07.Frequency domain methods for stability

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RP (s) = RP (s) +

are stabilised by RC provided that

is allowable. The Nyquist plot for RP obtained by

is shown in Figure 7.27. This is a pretty sophisticated Nyquist plot, but

nonetheless, it is one for an IBIBO stable system.

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22/10/2004

7.2 The relationship between the Nyquist contour and the Bode plot

299

	3								10
									0
									-10
	2						dB		-20
							dB		-30
									-30
									-40
	1								-50
									-1.5	-1	-0.5	0	0.5	1	1.5	2
Im	0										log ω
Im

									150
	-1								100
	-1								50
							deg		50
							deg		0
									0
	-2								-50
	-2							-100
								-100
								-150
	-4	-3	-2	-1	0	1			-1.5	-1	-0.5	0	0.5	1	1.5	2
	-4	-3	-2	-1	0	1					log ω
				Re

Figure 7.15 Nyquist and Bode plots for RL(s) = − 2 10

s +4s+3

Φmin ≈ 110◦. We note that there the upper phase margin is undeﬁned, as are the phase

crossover frequencies.

gain margin

For cases when RL is BIBO stable, we can o er an interpretation of the gain and phase margins in terms of closed-loop stability.

7.12 Proposition Let RL RH+∞ be a BIBO stable loop gain, and consider a unity gain feedback block diagram conﬁguration like that of Figure 7.12. Either of the following statements implies IBIBO stability of the closed-loop system:

(i)Kmin is undeﬁned and either

(a)Kmax > 1 or

(b)Kmax is undeﬁned.

(ii)Φmin is undeﬁned and either

(a)Φmax > 0.

(b)Φmax is undeﬁned.

Proof We use the Nyquist criterion for evaluating IBIBO stability. For a stable loop gain RL, the closed-loop system is IBIBO stable if and only if there are no encirclements of

+
−1 + i0. Note that the assumption that RL RH∞ be BIBO stable implies that there are		gain margin
no poles for RL on the imaginary axis, so the (∞, 0)-Nyquist contour is well-deﬁned and
bounded.
(i) In this case, all crossings of the imaginary axis will occur in the interval (−1, 0). This
precludes any encirclements of −1 + i0.
(ii) If (ii) holds, then all intersections at positive frequency of the (∞, 0)-Nyquist contour
with the unit circle in C will occur in the lower complex plane. Similarly, those crossings of
the unit disk at negative frequencies will take place in the upper complex plane. This clearly
precludes any encirclements of −1 + i0 which thus implies IBIBO stability.

300 7 Frequency domain methods for stability 22/10/2004

Often when one reads texts on classical control design, one simply sees mentioned “gain margin” and “phase margin” without reference to upper and lower. Typically, a result like Proposition 7.12 is in the back of the minds of the authors, and it is being assumed that the lower margins are undeﬁned. In these cases, there is a more direct link between the stability margins and actual stability—at least when the loop gain is itself BIBO stable—as evidenced by Proposition 7.12. The following examples demonstrate this, and also show that one cannot generally expect the converse of the statements in Proposition 7.12 to hold.

7.13 Examples 1. We consider

RL(s) = s2 + 4s + 3,

which we looked at in Example 7.11. In this case, the hypotheses of part (ii) of Proposition 7.12 are satisﬁed, and indeed one can see from the Nyquist criterion that the system is IBIBO stable.

2. Next we consider

RL(s) = −s2 + 4s + 3,

which we also looked at in Example 7.11. Here, the lower phase margin is deﬁned, so the hypotheses of part (ii) of Proposition 7.12 are not satisﬁed. In this case, the Nyquist criterion tells us that the system is indeed not BIBO stable.

3.The ﬁrst of the preceding examples illustrates how one can use the conditions on gain and phase margin of Proposition 7.12 to determine closed-loop stability when RL is BIBO stable. The second example gives a system which violates the su cient conditions of Proposition 7.12, and is indeed not IBIBO stable. The question then arises, “Is the condition (ii) of Proposition 7.12 necessary for IBIBO stability?” The answer is, “No,” and we provide an example which illustrates this.

We take	(1 − s)2(1 + 253s )3
RL(s) =							.
			s		s
	2(1 + s)2(1 +			)(1 +		)3
		100
				50

	This is a BIBO stable loop gain, and its Bode plot is shown in Figure 7.16. From the
	Bode plot we can see that there is a gain crossover frequency ωgc satisfying something
	like log ωgc = 0.8. The lower phase margin is deﬁned at this frequency, and it is roughly
	60◦. Thus the lower phase margin is deﬁned, and this loop gain is thus contrary to the
	hypotheses of part (ii) of Proposition 7.12. Now let us examine the Nyquist contour for the
	system. In Figure 7.17 we show the Nyquist contour, with the right plot showing a blowup
	around the origin. This is a pretty messy Nyquist contour, but a careful accounting of
	what is going on will reveal that there is actually a total of zero encirclements of −1 + i0.
	Thus the closed-loop system with loop gain RL is IBIBO stable. This shows that the
	converse of Proposition 7.12 is not generally true.
4.

