Nyquist Interpretation of the Large Gain Theorem*

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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Ryan James Caverly ∗ James Richard Forbes ∗∗ ∗∗ ∗ James Richard Forbes ∗∗ Ryan James Caverly Ryan Ryan James James Caverly Caverly ∗ James James Richard Richard Forbes Forbes ∗∗ ∗ ∗ Department of Aerospace Engineering, University of Michigan, ∗ Department of Aerospace Engineering, University of Michigan, Aerospace Engineering, University ∗ Department Ann Arbor,of 48109 USA (e-mail: [email protected]). Department ofMI Aerospace Engineering, University of of Michigan, Michigan, Arbor, MI 48109 USA (e-mail: [email protected]). ∗∗ Ann Ann Arbor, MI 48109 USA (e-mail: [email protected]). Department of Mechanical Engineering, McGill University, ∗∗ Ann Arbor, MI 48109 USA (e-mail: [email protected]). ∗∗ Department of Mechanical Engineering, McGill University, of McGill ∗∗ Department Montreal, QC, Canada, (e-mail:Engineering, [email protected]) Department of Mechanical Mechanical Engineering, McGill University, University, Montreal, QC, Canada, (e-mail: [email protected]) Montreal, QC, Canada, (e-mail: [email protected]) Montreal, QC, Canada, (e-mail: [email protected]) Abstract: This paper presents a proof of the Large Gain Theorem using the Nyquist Stability Abstract: This paper presents a proof of the Largeby Gain using the Nyquist Stability Abstract: This paper a the Gain Theorem using the Stability Criterion. The minimum gain constraint stipulated the Theorem Large Gain Theorem guarantees the Abstract: Thisminimum paper presents presents a proof proof of ofstipulated the Large Largeby Gain Theorem using the Nyquist Nyquist Stability Criterion. The gain constraint the Large Gain Theorem guarantees the Criterion. The minimum gain constraint stipulated by the Large Gain Theorem guarantees the open-loop transfer function encircles the point (−1, 0) exactly P times in the counterclockwise Criterion. The minimum gain constraint stipulated by the Large Gain Theorem guarantees the open-loop transfer encircles point (−1, 0) exactly P times the counterclockwise open-loop transfer function encircles the point 0) P in the direction, where P function is the number ofthe open-loop open right-half plane in poles. This guarantees open-loop transferP function encirclesof the point (−1, (−1, 0) exactly exactly P times times in the counterclockwise counterclockwise direction, where is the number open-loop open right-half plane poles. This guarantees direction, P of open-loop open plane This asymptoticwhere stability ofthe thenumber feedback system, even in right-half the presence of poles. an unstable open-loop direction, where P is isof the number of system, open-loop open right-half plane poles. This guarantees guarantees asymptotic stability the feedback even in the presence of an unstable open-loop asymptotic stability the feedback system, in the presence of an unstable open-loop transfer function. Theof Nyquist interpretation ofeven the Large Gain Theorem is compared to Nyquist asymptotic stability of the feedback system, even in the presence of an unstable open-loop transfer function. The Nyquist interpretation of the Large Gain Theorem is compared to Nyquist transfer function. The Nyquist interpretation of the Large Gain Theorem is compared to Nyquist interpretations of the Large Gain and Passivity Theorems. Applications of the Large Gain transfer function. The Nyquist interpretation of the Theorems. Large Gain Applications Theorem is compared to Nyquist interpretations of the Large Gain and Passivity of the Large Gain interpretations of the Large Gain and Passivity Theorems. Applications of the Large Gain Theorem are discussed and numerical examples illustrating the concept of minimum gain and interpretations of the Large Gain and Passivityillustrating Theorems.the Applications of the Large Gain Theorem discussed and concept of minimum gain and Theorem are discussed and numerical examples illustrating the concept of minimum gain and the Large are Gain Theorem arenumerical presented.examples Theorem are discussed and numerical examples illustrating the concept of minimum gain and the Large Gain Theorem are presented. the Large Gain Theorem presented. the Large Gain Theorem are are presented. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Large Gain Theorem, Nyquist Stability Criterion, linear systems, stability of Keywords: Large Gain Theorem, Nyquist Stability Keywords: Large Gain Gain Theorem, Theorem, Nyquiststability. Stability Criterion, Criterion, linear linear systems, systems, stability stability of of feedback interconnections, input-output Keywords: Large Nyquist Stability Criterion, linear systems, stability of feedback interconnections, input-output stability. feedback interconnections, input-output stability. feedback interconnections, input-output stability. 1. INTRODUCTION and a comparison to the Small Gain and Passivity The1. INTRODUCTION INTRODUCTION and aa comparison the Small Gain and Passivity The1. and to the Small Gain and Passivity Theorems. The goal ofto this is to bring the 1. INTRODUCTION and a comparison comparison to thework Small Gain andattention Passivityto Theorems. The goal of this work is to bring attention to the orems. The goal of this work is to bring attention to the Large Gain Theorem as a useful stability result. The robust stabilization of open-loop unstable systems orems. The goal of this work is to bring attention to the Large Gain Theorem as a useful stability result. The robust stabilization of open-loop unstable systems Large Gain Theorem as a useful stability result. The robust stabilization of open-loop unstable systems remains a challenging problem in modern control engiLarge Gain Theorem as a useful stability result. The robust stabilization of open-loop unstable systems This paper proceeds as follows. Preliminary definitions, remains challenging problem inattributed modern control control engiremains challenging modern engipaper neering. aa Until recently,problem this wasin to a lack of This remains aUntil challenging problem inattributed modern control engiThis paper proceeds as follows. Preliminary definitions, results, and proceeds theorems as arefollows. includedPreliminary in Section 2.definitions, Section 3 neering. recently, this was to a lack of This paper proceeds as follows. Preliminary definitions, neering. Until recently, this was attributed to a lack of results, and theorems are included in Section 2. Section 3 stability results pertaining to the feedback interconnection neering. Until recently, this was attributed to a lack of results, and theorems are included in Section 2. 