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Ellipsoidal Estimates of Reachable Sets for Control Systems with Nonlinear Terms *
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of 20th The International Federation of Congress Automatic Control Toulouse, France, July 9-14, 2017 The International Federation of Automatic Proceedings of the 20th World The International of Congress Automatic Control Control Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 15355–15360 Ellipsoidal Estimates of Reachable Sets for Ellipsoidal Ellipsoidal Estimates Estimates of of Reachable Reachable Sets Sets for for ⋆⋆ Control Systems with Nonlinear Terms Ellipsoidal Estimates of Reachable Sets for Control Systems with Nonlinear Terms Control Systems with Nonlinear Terms ⋆⋆ Control Systems with Nonlinear Terms Tatiana F. Filippova
Tatiana F. Filippova Tatiana Tatiana F. F. Filippova Filippova Tatiana F. Filippova Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Krasovskii Institute of and Ural Branch of Krasovskii Institute of Mathematics Mathematics and Mechanics, Mechanics, Ural Branch of Russian Academy of Sciences, Sofia Kovalevskaya str.Branch 16, of Krasovskii Institute of Mathematics and Mechanics, Ural Russian Academy of Sciences, Sofia Kovalevskaya str. 16, Russian Academy of Sciences, Sofia Kovalevskaya str. 16, Krasovskii Institute of Mathematics and Mechanics, Ural Branch Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) Russian 620990, AcademyRussian of Sciences, Sofia Kovalevskaya str. 16, of Ekaterinburg Federation (e-mail: Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) [email protected]) Russian 620990, AcademyRussian of Sciences, Sofia Kovalevskaya str. 16, Ekaterinburg Federation (e-mail: [email protected]) Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) Abstract: The problem of estimating reachable sets of nonlinear dynamical control systems Abstract: The The problem problem of of estimating estimating reachable reachable sets sets of of nonlinear nonlinear dynamical dynamical control control systems systems Abstract: with quadratic and with reachable uncertainty initial states is studied. We systems assume Abstract: The nonlinearity problem of estimating setsin nonlinear dynamical control with quadratic nonlinearity and with with reachable uncertainty inof initial states is studied. studied. We systems assume with quadratic nonlinearity and uncertainty in initial states is We assume Abstract: The problem of estimating sets of nonlinear dynamical control that the uncertainty is of a set-membership kind when we know only the bounding set for with quadratic nonlinearity and with uncertainty in initial states is studied. We assume that the uncertainty is of a set-membership kind when we know only the bounding set for for that the uncertainty is additional of aa set-membership set-membership kind when when wetheir know only the bounding set with quadratic nonlinearity and with uncertainty in initial states is the studied. We assume unknown items and any statistical information on behavior is not available. We that the uncertainty is of kind we know only bounding set for unknown items and any additional statistical information on their behavior is not available. We unknown items and any any additional statistical information on their behavior issets not available. that thehere uncertainty is additional of a allow set-membership kind when wetheir know only the bounding set We for present approaches that finding ellipsoidal estimates of behavior reachable which use the unknown items and information on not available. present here here approaches that allow allow statistical finding ellipsoidal ellipsoidal estimates of behavior reachableis sets which use We the present approaches that finding estimates of reachable sets which use the unknown items and any additional statistical information on their issets not which available. We special structure of nonlinearity of studied control system. The algorithms of constructing such present here approaches that allow finding ellipsoidal estimates of reachable use the special structure structure of nonlinearity nonlinearity of studied studied control system. The algorithms algorithms ofsets constructing such special of of control system. The of constructing such present here approaches that allow finding ellipsoidal estimates ofare reachable which usesuch the ellipsoidal set-valued estimates and numerical simulation results given. of special structure of nonlinearity of studied control system. The algorithms constructing ellipsoidal set-valued estimates and numerical simulation results are given. ellipsoidal set-valued estimates and numerical simulation results are given. special structure of nonlinearity of studied control system. The algorithms of constructing such ellipsoidal set-valued estimates and numerical simulation results are given. © 2017, IFAC (International Federation of Automaticsimulation Control) Hosting byare Elsevier rights reserved. ellipsoidal andNonlinear numerical results given.Ltd. All Keywords: set-valued Estimationestimates algorithms, control systems, Set-membership uncertainty, Keywords: Estimation algorithms, Nonlinear control systems, Set-membership uncertainty, Keywords: Estimation Estimation algorithms, Nonlinear Nonlinear control control systems, systems, Set-membership Set-membership uncertainty, uncertainty, Ellipsoidal calculus. algorithms, Keywords: Ellipsoidal calculus. Ellipsoidal calculus. Keywords: Estimation algorithms, Nonlinear control systems, Set-membership uncertainty, Ellipsoidal calculus. Ellipsoidal calculus. 1. INTRODUCTION developed approaches to researches in this area. The first 1. INTRODUCTION developed approaches to researches in this area. The first 1. INTRODUCTION developed approaches to this one is based on ellipsoidal calculus in (Kurzhanski and first Va1. INTRODUCTION developed approaches to researches researches in this area. area. The The first one is based on ellipsoidal calculus (Kurzhanski and Vaone is based on ellipsoidal calculus (Kurzhanski and Va1. INTRODUCTION approaches toand researches this area. The first lyi (1997); Kurzhanski Varaiyain (2014); Chernousko In this paper we deal with control systems with unknown developed one is based on ellipsoidal calculus (Kurzhanski and Valyi (1997); Kurzhanski and Varaiya (2014); Chernousko In this paper we deal with control systems with unknown lyi (1997); Kurzhanski and Varaiya (2014); Chernousko one is based on ellipsoidal calculus (Kurzhanski and Va(1994); Polyak et al. (2004)) and the second one uses the In this paper we deal with control systems with unknown but bounded uncertainties and we consider the case of a lyi (1997); Kurzhanski and Varaiya (2014); Chernousko In this paper we deal with control systems with unknown (1994); Polyak et al. (2004)) and the second one uses the but bounded uncertainties and we consider the case of aa lyi (1994); Polyak et al. (2004)) and the second one uses the (1997); Kurzhanski and Varaiya (2014); Chernousko interval analysis (Walter and Pronzato (1997); Kostousova but bounded uncertainties and we consider the case of In this paper we deal with control systems with unknown set-membership description of uncertain parameters and (1994); Polyak et al. (2004)) and the second one uses the but bounded uncertainties and we consider the case of a interval analysis (Walter and Pronzato (1997); Kostousova set-membership description of uncertain parameters and interval analysis and Pronzato (1997); Kostousova Polyak et(Walter al. (2004)) and the second one uses the (2013)). set-membership description of uncertain uncertain parameters and but bounded and we the problems case of a (1994); functions. Theuncertainties study of control andconsider estimation interval analysis (Walter and Pronzato (1997); Kostousova set-membership description of parameters and (2013)). functions. The study of control and estimation problems (2013)). analysis (Walter and Pronzato (1997); Kostousova functions. The study ofand control and estimation estimation problems set-membership description of uncertain parameters and interval follow hereThe thestudy ideas the basic theoretical schemes (2013)). functions. of control and problems Many applied problems are mostly nonlinear in their pafollow here the ideas and the basic theoretical schemes (2013)). Many applied problems are mostly nonlinear in their pafollow here the ideas and the basic theoretical schemes functions. The study of control and estimation problems proposed and developed in Kurzhanski (1974, 1977, 1991, follow here the ideas and the basic theoretical schemes Many applied mostly nonlinear in parameters and problems the set ofare states is usuproposed and developed in Kurzhanski (1974, 1977, 1991, Many applied problems arefeasible mostly system nonlinear in their their parameters and the set of feasible system states is usuproposed and developed in Kurzhanski (1974, 1977, 1991, follow here the ideas and the basic theoretical schemes 2010). The motivation of this study is to apply the setproposed and developed in Kurzhanski (1974, 1977, 1991, rameters and the set of feasible system states is usuMany applied problems are mostly nonlinear in their paally non-convex or even non-connected. The key issue in 2010). The motivation of this study is to apply the setrameters and the set of feasible system states is usuor even The key issue in 2010). The The motivation ofinnew this study isof tocontrol apply the set- ally proposed and developedto Kurzhanski (1974, 1977, 1991, membership approach classesis systems 2010). motivation of this study to apply the setally non-convex non-convex or set evenofnon-connected. non-connected. The key suitable issue in rameters and the feasible system is usunonlinear set-membership estimation is The tostates find membership approach to new classes of control systems ally non-convex or even non-connected. key issue in set-membership estimation is to find suitable membership approach to new classesand oftouncertainty. control systems 2010). Thesystems motivation of this study isof apply the setincluding with to nonlinearity We nonlinear membership approach new classes control systems nonlinear set-membership estimation is to find suitable ally non-convex or produce even non-connected. The key issue techniques, which related bounds for thesuitable set in of including systems with nonlinearity and uncertainty. We nonlinear set-membership estimation is to find which produce related bounds set of including systems systems with to nonlinearity and uncertainty. We membership approach classesand control systems emphasize the relevance ofnew research inofthis direction be- techniques, including with nonlinearity uncertainty. We techniques, which produce related bounds for the set of nonlinear set-membership estimation is computationally to for findthe suitable unknown system states without being too emphasize the relevance of research in this direction betechniques, which produce related bounds for the set of unknown system states without being too computationally emphasize the relevance of research in this direction beincluding systems with nonlinearity and uncertainty. We cause the characterization of parameter uncertainties in emphasize the relevance of research in this direction beunknown system states without being too computationally techniques, which produce related bounds for the set of demanding. In this paper the modified state estimation cause the characterization of parameter uncertainties in unknown system states without being too computationally In paper the modified estimation cause the the characterization characterization of parameter uncertainties in demanding. emphasize the relevance models of of research in this direction betraditional probabilistic requires assumptions on cause parameter uncertainties in demanding. In this this paper the modified state estimation unknown system states without being too state computationally approaches which use the special quadratic structure of traditional probabilistic models requires assumptions on demanding. In this paper the modified state estimation which the special quadratic structure of traditional probabilistic models requires assumptions on cause the characterization of parameter uncertainties in approaches mean, variances or probability density function of errors. traditional probabilistic models requires assumptions on approaches use the special quadratic of demanding. In studied thisuse paper the modified estimation nonlinearity which of control system andstate use structure also the admean, variances or probability density function of errors. approaches which use the special quadratic structure of nonlinearity of studied control system and use also the admean, variances or probability density function of errors. traditional probabilistic models requires assumptions on However in many applied areas ranged from engineering mean, variances or probability density function of errors. nonlinearity of studied control system and use also the adapproaches which use the special quadratic structure of vantages of ellipsoidal calculus are presented. The special However in many applied areas ranged from engineering nonlinearity of studied control system and use also the adof ellipsoidal calculus are presented. The special However in many applied areasdensity