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

˜
DC, one can easily produce such a formula by balancing polynomial coe cients, just as is
done in Theorem 13.2 and Proposition 13.10.

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

RC(s) = K 1 +	T s		1		.
	D	+				(13.5)
	τDs + 1		TI s

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

514

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If one defines

K =

a1b1 − a0

, TD =

a0 − a1b1 + a2b12

, TI

=

a1b1 − a0

, τD =

1

,

b12

b1(a1b1 − a0)

a0b1

b1

then R = RC, where RC is as in (13.5).

Proof

We compute

K(1 + TD )s2 + K(

1

+

1

)s +

K

τDTI

RC(s) =

τD

τD

TI

s2 +

1

s

τD

The lemma follows by setting

a2 = K(1 +

TD

), a1

= K(

1

+

1

), a0 =

K

, b1 =

1

,

τD

τD

TI

τDTI

τD

and solving for a0, a1, a2, and b1.

It is appropriate to employ a PID controller for first and second-order plants. To design

PID controllers using the machinery of Sections 13.2.1 and 13.2.2, we use Proposition 13.10

with k = 1 to take the integrator into account. Let us see explicitly how to do this for

na¨ıve PID

general first and second-order plants.

controllers

with positive

parameters

13.17 Proposition

Consider the three strictly proper plant transfer functions

1

ω02(τs + 1)

τs + 1

Rτ

(s) =

,

Rζ,ω0

,τ (s) =

, Rτ1,τ2,τ (s) =

,

s2 + 2ζω0s + ω02

τs + 1

(τ1s + 1)(τ2s + 1)

and the two polynomials

P1(s) = s3 + as2 + bs + c, P2(s) = s4 + as3 + bs2 + cs + d.

The following statements hold:

(i)

if

RP = Rτ

and if

RC(s) =

(aτ − 1)s2 + bτs + cτ − αs(τs + 1)

s2 + αs

(ii)

for α R, then the closed-loop polynomial of the interconnection of Figure 13.8 is P1;

if

RP = Rζ,ω0,τ and if

a2s2 + a1s + a0

RC(s) =

s2 + b1s

a2

=

b − cτ + dτ2 − ω02 + aτω02 − 2aω0ζ − 2τω03ζ + 4ω02ζ2

ω02(1 + τ2ω02 − 2τω0ζ)

a

=

c

dτ

a − bτ + cτ2 − dτ3 + τω02 − 2ω0ζ

1

ω02

−

ω02

−

1 + τ2ω02 − 2τω0ζ

a0 =

d

ω02

b1

=

a − bτ + cτ2 − dτ3 + τω02 − 2ω0ζ

1 + τ2ω02 − 2τω0ζ

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

22/10/2004			13.2 Synthesis using pole placement		515
(iii) if RP	= Rτ1,τ2,τ and if
RC(s) =		(a2/b2)s2 + (a1/b2)s + (a0/b2)
		s2 + (b1	/b2)s

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

	PID controllers of the form (13.5).
2.	Note that the three plants Rτ , Rζ,ω0,τ and Rτ1,τ2,τ cover all possible strictly proper first
	and second-order plants.
3.	The essential point is not so much the formulae themselves as their existence. That is to
	say, the main point is that for a first-order plant, a PID controller can be explicitly found
	that produces a desired third-order characteristic polynomial, and that for a second-
	order plant, a PID controller can be explicitly found that produces a desired fourth-order
	characteristic polynomial.
4.	In practice, one would not use the formulae of Proposition 13.17, but would simply
	design a controller of the type RC(s) = a2s2+a1s+a0	by enforcing a pole at s = 0 as in
	s(s+b1)
	Proposition 13.10.
5.	Note that for first-order plants, there is some freedom in the design of an appropriate PID
	controller (characterised by the presence of the parameter α in part (i) of the proposition).
	However, the PID controller of part (ii) is uniquely specified by the desired characteristic
	polynomial.

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)

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Converting this to the PID form of (13.5) using Lemma 13.16 gives
	580			2997			29			1
	K =		,	TD = −		,	TI =		, τD =		.
		49			8120			35		14

In Figure 13.12 we show the Nyquist plot for the loop gain RCRP . Note that this controller

	60						3
	40						2
	20						1
Im	0					Im	0
	-20						-1
	-40
							-2
	-60
	-40	-20	0	20	40	60		-2		-1	0	1	2	3
			Re								Re
		Figure 13.12		Nyquist plot for plant RP (s)				=	12	and controller
		RC(s) = 73s2+180s+200							s
				s(s+14)

has o the bat presented us with respectable gain and phase margins. If one wished, this controller could be used as a starting point for further refinements to the stability margins. In Figure 13.13 is shown the step response and the response to a step disturbance between

	1.4					1
	1.2					0.8
	1					0.6
(t)	0.8				(t)	0.4
Σ					Σ	0.2
1	0.6				1
	0.4					0
	0.2					-0.2
	0					-0.4
	2	4	6	8	10
			t

2	4	6	8	10
		t

Figure 13.13 Step response (top) and response to step disturbance (bottom)

which enters the system at the output. The response time is quite good, although the overshoot is a bit large, and could be improved with a larger derivative time, perhaps.