3 presents Nyquist interpretations of the Large Gain, Small stability results pertaining to the theoffeedback feedback interconnection and theorems are included in Section 2. Section Section 3 stability results pertaining to interconnection presents Nyquist interpretations of the Large Gain, Small of unstable systems. The work Georgiou et al. (1997) results, stability results pertaining to the feedback interconnection presents Nyquist interpretations of the Large Gain, Small Gain, and Passivity Theorems. Numerical examples of of unstable systems. The work of Georgiou et al. (1997) presents Nyquist interpretations ofNumerical the Large examples Gain, Small of unstable systems. The work of Georgiou et al. (1997) Gain, and Passivity Theorems. of first demonstrated that the open-loop gain of a feedback of unstable systems. Thethe work of Georgiou et al. (1997) Nyquist Gain, and Theorems. Numerical examples of andPassivity Bode plots obtained when satisfying first demonstrated that open-loop gain of feedback Gain, and Passivity Theorems. Numerical examplesthe of first demonstrated that open-loop feedback Nyquist and Bode plots obtained when satisfying the interconnection must bethe greater than gain one of to aa guarantee first demonstrated that the open-loop gain of a feedback Nyquist and Bode plots obtained when satisfying the Large Gain Theorem are in Section 4. Section 5 includes interconnection must be greater than one to guarantee Nyquist and Bode plots obtained when satisfying the interconnection must greater one guarantee Gain Theorem are in Section 4. Section includes input-output stability ifbe any systemthan within theto feedback in- Large interconnection must if beany greater than one tofeedback guarantee Large Gain are in 4. 5 includes of the Large Theorem. Section 5 presents input-output stability system within the in- applications Large Gain Theorem Theorem areGain in Section Section 4. Section Section 56 includes input-output if any system feedback inapplications of the Large Gain Theorem. Section 6 presents terconnectionstability is unstable. The Largewithin Gain the Theorem of Zainput-output stability if any system within the feedback inapplications of the Large Gain Theorem. Section 6 presents concluding remarks. Appendix A and Appendix Bpresents include terconnection is unstable. The Large Gain Theorem of Zaapplications of the Large Gain Theorem. Section 6 terconnection is unstable. The Large Gain Theorem of Zaconcluding remarks. Appendix A and Appendix B include hedzadeh et al. (2008) provides formal stability criteria terconnection is unstable. The Large Gainstability Theoremcriteria of Za- proofs concluding remarks. Appendix and Appendix B include of the minimum gain ofA nonminimum phase and hedzadeh et al. (2008) provides formal concluding remarks. Appendix A and Appendix B include hedzadeh et al. (2008) provides formal stability criteria proofs of the minimum gain of nonminimum phase and based on these notions and Bridgeman and Forbes (2015) hedzadeh et al. (2008) provides formal stability criteria proofs of the minimum gain of nonminimum phase cascaded systems, respectively. based on the theseresult notions and Bridgeman Bridgeman and Forbes Forbes (2015) proofs of systems, the minimum gain of nonminimum phase and and based on these notions and and (2015) cascaded respectively. modifies to incorporate nonzero initial condibased on the theseresult notions and Bridgeman and Forbes (2015) cascaded systems, respectively. modifies to incorporate nonzero initial condicascaded systems, respectively. modifies the result to incorporate nonzero initial conditions. Despite the appealing features of theinitial LargecondiGain modifies the result to incorporate nonzero tions. Despite the appealing appealing features of the the Large Gain 2. PRELIMINARIES tions. Despite the features of Gain Theorem, including the ability to assess the Large closed-loop tions. Despite the appealing features of the Large Gain 2. PRELIMINARIES Theorem, including the ability to to assess assess the closed-loop closed-loop 2. Theorem, including the ability the stability of a feedback interconnection of unstable systems, 2. PRELIMINARIES PRELIMINARIES Theorem, including the ability to assess the closed-loop stability of aaa feedback feedback interconnection of unstable unstable systems, stability of interconnection of systems, 2.1 Notation it remains relatively unknown stability result. This is stability of aa feedback interconnection of unstable it remains relatively unknown stability result.systems, This is is 2.1 Notation it remains aa relatively unknown stability result. partially due to a shortage of literature demonstrating 2.1 Notation Notation it remainsdue relatively unknown stability demonstrating result. This This is 2.1 partially to a shortage of literature partially due to a shortage of literature demonstrating its practicality, which is entirely spanned by Bridgeman In this paper, boldface letters represent matrices, script partially due to a shortage of literature demonstrating its practicality, which is entirely entirely spanned(2012); by Bridgeman