The above examples illustrate that one might wish to make the phase margins large and positive, and make the gain margins large. However, this is only a rough rule of thumb. In Exercise E7.10 the reader can work out an example where look at one stability margin while ignoring the other can be dangerous. The following example from the book of Zhou [1996] indicates that even when looking at both, they do not necessarily form a useful measure of stability margin.

22/10/2004

7.2 The relationship between the Nyquist contour and the Bode plot

301

	30
	20
dB	10
dB
	0
	-10
	-20	-1	0	1	2	3	4	5
		-1	0	1	2	3	4	5

log ω

	150
	100
deg	50
	0

	-50
	-100
	-150
	-1	0	1	2	3	4	5

log ω

		Figure 7.16		Bode plot for BIBO stable transfer function with pos-
		itive phase margins
	40
	20							2
	20
								1.5
Im								1
Im	0							0.5
	0

							Im	0
	-20							-0.5
								-1
								-1.5
	-40
	0	10	20	30	40	50		-1.5	-1	-0.5	0	0.5	1	1.5	2
			Re								Re

Figure 7.17 Nyquist contour for BIBO stable transfer function with positive phase margins

7.14 Example We take as our plant

2 − s RP (s) = 2s − 1,

and we consider two controllers, both of which may be veriﬁed to be stabilising:

	s + 33	s + 11	17 s2 +		3 s + 1
RC,1(s) = 1, RC,2(s) =	10	20	10		2	.
RC,1(s) = 1, RC,2(s) =	33 s + 1	11 s + 1 s2		+ 3 s + 17		.
	10	20		2	10

302

7 Frequency domain methods for stability

22/10/2004

This second controller is obviously carefully contrived, but let us see what it tells us. In Figure 7.18 are shown the Nyquist plots for the loop gain RC,1RP and RC,2RP . One can see

	1.5		1.5
	1		1
	0.5		0.5
Im	0	Im	0
	0		0
	-0.5		-0.5
	-1		-1

-2 -1.5 -1 -0.5 0		0.5	1	-2	-1.5	-1 -0.5 0	0.5	1
Re						Re
	1.5
	1
	0.5
Im	0
	-0.5
	-1
	-2	-1.5	-1 -0.5	0	0.5	1
			Re
Figure 7.18 Nyquist plots for plant RP (s) = 22s−−s1						with controller
RC,1 (top left), with controller RC,2 (top right), and both (bot-

tom)

that the gain and phase margins for the loop gain RC,2RP are at least as good as those for the loop gain RC,1RP , but that the Nyquist contour for RC,2RP passes closer to the critical point −1 + i0. This suggests that gain and phase margin may not be perfect indicators of stability margin.

With all of the above machinations about gain and phase margins out of the way, let us give perhaps a simpler characterisation of what these notions are trying to capture. If the objective is to stay far way from the point −1 + i0, then the following result tells us that the sensitivity function is crucial in doing this.

7.15 Proposition inf |−1 − RL(iω)| = kSLk−∞1.

ω>0

22/10/2004

7.3

Robust stability

303

Proof We compute

|− −

ω>0 |

ω>0

(iω)

1 + R

inf

(iω)

supω>0

−1

1 + RL(iω)

−1

k∞

as desired.

The results says, simply, that the point on the Nyquist contour which is closest to the point −1 + i0 is a distance kSLk−∞1 away. Thus, to increase the stability margin, one may wish to make the sensitivity function small. This is a reason for minimising the sensitivity function. We shall encounter others in Sections 8.5 and 9.3.

Let us illustrate Proposition 7.15 on the two loop gains of Example 7.14.

7.16 Example (Example 7.14 cont’d) We consider the plant transfer function RP and the two controller transfer functions RC,1 and RC,2 of Example 7.14. The magnitude Bode plots of the sensitivity function for the two loop gains are shown in Figure 7.19. As expected, the

	8								8
	6								6
dB	4							dB	4
dB								dB
	2								2
	0								0
	-1.5	-1	-0.5	0	0.5	1	1.5	2

log ω

-1.5

-1

-0.5

0.5

1.5

log ω

Figure 7.19 Magnitude bode plot for the sensitivity function with loop gain RC,1RP (left) and RC,2RP (right)

peak magnitude for the sensitivity with the loop gain RC,1RP is lower than that for RC,2RP , reﬂecting the fact that the Nyquist contour for the former is further from −1 + i0 than for the latter.

