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Homogeneous Generalisation of the Lur’e Problem and the Circle Criterion
Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC IFAC Conference Conference on Proceedings, 2nd on Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Available online at www.sciencedirect.com Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Proceedings, 2nd IFAC Conference on Guadalajara, Mexico, 20-22, 2018 Guadalajara, Mexico, June June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018
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IFAC PapersOnLine 51-13 (2018) 514–519
Homogeneous Generalisation of the Lur’e Homogeneous Generalisation of the Lur’e Homogeneous Generalisation of the Lur’e Problem and the Circle Criterion Problem and the Circle Criterion Homogeneous Generalisation of the Problem and the Circle CriterionLur’e ∗∗ Emanuel Rocha ∗ Jaime Moreno Fernando Casta˜ nos ∗∗ Problem and A. the Circle Criterion Emanuel Rocha ∗ Jaime A. Moreno ∗∗ Fernando Casta˜ nos
∗ ∗∗ ∗ Emanuel Rocha n Emanuel Rocha ∗ Jaime Jaime A. A. Moreno Moreno ∗∗ Fernando Fernando Casta˜ Casta˜ nos os ∗ ∗ ∗ ∗∗ Departamento de Control Autom´ a tico, Cinvestav-IPN, ∗ Emanuel Rocha Jaime A. Moreno Fernando Casta˜ nos ∗ Departamento de Control Autom´ atico, Cinvestav-IPN, ∗ 07360 Ciudad de M´ e xico, M´ e xico (e-mail: {erocha, ∗ Departamento de Control Autom´ a tico, Cinvestav-IPN, Departamento atico, Cinvestav-IPN, 07360 Ciudadde deControl M´exico,Autom´ M´exico (e-mail: {erocha, ∗ 07360 Ciudad fcastanos}@ctrl.cinvestav.mx). de M´ M´ (e-mail: {erocha, Departamento de atico, Cinvestav-IPN, 07360 Ciudad deControl M´eexico, xico,Autom´ M´eexico xico (e-mail: {erocha, fcastanos}@ctrl.cinvestav.mx). ∗∗ de Ingenier´ ıa,deUniversidad Nacional Aut´ o{erocha, noma de M´exico, fcastanos}@ctrl.cinvestav.mx). ∗∗ Instituto 07360 Ciudad M´ e xico, M´ e xico (e-mail: fcastanos}@ctrl.cinvestav.mx). Instituto de Ingenier´ıa, Universidad Nacional Aut´ onoma de M´exico, ∗∗ 04510 Ciudad de M´ e xico, M´ e xico (e-mail: [email protected]) ∗∗ Instituto de Ingenier´ ıa, Universidad Nacional Aut´ o Instituto de de Ingenier´ ıa, M´ Universidad Nacional Aut´ onoma noma de de M´ M´eexico, xico, 04510 Ciudad M´efcastanos}@ctrl.cinvestav.mx). xico, exico (e-mail: [email protected]) ∗∗ 04510 Ciudad de M´ e xico, M´ e xico (e-mail: [email protected]) Instituto de de Ingenier´ ıa, M´ Universidad Nacional Aut´ onoma de M´exico, 04510 Ciudad M´exico, exico (e-mail: [email protected]) 04510 Ciudad de M´exico, M´ exico (e-mail: [email protected]) Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. On the one hand, many of the properties for which linear systems areclass useful the systems. analysis Abstract. Homogeneous systems are an interesting generalisation of of linear Abstract. Homogeneous are an interesting generalisation of the the of in linear On the one hand, manysystems of the properties for which linear systems areclass useful in the systems. analysis of more complex systems are shared by their homogeneous counterparts. On the other hand, On the one hand, many of the properties for which linear systems are useful in the analysis Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. On the one hand,systems many ofarethe properties for homogeneous which linear systems are useful in the analysis of more complex shared by their counterparts. On the other hand, they exhibit a considerably larger set by of behaviours. Inlinear this article a generalisation of the Lur’e of more complex systems are shared by their homogeneous counterparts. On the other hand, On the one hand, many of the properties for which systems are useful in the analysis of more complex systems are shared their homogeneous counterparts. On the other hand, they exhibit a considerably larger set of behaviours. In this article a generalisation of the Lur’e problem of absolute stability is presented as a result of an analysis on homogeneous systems. they exhibit a considerably larger set of behaviours. In this article a generalisation of the Lur’e of more complex systems are their counterparts. On the other hand, they exhibit a considerably larger set by of behaviours. a generalisation of the Lur’e problem of absolute stability isshared presented as ahomogeneous resultInofthis an article analysis on homogeneous systems. Moreover, a solution to this problem is introduced as the homogeneous extension of the circle problem of absolute stability is presented as a result of an analysis on homogeneous systems. they exhibit a considerably set of behaviours. Inof this article a generalisation of the the Lur’e problem ofaabsolute stability is presented as a result anhomogeneous analysis on homogeneous systems. Moreover, solution to thislarger problem is introduced as the extension of circle criterion.ofa Moreover, a solution to this problem is introduced as the homogeneous extension of the circle problem absolute stability is presented as a result of an analysis on homogeneous systems. Moreover, solution to this problem is introduced as the homogeneous extension of the circle criterion. criterion. Moreover, a solution to this problemof isAutomatic introduced as the homogeneous extension of the circle criterion. © 2018, IFAC (International Federation Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Absolute stability, Homogeneous systems, Lur’e problem, Circle criterion. criterion. Keywords: Absolute stability, Homogeneous systems, Lur’e problem, Circle criterion. Keywords: Absolute Absolute stability, stability, Homogeneous Homogeneous systems, systems, Lur’e Lur’e problem, problem, Circle Circle criterion. criterion. Keywords: 1. INTRODUCTION Keywords: Absolute stability, Homogeneous systems, Lur’e problem, criterion. trol (Levant, 2005), Circle also have robustness properties that 1. INTRODUCTION trol (Levant, 2005), also have robustness properties that make them particularly attractive. 1. INTRODUCTION INTRODUCTION trol (Levant, 2005), also have robustness properties properties that that 1. trol (Levant, 2005), also have robustness make them particularly attractive. To this day, the impossibility of constructing a general make them particularly attractive. 1. INTRODUCTION trol (Levant, 2005), also have robustness properties make them particularly attractive. has To this day, the impossibility of constructing a general An important result in the theory of linear systems that theory nonthe linear systems isof clear, which compels us An important result in attractive. the theory of linear systems has To thisof day, impossibility constructing aa general them To this impossibility constructing general its roots in particularly theresult studyin ofthe thetheory stability of a feedback loop theory ofday, nonthe linear systems isofclear, which compels us make An important result in the theory of linear systems has An important of linear systems has to focus our on certain classes of systems and roots in the study of the stability of a feedback loop theory ofday, nonattention linear systems isofclear, clear, which compels us its To thisof the impossibility constructing a general theory non linear systems is which compels us consisting ofthe a result nominal linear systemofinof the forward path to focus our attention on certain classes of systems and its roots in study of the stability a feedback loop An important in the theory linear systems has its roots in the study of the stability of a feedback loop to develop models and control algorithms adapted to each of a nominal linear system in the forward path focusofour our attention on certain certain classes of systems systems and theory non linear is clear, which compels us consisting to develop focus attention on classes of and and a non linear memoryless function in the feedback models andsystems control algorithms adapted to each consisting aa nominal system in forward path its roots inof study oflinear the stability a feedback loop consisting ofthe nominal linear system inofthe the forward path class. and a non linear memoryless function in the feedback to develop models and control algorithms adapted to each to focus our attention on certain classes adapted of systems and path, develop models and control algorithms to each other thelinear analysis of the stability of Lur’e class. and memoryless function in feedback consisting oflinear a words, nominal system in the forward path and aain non linear memoryless function in the the feedback innon other words, the analysis of the stability of Lur’e class. to develop control adapted to each path, class. systems (Lur’e, 1944). The focus models of this and article is onalgorithms the class of homogeneous path, in other words, the analysis of the stability of Lur’e and a non linear memoryless function in the feedback path, in other words, the analysis of the stability of Lur’e The focus of this article is on the class of homogeneous systems (Lur’e, 1944). class. systems, final objective to model and control systems (Lur’e, 1944). The focusthough of this thisthe article is on on the theis class of homogeneous homogeneous other words, analysis of the class stability of linear Lur’e systems (Lur’e, 1944). The focus of article is of Whenin such a system isthe stable for a whole of non systems, though the final objective isclass to model and control path, systems that need not necessarily be homogeneous. In a When such a system is stable for a whole class of non linear systems, though the final objective is to model and control systems (Lur’e, 1944). The focus of need thisthe article is on theis class of homogeneous systems, though finalnecessarily objective tohomogeneous. model and control we speak of absolute stability. From the day of its systems that not be In a functions, When such suchwe system isabsolute stable for for wholeFrom class the of non non linear When aa system stable aa whole class of sense, wethat strive to some of the speak ofis stability. daylinear of its systems need not necessarily betoaccomplishments homogeneous. In of systems, though theextend finalnecessarily objective is model and control systems need not be homogeneous. In aa functions, statement, the Lur’e problem has been widely studied by sense, wethat strive to extend some of the accomplishments of functions, we speak of absolute stability. From the day of When suchwe a system stable for a whole class the ofstudied non functions, speak absolute stability. daylinear of its its the linear control theory with regard toaccomplishments the modelling In and statement, the Lur’eofisproblem has been From widely by sense, we strive to extend some of the accomplishments of systems that need not necessarily be homogeneous. a sense, we strive to extend some of the of control community because it relates to the analysis the linear control theory with regard to the modelling and the statement, the Lur’e problem has been widely studied by functions, we speak absolute stability. From the day of its statement, the Lur’eofproblem has been widely studied by control of complex physical systems that might eventually the control community because it relates to the analysis the linear control theory with regard to the modelling and sense, we totheory extend some of the of of the linear control with regard toaccomplishments the modelling and the stability of systems with uncertainties. Moreover, control of strive complex physical systems that might eventually the control community because it relates to the analysis statement, the Lur’e problem has been widely studied by the control community because it relates to the analysis be non linear. of the stability of systems with uncertainties. Moreover, control of complex physical systems that might eventually the linear with regardthat to the modelling and it control of control complextheory physical systems might eventually has a strong connection to the passivity theory and be non linear. of the stability of systems with uncertainties. Moreover, the control community because it relates to the analysis of the stability of systems with uncertainties. Moreover, it has a strong connection to the passivity theory and be non linear. control of complex physical systems thatappears might eventually be linear. rolestability instrong the design of non observers, (see, and for Thenon family of homogeneous systems to be an its it to the passivity theory the ofconnection systems with uncertainties. Moreover, it has connection to linear the passivity theory its has roleaa instrong the design of non linear observers, (see, and for Thenon family of homogeneous systems appears to be an of be linear. instance, Brogliato and Heemels (2009)). A solution to the interesting alternative because, on the one hand, it strictly its role in the design of non linear observers, (see, for The family of homogeneous systems appears to be an has strong connection to linear the passivity theory its rolea inBrogliato the design non observers, (see, for The familyalternative of homogeneous appears be an it instance, and of Heemels (2009)). A solution to and the interesting because,systems on the one hand, to it strictly absolute stability problem is given by the circle criterion contains the class of linear systems and, on the other hand, instance, Brogliato and Heemels (2009)). A solution to the interesting alternative because, on the one hand, it strictly its role in the design of non linear observers, (see, for