to 0. Thus, even though we are not immediately in the form of Figure 11.1, we can write the characteristic polynomial is the form of a sum of two polynomials, one with a positive coe cient. Note that this may not always be possible, but sometimes it is.

In each of the preceding three examples, we arrive at a characteristic polynomial that is of the form P1 + KP2 where deg(P2) < deg(P1). It is such polynomials that are of interest to us in this chapter.

11.1.2 Definitions and general properties Based on the discussion of the preceding section, we make the following definition.

11.2Definition Let N, D R[s] have the following properties:

(i)D is monic;

(ii)either N or −N is monic;

(iii)deg(N) < deg(D).

A (N, D)-polynomial family is a family of polynomials of the form P(N, D) = {D + KN | K ≥ 0}. For a fixed K ≥ 0 we shall denote PK = D + KN R[s]. The root-locus of an (N, D)-polynomial family is the subset

RL(P(N, D)) = {z C | PK (z) = 0 for some K ≥ 0}

of the complex plane.1 A (N, D)-polynomial family is Hurwitz if there exists K ≥ 0 so that PK is Hurwitz.

11.3 Remark Note that we do ask for D to be monic and either N or −N to be monic when defining a (N, D)-polynomial family. That D should be taken as monic seems natural. As for N, clearly one can make either N or −N monic by simply redefining K if necessary.

witz is exactly equivalent to the problem of determining whether a SISO linear system Σ = (A, b, ct, D) admits a stabilising static output feedback controller (see Exercise E11.2). Therefore, to determine whether a (N, D)-polynomial family is Hurwitz is as di cult as proving the existence of stabilising static output feedback. This problem, as was noted at the beginning of Section 10.2.2 is NP hard. Thus we cannot expect to solve this problem is an easily computable manner. However, we shall provide a rough description of the rootlocus for a (N, D)-polynomial family which suggests that you might be able to numerically ascertain whether such a family is Hurwitz.

Note that the root-locus consists of collections of roots of polynomials that depend continuously, in this case linearly, on a parameter K. Let us record some properties of the roots of such parameterised polynomials.

1Note that S denotes the closure of the set S, meaning the set and all of its limit points.

with coe cients di erentiable functions of the parameter K R. For K R, denote by {z1(K), . . . , zn(K)} the roots of P (s, K).

(i)The functions R 3 K 7→zi(K) C, i = 1, . . . , n, may be chosen to be continuous.

(ii)If the roots zi(K), i = 1, . . . , n, are chosen to be continuous functions, then, if zi(K0) is a root of multiplicity one for P (s, K0) then there exists > 0 so that zi|[K0− , K0+ ] so that

(a)fi(K0) = zi(K0) and

(b)P (zi(K), K) = 0.

More succinctly, near nondegenerate roots of P (s, K0), the roots are di erentiable functions of the parameter K.

11.5 Remark With the notation of the lemma, we may write the roots of P (s, K) as {z1(K), . . . , zn(K)}. We shall do this frequently, and we always make the assumption that this is done in such a way that the functions K 7→zi(K), i = 1, . . . , n, are continuous.

We do not prove this lemma, although part (ii) is easily proved using the implicit function theorem (see Exercise E11.3). Let us illustrate the lemma with a simple example.

Figure 11.2 we plot the locus of roots for this polynomial as K runs from −4 to 4. Note that

-1

-2

Re

Figure 11.2 Locus of roots for P(s, K) = s2 + K

if K varies slightly, the location of the roots also change only slightly. What’s more, as long as K 6= 0 the roots are distinct, and the locus of roots near such values of K are smooth

curves in the complex plane (lines in this case). However, for the repeated root when K = 0, the character of the locus of roots is not smooth near this repeated root. Indeed, at the origin where this repeated root lies, the locus of roots has an intersection. This will typically be the case, and constitutes one of the more challenging aspects of making root-locus plots.

11.2 Properties of the root-locus

In this section we do two things. First we provide a list of provable properties of the root-locus of a (N, D)-polynomial family. These are presented with proofs as these do not seem to be part of the standard discussion of the root-locus method. The presentation in this first part of the section produces for us a good understanding of what is known and unknown about the nature of the root-locus. The next thing we do is put this understanding together to develop a methodology for graphically producing the root-locus for a (N, D)- polynomial family. It is this graphical method which forms the bulk of the presentation on the root-locus method in most standard texts.