ranged fromitengineering engineering mean, variances orapplied probability function errors. vantages problems in many physics to economical modeling isofdifficult However in areas ranged from vantages of ellipsoidal calculus are presented. The special nonlinearity of studied control system and use also the adcase when the quadratic form in the equations of dynamics problems in physics to economical modeling it is difficult vantages ofthe ellipsoidal calculus are presented. The special when quadratic form in the equations of dynamics problems in many physics to economical economical modeling itengineering is difficult difficult However in applied areas from to get necessary statistical dataranged for such description of case problems in physics to modeling it is case when the quadratic form inbe thenot equations of dynamics vantages ofthe ellipsoidal calculus are presented. The special of the controlled system mayin positiveof definite is to get necessary statistical data for such description of case when quadratic form the equations dynamics of the controlled system may be not positive definite is to get necessary statistical data for such description of problems in physics to economical modeling it is difficult theget model (Apreutesei (2009); August and Koeppl (2012); to necessary statistical data for such description of case of the controlled system may not positive definite is when the The quadratic form inbe the equations dynamics studied here. studies in this direction areofmotivated the model (Apreutesei (2009); August and Koeppl (2012); of the controlled system may be not positive definite is studied here. The studies in this direction are motivated the model (Apreutesei (2009); August and Koeppl (2012); to get necessary statistical data for such description of Boscain et al. (2013); Ceccarelli et al. (2004)). As an alterthe model (Apreutesei (2009); August and Koeppl (2012); studied here. The studies in this direction are motivated of the controlled system may be not positive definite is also by applications related, e.g., to satellite control probBoscain et al. (2013); Ceccarelli et al. (2004)). As an alterstudied here. The studies in this direction are motivated by applications related, e.g., to satellite control probBoscain eta(Apreutesei al. (2013); Ceccarelli Ceccarelli et description al. (2004)). (2004)). As an(2012); alter- also the model (2009); August and Koeppl native toet stochastic approach to of unknown Boscain al. (2013); et al. As an alteralso by applications related, e.g., to satellite control probstudied here. The studies in this direction are motivated lems (Kuntsevich and Volosov (2015)) where the attainnative to a stochastic approach to description of unknown by applications related, e.g.,(2015)) to satellite control problems (Kuntsevich and Volosov where the native to toetaa al. stochastic approach to description of unknown unknown Boscain (2013); Ceccarelli et description al. (2004)). model As an alterparameters and functions, the bounded-error char- also native stochastic approach to of lems by (Kuntsevich and Volosov (2015)) where the attainattainalso applications related, e.g.,(2015)) to satellite control probability sets of the and control systems werewhere investigated for parameters and functions, the bounded-error model charlems (Kuntsevich Volosov the attainability sets of the control systems were investigated for parameters and functions, the bounded-error model charnative to a stochastic approach to description of unknown acterization and (i.e.,functions, in the frameworks of the set-membership parameters the bounded-error model char- lems ability sets of the control systems were for (Kuntsevich and Volosov (2015)) where the attainnonstationary linear and some classes ofinvestigated nonlinear disacterization (i.e., in the frameworks of the set-membership ability sets of the control systems were investigated for nonstationary linear and some classes of nonlinear disacterization (i.e., in the frameworks of the set-membership parameters and functions, the bounded-error model charapproach) and related theoretical and applied investigaacterization (i.e., in the frameworks of the set-membership nonstationary linear and some classes of nonlinear disability sets of the control systems were investigated for crete systems under bounded parametric (multiplicative) approach) and related theoretical and applied investiganonstationary linear and some classes of nonlinear dissystems under parametric approach) and related theoretical and applied investigaacterization (i.e.,related the theoretical frameworks of the set-membership tions continue toin develop intensively (Kurzhanski and crete approach) and and applied investigacreteadditive systems disturbances. under bounded parametric (multiplicative) nonstationary linear bounded and some classes of(multiplicative) nonlinear disand The suggested algorithms for tions continue to develop intensively (Kurzhanski and crete systems under bounded parametric (multiplicative) disturbances. The suggested algorithms tions (1997); continue to develop develop intensively (Kurzhanski and and approach) andKurzhanski related theoretical and(2014); applied investigaValyi and Varaiya Mazurenko tions continue to intensively (Kurzhanski and and additive additive disturbances. The suggested algorithms for for crete systems under bounded parametric (multiplicative) solving these disturbances. problems allow efficient parallelization of Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko and additive The suggested algorithms for solving these problems allow efficient parallelization of Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko tions continue to develop intensively (Kurzhanski and (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko solving these problems efficient of and additive disturbances. The suggested algorithms for computations and may allow also provide aparallelization solution to the (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. solving these problems allow efficient parallelization of computations and may also provide a solution to the (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko (1996); Milanese and Vicino (1991); Chernousko (1994); (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. computations and may also provide a solution to the solving these problems allow efficient parallelization of related control synthesis problems. It should be noted that (1996); Milanese and Vicino (1991); Chernousko (1994); computations and may also provide a solution to the control synthesis It be (1996); et Milanese andSchweppe Vicino (1991); Chernousko (1994); (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. related Polyak al. (2004); (1973); Gusev (2016)). (1996); Milanese and Vicino (1991); Chernousko (1994); related control synthesis problems. It should should be noted noted that computations may problems. also a solution to that the the casecontrol of anand arbitrary (notprovide necessarily positively dePolyak et al. (2004); Schweppe (1973); Gusev (2016)). related synthesis problems. It should be noted that casecontrol of an arbitrary (not necessarily positively dePolyak et et al. (2004); (2004); Schweppe (1973);Chernousko Gusev (2016)). (2016)). (1996); Milanese andSchweppe Vicino (1991); (1994); the Polyak al. Gusev an arbitrary (not necessarily positively derelated problems. It should be noted that fined,case andof possibly degenerate) function entering the case of ansynthesis arbitrary (not quadratic necessarily positively deThe solution of many control (1973); and estimation problems the fined, and possibly degenerate) quadratic function entering Polyak et al. (2004); Schweppe (1973); Gusev (2016)). The solution of many control and estimation problems fined, and possibly degenerate) quadratic function entering the case of an arbitrary (not necessarily positively destate velocities of a dynamical system is much more The solution solution of many many control and estimation estimation problems under uncertainty involves constructing reachableproblems sets and fined, and possibly degenerate) quadratic function entering The of control and the state velocities of aa dynamical system is much more under uncertainty involves constructing reachable sets and the state velocities of dynamical system is much more fined, and possibly degenerate) quadratic function entering difficult to analyze and requires more detailed analysis, under uncertainty involves constructing reachable sets and The ofFor many control estimation problems their solution analogs. models withand linear dynamics under statetovelocities of a dynamical system is much more under uncertainty involves constructing reachable sets and the difficult analyze and requires more detailed analysis, their analogs. For models with linear dynamics under difficult to analyze and requires more detailed analysis, state velocities dynamical system is much more which is to the subjectof ofa this paper. Thisdetailed study continues their set-membership analogs. Forinvolves models with linear linear dynamics under under uncertainty constructing sets and the difficult analyze and requires more analysis, such uncertainty therereachable are several contheir analogs. For models with dynamics under which is the subject of this paper. This study continues such set-membership uncertainty there are several conwhich is the subject of this paper. This study continues difficult to analyze and requires more detailed analysis, the researches Filippova (2014); Filippova and Matviychuk such set-membership uncertainty there are several contheir analogs. For models with linear dynamics under structive approaches which allow finding effective estiwhich is the subject of this paper. This study continues such set-membership uncertainty there areeffective several esticon- the researches Filippova (2014); Filippova and Matviychuk structive approaches which allow finding the researches Filippova Filippova Matviychuk is the and subject of (2014); this paper. This and study continues (2014, 2015) considers a more complicated case, when structive approachessets. which allow finding effective estisuch set-membership uncertainty areeffective several con- which mates of reachable We allow note there here two of the most the researches Filippova (2014); Filippova and Matviychuk structive approaches which finding esti(2014, 2015) and considers aa more complicated case, when mates of reachable sets. We note here two of the most (2014, 2015) and considers more complicated case, when the researches Filippova (2014); Filippova and Matviychuk quadratic forms that determine the nonlinearity in the mates of reachable sets. We note here two of the most structive approaches which allow finding effective esti2015) and considers a more complicated case, in when mates of reachable sets. We note here two of the most (2014, the quadratic forms that determine the nonlinearity the ⋆ the quadratic forms that determine the nonlinearity in the (2014, 2015) and considers a more complicated case, when system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF mates of reachable sets. We note here two of the most the quadratic forms that determine the nonlinearity in the ⋆ The research was supported by Russian Science Foundation (RSF system may be degenerate, this case is both of theoretical ⋆ The research Project No.16-11-10146) system may be degenerate, this case is both of theoretical was supported by Russian Science Foundation (RSF ⋆ the quadratic forms that determine the nonlinearity in the system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF Project No.16-11-10146) ⋆ Project No.16-11-10146) system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF Project No.16-11-10146) Project No.16-11-10146) Copyright © 2017 IFAC 15925 Copyright © 2017, 2017 IFAC 15925 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 15925 Copyright © 2017 2017 IFAC IFAC 15925 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 15925 10.1016/j.ifacol.2017.08.2460
Proceedings of the 20th IFAC World Congress 15356 Tatiana F. Filippova / IFAC PapersOnLine 50-1 (2017) 15355–15360 Toulouse, France, July 9-14, 2017
and applied importance. Here we also propose a procedure for constructing ellipsoidal estimates for reachable sets that simultaneously takes into account all the complicating factors (uncertainty in the initial data, nonlinearity and functional disturbances in the right-hand sides of the system differential equations with related ellipsoidal constraints on unknown values). All these new feature constitute the novelty of results presented here. Examples and numerical simulations based on the procedures of setvalued approximations of trajectory tubes and reachable sets are also presented. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory.
The main problem of the paper is to find the external ellipsoidal estimate E(a+ (t), Q+ (t)) (with respect to the inclusion of sets) of the reachable set X(t) (t0 < t ≤ T ) by using the analysis of a special type of nonlinear control systems with uncertain initial data. The approach presented here develops and refines previous research in this area. Examples and numerical results related to procedures of set-valued approximations of trajectory tubes and reachable sets are given. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory.