instance, Brogliato and Heemels (2009)). A solution to the The family of homogeneous systems appears to be an interesting because, on the it strictly stability problem is given by the circle criterion contains thealternative class of linear systems and,one on hand, the other hand, absolute from the state-space representation perspective, is it makes the possible to analyse behaviours that are truly non which, absolute stability problem is given by the circle criterion contains the class of linear systems and, on the other hand, instance, Brogliato and Heemels (2009)). A solution to the absolute stability problem is given by the circle criterion interesting alternative because, on the one hand, it strictly contains class of linear systems and, on the other hand, from the state-space representation perspective, is it makes possible to analyse behaviours that are truly non which, a consequence of the Kalman-Yakubovich-Popov (KYP) linear by using a set of analytic tools that have been under which, from the state-space representation perspective, is it makes possible to analyse behaviours that are truly non stability problem isrepresentation given by the circle criterion which, from theofstate-space perspective, is contains class of linear systems and, on theare other hand, it makes possible to analyse behaviours that truly non absolute a consequence the Kalman-Yakubovich-Popov (KYP) linear by the using a set of analytic tools that have been under lemma, hence its connection to the passivity property of development over the years, see a recent survey by Kawski a consequence of the Kalman-Yakubovich-Popov (KYP) linear by using a set of analytic tools that have been under from representation perspective, is alemma, consequence ofstate-space the Kalman-Yakubovich-Popov (KYP) it makes to behaviours that arebeen truly non which, linear by possible using a set ofyears, analytic tools that have under hencethe its connection to the passivity property of development over theanalyse see a recent survey by Kawski (2015),byand the therein. lemma, hence its its connection to the the passivity passivity property property of development over theofyears, years, see tools recent survey by alemma, consequence of connection the Kalman-Yakubovich-Popov (KYP) hence to of linear using areferences set analytic thatsurvey have been under systems. development over the see aa recent by Kawski Kawski systems. (2015), and the references therein. systems. (2015), andathe the references therein. its connection the passivity property of systems. development over the years,therein. see a can recent survey by Kawski (2015), and references Absolutehence stability is, without to a doubt, a paramount propIn general, homogeneous model approximate a given lemma, Absolute stability is, without a doubt, a paramount propIn general, a homogeneous model can approximate a given systems. (2015), and the references therein. erty that has permeated the control theory literature and system withaa better precision than can a linear model (Hermes, Absolute stability is, without without doubt, paramount propIn general, homogeneous model approximate given Absolute is, aa doubt, aa paramount In general, homogeneous model approximate aa given erty that stability has permeated the control theory literaturepropand system with better precision than can a linear model (Hermes, whose extension tois,more general systems isliterature an attractive 1991). It is aconvenient to use homogeneous models when, erty that has permeated the control theory and system with better precision than a linear model (Hermes, Absolute stability without a doubt, a paramount property that has permeated the control theory literature and In general, homogeneous model can approximate a given system with better precision than a linear model (Hermes, extension to more general systems is an attractive 1991). It is convenient to use homogeneous models when, whose possibility. For Lur’e-like systems in which the nominal for instance, the linear approximation of the system under whose extension to more general systems is an attractive 1991). It is convenient to use homogeneous models when, that has permeated control and whose extension to more the general systems isliterature an system with better precision a linear model (Hermes, 1991). It is convenient to usethan homogeneous models when, possibility. For Lur’e-like systems intheory which theattractive nominal for instance, the linear approximation of the system under erty is nonlinear, theorem by systems Hill andisMoylan (1976) analysis not preserve properties such part possibility. For systems in the nominal for instance, the linear approximation of the the system when, under tothe more general an possibility. For Lur’e-like Lur’e-like systems in which which theattractive nominal 1991). It does is convenient to usefundamental homogeneous models for instance, the linear approximation of system under part is extension nonlinear, the theorem by Hill and Moylan (1976) analysis does not preserve fundamental properties such whose a set sufficient conditions absolute as controllability or stabilisability, but they are instead part is nonlinear, nonlinear, the theorem theorem by Hill Hill and Moylan (1976) analysis doesthe not preserve fundamental properties such provides possibility. Forof systems infor the stability. nominal part is the by and Moylan (1976) for instance, linear approximation of the system under analysis does not preserve fundamental properties such provides a set of Lur’e-like sufficient conditions forwhich absolute stability. as controllability or stabilisability, but they are instead However, this result is not a constructive one in the sense preserved in homogeneous approximations of higher order. provides a set of sufficient conditions for absolute stability. as controllability or stabilisability, but they are instead part is nonlinear, the theorem by Hill and Moylan provides a set of sufficient conditions for absolute stability. analysis does not preserve fundamental properties such as controllability or stabilisability, but they are instead this result is not a constructive one in the(1976) sense preserved in homogeneous approximations of higher order. However, that the storage function must be specified a priori. FurHowever, this result is not a constructive one in the sense preserved in homogeneous approximations of higher order. set of sufficient for absolute stability. However, this result is notconditions a constructive onea in the sense as controllability or stabilisability, instead preserved in homogeneous approximations of higher order. that the astorage function must be specified priori. FurFurthermore, homogeneous systems but give they rise toare more am- provides thermore, it strongly restricts the class of possible storage that the storage function must be specified a priori. FurFurthermore, homogeneous systems give rise to more amHowever, this result is not a constructive one in the sense that the storage function mustthe beclass specified a priori. Furpreserved in homogeneous approximations of higher order. thermore, it strongly restricts of possible storage bitious objectives of control, such as finite-time stabilisaFurthermore, homogeneous systems give rise to more amFurthermore, homogeneous rise to more am- functions. For example, inmust thethe linear case, only quadratic thermore, it strongly restricts the class of possible storage bitious objectives of control,systems such asgive finite-time stabilisathat the storage function be specified a priori. Furthermore, it strongly restricts class of possible storage functions. For example, in the linear case, only quadratic tion, see Bhat and Bernstein (1997), which is not possible bitious objectives of control, such as finite-time stabilisaFurthermore, riseis to am- storage bitious control,systems such asgive finite-time stabilisafunctions can restricts bein considered. functions. For example, the linear case, only quadratic tion, seeobjectives Bhathomogeneous andofBernstein (1997), which notmore possible thermore, it strongly the class of possible storage functions. For example, in the linear case, only quadratic storage functions can be considered. in a purely linear Homogeneous controllers, which tion, seeobjectives Bhat andcontext. Bernstein (1997), which is not notstabilisapossible bitious of control, such as which finite-time tion, see Bhat and Bernstein (1997), is possible be considered. in a purely linear context. Homogeneous controllers, which storage For example, the linear case, only storage functions can bein considered. In this functions article, wecan propose a generalisation of quadratic the circle areaa developed within the framework of sliding mode con- functions. in purely linear context. Homogeneous controllers, which tion, see Bhat and Bernstein (1997), which is not possible in purely linear context. Homogeneous controllers, which In this functions article, wecan propose a generalisation of the circle are developed within the framework of sliding mode con- storage be considered. criterion different from the one given by the Hill-Moylan In this article, we propose a generalisation of the are developed within the framework of sliding mode con The In this article, wefrom propose a generalisation the circle circle in a developed purely linear context. Homogeneous controllers, are within the framework sliding modewhich conauthors gratefully acknowledge the of financial support from criterion different the one given by the of Hill-Moylan The theorem. This criterion allows for a larger class of authors gratefully acknowledge the financial support from criterion different from the one given by the Hill-Moylan In this article, wefrom propose a generalisation of thefeasible circle The criterion different the one given by the Hill-Moylan are developed within framework offinancial slidingsupport mode conCONACyT CVU 627584. the theorem. This criterion allows for a larger class of feasible authors gratefully acknowledge the from The authors gratefully CONACyT CVU 627584. acknowledge the financial support from theorem. different This criterion criterion allows forgiven a larger larger class of feasible feasible criterion from the one by the Hill-Moylan theorem. This allows for a class of CONACyT CVU 627584. CONACyT CVU 627584. acknowledge the financial support from The authors gratefully Proceedings, 2nd627584. IFAC Conference on 514 theorem. This criterion allows for a larger class of feasible CONACyT CVU
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storage functions, making it possible to consider nonquadratic ones in the analysis of Lur’e systems.
forthcoming section.
On the other hand, we focus on nominal systems and storage functions which are homogeneous, allowing a more constructive approach to the problem of finding the appropriate storage functions. It is possible, in principle, to use tools such as sum of squares (SOS) and P´ olya’s Theorem, to name a few.
Theorem 1. [Rosier (1992)] Let f be a vector field on Rn such that the origin is a locally asymptotically stable (AS) equilibrium point. Assume that f ∈ nτ for some r ∈ (0, ∞)n . Then, for any positive integer p and any m > p·maxi {ri }, there exists a strict Lyapunov function V for system x˙ = f (x), which is δεr −homogeneous of degree m and of class C p . As a direct consequence, the timederivative V˙ = ∇V f ∈ Hm+τ .
The present article is organized as follows: Section 2 provides the necessary concepts for the rest of the document. Section 3 presents the problem of absolute stability in its classic form, as a reminder, and its generalisation to a class of homogeneous systems. Section 4 contains the main result of the article, which is the solution to the absolute stability problem by using a generalisation of the circle criterion. An example of an application of this result is presented in Section 5 and it is subject to discussion in Section 6. Finally, the conclusions of this work are given in Section 7.
3. PROBLEM STATEMENT The solution to the Lur’e problem of absolute stability has a close relationship with a number of conjectures that either have been refuted by counterexamples or have not yet been confirmed. Nonetheless, these conjectures were the starting point from which the solution was reached. Let us, as a reminder, write Aizerman’s and Kalman’s, two of the most representative and well-known of these conjectures.
2. PRELIMINARIES
Consider the following feedback loop (Lur’e system). x˙ = Ax + Bu, (3) y = Cx (4) u = −ψ(t, y) (5)
In this section, the concepts of homogeneous function, homogeneous vector field, and homogeneous Lyapunov function are presented. The term “homogeneity” throughout this work refers to the concept of weighted homogeneity (Sepulchre and Aeyels, 1996; Bacciotti and Rosier, 2006; Gr¨ une, 2000). Let us start with some basic definitions:
where x ∈ Rn , u ∈ R and y ∈ R are, respectively, the state vector, the input and the output of the nominal system. Matrices A, B, and C are constant. The feedback interconnection (5) is given by a non-linear memoryless function which satisfies a sector condition, this is ψ ∈ [k1 , k2 ], and it is said that ψ belongs to the sector [k1 , k2 ], if it satisfies (ψ(t, y) − k1 y) (ψ(t, y) − k2 y) ≤ 0 (6) ∀ t ∈ R+ , ∀ y ∈ R, where k1 , k2 are such that k2 > k1 . The sector can also be defined by [k1 , ∞] if (7) y (ψ(t, y) − k1 y) ≥ 0 ∀ t ∈ R+ , ∀ y ∈ R.