Bridgeman this paper, boldface letters represent script its which is spanned by In this paper, boldface represent matrices, script andpracticality, Forbes (2015); Vasegh and Ghaderi Ghaderi In letters and simple lettersmatrices, denote scalars. its which is entirely spanned(2012); by Bridgeman In this denote paper,operators, boldface letters letters represent matrices, script andpracticality, Forbes (2015); Vasegh and Ghaderi Ghaderi letters denote operators, and simple letters denote scalars. and Forbes (2015); Vasegh and Ghaderi (2012); Ghaderi letters denote operators, and simple letters denote scalars. Vasegh Caverly and Forbes (2016). Proofs of Summation points within block diagrams are positive and Forbes (2015); Vasegh and Ghaderi (2012);Proofs Ghaderi letters denote operators, and simple letters denote scalars. and Vasegh (2015); Caverly and Forbes (2016). of Summation points within block diagrams are positive 2 and Vasegh (2015); Caverly and (2016). Proofs of points within block diagrams positive the Large Gain Theorem employ input-output theory Zaunless otherwise Recall y ∈ L2are y222 = and Vasegh (2015); Caverly and Forbes Forbes (2016).theory ProofsZaof Summation Summation pointsnoted. within blockthat diagrams areif positive the Large Gain Theorem employ input-output unless unless noted. Recall that y ∈ L if y the Large Gain Theorem employ input-output theory Za2 ∞ T otherwise 2 hedzadeh et al. (2008); Bridgeman and Forbes (2015), 2 = otherwise that = 2 if y the Large Gain employ input-output theory Za- unless  ∞ yT (t)y(t)dt ∞, andRecall similarly = otherwise< noted. noted. Recall that yyy ∈∈ ∈LL L y2T 22 hedzadeh et not al.Theorem (2008); offer Bridgeman and physical Forbes (2015), 2e2 ifif y 0∞ yT (t)y(t)dt 22 = hedzadeh et al. (2008); Bridgeman and Forbes (2015), which does always the same insight  < ∞, and similarly y ∈ L if y ∞ 2e 2 hedzadeh et al. (2008); Bridgeman and Forbes (2015), 2T + T y (t)y(t)dt < ∞, and similarly y ∈ L if y = 0 2e  which does control not always always offer the same same physicalstability insight 0∞ yT (t)y(t)dt 2T for (t)yT (t)dt<<∞, ∞,and T ∈similarly R+ , where (t) if=y y(t) y ∈ yL = T 2e which does not the physical insight as classical theory.offer Popular input-output 2T for + , where yT (t) = y(t) yyT (t)y < ∞, T ∈ R which does control not always offer the same physicalstability insight 000∞ T (t)dt ∞ T (t)y (t)dt < ∞, T ∈ R , where y (t) = y(t) for as classical theory. Popular input-output T + Tand T∞ (t) = 0 for t > T . The H norm of the ≤ T y 0≤ ytT T as classical control theory. Popular input-output stability (t)y (t)dt < ∞, T ∈ R , where y (t) = y(t) for theorems, such as the Small Gain and Passivity Theorems, Tand y (t) = 0 for t > T . The H T as classicalsuch control theory. input-output stability 000≤ norm of tT≤ function T T (t) =is0G(s) ∞ theorems, as the the SmallPopular Gain and Passivity Theorems, Theorems, for T .. sup The H norm the T |G(jω)| = γthe G. theorems, such as Small Gain and Passivity have Nyquist interpretations, which motivates a Nyquist transfer ∞ T= (t) =is0G(s) for tt > > Theω∈R H∞ norm of of the 0≤ ≤ tt ≤ ≤ function T and and yyTTG(s) ∞ theorems, such as the Small Gain and Passivity Theorems, transfer G(s) = sup |G(jω)| γ G .. ω∈R |G(jω)| = have Nyquist interpretations, which motivates a Nyquist ∞ transfer function G(s) is G(s) = sup = γ G have Nyquist interpretations, which motivates aa Nyquist interpretation of the Large Gain Theorem. transfer function G(s) is G(s)∞ = supω∈R ω∈R |G(jω)| = γG . have Nyquist interpretations, which motivates Nyquist ∞ interpretation of of the the Large Large Gain Gain Theorem. Theorem. interpretation Problem Statement interpretation of the Large Theorem. The novel contributions of Gain this paper are a proof of the 2.2 2.2 Problem Statement 2.2 The novel contributions of this paper are a proof of the 2.2 Problem Problem Statement Statement The of this paper are aa proof of the Largenovel Gaincontributions Theorem using the Nyquist Stability Criterion The novel contributions of this paper are proof of the Large Gain Gain Theorem Theorem using using the the Nyquist Nyquist Stability Stability Criterion Criterion Consider the negative feedback interconnection of G 1 : Large Large Gain Theorem using the Nyquist Stability Criterion Consider the and negative ofFig. G 1 :: Consider negative feedback interconnection G  This work was supported in part by the Natural Sciences and L G 2 : feedback L2e → Linterconnection 2e → Lthe 2e 2e , pictured in of Consider the negative feedback interconnection ofFig. G 11 1. :  L L and G L L in 1. 2e → 2e and 2are:: linear 2e → 2e ,, pictured This work was supported in part by the Natural Sciences and  L → L and G L → L pictured in Fig. 1. Assume G G time-invariant (LTI) single2e → L2e Engineering Council in of Canada’s Postgraduate Scholarship This workResearch was supported part by the Natural Sciences and 1 and G 2 2  L L2e L2e in Fig. 1. 2e 2e 2are: linear 2e → 2e , pictured This workResearch was supported in part by the Natural Sciences and Assume G and G time-invariant (LTI) singleEngineering Council of Canada’s Postgraduate Scholarship 1 2 Assume G are time-invariant (LTI) singleinput single-output systems, with transfer program. Engineering Research Council of Canada’s Postgraduate Scholarship Assume G 11 and and G G 22 (SISO) are linear linear time-invariant (LTI)function singleEngineering Research Council of Canada’s Postgraduate Scholarship input single-output (SISO) systems, with transfer function program. input single-output (SISO) systems, with transfer function program. input single-output (SISO) systems, with transfer function program. Copyright © 2017, 2017 IFAC 3669Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 3669 Copyright ©under 2017 responsibility IFAC 3669Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 3669 10.1016/j.ifacol.2017.08.702