2. PROBLEM FORMULATION
We will need some auxiliary constructions and results which will be used in the following.
In this section we introduce the following basic notations. Let Rn be the n–dimensional Euclidean space, Rn×n stands for the set of all n × n–matrices and x′ y be the usual inner product of x, y ∈ Rn with prime as a transpose, �x� = (x′ x)1/2 , compRn denotes the set of all compact subsets of Rn , h(A, B) is the Hausdorff distance between A, B ∈ compRn . We denote as B(a, r) the ball in Rn , B(a, r) = {x ∈ Rn : �x − a� ≤ r}, I is the identity n × n-matrix. Denote by E(a, Q) the ellipsoid in Rn , E(a, Q) = {x ∈ Rn : (Q−1 (x−a), (x−a)) ≤ 1} with center a ∈ Rn and symmetric positive definite n × n–matrix Q, for any n × n–matrix M = {mij } denote T r(M ) =
i=n
mii .
i=1
Consider the following nonlinear control system x˙ = A(t)x + f (x)d + u(t), x0 ∈ X0 , t0 ≤ t ≤ T,
(1)
where x, d ∈ Rn , �x� ≤ K (K > 0), the n × n–matrix A(t) is assumed to be continuous on t ∈ [t0 , T ] and f (x) = x′ Bx is scalar function, with a symmetric n × n–matrix B, u(t) ∈ U, U ⊂ Rm for a.e. t ∈ [t0 , T ]. (2) We will assume that X0 in (1) is an ellipsoid, X0 = E(a, Q), with a symmetric and positive definite matrix Q and with a center a. Let the absolutely continuous function x(t) = x(t, u(·), t0 , x0 ) be a solution to (1). We will study the solutions of the system (1)–(2) in the framework of the theory of uncertain dynamical systems through the techniques of trajectory tubes of related differential inclusions (Kurzhanski and Filippova (1993); Kurzhanski and Varaiya (2014)): X(·) = X(·; t0 , X0 ) = { x(·) = (3) x(·, u(·), t0 , x0 ) | x0 ∈ X0 , u(·) ∈ U }. The reachable set X(t) of the system (1) at time t (t0 < t ≤ T ) is defined as the cross-section of the trajectory tube (3) of the system (1)-(2) X(t) = X(t; t0 , X0 ) =
3. PRELIMINARIES
3.1 System Dynamics Nonlinearity Defined by a Positive Definite Quadratic Form Consider the nonlinear control system x˙ = A(t)x + f (x)d + u(t), x0 ∈ X0 = E(a0 , Q0 ),
t0 ≤ t ≤ T,
(4)
where x ∈ Rn , �x� ≤ K (K > 0), A(t) ∈ Rn×n is a given ˆ d, a0 , a ˆ are given continuous matrix, u(t) ∈ U = E(ˆ a, Q); n-vectors, a scalar function f (x) has a form f (x) = x′ Bx, ˆ are symmetric and positive definite. matrices B, Q0 , Q Denote the maximal eigenvalue of the matrix B 1/2 Q0 B 1/2 by k 2 , it is easy to see this k 2 is the smallest number for which the inclusion X0 ⊆ E(a0 , k 2 B −1 ) is true. The following result describes the external ellipsoidal estimate of the reachable set X(t) = X(t; t0 , X0 ) (t0 ≤ t ≤ T ). Theorem 1. (Filippova (2009)). Assume that X0 = E(a, k 2 (B −1 )) with some k > 0, then for all σ > 0 and for X(t0 + σ) = X(t0 +σ, t0 , X0 ) we have the following upper estimate X(t0 + σ) ⊆ E(a+ (σ), Q+ (σ)) + o(σ)B(0, 1), (5) where σ −1 o(σ) → 0 when σ → +0 and a+ (σ) = a(σ) + σˆ a, a(σ) = a + σ(A0 a + a′ Bad + k 2 d), ˆ Q+ (σ) = (p−1 + 1)Q(σ) + (p + 1)σ 2 Q,
(6)
Q(σ) = k22 (I + σR)B −1 (I + σR)′ , R = A + 2da′ B. and where p is the unique positive root of the equation n n 1 = p + αi p(p + 1) i=1 with αi ≥ 0 (i = 1, ..., n) being the roots of the following ˆ = 0. equations |Q(σ) − ασ 2 Q|
{ x ∈ Rn : ∃ x0 ∈ X0 , ∃ u(·) ∈ U such that
The following result presents the continuous-type version of the Theorem 1. Theorem 2. (Filippova (2010)). The inclusion is true for any t ∈ [t0 , T ]
x = x(t) = x(t; u(·), x0 ) }, t0 < t ≤ T.
X(t; t0 , X0 ) ⊆ E(a+ (t), r+ (t)B −1 ), 15926
(7)
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Tatiana F. Filippova / IFAC PapersOnLine 50-1 (2017) 15355–15360
4. MAIN RESULTS
E(a *(t),Q *(t))
4.1 Control Systems with Two Positive Definite Quadratic Forms
(t)
t
15357
0.4 0.2 0
Consider the following control system with nonlinearity defined by two different positive definite quadratic forms in the dynamical equations and with uncertain initial state x0 ,
1.5 1 0.5
x2
x˙ = Ax + f (1) (x)d(1) + f (2) (x)d(2) + u(t), ˆ t0 ≤ t ≤ T, x0 ∈ X0 , u(t) ∈ U = E(ˆ a, Q),
2.5 2
0
1.5 -0.5
1 0.5
-1
0 -1.5
where d(1) and d(2) are n-vectors and f (1) , f (2) are scalar functions,
x1
-0.5
f (1) (x) = x′ B (1) x, f (2) (x) = x′ B (2) x,
-1
Fig. 1. Trajectory tube X(t) and its upper estimating tube E(a∗ (t), Q∗ (t)). where functions a+ (t), r+ (t) are the solutions of the following system of ordinary differential equations a˙ + (t) = A0 a+ (t) + ((a+ (t))′ Ba+ (t) + ˆ, t0 ≤ t ≤ T, r+ (t))d + a + ′ r˙ (t) = max l 2r+ (t)B 1/2 (A0 + �l�=1
2d(a+ (t))′ B)B −1/2 + ˆ 1/2 ) l + q(r+ (t))r+ (t), q −1 (r+ (t))B 1/2 QB ˆ 1/2 , q(r) = ((nr)−1 Tr(B Q))
(8)
with initial state a+ (t0 ) = a0 ,
r+ (t0 ) = k 2 .
3.2 Numerical Simulation Consider the example which shows that in nonlinear case the reachable sets may lose their convexity with increasing time t > t0 . Nevertheless the related external estimates calculated on the base of results of Theorems 1–2 are ellipsoidal-valued (and therefore convex) and contain the reachable sets of the system (4). Example 1. Consider the following control system x˙ 1 = x2 + x21 + x22 + u1 , x˙ 2 = 0.5x1 + u2 ,
(10)
(9)
Here we take x0 ∈ X0 = B(0, 1), 0 ≤ t ≤ 0.4 and U = B(0, 0.1). The reachable sets X (t) and their external ellipsoidal estimates E(a∗ (t), Q∗ (t)) are shown in Figure 1. This example is included here to illustrate the main approach, more details on numerical algorithms basing on Theorem 1 and Theorem 2 and producing the external ellipsoidal tube E + (t) = E(a∗ (t), Q∗ (t)) under different assumptions on system parameters may be found in Filippova (2012); Filippova and Matviychuk (2014).
with symmetric and positive definite matrices B (1) , B (2) . (1) We assume also that di = 0 for i = k + 1, . . . , n and (2) dj = 0 for j = 1, . . . , k where k (1 ≤ k ≤ n) is fixed. This assumption means that the first k equations of the system (10) contain only the nonlinear function f (1) (x) (with some (1) constant coefficients di ) while f (2) (x) is included only in the equations with numbers k + 1, . . . , n. Remark 1. In general case, these two positive quadratic functions can not be combined into one positive form because the coefficients (coordinates of vectors d(1) and d(2) ) are arbitrary, therefore some of them may be negative or zeros. As a consequence we cannot apply here the approaches and related results of previous investigations and need to find new techniques which will allow us to do this and to get related estimates of reachable sets of the system considered here. We will assume as before that X0 in (10) is an ellipsoid, X0 = E(a0 , Q0 ). We need first the following auxiliary result, its proof follows directly from the definition of eigenvalues of matrix operators and was obtained earlier in Filippova and Matviychuk (2014), therefore the proof is not included here. Lemma 1. (Filippova and Matviychuk (2014)). The inclusion is true X0 ⊆ E(a, k12 (B (1) )−1 ) E(a, k22 (B (2) )−1 ) (11)
where ki2 is the maximal eigenvalue of the matrix (B (i) )1/2 Q(B (i) )1/2 (i = 1, 2). The following equalities are true max z ′ B (2) z = k12 λ212 , z ′ B (1) z≤k12 (12) max z ′ B (1) z = k22 λ221 , z ′ B (2) z≤k22
where λ212 (1) −1/2
(B ) spectively.
and λ221 are maximal eigenvalues of matrices B (2) (B (1) )−1/2 and (B (2) )−1/2 B (1) (B (2) )−1/2 re-
The following new result produces the upper estimate in the case under consideration which is more general than the problem studied in Filippova (2009, 2010, 2012) and also produces better (in comparison with results of Filippova and Matviychuk (2014, 2015)) ellipsoidal estimates of reachable set X(t) because this new ellipsoidal estimates of the reachable set also take into account and absorb the ellipsoidal constraints on control functions or
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functional disturbances (noises) which may be presented in the system. Theorem 3. For all σ > 0 and for X(t0 + σ) = X(t0 + σ, t0 , X0 ) we have the following upper estimate X(t0 + σ) ⊆ (13) E(a(1) (σ), Q(1) (σ)) E(a(2) (σ), Q(2) (σ)) where σ
−1
+ o(σ)B(0, 1), o(σ) → 0 when σ → +0 and a(1) (σ) = a(σ) + σk12 λ212 d(2) + σˆ a,
of trajectory tube of the nonlinear control system using a stepwise procedure, convenient for computer modeling. We emphasize that in (Filippova and Matviychuk (2014, 2015)) we studied the dynamic system of a more special type, with a nonlinearity in dynamics and with uncertainty in initial data, but control functions were not subjected to a joint estimation procedure. So the result of Theorem 3 may be considered as a further generalization of previous estimation schemes. Remark 3. Important problems connected with conservativeness of constructed ellipsoidal estimates and of related algorithmic procedures are beyond the scope of this paper and will be studied in the future (see also related discussions, e.g., in Polyak et al. (2004); Asselborn et al. (2013); Kishida and Braatz (2015)).
a(2) (σ) = a(σ) + σk22 λ221 d(1) + σˆ a, a(σ) = (I + σA)a + σa′ B (1) ad(1) + σa′ B (2) ad(2) , Q(1) (σ) = (p−1 1 + 1)(I+ σR)k12 (B (1) )−1 (I + σR)′
(14)
ˆ +(p1 + 1)σ 2 (||d(2) ||2 k14 λ412 · I + Q),
Consider the general case x˙ = A(t)x + x′ Bx · d + u(t), t0 ≤ t ≤ T, ˆ x0 ∈ X0 = E(a0 , Q0 ), u(t) ∈ U = E(ˆ a, Q).
