A Bode Plot Characterization of All Stabilizing Controllers

2650 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010

A Bode Plot Characterization of All Stabilizing Controllers

L. H. Keel and Shankar P. Bhattacharyya

Abstract—In this technical note, we consider continuous-time controlsystems and present a new characterization of the Nyquist criterion interms of Bode plots of the plant and the controller. This gives a non-parametric, and model independent characterization of arbitrary orderstabilizing controllers. The result shows that the frequency response ofany stabilizing controller must satisfy constraints on its magnitude, phase,and rate of change of phase at certain frequencies that are imposed by thefrequency response of the inverse plant.

Index Terms—All stabilizing controllers, Bode plots, Nyquist criterion.

I. INTRODUCTION

The Nyquist criterion [1] provides a powerful test for closed-loopstability in terms of open-loop measured data. When applied to a plant-controller pair however, it requires the testing of the combined transferfunction. This is not convenient in some synthesis and design prob-lems, where explicit conditions are required on the controller to be de-signed, in terms of given plant data. In this technical note, we developnew criteria for controller design to precisely address and fix the aboveproblems. This is done by interpreting the Nyquist criterion via separateBode plots [2] of the plant and the controllers. This result shows that thefrequency response of the inverse plant imposes constraints on the mag-nitude, phase, and rate of change of phase of the controller at certainfrequencies. We also show that such conditions can easily be extendedto performance requirements such as gain and phase margin specifica-tions. These provide useful results for controller design as shown, byexamples.

Some recent related results on this problem are as follows. The de-sign of fixed order controllers for discrete-time systems was discussedin [3]. In [4], the use of quantifier elimination (QE) techniques to dealwith the fixed order controller design problem was proposed. In [5], theD-decomposition technique [6] was applied to design fixed order con-trollers. We mention the related works of Hara [7], Ikeda [8], [9] andJayasuriya [10]. In Hara [7] a frequency dependent version of the KYPLemma is developed and used for synthesis. Ikeda [8], [9] advocates amodel free approach to design. In Shafai [11] qualitative robust control(QRC) was suggested as a procedure for designing compensators froma qualitative model of the plant. In Jayasuriya [10] a quantitative feed-back theory (QFT) approach to design is discussed, wherein robust-ness bounds are imposed at various frequencies that are relevant to loopshaping. Parameter space methods advocated by Ackermann [12] andSiljak [13] are practical and effective methods in industrial practice.Indeed, dealing with a family of plants in parameter space rather than

Manuscript received February 16, 2010; revised July 06, 2010; July 20, 2010,and July 21, 2010; accepted July 27, 2010. Date of publication August 16, 2010;date of current version November 03, 2010. This work was supported in part byDOD Grant W911NF-08-0514, NSF Grant CMMI-0927664, and NSF GrantCMMI-0927652. Recommended by Associate Editor P. Tsiotras.

L. H. Keel is with the Department of Electrical and Computer Engineering,Tennessee State University, Nashville, TN 37203 USA (e-mail: [email protected]).

S. P. Bhattacharyya is with the Department of Electrical and Computer En-gineering, Texas A&M University, College Station, TX 77843 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this technical note are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2010.2067390

Fig. 1. Unity feedback system.

a single plant was the first step in robust parametric control [14]–[17].In [18], a method to design three term controllers based directly on fre-quency domain test data was introduced. The attraction of data basedmethods is that direct design based on test data is at least as reliable asthat based on models. In Richardson, Anderson, and Bose [19], it wasshown that controllability indices in a minimal state space realizationof a real rational transfer function matrix may be calculated from eval-uations of the transfer function matrix at a sufficient number of discretepoints in the frequency domain. It will be seen that the new methods de-veloped here also involve various frequencies where specific conditionsmust hold. The method presented assumes the frequency response dataof the plant is available for a sufficient range of frequencies. The resultis applicable to stable and unstable systems just as the Nyquist criterionis. If the open loop transfer function �� is known, the required Bodeplot data can be obtained by evaluating ��. If the open loop systemis stable, the Bode plot data can be determined experimentally as thesteady state sinusoidal frequency response over a wide range of fre-quencies. However, it is impossible to directly measure the frequencyresponse of an unstable system. It must be obtained after stabilizingit with a known controller, measuring the corresponding frequency re-sponse and then computing it by “dividing” out the known controller.We believe that it is impossible to develop measurement-only basedidentification or synthesis methods for unstable systems unless onesuch controller is known by experimental or other methods and thismay be regarded as a “limitation” of such methods.