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the Nyquist plot of L(s) encircles the point (−1, 0) exactly P times CCW. 3. NYQUIST INTERPRETATIONS

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3.1 Nyquist Interpretation of the Large Gain Theorem

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Fig. 1. Block diagram of the negative feedback system involving G 1 and G 2 .

representations G1 (s) and G2 (s), respectively. The characteristic equation of the feedback system is F (s) = 1 + L(s) = 0, where L(s) = G1 (s)G2 (s). The goal is to assess the stability of the feedback system using properties of G1 (s) and G2 (s).

2.3 Theorems and Definitions Definition 1. (Minimum Gain Bridgeman and Forbes (2015)) A causal system, G : L2e → L2e , has minimum gain 0 ≤ ν < ∞ if there exists a finite β such that Gu2T − ν u2T ≥ β, ∀u ∈ L2e , ∀T ∈ R+ . The original presentation of minimum gain, which does not account for nonzero initial conditions, is found in Zahedzadeh et al. (2008). Lemma 1. (Nonminimum Phase Minimum Gain) A nonminimum phase causal LTI SISO system has minimum gain ν = 0. Proof : See Appendix A. Lemma 2. (Minimum Gain of Cascaded Systems) A cascaded system, G 3 : L2e → L2e , has minimum gain ν3 = ν1 ν2 , where y2 = G 3 u1 , y1 = u2 = G 1 u1 , y2 = G 2 u2 , and ν1 and ν2 are minimum gains of G 1 and G 2 , respectively. Proof : See Appendix B. Theorem 1. (The Large Gain Theorem Zahedzadeh et al. (2008); Bridgeman and Forbes (2015)) Consider the negative feedback interconnection of the systems G 1 : L2e → L2e and G 2 : L2e → L2e , shown in Fig. 1 and defined     T T T T as y = yT , y1 = G 1 u1 , y2 = G 2 u2 , r = rT , 1 y2 1 r2 u1 = r1 − y2 , u2 = r2 + y1 . If G 1 and G 2 have strictly positive minimum gains of ν1 and ν2 , respectively, satisfying 1 < ν1 ν2 < ∞, then the closed-loop system, y = Gr is input-output stable. Theorem 2. (Principle of the Argument Skogestad and Postlethwaite (2005)) Let D be a closed clockwise (CW) contour with no self intersections and F (s) be a proper rational transfer function. Suppose F (s) has no poles or zeros on D, but D encloses P poles and Z zeros of F (s). The number of counterclockwise (CCW) encirclements of the origin of the F (s)-plane is NCCW = P − Z. Theorem 3. (The Nyquist Stability Criterion Doyle et al. (1992)) Suppose L(s) = G1 (s)G2 (s) has P open right-half plane (ORHP) poles and assume L(s) does not contain any unstable pole-zero cancellations. Construct the Nyquist plot of L(s), indenting to the right around poles on the imaginary axis when constructing the Nyquist contour. The feedback system is asymptotically stable if and only if