13.3 Two controller configurations

In the preceding section we considered the achieving of a specified closed-loop characteristic polynomial using a unity gain feedback configuration as in Figure 13.8. Next we

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13.3 Two controller configurations

517

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

rˆ(s)					R (s)			RP (s)		yˆ(s)
	−

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.

(i)	R is implementable;
		R	+
(ii)	R,		RH∞ and the strictly nonminimum phase zeros of R and RP	agree, including
		RP

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.

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13 Advanced synthesis, including PID synthesis

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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

uˆ

y/ˆ rˆ

R

=

=

RH∞+ .

rˆ

y/ˆ uˆ

RP

Thus (ii) holds.

N

R

+

R

(ii) = (iii) As R =

obviously D is Hurwitz so (iii a) holds. Since

RH∞,

must

D

RP

RP

be proper. Since

R

NDP

=

,

(13.6)

RP

DNP

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 ),

from which (iii b) follows. If R,		R	RH	+	+∞ it follows from (13.6) that all roots of NP	in
		RP

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

(iii) = (i) Suppose that R has c.f.r. (N, D) satisfying the conditions of (iii).		Let F be
˜	˜	˜

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

	D DP + NfbNP = P1P2.				(13.7)
˜		N		Nfb
We also take N = Dfb = NP2	. We now claim that if we take R =	D	and Rfb =	Dfb	, then

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

	T1 =	RP R Rfb	,	T2 =	RP R
		1 + RP R Rfb			1 + RP R Rfb
	T3 =	RP Rfb	,	T4 =	R Rfb
		1 + RP R Rfb			1 + RP R Rfb
	T5 =	RP	,	T6 =	R
		1 + RP R Rfb			1 + RP R Rfb

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

˜	˜	2	(13.8)
D DfbDP + N NfbNP = NP2	(D DP + NfbNP ) = NP1P2 .

˜

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

˜

NNP P2 = NP Dfb = T2, P1P2 P1P2

22/10/2004		13.3 Two controller configurations								519
									˜
proper. In doing this, it will be helpful to have a bound on deg(NP2), which we now obtain.
We suppose that deg(P1) = k so that deg(P2) = 2n − k − 1. Now,
					˜
		deg(P1) = deg(D) + deg(NP )
		= deg(D) + deg(NP ) − deg(F )
		≥ deg(DP ) + deg(N) − deg(F )
		˜
		= n + deg(N),
using (iii b). Therefore
										˜
		deg(P2) = 2n − 1 − deg(P1) ≤ 2n − 1 − n − deg(N)
	=	˜								(13.9)
		deg(NP2) ≤ n − 1.
Now we proceed with our calculations.
1.					˜				this gives
	T1: The numerator is NP N Nfb. Since N = NP2
		T1 =			NP Nfb		.
					P1P2
	We have deg(P1P2) = 2n − 1 and deg(NP ) ≤ n. Since Nfb is obtained from (13.7),
	deg(Tfb) ≤ n − 1. From this we deduce that T1					is proper.
2.									˜
	T2: The numerator of T2 is NP N Dfb. Since N = NP2 this gives
		T2 =		NP Dfb				.		(13.10)
					P1P2
	Again, deg(P1P2) = 2n − 1 and deg(NP ) ≤ n. In (13.9) we showed that deg(Dfb) ≤
3.	n − 1, thus T2 is proper.
	T3: The numerator of T3 is NP NfbD , giving
		T3 =	NP NfbD
					˜	2
					NP1P2
	We have deg(NP ) ≤ n and deg(P1P2) = 2n − 1. Since D									and Nfb satisfy (13.7),
	deg(D ), deg(Nfb) ≤ n − 1.									finish
4.					˜				this gives
	T4: The numerator is DP N Nfb. Since N = NP2
		T4 =		DP Nfb				.
					P1P2
	We have deg(DP ) = n and deg(P1P2) = 2n − 1. Since Nfb satisfies (13.7), deg(Nfb) ≤
	n − 1. This shows that T4 is proper.
5.					˜				this gives
	T5: The numerator is NP D Dfb. Since Dfb = NP2

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.

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6.			˜			this gives
	T6: The numerator is DP DfbN . Since N = NP2
	T6 =		DP Dfb		.
			P1P2
	We have deg(DP ) = n and deg(P1P2) = 2n − 1. By (13.9) deg(Dfb) ≤ n − 1, showing
	that T6 is proper.
7.	T7: The numerator is DP D Nfb, giving
	T7 =	DP D Nfb
			˜	2
			NP1P2
	We have deg(DP ) = n and deg(P1P2) = 2n − 1.					Since D	and Nfb satisfy (13.7) we
	have deg(D ), deg(Nfb) ≤ n − 1.

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

function R. This means that R is implementable.
13.22 Remarks 1. The conditions R,	R		+
			RH∞ are actually the weakest possible for a
	RP

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

transfer function from the reference rˆ to the input uˆ to the plant is exactly	R	(cf. the
	RP

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.