11.2.1 A rigorous discussion of the root-locus We follow here the paper of Krall [1961], starting with a (N, D)-polynomial family P(N, D). The essential idea is that when K = 0 the roots of PK are obviously those of D. Suppose that deg(D) = n so that P0 has n roots, if one counts multiplicities. Now, as K increases, Lemma 11.4 suggests that these n roots should move about continuously in the complex plane. Now suppose that m = deg(N). What will turn out to happen is that m of the n roots of PK will start at the roots for D = P0 and end up at the roots of N as K → ∞. One must then account for the remaining n − m roots, which, it turns out, shoot o to infinity in predictable ways.

The following result gives a simple characterisation of the centre of gravity in terms of the roots of N and D.

and ζ1, . . . , ζm are the roots for N.

Proof For a monic polynomial

P (s) = sn + an−1sn−1 + · · · + a1s + a0,

one may easily show that the sum of the roots of P is equal to −an−1 (see Exercise EC.3). Thus the sum of the roots for D is pn−1, and the sum of the roots for N is qm−1 is N is monic and −qn−1 is −N is monic.

Now, through the centre of gravity we construct n − m rays in the complex plane. We denote these rays by α1, . . . , αn−m and define them by

We call these rays the asymptotes for P(N, D). In Figure 11.3 we show the asymptotes

Figure 11.3 Asymptotes if CG(P(N, D)) = 1 + i0 and n −m = 3, and when N is monic (left) and −N is monic (right)

when the centre of gravity is at 1 + i0 and when n − m = 3. Note that generally the asymptotes will depend on whether N or −N is monic.

With the notion of asymptotes in place, we may state the following result which indicates in what manner some of the roots of PK go to infinity.

11.8 Proposition Let P(N, D) be a (N, D)-polynomial family and denote the roots of PK = D + KN by {z1(K), . . . , zn(K)}. Then there exists distinct j1, . . . , jn−m {1, . . . , n} so that

Proof The bulk of the proof is contained in the following result.

1 Lemma The proposition holds when CG(P(N, D)) = 0 + i0.

Proof In this case we can write

PK (s) = sn +asn−1 +pn−2sn−2 +· · ·+p1s+p0 −Keiθ sm +asm−1 +qm−2sm−2 +· · ·+q1s+q0 ,

where K ≥ 0 and θ {0, −π}. The main point is that since CG(P(N, D)) = 0+i0, the next to highest coe cients of D and N must be equal if N is monic. Now fix k {1, . . . , n − m} and make the substitution

s = weiθk , θk = 2kπ + θ , w C, n − m

for some coe cients a0, . . . , an−2, b0, . . . , bm−2 whose exact character are not of much interest to us. Now define

Now make the substitution ξ = wρ C so that, with our previous substitution for s, we have

Now let be the circle of radius centred at 1+i0, with as previously defined. Since < 13 , the real part of ξk C is positive on for k = 1, . . . , n − m. Note that ξn−m − 1 vanishes

Therefore, for ξ we have

|f(ξ)| > (1 − )m−1 · 13 · 1 · > 13 m .

Thus there exists K0 (redefine this if necessary) so that for all K > K0, f has only the zero at 1 + i0 within . By definition of M, for ξ we have

|g(ξ)| ≤ Mρ2 = 13 m .

Therefore, for ξ we have |f(ξ)| > |g(ξ)|. From Rouch´es theorem, Theorem D.6, we may conclude that the number of zeros for f + g within is the same as the number of zeros f, namely 1, provided that K > K0. This shows that there is a zero ξ0 inside so that

With the lemma in hand, it is now comparatively straightforward to prove the proposition. Let us proceed in the case when N is monic so that

CG(P(N, D)) = −pn−q − qn−1 . n − m

Making the substitution s = w + CG(P(N, D)) gives

n−2

PK (w + CG(P(N, D))) = wn + nqn−1 − mpn−1 wn−1 + X ajwj+ n − m

j=0

m−2

K wm + nqn−1 − mpn−1 wm−1 + X bjwj , n − m

j=0

for some coe cients a0, an−2, b0, . . . , bm−2 whose exact form is not of particular interest to us. Now, by Lemma 1, there is a collection of roots {ωj1 (K), . . . , ωjn−m (K)} to this polynomial that satisfy

lim ωjk (K) − K1/(n−m)e2kπi/(n−m) = 0.

K→∞

These clearly give rise to the roots {zj1 (K), . . . , zjn−m (K)} as given in the statement of the proposition.

