IN 227 Control Systems Design

Lecture 20

Instructor: G R Jayanth Department of Instrumentation and Applied Physics

Ph: 22933197 E-mail: jayanth@isu.iisc.ernet.in

Control of unstable plants • Unstable plants possess one or more poles in the right half of the complex plane (RHP).

A plant that possesses one unstable pole at s=a can be written as PU(s)=P1(s)/(s-a), where, P1(s) does not contain any pole in the RHP.

• The way an unstable pole would affect the closed-loop dynamics can be seen by writing PU(s)=[(s+a)/(s-a)][PMP(s)] where, PMP(s)=P1(s)/(a+s), is a minimum phase “well behaved” plant model and (s+a)/(s-a) is a Blaschke product. Let us now assume the a well behaved controller C(s), i.e., a controller with no right half plane poles, is used to control the plant and the resulting loop gain is LU(s)=C(s)PU(s). For reference, we shall also define the minimum phase counterpart LMP(s)=C(s)PMP(s). Thus, LU(s)=[(s+a)/(s-a)]LMP(s).

• We can easily draw the bode plot of the minimum phase part, viz., LMP(jω). We assume, for the sake of illustration, that LMP(jω) contains an integrator so that the phase of LMP(jω) is -90˚ and the gain roll-off is -20dB/decade at low frequencies. The gain characteristics of LU(jω) is the same as that of LMP(jω) since |(a+jω)/(a-jω)|=1 at all frequencies (Fig. top right). The gain crosses over at ωgc. The phase of LMP(s) gradually decreases further and in general crosses -π at a certain frequency. The Blaschke product (a-jω)/(a+jω) possesses a phase lag of -π+2tan-1(ω/a). Thus, the overall phase characteristic starts at -3π/2, increase initially, cross-over at ωpcl, continue increasing until the phase lag of LMP(s) begins to dominate. At this point the phase crosses over again at a higher frequency ωpcu, and continues to decrease beyond ωpcu (Fig. top right).

• In contrast with the general phase characteristics of minimum phase plants, the phase characteristics above appear unfamiliar. For example, how can we define the gain margin when we have two phase cross-over frequencies? Does the conventional phase margin, defined as the phase at ωgc, carry the same significance as in minimum phase systems? How does the position of ωgc relative to ωpcl and ωpcu affect stability?

• Bode plots seldom answer questions related to stability independently: they are merely tools for achieving control performance. We need to transfer the open-loop system into the Nyquist domain and examine its stability.

20log |L|

logω

ωgc

20log |LU| (=20log |LMP|)

logω

ωpcu

12tan ( / )U MPL L a

-π

-20 dB/decade

ωpcl

-3π/2

• Before answering the questions posed in the previous slide, we first need to develop the condition under which an open-loop system with one unstable pole would result in a stable closed loop system: The denominator transfer function of the overall closed-loop system is

, where, LMP=NMP/DMP.

• We note that 1+LU has a pole in the RHP. Thus, when the s-variable is taken round the contour C once in the clock-wise direction, the term 1/(s-a) encircles the critical point -1+j0 once in the counter-clockwise direction. Therefore, if the numerator polynomial of 1+LU does not possess any zeros on the RHP, i.e., the closed-loop system is stable, then 1+LU should encircle the critical point in the counter clockwise direction exactly once. If the net encirclement is found to be zero, then 1+LU has to possess a zero within C whose clockwise encirclement cancels that of 1/(s-a). More generally, number of unstable poles=number of clockwise encirclements+1.

• In analyzing the stability for LU, we have to look at four cases: (a) ωgc <ωpcl (b) ωpcl < ωgc < ωpcu (c) ωgc >ωpcu

(d) the phase lag does not cross –π radians at any frequency. The Nyquist contour C is shown on the right. Since we have an integrator, and hence a singularity at the origin, we have to introduce an infinitesimally small kink in the contour at s=0 to avoid the singularity (viz., a semicircle of radius r→0).