Q(2) (σ) = (p−1 2 + 1)(I+ σR)k22 (B (2) )−1 (I + σR)′ + ˆ (p2 + 1)σ 2 (||d(1) ||2 k24 λ421 · I + Q), (1) ′
(1)
(2) ′
(2)
R = A + 2d a B + 2d a B and p1 , p2 are the unique positive solutions of related algebraic equations n 1 n p1 +αi = p1 (p1 +1) , i=1 (15) n 1 n = p2 +βi p2 (p2 +1) i=1
with αi , βi ≥ 0 (i = 1, ..., n) being the roots of the following equations det((I + σR)k12 (B (1) )−1 (I + σR)′ − ασ 2 ||d(2) ||2 k14 λ412 · I) = 0, det((I + σR)k22 (B (2) )−1 (I + σR)′ − βσ
2
||d(1) ||2 k24 λ421
4.2 Generalization of the Estimation Approach
(16)
ˆ and Q0 are symmetric, We assume here that matrices B, Q ˆ Q and Q0 are positive definite. This assumption produces more general case because we do not assume here the positive definiteness of the matrix B in the nonlinear term of the right-hand side of dynamic equations (19). This new setting generalizes previous results and is motivated also by applied problems (e.g., Kuntsevich and Volosov (2015)). Using well-known diagonalization procedures of matrix analysis (Bellman (1997)) we can find the non-degenerate n × n–matrix Z of transformation z = Zx (x, z ∈ Rn ) of the state space Rn under which the system (19) will take the form z˙ = A∗ (t)z + z ′ B ∗ z · d∗ + w(t), t0 ≤ t ≤ T, z0 ∈ Z0 = E(a∗0 , Q∗0 ), ˆ ∗ ), w(t) ∈ W = E(ˆ a∗ , Q
· I) = 0.
Proof. The funnel equation (Kurzhanski and Valyi (1997); Kurzhanski and Varaiya (2014); Kurzhanski and Filippova (1993)) describing the time evolution of the reachable set X(t) = X(t, t0 , X0 ) of the system (10) has the following form lim σ −1 h(X(t + σ, t0 , X0 ), {x+ σ→+0 x∈X(t,t0 ,X0 ) (17) ˆ σ(Ax + f (1) (x)d(1) + f (2) (x)d(2) + E(ˆ a, Q))} = 0, X(t0 , t0 , X0 ) = X0 , t0 ≤ t ≤ T. Therefore we have the inclusion X(t0 + σ) ⊆ {x + σ(Ax + f (1) (x)d(1) + x∈X0 (18) ˆ a, Q))} + o(σ)B(0, 1), f (2) (x)d(2) + E(ˆ where σ −1 o(σ) → 0 when σ → +0. Estimates (13)-(16) are derived from the inclusion (18) according to the scheme of reasoning in the proof of Theorem 1 in (Filippova and Matviychuk (2014)) with the necessary adjustments caused by more complicated structure of nonlinearity in the considered system (10). Remark 2. The significance of the above result is that it allows to find upper ellipsoidal estimates for the construction
(19)
(20)
where B ∗ = diag{b∗1 , . . . b∗n } with b∗i (i = 1, . . . , n) being the eigenvalues of the matrix B ∗ . We may assume without loss of generality that b∗i = α2i (i = 1, . . . , s) and b∗i = −βi2 (i = i + 1, . . . , n). Denote f (1) (z) =
s
i=1 (1)
d
α2i zi2 , f (2) (z) =
n
i=s+1 ∗
βi2 zi2 ,
(21)
= d∗ , d(2) = −d ,
and rewrite the system (20) as z˙ = A∗ (t)z + f (1) (z) · d(1) + f (2) (z) · d(2) + w(t), z0 ∈ Z0 = E(a∗0 , Q∗0 ), t0 ≤ t ≤ T, ˆ ∗ ). w(t) ∈ W = E(ˆ a∗ , Q
(22)
We see here that the quadratic functions f (1) (z) and f (2) (z) in (22) are of the same form as in (10) except the property of their positive definiteness because in general case both functions f (i) (z) (i = 1, 2) are only positive semidefinite quadratic forms. To avoid this problem, we modify the system (22) as follows. Let λ > 0 be a small parameter and let
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(1)
s
i=1
α2i zi2 + λ2 ·
n
zi2 , i=s+1 (23) s n (2) fλ (z) = λ2 · zi2 + βi2 zi2 . i=1 i=s+1 assume that all parameters α2i (i = 1, . . . , s) and fλ (z) =
E(a(1)( ),Q (1)( ))
z˙ = A∗ (t)z +
z0 ∈
(2) · d(1) + fλ (z) Z0 = E(a∗0 , Q∗0 ),
2.5
t
· d(2) + w(t), (24)
ˆ ∗ ), t0 ≤ t ≤ T, w(t) ∈ W = E(ˆ a∗ , Q where functions quadratic forms.
(i) fλ (z)
(i = 1, 2) are positive definite
Using results Filippova (2014), we may conclude that if we find the external ellipsoidal estimates for the modified system (24) then they will be close to external ellipsoidal estimate of the original system (20) (and therefore (19)) in the Hausdorff metric for small λ > 0. We may formulate now the following scheme that produces the external estimate of trajectory tube X(t) of the system (24) with given accuracy. Algorithm. Subdivide the time segment [t0 , T ] into subsegments [ti , ti+1 ] where ti = t0 + ih (i = 1, . . . , m), h = (T − t0 )/m, tm = T . (1) Given Z0 = E(a∗0 , Q∗0 ), take σ = h and define ellipsoids E(a(1) (σ), Q(1) (σ)) and E(a(2) (σ), Q(2) (σ)) from Theorem 3. (2) Find the smallest (with respect to some criterion Chernousko (1994); Kurzhanski and Valyi (1997)) ellipsoid E(a∗ , Q∗ ) which contains the intersection E(a∗ , Q∗ ) ⊇ E(a(1) (σ), Q(1) (σ)) E(a(2) (σ), Q(2) (σ)). (3) From Theorem 1 find the ellipsoid E(a1 , Q1 ) which is the upper estimate of the sum Chernousko (1994); Kurzhanski and Valyi (1997) of two ellipsoids, E(a∗ , Q∗ ) and σE(g, G): E(a∗ , Q∗ ) + σE(g, G) ⊆ E(a1 , Q1 ). (4) Consider the system on the next subsegment [t1 , t2 ] with E(a1 , Q1 ) as the initial ellipsoid at instant t1 . (5) Next steps continue iterations 1-3. At the end of the process we will get the external estimate E(a(t), Q(t)) of the tube X(t) with accuracy tending to zero when m → ∞. Remark 4. The algorithm presented here is more complicated in comparison with the related algorithm of Filippova and Matviychuk (2015) because it contains additional intermediate step of finding the ellipsoid E(a1 , Q1 ) due to necessity to take into account the ellipsoidal constraints on controls. 4.3 Example The following example illustrates the main procedure of the above algorithm. Example 2. Consider the following control system with two quadratic forms in its dynamical equations:
E(a(2)( ),Q (2)( )) E(a1 ,Q 1)
X(t0+ )
We can βi2 (i = s + 1, . . . , n) are positive, otherwise instead of related zeros we may add small positive terms in the same way as before. So instead of the system (22) we have (1) fλ (z)
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0.2 0.15 0.1 0.05 0 2.5
2 1.5 1 0.5
E(a,k12(B(1))-1) 2
1.5
1
0
E(a,k 22(B(2)) )
X0 0.5
x2
0
-0.5
x1
-1
-1.5 -0.5
-1 -1
-2 -1.5
-2
-2.5 -2.5
Fig. 2. Steps of the Algorithm of ellipsoidal estimating process for the trajectory tube X(t). x˙ 1 = 1.6x1 + x21 + 2x22 + u1 , x˙ 2 = 1.5x2 + 2.1x21 + x22 + u2 , (25) x0 ∈ X0 , t0 ≤ t ≤ T. Here t0 = 0, T = 0.3, X0 = B(0, 1) and U = B(0, 0.5). Several steps of the algorithm of external ellipsoidal estimating the reachable set X(t) with the resulting ellipsoidal tube E(a(t), Q(t)) are shown in Fig. 2. 5. CONCLUSIONS The paper deals with the problems of state estimation for uncertain dynamical control systems for which we assume that the initial system state is unknown but bounded with given constraints. Basing on the results of ellipsoidal calculus developed earlier for linear uncertain systems we present the modified state estimation approaches which use the special quadratic structure of nonlinearity of the control system and allow to construct the external ellipsoidal estimates of reachable sets. The special case when the quadratic form in the equations of dynamics of the controlled system may be not positive definite is studied. Examples and numerical results related to procedures of set-valued approximations of trajectory tubes and reachable sets are also presented. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory. REFERENCES Apreutesei, N. C. (2009). An optimal control problem for a prey-predator system with a general functional response. Appl. Math. Lett., 22(7), 1062-1065. Asselborn, L., Groß, D. and Stursberg, O. (2013). Control of uncertain nonlinear systems using ellipsoidal reachability calculus. In: Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems, Toulouse, France, September 4-6, 2013. IFAC, 50–55. August, E., Lu, J. and Koeppl, H.(2012). Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters. In: Proceedings of the 2012 American Control Conference, 27-29 June 2012, Montreal, QC. IEEE Computer Soc., 1488-1493.
