1Nyquist Meets Bode: Frequency Domain ControllerDesignL.H. Keel and S.P. BhattacharyyaAbstract—In this paper, we first present a new characterizationof the Nyquist criterion in terms of Bode plots of the plant and thecontroller. This gives a nonparametric, and model independentcharacterization of arbitrary order stabilizing controllers. Theresult shows that the frequency response of any stabilizingcontroller must satisfy constraints on its magnitude, phase, andrate of change of phase at certain frequencies that are imposedby the frequency response of the inverse plant. We then show,by examples, that detailed and new results on controller designcan be obtained for continuous and discrete time systems usingthese frequency domain criteria.I. INTRODUCTIONTHE Nyquist criterion [1] provides a powerful test forclosed-loop stability in terms of open-loop measureddata. When applied to a plant-controller pair however, itrequires the testing of the combined transfer function. Thisis not convenient in some synthesis and design problems,where explicit conditions are required on the controller tobe designed, in terms of given plant data. In this paper, wedevelop new criteria for controller design to precisely addressand fix the above problems. This is done by interpreting theNyquist criterion via separate Bode plots [2] of the plant andthe controllers. This result shows that the frequency responseof the inverse plant imposes constraints on the magnitude,phase and rate of change of phase of the controller at certainfrequencies. We also show that such conditions can easily beextended to meet performance requirements such as gain andphase margin specifications. These provide useful results forcontroller design as shown in the paper, by examples.Our results reflect the resurgence of interest in fixed and loworder control design problems. This is in general a much moredifficult problem than design with controllers of unconstrainedorder. Some of the recent results on this problem are asfollows. The design of fixed order controllers for discrete-timesystems was discussed in [3]. In [4], the use of quantifier elim-ination (QE) techniques to deal with the fixed order controllerdesign problem was proposed. In [5], the D-decompositiontechnique [6] was applied to design fixed order controllers.The results introduced here also reflect the recent attempt touse classical control ideas in control theory with a “moderntwist”. In this general philosophy, we should mention theThis work was supported by DOD Grant W911NF-08-0514 and NSF GrantCMMI-0927664.L.H. Keel is with Department of Electrical & Computer Engineer-ing, Tennessee State University, Nashville, TN 37203, USA. Email:[email protected]. Bhattacharyya is with Department of Electrical & Computer Engi-neering, Texas A&M University, College Station, TX 77843, USA. Email:[email protected] works of Hara [7], Ikeda [8], [9] and Jayasuriya [11].In Hara [7] a frequency dependent version of the KYP Lemmais developed and used for synthesis. Ikeda [8], [9] advocatesa model free approach to design. In Shafai [10] QualitativeRobust Control (QRC) was suggested and a procedure fordesigning fuzzy theory based compensators from a qualitativeplant model was developed. In Jayasuriya [11] a QuantitativeFeedback Theory (QFT) approach to design is discussed,wherein robustness bounds are imposed at various frequenciesthat are relevant to loop shaping. Parameter space methodsadvocated by Ackermann [12] and also by˘Siljak [13] aresome of the most practical and effective methods in industrialpractice. Indeed, dealing with a family of plants in parameterspace rather than a single plant was the first step in robustparametric control [14], [15], [16], [17]. In [18], a parameterspace method to design three term controllers was introduced.The most notable feature of this method is that it relies onlyupon frequency domain test data of the plant. The attractionof data based methods is the belief that direct design basedon test data is at least as reliable as that based on models. Forexample, an interesting observation was made by Richardson,Anderson, and Bose that showed the controllability indices ina minimal state space realization of a real rational transferfunction matrix may be calculated from evaluations of thetransfer function matrix at a sufficient number of discretepoints in the frequency domain [19]. It will be seen that thenew methods developed here also involve various frequencieswhere specific conditions must hold.II. NOTATION AND PRELIMINARIESLet us begin by considering a continuous time, linear timeinvariant finite dimensional single input single output plantG described by a rational proper transfer function G(s) withp+open right half plane (RHP) poles, in a unity feedbackconfiguration as shown in Fig. 1.+−GFig. 1. A unity feedback systemLet G(jω) be the frequency response of the plant and letωi, i = 0, 1, 2, · · · , k + 1 with ω0= 0 and ωk+1= ∞denote the frequencies where the Nyquist plot of G(s) cutsthe negative real axis of the complex plane. In other words,2these frequencies are the solutions of the following equation:6G(jω) = nπ, for n = ±1, ±3, ±5, · · · . (1)Define the setΩ = {ω0, ω1, · · · , ωk, ωk+1} (2)where0 =: ω0< ω1< ω2< · · · < ωk< ωk+1:= ∞ (3)and ω0and ωk+1are included only if they satisfy the aboveangle condition. Introduce the corresponding sequence ofintegers{i0, i1, i2, · · · , ik, ik+1} (4)whereit:= 0, if |G(jωt)| < 1 (5)and otherwise,it:=+1, ifddω6G(jω)ω= ωt> 00, ifddω6G(jω)ω= ωt= 0−1, ifddω6G(jω)ω= ωt< 0(6)for t = 0, 1, 2, · · · , k + 1.Remark 1: It is easy to see that the Nyquist plot of G(s)cuts the negative real axis at the frequencies where (1) holds.The cuts are to the left of −1 + j0 when |G(jωt)| > 1. Theconditions in (6) along with |G(jωt)| > 1 indicate that it=+1 when the Nyquist plot cuts the negative real axis to the leftof −1 + j0 downward, corresponding to a counterclockwiseencirclement, and it= −1 when the plot cuts the negativereal axis to the left of −1 + j0 upward, corresponding to aclockwise encirclement of −1 + j0.III. A BODE EQUIVALENT OF THE NYQUIST CRITERIONSuppose first that the plant G has no imaginary axis poles.We assume as usual that the Nyquist contour is traversed inthe clockwise direction, that is with ω increasing.Lemma