7.3 Robust stability

We now return to the uncertainty representations of Section 4.5. Let us recall here the

¯	< ∞,
basic setup. Given a nominal plant RP and a rational function Wu satisfying kWuk∞	< ∞,
¯
we denote by P×(RP , Wu) the set of plants satisfying
¯
RP = (1 + Wu)RP ,

k k ≤ ¯

where ∞ 1. We also denote by P+(RP , Wu) the set of plants satisfying

RP = RP + Wu,

where k k∞ ≤ 1. This is not quite a complete deﬁnition, since in Section 4.5 we additionally

imposed the requirement that the plants in P

, W

) and P

, W

) have the same

number of unstable poles as does RP .

304

7 Frequency domain methods for stability

22/10/2004

rˆ(s)	RC (s)	RP (s)	yˆ(s)
	−

Figure 7.20 Unity gain feedback loop for robust stability

Now we, as usual, consider the unity gain feedback loop of Figure 7.20. We wish to design a controller RC which stabilises a whole set of plants. We devote this section to a precise formulation of this problem for both uncertainty descriptions, and to providing useful necessary and su cient conditions for our problem to have a solution.

Let us be precise about this, as it is important that we know what we are saying.

¯		< ∞.	A
7.17 Deﬁnition Let RP , Wu R(s) be a proper rational functions with kWuk∞		< ∞.	A
¯	¯
controller RC R(s) provides robust stability for P×(RP , Wu) (resp. P+(RP , Wu)) if
		¯
the feedback interconnection of Figure 7.20 is IBIBO stable for every RP P×(RP , W )u)
¯
(resp. for every RP P+(RP , Wu)).

Now we provide conditions for determining when a controller is robustly stabilising, breaking our discussion into that for multiplicative, and that for additive uncertainty.

7.3.1 Multiplicative uncertainty The following important theorem gives simple conditions on when a controller is robustly stabilising. It was ﬁrst stated by Doyle and Stein [1981] with the ﬁrst complete proof due to Chen and Desoer [1982].

be a proper rational functions with kWuk∞ < ∞, and

7.18 Theorem Let RP , Wu R(s)

IBIBO stable in the interconnection of

suppose that RC R(s) renders the nominal plant RP

, Wu) if and only if

∞ < 1,

Figure 7.20. Then RC provides robust stability for P×(RP

WuTL

loop gain RC R¯P .

function, for the

where TL is the complementary sensitivity function, or closed-loop transfer

Proof

Let RC R(s).

, Wu) denote RL(RP ) = RC RP .

By deﬁnition

For RP P×(RP

of P

, W

and R

share the same imaginary axis poles for every

) it follows that R

RP P×(RP , Wu). For r, R > 0 and for RP

P×(RP

, Wu) let NR,r(RP ) be the (R, r)-

Nyquist contour for RL(RP ).

Now we make a simple computation:

Wu(iω L

< 1

WuTL

∞

Wu(iω)R¯L(iω)

< 1,

)T (iω)

< 1,

1 + R¯L(iω)

(iω)R¯

(iω)

< 1 + R¯

(iω) , ω

Wu(iω)RL(iω)

−

RL(

−

iω) ,

ω .

This gives a

simple

interpretation of the condition

W T

∞

< 1.

We note that

−

− −

(iω). Thus the condition

RL(iω)

is the distance from the point

1+i0 to

the point R

22/10/2004					7.3 Robust stability		305
	¯		< 1 is equivalent to the condition that the open disk of radius			¯
	W2TL	∞				Wu(iω)RL(iω)
		∞	¯		(iω) not contain the point 1 + i0. This is depicted in Figure 7.21. It is this
centred at R				L
				L
					−

−1 + i0

¯ ¯

( ) ¯ ( )

¯Wu iω RL iω ¯

¯ ( )

RL iω

Figure 7.21

Interpretation of robust stability condition for multi-

plicative uncertainty

interpretation of the condition

∞ < 1 we shall employ in the proof.

W2TL

have

< 1. Note that for ω

First suppose that WuT¯L

∞

R and for RP

P×(R¯P , Wu) we

RL(RP )(iω) = RC (iω)RP (iω) = (1 + Δ(iω)Wu(iω))RC (iω)R¯P (iω).

Thus the point RL(RP )(iω) lies in the closed disk of radius

with centre

Wu(iω)RL(iω)

Nyquist contour N

) that

RL(iω). From Figure 7.21 we infer that the point of the the

R,r

are the image of points on the imaginary axis will follow the

points on the Nyquist contour

NR,r(RP ) while remaining on the same “side” of the point −1 + i0.