It is worthwhile to mention that the results presented are admittedlyequivalent to model based methods in a mathematical sense becausethey are solutions to the same problem. Nevertheless, it is useful toknow that identified models need not be produced as an intermediatestep in the design process. This is especially significant in view of thefact that identification involves knowledge of the order of the systemwhereas our methods do not require any knowledge of the system order.

II. NOTATION AND PRELIMINARIES

Let us begin by considering a continuous-time, linear time-invariantfinite dimensional single-input single-output plant � described by arational proper transfer function �� with �� open right half plane(RHP) poles, in a unity feedback configuration as shown in Fig. 1.

Let �� be the frequency response of the plant and let ��, � �� with �� and �� denote the frequencieswhere the Nyquist plot of �� cuts the negative real axis of the com-plex plane. In other words, these frequencies are the solutions of thefollowing equation:

� �� (1)

Define the set � �� where � ��

�� and �� and �� are included onlyif they satisfy the above angle condition. Introduce the correspondingsequence of integers �� where

�� (2)

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 2651

and otherwise

��

�� (3)

III. BODE EQUIVALENT OF THE NYQUIST CRITERION

Suppose first that the plant � has no imaginary axis poles. We as-sume as usual that the Nyquist contour is traversed in the clockwisedirection, that is with � increasing.

Lemma 1: Under the assumption that the plant � has no imaginaryaxis poles, let denote the net number of counterclockwise encir-clement of � � �� by the Nyquist plot of ��. Then

� ��

�

��

�� (4)

Proof: It is easy to see that the Nyquist plot of �� cuts thenegative real axis at the frequencies where (1) holds. The cuts are tothe left of � � �� when �� . The conditions in (3) alongwith �� indicate that �� when the Nyquist plot cutsthe negative real axis to the left of � � �� downward, correspondingto a counterclockwise encirclement, and �� when the plot cutsthe negative real axis to the left of � � �� upward, correspondingto a clockwise encirclement of � � ��. The factor 2 accounts for thenegative values of� where these real axis cuts occur. Thus, the formula(4) is the count of the net number of counterclockwise encirclementsof � � �� by the Nyquist plot of ��.

We can now state the well-known Nyquist criterion for stability ofthe feedback system. The proof is omitted.

Theorem 1: Under the assumption that the plant has no imaginaryaxis poles and � open RHP poles �� , the unity feedbacksystem in Fig. 1 is stable iff

�� (5)

Remark 1: The above result is the Bode plot equivalent of theNyquist criterion. It is applicable to stable and unstable open loopsystems. If �� data is available, �� may be computed andthe above relationship verified, provided � is known. For unstablesystems, �� may be obtained if the transfer function �� isknown. If � is stable �� may be experimentally measured. Directexperimental measurement of �� for an unstable � is not possibleand in this case �� can only be obtained if a known stabilizingcontroller is available and the corresponding closed loop frequencyresponse can be measured. We emphasize that the knowledge of �

is also required for the Nyquist criterion. Moreover, knowledge ofone stabilizing controller, which is required anyway for an unstablesystem, is sufficient to obtain �. This is easy to show and was alsoestablished in Theorem 1 of [18].

Now consider the case when the plant � has imaginary axis poles.This includes the important class of systems with one or more integra-tors that is required for a system to track, say, steps and ramps.

A. Plants with Poles at the Origin

Let �� be the number of poles at the origin, and let �� denote thecorresponding number of encirclements in the counterclockwise direc-tion by the Nyquist plot of �� at � � �. Note that here

�� (6)

As typically done in Nyquist theory, we use right indentation of theNyquist �-contour when the contour approaches imaginary axis polesto avoid singularities (see Fig. 2).

Fig. 2. �—contour for Nyquist plot.

1) �� is odd: In this case, the Nyquist plot starts from the negativeor positive imaginary axis as � increases from zero, depending uponthe values of ��. Furthermore, the Nyquist plot turns 180� clockwisefor every pole at the origin. The first clockwise half circle is located inthe LHP or RHP depending upon the sign of ��. Let us consider,for illustration, the case when �� . If �� , the clockwisehalf circle is located in the LHP and it results in a negative real axiscut to the left of � � ��. Since this cut is upward, we have �� .On the other hand, the clockwise half circle is located in the RHP for�� and this results in no negative real axis cut, that is, �� .From such considerations, we derive the following general formulas:For �� odd

��

�if ��

� � ��

�if ��

(7)

2) �� is even: For the case of a plant with an even number of polesat the origin, the Nyquist plot begins from the negative or positive realaxis as � increases from zero, depending upon the value of ��. Withconsiderations similar to the previous case we derive the following gen-eral conditions: For �� even

��

� �

�� if �� and

�

��

��

� �

�otherwise.