Definition 1. A mapping δεr x = (εr1 x1 , . . . , εrn xn )T , ∀ ε > 0, ∀x ∈ Rn \ {0} is said to be a dilation on Rn , where r = (r1 , r2 , . . . , rn )T and 0 < ri < ∞, i = 1, 2, . . . , n. Definition 2. A function h : Rn → R is said to be a homogeneous function of degree σ ∈ R with respect to δεr x, this is denoted as h ∈ Hσ , if h(δεr x) = εσ h(x). Definition 3. A system x˙ = f (x, u) (1) where u ∈ R is the input and x ∈ Rn is the state vector, is said to be a homogeneous system of degree τ with respect to δεr x and εs u or, equivalently, f ∈ nτ if f (δεr x, εs u) = ετ δεr f (x, u). (2) n ∂ Notice that a vector field f (x, u(x)) = j=1 fj ∂xj is homogeneous of degree τ if and only if each component fj is a homogeneous function of degree τ + rj with respect to the dilation δεr x. Therefore, in particular, a linear vector field (as in the linear system x˙ = Ax+Bu) is homogeneous of zero degree with respect to the standard dilation (this is r = (1, 1, . . . , 1)T ).
Consider now the case ψ(t, y) = ky, k ∈ [k1 , k2 ] in system (3)-(5). That is, ψ is a linear function in the sector [k1 , k2 ]. Suppose that the resulting system is AS for all k ∈ [k2 , k2 ]. Aizerman’s conjecture (Aizerman, 1949) states that the feedback loop is absolutely stable, this is, the origin is globally AS (GAS) for any non linearity ψ ∈ [k1 , k2 ]. Furthermore, Kalman’s conjecture (Kalman, 1957) states, broadly speaking, that the system (3)-(5) is absolutely stable if the following conditions are fulfilled: K.I. The function ψ(y) is differentiable and such that k1 < ψ (y) < k2 . K.II. All linear feedback loops with ψ(y) = ky, k ∈ [k1 , k2 ] are AS.
Definition 4. A class C 1 proper positive-definite homogeneous function V : Rn → R+ is said to be a strict homogeneous Lyapunov function for the system x˙ = f (x) if ∂ ∇V (x)f (x) < 0, x = 0, where ∇V (x) = V (x). ∂x
As it is well-known, both conjectures are false. However, their importance lies in the theoretical point of view, since they lead to the question ‘What additional properties does the linear system have to satisfy in order for the conjectures to be true?’ An answer to this question is provided by the circle criterion in the frequency domain, which relates to the KYP lemma in the state-space formulation.
Apart from the previous definitions, let us introduce the following stability theorem to which we will refer in a 515
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The background presented thus far makes it possible to state the absolute stability problem for system (8)-(10) as follows: Definition 6. Suppose that ψ in (10) satisfies a homogeneous sector condition. The closed loop system (8)-(10) is said to be absolutely stable if the origin is a GAS equilibrium.
3.1 Homogeneous extension of Lur’e problem
X
+RPRJHQHRXV V\VWHP
\
Sufficient conditions for absolute stability of system (8)(10) are presented in the following section.
ψ W
4. HOMOGENEOUS CIRCLE CRITERION
Figure 1. Lur’e-like system with a homogeneous block in the forward path.
Proposition 7. The closed-loop system (8)-(10) is absolutely stable if
Consider a generalisation of the Lur’e system (3)-(5): x˙ = f (x) + g(x)u, y = h(x) u = −ψ(t, y)
(i) ψ ∈ [k1 , ∞]φ and there exist a homogeneous Lyapunov function V (x) ∈ Hm , m > max1≤i≤n ri , real functions M (x), and L(x) such that L2 (x) is positive definite, satisfying the following equations (13) ∇V (x) (f (x) − k1 g(x)φ(h(x)) = −L2 (x) 2 ∇V (x)g(x) = M (x)φ(h(x)) (14) (ii) ψ ∈ [k1 , k2 ]φ , with k = k2 − k1 , and there exist a homogeneous Lyapunov function V (x) ∈ Hm , m > max1≤i≤n ri , L : Rn → Rη , such that LT (x)L(x) is a positive definite function, and M : Rn → Rη for some value of η, which satisfy ∇V (x) (f (x) − k1 g(x)φ(h(x)) = −LT (x)L(x) (15)
(8) (9) (10)
as it is shown in Fig. 1, where x = (x1 , . . . , xn ) ∈ Rn represents the states of the block in the forward path, whereas u and y, the input and output of the block, respectively, are scalar quantities. The vector field on the right hand side of (8) is considered to be continuous in x and homogeneous of degree τ ∈ R with respect to the dilations δεr x and εs u. Notice that, since the homogeneous feedforward block is affine in the input, it is required that f (x) ∈ nτ and g(x) ∈ nτ −s . Furthermore, h(x) in (9) is a continuous homogeneous function of degree σ > 0 with respect to δεr x (i.e. h ∈ Hσ ).
∇V (x)g(x) = kφ(h(x))M T (x)M (x) − 2LT (x)M (x) (16)
Notice that the difference between this Proposition and the conditions for absolute stability derived from the dissipativity result by Hill and Moylan is the introduction of the homogeneous function or array of functions M (x). In Section 6 this difference is further developed.
Finally, the feedback interconnection (10) is given by a memoryless function satisfying a homogeneous sector condition.
This new sector condition is proposed by following the logic of Aizerman’s conjecture, in such a way that the feedback function ψ is chosen in order for the closed loop to preserve the homogeneity of degree τ of the block in the forward path. This is achieved by making ψ(t, y) = γφ(y), γ ∈ R, with φ(y) a homogeneous function of degree s/σ with respect to the dilation εy, which implies u(δεr x) = εs u(x). Therefore, from (2), the δεr −homogeneity of degree τ is preserved. Definition 5. Let ψ : R+ ×R → R. It is said that ψ belongs to the homogeneous sector
Proof. (i) The derivative of V along the trajectories of system (8)-(10) is given by V˙ = ∇V (f − gψ(t, h)) = ∇V (f − k1 gφ(h)) − ∇V g (ψ(t, h) − k1 φ(h)) = −L2 − M 2 φ(h) (ψ(t, h) − k1 φ(h)) The fact that ψ ∈ [k1 , ∞]φ , i.e. φ(h) (ψ(t, h) − k1 φ(h)) ≥ 0, implies V˙ ≤ −L2 (x) < 0. This allows us to conclude that the origin is a GAS equilibrium of the closed-loop system.
• [k1 , k2 ]φ , if ψ(t, y) satisfies
(ψ(t, y) − k1 φ(y)) (ψ(t, y) − k2 φ(y)) ≤ 0 ∀ t ∈ R+ , ∀ y ∈ R,
(11)
(ii) The derivative of V along the trajectories of system (8)-(10) is given by
where k1 , k2 are constants satisfying k1 < k2 , and φ is a homogeneous function of degree s/σ, σ > 0, with respect to ε > 0.
V˙ = ∇V (f − gψ(t, h)) = ∇V (f − k1 gφ(h)) − ∇V g (ψ(t, h) − k1 φ(h)) = −LT L − kφ(h)M T M − 2LT M (ψ(t, h) − k1 φ(h))
• [k1 , ∞]φ , if ψ(t, y) satisfies
φ(y) (ψ(t, y) − k1 φ(y)) ≥ 0 ∀ t ∈ R+ , ∀ y ∈ R.
= −L − M (ψ(t, h) − k1 φ(h))2
(12)
− M T M (k2 φ(h) − ψ(t, h)) (ψ(t, h) − k1 φ(h))
where k1 ∈ R and φ is a homogeneous function of degree s/σ, σ > 0, with respect to ε > 0.
where · is the Euclidean norm. Given that ψ ∈ [k1 , k2 ]φ , V˙ ≤ −L − M (ψ(t, h) − k1 φ(h))2 < 0. This allows us 516
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to conclude that the origin is a GAS equilibrium of the closed-loop system.
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stability of the STA (ρ = 1/2), tools of nonsmooth analysis can be used as those in Shevitz and Paden (1994), Clarke (1990), and Bacciotti and Ceragioli (1999).
5. EXAMPLES Example 1. Consider the feedback loop ρ x˙ 1 = −a x1 + x2 , Σ : x˙ 2 = −b x1 2ρ−1 + u y = h(x) = x2 u = −ψ(t, y), ψ ∈ [k1 , ∞]φ φ(y) = y
2ρ−1 ρ
Example 2. Let us consider now the same system with a different output, as follows ρ x˙ 1 = −a x1 + x2 , 2ρ−1 Σ : x˙ 2 = −b x1 (22) +u y = h(x) = x1 (23) u = −ψ(t, y), ψ ∈ [k1 , k2 ]φ 2ρ−1 φ(y) = y (24)
(17) (18) (19)
where · q = | · |q sign(·), q ≥ R. The vector field x˙ = T (x˙ 1 , x˙ 2 ) = f (x)+g(x)u is homogeneous of degree τ = (ρ− 1)/ρ with respect to the homogeneity weights (r1 , r2 ) = (1/ρ, 1) and s = (2ρ−1)/ρ. Moreover, the output y = h(x) is a homogeneous function of degree σ = 1.
The vector field is again homogeneous of degree τ = (ρ − 1)/ρ with respect to the homogeneity weights (r1 , r2 ) = (1/ρ, 1) and s = (2ρ − 1)/ρ. However, the output y = h(x) = x1 is a homogeneous function of degree σ = 1/ρ. Conditions (15) and (16) can be used in order to find a region in the space of parameters a, b for which there exist k1 and k2 such that the feedback loop (22)-(24) is absolutely stable. Following a similar approach to that of the previous example, the following homogeneous Lyapunov candidate function (Moreno et al., 2014) is proposed: γ2 γ1 |x1 |3ρ − x1 2ρ x2 + |x2 |3 . V = 3ρ 3
The case ρ = 1/2 corresponds to the Super Twisting algorithm (STA) (Levant, 1993), which has properties of robustness, precision and convergence to the origin in finite time. This algorithm is the basis for the construction of controllers, observers (Davila et al., 2005), and differentiators (Levant, 1998), in the frame of higher order slidingmode control. The sufficient conditions for stability introduced in the previous section can be used in order to find a region in the space of parameters a, b for which there exists k1 such that the closed-loop system (17)-(19) is absolutely stable. Moreover, the values of k1 can also be found following this strategy.
In order for this function to be positive definite, the inequalities γ1 − 2ρc3/2 > 0
αβ <
Let us find a system of inequalities that guarantees the conditions in (15) and (16) are met. On the one hand, condition (16) is satisfied if
M 2 (x). 1−ρ
This is achieved by making M 2 (x) = |x2 | ρ . On the other hand, we find that condition (13) is fulfilled if −∇V (f − k1 gφ(h)) = L2 is a positive definite function, this is
= ab|x1 |3ρ−1 + k1 |x2 |
3ρ−1 ρ
cp p c−q q α + β . p q
The inequalities in (25) are equivalent to γ1 > 2ρc3/2 and c3 > γ12 . Thus, a necessary and sufficient condition for the positivity of V is that γ2 γ12 > 4ρ2 .