• We see from the Nyquist plots of LU (figs. below) that for cases (a), (c) and (d), the critical point is encircled once in the clockwise direction, and thus, the closed loop system would possesses two unstable poles. Only for case (b), i.e., when the grain crossover frequency is between the two phase crossover frequencies (ωpcl < ωgc < ωpcu ), is the critical point encircled in the counter clockwise direction once and thus, the closed-loop system would be stable.

Closed loop stability of unstable plants

( ) ( )1 1

( )

MP MPU MP

MP

s a D s a Ns aL L

s a s a D

R→∞

C

0

jω

σ a

s a

s

C

0

jω

σ a

s

-1+j0 -1+j0 -1+j0

Case (a) Case (b) Case (c) Case (d)

-1+j0

Theoretical minimum gain cross-over frequency for specified PM

• We shall now work out the minimum gain cross-over frequency necessary to achieve a specified phase margin PM.

• Let LU cross 0 dB at ωgc and let the slope of the magnitude characteristic in the neighborhood of ωgc be -40α db/decade. This is also the slope of LMP. Bode’s gain-phase relationship tells us that the phase of LMP at any frequency ω is determined predominantly by the slope of the magnitude characteristic near ω alone, regardless of the actual shape of the magnitude characteristic of LMP at other frequencies. Thus, the phase of LMP at ωgc is -απ rad. Consequently, the net phase of the loop gain LU at ωgc is

To satisfy the prescribed phase margin PM, we note that . Equating these two we get

• Since tan(θ) is an increasing function of θ, we note from the equation above that for a given PM, the gain cross-over frequency is an increasing function of α. Thus, the lowest gain cross-over frequency is obtained for the smallest value of α. However, since the low frequency loop gain needs to be greater than 0dB, we require that α>0 in order for the magnitude characteristic to cross the 0dB line. Thus, for the given phase margin PM, the absolute maximum gain cross-over frequency is obtained when α~0 and is given by

• We see, therefore, that the minimum possible value of the gain cross-over frequency ωgc is theoretically limited to a.tan(PM/2) due to the RHP pole at s=a.

20log |L|

logω

ωgc

-40α dB/decade

logω

ωpcu -π ωpcl

-3π/2

1( ) ( ) 2 tan ( / )gc gc gcj a j a a

( )U gcL PM

tan2

gc PM

a

PM

tan2

gc

PMa

• Since we have two phase cross-over frequencies, we have two corresponding gain margins GML=20log|LU(jωpcl)| and GMU=-20log|LU(jωpcu)|. For stability, we require GML,GMU>0dB.

• Since ωpcl < ωgc, the minimum gain cross-over frequency necessary is decided by the specified GML. We shall therefore focus on achievement of the specified GML and PM.

• By definition, . Assuming that the slope of LU remains approximately -40α db/decade between ωgc and ωpcl, and thus, the phase of LMP remains at –απ at

ωpcl,we have, or equivalently,

• Since the slope of LU remains approximately -40α db/decade between ωgc and ωpc, (fig. below right), we have .Using the fact that and , we obtain,

• The last equation above is a transcendental equation in α. It is solved numerically to obtain α and subsequently used to evaluate ωgc. The figure below sketches the minimum value of ωgc (normalized with respect to a) for different phase and gain margin requirements.

Theoretical minimum gain cross-over frequency to satisfy the specified GM and PM

20log |L|

logω -40α dB/decade

logω

ωpcu -π

GML

ωpcl

-3π/2

PM

GMU

( )U pclL

12tan ( )pcl a tan( )2

pcl

a

ωgc

20log |LU|

logω

ωgc

-40α dB/decade

logω

ωpcl

UL

-π

GML

40 log gc pclGML tan( 2)pcl a

tan ( ) 2gc a PM tan ( ) 2log

40 tan( 2)

PMGML

GML(dB)

gc

a

tan( 2)PM

5dB 10dB

2

5

PM=30° 40°

20°

Consequences of the theoretical limits

• The theoretical limits to the minimum gain crossover frequency necessary to achieve the specified stability margins pose corresponding challenges to the control of unstable systems. Some of them are highlighted below.

