IN 227 Control Systems Design

Lecture 19

Instructor: G R Jayanth

Department of Instrumentation and Applied Physics

Ph: 22933197

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

Control of non-minimum phase systems

In all the control design techniques we have discussed, the plant that we intend to control took a back seat. Beyond the fact that it imposed some limits on our ambitions as control engineers, we assumed that it caused no further inconveniencesthere was a significant bandwidth, viz., the control bandwidth, which could be made arbitrarily large ,wherein we could maximize the loop gain by any desired amount and thus achieve the goal of control, namely x(t)=r(t).

The plant was intentionally assumed to be cooperative because we wanted to shine the spotlight on the different control techniques that could be employed to control a general plant. Cooperative plants possess all their poles and zeros in the left half of the complex plane and do not possess any delays. However, there exist a class of linear plants that refuse to be cooperative and demand special attention by virtue of the inconveniences they cause to conventional controllers: they impose fundamental theoretical limits to the achievable loop gain and the gain cross-over frequency. This lecture is devoted to studying the control of such plants.

These plants could be afflicted with three types of maladies (a) time delay (b) right-half plane zeros and (c) unstable dynamics.

There is not much connecting these three types of plants except that they are all non-minimum phase plants. For minimum-phase plants, when we specify the gain of a plant, we also uniquely fix its phase. However, a non minimum-phase plant possesses lot higher phase lag for the same gain characteristics.

Example: Consider a minimum phase system 1/(j+1) and a non-minimum phase system 1/(j-1). The magnitude bode plots for both are exactly the same: they starts at 0 dB and begin to roll off at 20dB/decade beyond =1rad/s. However, when we look at the phase plots, the former starts at 0 deg and tends to -90 deg. at higher frequencies, while the latter begins at -180 deg. and ends at -90 deg.

This lecture is devoted to studying the control of plants with delays and with right half plane zeros.

When we are dealing with such weird plants, all our traditional compasses of stability in Bode plot, namely the gain and phase margins, either break down or may require to be interpreted with care. Hence in analyzing the control of non-minimum phase systems, we step back and check for stability using Nyquist stability theory, before we transfer the system to the familiar domain for design proposes, viz., Bode plots.

Time-delay is ubiquitous in process industries and other systems where it takes finite time for a particular event to reach the sensor. This may be because the event-to-be-sensed may take time to physically or chemically move to the sensor. For example, in a steel plant, the thickness of a steel sheet can be measured only after it has cooled down, but the thickness can be controlled only when it is hot. Thus, there is a finite time lag between control and measurement during which the sheet cools. Of course, one may choose to be more philosophical and say that time delays are inescapable facts of life thanks to Einsteins special theory of relativity. Whether the delay is of practical consequence or not is a matter that engineers have to decide.

Assume that a certain system has a time delay T in addition to certain dynamics P0(s). Since the Laplace transform of a delay is e-sT, the transfer function of the plant without delay (P0(s)) gets multiplied by the term e

-sT. Thus, the control block-diagram of a 1 DOF control system with delay appears as shown on the right.

It is not intuitively clear from the block-diagram why the time delay could cause any inconvenience. Thus, to understand this, we turn to the mathematical tools, in particular, Nyquist stability theory.

As explained before, we first check if Nyquist stability criterion can be uncritically applied: the closed-loop transfer function is X/R=CP0e

-sT /(1+CP0e-sT). The

denominator is 1+CG/esT. The zeros of esT which are the poles of the open-loop system exist at Re(s)-. Thus, if P0 (s) is stable, then there are no open-loop poles on the RHP. Furthermore, . Thus, we can employ the same Nyquist path (C) as we did for ordinary P0 (s) , viz., a semicircle centered at the origin encompassing the entire RHP. All the points on the curved part of C collapse to the origin. For stable closed-loop system, we need to evaluate CP0e

-sT along s=j.