Since Wu RH∞ and

since

have the same poles on iR and the same number of

is allowable, RL(RP ) and RL

poles in C+. Thus by choosing r0 < 0 su ciently small and for R0 > 0 su ciently large, the

− ¯ number of clockwise encirclements of 1 + i0 by NR,r(RP ) will equal the same by NR,r(RP )

for all R > R0 and r < r0. From Theorem 7.6 we conclude IBIBO stability of the closed-loop

system with loop gain RL(RP ).

Now suppose that

∞ ≥ 1. As depicted in Figure 7.21, this implies the existence

WuTL

open disk of radius

(iω¯) contains the

of ω¯

0 so that the

(iω¯)R

(iω¯) centred at R

≥

. We claim

closed disk of

radius r centred at s

point −1 + i0. Denote by D(s0, r) be the

that for each ω > 0, the map from

P×

(R¯

¯ u

(iω)R¯

(iω)

deﬁned by

, W

) to D R¯

(iω),

Wu)RP

RC (iω)RP (

= (1 +

7→

iω)

(7.4)

is surjective. The following lemma helps us to establish this fact.

1 Lemma For any ω¯ ≥ 0 and for any θ (−π, π] there exists a function Gθ RH+∞ with the properties

306

7 Frequency domain methods for stability

22/10/2004

(i)]Gθ(iω¯) = θ and

(ii)the map ω 7→G|θ(iω)| has a maximum at ω = ω¯.

Proof

Clearly if θ = 0 we may simply deﬁne Gθ(s) = 1. Thus suppose that θ 6= 0. Consider

the rational function

ω2

Tζ,ω0 (s) =

s2 + 2ζω0s + ω02

ζ, ω0

> 0. In Exercise E4.6 it was shown that for

< ζ < 1 the function ω

7→ |

√

T (iω)

achieves a unique maximum at ωmax = ω0

1 − 2ζ2. Furthermore, the phase at this maxi-

−

Thus, as ζ

varies in the interval ( 1

, 1), the phase at

mum is given by atan2(ζ,

1 2ζ

√2

between

−4

and 0.

the frequency ωmax varies p

Now, given θ (−π, π], deﬁne

θ˜ = (5

2π−θ

, θ > 0.

θ < 0

−

Thus θ is guaranteed to live in the interval (−4

, 0). Therefore, there exists ζ (

√2

, 1) so

that θ˜ = atan2(ζ, p

). Now deﬁne ω0

5so that ωmax = ω¯. Now deﬁne

1 − 2ζ2

(Tζ,ω10 0

θ > 0.

Gθ =

Tζ,ω0

θ < 0

This function Gθ, it is readily veriﬁed, does the job.

We now resume showing that the map deﬁned by (7.4) is surjective. It clearly su ces

lies

to show that the image of every point on the boundary of D RL(iω), Wu(iω)RL(iω)

in the image of the map, since all other points can then be obtained by scaling

For

(

π, π] let

−

iφ

sφ = RL(iω) +

Wu(iω)RL(iω)

s = R (iω)R

(iω) =

(iω)R

(iω)

be a point on the boundary of D RL(iω),

R (iω) + Δ(iω)W (iω)R (iω)

for an appropriate choice of allowable

. Thus

RH∞+ should necessarily satisfy

iφ

Δ(iω)Wu(iω)RL(iω) = Wu(iω)RL(iω) e

It follows that

Δ(iω)

= 1 and that

]

(iω)R¯

Δ(iω) +

(iω)) = φ. Therefore, deﬁne

− ¯

θ = φ ](Wu(iω)RL(iω)).

If Gθ is as deﬁned above, then deﬁning

Δ(s) = Gθ(s)

|Gθ(iω)|

does the job.

The remainder of the proof is now straightforward. Since the map deﬁned by (7.4) is

surjective, we conclude that there exists an allowable so that if RP = (1 + Wu)RP we have

RL(RP )(iω¯) = RC (iω¯)RP (iω¯) = −1 + i0.

This implies that the Nyquist contour NR,r(RP ) for R su ciently large passes through the point −1 + i0. By Theorem 7.6 the closed-loop system is not IBIBO stable.

22/10/2004 7.3 Robust stability 307

The proof of the above theorem is long-winded, but the idea is, in fact, very simple. Indeed, the essential observation, repeated here outside the conﬁnes of the proof, is that the condition kWuTLk ∞ < 1 is equivalent to the condition, depicted in Figure 7.21, that,

for each frequency ω, the open disk of radius			W		¯		(iω)	¯		(iω), not
				u	(iω)R	L		and centred at R	L
contain the point −1 + j0.
One may also “reverse engineer” this	problem as well. The idea here is as follows.
	¯
		and a controller RC which IBIBO stabilises
Suppose that we have our nominal plant RP

the closed-loop system of Figure 7.20. At this point, we have not speciﬁed a set of plants over which we want to stabilise. Now we ask, “What is the maximum size of the allowable

if the perturbed plant is to be stabilised by RC ?” The following result

perturbation to RP

makes precise this vague question, and provides its answer.