Let us see how this works in a simple example.

expect there to be 2 roots that shoot o to infinity. According to Proposition 11.8(i), in the

√ √ √ √

limit as K → ∞ these roots should behave like Keπi/2 = i K and Ke3πi/2 = −i K as K gets large. Not only do the roots behave like this as K gets large, they are exactly given by these expressions for all K!

When N = −1 we have P (s) = s2 − K. In this case, Proposition 11.8(ii) predicts that

K √ √ √ √

as K → ∞ the two roots of PK behave like Keπi = − K and Ke2πi = + K. Again, these happen to be the exact values of the roots. Clearly we do not expect this to generally be the case.

Now let us consider what happens to those remaining m roots of PK as K varies.

11.10 Proposition Let P(N, D) be a (N, D) polynomial family with {z1(K), . . . , zn(K)} the roots of PK for K ≥ 0. Also denote {ζ1, . . . , ζm} as the roots for N. There exists distinct

j1, . . . , jm {1, . . . , n} so that for each k = 1, . . . , m, the curve K 7→zjk (K) is a continuous curve starting from zjk (0) and satisfying limK→∞ zjk (K) = ζk.

Proof By part (i) of Lemma 11.4 we know that the curve K 7→zj(K) is continuous for every j = 1, . . . , n. Now fix k {1, . . . , m} and suppose that the multiplicity of the root ζk is `. Choose > 0 and let be the circle of radius centred at ζk. Take su ciently small that there are no roots of N within but zk. The number of zeros of PK within is given by the Principle of the Argument to be

Since is bounded, both polynomials N and D are bounded on . Therefore, we may choose K su ciently large that K1 |D(s)| < |N(s)| for s . Therefore we see that the number of zeros of PK within is given by ` + δ where δ may be made as small as we please by increasing K. Thus we conclude that for K su ciently large, PK has ` zeros in . As this holds for all > 0, we conclude that ` zeros of PK limit to ζk as K → ∞. As this is true for all roots ζk for N, this proves the proposition.

Let us illustrate this proposition in another simple example.

11.11 Example We take (N(s), D(s)) = (s + 1, s2) so that PK (s) = s2 + Ks + K. Note that CG(P(N, D)) = 1 + i0. Since n − m = 1 and N is monic, by part (i) of Proposition 11.8 we expect to see one of the roots approaching 1 + Keπi = 1 − K as K → ∞. The other root should start at one of the roots for D and end up in the root ζ = −1 for N. We compute the roots to be

q

−K2 ± (K2 )2 − K,

and the root-locus is shown in Figure 11.4. Note that one of the roots does indeed start out at 0 + i0, a root for D, and moves continuously toward −1 + i0, a root for N. In the root-locus of Figure 11.4, the path taken by this root is not unique. It can start at 0 + i0

and go down along the semi-circle, then turn right toward −1 + i0, or it can go up along the other semi-circle, and then again turn right toward −1 + i0. Note that Proposition 11.10 does not preclude this ambiguity. Also, we see in Figure 11.4 that one of the roots is going o to −∞ as predicted.

Let us see if we can verify the limiting behaviour analytically. For K large, both roots of PK are real. Let us write

q q

z1(K) = −K2 − (K2 )2 − K, z2(K) = −K2 + (K2 )2 − K

=−1 + K2 − (K2 ) 1 − 12 K4 − 18 (K4 )2 + · · ·

=K1 + · · · ,

so that limK→∞(z1(K) − (1 − K)) = 0 as desired. Similarly we have

=1 − K2 + (K2 ) 1 − 12 K4 − 18 (K4 )2 + · · ·

=− K1 + · · · ,

Now we wish to examine the way that the root-locus passes through certain points on the root-locus. Let z0 RL(P(N, D)) so that z0 is a root of PK0 for some K0 ≥ 0. If the multiplicity of z0 is one, then the root-locus passes di erentiably through z0. Denote by

zi(K) that root for PK for which zi(K0) = z0. We define the arrival angle of RL(P(N, D)) at z0 to be the angle θa(z0) (−π, π] for which

lim zi(K) − z0 = αaeiθa(z0)

K↑K0 K − K0

for some suitable αa > 0. This make precise the intuitive notion of arrival angle. Similarly, the departure angle of RL(P(N, D)) at z0 is the angle θd(z0) (−π, π] for which

lim zi(K) − z0 = αdeiθd(z0)

K↓K0 K − K0

for some suitable αd > 0. Since the root-locus is di erentiable at z0, one can easily see that we must have θd(z0) {θa(z0) + π, θa(z0) − π}. When z0 has multiplicity ` > 1, things are not so transparent since there will be more than one arrival and departure angle. In this case we have ` roots zj1 (K), . . . , zj` (K) which can pass through z0 when K0. We may define

` arrival angles, θa,1(z0), . . . , θa,`(z0), by

for some suitable αd,1, . . . , αd,` > 0. For any multiplicity of the root z0, let us denote by Θa(z0) the collection of all arrival angles, and by Θd(z0) the collection of all departure angles.