Control of systems with time-delay

0( )sTP s e

r

d

-

u

e

( )C sx

0lim 0sT

sCPe

R

C

0

j

Since we know how to plot CP0(j), we focus on the effect of e-iT. We note that e-iT

=cosT-isinT. Thus, the magnitude of e-iT is one, regardless of frequency. The phase is tan-1(-sinT/cosT)= -T. Since e-iT multiplies CP0(j), we see that at each frequency , the effect of e-iT is to rotate the phasor CP0(j) by an angle T without affecting the gain. At frequencies close to zero, the effect is barely noticeable. At higher frequencies, the phasors get rotated by larger angles. For >>2/T, the phasor gets rotated several cycles. In particular, we note that the phase of the transfer function CP0 e

-iT becomes -180 at a lesser frequency 1 than that of CP0 (which is defined to happen at 0). The gain of CP0 at 1 however, is much higher than the gain at 0. Thus, a time delay has the effect of reducing the gain margin of the overall system. If the delay T is larger, then the phasor at an even lesser frequency gets rotated to align along the negative real-axis. At such a small frequency, the gain would in general be even greater. If T is sufficiently large, then it will rotate a phasor at such a small frequency that at that frequency the plant gain is greater than 1. In this situation the system becomes unstable. Thus, we see that delays can destabilize closed-loop systems that were originally stable.

Since we can apply Nyquist stability principles exactly as before for a system with time delay, we can safely transfer it to the Bode domain. Here the effect of the delay is to add a phase lag T without affecting the gain curve (since |e-iT|=1). Since Bode plots are log plots, we see that the phase of e-iT is an negative exponential function.

For the sake of illustrating the influence of this phase lag, we choose a first order system P0(s)=100/(s+10). We see that the phase of this function tends to -90 as . Thus, this plant possesses no phase cross-over and thus can be designed to achieve high gain over arbitrarily large bandwidths.

However if we add a delay of just 0.01s, i.e., P0(s)=100e-0.01s/(s+10), the phase of the

open-loop system crosses -180. We now have to ensure that the gain of the open loop system is less than 0dB at this frequency . Thus a system which can be designed to possess arbitrarily high bandwidth without delay gets limited in bandwidth due to the delay.

Bode plots make it abundantly clear how just phase lag addition can limit control performance.

Control of systems with time-delay

CP0 (j)

=0 =

-1

-T

CP0 (j)e-iT

1

0

CP0 (j1)e-i

1T

10-1

100

101

102

-10

0

10

20

10-1

100

101

102

-300

-200

-100

0

-180

Bode plot of P0 (s)=100/(s+10)

Bode plot of G(s)=100e-0.01s/(s+10)

-20

-10

0

10

20

Magnitu

de (

dB

)

10-1

100

101

102

103

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

If the delay is small enough that the phase cross-over frequency is close to the gain cross-over frequency, we can continue to use the same design tools as for conventional control systems in order to control the plants. However, since we traditionally deal with rational functions of , and e-iT is not a rational function, it would be convenient to approximate e-iT by rational functions for design purposes. However, unlike e-i whose phase tends to negative infinity, the phase of any rational function will settle at some integral multiple of -90. Thus, we need to note that rational functions can approximate e-iT only for finite frequency range.

We note that truncated Taylors series expansion of e-iT (as 1- T+(T)2/2!-..up to nth degree) will not work because any truncation will result in polynomial of whose magnitude also varies along with its phase This is a poor approximation of e-iT because the magnitude of e-iT is constant.

Pade approximants: The most popular approximants for e-iT were proposed by H. Pade. The general nth degree approximant is given by

For n=1, we see that . For n=2, we have and so on.

We see that Pade approximants, unlike, Taylors series truncation, yield rational functions with unity gain at all frequencies, and thus, correctly reflect the effect of e-iT on the magnitude plots. Depending on the extent of accuracy required, we use one of the approximants in plotting phase plots.

It is worth pointing out that other approximants are possible. For example, is another valid nth degree rational function approximant for e-iT.

Control of systems with time-delay

2 3

2 3

( ) ( 1) ( ) ( 1)( 2)1 ...

2 2! 2 (2 1) 3! 2 (2 1)(2 2)( )

( ) ( 1) ( ) ( 1)( 2)1 ...

2 2! 2 (2 1) 3! 2 (2 1)(2 2)

sT sT n n sT n n n

n n n n nP s

sT sT n n sT n n n

n n n n n

2

2

( )1

2 12( )( )

12 12

sT sT

P ssT sT

12

( )

12

n

n

sT

nQ s

sT

n

12( )

12

sT

P ssT

As we noted before, the use of conventional control techniques is fine if the phase cross-over is not seriously affected by the delay. However, if it does get reduced dramatically, conventional design techniques lead to very poor performance. This is illustrated below.

Example 1: Suppose in the previous delay-free plant that we chose (P0(s)=100/(s+10)) we incorporate a time delay of 2s, i.e., P0(s)=100e

-2s/(s+10). We are required to develop feedback controllers for this plant.