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Bellman, R.(1997). Introduction to Matrix Analysis: Second Edition. Classics in Applied Mathematics (Book 19). SIAM, Philadelphia. Bertsekas, D. P.(1995). Dynamic Programming and Optimal Control. Athena Scientific, V.I,II. Belmont, MA. Blanchini, F. and Miani, S. (2015). Set-Theoretic Methods in Control. Systems & Control: Foundations & Applications. Birkh¨ auser, Basel. Boscain, U., Chambrion, T. and Sigalotti, M.(2013). On some open questions in bilinear quantum control. In: European Control Conference (ECC), Jul 2013, Zurich, Switzerland, IEEE, 2080-2085. Ceccarelli, N., Di Marco, M., Garulli, A. and Giannitrapani, A. (2004). A set theoretic approach to path planning for mobile robots. In: Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, December 14-17, 147-152. Chernousko, F. L. (1994). State Estimation for Dynamic Systems. CRC Press. Boca Raton. Filippov, A. F. (1988). Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers. Dordrecht. Filippova, T. F. (2009). Estimates of trajectory tubes in control problems under uncertainty. In: Proc. of the 4th International Conf. ”Physics and Control - 2009”, PhysCon2009, Italy, Catania, 1-4 September, 1–6. Filippova, T. F. (2010). Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system. Proceedings of the Steklov Institute of Mathematics, 271(1, Supplement), 75–84. Filippova, T. F. (2012). Set-valued dynamics in problems of mathematical theory of control processes. International Journal of Modern Physics B (IJMPB), 26(25), 1–8. Filippova, T. F. (2014). Estimates of reachable sets of control systems with nonlinearity and parametric perturbations. Trudy Inst. Mat. i Mekh. UrO RAN, 20 (4), 287–296. Filippova, T. F. and Matviychuk, O. G. (2014). Algorithms of estimating reachable sets of nonlinear control systems with uncertainty. In: Proceedings of the 7th Chaotic Modeling and Simulation International Conference (Lisbon, Portugal: 7-10 June, 2014). Published by ISAST: International Society for the Advancement of Science and Technology, Christos H Skiadas (Ed.), 115–124. Filippova, T. F. and Matviychuk, O. G. (2015). Algorithms of estimating reachable sets of nonlinear control systems with uncertainty. Journal of Chaotic Modeling and Simulation (CMSIM). No.3. P. 205-214. Gusev, M. I. Application of penalty function method to computation of reachable sets for control systems with state constraints. AIP Conference Proceedings, 1773(050003), 1–8. H¨ ackl, G. (1996). Reachable sets, control sets and their computation. Augsburger MathematischNaturwissenschaftliche Schriften, 7. PhD Thesis, University of Augsburg, Augsburg. Kishida, M. and Braatz, R. D. (2015). Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties. Optim Eng, 16, 695-711. Kostousova, E. K. (2013). On polyhedral estimates for trajectory tubes of differential systems with a bilinear uncertainty Springer Proc. in Mathematics & Statistics,
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IFAC PapersOnLine 50-1 (2017) 15355–15360 Ellipsoidal Estimates of Reachable Sets for Ellipsoidal Ellipsoidal Estimates Estimates of of Reachable Reachable Sets Sets for for ⋆⋆ Control Systems with Nonlinear Terms Ellipsoidal Estimates of Reachable Sets for Control Systems with Nonlinear Terms Control Systems with Nonlinear Terms ⋆⋆ Control Systems with Nonlinear Terms Tatiana F. Filippova
Tatiana F. Filippova Tatiana Tatiana F. F. Filippova Filippova Tatiana F. Filippova Krasovskii Institute of Mathematics and Mechanics, Ural Branch of Krasovskii Institute of and Ural Branch of Krasovskii Institute of Mathematics Mathematics and Mechanics, Mechanics, Ural Branch of Russian Academy of Sciences, Sofia Kovalevskaya str.Branch 16, of Krasovskii Institute of Mathematics and Mechanics, Ural Russian Academy of Sciences, Sofia Kovalevskaya str. 16, Russian Academy of Sciences, Sofia Kovalevskaya str. 16, Krasovskii Institute of Mathematics and Mechanics, Ural Branch Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) Russian 620990, AcademyRussian of Sciences, Sofia Kovalevskaya str. 16, of Ekaterinburg Federation (e-mail: Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) [email protected]) Russian 620990, AcademyRussian of Sciences, Sofia Kovalevskaya str. 16, Ekaterinburg Federation (e-mail: [email protected]) Ekaterinburg 620990, Russian Federation (e-mail: [email protected]) Abstract: The problem of estimating reachable sets of nonlinear dynamical control systems Abstract: The The problem problem of of estimating estimating reachable reachable sets sets of of nonlinear nonlinear dynamical dynamical control control systems systems Abstract: with quadratic and with reachable uncertainty initial states is studied. We systems assume Abstract: The nonlinearity problem of estimating setsin nonlinear dynamical control with quadratic nonlinearity and with with reachable uncertainty inof initial states is studied. studied. We systems assume with quadratic nonlinearity and uncertainty in initial states is We assume Abstract: The problem of estimating sets of nonlinear dynamical control that the uncertainty is of a set-membership kind when we know only the bounding set for with quadratic nonlinearity and with uncertainty in initial states is studied. We assume that the uncertainty is of a set-membership kind when we know only the bounding set for for that the uncertainty is additional of aa set-membership set-membership kind when when wetheir know only the bounding set with quadratic nonlinearity and with uncertainty in initial states is the studied. We assume unknown items and any statistical information on behavior is not available. We that the uncertainty is of kind we know only bounding set for unknown items and any additional statistical information on their behavior is not available. We unknown items and any any additional statistical information on their behavior issets not available. that thehere uncertainty is additional of a allow set-membership kind when wetheir know only the bounding set We for present approaches that finding ellipsoidal estimates of behavior reachable which use the unknown items and information on not available. present here here approaches that allow allow statistical finding ellipsoidal ellipsoidal estimates of behavior reachableis sets which use We the present approaches that finding estimates of reachable sets which use the unknown items and any additional statistical information on their issets not which available. We special structure of nonlinearity of studied control system. The algorithms of constructing such present here approaches that allow finding ellipsoidal estimates of reachable use the special structure structure of nonlinearity nonlinearity of studied studied control system. The algorithms algorithms ofsets constructing such special of of control system. The of constructing such present here approaches that allow finding ellipsoidal estimates ofare reachable which usesuch the ellipsoidal set-valued estimates and numerical simulation results given. of special structure of nonlinearity of studied control system. The algorithms constructing ellipsoidal set-valued estimates and numerical simulation results are given. ellipsoidal set-valued estimates and numerical simulation results are given. special structure of nonlinearity of studied control system. The algorithms of constructing such ellipsoidal set-valued estimates and numerical simulation results are given. © 2017, IFAC (International Federation of Automaticsimulation Control) Hosting byare Elsevier rights reserved. ellipsoidal andNonlinear numerical results given.Ltd. All Keywords: set-valued Estimationestimates algorithms, control systems, Set-membership uncertainty, Keywords: Estimation algorithms, Nonlinear control systems, Set-membership uncertainty, Keywords: Estimation Estimation algorithms, Nonlinear Nonlinear control control systems, systems, Set-membership Set-membership uncertainty, uncertainty, Ellipsoidal calculus. algorithms, Keywords: Ellipsoidal calculus. Ellipsoidal calculus. Keywords: Estimation algorithms, Nonlinear control systems, Set-membership uncertainty, Ellipsoidal calculus. Ellipsoidal calculus. 1. INTRODUCTION developed approaches to researches in this area. The first 1. INTRODUCTION developed approaches to researches in this area. The first 1. INTRODUCTION developed approaches to this one is based on ellipsoidal calculus in (Kurzhanski and first Va1. INTRODUCTION developed approaches to researches researches in this area. area. The The first one is based on ellipsoidal calculus (Kurzhanski and Vaone is based on ellipsoidal calculus (Kurzhanski and Va1. INTRODUCTION approaches toand researches this area. The first lyi (1997); Kurzhanski Varaiyain (2014); Chernousko In this paper we deal with control systems with unknown developed one is based on ellipsoidal calculus (Kurzhanski and Valyi (1997); Kurzhanski and Varaiya (2014); Chernousko In this paper we deal with control systems with unknown lyi (1997); Kurzhanski and Varaiya (2014); Chernousko one is based on ellipsoidal calculus (Kurzhanski and Va(1994); Polyak et al. (2004)) and the second one uses the In this paper we deal with control systems with unknown but bounded uncertainties and we consider the case of a lyi (1997); Kurzhanski and Varaiya (2014); Chernousko In this paper we deal with control systems with unknown (1994); Polyak et al. (2004)) and the second one uses the but bounded uncertainties and we consider the case of aa lyi (1994); Polyak et al. (2004)) and the second one uses the (1997); Kurzhanski and Varaiya (2014); Chernousko interval analysis (Walter and Pronzato (1997); Kostousova but bounded uncertainties and we consider the case of In this paper we deal with control systems with unknown set-membership description of uncertain parameters and (1994); Polyak et al. (2004)) and the second one uses the but bounded uncertainties and we consider the case of a interval analysis (Walter and Pronzato (1997); Kostousova set-membership description of uncertain parameters and interval analysis and Pronzato (1997); Kostousova Polyak et(Walter al. (2004)) and the second one uses the (2013)). set-membership description of uncertain uncertain parameters and but bounded and we the problems case of a (1994); functions. Theuncertainties study of control andconsider estimation interval analysis (Walter and Pronzato (1997); Kostousova set-membership description of parameters and (2013)). functions. The study of control and estimation problems (2013)). analysis (Walter and Pronzato (1997); Kostousova functions. The study ofand control and estimation estimation problems set-membership description of uncertain parameters and interval follow hereThe thestudy ideas the basic theoretical schemes (2013)). functions. of control and problems Many applied problems are mostly nonlinear in their pafollow here the ideas and the basic theoretical schemes (2013)). Many applied problems are mostly nonlinear in their pafollow here the ideas and the basic theoretical schemes functions. The study of control and estimation problems proposed and developed in Kurzhanski (1974, 1977, 1991, follow here the ideas and the basic theoretical schemes Many applied mostly nonlinear in parameters and problems the set ofare states is usuproposed and developed in Kurzhanski (1974, 1977, 1991, Many applied problems arefeasible mostly system nonlinear in their their parameters and the set of feasible system states is usuproposed and developed in Kurzhanski (1974, 1977, 1991, follow here the ideas and the basic theoretical schemes 2010). The motivation of this study is to apply the setproposed and developed in Kurzhanski (1974, 1977, 1991, rameters and the set of feasible system states is usuMany applied problems are mostly nonlinear in their paally non-convex or even non-connected. The key issue in 2010). The motivation of this study is to apply the setrameters and the set of feasible system states is usuor even The key issue in 2010). The The motivation ofinnew this study isof tocontrol apply the set- ally proposed and developedto Kurzhanski (1974, 1977, 1991, membership approach classesis systems 2010). motivation of this study to apply the setally non-convex non-convex or set evenofnon-connected. non-connected. The key suitable issue in rameters and the feasible system is usunonlinear set-membership estimation is The tostates find membership approach to new classes of control systems ally non-convex or even non-connected. key issue in set-membership estimation is to find suitable membership approach to new classesand oftouncertainty. control systems 2010). Thesystems motivation of this study isof apply the setincluding with to nonlinearity We nonlinear membership approach new classes control systems nonlinear set-membership estimation is to find suitable ally non-convex or produce even non-connected. The key issue techniques, which related bounds for thesuitable set in of including systems with nonlinearity and uncertainty. We nonlinear set-membership estimation is to find which produce related bounds set of including systems systems with to nonlinearity and uncertainty. We membership approach classesand control systems emphasize the relevance ofnew research inofthis direction be- techniques, including with nonlinearity uncertainty. We techniques, which produce related bounds for the set of nonlinear set-membership estimation is computationally to for findthe suitable unknown system states without being too emphasize the relevance of research in this direction betechniques, which produce related bounds for the set of unknown system states without being too computationally emphasize the relevance of research in this direction beincluding systems with nonlinearity and uncertainty. We cause the characterization of parameter uncertainties in emphasize the relevance of research in this direction beunknown system states without being too computationally techniques, which produce related bounds for the set of demanding. In this paper the modified state estimation cause the characterization of parameter uncertainties in unknown system states without being too computationally In paper the modified estimation cause the the characterization characterization of parameter uncertainties in demanding. emphasize the relevance models of of research in this direction betraditional probabilistic requires assumptions on cause parameter uncertainties in demanding. In