• (1) Enhanced effect of measurement noise: The fact that the gain cross-over frequency has a lower limit indicates that even if we desire the benefits of feedback only to a relatively small frequency ω relative to the value of the unstable pole a, we have to keep the magnitude of the loop gain LU high up to significantly higher frequencies (usually several multiples of a). Thus, bandwidth of the transmission function relating the sensor noise to the output (-LU/(1+LU)) also needs to be typically several times a. Thus, we have to let in much more measurement noise, when compared to minimum phase plants, to reap the same benefits of feedback.

• (2) Plants with RHP pole and RHP zero: Suppose the plant possesses a RHP zero in addition to an RHP pole, i.e.,

the plant transfer function is of the form PNMP=(a-s)P1/(s-b), where P1 is a minimum phase transfer function,

then, if we use a minimum phase controller C, we can write the resulting non minimum phase loop gain

LNMP=CPNMP as , where is the minimum phase counterpart of the non

minimum phase loop gain. As we saw in the previous lecture, for the specified gain and phase margins, the term

imposes an upper limit on the gain crossover frequency ωgc to a value around a. Likewise, for the same gain and

phase margins, the term , imposes a lower limit on the gain cross-over frequency ωgc to a value around b.

Thus, if b>a, then it is impossible to design a controller that satisfies the specified gain and phase margins.

• Example: Suppose we desire a gain margin of 10dB and a phase margin of 30°: based on the graphs provided in the previous lecture, we see that the upper limit to the gain cross-over frequency is ωgc≤0.2a. To achieve the same specifications, we see from the sketch in the previous slide that the lower limit should be ωgc≥5b. Thus, if 5b>0.2a or b>0.04a, it is impossible to achieve the specified gain and phase margin requirements.

.NMP MP

a s s bL L

a s s b

1

( )( )MP

CPL

s a s b

a s

a s

s b

s b

Bode Sensitivity Integrals • The third consequence is that an unstable plant, even without a RHP zero, imposes

fundamental limits on the gain and phase margins in practical control systems. The controller in practical control systems possess two subsystems: the electronic subsystem Ce where the control law is implemented, and the physical subsystem Cp, also called the “actuator” (ex. motors, solenoid valves, etc…), which transforms electrical energy into changes in physical state. While the electronic subsystem can typically be designed to possess very high bandwidth, the physical subsystem is fundamentally limited in the frequency range over which it operates.

• We shall now explore the limits to achievable stability margins imposed by a very real practical constraint, viz., limited actuator bandwidth. This can be done in a very transparent manner by means of Bode sensitivity integrals.

• Bode sensitivity integrals: All the benefits of feedback are tied to achieving a high loop gain L(jω). The consequence of high L(jω) is the reduction in the sensitivity function S(jω)=1/(1+L(jω)), with resulting good rejection of disturbances and good robustness to variations in plant parameters. As control engineers, therefore, we desire zero sensitivity at all frequencies. However, Bode sensitivity integrals impose constraints on the achievable reduction in sensitivity. In particular, if the loop gain L is minimum phase, then the sensitivity function satisfies the first equation below. If L is open-loop unstable, then the corresponding sensitivity function satisfies the second equation below. Both of them can be derived from Cauchy’s theorem.

NMPPr

-

u e ( )eC s

x ( )pC s

Controller C(s)

0

0

ln ( ) 0

ln ( ) Re( )i

i

S j d

S j d p

For minimum phase L

For open-loop unstable L

(pi are the open loop poles in the RHP)

Bode sensitivity integrals

The sensitivity “dirt” • Bode sensitivity integrals are conservation laws: they state that the net area under the

log|S| curve is conserved. Thus, no matter what controller is used, it is impossible to reduce sensitivity at all frequencies. If our controller reduces sensitivity in a certain part of the frequency domain (ω< ω0 in the fig. right), then it will inevitably increase sensitivity in other parts of the frequency domain. Thus, log|S| is informally called sensitivity dirt (Ref. 2): we would like to have as little of it as possible, but if we dig a trench to remove it somewhere, the removed dirt has to be piled up or spread around somewhere else.