Solution: The bode plot of the open-loop system shows that the phase cross-over frequency is significantly less than the gain cross-over frequency. We also note that the gain margin0. For this controller we see that the phase margin is >0. However, in using this control, we have dramatically reduced the control bandwidth (by more than a factor of 100!). Thus, the step response shows very slow transience (transience slower than the speed of the open-loop system). Furthermore, since the closed-loop system is given by X/R={CP0/(1+CP0e

-sT)}e-sT, we note that we will always have a delay T in the response of the system.

We see that feedback controllers perform pretty poorly in the presence of delays. Further, there appears to be no way for us to substantially improve the performance.

We shall soon see that the performance of these systems is theoretically limited.

Control of systems with time-delay

10-1

100

101

102

-10

0

10

20

10-1

100

101

102

-300

-200

-100

0

-180

Phase cross-over

Gain cross-over

100

102

-100

-50

0

50

100

102

-300

-200

-100

0

PM

0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

Log()

Time (s)

x(t)

A nonminimum phase plant, i.e., one with a right half plane (RHP) zero has the following form: PNMP(s)=(a-s)P1(s), where, P1(s) does not contain any non-mimimum phase terms.

The only way to study the effect of a transfer function on closed-loop stability is by means of Nyquist plots. We note the plant PNMP(s) has no poles in the RHP. The presence of a zero of PNMP(s) in the RHP does not in any way cause problems for applying Nyquist stability theory to search for closed-loop poles in the RHP. Thus, the rules remain unchanged: the system is stable if CPNMP(j) does not encircle -1. Therefore, Bode plots can also be used without any modification.

The way a non-minimum phase term would affect the closed-loop dynamic can be seen by writing PNMP(s)=[(a-s)/(a+s)][PMP(s)] where, PMP(s)=P1(s)(a+s). We know how to draw the bode plot of the minimum phase part, viz., PMP(s). We therefore examine the other term (a-s)/(a+s). This term, and the product of such terms (if there happen to be multiple RHP zeros) are called Blaschke products.

We note that the gain of this term is 1 at all frequencies. Thus the Blaschke product does not affect the magnitude characteristics of the plant. However, the phase of this term is given by -2tan-1(/a): it starts at 0 at =0, becomes -90 at =a and tends to -180 as . Thus, the term adds a phase lag of -180, with the corner frequency being =a . Since all causal physical systems display increased phase lag with frequency, a RHP zero mandatorily causes the phase of the system to cross -180 a bit beyond =a . Thus, an RHP zero affects control performance the same way that time-delay does-it reduces the phase cross-over frequency.

Note that we cannot cancel an RHP zero with a controller pole at =a. This is because in practice, the cancellation can never be perfect. Thus, a branch of the root locus would exist between the zero and the pole, and as the result, the closed-loop system will be unstable.

Control of systems with right half plane zeros

a

Slope=-20dB/decade

Log()

20

Lo

g(M

ag.)

P

ha

se

(d

eg.)

0 -90

-180

Bode plot of (a-s)/(a+s) ~0.1a

~10a

Fundamental upper limit to the gain cross-over frequency for systems with RHP zeros

In order illustrate fundamental limits to achievable benefits of feedback imposed by the nonminimum phase plant, we shall consider the design of a control system where we have specified that the loop gain should possess a phase margin of PM radians and a certain gain margin GM dB.

It is assumed first that a minimum phase, i.e., well behaved controller C(s) is used to control the nonminimum phase plant PNMP(s). The resulting loop gain is given by LNMP=CPNMP. We need to shape LNMP to achieve the desired performance and stability specifications. For reference, we shall also consider the minimum phase counterpart LMP=CPMP whose only difference from LNMP in that it is not multiplied by the Blaschke product (a-s)/(a+s). Since the Blaschke product does not affect the magnitude characteristics, LNMP has the same magnitude characteristics as that of the minimum phase counterpart. This is sketched on the right. The phase of the loop gain LNMP at any frequency , however, is lesser than the phase of LMP by the amount 2tan

-1(/a), i.e.,

Let LNMP cross 0 dB at gc and let the slope of the magnitude characteristic in the neighborhood of gc be -40 db/decade. Bodes gain-phase relationship tells us that the phase of the minimum phase loop gain LMP at any frequency is determined predominantly by the slope of the magnitude characteristic near alone, regardless of the actual shape of the magnitude characteristic of LMP at other frequencies. Thus, the phase of LMP at gc is - rad. Consequently, the net phase of the loop gain LNMP at gc is

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

Since cot() is a decreasing function of , we note from the equation above that, for a given PM, the gain cross-over frequency is a decreasing function of . Thus, the highest gain cross-over frequency is obtained for the smallest value of . However, since the low frequency loop gain needs to be greater than 0dB, we require that >0 in order for gain to cross the 0dB line.