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521

13.23 Example Let RP (s) =	s2−2		. One can verify that R(s) =
+	s	−1
Indeed, clearly R RH∞ and we also have
		R(s)		=		1 − s2	RH+ .
		RP (s) s2 + 2s + 2						∞
Since we have						RC(s)RP (s)
			R(s) =					,
					1 + RC(s)RP (s)

we can solve this for RC to get

2 2−s	is implementable.
s +2s+2

RC(s) =	R	=		1 − s2	.
	(R − 1)RP
			s(s + 3)

Note that although R is implementable, the interconnection of Figure 13.8 is not IBIBO stable. Thus we see that implementability is di erent from an IBIBO interconnection of the

form Figure 13.8.
13.3.2 Implementations that meet design considerations	As we saw in Sec-

tion 13.2.2, one often wants to enforce more than IBIBO stability on one’s feedback loop. Let us see how these considerations can be enforced in the two controller configuration of Figure 13.14.

13.4 Synthesis using controller parameterisation

In Section 10.3 we described the set of proper controllers that stabilise a proper plant RP , and the parameterisation came to us in terms of a free function in RH+∞. In this section we turn to using this parameterisation to provide a useful guide to controller design. We will be concerned with the standard plant/controller unity feedback loop, and we reproduce this in Figure 13.15 for easy reference.

rˆ(s)	RC (s)	RP (s)	yˆ(s)
	−

Figure 13.15 Interconnection for studying controller parameterisation

13.4.1 Properties of the Youla parameterisation We first recall the Youla parameterisation of Theorem 10.37. Given a proper plant RP , the set Spr(RP ) of proper controllers that render the interconnection of Figure 13.15 IBIBO stable is given by

		Spr(RP ) =	ρ1	+ θP2			θ admissible .
			ρ2	−	θP1
Recall that (P	, P	) is a coprime fractional representative of R						P	and (ρ	, ρ	) is a coprime
1	2								1	2

factorisation of (P1, P2). Also recall that the collection of admissible functions θ are defined by

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1.	θ 6=	ρ2
				, and
		P1							6
2.	s→∞		2		(s)	−	1	(s)
	lim		ρ				θ(s)P		= 0.

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

	ρ1	+ θP2
Spr(RP , Cdes) = ρ2		−	θP1		θ admissible and θ has all poles in Cdes .
Proof

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.

Exercises for Chapter 13 523 524 13 Advanced synthesis, including PID synthesis 22/10/2004

Exercises

rˆ(s)

R (s)

RP (s)

yˆ(s)

E13.1

In this exercise, you will show that the Ziegler-Nicols tuning method cannot be

−

applied, even in some quite simple cases.

a

Nfb(s)

(a) Show that for any first-order plant, say RP = s+b , both of the Ziegler-Nicols

methods will not yield PID parameter values.

Dff(s)

(b) Show that there exists a second-order plant, say of the form RP (s) =

a

, so

(a)

s2+bs+c

that Assumption 13.1 will not be satisfied.

E13.2

In this exercise, you will apply the Ziegler-Nicols tuning method to design PID

controllers for the plant transfer function

rˆ(s)

Nff(s)

Nfb(s)

RP (s)

yˆ(s)

1

Nfb(s)

Dff(s)

−

RP (s) =

.

s3 + s2 + 2s + 1

First, the first method.

Nfb(s)

(a) Produce the step response numerically using Proposition 3.40.

Dff(s)

(b) Graphically (or with the computer, if you can) determine the values of σ and τ,

(b)

and so determine the parameters K, TI , and TD for the three cases of a P, PI,

and PID controller.

Figure E13.1

Alternate two controller configuration implementa-

(c) Using Proposition 3.40, determine the closed-loop step response.

tions

(d)Comment on the performance of the three controllers. Now the second method.

(e)Verify that Assumption 13.1 is satisfied for the plant transfer function RP .

Hint: A good way to do this might be to use a computer package to solve the equation obtained by setting the real part of the poles to zero.

(f)Determine by trial and error (or with the computer, if you can) the value Ku of the proportional gain that makes two poles of the transfer function negative, and the period of the oscillatory part of the step response. Use these numbers to give the corresponding Ziegler-Nicols values for the parameters K, TI , and TD for the three cases of a P, PI, and PID controller.

(g)Using Proposition 3.40, determine the closed-loop step response.

(h)Comment on the performance of the three controllers.

E13.3 Exercises on controlling state examples using pole placement on output.

E13.4 Let RP be a proper plant and let R I (RP ). Suppose that the feedback and feedforward controllers giving R as the closed-loop transfer function in Figure 13.14 are Rfb and R , respectively. Let (N, D), (Nfb, Dfb), and (N , D ) be the c.f.r.’s for R, Rfb, and R , respectively.

(a)Show that all interconnections of Figure E13.1 have closed-loop transfer function R. Note that the recipe of Theorem 13.21 gives N = Dfb, so let us assume that this is the case.

(b)Show that the interconnection (a) of Figure E13.1 will not generally be IBIBO stable.

(c)Show that the interconnection (b) of Figure E13.1 will not generally be IBIBO stable.