• In the case of minimum phase systems, the amount of dirt that gets piled up elsewhere is the same as the amount of dirt that got removed somewhere. However, in the case of open loop unstable systems, the amount of dirt that needs to be piled up is greater by the amount π∑pi than the amount of dirt that was removed (Fig. on the right). If the loop gain LU possesses a single unstable pole at s=a, then the excess sensitivity dirt that needs to be distributed is πa.

• Once we have removed sensitivity dirt up to a certain frequency ω0 in the interest of reaping the benefits of feedback, we need to decide on the strategy to distribute this dirt plus the excess dirt πa: do we distribute them uniformly over the reminder of the frequency domain or do we make a tall heap, i.e., greatly increase sensitivity, around some frequency ω1?

• To decide this, we need to understand the result of increased sensitivity: if we choose to make a tall heap at some frequency ω1, we note that log|S(jω1)|>>1. Thus, or equivalently . We note from the Nyquist plot shown on the right that for any loop gain L(jω), 1+L(jω) is the complex number starting at the critical point (-1+j0) and ending at the tip of the complex number L. Thus, , describes a circle of radius 1/K centered at the critical point.

• As seen from the schematic on the right, if the loop gain assumes values on the circle

, then the maximum phase margin achievable is PMmax~2sin-1(1/2K) and the maximum gain margin achievable is GMmax=1/(1-1/K).

• Thus, if K>>1, then PMmax~sin-1(1/K)~0 and GMmax~0dB. Thus, high sensitivity is synonymous with small phase and gain margins.

Log|S

|

ω

A2: Area of

heightened sensitivity

A1: Area of

reduced sensitivity

A1+A2=constant

Log|S

|

ω

Unstable L

Minimum phase L

11 [1 ( )] ( 1)L j K

11 ( ) 1L j K

11 ( ) . 1L j const K

Log|S

|

ω

Tall heap?

Thin layer?

-1+j0

1/K PMmax

1/GMmax

11 ( ) 1L j K

L(jω) 1+L(jω)

-1+j0

Radius 1/K Im(L)

Re(L)

ω0 ω1

ω0

ω0

(1-1/K)

Unit circle

Distributing the sensitivity dirt • We conclude, from the previous slide, that in the interest of stability, we need to minimize the

height of sensitivity dirt in the region ω>ω0. Since we have an infinitely large region from ω0 to ∞ , to spread the dirt, we may be tempted to spread it out as an infinitesimally thin layer at all frequencies, and thus, ensure that sensitivity is small everywhere. This procedure, however, assumes that it is possible to control the thickness of the dirt layer over all frequencies from ω0

to ∞ .In practice, though, all actuators possess a “bandwidth”, i.e., a frequency Ωa beyond which it is not possible to control their dynamic performance. Indeed their gain attenuates greatly for all larger frequencies and causes the loop gain L to tend to zero.

• Thus, the shape of log|S| cannot be controlled for ω>Ωa, and consequently, the area enclosed

by it is a constant (δ), i.e., . Since , we

conclude that . In other words, contrary to the theoretical

expectations, the frequency region available to spread the dirt is limited to Ωa.

• Since we would be reducing the dirt for frequencies between 0 and ω0 by the amount A1, we

note that . Thus, we see that . In other words, the

amount of dirt that we need to spread between ω0 and Ωa is πa-δ+A1. This should be done in

such a way that the maximum height of the dirt pile is minimized. A moment’s thought reveals

that the only way to do this is to distribute the dirt equally over the entire frequency range,

i.e., log|S(jω)|=constant for ω0<ω<Ωa as shown on the right.