(8)

B. Plants With Poles on the Imaginary Axis

Let the denominator of the plant transfer function be

��

�

�

�

��

�

�

�

� � � ��

�

�

�

(9)

that is, the plant has �� pairs of poles at �� for � � � �� . De-fine integer quantities �� for � � � �� which denote the numberof corresponding counterclockwise encirclement by the Nyquist plotof the point � � �� as follows. The verification of these is left to thereader and is based on arguments outlined in the previous cases.

• �� is complex and �� is odd

�� if � � ��

��

�� if � � ��

� �� (10)

• �� is complex and �� is even

�� (11)


• �� is real and �� is odd,

��

�� if � �� and�

��

��

�� otherwise

(12)

• �� is real and �� is even,

��

�� if � �� or � ��

��

��

�� otherwise/

(13)

Theorem 1 can now be restated without restrictions on the location ofpoles of the plant.

Theorem 2: The unity feedback system in Fig. 1 is stable iff

��

�

��

��

�

��

�� (14)

where � is the number of open RHP poles of the plant �.

IV. STABILIZING CONTROLLER CHARACTERIZATION

We now consider a finite dimensional rational proper controller withfrequency response �� and with �� RHP poles and ask when itcan stabilize a finite dimensional rational, proper plant with frequencyresponse � ��.

Define the set of distinct non-negative frequencies

�� (15)

with � �� satisfying the phasecondition

� �� (16)

and the magnitude condition

�� (17)

Note that these are the frequencies where the Nyquist plot of� ��intersects the negative real axis to the left of �� .

For the case when � �� has no imaginary axis poles, we intro-duce the integers �� for � � ��

��

�� if �

��

��

��

��

�� if �

��

��

��

��

�� if �

��

��

��

��

The case when � �� has imaginary axis poles can be treated byusing the formulas (7), (8), and (10)–(13).

The Nyquist criterion can now be restated as follows.Theorem 3: The controller stabilizes the plant � if and only if

��

�

��

��

�

��

�� (18)

A. Gain and Phase Margins

The performance of a controller is often determined by the closed-loop stability margins it provides. The gain margin is such a perfor-mance measure. To compute it, define the distinct frequency set (phasecrossover frequencies)

�� (19)

for � � �� with �� and �� . Let us denotemagnitudes measured in decibels and define

��

��

The upper (lower) gain margin is the smallest increase (decrease) ingain measured in decibels that destabilizes the closed-loop.

Remark 2: If is a stabilizing controller, the upper gain margindenoted by ��

�� is

��

� �� (20)

The lower gain margin denoted by �� is

��

� �� (21)

The phase margin is also an important performance measure. Let�� be the set of gain crossover frequencies

�� (22)

and define

��

��

Remark 3: If is a stabilizing controller, the phase margin is � �� where

��

��

� ��

��

��

� ��

V. EXAMPLES

In this section, we illustrate the usefulness of the previous results incontroller design by examples.

Example 1: Consider a plant with 2 RHP poles and known fre-quency response � �� as shown in Fig. 3.

In this example, we examine whether the plant under considerationcan be stabilized by integral controllers with various structures. Forconvenience we write �� . For any integral controller,it is important to note that �� . Thus, we have �� . On the other hand, � �� where �� and�� are numbers of RHP poles and zeros of the controller, respectively.However, � �� quickly jumps to �� because ofthe pole at the origin (an integrator). This results in

�

��

��

��

��

��

�

Note that � �� , so we have

�� when ��

�� otherwise.(23)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 11, NOVEMBER 2010 2653

Fig. 3. Frequency response of the plant considered (Example 1).

We also note that �� because �� forany proper ��. We now consider integral controllers with variousstructures.

1) Pure Integral Control

��

�� (24)

From (23), we have �� . Clearly, it is impossible to achieve�� that is required for stabilization.