Consider the feedback system described in Section 2.2 involving G1 (s) and G2 (s), whose minimum gains satisfy 1 < ν1 ν2 < ∞. Since 1 < ν1 ν2 < ∞, it is known from Lemma 2 that 1 < νL < ∞, where νL is a minimum gain of L(s). As a result of Lemma 1, L(s) does not have any nonminimum phase zeros. Applying the Principle of the Argument with a contour D that encircles the entire closed right-half plane (CRHP), NCCW,0 = P follows, where NCCW,0 is the number of CCW encirclements the Nyquist plot of L(s) makes about the origin and P is the number of ORHP poles of L(s). Since 1 < |L(jω)| < ∞, the Nyquist plot of L(s) cannot lie inside a unit disk centered at the origin and, as shown in Fig. 2(a), any encirclements of the origin are also encirclements of the point (−1, 0). To be clear, NCCW,−1 = NCCW,0 = P , where NCCW,−1 is the number of CCW encirclements about the point (−1, 0). By the Nyquist Stability Criterion, the feedback system is asymptotically stable, since NCCW,−1 = P . 3.2 Nyquist Interpretation of the Small Gain Theorem The Small Gain Theorem guarantees the input-output stability of the feedback system described in Section 2.2 provided 0 < γ1 γ2 < 1, where G1 (s)∞ = γ1 and G2 (s)∞ = γ2 . From this, it can be shown that 0 < L(s)∞ < 1. This implies that the Nyquist plot of L(s) resides strictly within a unit disk centered at the origin, as seen in Fig. 2(b), which ensures that no encirclements of (−1, 0) are made. Given the finite H∞ norm of L(s), the open-loop transfer function has no ORHP poles, and it can be concluded by the Nyquist Stability Criterion that the feedback system is asymptotically stable. 3.3 Nyquist Interpretation of the Passivity Theorem The Passivity Theorem guarantees the input-output stability of the feedback system described in Section 2.2 provided G1 (s) and G2 (s) are both passive and either G1 (s) or G2 (s) is input strictly passive (ISP) with finite gain. Without loss of generality, it is assumed that G1 (s) is passive and G2 (s) is ISP with finite gain. It can be shown that −π/2 ≤ ∠G1 (jω) ≤ π/2 and −π/2 < ∠G2 (jω) < π/2 (Marquez, 2003, p. 214). The phase angles of cascaded systems add, which leads to −π < ∠L(jω) < π, as shown in Fig. 2(c). The Nyquist plot of L(s) does not cross the negative real axis and never encircles the point (−1, 0). By the definition of passive systems (Marquez, 2003, p. 214), G1 (s), G2 (s), and L(s) have no ORHP poles. Therefore, the Nyquist Stability Criterion ensures asymptotic stability of the feedback system. 3.4 Discussion The analyses in this section show that the Large Gain and Small Gain Theorems are complementary. The Small Gain Theorem relies on the open-loop transfer function Nyquist

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Fig. 2. Plots of the regions (shaded pink) that the Nyquist plot of L(s) cannot lie based on the conditions set forth in the (a) Large Gain Theorem, (b) Small Gain Theorem, and (c) Passivity Theorem. Example Nyquist plots of L(s) that satisfy the criteria stipulated by these theorems are illustrated in blue. plot lying within a unit disk centered at the origin, while the Large Gain Theorem requires the open-loop system Nyquist plot reside outside a unit disk. Moreover, the Small Gain Theorem and Large Gain Theorem cannot account for unstable systems and nonminimum phase systems, respectively.

When k = 1, L(s) has no ORHP poles and its Nyquist plot, shown in Fig. 4(a), does not encircle (−1, 0). When k = −1, L(s) has three ORHP poles and its Nyquist plot, shown in Fig. 4(b), has three CCW encirclements of (−1, 0). In both cases, closed-loop stability is guaranteed by the Nyquist Stability Criterion, since NCCW,−1 = P .

The Small Gain and Large Gain Theorems do not exploit information about the phase of the open-loop transfer function. Both theorems guarantee stability by making use of the open-loop transfer function gain. In contrast, the Passivity Theorem relies on the phase of the openloop transfer function, rather than the gain of the openloop transfer function, to assure closed-loop asymptotic stability. Note that the Large Gain Theorem guarantees an infinite upper gain margin, a finite lower gain margin, and infinite phase margin. The Small Gain Theorem ensures a finite upper gain margin, an infinite lower gain margin, and infinite phase margin. An infinite gain margin and finite phase margin are guaranteed by the Passivity Theorem.

Note that a linear matrix inequality (LMI) formulation from Bridgeman and Forbes (2015) was used to numerically solve for ν1 and ν2 in the previous examples.

4. NUMERICAL EXAMPLES 4.1 Example 1: Minimum Gain Consider the LTI SISO systems given by 2s2 + 30s + 4 10s + 2 G1 (s) = , G2 (s) = 2 , s + 3k s + ks + 2

k = ±1.