We are now ready to state the character of some of the arrival and departure angles, namely the departure angles for the roots of D and the arrival angles for the roots of N. In practice, these are the most helpful to know in terms of being able to produce the root-locus for an (N, D)-polynomial family.

11.12 Proposition Let P(N, D) be an (N, D)-polynomial family and let {z1, . . . , zn} be the roots of D and {ζ1, . . . , ζm} be the roots of N.

(i)If zj has multiplicity `, then Θd(zj) is the following collection of ` angles:

(a)if N is monic, take

(ii) If ζj has multiplicity `, then Θa(ζj) is the following collection of ` angles:

Proof Let us for convenience write PK = D − KeiθN and assume that N is monic. Thus, we recover the monic cases for N and −N by taking θ = π and θ = 0, respectively. We then write

nm

YY

(i) For j {1, . . . , n}, let ` be the multiplicity of zj. For k {1, . . . , `} let zjk (K) be a root of PK which approaches zj as K → 0. Note that

Taking complex logarithms of both sides, using (11.4), we obtain

mn

XX

where k Z. Therefore,

mn

In the limit as K → 0, the result follows, noting the relation between θ and N or −N being

11.13 Example (Example 11.11 cont’d) We again take (N(s), D(s)) = (s+1, s2). In this case we have a root z1 = 0 for D of multiplicity 2 and a root ζ1 = −1 for N of multiplicity 1. Since N is monic, Proposition 11.12 predicts that

The next result tells us that we should expect to see certain parts of the real axis within the root-locus.

(i) Suppose that N is monic and let x0 R C. Then x0 RL(P(N, D)) if and only if the set

{j | Re(zj) > x0} {j | Re(ζj) > x0}

has odd cardinality.2

(ii) Suppose that −N is monic and let x0 R C. Then x0 RL(P(N, D)) if and only if the set

{j | Re(zj) > x0} {j | Re(ζj) > x0}

has even cardinality.

Proof (i) First note that since the roots of PK come in complex conjugate pairs, roots with nonzero imaginary part will always contribute an even number to the cardinality of the sets

Taking complex logarithms, and using the fact that D and N can be written as in (11.4), gives

mn

XX

for k Z. Clearly, the sums can be taken as being over real roots since the terms corresponding to a complex root for N or D and its conjugate will cancel from (11.5). If the cardinality of the set

{j | Re(zj) > x0} {j | Re(ζj) > x0}

is odd, then we will have

mn

XX

for some r Z. Thus this part of the proposition follows.

(ii) The proof here goes just as in part (i), except that we write D(x0) − KN(x0) = 1

11.15 Example (Example 11.11 cont’d) We again take (N(s), D(s)) = (s + 1, s2). The roots of D are {z1 = 0, z2 = 0} and the roots of N are {ζ1 = −1}. Thus, if x0 R C, there are an odd number of zeros for both D and N to the right of x0 is and only if x0 < −1. Then Proposition 11.14 tells us that

typically presented in classical texts for producing the root-locus. This rather ingenious technique was that developed by Evans [1948, 1950]. For us, this is simply a matter of applying the results of the preceding section, and we shall only enumerate the steps one typically takes in such a construction.

1.Compute the roots {z1, . . . , zn} for D and place an X at the location of each root in C.

2.Compute the roots {ζ1, . . . , ζm} for N and place an O at the location of each root in C.

3.Compute the centre of gravity using (11.1) or Lemma 11.7.

4.Draw the n − m asymptotes using (11.2).

5.Use Proposition 11.14 to determine RL(P(N, D)) ∩ R.

6.Use Proposition 11.12 to determine how the root-locus departs from the roots of D.

7.Use Proposition 11.12 to determine how the root-locus arrives at the roots of N.

8.If you are lucky, you can give a reasonable guess as to how the root locus behaves. For filling in the gaps in a root-locus plot, a useful property of the root-locus is that it is

invariant under complex conjugation.