7.19 Proposition

proper. Also, suppose that the interconnection

Let RP , RC R(s) with RP

be the closed-loop transfer function

of Figure 7.20 is IBIBO stable with RP = RP . Let TL

. The following statements hold:

for the loop gain RL = RC RP

−1

(i)

for P×(R¯P , Wu);

∞

then RC provides robust stability

if Wu RH∞ has the property that kWuk∞ < TL

robustly

−1

P u

(ii)

≥

P×

for any β

∞

there exists Wu RH∞ satisfying kWuk = β so that RC does not

stabilise

(R , W ).

Proof (i) This follows directly from Theorem 7.18.

(ii) This part of the result too follows from Theorem 7.18. Indeed, let ω¯ > 0 be a frequency

for which

One can readily deﬁne Wu

TL(iω¯)

TL ∞.

RH∞ so that |Wu(iω¯)| =

β. It then follows that W

1, so we may apply Theorem 7.18.

k∞

∞

≥

Roughly, the proposition tells us

that as long as we choose the uncertainty weight W

−1

that its H

-norm is bounded by

, then we are guaranteed that R

will provide robust

∞

-norm of W

−1

stability. If the H

exceeds

, then it is possible, but not certain, that R

∞

will not robustly stabilise. In this latter case, we must check the condition of Theorem 7.18. Let us illustrate the concept of robust stability for multiplicative uncertainty with a fairly

simple example.

7.20 Example We take as our nominal plant the model for a unit mass. Thus

RP (s) = s2 .

The PID control law given by

1 RC (s) = 1 + 2s + s

may be veriﬁed to stabilise the nominal plant; the Nyquist plot is shown in Figure 7.22. To model the plant uncertainty, we choose

Wu(s) = s + 1, a > 0,

which has the desirable property of not tailing o to zero as s → ∞; plant uncertainty will generally increase with frequency. We shall determine for what values of a the controller RC

provides robust stability for P×(RP , Wu).

308

7 Frequency domain methods for stability

22/10/2004

	15
	10
	5
Im	0
	-5
	-10
	-10	-5	0	5	10	15
			Re

Figure 7.22 Nyquist plot for PID controller and nominal plant

According to Theorem 7.18 we should determine for which values of a the inequality

¯ ¯

WuTL ∞ < 1 is satisﬁed. To determine WuTL ∞ we merely need to produce magnitude

Bode plots for WuTL for various values of a, and determine for which values of a the maximum

magnitude does not exceed 0dB. In Figure 7.23 is shown the magnitude Bode plot for WuTL

0								0
0
-10								-10
-10
dB-20							dB	-20
dB-20							dB
-30								-30
-30
-40								-40
-40
-1.5	-1	-0.5	0	0.5	1	1.5	2

log ω

-1.5

-1

-0.5

0.5

1.5

log ω

Figure 7.23 The magnitude Bode plot for WuTL when a = 1 (left) and when a = 34 (right)

with a = 1. We see that we exceed 0dB by about 2.5dB. Thus we should reduce a by a factor K having the property that 20 log K = 2.5 or K ≈ 1.33. Thus a ≈ 0.75. Thus let us take a = 34 for which the magnitude Bode plot is produced in Figure 7.23. The magnitude is bounded by 0dB, so we are safe, and all plants of the form

			RP (s) = 1 +	3 sΔ(s)	1
				4	1
				s + 1	s2
will be stabilised by RC if			is allowable. For example, the Nyquist plot for RP			when
=	ss+2−1	(which, it can be checked, is allowable), is shown in Figure 7.24.

7.3.2 Additive uncertainty Now let us state the analogue of Theorem 7.18 for additive uncertainty.

22/10/2004						7.3	Robust stability				309
					6
					4
					2
				Im	0
					0
					-2
					-4
					-2	0	2	4	6		8
							Re
		Figure 7.24		Nyquist plot for perturbed plant under multiplicative
			uncertainty
7.21 Theorem Let			¯			be a proper rational functions with kWuk∞ < ∞, and
7.21 Theorem Let			RP , Wu R(s)			be a proper rational functions with kWuk∞ < ∞, and
suppose that RC			R(s)	renders the nominal plant					¯	IBIBO stable in the interconnec-
suppose that RC			R(s)	renders the nominal plant					RP	IBIBO stable in the interconnec-
tion of Figure 7.20. Then RC provides robust								stability		for	¯
tion of Figure 7.20. Then RC provides robust								stability		for	P+(RP , Wu) if and only if
	¯			¯							¯
	WuRC SL	∞ < 1, where SL is the sensitivity function for the loop gain RC RP .