• Thus the minimum height of the dirt pile is , or equivalently,

Log|S

|

ω ω0 Ωa

0 0

ln ( ) ln ( ) ln ( )a

a

S j d S j d S j d

ln ( )

a

S j d

0

ln ( ) .a

S j d const a

Loop shape cannot be

controlled beyond this

frequency

1

1

0

ln ( )S j d A

A1: Area of

reduced sensitivity

0

1ln ( )a

S j d a A

Log|S

|

ω ω0 Ωa

A1

A2= πa-δ+A1

δ

Log|Smin|

1min

0

lna

a AS

1

0

mina

a A

S e

Maximum achievable stability margins

• The final equation on the previous slide, viz., specifies fundamental limits to the minimum

achievable sensitivity in the region ω0<ω<Ωa and thus sets upper limits to the achievable gain and phase margins for

the specified actuator bandwidth Ωa and the specified amount of feedback necessary (A1) and the frequency ω0 up to

which it is necessary.

• We note from this equation that the absolute least sensitivity |Smin| is reached when the expected benefits of

feedback and the frequency range of expectation are both small relative to a and Ωa, i.e., A1/πa, ω0/Ωa<<1

and is given by

• For the sake of illustration, let us assume that the amount of sensitivity dirt for ω>Ωa is negligible, i.e., δ~0. Thus, we

get . Suppose the speed of the actuator used to control the system is comparable to the rate at which

the open-loop plant response diverges, i.e., , then we note that . For this case, the maximum

achievable phase margin is 2.5° and the maximum possible gain margin is 0.86dB!

• Suppose we pick an actuator that is half as fast as the unstable plant, i.e., , the we note that

Thus, for this case, the maximum phase margin possible is 0.1° and the maximum gain margin possible is 0.04dB!

• It is worth noting the startlingly low values of stability margins for both the cases above: if the actuator is not fast enough, the resulting control system is, for all practical purposes, unstable. It is also worth noting that, in our estimates above, nowhere did we make any assumption about the specific dynamics of the plant or the structure of the controller. All we used was just one little piece of information: the ratio of the speed of the unstable pole to that of the actuator, i.e., . This was sufficient, with the help of Bode integrals, to work out the best stability margins possible, regardless of the specific controller we might choose to use.

1min

0

expa

a AS

min expa

aS

min expa

aS

a a min exp 23.14S

2a a min exp 2 534S

/ aa

Some comments on Bode integrals

• Bode integrals are of fundamental importance to control engineers because of their generality: they enable us to work out the best possible stability margins, with the barest minimum information about the system to be controlled. Thus, they can quickly reveal the limits of achievable performance and stability even before the plant has been fully identified and a specific controller is designed.

• Any practical controller can only do as well as the best possible. Since the shape of the sensitivity function that guarantees minimum sensitivity for ω0<ω<Ωa cannot be readily constructed using simple pole-zero patterns for the controller, the practical controller, constructed with just a few poles and zeros in the interest of minimizing complexity, will probably do much worse than the best possible estimate.

• The actuator bandwidth is a fundamental practical limit, if not a theoretical one. Thus, the integrals provide a way to incorporate this limit into calculations of the best possible stability specifications and the desired benefits of feedback possible for a given actuator.

• In some sense, the Bode Sensitivity Integrals, in combination with the “Bode gain-phase relationship” and “ideal Bode characteristic” occupy the same place in control theory as Newton’s laws do in mechanics: they specify the “laws” that the control systems cannot disobey, and are therefore invaluable in computing the limits to the benefits of feedback and the minimum price that we should pay for enjoying these benefits.

• Controllers that are designed without paying adequate attention to these fundamental limits, especially for controlling unstable plants, are likely to have disastrous consequences (see Ref. 2). One of them (Chernobyl nuclear meltdown) is one of the worst industrial disasters in the history of humanity. If other such disasters are to be avoided, it is only prudent for control engineers to develop intimate working knowledge and familiarity with Bode’s laws.

References

(1) Marcel Sidi, Design of Robust Control systems From Classical to Modern Practical Approaches, Kreiger Publishing Co. FL, USA, (2001)

(2) Gunter Stein, “Respect the Unstable”, IEEE Control Systems Magazine, pp. 12-25 (August 2003)