Let us find a system of inequalities that guarantees the conditions in (13) and (14) are met. On the one hand, condition (16) is satisfied if
L2 = (a x1 ρ − x2 ) b x1 2ρ−1 2ρ−1 x2 + b x1 2ρ−1 + k1 x2 ρ
(25)
Therefore, 1 1 γ1 − 2ρc3/2 |x1 |3ρ + γ2 − c−3 |x2 |3 . V ≥ 3ρ 3
This function is positive definite if the constant b is positive.
2ρ−1 ρ
γ2 − c−3 > 0
have to be satisfied, for some c > 0. This is due to the fact that for any real numbers α > 0, β > 0, c > 0, p > 1, and q > 1 with p−1 + q −1 = 1, the following inequality (Young’s inequality) holds
Theorem 1 suggests a homogeneous Lyapunov candidate function. In order for the function to be of class C 1 , we make m equal to 2. Given the homogeneity weights, the following function appears as a natural candidate: b 1 V = |x1 |2ρ + |x2 |2 (20) 2ρ 2
∇V (x)g(x) = x2 = x2
and
∇V (x)g(x) = − x1 2ρ + γ2 x2 2 T = k x1 2ρ−1 M (x) − 2L(x) M (x).
This can be achieved by making |x1 |1/2 1−2ρ M (x) = k −1/2 γ21/2 |x1 | 2 |x2 | , and 0 4ρ−1 k 1/2 x1 2 1/2 1/2 2ρ−1 2ρ−1 γ k L(x) = 2 2 2 x1 |x2 | + |x1 | x2 . 2 L3 (x)
(21)
,
which is, clearly, positive definite if a > 0 and k1 > 0. Therefore, it is possible to assert the absolute stability of system (17)-(19) for such values of a, b, and k1 . It should be noted that the Lyapunov function (20) is of class C 1 if ρ > 1/2. In order to conclude on the absolute 517
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On the other hand, we find that condition (15) is fulfilled if L23 = −∇V (f − k1 gφ(h)) − L21 − L22 is a positive definite function, this is L23 = (a x1 ρ − x2 ) γ1 x1 3ρ−1 − 2ρ|x1 |2ρ−1 x2 + (b + k1 ) x1 2ρ−1 γ2 x2 2 − x1 2ρ kγ2 |x1 |2ρ−1 |x2 |2 − γ12 k|x1 |4ρ−1 − 2 kγ2 x1 2ρ−1 x2 2 + 2 T x1 ρ x1 ρ = |x1 |2ρ−1 Q(x) , x2 x2 where
(a) Case ρ = 1
Figure 2. Regions of absolute stability in the space of parameters for different values of ρ and k = 2.
Q11 Q12 Q12 Q22 (x) γ1 + 2ρa aγ1 − (b + k1 + k) − 2 = , γ1 + 2ρa Q22 (x) − 2 with k k 0 Q22 (x) = 2ρ − γ2 − b + k1 + x1 x2 . 2 2 Q(x) =
(b) Case ρ = 1/2
Therefore, the circle criterion brings a two-way connection between the solution to the Lur’e problem and the concept of passivity, which is a recurring theme in the literature of non linear systems theory. However, the generalisation that we propose does not have such an immediate connection to conditions for passivity as those of Hill-Moylan; the difference comes from the introduction of the term M (x). Making M (x) = 1, that is if conditions (15), (16) were instead ∇V (x) (f (x) − k1 g(x)φ(h(x))) = −L2 (x) (27) ∇V (x)g(x) = kφ(h(x)) − 2L(x), (28)
Observe that for x1 x2 < 0, L23 (x) is positive if and only if Q11 > 0 and 2ρ − γ2 (b + k1 + k) > 0.
Notice also that L23 (x) is positive if and only if Q(x) is a positive definite matrix for x1 x2 > 0 or, equivalently, Q11 > 0 and 2 γ1 + 2ρa Q11 (2ρ + γ2 (b + k1 )) > . 2
then, the conditions are equivalent to the cited Hill-Moylan conditions of passivity for the system x˙ = f (x) + g(x)(¯ u − k1 φ(h(x))) y¯ = kφ(h(x)) + u ¯, which is a loop transformation as those found in the literature of nonlinear systems regarding the absolute stability problem (Vidyasagar, 2002) (Khalil, 1996). However, if these conditions were satisfied there would be an undesirable restriction on the degree of homogeneity of V (x).
In summary, system (17)-(19) is absolutely stable if there exist k1 ∈ R, k ≥ 0, γ1 , and γ2 such that following inequalities are satisfied: γ2 γ12 > 4ρ2 Q11 = aγ1 − (b + k1 + k) > 0 (26) 1 2 Q11 (2ρ + γ2 (b + k1 )) > (γ1 + 2ρa) 4 2ρ − γ2 (b + k1 + k) > 0
To illustrate this, let us take the case of Example 2. The conditions in (27) would force the homogeneity degree of V (x) (which we denote as m) to be equal to 2s − τ . This is because equation (28) and the fact that V , g, φ(h), and L are δεr homogeneous of degrees m, τ − s, s, and s, respectively, imply that the homogeneity degree of ∇V (x)g(x) is equal to that of φ(h(x)). This is m+τ −s = s. In this case s = (2ρ − 1)/ρ, and τ = (ρ − 1)/ρ. Therefore, the degree of homogeneity of V would necessarily be equal to 2s − τ = (3ρ − 1)/ρ.
Notice that when the inequalities in (26) are true, L2 (x) is positive definite only if ρ = 1/2. For other values of ρ, it is positive semidefinite. However, for the case in which ψ does not depend directly on t, asymptotic stability still holds due to the invariance principle. A plot of the region of parameters that renders system (17)-(19) absolutely stable can be found using inequalities (26), as shown in Fig. 2.
As the interest of this article is to work with functions as those described by Theorem 1, the recently found value of m allows to propose functions of class C 1 only if ρ > 2/3 (because this requires m to be greater than max{1/ρ, 1}, therefore (3ρ − 1)/ρ > 1/ρ which implies ρ > 2/3). Introducing M (x) allows to choose an arbitrary homogeneity degree for the function V (x), which translates to V being differentiable an arbitrary number of times.
6. DISCUSSION As it was stated in Section 3, the circle criterion that provides a solution to the Lur’e problem has a close relationship with the prominent KYP lemma. This lemma, in turn, offers a tool in the state-space to find out whether or not a system is (strictly) positive real, a property intimately related to the concept of passivity.
Equally remarkable is the fact that the homogeneous sector condition introduced in Section 3 enables the analysis of closed-loop systems with feedback functions having different properties, for instance, non-Lipschitz functions at 518
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Figure 3. Feedback functions ψ in different sectors. zero or functions that do not cross the origin, as shown in Fig. 3.
Gr¨ une, L. (2000). Homogeneous state feedback stabilization of homogenous systems. SIAM Journal on Control and Optimization, 38(4), 1288–1308. Hermes, H. (1991). Nilpotent and high-order approximations of vector field systems. SIAM review, 33(2), 238– 264. Hill, D. and Moylan, P. (1976). The stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control, 21(5), 708–711. Jerbi, H., Kharrat, T., and Kallel, W. (2013). Some results on stability and stabilization of homogeneous time-varying systems. Mediterranean Journal of Mathematics, 1–12. Kalman, R.E. (1957). Physical and mathematical mechanisms of instability in nonlinear automatic control systems. Trans. ASME, 79(3), 553–566. Kawski, M. (2015). Homogeneity in control: Geometry and applications. In European Control Conference (ECC), 2015, 2449–2457. IEEE. Khalil, H.K. (1996). Noninear systems. Prentice-Hall, New Jersey, 2(5). Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247–1263. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), 379–384. Levant, A. (2005). Homogeneity approach to high-order sliding mode design. Automatica, 41(5), 823–830. Lur’e, AI y Postnikov, V. (1944). On the theory of stability of control systems. Applied mathematics and mechanics, 8(3), 246–248. Moreno, J., S´anchez, T., and Cruz-Zavala, E. (2014). Una funci´on de Lyapunov suave para el algoritmo supertwisting. 16th CLCA, 182–187. Peuteman, J. and Aeyels, D. (1999). Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast timevarying. SIAM journal on control and optimization, 37(4), 997–1010. Rosier, L. (1992). Homogeneous Lyapunov function for homogeneous continuous vector field. Systems & Control Letters, 19(6), 467–473. Sepulchre, R. and Aeyels, D. (1996). Homogeneous Lyapunov functions and necessary conditions for stabilization. Mathematics of Control, Signals and Systems, 9(1), 34–58. Shevitz, D. and Paden, B. (1994). Lyapunov stability theory of nonsmooth systems. IEEE Transactions on automatic control, 39(9), 1910–1914. Vidyasagar, M. (2002). Nonlinear systems analysis. SIAM.