Proof We adopt the notation of the ﬁrst paragraph of the proof of Theorem 7.18. The following computation, mirroring the similar one in Theorem 7.18, is the key to the proof:

Wu(iω)RC (iω)S¯L(iω) < 1, ω R

WuRC SL

∞ < 1

Wu(iω)RC (iω)

< 1, ω

1 + R¯L(iω)

(iω)R

(iω)

< 1 + R¯

(iω) , ω

Wu(iω)RC (iω)

RL(

− −

¯ iω)

ω .

The punchline here is thus that the condition

WuRC S¯L

< 1 is equivalent to the con-

|Wu(iω)RC (iω)| with centre RL(iω). This is

dition that the point −1 + i0 not be contained, for any

∞

, in the open disk of radius

¯	depicted in Figure 7.25.

The remainder of the proof now very much follows that of Theorem 7.18, so we can

P+(R¯P , Wu) we have

L ∞

< 1.

For ω

and for R

safely omit some details. First assume that

RL(RP )(iω) = RC (iω)RP (iω) = RL(iω) + Δ(iω)RC (iω)Wu(iω).

Thus the point RL(RP )(iω) lies in the closed disk of radius |Wu(iω)RC (iω)| with centre

RL(iω). We may now simply repeat the argument of Theorem 7.18, now using Figure 7.25 rather than Figure 7.21, to conclude that the closed-loop system with loop gain RL(RP ) is IBIBO stable.

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7 Frequency domain methods for stability

22/10/2004

−1 + i0

|Wu(iω)RC (iω)|

RL(iω)

Figure 7.25 Interpretation of robust stability condition for addi-

tive uncertainty

Now suppose that kWuRC k∞ ≥ 1.

Thus there exists ω¯ ≥ 0 so that |Wu(iω¯)RC (iω¯)| ≥

Thus by

Figure 7.25,

follows

that −1 + i0 is contained

the open

disk

radius |Wu(iω¯), RC (iω¯)|

with

centre

the

proof

Theorem 7.18,

RL(iω¯). As in

may

employ

Lemma 1

that proof to show that

the

map

from P+(RP , Wu)

D RL(iω), |Wu(iω)RC (iω)|

deﬁned by

RP = RP + Wu 7→RC (iω)RP (iω)

is surjective for each ω > 0. In particular, it follows that there exists an allowable

giving

P+(RP , Wu) so that RL(RP )(iω¯) = −1 + i0. IBIBO instability now follows from

Theorem 7.6.

Again, the details in the proof are far more complicated than is the essential idea. This essential idea is that for each ω R the open disk of radius |Wu(iω)RC (iω)| and centre

RL(iω) should not contain the point −1 + i0. This is what is depicted in Figure 7.25.

We also have the following, now hopefully obvious, analogue of Proposition 7.19.

proper. Also, suppose that the interconnection

7.22 Proposition Let RP , RC R(s) with RP

be the sensitivity function for the loop

of Figure 7.20 is IBIBO stable with RP = RP . Let SL

gain RL = RC RP . The following statements hold:

k∞

−1

(i)

stability for P+(R¯P , Wu);

RC SL

∞

then RC provides robust

if Wu

RH∞ has the property that

(ii)

for any β ≥

−1

there exists Wu

RC SL

∞

RH∞ satisfying kWuk = β so that RC does

not robustly

stabilise

, W

P+

An example serves to illustrate the ideas for this section.

as Wu = (s + 1)2

22/10/2004

7.4 Summary

311

7.23 Example (Example 7.20 cont’d) We carry on look at the nominal plant transfer function

RP (s) = s2 which is stabilised by the PID controller

1 RC (s) = 1 + 2s + s.

Note that we may no longer use our Wu from Example 7.20 as our plant uncertainty model.

Indeed, since RC is improper, Wu is proper but not strictly proper, and SL is proper but not

strictly proper (one can readily compute that this is so), it follows that WuRC SL is improper. Thus we modify Wu to

to model the plant uncertainty. Again, our objective will be to determine the maximum

		¯
value of a so that RC provides robust stability for P+(RP , Wu). Thus we should ﬁnd the
maximum value for a so that	WuRC S¯L ∞ < 1. In Figure 7.26 is shown the magnitude

	0								0
	0
	-20								-20
	-20
	-40								-40
dB	-40							dB
	-60								-60
	-60
	-80								-80
	-80
	-100								-100
	-100
	-120								-120
	-120
	-1.5	-1	-0.5	0	0.5	1	1.5	2

log ω

-1.5

-1

-0.5

0.5

1.5

log ω

Figure 7.26 The magnitude Bode plot for WuRC SL when a = 1 (left) and when a = 12 (right)

Bode plot for WuRC SL when a = 1. From this plot we see that we ought to reduce a by a factor K having the property that 20 log K = 6 or K ≈ 2.00. Thus we take a = 12 , and in Figure 7.26 we see that with this value of a we remain below the 0dB line. Thus we are guaranteed that all plants of the form

7.4 Summary

In this chapter we have provided a graphical method for analysing the closed-loop stability of a single loop feedback interconnection. You should understand the following things.