(1) Bode magnitude plots of G1 (s) and G2 (s) with k = 1 are shown in Fig. 3. From Fig. 3 it can be gleaned that for k = 1 G1 (s) has an H∞ norm of γ1 = 10 and a least conservative minimum gain of ν1 = 2/3, while G2 (s) has an H∞ norm of γ2 = 30 and a least conservative minimum gain of ν1 = 2. Note that the least conservative minimum gains of G1 (s) and G2 (s) remain unchanged for k = −1. 4.2 Example 2: The Large Gain Theorem Consider the open-loop transfer function defined by L(s) = G1 (s)G2 (s), where G1 (s) and G2 (s) are defined in (1). The feedback system is input-output stable by the Large Gain Theorem, since 1 < ν1 ν2 < ∞. In this example, the feedback system has a lower gain margin of 3/4.

5. APPLICATIONS 5.1 Robust Controller Design The few papers that exist on applications of the Large Gain Theorem primarily focuses on its use in designing robust controllers Bridgeman and Forbes (2015); Vasegh and Ghaderi (2012); Ghaderi and Vasegh (2015); Caverly and Forbes (2016). A benefit of using the Large Gain Theorem over the Small Gain Theorem to guarantee robust stability is that robust stability can be guaranteed for systems whose uncertainty is unbounded, which can include unstable uncertainty. The work of Bridgeman and Forbes (2015) includes a numerical example where an uncertain plant and controller are selected that satisfy the Large Gain Theorem. A controller synthesis method is not presented in Bridgeman and Forbes (2015), but instead a simple illustrating example of a robustly stabilizing controller is provided. The chosen controller robustly stabilizes the uncertain plant and is shown to successfully track a desired signal. Feedforward controllers are presented in Vasegh and Ghaderi (2012); Ghaderi and Vasegh (2015), which stabilize certain classes of nonlinear systems that have known minimum gain by satisfying the Large Gain Theorem. Robust controller synthesis methods using the Large Gain Theorem are presented in Caverly and Forbes (2016). Consider a general robust control problem, shown in Fig. 5(a), where G is the nominal plant, ∆ is the uncertainty with nonzero minimum gain 0 < ν∆ < ∞, and G c is the controller to be designed. A robustly stabilizing controller can

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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ryan James Caverly et al. / IFAC PapersOnLine 50-1 (2017) 3606–3611

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ω (rad/s) Fig. 3. Bode magnitude plot of G1 (s) and G2 (s) with k = 1. be designed such that the nominal closed-loop system G CL , shown in Fig. 5(b), has minimum gain strictly greater than 1/ν∆ , thus satisfying the Large Gain Theorem. Controller synthesis methods are found in Caverly and Forbes (2016) that solve this problem for the case of full-state feedback. Due to the Large Gain Theorem’s complementary nature to the Small Gain Theorem, it can potentially be used to design a controller similar to an H∞ controller that can be used in situations that H∞ control may not be practical, or even possible, to implement. For example, in cases where the H∞ norm of the uncertainty is quite large, it might be impractical to implement an H∞ controller. If the minimum gain of the uncertainty is also large, it may be possible to design a robustly stabilizing controller using the Large Gain Theorem. Furthermore, it can be shown that controllers designed using the Large Gain Theorem can outperform H∞ controllers, since they rely on different conditions to guarantee robust stability. 5.2 Robust Performance Consider a plant subject to multiplicative uncertainty, whose uncertain open-loop transfer function is given by Lp (s) = L(s) (1 + W2 (s) · ∆) , where |∆| ≤ 1.

Assuming that L(s) leads to an asymptotically stable nominal closed-loop system, the robust performance condition is stated in Guzzella (2011); Doyle et al. (1992) as |W2 (s)T (s)| + |W1 (s)S(s)|∞ < 1, (2) where T (s) = L(s)/(1 + L(s)) and S(s) = 1/(1 + L(s)). The robust performance condition of (2) requires that at each frequency ω the inequality |W2 (jω)T (jω)| + |W1 (jω)S(jω| < 1 is satisfied, or equivalently |W2 (jω)L(jω)| + |W1 (jω)| < |1 + L(jω)| (3) is satisfied. The condition in (3) can be graphically interpreted as a requirement that a disk of radius |W1 (jω)| centered never intersect a disk of radius |W2 (jω)L(jω)| centered at L(jω), for all frequencies ω ∈ R. This interpretation is illustrated in Fig. 6. The robust performance condition can also be written in terms of minimum gain as |W1 (s)| + |W2 (s)L(s)|∞ < ν (1 + L(s)) , (4) or equivalently     1 + W1−1 (s)W2 (s)L(s)∞ < ν W1−1 (s) (1 + L(s)) , (5)