this this paper the modified state estimation unknown system states without being too state computationally approaches which use the special quadratic structure of traditional probabilistic models requires assumptions on demanding. In this paper the modified state estimation which the special quadratic structure of traditional probabilistic models requires assumptions on cause the characterization of parameter uncertainties in approaches mean, variances or probability density function of errors. traditional probabilistic models requires assumptions on approaches use the special quadratic of demanding. In studied thisuse paper the modified estimation nonlinearity which of control system andstate use structure also the admean, variances or probability density function of errors. approaches which use the special quadratic structure of nonlinearity of studied control system and use also the admean, variances or probability density function of errors. traditional probabilistic models requires assumptions on However in many applied areas ranged from engineering mean, variances or probability density function of errors. nonlinearity of studied control system and use also the adapproaches which use the special quadratic structure of vantages of ellipsoidal calculus are presented. The special However in many applied areas ranged from engineering nonlinearity of studied control system and use also the adof ellipsoidal calculus are presented. The special However in many applied areasdensity ranged fromitengineering engineering mean, variances orapplied probability function errors. vantages problems in many physics to economical modeling isofdifficult However in areas ranged from vantages of ellipsoidal calculus are presented. The special nonlinearity of studied control system and use also the adcase when the quadratic form in the equations of dynamics problems in physics to economical modeling it is difficult vantages ofthe ellipsoidal calculus are presented. The special when quadratic form in the equations of dynamics problems in many physics to economical economical modeling itengineering is difficult difficult However in applied areas from to get necessary statistical dataranged for such description of case problems in physics to modeling it is case when the quadratic form inbe thenot equations of dynamics vantages ofthe ellipsoidal calculus are presented. The special of the controlled system mayin positiveof definite is to get necessary statistical data for such description of case when quadratic form the equations dynamics of the controlled system may be not positive definite is to get necessary statistical data for such description of problems in physics to economical modeling it is difficult theget model (Apreutesei (2009); August and Koeppl (2012); to necessary statistical data for such description of case of the controlled system may not positive definite is when the The quadratic form inbe the equations dynamics studied here. studies in this direction areofmotivated the model (Apreutesei (2009); August and Koeppl (2012); of the controlled system may be not positive definite is studied here. The studies in this direction are motivated the model (Apreutesei (2009); August and Koeppl (2012); to get necessary statistical data for such description of Boscain et al. (2013); Ceccarelli et al. (2004)). As an alterthe model (Apreutesei (2009); August and Koeppl (2012); studied here. The studies in this direction are motivated of the controlled system may be not positive definite is also by applications related, e.g., to satellite control probBoscain et al. (2013); Ceccarelli et al. (2004)). As an alterstudied here. The studies in this direction are motivated by applications related, e.g., to satellite control probBoscain eta(Apreutesei al. (2013); Ceccarelli Ceccarelli et description al. (2004)). (2004)). As an(2012); alter- also the model (2009); August and Koeppl native toet stochastic approach to of unknown Boscain al. (2013); et al. As an alteralso by applications related, e.g., to satellite control probstudied here. The studies in this direction are motivated lems (Kuntsevich and Volosov (2015)) where the attainnative to a stochastic approach to description of unknown by applications related, e.g.,(2015)) to satellite control problems (Kuntsevich and Volosov where the native to toetaa al. stochastic approach to description of unknown unknown Boscain (2013); Ceccarelli et description al. (2004)). model As an alterparameters and functions, the bounded-error char- also native stochastic approach to of lems by (Kuntsevich and Volosov (2015)) where the attainattainalso applications related, e.g.,(2015)) to satellite control probability sets of the and control systems werewhere investigated for parameters and functions, the bounded-error model charlems (Kuntsevich Volosov the attainability sets of the control systems were investigated for parameters and functions, the bounded-error model charnative to a stochastic approach to description of unknown acterization and (i.e.,functions, in the frameworks of the set-membership parameters the bounded-error model char- lems ability sets of the control systems were for (Kuntsevich and Volosov (2015)) where the attainnonstationary linear and some classes ofinvestigated nonlinear disacterization (i.e., in the frameworks of the set-membership ability sets of the control systems were investigated for nonstationary linear and some classes of nonlinear disacterization (i.e., in the frameworks of the set-membership parameters and functions, the bounded-error model charapproach) and related theoretical and applied investigaacterization (i.e., in the frameworks of the set-membership nonstationary linear and some classes of nonlinear disability sets of the control systems were investigated for crete systems under bounded parametric (multiplicative) approach) and related theoretical and applied investiganonstationary linear and some classes of nonlinear dissystems under parametric approach) and related theoretical and applied investigaacterization (i.e.,related the theoretical frameworks of the set-membership tions continue toin develop intensively (Kurzhanski and crete approach) and and applied investigacreteadditive systems disturbances. under bounded parametric (multiplicative) nonstationary linear bounded and some classes of(multiplicative) nonlinear disand The suggested algorithms for tions continue to develop intensively (Kurzhanski and crete systems under bounded parametric (multiplicative) disturbances. The suggested algorithms tions (1997); continue to develop develop intensively (Kurzhanski and and approach) andKurzhanski related theoretical and(2014); applied investigaValyi and Varaiya Mazurenko tions continue to intensively (Kurzhanski and and additive additive disturbances. The suggested algorithms for for crete systems under bounded parametric (multiplicative) solving these disturbances. problems allow efficient parallelization of Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko and additive The suggested algorithms for solving these problems allow efficient parallelization of Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko tions continue to develop intensively (Kurzhanski and (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko solving these problems efficient of and additive disturbances. The suggested algorithms for computations and may allow also provide aparallelization solution to the (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. solving these problems allow efficient parallelization of computations and may also provide a solution to the (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. Valyi (1997); Kurzhanski and Varaiya (2014); Mazurenko (1996); Milanese and Vicino (1991); Chernousko (1994); (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. computations and may also provide a solution to the solving these problems allow efficient parallelization of related control synthesis problems. It should be noted that (1996); Milanese and Vicino (1991); Chernousko (1994); computations and may also provide a solution to the control synthesis It be (1996); et Milanese andSchweppe Vicino (1991); Chernousko (1994); (2012); Sinyakov (2015); Bertsekas (1995); Milanese et al. related Polyak al. (2004); (1973); Gusev (2016)). (1996); Milanese and Vicino (1991); Chernousko (1994); related control synthesis problems. It should should be noted noted that computations may problems. also a solution to that the the casecontrol of anand arbitrary (notprovide necessarily positively dePolyak et al. (2004); Schweppe (1973); Gusev (2016)). related synthesis problems. It should be noted that casecontrol of an arbitrary (not necessarily positively dePolyak et et al. (2004); (2004); Schweppe (1973);Chernousko Gusev (2016)). (2016)). (1996); Milanese andSchweppe Vicino (1991); (1994); the Polyak al. Gusev an arbitrary (not necessarily positively derelated problems. It should be noted that fined,case andof possibly degenerate) function entering the case of ansynthesis arbitrary (not quadratic necessarily positively deThe solution of many control (1973); and estimation problems the fined, and possibly degenerate) quadratic function entering Polyak et al. (2004); Schweppe (1973); Gusev (2016)). The solution of many control and estimation problems fined, and possibly degenerate) quadratic function entering the case of an arbitrary (not necessarily positively destate velocities of a dynamical system is much more The solution solution of many many control and estimation estimation problems under uncertainty involves constructing reachableproblems sets and fined, and possibly degenerate) quadratic function entering The of control and the state velocities of aa dynamical system is much more under uncertainty involves constructing reachable sets and the state velocities of dynamical system is much more fined, and possibly degenerate) quadratic function entering difficult to analyze and requires more detailed analysis, under uncertainty involves constructing reachable sets and The ofFor many control estimation problems their solution analogs. models withand linear dynamics under statetovelocities of a dynamical system is much more under uncertainty involves constructing reachable sets and the difficult analyze and requires more detailed analysis, their analogs. For models with linear dynamics under difficult to analyze and requires more detailed analysis, state velocities dynamical system is much more which is to the subjectof ofa this paper. Thisdetailed study continues their set-membership analogs. Forinvolves models with linear linear dynamics under under uncertainty constructing sets and the difficult analyze and requires more analysis, such uncertainty therereachable are several contheir analogs. For models with dynamics under which is the subject of this paper. This study continues such set-membership uncertainty there are several conwhich is the subject of this paper. This study continues difficult to analyze and requires more detailed analysis, the researches Filippova (2014); Filippova and Matviychuk such set-membership uncertainty there are several contheir analogs. For models with linear dynamics under structive approaches which allow finding effective estiwhich is the subject of this paper. This study continues such set-membership uncertainty there areeffective several esticon- the researches Filippova (2014); Filippova and Matviychuk structive approaches which allow finding the researches Filippova Filippova Matviychuk is the and subject of (2014); this paper. This and study continues (2014, 2015) considers a more complicated case, when structive approachessets. which allow finding effective estisuch set-membership uncertainty areeffective several con- which mates of reachable We allow note there here two of the most the researches Filippova (2014); Filippova and Matviychuk structive approaches which finding esti(2014, 2015) and considers aa more complicated case, when mates of reachable sets. We note here two of the most (2014, 2015) and considers more complicated case, when the researches Filippova (2014); Filippova and Matviychuk quadratic forms that determine the nonlinearity in the mates of reachable sets. We note here two of the most structive approaches which allow finding effective esti2015) and considers a more complicated case, in when mates of reachable sets. We note here two of the most (2014, the quadratic forms that determine the nonlinearity the ⋆ the quadratic forms that determine the nonlinearity in the (2014, 2015) and considers a more complicated case, when system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF mates of reachable sets. We note here two of the most the quadratic forms that determine the nonlinearity in the ⋆ The research was supported by Russian Science Foundation (RSF system may be degenerate, this case is both of theoretical ⋆ The research Project No.16-11-10146) system may be degenerate, this case is both of theoretical was supported by Russian Science Foundation (RSF ⋆ the quadratic forms that determine the nonlinearity in the system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF Project No.16-11-10146) ⋆ Project No.16-11-10146) system may be degenerate, this case is both of theoretical The research was supported by Russian Science Foundation (RSF Project No.16-11-10146) Project No.16-11-10146) Copyright © 2017 IFAC 15925 Copyright © 2017, 2017 IFAC 15925 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 15925 Copyright © 2017 2017 IFAC IFAC 15925 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 15925 10.1016/j.ifacol.2017.08.2460
Proceedings of the 20th IFAC World Congress 15356 Tatiana F. Filippova / IFAC PapersOnLine 50-1 (2017) 15355–15360 Toulouse, France, July 9-14, 2017
and applied importance. Here we also propose a procedure for constructing ellipsoidal estimates for reachable sets that simultaneously takes into account all the complicating factors (uncertainty in the initial data, nonlinearity and functional disturbances in the right-hand sides of the system differential equations with related ellipsoidal constraints on unknown values). All these new feature constitute the novelty of results presented here. Examples and numerical simulations based on the procedures of setvalued approximations of trajectory tubes and reachable sets are also presented. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory.