1. You need to understand what the Nyquist criterion, in the form of Theorem 7.6, is saying.

312

7 Frequency domain methods for stability

22/10/2004

	15
	10
	5
Im	0
	-5
	-10
	-5	0	5	10	15	20
			Re

Figure 7.27 Nyquist plot for perturbed plant under additive uncertainty

2.Drawing Nyquist plots can be a bit tricky. The essential idea is that one should focus on the image of the various parts of the Nyquist contour R,r. In particular, one should focus on the image of the imaginary axis for very large and very small frequencies. From there one can try to understand of something more subtle is happening.

3.The gain and phase margins are sometimes useful measures of how close a system is to being unstable.

Exercises for Chapter 7	313
Exercises

E7.1 For the following rational functions R and contours C, explicitly verify the Principle

of the Argument by determining the number of encirclements of the origin by the

image of C under R.

(b)

R(s) = s and C =

(a)

R(s) =

and C =

eiθ

− π < θ ≤ π .

(c)

iθ−

≤

R(s) = s2+1 and C

π < θ

eiθ

π .

−

≤

iθ

Let F (s) =

ln s, and

π < θ π .

−

≤

E7.2

consider the contour C =

π < θ

π . Does the

Principle of the Argument hold for the image of C

under F ? Why or why not?

E7.3 For the following loop gains,

(a)

RL(s) =

, α > 0,

s2(s+1)

(b)

RL(s) =

, α > 0, and

s(s−1)

(c)

α(s+1)

RL(s) =

, α > 0,

s(s−1)

do the following.

1.Determine the Nyquist contour for the following loop gains which depend on a parameter α satisfying the given conditions. Although the plots you produce may be computer generated, you should make sure you provide analytical explanations for the essential features of the plots as they vary with α.

2.Draw the unity gain feedback block diagram which has RL as the loop gain.

3.For the three loop gains, use the Nyquist criterion to determine conditions on α for which the closed-loop system is IBIBO stable.

4.Determine IBIBO stability of the three closed-loop systems using the Routh/Hurwitz criterion, and demonstrate that it agrees with your conclusions using the Nyquist criterion.

In this next exercise you will investigate some simple ways of determining how to “close” a Nyquist contour for loop gains with poles on the imaginary axis.

E7.4 Let RL R(s) be a proper loop gain with a pole at iω0 of multiplicity k. For concreteness, suppose that ω0 ≥ 0. If ω0 6= 0, this implies that −iω0 is also a pole of multiplicity k. Thus we write

	((s2			+ω2)k R(s), ω0		= 0,
		1	R(s),		ω0	= 0
		k	R(s),		ω0	= 0
RL(s) =		s
RL(s) =				1		6
				0		6

where iω0 is neither a pole nor a zero for R.

(a) Show that

lim RL(iω0 + reiθ) = rak e−ikθ + O(r1−k).

r→0

where a is either purely real or purely imaginary. Give the expression for a.

(b) For ω0 = 0 show that

ω→ω0,− ]	L	(iω)	−	π	, 0,	π	, π .
				2		2
lim	R

Denote θ0 = limω→ω0,− ]RL(iω).

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7 Frequency domain methods for stability

22/10/2004

(c)Determine a relationship between θ0, and k and a.

(d)Determine the relationship between

lim ]RL(iω)

ω→ω0,+

and θ0. This relationship will depend on k.

(e)Conclude that for a real rational loop gain, the “closing” of the Nyquist contour is always in the clockwise direction and the closing arc subtends an angle of kπ.

E7.5 Let (N, D) be a proper SISO linear system in input/output form, and let n = deg(D) and m = deg(N).

(a)Determine limω→∞ ]TN,D(iω) and limω→∞ ]TN,D(iω).

(b)Comment on part (a) as it bears on the Nyquist plot for the system.

E7.6 Consider the SISO linear system (N(s), D(s)) = (1, s2 + 1) in input/output form.

(a)Sketch the Nyquist contour for the system, noting that the presence of imaginary axis poles means that the (∞, r)-Nyquist contour is not bounded as r → 0.

Consider the SISO linear system (N (s), D (s)) = (1, s2 + s + 1) which now has a bounded Nyquist contour for > 0.