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Fig. 4. Nyquist plots of the numerical example L(s) with (a) k = 1 and (b) k = −1. The Nyquist plots are generated on a logarithmic scale, where the gridlines are red, except for the 0 dB gridline, which is black Andresen (2009). The point (−1, 0) is labeled by a black circular marker.   where ν (1 + L(s)) and ν W1−1 (s) (1 + L(s)) are minimum gains of the transfer functions 1 + L(s) and W1−1 (s) (1 + L(s)), respectively. The robust performance condition of (4) is equivalent to the condition of (2), which is shown next. From Zahedzadeh et al. (2008) it is known that ν (G(s)) ≤ |G(jω)| ≤ G(s)∞ . Using this property on both sides of (4) yields (6) |W1 (jω)| + |W2 (jω)L(jω)| < |1 + L(jω)| , which is identical to the condition in (3). It can be shown that (5) is equivalent to (2) in a similar manner. The inequalities of (4) and (5) present alternative robust performance conditions to that of (2), which may lead to the feasibility of problems that were otherwise intractable. Both forms of (4) and (5) are presented, since one form may be more convenient than the other, depending on the problem at hand. 5.3 Zero Shaping As stated in Lemma 1, a nonminimum phase causal LTI SISO system has minimum gain ν = 0. This result can be taken advantage of to design controllers that guarantee minimum phase closed-loop zeros, by adding a constraint of nonzero minimum gain to the desired controller synthesis method. Closed-loop zeros have a tremendous impact on closed-loop performance, and in particular, nonmini-

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Im

∆ p w

q z

G u

y

|W1 (jω)|

∆ p w

q G CL

Re z

−1

Gc (a)

L(jω)

|1 + L(jω)|

(b)

Fig. 5. Block diagram of (a) a general robust control problem and (b) a closed-loop system with uncertainty. mum phase zeros can cause severe limitations to closedloop performance, as discussed in Hoagg and Bernstein (2007). As such, controller synthesis methods that can avoid nonminimum phase closed-loop zeros are attractive. Static output feedback controller synthesis methods are presented in Caverly and Forbes (2017) that force closed-loop blocking and transmission zeros to be minimum phase, while minimizing the H∞ norm of the closed-loop system. The constraint of nonzero minimum gain is enforced using LMI formulations of the Minimum Gain Lemma and Modified Minimum Gain Lemma, found in Bridgeman and Forbes (2015) and Caverly and Forbes (2016), respectively.

|W2 (jω)L(jω)|

Fig. 6. Nyquist plot illustrating the geometric interpretation of the robust performance condition stated in (3), inspired by Guzzella (2011). The Nyquist plot of L(s) is drawn in blue with the arrow indicating the direction of the plot with increasing frequency.

The concept used in Caverly and Forbes (2017) can potentially be used to augment the design of other controllers, such as H2 or H∞ controllers, in order to provide a guarantee of minimum phase closed-loop zeros. This is similar to the additional LMI constraints that are appended to the design of H2 or H∞ controllers in Chilali and Gahinet (1996); Chilali et al. (1999) to place closed-loop poles within desired LMI regions of the complex plane.

robust performance. A more subtle application of the concept of minimum gain is the synthesis of controllers that guarantee minimum phase closed-loop zeros. This application exploits the fact that systems with nonzero minimum gain have minimum phase zeros. The applications of the Large Gain Theorem discussed in this paper are preliminary in nature and are far from exhaustive. It is expected that many more interesting applications will surface once the Large Gain Theorem becomes better known within the control systems community.

6. CONCLUSION

ACKNOWLEDGEMENTS

The Large Gain Theorem is a little-known input-output stability result that is similar, but complementary, to the Small Gain Theorem. For this reason, the Large Gain Theorem is not suited to replace the Small Gain Theorem, but instead has the potential to be used alongside the Small Gain Theorem to solve a large class of problems.

The authors would like to thank Dr. Leila J. Bridgeman and Mr. Alex Walsh for insightful discussions and useful comments that greatly strengthened this paper, as well as Prof. Dennis S. Bernstein for agreeing to sit on the first author’s Ph.D. committee, thus providing additional motivation to pursue this topic of research.

It was shown in this paper that the constraint on minimum gain stated in the Large Gain Theorem ensures that the Nyquist plot of the open-loop transfer function encircles the point (−1, 0) exactly P times in the CCW direction, where P is the number of ORHP poles of the open-loop transfer function. In other words, the Large Gain Theorem accounts for feedback interconnections of unstable systems by guaranteeing the exact number of encirclements needed for closed-loop asymptotic stability. As discussed in Section 5, the Large Gain Theorem and the concept of minimum gain have intriguing applications. The design of robust controllers is a natural application of the Large Gain Theorem, due to its complementary nature to the Small Gain Theorem, which is used extensively in the synthesis of robust controllers. It was also shown that a minimum gain condition can be derived to guarantee

REFERENCES Andresen, T. (2009). Nyquist plot with logarithmic amplitudes. URL http://www.mathworks.com/ matlabcentral/fileexchange/7444. Bridgeman, L.J. and Forbes, J.R. (2015). The minimum gain lemma. Int. J. of Robust Nonlin., 25(14), 2515– 2531. Caverly, R.J. and Forbes, J.R. (2016). Robust controller design using the large gain theorem: The full-state feedback case. In Amer. Contr. Conf., 3832–3837. IEEE, Boston, MA. Caverly, R.J. and Forbes, J.R. (2017). Regional pole and zero placement with static output feedback via the minimum gain lemma. In Amer. Contr. Conf. IEEE, Seattle, WA.