2) Proportional-Integral (PI) Control

��

�� (25)

a) PI Control with a minimum phase zero �� : from (23),we have �� . Consequently, �� is impossible.

b) PI Control with a nonminimum phase zero �� : from(23), we have �� . Note that � �� runs monotonicallyfrom � to 0 for � � �� . Thus, � �� cannot intersect� �� for � � �� . Thus, �� isnot possible to attain.

Therefore, the plant cannot be stabilized by a PI controller.3) PID Control

��

� ��

�� (26)

Observing the high frequency slope of ��, weconclude that the relative degree of the plant is 2. Thus,�� and �� .

a) PID Control with two minimum phase zeros �� : from(23), we have �� . Thus, �� is not possible.

b) PID Control with one nonminimum phase zero �� :from (23), we have �� . In this case, � �� increasesfrom �� and decreases back to �� as � runs from �� to

�. Thus, it cannot intersect � ��. Thus, �� is not possible.

c) PID Control with two nonminimum phase zeros �� :from (23), we have �� . Thus, �� is not possible.

Therefore, the plant cannot be stabilized by a PID controller.4) PI control with a real pole

��

�� (27)

a) When � � � (a stable pole) closed-loop stability requires�� , equivalently �� . The phase of a such con-troller decreases monotonically from �� to�� for� �� . Such a phase plot can cut � �� only down-ward. Thus, �� cannot be achieved.

b) When� � � (an unstable pole) closed-loop stability requires�� . Since �� can only be �1 with �� , it requiresthat � �� cuts � �� at least three times. Dueto the behavior of these two phase functions, it is impossible.

Therefore, the plant cannot be stabilized by a controller of thegiven structure.

5) PID with a real pole

��

� ��

�� (28)

a) When � � � (a stable pole): The stability condition requires�� , equivalently �� . Clearly, � �� for � �� moves from �� to 0 and cannot cut � ��.Thus, �� cannot be achieved.

b) When � � � (an unstable pole): The stability conditionrequires �� (i.e., �� ) and �� . This requires� �� to cut � �� at least five times, which isnot possible, due to the behavior of the two phase plots

Therefore, the plant cannot be stabilized by a PID controller witha real pole, either.

6) Second Order Controllers with an integrator:

��

� � ��

(29)

a) When �� (stable controllers): The condition �� leads �� and �� . Although � �� cuts� �� for � � �� once, the cut will be down-ward. Thus, �� is not possible.

b) When �� (one unstable pole): We need �� to have�� . To obtain �� for closed-loop stability, weneed that � �� cuts � �� for � � �� at least three times. However, it is easy to see that � ��cannot cut � �� at all. Thus, �� is notpossible.

c) When �� (two unstable poles): Similarly, the closed-loop stability requires that �� and �� , and it isnot possible to obtain.

The above analysis shows that the plant requires a controller with astructure more complicated than (29). To the best of our knowledge,such informative analysis would not be possible without Theorem 3 ofthis technical note.

The following example illustrates the gain and phase margin com-putations.


Fig. 4. Illustrating gain and phase margins (Example 2).

Example 2: Consider a plant with 2 RHP poles and let the frequencyresponse of the system � �� be known. Consider the test controller

��

��

Since�� and�� , we have �� .From Fig. 4, we have

��

and

� �

� � � � � �

and therefore the closed-loop system is stable. The gain and phase mar-gins are shown in Fig. 4.

VI. CONCLUSION

In this note, we have presented new interpretations and alternativesto the traditional Nyquist criterion which are useful for both synthesisand analysis of controllers. The result shows how traditional Bode plotdata augmented by rates of change of phase, of the inverse plant andcontroller are to be related in specific ways to ensure closed loop sta-bility as well as gain and phase margins. The fact that every stabilizingcontroller regardless of order, must have this relationship to the plantcan serve as a useful tool for analysis and synthesis as we have shownby examples.

We also note that it is practically impossible to measure the completefrequency response of a system ranging from 0 to infinite frequency.Nevertheless frequency response methods based on Bode plots haveproved very useful in control engineering. Our results clearly show thatdetailed frequency response is needed only in a low frequency rangedetermined by the characteristics of the controller which in turn deter-mine the range of crossover frequencies. For most plants, magnitudeand rate of change of phase decrease monotonically after a low fre-quency range. Rough information only is needed outside this range. Itis also our hope that these results will ultimately lead to design methodsbased on measurements at a finite set of frequencies as in [20] and [21].

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A Bode Plot Characterization of All Stabilizing Controllers

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