The main problem of the paper is to find the external ellipsoidal estimate E(a+ (t), Q+ (t)) (with respect to the inclusion of sets) of the reachable set X(t) (t0 < t ≤ T ) by using the analysis of a special type of nonlinear control systems with uncertain initial data. The approach presented here develops and refines previous research in this area. Examples and numerical results related to procedures of set-valued approximations of trajectory tubes and reachable sets are given. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory.
2. PROBLEM FORMULATION
We will need some auxiliary constructions and results which will be used in the following.
In this section we introduce the following basic notations. Let Rn be the n–dimensional Euclidean space, Rn×n stands for the set of all n × n–matrices and x′ y be the usual inner product of x, y ∈ Rn with prime as a transpose, �x� = (x′ x)1/2 , compRn denotes the set of all compact subsets of Rn , h(A, B) is the Hausdorff distance between A, B ∈ compRn . We denote as B(a, r) the ball in Rn , B(a, r) = {x ∈ Rn : �x − a� ≤ r}, I is the identity n × n-matrix. Denote by E(a, Q) the ellipsoid in Rn , E(a, Q) = {x ∈ Rn : (Q−1 (x−a), (x−a)) ≤ 1} with center a ∈ Rn and symmetric positive definite n × n–matrix Q, for any n × n–matrix M = {mij } denote T r(M ) =
i=n
mii .
i=1
Consider the following nonlinear control system x˙ = A(t)x + f (x)d + u(t), x0 ∈ X0 , t0 ≤ t ≤ T,
(1)
where x, d ∈ Rn , �x� ≤ K (K > 0), the n × n–matrix A(t) is assumed to be continuous on t ∈ [t0 , T ] and f (x) = x′ Bx is scalar function, with a symmetric n × n–matrix B, u(t) ∈ U, U ⊂ Rm for a.e. t ∈ [t0 , T ]. (2) We will assume that X0 in (1) is an ellipsoid, X0 = E(a, Q), with a symmetric and positive definite matrix Q and with a center a. Let the absolutely continuous function x(t) = x(t, u(·), t0 , x0 ) be a solution to (1). We will study the solutions of the system (1)–(2) in the framework of the theory of uncertain dynamical systems through the techniques of trajectory tubes of related differential inclusions (Kurzhanski and Filippova (1993); Kurzhanski and Varaiya (2014)): X(·) = X(·; t0 , X0 ) = { x(·) = (3) x(·, u(·), t0 , x0 ) | x0 ∈ X0 , u(·) ∈ U }. The reachable set X(t) of the system (1) at time t (t0 < t ≤ T ) is defined as the cross-section of the trajectory tube (3) of the system (1)-(2) X(t) = X(t; t0 , X0 ) =
3. PRELIMINARIES
3.1 System Dynamics Nonlinearity Defined by a Positive Definite Quadratic Form Consider the nonlinear control system x˙ = A(t)x + f (x)d + u(t), x0 ∈ X0 = E(a0 , Q0 ),
t0 ≤ t ≤ T,
(4)
where x ∈ Rn , �x� ≤ K (K > 0), A(t) ∈ Rn×n is a given ˆ d, a0 , a ˆ are given continuous matrix, u(t) ∈ U = E(ˆ a, Q); n-vectors, a scalar function f (x) has a form f (x) = x′ Bx, ˆ are symmetric and positive definite. matrices B, Q0 , Q Denote the maximal eigenvalue of the matrix B 1/2 Q0 B 1/2 by k 2 , it is easy to see this k 2 is the smallest number for which the inclusion X0 ⊆ E(a0 , k 2 B −1 ) is true. The following result describes the external ellipsoidal estimate of the reachable set X(t) = X(t; t0 , X0 ) (t0 ≤ t ≤ T ). Theorem 1. (Filippova (2009)). Assume that X0 = E(a, k 2 (B −1 )) with some k > 0, then for all σ > 0 and for X(t0 + σ) = X(t0 +σ, t0 , X0 ) we have the following upper estimate X(t0 + σ) ⊆ E(a+ (σ), Q+ (σ)) + o(σ)B(0, 1), (5) where σ −1 o(σ) → 0 when σ → +0 and a+ (σ) = a(σ) + σˆ a, a(σ) = a + σ(A0 a + a′ Bad + k 2 d), ˆ Q+ (σ) = (p−1 + 1)Q(σ) + (p + 1)σ 2 Q,
(6)
Q(σ) = k22 (I + σR)B −1 (I + σR)′ , R = A + 2da′ B. and where p is the unique positive root of the equation n n 1 = p + αi p(p + 1) i=1 with αi ≥ 0 (i = 1, ..., n) being the roots of the following ˆ = 0. equations |Q(σ) − ασ 2 Q|
{ x ∈ Rn : ∃ x0 ∈ X0 , ∃ u(·) ∈ U such that
The following result presents the continuous-type version of the Theorem 1. Theorem 2. (Filippova (2010)). The inclusion is true for any t ∈ [t0 , T ]
x = x(t) = x(t; u(·), x0 ) }, t0 < t ≤ T.
X(t; t0 , X0 ) ⊆ E(a+ (t), r+ (t)B −1 ), 15926
(7)
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4. MAIN RESULTS
E(a *(t),Q *(t))
4.1 Control Systems with Two Positive Definite Quadratic Forms
(t)
t
15357
0.4 0.2 0
Consider the following control system with nonlinearity defined by two different positive definite quadratic forms in the dynamical equations and with uncertain initial state x0 ,
1.5 1 0.5
x2
x˙ = Ax + f (1) (x)d(1) + f (2) (x)d(2) + u(t), ˆ t0 ≤ t ≤ T, x0 ∈ X0 , u(t) ∈ U = E(ˆ a, Q),
2.5 2
0
1.5 -0.5
1 0.5
-1
0 -1.5
where d(1) and d(2) are n-vectors and f (1) , f (2) are scalar functions,
x1
-0.5
f (1) (x) = x′ B (1) x, f (2) (x) = x′ B (2) x,
-1
Fig. 1. Trajectory tube X(t) and its upper estimating tube E(a∗ (t), Q∗ (t)). where functions a+ (t), r+ (t) are the solutions of the following system of ordinary differential equations a˙ + (t) = A0 a+ (t) + ((a+ (t))′ Ba+ (t) + ˆ, t0 ≤ t ≤ T, r+ (t))d + a + ′ r˙ (t) = max l 2r+ (t)B 1/2 (A0 + �l�=1
2d(a+ (t))′ B)B −1/2 + ˆ 1/2 ) l + q(r+ (t))r+ (t), q −1 (r+ (t))B 1/2 QB ˆ 1/2 , q(r) = ((nr)−1 Tr(B Q))
(8)
with initial state a+ (t0 ) = a0 ,
r+ (t0 ) = k 2 .