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Chilali, M. and Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach. IEEE T. Automat. Contr., 41(3), 358–367. Chilali, M., Gahinet, P., and Apkarian, P. (1999). Robust pole placement in LMI regions. IEEE T. Automat. Contr., 44(12), 2257–2270. Doyle, J.C., Francis, B.A., and Tannenbaum, A.R. (1992). Feedback Control Theory. Dover, Mineola, New York. Georgiou, T.T., Khammash, M., and Megretski, A. (1997). On a large-gain theorem. Syst. Control Letters, 32(4), 231–234. Ghaderi, A. and Vasegh, N. (2015). Input-output stabilizing controller synthesis for SISO T-S fuzzy systems by applying large gain theorem. Int. J. Fuzzy Syst., 1–7. Guzzella, L. (2011). Analysis and Synthesis of Single-Input Single-Output Control Systems. vdf Hochschulverlag AG. Hoagg, J.B. and Bernstein, D.S. (2007). Nonminimumphase zeros - much to do about nothing - classical control - revisited part II. IEEE Contr. Syst. Mag., 27(3), 45–57. Marquez, H.J. (2003). Nonlinear Control Systems: Analysis and Design. Wiley, Hoboken, NJ. Skogestad, S. and Postlethwaite, I. (2005). Multivariable Feedback Control: Analysis and Design. WileyInterscience, 2 edition. Vasegh, N. and Ghaderi, A. (2012). Stabilizing a class of nonlinear systems by applying large gain theorem. In Int. Conf. Syst. Th. Contr. Comp., 1–4. Sinaia, Romania. Zahedzadeh, V., Marquez, H.J., and Chen, T. (2008). On the input-output stability of nonlinear systems: Large gain theorem. In Amer. Contr. Conf., 3440–3445. IEEE, Seattle, WA.

˜ OC w, y(s) = COC (s1 − AOC )−1 x0 + COC (s1 − AOC )−1 B (A.2) ˜ OC is the “BOC ” matrix associated with the transwhere B fer function G(s)ˆ u(s) in observable canonical form. The transfer function G(s)ˆ u(s) has the same poles as G(s), but one less zero (or two less zeros if ω0 = 0). The order of the numerator of G(s)ˆ u(s) is one (or two) less than that of G(s), which makes it strictly proper and explains why the matrix DOC is no longer present in (A.2). The inequality of (A.1) must hold for all initial conditions, ˜ OC w is chosen, which so the initial condition x0 = −B leads to y(s) = 0 and y(t) = 0. For the chosen input, u2T −→ ∞ as T −→ ∞. By increasing T , the value of ν in (A.1) becomes arbitrarily small, and therefore the minimum gain must be ν = 0 for β to be finite.  Appendix B. PROOF OF LEMMA 2 Consider the cascaded systems G 1 and G 2 with minimum gains ν1 and ν2 , respectively, that satisfy G 1 u1 2T − ν1 u1 2T ≥ β1 , (B.1) G 2 u2 2T − ν2 u2 2T ≥ β2 , (B.2) where u1 is the input to G 1 and u2 = y1 = G 1 u1 is the input to G 2 . Substituting u2 = y1 = G 1 u1 and (B.1) into (B.2) gives G 2 y1 2T − ν1 ν2 u1 2T ≥ ν2 β1 + β2 , which can be rewritten as G 3 u1 2T − ν3 u1 2T ≥ β3 , where G 2 y1 = G 3 u1 , ν3 = ν1 ν2 , and β3 = ν2 β1 + β2 . Therefore, G 3 has minimum gain ν3 = ν1 ν2 . 

Appendix A. PROOF OF LEMMA 1 A proof of ν = 0 is found in Zahedzadeh et al. (2008) for stable, nonminimum phase LTI SISO systems without consideration of initial conditions. The proof using the definition of minimum gain from Bridgeman and Forbes (2015) is presented in this appendix. A preliminary version of this proof appears in Caverly and Forbes (2016). Consider a causal LTI SISO system represented by the biproper transfer function G(s) and the state-space realization (A, B, C, D). Assume that the numerator and denominator of G(s) have no common roots, which implies that the pair (A, B) is controllable and the pair (A, C) is observable. Also, assume that the system has a pair of complex conjugate nonminimum phase zeros at s = σ0 ± jω0 , where σ0 ∈ R+ and ω0 ∈ R. Since (A, C) is observable, the system can be written in the observable canonical form (AOC , BOC , COC , DOC ). The expression for y(s) = L {y(t)} is y(s) = COC (s1 − AOC )−1 x0 + G(s)u(s). The definition of minimum gain can be rewritten for the system as y2T − ν u2T ≥ β.

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(A.1)

The inequality in (A.1) must hold for all u ∈ L2e for the system to have minimum gain 0 ≤ ν < ∞. The input is chosen as u(t) = wˆ u(t) = weσ0 t if ω0 = 0, or σ0 t u(t) = wˆ u(t) = we sin(ω0 t) if ω0 = 0, where w ∈ R. Substituting either expression into y(s) yields 3674