It is also important to emphasise the relevance of this result concerning the analysis of complex non-linear and non-homogeneous systems exhibiting properties such as finite-time stabilisation, as long as they can be represented by the generalised Lur’e system that is proposed. Moreover, this is also a tool for the study of the stability of time-varying homogeneous systems, see, for instance, Jerbi et al. (2013) and Peuteman and Aeyels (1999). 7. CONCLUSIONS The aim of this paper is to develop a set of tools in order to analyse the stability of a class of systems that can be modelled by the feedback loop (8)-(10), in similar terms as those that offer solutions to the Lur’e problem. The main contributions presented are the statement of homogeneous generalisations of the Lur’e problem and the circle criterion. Contrary to the original circle criterion, the conditions presented in this article do not have an immediate connection to the results of passivity and dissipativity, which motivates further research. REFERENCES Aizerman, M.A. (1949). On a problem concerning the stability “in the large” of dynamical systems. Uspekhi matematicheskikh nauk, 4(4), 187–188. Bacciotti, A. and Ceragioli, F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM: Control, Optimisation and Calculus of Variations, 4, 361–376. Bacciotti, A. and Rosier, L. (2006). Liapunov functions and stability in control theory. Springer Science & Business Media. Bhat, S.P. and Bernstein, D.S. (1997). Finite-time stability of homogeneous systems. In American Control Conference, 1997. Proceedings of the 1997, volume 4, 2513–2514. IEEE. Brogliato, B. and Heemels, W.M.H. (2009). Observer design for Lur’e systems with multivalued mappings: A passivity approach. IEEE Transactions on Automatic Control, 54(8), 1996–2001. Clarke, F.H. (1990). Optimization and nonsmooth analysis. SIAM. Davila, J., Fridman, L., and Levant, A. (2005). Secondorder sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11), 1785– 1789. 519
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Homogeneous Generalisation of the Lur’e Homogeneous Generalisation of the Lur’e Homogeneous Generalisation of the Lur’e Problem and the Circle Criterion Problem and the Circle Criterion Homogeneous Generalisation of the Problem and the Circle CriterionLur’e ∗∗ Emanuel Rocha ∗ Jaime Moreno Fernando Casta˜ nos ∗∗ Problem and A. the Circle Criterion Emanuel Rocha ∗ Jaime A. Moreno ∗∗ Fernando Casta˜ nos
∗ ∗∗ ∗ Emanuel Rocha n Emanuel Rocha ∗ Jaime Jaime A. A. Moreno Moreno ∗∗ Fernando Fernando Casta˜ Casta˜ nos os ∗ ∗ ∗ ∗∗ Departamento de Control Autom´ a tico, Cinvestav-IPN, ∗ Emanuel Rocha Jaime A. Moreno Fernando Casta˜ nos ∗ Departamento de Control Autom´ atico, Cinvestav-IPN, ∗ 07360 Ciudad de M´ e xico, M´ e xico (e-mail: {erocha, ∗ Departamento de Control Autom´ a tico, Cinvestav-IPN, Departamento atico, Cinvestav-IPN, 07360 Ciudadde deControl M´exico,Autom´ M´exico (e-mail: {erocha, ∗ 07360 Ciudad fcastanos}@ctrl.cinvestav.mx). de M´ M´ (e-mail: {erocha, Departamento de atico, Cinvestav-IPN, 07360 Ciudad deControl M´eexico, xico,Autom´ M´eexico xico (e-mail: {erocha, fcastanos}@ctrl.cinvestav.mx). ∗∗ de Ingenier´ ıa,deUniversidad Nacional Aut´ o{erocha, noma de M´exico, fcastanos}@ctrl.cinvestav.mx). ∗∗ Instituto 07360 Ciudad M´ e xico, M´ e xico (e-mail: fcastanos}@ctrl.cinvestav.mx). Instituto de Ingenier´ıa, Universidad Nacional Aut´ onoma de M´exico, ∗∗ 04510 Ciudad de M´ e xico, M´ e xico (e-mail: [email protected]) ∗∗ Instituto de Ingenier´ ıa, Universidad Nacional Aut´ o Instituto de de Ingenier´ ıa, M´ Universidad Nacional Aut´ onoma noma de de M´ M´eexico, xico, 04510 Ciudad M´efcastanos}@ctrl.cinvestav.mx). xico, exico (e-mail: [email protected]) ∗∗ 04510 Ciudad de M´ e xico, M´ e xico (e-mail: [email protected]) Instituto de de Ingenier´ ıa, M´ Universidad Nacional Aut´ onoma de M´exico, 04510 Ciudad M´exico, exico (e-mail: [email protected]) 04510 Ciudad de M´exico, M´ exico (e-mail: [email protected]) Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. On the one hand, many of the properties for which linear systems areclass useful the systems. analysis Abstract. Homogeneous systems are an interesting generalisation of of linear Abstract. Homogeneous are an interesting generalisation of the the of in linear On the one hand, manysystems of the properties for which linear systems areclass useful in the systems. analysis of more complex systems are shared by their homogeneous counterparts. On the other hand, On the one hand, many of the properties for which linear systems are useful in the analysis Abstract. Homogeneous systems are an interesting generalisation of the class of linear systems. On the one hand,systems many ofarethe properties for homogeneous which linear systems are useful in the analysis of more complex shared by their counterparts. On the other hand, they exhibit a considerably larger set by of behaviours. Inlinear this article a generalisation of the Lur’e of more complex systems are shared by their homogeneous counterparts. On the other hand, On the one hand, many of the properties for which systems are useful in the analysis of more complex systems are shared their homogeneous counterparts. On the other hand, they exhibit a considerably larger set of behaviours. In this article a generalisation of the Lur’e problem of absolute stability is presented as a result of an analysis on homogeneous systems. they exhibit a considerably larger set of behaviours. In this article a generalisation of the Lur’e of more complex systems are their counterparts. On the other hand, they exhibit a considerably larger set by of behaviours. a generalisation of the Lur’e problem of absolute stability isshared presented as ahomogeneous resultInofthis an article analysis on homogeneous systems. Moreover, a solution to this problem is introduced as the homogeneous extension of the circle problem of absolute stability is presented as a result of an analysis on homogeneous systems. they exhibit a considerably set of behaviours. Inof this article a generalisation of the the Lur’e problem ofaabsolute stability is presented as a result anhomogeneous analysis on homogeneous systems. Moreover, solution to thislarger problem is introduced as the extension of circle criterion.ofa Moreover, a solution to this problem is introduced as the homogeneous extension of the circle problem absolute stability is presented as a result of an analysis on homogeneous systems. Moreover, solution to this problem is introduced as the homogeneous extension of the circle criterion. criterion. Moreover, a solution to this problemof isAutomatic introduced as the homogeneous extension of the circle criterion. © 2018, IFAC (International Federation Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Absolute stability, Homogeneous systems, Lur’e problem, Circle criterion. criterion. Keywords: Absolute stability, Homogeneous systems, Lur’e problem, Circle criterion. Keywords: Absolute Absolute stability, stability, Homogeneous Homogeneous systems, systems, Lur’e Lur’e problem, problem, Circle Circle criterion. criterion. Keywords: 1. INTRODUCTION Keywords: Absolute stability, Homogeneous systems, Lur’e problem, criterion. trol (Levant, 2005), Circle also have robustness properties that 1. INTRODUCTION trol (Levant, 2005), also have robustness properties that make them particularly attractive. 1. INTRODUCTION INTRODUCTION trol (Levant, 2005), also have robustness properties properties that that 1. trol (Levant, 2005), also have robustness make them particularly attractive. To this day, the impossibility of constructing a general make them particularly attractive. 1. INTRODUCTION trol (Levant, 2005), also have robustness properties make them particularly attractive. has To this day, the impossibility of constructing a general An important result in the theory of linear systems that theory nonthe linear systems isof clear, which compels us An important result in attractive. the theory of linear systems has To thisof day, impossibility constructing aa general them To this impossibility constructing general its roots in particularly theresult studyin ofthe thetheory stability of a feedback loop theory ofday, nonthe linear systems isofclear, which compels us make An important result in the theory of linear systems has An important of linear systems has to focus our on certain classes of systems and roots in the study of the stability of a feedback loop theory ofday, nonattention linear systems isofclear, clear, which compels us its To thisof the impossibility constructing a general theory non linear systems is which compels us consisting ofthe a result nominal linear systemofinof the forward path to focus our attention on certain classes of systems and its roots in study of the stability a feedback loop An important in the theory linear systems has its roots in the study of the stability of a feedback loop to develop models and control algorithms adapted to each of a nominal linear system in the forward path focusofour our attention on certain certain classes of systems systems and theory non linear is clear, which compels us consisting to develop focus attention on classes of and and a non linear memoryless function in the feedback models andsystems control algorithms adapted to each consisting aa nominal system in forward path its roots inof study oflinear the stability a feedback loop consisting ofthe nominal linear system inofthe the forward path class. and a non linear memoryless function in the feedback to develop models and control algorithms adapted to each to focus our attention on certain classes adapted of systems and path, develop models and control algorithms to each other thelinear analysis of the stability of Lur’e class. and memoryless function in feedback consisting oflinear a words, nominal system in the forward path and aain non linear memoryless function in the the feedback innon other words, the analysis of the stability of Lur’e class. to develop control adapted to each path, class. systems (Lur’e, 1944). The focus models of this and article is onalgorithms the class of homogeneous path, in other words, the analysis of the stability of Lur’e and a non linear memoryless function in the feedback path, in other words, the analysis of the stability of Lur’e The focus of this article is on the class of homogeneous systems (Lur’e, 1944). class. systems, final objective to model and control systems (Lur’e, 1944). The focusthough of this thisthe article is on on the theis class of homogeneous homogeneous other words, analysis of the class stability of linear Lur’e systems (Lur’e, 1944). The focus of article is of Whenin such a system isthe stable for a whole of non systems, though the final objective isclass to model and control path, systems that need not necessarily be homogeneous. In a When such a system is stable for a whole class of non linear systems, though the final objective is to model and control systems (Lur’e, 1944). The focus of need thisthe article is on theis class of homogeneous systems, though finalnecessarily objective tohomogeneous. model and control we speak of absolute stability. From the day of its systems that not be In a functions, When such suchwe system isabsolute stable for for wholeFrom class the of non non linear When aa system stable aa whole class of sense, wethat strive to some of the speak ofis stability. daylinear of its systems need not necessarily betoaccomplishments homogeneous. In of systems, though theextend finalnecessarily objective is model and control systems need not be homogeneous. In aa functions, statement, the Lur’e problem has been widely studied by sense, wethat strive to extend some of the accomplishments of functions, we speak of absolute stability. From the day of When suchwe a system stable for a whole class the ofstudied non functions, speak absolute stability. daylinear of its its the linear control theory with regard toaccomplishments the modelling In and statement, the Lur’eofisproblem has been From widely by sense, we strive to extend some of the accomplishments of systems that need not necessarily be homogeneous. a sense, we strive to extend some of the of control community because it relates to the analysis the linear control theory with regard to the modelling and the statement, the Lur’e problem has been widely studied by functions, we speak absolute stability. From the day of its statement, the Lur’eofproblem has been widely studied by control of complex physical systems that might eventually the control community because it relates to the analysis the linear control theory with regard to the modelling and sense, we totheory extend some of the of of the linear control with regard toaccomplishments the modelling and the stability of systems with uncertainties. Moreover, control of strive complex physical systems that might eventually the control community because it relates to the analysis statement, the Lur’e problem has been widely studied by the control community because it relates to the analysis be non linear. of the stability of systems with uncertainties. Moreover, control of complex physical systems that might eventually the linear with regardthat to the modelling and it control of control complextheory physical systems might eventually has a strong connection to the passivity theory and be non linear. of the stability of systems with uncertainties. Moreover, the control community because it relates to the analysis of the stability of systems with uncertainties. Moreover, it has a strong connection to the passivity theory and be non linear. control of complex physical systems thatappears might eventually be linear. rolestability instrong the design of non observers, (see, and for Thenon family of homogeneous systems to be an its it to the passivity theory the ofconnection systems with uncertainties. Moreover, it has connection to linear the passivity theory its has roleaa instrong the design of non linear observers, (see, and for Thenon family of homogeneous systems appears to be an of be linear. instance, Brogliato and Heemels (2009)). A solution to the interesting alternative because, on the one hand, it strictly its role in the design of non linear observers, (see, for The family of homogeneous systems appears to be an has strong connection to linear the passivity theory its rolea inBrogliato the design non observers, (see, for The familyalternative of homogeneous appears be an it instance, and of Heemels (2009)). A