The last step is in some sense the most crucial. It is possible that one can do the steps preceding it, and still get the root-locus wrong. Some experience is typically involved in knowing how a “typical” root-locus diagram looks, and then extrapolating this to a given example. Thankfully, computers typically do a good job with producing root-locus plots. These can run into problems when there are repeated roots of PK , however. Thus one should check that a computer produced root-locus has the essential properties, mainly the correct number of branches.

Let us go through the steps outlined above in an example to see how well it works.

11.17 Example We take (N(s), D(s) = (s + 1, s4 + 6s3 + 14s2 + 16s + 8). 1. The roots for D are {z1 = −2, z2 = −2, z3 = −1 − i, z4 = −1 + i}.

2.The roots for N are {ζ1 = −1}.

3.CG(P(N, D)) = −53 .

4.The asymptotes are given by

We show these asymptotes in Figure 11.5.

5.The point on the real-axis which lie on the root-locus are those points x < −2.

6.The departure angles from the roots of D are

Θd(z1) = Θd(z2) = {0, −P i}, Θd(z3) = {−π2 }, Θd(z4) = {−32π }.

These departure angles are shown in Figure 11.6

7. The arrival angles from the roots of N are

Θa(ζ1) = {π}.

8.We play “connect the dots,” hoping that we will arrive at a decent approximation of the actual root-locus. The actual root locus, along with the skeleton produced by our procedure, is shown in Figure 11.7. Note that without the knowledge of which roots of D go where as K → ∞, it is in actuality di cult to know the details of the character of

the root locus. Nevertheless, in simple examples one can often figure out the character of the root locus.

11.3 Design based on the root-locus

The root-locus method can be thought of as a design technique. As such, it is most useful for plants that can be stabilised using simple controllers. For such plants, the rootlocus method allows one to evaluate a one-parameter family of controllers by investigating the root-locus plot as the parameter varies. This assumes that one can put the system into a form where as the parameter varies we produce an (N, D)-polynomial family. We argued in Section 11.1.1 that this can sometimes be done, although it will by no means always be the case.

11.3.1 Location of closed-loop poles using root-locus For systems that have simple enough behaviour, one can attempt to determine the quality of the performance of the system based on the location of the transfer function poles in the complex plane. For example, as we

Figure 11.6 Departure angles from roots of D(s) = s4+6s3+14s2+ 16s + 8 (left) and arrival angle from the roots of N(s) = s + 1 (right).

saw in Section 8.2, first and second-order transfer functions have performance attributes that are easily related to pole locations. Sometimes in practice one is able to design a controller that makes a system behave similarly to one of these two simple transfer functions, so one can use them as a basis for control design. With this as flimsy justification, we deal with interconnections like that depicted in Figure 11.8, and make the following assumption.

In this section we will assume that the closed-loop interconnection of Figure 11.8 is designed so that for some values of K, the closed-loop transfer function is stable, and has a pair of complex conjugate poles whose real part is larger than that of all other poles. We call these complex conjugate poles the dominant poles.

The idea is that we pretend the dominant poles allow us to think of the closed-loop system as being second-order, and we base our choice of K on this consideration. In practice, one may wish to find a second-order transfer function that well approximates the system by, say, matching Bode plots as best one can.

Our approach is to carefully observe the relationship between system performance and pole location for second-order transfer functions. Thus we let

ω2

Tζ,ω0 (s) = s2 + 2ζω00s + ω02 .

For this transfer function, let us list some of the more important performance measures, some approximately, in terms of the parameters ζ and ω0.

11.18 Performance measures for second-order transfer functions Consider the transfer function

Tζ,ω0 .

1.

11.3.2 Root sensitivity in root-locus See Bishop and Dorf.

(N(s), D(s) = (s + 1, s4 + 6s3 + 14s2 + 16s + 8).

Figure 11.8 Interconnection for investigating closed-loop pole locations

11.4 The relationship between the root-locus and the Nyquist contour

It turns out that there are some rather unobvious connections between the root-locus

and the Nyquist contour. This is explained in a MIMO setting in [MacFarlane 1982], and cite info we only look at the SISO case here.