2tan( )NMP MPL L a

1( ) 2 tan ( )NMP gc gcL a

20log |L|

log

gc

20log |LNMP| (=20log |LMP|)

-40 dB/decade

log

pc

NMPL

-

PM

GM

( )NMP cL PM

1 (1 )tan ( )2

gc PM

a

12tan ( )NMP MPL L a

cot2

gc PM

a

Thus, for the given phase margin PM, the absolute maximum gain cross-over frequency is obtained when ~0 and is given by

Thus, we see that the maximum possible value of the gain cross-over frequency gc is theoretically limited to a.cot(PM/2) due to the RHP zero at s=a.

Likewise, we can derive the theoretical limit to the maximum gain cross-over frequency for which the specified phase margin PM and the gain margin GM are simultaneously satisfied. This is done below:

GM is the plant gain at the phase cross-over frequency pc , i.e., where the phase is -. Assuming that the slope of LNMP remains approximately -40 db/decade between gc and pc, and thus, the phase of LMP remains at , we have, or equivalently,

Since the slope of LNMP remains approximately -40 db/decade between gc and pc, (fig. below right), we have

. Using the fact that and , we obtain,

The last equation above is a transcendental equation in . It is solved numerically to obtain and subsequently used to evaluate gc. The figure below (left) plots the values of gc (normalized with respect to a) for different phase and gain margin requirements.

Fundamental upper limit to the gain cross-over frequency for systems with RHP zeros

max

cot 2gc a PM

12tan ( )pc a

log40

gc

pc

GM

cot( 2)pc a

20log |LNMP|

log

gc

-40 dB/decade

log

pc

NMPL

-

GM

cot( 2)pc a cot ( ) 2gc a PM tan ( ) 2

log40 tan( 2)

PMGM

gc

a

From Ref. [1]

The analysis of the previous page indicates that for a specified stability margin GM and PM, the maximum possible gain cross-over frequency is limited, typically to a small fraction of a, where a is the location of the RHP zero. We shall now examine the consequences of this limit on the maximum achievable loop gain.

We start by noting that if we want the phase margin to be PM, then the maximum phase-lag that we can allow for the loop gain LNMP is -+PM for all frequencies

We note that the non-minimum phase terms result in poor closed-loop performance with respect to rejection of disturbances and parameter variations.

Good tracking is one place that open-loop control can help substantially. If we simply invert the undelayed part of the plant by means of a feedforward controller (i.e., use the strategy schematically shown on the right) without using feedback control we can at least achieve excellent dynamic tracking performance, even if the delay of T seconds between the input and the output continues to exist.

Now, if we know that when f-1() cannot be directly implemented, we can use high gain-based open-loop model-inversion. The corresponding block-diagram, which was discussed in Lecture #5, is shown on the right.

However, we know that open-loop control is sensitive to drifts in plant parameters. Thus, we use feedback only to correct for the plant inversion that was already achieved by open-loop control. In other words, we compare the estimate of the output of our feed forward controller (z) with the actual output (x) and the error is fed back. Thus, feedback plays a weak role in the overall control system: it is active only when there is difference between the nominal models of the plant and the actual one. The resulting structure of the control system is called Smith Predictor or Smith Controller. We note that since conventional controllers did a poor job of controlling a plant with delay, we are forced to model the plant as accurately as we possibly can, and subsequently employ open-loop control to invert the model. Only errors in modeling are compensated by means of feedback.

The transfer function of the control system is given by

Control of systems with time-delay

f r(t)

x(t)

1f

Plant

sTe

h f r

x

f

-

u

z

e +

sTe

( )C s 0 ( )P s

x

0 ( )P s

-

u sTe

sTe

r +

z

-

-

0

0 0 0

( ) ( )( )

( ) 1 ( ) ( ) ( )[ ]

sT

sT sT

C s P s eX s

R s C s P s C s Pe Pe

Feed forward based input tracking

0 10 20 30 40 500

0.5

1

1.5

With smith predictor

Conventional control Comparison of

step responses

Smith controller

References

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

(2) G C Goodwin, S E Graebe, M E Salgado, Control System design, Prentice-Hall of India (2001)