solution to and the interesting because,systems on the one hand, to it strictly absolute stability problem is given by the circle criterion contains the class of linear systems and, on the other hand, instance, Brogliato and Heemels (2009)). A solution to the interesting alternative because, on the one hand, it strictly its role in the design of non linear observers, (see, for instance, Brogliato and Heemels (2009)). A solution to the The family of homogeneous systems appears to be an interesting because, on the it strictly stability problem is given by the circle criterion contains thealternative class of linear systems and,one on hand, the other hand, absolute from the state-space representation perspective, is it makes the possible to analyse behaviours that are truly non which, absolute stability problem is given by the circle criterion contains the class of linear systems and, on the other hand, instance, Brogliato and Heemels (2009)). A solution to the absolute stability problem is given by the circle criterion interesting alternative because, on the one hand, it strictly contains class of linear systems and, on the other hand, from the state-space representation perspective, is it makes possible to analyse behaviours that are truly non which, a consequence of the Kalman-Yakubovich-Popov (KYP) linear by using a set of analytic tools that have been under which, from the state-space representation perspective, is it makes possible to analyse behaviours that are truly non stability problem isrepresentation given by the circle criterion which, from theofstate-space perspective, is contains class of linear systems and, on theare other hand, it makes possible to analyse behaviours that truly non absolute a consequence the Kalman-Yakubovich-Popov (KYP) linear by the using a set of analytic tools that have been under lemma, hence its connection to the passivity property of development over the years, see a recent survey by Kawski a consequence of the Kalman-Yakubovich-Popov (KYP) linear by using a set of analytic tools that have been under from representation perspective, is alemma, consequence ofstate-space the Kalman-Yakubovich-Popov (KYP) it makes to behaviours that arebeen truly non which, linear by possible using a set ofyears, analytic tools that have under hencethe its connection to the passivity property of development over theanalyse see a recent survey by Kawski (2015),byand the therein. lemma, hence its its connection to the the passivity passivity property property of development over theofyears, years, see tools recent survey by alemma, consequence of connection the Kalman-Yakubovich-Popov (KYP) hence to of linear using areferences set analytic thatsurvey have been under systems. development over the see aa recent by Kawski Kawski systems. (2015), and the references therein. systems. (2015), andathe the references therein. its connection the passivity property of systems. development over the years,therein. see a can recent survey by Kawski (2015), and references Absolutehence stability is, without to a doubt, a paramount propIn general, homogeneous model approximate a given lemma, Absolute stability is, without a doubt, a paramount propIn general, a homogeneous model can approximate a given systems. (2015), and the references therein. erty that has permeated the control theory literature and system withaa better precision than can a linear model (Hermes, Absolute stability is, without without doubt, paramount propIn general, homogeneous model approximate given Absolute is, aa doubt, aa paramount In general, homogeneous model approximate aa given erty that stability has permeated the control theory literaturepropand system with better precision than can a linear model (Hermes, whose extension tois,more general systems isliterature an attractive 1991). It is aconvenient to use homogeneous models when, erty that has permeated the control theory and system with better precision than a linear model (Hermes, Absolute stability without a doubt, a paramount property that has permeated the control theory literature and In general, homogeneous model can approximate a given system with better precision than a linear model (Hermes, extension to more general systems is an attractive 1991). It is convenient to use homogeneous models when, whose possibility. For Lur’e-like systems in which the nominal for instance, the linear approximation of the system under whose extension to more general systems is an attractive 1991). It is convenient to use homogeneous models when, that has permeated control and whose extension to more the general systems isliterature an system with better precision a linear model (Hermes, 1991). It is convenient to usethan homogeneous models when, possibility. For Lur’e-like systems intheory which theattractive nominal for instance, the linear approximation of the system under erty is nonlinear, theorem by systems Hill andisMoylan (1976) analysis not preserve properties such part possibility. For systems in the nominal for instance, the linear approximation of the the system when, under tothe more general an possibility. For Lur’e-like Lur’e-like systems in which which theattractive nominal 1991). It does is convenient to usefundamental homogeneous models for instance, the linear approximation of system under part is extension nonlinear, the theorem by Hill and Moylan (1976) analysis does not preserve fundamental properties such whose a set sufficient conditions absolute as controllability or stabilisability, but they are instead part is nonlinear, nonlinear, the theorem theorem by Hill Hill and Moylan (1976) analysis doesthe not preserve fundamental properties such provides possibility. Forof systems infor the stability. nominal part is the by and Moylan (1976) for instance, linear approximation of the system under analysis does not preserve fundamental properties such provides a set of Lur’e-like sufficient conditions forwhich absolute stability. as controllability or stabilisability, but they are instead However, this result is not a constructive one in the sense preserved in homogeneous approximations of higher order. provides a set of sufficient conditions for absolute stability. as controllability or stabilisability, but they are instead part is nonlinear, the theorem by Hill and Moylan provides a set of sufficient conditions for absolute stability. analysis does not preserve fundamental properties such as controllability or stabilisability, but they are instead this result is not a constructive one in the(1976) sense preserved in homogeneous approximations of higher order. However, that the storage function must be specified a priori. FurHowever, this result is not a constructive one in the sense preserved in homogeneous approximations of higher order. set of sufficient for absolute stability. However, this result is notconditions a constructive onea in the sense as controllability or stabilisability, instead preserved in homogeneous approximations of higher order. that the astorage function must be specified priori. FurFurthermore, homogeneous systems but give they rise toare more am- provides thermore, it strongly restricts the class of possible storage that the storage function must be specified a priori. FurFurthermore, homogeneous systems give rise to more amHowever, this result is not a constructive one in the sense that the storage function mustthe beclass specified a priori. Furpreserved in homogeneous approximations of higher order. thermore, it strongly restricts of possible storage bitious objectives of control, such as finite-time stabilisaFurthermore, homogeneous systems give rise to more amFurthermore, homogeneous rise to more am- functions. For example, inmust thethe linear case, only quadratic thermore, it strongly restricts the class of possible storage bitious objectives of control,systems such asgive finite-time stabilisathat the storage function be specified a priori. Furthermore, it strongly restricts class of possible storage functions. For example, in the linear case, only quadratic tion, see Bhat and Bernstein (1997), which is not possible bitious objectives of control, such as finite-time stabilisaFurthermore, riseis to am- storage bitious control,systems such asgive finite-time stabilisafunctions can restricts bein considered. functions. For example, the linear case, only quadratic tion, seeobjectives Bhathomogeneous andofBernstein (1997), which notmore possible thermore, it strongly the class of possible storage functions. For example, in the linear case, only quadratic storage functions can be considered. in a purely linear Homogeneous controllers, which tion, seeobjectives Bhat andcontext. Bernstein (1997), which is not notstabilisapossible bitious of control, such as which finite-time tion, see Bhat and Bernstein (1997), is possible be considered. in a purely linear context. Homogeneous controllers, which storage For example, the linear case, only storage functions can bein considered. In this functions article, wecan propose a generalisation of quadratic the circle areaa developed within the framework of sliding mode con- functions. in purely linear context. Homogeneous controllers, which tion, see Bhat and Bernstein (1997), which is not possible in purely linear context. Homogeneous controllers, which In this functions article, wecan propose a generalisation of the circle are developed within the framework of sliding mode con- storage be considered. criterion different from the one given by the Hill-Moylan In this article, we propose a generalisation of the are developed within the framework of sliding mode con The In this article, wefrom propose a generalisation the circle circle in a developed purely linear context. Homogeneous controllers, are within the framework sliding modewhich conauthors gratefully acknowledge the of financial support from criterion different the one given by the of Hill-Moylan The theorem. This criterion allows for a larger class of authors gratefully acknowledge the financial support from criterion different from the one given by the Hill-Moylan In this article, wefrom propose a generalisation of thefeasible circle The criterion different the one given by the Hill-Moylan are developed within framework offinancial slidingsupport mode conCONACyT CVU 627584. the theorem. This criterion allows for a larger class of feasible authors gratefully acknowledge the from The authors gratefully CONACyT CVU 627584. acknowledge the financial support from theorem. different This criterion criterion allows forgiven a larger larger class of feasible feasible criterion from the one by the Hill-Moylan theorem. This allows for a class of CONACyT CVU 627584. CONACyT CVU 627584. acknowledge the financial support from The authors gratefully Proceedings, 2nd627584. IFAC Conference on 514 theorem. This criterion allows for a larger class of feasible CONACyT CVU
2405-8963 © 2018, IFACConference (International Federation of Automatic Control) Proceedings, 2nd IFAC on 514 Hosting by Elsevier Ltd. All rights reserved. Modelling, and Control of Nonlinear Proceedings, 2nd IFAC on 514 Peer reviewIdentification under of International Federation of Automatic Proceedings, 2nd responsibility IFAC Conference Conference on 514Control. Modelling, Identification and Control of Nonlinear Systems 10.1016/j.ifacol.2018.07.331 Modelling, Identification and of Modelling, Identification and Control Controlon of Nonlinear Nonlinear Systems Proceedings, 2nd IFAC Conference 514 Guadalajara, Mexico, June 20-22, 2018 Systems Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear
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storage functions, making it possible to consider nonquadratic ones in the analysis of Lur’e systems.
forthcoming section.
On the other hand, we focus on nominal systems and storage functions which are homogeneous, allowing a more constructive approach to the problem of finding the appropriate storage functions. It is possible, in principle, to use tools such as sum of squares (SOS) and P´ olya’s Theorem, to name a few.
Theorem 1. [Rosier (1992)] Let f be a vector field on Rn such that the origin is a locally asymptotically stable (AS) equilibrium point. Assume that f ∈ nτ for some r ∈ (0, ∞)n . Then, for any positive integer p and any m > p·maxi {ri }, there exists a strict Lyapunov function V for system x˙ = f (x), which is δεr −homogeneous of degree m and of class C p . As a direct consequence, the timederivative V˙ = ∇V f ∈ Hm+τ .
The present article is organized as follows: Section 2 provides the necessary concepts for the rest of the document. Section 3 presents the problem of absolute stability in its classic form, as a reminder, and its generalisation to a class of homogeneous systems. Section 4 contains the main result of the article, which is the solution to the absolute stability problem by using a generalisation of the circle criterion. An example of an application of this result is presented in Section 5 and it is subject to discussion in Section 6. Finally, the conclusions of this work are given in Section 7.
3. PROBLEM STATEMENT The solution to the Lur’e problem of absolute stability has a close relationship with a number of conjectures that either have been refuted by counterexamples or have not yet been confirmed. Nonetheless, these conjectures were the starting point from which the solution was reached. Let us, as a reminder, write Aizerman’s and Kalman’s, two of the most representative and well-known of these conjectures.
2. PRELIMINARIES
Consider the following feedback loop (Lur’e system). x˙ = Ax + Bu, (3) y = Cx (4) u = −ψ(t, y) (5)
In this section, the concepts of homogeneous function, homogeneous vector field, and homogeneous Lyapunov function are presented. The term “homogeneity” throughout this work refers to the concept of weighted homogeneity (Sepulchre and Aeyels, 1996; Bacciotti and Rosier, 2006; Gr¨ une, 2000). Let us start with some basic definitions:
where x ∈ Rn , u ∈ R and y ∈ R are, respectively, the state vector, the input and the output of the nominal system. Matrices A, B, and C are constant. The feedback interconnection (5) is given by a non-linear memoryless function which satisfies a sector condition, this is ψ ∈ [k1 , k2 ], and it is said that ψ belongs to the sector [k1 , k2 ], if it satisfies (ψ(t, y) − k1 y) (ψ(t, y) − k2 y) ≤ 0 (6) ∀ t ∈ R+ , ∀ y ∈ R, where k1 , k2 are such that k2 > k1 . The sector can also be defined by [k1 , ∞] if (7) y (ψ(t, y) − k1 y) ≥ 0 ∀ t ∈ R+ , ∀ y ∈ R.