(b)Show, using the computer, that the Nyquist contour for (N , D ) approaches that for the system of part (a) as → 0.

E7.7 Let (N, D) be a SISO linear system in input/output form with deg(N) = 0 and let

ω¯ = max {Im(p) | p is a root of D} .

Answer the following questions.

(a)Show that |TD,N (iω)| is a strictly decreasing function of ω for |ω| > ω¯.

(b)Comment on part (a) as it bears on the Nyquist plot for the system.

E7.8 Formulate and prove a condition for IBIBO stability for each of the two interconnected systems in Figure E7.1, using the Principle of the Argument along the lines of the

rˆ(s)		RC (s)	RP (s)	yˆ(s)
rˆ(s)	−	RC (s)	RP (s)	yˆ(s)
	−

RS (s)

rˆ(s)	RI (s)	RC (s)	RP (s)	yˆ(s)
		−

Figure E7.1 Alternate conﬁgurations for Nyquist criterion

Nyquist criterion of Theorem 7.6.

Exercises for Chapter 7			315
E7.9 Let
	ω02
RP (s) =		, RC (s) = 1.
	s2 + 2ζω0s
Plot the upper phase margin of the closed-loop system as a function of ζ.

In the next exercise you will investigate gain and phase margins for a simple plant and controller. You will see that it is possible to have one stability margin be large while the other is small. This exercise is taken from the text of [Zhou 1996].

E7.10 Take

a − s s + b RP (s) = as − 1, RC (s) = bs + 1,

where a > 1 and b > 0. Consider these in the standard unity gain feedback loop.

(a)Use the Routh/Hurwitz criterion to show that the closed-loop system is IBIBO stable if b (a1 , a).

Take b = 1.

(b)Show that Kmin = a1 , Kmax = a, Φmin is undeﬁned, and Φmax = arcsin aa22−+11 .

(c)Comment on the nature of the stability margins in this case.

Fix a and take b (a1 , a).
(d) Show that	1
lim Kmin =	1	,	lim Kmax = a2,	lim Φmax = 0.
lim Kmin =	a2	,	lim Kmax = a2,	lim Φmax = 0.
b→a	a2		b→a	b→a

(e)Comment on the stability margins in the previous case.

(f)Show that

b→a1	b→a1	b→a1	a2	+ 1
lim Kmin = 1,	lim Kmax = 1,	lim Φmax = 2 arcsin	a2	− 1	.

(g) Comment on the stability margins in the previous case.

In this chapter we have used the Nyquist criterion to assess IBIBO stability of input/output feedback systems. It is also possible to use the Nyquist criterion to assess stability of static state feedback, and the following exercise indicates how this is done.

E7.11 Let Σ = (A, b, ct, D) be a SISO linear system.

(a)Let f Rn. Show that if Σ is controllable then the closed-loop system Σf is internally asymptotically stable if and only if

ft(sIn − A)−1b			RH+ .
1 + ft(sIn	−	A)−1b	∞
	−

Thus, we may consider closed-loop stability under static state feedback to be equivalent to IBIBO stability of the interconnection in Figure E7.2. The Nyquist criterion can then be applied, as indicated in the following problem.

(b)Using part (a), prove the following result.

Proposition Let Σ = (A, b, ct, 01) be a controllable SISO linear system and let np be the number of eigenvalues of A in C+. For a state feedback vector f Rn deﬁne the loop gain Rf (s) = ft(sIn−A)−1b. Assuming that (A, f) is observable, the following statements are equivalent:

316

7 Frequency domain methods for stability

22/10/2004

r s

)

(sI n

−1

y s

)

−

ˆ(

−

Figure E7.2 A feedback interconnection for static state feedback

(i)the matrix A − bft is Hurwitz;

(ii)there exists r0, R0 > 0 so that the image of R,r under Rf encircles the point −1 + i0 np times in the clockwise direction for every r < r0 and R > R0.

To be concrete, consider the situation where

A =	−1	1	,	b =	1 .
	0	1			0

For this system, answer the following questions.

(d)How many eigenvalues are there for A in C+?

(e)Plot the Nyquist contour for Figure E7.2 and verify that the Nyquist criterion of part (b) holds.

E7.12 For the plant uncertainty set

RP = ¯ , k k∞ ≤ 1,

1 + WuRP

use the idea demonstrated in the proofs of Theorems 7.18 and 7.21 to state and prove a theorem providing conditions for robust stability.

E7.13 For the plant uncertainty set

	¯
RP =	RP	,	k k∞ ≤ 1,
	1 + Wu

use the idea demonstrated in the proofs of Theorems 7.18 and 7.21 to state and prove a theorem providing conditions for robust stability.

Exercises for Chapter 7

317

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