11.4.1 The symmetry between gain and frequency We begin with a proper rational function R with c.f.r. (N, D) and canonical minimal realisation Σ = (A, b, ct, D). We next place R, represented by its canonical minimal realisation, into a positive feedback loop with feedback gain k−1 as shown in the upper block diagram in Figure 11.9. There are a few things to note here: (1) there is no negative sign where the feedback enters the summer with the reference r, (2) except for the sign, the block diagram is the same as that for static output feedback as studied in Section 6.4.2, and (3) the top block diagram in Figure 11.9 is equivalent to the bottom block diagram in the same figure. The second of these block diagrams has the advantage of making apparent a symmetry that exists between the parameter s and the parameter k. It is this symmetry which we study here.

(sI n − A)−1

rˆ(s) (kI 1 − D)−1 yˆ(s)

Figure 11.9 Equivalent block diagrams for studying the relationship between the root-locus and the Nyquist contour

Let us write

K(s) = ct(sIn − A)−1b + D R(s)1×1

S(k) = ct(kI1 − D)−1b + A R(k)n×n.

The following lemma tells us how we should interpret these matrices of rational functions.

11.19 Lemma The following statements hold:

(i)K(s) is the open-loop transfer function from r to y for the uppermost block diagram in Figure 11.9, i.e., the transfer function if the feedback loop is snipped;

(ii)S(k) is the “A-matrix” for the closed-loop system in Figure 11.9, i.e., if the closed-loop

Proof The first assertion is clear. The second follows from the same computations that gave the closed-loop system under static output feedback in Section 6.4.2.

Note that the closed-loop system is internally stable if and only if spec(S(k)) C−. We wish to understand a similar relationship between closed-loop stability and K(s). The following lemma gives us the essence of this relationship

11.20 Lemma If s C \ spec(A) and k C \ spec(D), then the following statements are equivalent:

(i)det(sIn − S(k)) = 0;

(ii)det(kI1 − K(s)) = 0.

22/10/2004 11.4 The relationship between the root-locus and the Nyquist contour 465

Proof First note that if s and k are as hypothesised, then S(k) and K(s) are well-defined. Since Σ is assumed controllable and observable, the poles of the closed-loop transfer function for the system in Figure 11.9 are exactly the eigenvalues of S(k), as a consequence of Lemma 11.19(ii). However, the closed-loop transfer function is

From this we have as a consequence the test for closed-loop stability in terms of K(s).

11.21 Corollary spec(S(k)) C− if and only if spec(K(s)) C−.

Proof The closed-loop system of Figure 11.9 is BIBO stable if and only if spec(S(k)) C− if and only if the poles of the closed-loop transfer function lie in C−. The result then follows from Lemma 11.20.

What we have done to this point how the gain k C and the frequency s C have a symmetric relationship in determining the closed-loop stability of the system in Figure 11.9. Now let us see how this investigation bears fruit.

11.4.2 The characteristic gain and the characteristic frequency functions What we saw in the preceding discussion was that the two meromorphic functions

Fk(s) = det(sIn − S(k)) R(s)

Fs(k) = det(kI1 − K(s)) R(k).

With this as motivation we have the following result giving us the precise manner in which the gain and frequency are related. First we make two important definitions. To do so, the reader will wish to recall our discussion in Section D.5 on algebraic functions and Riemann surfaces.

466 11 Ad hoc methods I: The root-locus method 22/10/2004

Exercises

E11.1 Let P(N, D) = {D + KN | K ≥ 0} be a (N, D)-polynomial family.

(a)Show that there exists RL R[s] so that the closed-loop characteristic polynomial for the interconnection of Figure 11.1 is D + KN. Explicitly give RL in terms of N and D.

(b)Provide RL for the (N, D)-polynomial family of Example 11.1–3.

In Section 11.1.1 we saw that the problem of static output feedback leads naturally to a (N, D)-polynomial family. In the next exercise, you will show that the converse also happens, i.e., that a (N, D)-polynomial family leads naturally to a static output feedback problem.

E11.2 Let P(N, D) = {D + KN | K ≥ 0} be a (N, D)-polynomial family. Show that there exists a SISO linear system Σ = (A, b, ct, D) so that the closed-loop characteristic polynomial for the static output feedback interconnection of Figure E11.1 is

D

Figure E11.1 Static output feedback and (N, D)-polynomial families

exactly PK .

E11.3 Prove part (ii) of Lemma 11.4 using the implicit function theorem.

The development of the root-locus properties for an (N, D)-polynomial family assumed that deg(N) < deg(D). It is also possible to proceed when confronted with the case when deg(N) > deg(D).

E11.4 Let N, D R[s] and assume that D is monic, that either N or −N is monic, and that deg(N) > deg(D). Define

that RL(P(N, D)) = RL(P(N, D)).