Definition 1. A mapping δεr x = (εr1 x1 , . . . , εrn xn )T , ∀ ε > 0, ∀x ∈ Rn \ {0} is said to be a dilation on Rn , where r = (r1 , r2 , . . . , rn )T and 0 < ri < ∞, i = 1, 2, . . . , n. Definition 2. A function h : Rn → R is said to be a homogeneous function of degree σ ∈ R with respect to δεr x, this is denoted as h ∈ Hσ , if h(δεr x) = εσ h(x). Definition 3. A system x˙ = f (x, u) (1) where u ∈ R is the input and x ∈ Rn is the state vector, is said to be a homogeneous system of degree τ with respect to δεr x and εs u or, equivalently, f ∈ nτ if f (δεr x, εs u) = ετ δεr f (x, u). (2) n ∂ Notice that a vector field f (x, u(x)) = j=1 fj ∂xj is homogeneous of degree τ if and only if each component fj is a homogeneous function of degree τ + rj with respect to the dilation δεr x. Therefore, in particular, a linear vector field (as in the linear system x˙ = Ax+Bu) is homogeneous of zero degree with respect to the standard dilation (this is r = (1, 1, . . . , 1)T ).
Consider now the case ψ(t, y) = ky, k ∈ [k1 , k2 ] in system (3)-(5). That is, ψ is a linear function in the sector [k1 , k2 ]. Suppose that the resulting system is AS for all k ∈ [k2 , k2 ]. Aizerman’s conjecture (Aizerman, 1949) states that the feedback loop is absolutely stable, this is, the origin is globally AS (GAS) for any non linearity ψ ∈ [k1 , k2 ]. Furthermore, Kalman’s conjecture (Kalman, 1957) states, broadly speaking, that the system (3)-(5) is absolutely stable if the following conditions are fulfilled: K.I. The function ψ(y) is differentiable and such that k1 < ψ (y) < k2 . K.II. All linear feedback loops with ψ(y) = ky, k ∈ [k1 , k2 ] are AS.
Definition 4. A class C 1 proper positive-definite homogeneous function V : Rn → R+ is said to be a strict homogeneous Lyapunov function for the system x˙ = f (x) if ∂ ∇V (x)f (x) < 0, x = 0, where ∇V (x) = V (x). ∂x
As it is well-known, both conjectures are false. However, their importance lies in the theoretical point of view, since they lead to the question ‘What additional properties does the linear system have to satisfy in order for the conjectures to be true?’ An answer to this question is provided by the circle criterion in the frequency domain, which relates to the KYP lemma in the state-space formulation.
Apart from the previous definitions, let us introduce the following stability theorem to which we will refer in a 515
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The background presented thus far makes it possible to state the absolute stability problem for system (8)-(10) as follows: Definition 6. Suppose that ψ in (10) satisfies a homogeneous sector condition. The closed loop system (8)-(10) is said to be absolutely stable if the origin is a GAS equilibrium.
3.1 Homogeneous extension of Lur’e problem
X
+RPRJHQHRXV V\VWHP
\
Sufficient conditions for absolute stability of system (8)(10) are presented in the following section.
ψ W
4. HOMOGENEOUS CIRCLE CRITERION
Figure 1. Lur’e-like system with a homogeneous block in the forward path.
Proposition 7. The closed-loop system (8)-(10) is absolutely stable if
Consider a generalisation of the Lur’e system (3)-(5): x˙ = f (x) + g(x)u, y = h(x) u = −ψ(t, y)
(i) ψ ∈ [k1 , ∞]φ and there exist a homogeneous Lyapunov function V (x) ∈ Hm , m > max1≤i≤n ri , real functions M (x), and L(x) such that L2 (x) is positive definite, satisfying the following equations (13) ∇V (x) (f (x) − k1 g(x)φ(h(x)) = −L2 (x) 2 ∇V (x)g(x) = M (x)φ(h(x)) (14) (ii) ψ ∈ [k1 , k2 ]φ , with k = k2 − k1 , and there exist a homogeneous Lyapunov function V (x) ∈ Hm , m > max1≤i≤n ri , L : Rn → Rη , such that LT (x)L(x) is a positive definite function, and M : Rn → Rη for some value of η, which satisfy ∇V (x) (f (x) − k1 g(x)φ(h(x)) = −LT (x)L(x) (15)
(8) (9) (10)
as it is shown in Fig. 1, where x = (x1 , . . . , xn ) ∈ Rn represents the states of the block in the forward path, whereas u and y, the input and output of the block, respectively, are scalar quantities. The vector field on the right hand side of (8) is considered to be continuous in x and homogeneous of degree τ ∈ R with respect to the dilations δεr x and εs u. Notice that, since the homogeneous feedforward block is affine in the input, it is required that f (x) ∈ nτ and g(x) ∈ nτ −s . Furthermore, h(x) in (9) is a continuous homogeneous function of degree σ > 0 with respect to δεr x (i.e. h ∈ Hσ ).
∇V (x)g(x) = kφ(h(x))M T (x)M (x) − 2LT (x)M (x) (16)
Notice that the difference between this Proposition and the conditions for absolute stability derived from the dissipativity result by Hill and Moylan is the introduction of the homogeneous function or array of functions M (x). In Section 6 this difference is further developed.
Finally, the feedback interconnection (10) is given by a memoryless function satisfying a homogeneous sector condition.
This new sector condition is proposed by following the logic of Aizerman’s conjecture, in such a way that the feedback function ψ is chosen in order for the closed loop to preserve the homogeneity of degree τ of the block in the forward path. This is achieved by making ψ(t, y) = γφ(y), γ ∈ R, with φ(y) a homogeneous function of degree s/σ with respect to the dilation εy, which implies u(δεr x) = εs u(x). Therefore, from (2), the δεr −homogeneity of degree τ is preserved. Definition 5. Let ψ : R+ ×R → R. It is said that ψ belongs to the homogeneous sector
Proof. (i) The derivative of V along the trajectories of system (8)-(10) is given by V˙ = ∇V (f − gψ(t, h)) = ∇V (f − k1 gφ(h)) − ∇V g (ψ(t, h) − k1 φ(h)) = −L2 − M 2 φ(h) (ψ(t, h) − k1 φ(h)) The fact that ψ ∈ [k1 , ∞]φ , i.e. φ(h) (ψ(t, h) − k1 φ(h)) ≥ 0, implies V˙ ≤ −L2 (x) < 0. This allows us to conclude that the origin is a GAS equilibrium of the closed-loop system.
• [k1 , k2 ]φ , if ψ(t, y) satisfies
(ψ(t, y) − k1 φ(y)) (ψ(t, y) − k2 φ(y)) ≤ 0 ∀ t ∈ R+ , ∀ y ∈ R,
(11)
(ii) The derivative of V along the trajectories of system (8)-(10) is given by
where k1 , k2 are constants satisfying k1 < k2 , and φ is a homogeneous function of degree s/σ, σ > 0, with respect to ε > 0.
V˙ = ∇V (f − gψ(t, h)) = ∇V (f − k1 gφ(h)) − ∇V g (ψ(t, h) − k1 φ(h)) = −LT L − kφ(h)M T M − 2LT M (ψ(t, h) − k1 φ(h))
• [k1 , ∞]φ , if ψ(t, y) satisfies
φ(y) (ψ(t, y) − k1 φ(y)) ≥ 0 ∀ t ∈ R+ , ∀ y ∈ R.
= −L − M (ψ(t, h) − k1 φ(h))2
(12)
− M T M (k2 φ(h) − ψ(t, h)) (ψ(t, h) − k1 φ(h))
where k1 ∈ R and φ is a homogeneous function of degree s/σ, σ > 0, with respect to ε > 0.
where · is the Euclidean norm. Given that ψ ∈ [k1 , k2 ]φ , V˙ ≤ −L − M (ψ(t, h) − k1 φ(h))2 < 0. This allows us 516
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to conclude that the origin is a GAS equilibrium of the closed-loop system.
517
stability of the STA (ρ = 1/2), tools of nonsmooth analysis can be used as those in Shevitz and Paden (1994), Clarke (1990), and Bacciotti and Ceragioli (1999).
5. EXAMPLES Example 1. Consider the feedback loop ρ x˙ 1 = −a x1 + x2 , Σ : x˙ 2 = −b x1 2ρ−1 + u y = h(x) = x2 u = −ψ(t, y), ψ ∈ [k1 , ∞]φ φ(y) = y
2ρ−1 ρ
Example 2. Let us consider now the same system with a different output, as follows ρ x˙ 1 = −a x1 + x2 , 2ρ−1 Σ : x˙ 2 = −b x1 (22) +u y = h(x) = x1 (23) u = −ψ(t, y), ψ ∈ [k1 , k2 ]φ 2ρ−1 φ(y) = y (24)
(17) (18) (19)
where · q = | · |q sign(·), q ≥ R. The vector field x˙ = T (x˙ 1 , x˙ 2 ) = f (x)+g(x)u is homogeneous of degree τ = (ρ− 1)/ρ with respect to the homogeneity weights (r1 , r2 ) = (1/ρ, 1) and s = (2ρ−1)/ρ. Moreover, the output y = h(x) is a homogeneous function of degree σ = 1.
The vector field is again homogeneous of degree τ = (ρ − 1)/ρ with respect to the homogeneity weights (r1 , r2 ) = (1/ρ, 1) and s = (2ρ − 1)/ρ. However, the output y = h(x) = x1 is a homogeneous function of degree σ = 1/ρ. Conditions (15) and (16) can be used in order to find a region in the space of parameters a, b for which there exist k1 and k2 such that the feedback loop (22)-(24) is absolutely stable. Following a similar approach to that of the previous example, the following homogeneous Lyapunov candidate function (Moreno et al., 2014) is proposed: γ2 γ1 |x1 |3ρ − x1 2ρ x2 + |x2 |3 . V = 3ρ 3
The case ρ = 1/2 corresponds to the Super Twisting algorithm (STA) (Levant, 1993), which has properties of robustness, precision and convergence to the origin in finite time. This algorithm is the basis for the construction of controllers, observers (Davila et al., 2005), and differentiators (Levant, 1998), in the frame of higher order slidingmode control. The sufficient conditions for stability introduced in the previous section can be used in order to find a region in the space of parameters a, b for which there exists k1 such that the closed-loop system (17)-(19) is absolutely stable. Moreover, the values of k1 can also be found following this strategy.
In order for this function to be positive definite, the inequalities γ1 − 2ρc3/2 > 0
αβ <
Let us find a system of inequalities that guarantees the conditions in (15) and (16) are met. On the one hand, condition (16) is satisfied if
M 2 (x). 1−ρ
This is achieved by making M 2 (x) = |x2 | ρ . On the other hand, we find that condition (13) is fulfilled if −∇V (f − k1 gφ(h)) = L2 is a positive definite function, this is
= ab|x1 |3ρ−1 + k1 |x2 |
3ρ−1 ρ
cp p c−q q α + β . p q
The inequalities in (25) are equivalent to γ1 > 2ρc3/2 and c3 > γ12 . Thus, a necessary and sufficient condition for the positivity of V is that γ2 γ12 > 4ρ2 .
Let us find a system of inequalities that guarantees the conditions in (13) and (14) are met. On the one hand, condition (16) is satisfied if
L2 = (a x1 ρ − x2 ) b x1 2ρ−1 2ρ−1 x2 + b x1 2ρ−1 + k1 x2 ρ
(25)
Therefore, 1 1 γ1 − 2ρc3/2 |x1 |3ρ + γ2 − c−3 |x2 |3 . V ≥ 3ρ 3
This function is positive definite if the constant b is positive.
2ρ−1 ρ
γ2 − c−3 > 0
have to be satisfied, for some c > 0. This is due to the fact that for any real numbers α > 0, β > 0, c > 0, p > 1, and q > 1 with p−1 + q −1 = 1, the following inequality (Young’s inequality) holds
Theorem 1 suggests a homogeneous Lyapunov candidate function. In order for the function to be of class C 1 , we make m equal to 2. Given the homogeneity weights, the following function appears as a natural candidate: b 1 V = |x1 |2ρ + |x2 |2 (20) 2ρ 2
∇V (x)g(x) = x2 = x2
and
∇V (x)g(x) = − x1 2ρ + γ2 x2 2 T = k x1 2ρ−1 M (x) − 2L(x) M (x).
This can be achieved by making |x1 |1/2 1−2ρ M (x) = k −1/2 γ21/2 |x1 | 2 |x2 | , and 0 4ρ−1 k 1/2 x1 2 1/2 1/2 2ρ−1 2ρ−1 γ k L(x) = 2 2 2 x1 |x2 | + |x1 | x2 . 2 L3 (x)
(21)
,
which is, clearly, positive definite if a > 0 and k1 > 0. Therefore, it is possible to assert the absolute stability of system (17)-(19) for such values of a, b, and k1 . It should be noted that the Lyapunov function (20) is of class C 1 if ρ > 1/2. In order to conclude on the absolute 517
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On the other hand, we find that condition (15) is fulfilled if L23 = −∇V (f − k1 gφ(h)) − L21 − L22 is a positive definite function, this is L23 = (a x1 ρ − x2 ) γ1 x1 3ρ−1 − 2ρ|x1 |2ρ−1 x2 + (b + k1 ) x1 2ρ−1 γ2 x2 2 − x1 2ρ kγ2 |x1 |2ρ−1 |x2 |2 − γ12 k|x1 |4ρ−1 − 2 kγ2 x1 2ρ−1 x2 2 + 2 T x1 ρ x1 ρ = |x1 |2ρ−1 Q(x) , x2 x2 where
(a) Case ρ = 1
Figure 2. Regions of absolute stability in the space of parameters for different values of ρ and k = 2.
Q11 Q12 Q12 Q22 (x) γ1 + 2ρa aγ1 − (b + k1 + k) − 2 = , γ1 + 2ρa Q22 (x) − 2 with k k 0 Q22 (x) = 2ρ − γ2 − b + k1 + x1 x2 . 2 2 Q(x) =
(b) Case ρ = 1/2
Therefore, the circle criterion brings a two-way connection between the solution to the Lur’e problem and the concept of passivity, which is a recurring theme in the literature of non linear systems theory. However, the generalisation that we propose does not have such an immediate connection to conditions for passivity as those of Hill-Moylan; the difference comes from the introduction of the term M (x). Making M (x) = 1, that is if conditions (15), (16) were instead ∇V (x) (f (x) − k1 g(x)φ(h(x))) = −L2 (x) (27) ∇V (x)g(x) = kφ(h(x)) − 2L(x), (28)
Observe that for x1 x2 < 0, L23 (x) is positive if and only if Q11 > 0 and 2ρ − γ2 (b + k1 + k) > 0.
Notice also that L23 (x) is positive if and only if Q(x) is a positive definite matrix for x1 x2 > 0 or, equivalently, Q11 > 0 and 2 γ1 + 2ρa Q11 (2ρ + γ2 (b + k1 )) > . 2
then, the conditions are equivalent to the cited Hill-Moylan conditions of passivity for the system x˙ = f (x) + g(x)(¯ u − k1 φ(h(x))) y¯ = kφ(h(x)) + u ¯, which is a loop transformation as those found in the literature of nonlinear systems regarding the absolute stability problem (Vidyasagar, 2002) (Khalil, 1996). However, if these conditions were satisfied there would be an undesirable restriction on the degree of homogeneity of V (x).
In summary, system (17)-(19) is absolutely stable if there exist k1 ∈ R, k ≥ 0, γ1 , and γ2 such that following inequalities are satisfied: γ2 γ12 > 4ρ2 Q11 = aγ1 − (b + k1 + k) > 0 (26) 1 2 Q11 (2ρ + γ2 (b + k1 )) > (γ1 + 2ρa) 4 2ρ − γ2 (b + k1 + k) > 0
To illustrate this, let us take the case of Example 2. The conditions in (27) would force the homogeneity degree of V (x) (which we denote as m) to be equal to 2s − τ . This is because equation (28) and the fact that V , g, φ(h), and L are δεr homogeneous of degrees m, τ − s, s, and s, respectively, imply that the homogeneity degree of ∇V (x)g(x) is equal to that of φ(h(x)). This is m+τ −s = s. In this case s = (2ρ − 1)/ρ, and τ = (ρ − 1)/ρ. Therefore, the degree of homogeneity of V would necessarily be equal to 2s − τ = (3ρ − 1)/ρ.
Notice that when the inequalities in (26) are true, L2 (x) is positive definite only if ρ = 1/2. For other values of ρ, it is positive semidefinite. However, for the case in which ψ does not depend directly on t, asymptotic stability still holds due to the invariance principle. A plot of the region of parameters that renders system (17)-(19) absolutely stable can be found using inequalities (26), as shown in Fig. 2.
As the interest of this article is to work with functions as those described by Theorem 1, the recently found value of m allows to propose functions of class C 1 only if ρ > 2/3 (because this requires m to be greater than max{1/ρ, 1}, therefore (3ρ − 1)/ρ > 1/ρ which implies ρ > 2/3). Introducing M (x) allows to choose an arbitrary homogeneity degree for the function V (x), which translates to V being differentiable an arbitrary number of times.
6. DISCUSSION As it was stated in Section 3, the circle criterion that provides a solution to the Lur’e problem has a close relationship with the prominent KYP lemma. This lemma, in turn, offers a tool in the state-space to find out whether or not a system is (strictly) positive real, a property intimately related to the concept of passivity.
Equally remarkable is the fact that the homogeneous sector condition introduced in Section 3 enables the analysis of closed-loop systems with feedback functions having different properties, for instance, non-Lipschitz functions at 518
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Figure 3. Feedback functions ψ in different sectors. zero or functions that do not cross the origin, as shown in Fig. 3.
Gr¨ une, L. (2000). Homogeneous state feedback stabilization of homogenous systems. SIAM Journal on Control and Optimization, 38(4), 1288–1308. Hermes, H. (1991). Nilpotent and high-order approximations of vector field systems. SIAM review, 33(2), 238– 264. Hill, D. and Moylan, P. (1976). The stability of nonlinear dissipative systems. IEEE Transactions on Automatic Control, 21(5), 708–711. Jerbi, H., Kharrat, T., and Kallel, W. (2013). Some results on stability and stabilization of homogeneous time-varying systems. Mediterranean Journal of Mathematics, 1–12. Kalman, R.E. (1957). Physical and mathematical mechanisms of instability in nonlinear automatic control systems. Trans. ASME, 79(3), 553–566. Kawski, M. (2015). Homogeneity in control: Geometry and applications. In European Control Conference (ECC), 2015, 2449–2457. IEEE. Khalil, H.K. (1996). Noninear systems. Prentice-Hall, New Jersey, 2(5). Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58(6), 1247–1263. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica, 34(3), 379–384. Levant, A. (2005). Homogeneity approach to high-order sliding mode design. Automatica, 41(5), 823–830. Lur’e, AI y Postnikov, V. (1944). On the theory of stability of control systems. Applied mathematics and mechanics, 8(3), 246–248. Moreno, J., S´anchez, T., and Cruz-Zavala, E. (2014). Una funci´on de Lyapunov suave para el algoritmo supertwisting. 16th CLCA, 182–187. Peuteman, J. and Aeyels, D. (1999). Averaging results and the study of uniform asymptotic stability of homogeneous differential equations that are not fast timevarying. SIAM journal on control and optimization, 37(4), 997–1010. Rosier, L. (1992). Homogeneous Lyapunov function for homogeneous continuous vector field. Systems & Control Letters, 19(6), 467–473. Sepulchre, R. and Aeyels, D. (1996). Homogeneous Lyapunov functions and necessary conditions for stabilization. Mathematics of Control, Signals and Systems, 9(1), 34–58. Shevitz, D. and Paden, B. (1994). Lyapunov stability theory of nonsmooth systems. IEEE Transactions on automatic control, 39(9), 1910–1914. Vidyasagar, M. (2002). Nonlinear systems analysis. SIAM.
It is also important to emphasise the relevance of this result concerning the analysis of complex non-linear and non-homogeneous systems exhibiting properties such as finite-time stabilisation, as long as they can be represented by the generalised Lur’e system that is proposed. Moreover, this is also a tool for the study of the stability of time-varying homogeneous systems, see, for instance, Jerbi et al. (2013) and Peuteman and Aeyels (1999). 7. CONCLUSIONS The aim of this paper is to develop a set of tools in order to analyse the stability of a class of systems that can be modelled by the feedback loop (8)-(10), in similar terms as those that offer solutions to the Lur’e problem. The main contributions presented are the statement of homogeneous generalisations of the Lur’e problem and the circle criterion. Contrary to the original circle criterion, the conditions presented in this article do not have an immediate connection to the results of passivity and dissipativity, which motivates further research. REFERENCES Aizerman, M.A. (1949). On a problem concerning the stability “in the large” of dynamical systems. Uspekhi matematicheskikh nauk, 4(4), 187–188. Bacciotti, A. and Ceragioli, F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESAIM: Control, Optimisation and Calculus of Variations, 4, 361–376. Bacciotti, A. and Rosier, L. (2006). Liapunov functions and stability in control theory. Springer Science & Business Media. Bhat, S.P. and Bernstein, D.S. (1997). Finite-time stability of homogeneous systems. In American Control Conference, 1997. Proceedings of the 1997, volume 4, 2513–2514. IEEE. Brogliato, B. and Heemels, W.M.H. (2009). Observer design for Lur’e systems with multivalued mappings: A passivity approach. IEEE Transactions on Automatic Control, 54(8), 1996–2001. Clarke, F.H. (1990). Optimization and nonsmooth analysis. SIAM. Davila, J., Fridman, L., and Levant, A. (2005). Secondorder sliding-mode observer for mechanical systems. IEEE Transactions on Automatic Control, 50(11), 1785– 1789. 519