3.2 Numerical Simulation Consider the example which shows that in nonlinear case the reachable sets may lose their convexity with increasing time t > t0 . Nevertheless the related external estimates calculated on the base of results of Theorems 1–2 are ellipsoidal-valued (and therefore convex) and contain the reachable sets of the system (4). Example 1. Consider the following control system x˙ 1 = x2 + x21 + x22 + u1 , x˙ 2 = 0.5x1 + u2 ,
(10)
(9)
Here we take x0 ∈ X0 = B(0, 1), 0 ≤ t ≤ 0.4 and U = B(0, 0.1). The reachable sets X (t) and their external ellipsoidal estimates E(a∗ (t), Q∗ (t)) are shown in Figure 1. This example is included here to illustrate the main approach, more details on numerical algorithms basing on Theorem 1 and Theorem 2 and producing the external ellipsoidal tube E + (t) = E(a∗ (t), Q∗ (t)) under different assumptions on system parameters may be found in Filippova (2012); Filippova and Matviychuk (2014).
with symmetric and positive definite matrices B (1) , B (2) . (1) We assume also that di = 0 for i = k + 1, . . . , n and (2) dj = 0 for j = 1, . . . , k where k (1 ≤ k ≤ n) is fixed. This assumption means that the first k equations of the system (10) contain only the nonlinear function f (1) (x) (with some (1) constant coefficients di ) while f (2) (x) is included only in the equations with numbers k + 1, . . . , n. Remark 1. In general case, these two positive quadratic functions can not be combined into one positive form because the coefficients (coordinates of vectors d(1) and d(2) ) are arbitrary, therefore some of them may be negative or zeros. As a consequence we cannot apply here the approaches and related results of previous investigations and need to find new techniques which will allow us to do this and to get related estimates of reachable sets of the system considered here. We will assume as before that X0 in (10) is an ellipsoid, X0 = E(a0 , Q0 ). We need first the following auxiliary result, its proof follows directly from the definition of eigenvalues of matrix operators and was obtained earlier in Filippova and Matviychuk (2014), therefore the proof is not included here. Lemma 1. (Filippova and Matviychuk (2014)). The inclusion is true X0 ⊆ E(a, k12 (B (1) )−1 ) E(a, k22 (B (2) )−1 ) (11)
where ki2 is the maximal eigenvalue of the matrix (B (i) )1/2 Q(B (i) )1/2 (i = 1, 2). The following equalities are true max z ′ B (2) z = k12 λ212 , z ′ B (1) z≤k12 (12) max z ′ B (1) z = k22 λ221 , z ′ B (2) z≤k22
where λ212 (1) −1/2
(B ) spectively.
and λ221 are maximal eigenvalues of matrices B (2) (B (1) )−1/2 and (B (2) )−1/2 B (1) (B (2) )−1/2 re-
The following new result produces the upper estimate in the case under consideration which is more general than the problem studied in Filippova (2009, 2010, 2012) and also produces better (in comparison with results of Filippova and Matviychuk (2014, 2015)) ellipsoidal estimates of reachable set X(t) because this new ellipsoidal estimates of the reachable set also take into account and absorb the ellipsoidal constraints on control functions or
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functional disturbances (noises) which may be presented in the system. Theorem 3. For all σ > 0 and for X(t0 + σ) = X(t0 + σ, t0 , X0 ) we have the following upper estimate X(t0 + σ) ⊆ (13) E(a(1) (σ), Q(1) (σ)) E(a(2) (σ), Q(2) (σ)) where σ
−1
+ o(σ)B(0, 1), o(σ) → 0 when σ → +0 and a(1) (σ) = a(σ) + σk12 λ212 d(2) + σˆ a,
of trajectory tube of the nonlinear control system using a stepwise procedure, convenient for computer modeling. We emphasize that in (Filippova and Matviychuk (2014, 2015)) we studied the dynamic system of a more special type, with a nonlinearity in dynamics and with uncertainty in initial data, but control functions were not subjected to a joint estimation procedure. So the result of Theorem 3 may be considered as a further generalization of previous estimation schemes. Remark 3. Important problems connected with conservativeness of constructed ellipsoidal estimates and of related algorithmic procedures are beyond the scope of this paper and will be studied in the future (see also related discussions, e.g., in Polyak et al. (2004); Asselborn et al. (2013); Kishida and Braatz (2015)).
a(2) (σ) = a(σ) + σk22 λ221 d(1) + σˆ a, a(σ) = (I + σA)a + σa′ B (1) ad(1) + σa′ B (2) ad(2) , Q(1) (σ) = (p−1 1 + 1)(I+ σR)k12 (B (1) )−1 (I + σR)′
(14)
ˆ +(p1 + 1)σ 2 (||d(2) ||2 k14 λ412 · I + Q),
Consider the general case x˙ = A(t)x + x′ Bx · d + u(t), t0 ≤ t ≤ T, ˆ x0 ∈ X0 = E(a0 , Q0 ), u(t) ∈ U = E(ˆ a, Q).
Q(2) (σ) = (p−1 2 + 1)(I+ σR)k22 (B (2) )−1 (I + σR)′ + ˆ (p2 + 1)σ 2 (||d(1) ||2 k24 λ421 · I + Q), (1) ′
(1)
(2) ′
(2)
R = A + 2d a B + 2d a B and p1 , p2 are the unique positive solutions of related algebraic equations n 1 n p1 +αi = p1 (p1 +1) , i=1 (15) n 1 n = p2 +βi p2 (p2 +1) i=1
with αi , βi ≥ 0 (i = 1, ..., n) being the roots of the following equations det((I + σR)k12 (B (1) )−1 (I + σR)′ − ασ 2 ||d(2) ||2 k14 λ412 · I) = 0, det((I + σR)k22 (B (2) )−1 (I + σR)′ − βσ
2
||d(1) ||2 k24 λ421
4.2 Generalization of the Estimation Approach
(16)
ˆ and Q0 are symmetric, We assume here that matrices B, Q ˆ Q and Q0 are positive definite. This assumption produces more general case because we do not assume here the positive definiteness of the matrix B in the nonlinear term of the right-hand side of dynamic equations (19). This new setting generalizes previous results and is motivated also by applied problems (e.g., Kuntsevich and Volosov (2015)). Using well-known diagonalization procedures of matrix analysis (Bellman (1997)) we can find the non-degenerate n × n–matrix Z of transformation z = Zx (x, z ∈ Rn ) of the state space Rn under which the system (19) will take the form z˙ = A∗ (t)z + z ′ B ∗ z · d∗ + w(t), t0 ≤ t ≤ T, z0 ∈ Z0 = E(a∗0 , Q∗0 ), ˆ ∗ ), w(t) ∈ W = E(ˆ a∗ , Q
· I) = 0.
Proof. The funnel equation (Kurzhanski and Valyi (1997); Kurzhanski and Varaiya (2014); Kurzhanski and Filippova (1993)) describing the time evolution of the reachable set X(t) = X(t, t0 , X0 ) of the system (10) has the following form lim σ −1 h(X(t + σ, t0 , X0 ), {x+ σ→+0 x∈X(t,t0 ,X0 ) (17) ˆ σ(Ax + f (1) (x)d(1) + f (2) (x)d(2) + E(ˆ a, Q))} = 0, X(t0 , t0 , X0 ) = X0 , t0 ≤ t ≤ T. Therefore we have the inclusion X(t0 + σ) ⊆ {x + σ(Ax + f (1) (x)d(1) + x∈X0 (18) ˆ a, Q))} + o(σ)B(0, 1), f (2) (x)d(2) + E(ˆ where σ −1 o(σ) → 0 when σ → +0. Estimates (13)-(16) are derived from the inclusion (18) according to the scheme of reasoning in the proof of Theorem 1 in (Filippova and Matviychuk (2014)) with the necessary adjustments caused by more complicated structure of nonlinearity in the considered system (10). Remark 2. The significance of the above result is that it allows to find upper ellipsoidal estimates for the construction
(19)
(20)
where B ∗ = diag{b∗1 , . . . b∗n } with b∗i (i = 1, . . . , n) being the eigenvalues of the matrix B ∗ . We may assume without loss of generality that b∗i = α2i (i = 1, . . . , s) and b∗i = −βi2 (i = i + 1, . . . , n). Denote f (1) (z) =
s
i=1 (1)
d
α2i zi2 , f (2) (z) =
n
i=s+1 ∗
βi2 zi2 ,
(21)
= d∗ , d(2) = −d ,
and rewrite the system (20) as z˙ = A∗ (t)z + f (1) (z) · d(1) + f (2) (z) · d(2) + w(t), z0 ∈ Z0 = E(a∗0 , Q∗0 ), t0 ≤ t ≤ T, ˆ ∗ ). w(t) ∈ W = E(ˆ a∗ , Q
(22)
We see here that the quadratic functions f (1) (z) and f (2) (z) in (22) are of the same form as in (10) except the property of their positive definiteness because in general case both functions f (i) (z) (i = 1, 2) are only positive semidefinite quadratic forms. To avoid this problem, we modify the system (22) as follows. Let λ > 0 be a small parameter and let
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(1)
s
i=1
α2i zi2 + λ2 ·
n
zi2 , i=s+1 (23) s n (2) fλ (z) = λ2 · zi2 + βi2 zi2 . i=1 i=s+1 assume that all parameters α2i (i = 1, . . . , s) and fλ (z) =
E(a(1)( ),Q (1)( ))
z˙ = A∗ (t)z +
z0 ∈
(2) · d(1) + fλ (z) Z0 = E(a∗0 , Q∗0 ),
2.5
t
· d(2) + w(t), (24)
ˆ ∗ ), t0 ≤ t ≤ T, w(t) ∈ W = E(ˆ a∗ , Q where functions quadratic forms.
(i) fλ (z)
(i = 1, 2) are positive definite
Using results Filippova (2014), we may conclude that if we find the external ellipsoidal estimates for the modified system (24) then they will be close to external ellipsoidal estimate of the original system (20) (and therefore (19)) in the Hausdorff metric for small λ > 0. We may formulate now the following scheme that produces the external estimate of trajectory tube X(t) of the system (24) with given accuracy. Algorithm. Subdivide the time segment [t0 , T ] into subsegments [ti , ti+1 ] where ti = t0 + ih (i = 1, . . . , m), h = (T − t0 )/m, tm = T . (1) Given Z0 = E(a∗0 , Q∗0 ), take σ = h and define ellipsoids E(a(1) (σ), Q(1) (σ)) and E(a(2) (σ), Q(2) (σ)) from Theorem 3. (2) Find the smallest (with respect to some criterion Chernousko (1994); Kurzhanski and Valyi (1997)) ellipsoid E(a∗ , Q∗ ) which contains the intersection E(a∗ , Q∗ ) ⊇ E(a(1) (σ), Q(1) (σ)) E(a(2) (σ), Q(2) (σ)). (3) From Theorem 1 find the ellipsoid E(a1 , Q1 ) which is the upper estimate of the sum Chernousko (1994); Kurzhanski and Valyi (1997) of two ellipsoids, E(a∗ , Q∗ ) and σE(g, G): E(a∗ , Q∗ ) + σE(g, G) ⊆ E(a1 , Q1 ). (4) Consider the system on the next subsegment [t1 , t2 ] with E(a1 , Q1 ) as the initial ellipsoid at instant t1 . (5) Next steps continue iterations 1-3. At the end of the process we will get the external estimate E(a(t), Q(t)) of the tube X(t) with accuracy tending to zero when m → ∞. Remark 4. The algorithm presented here is more complicated in comparison with the related algorithm of Filippova and Matviychuk (2015) because it contains additional intermediate step of finding the ellipsoid E(a1 , Q1 ) due to necessity to take into account the ellipsoidal constraints on controls. 4.3 Example The following example illustrates the main procedure of the above algorithm. Example 2. Consider the following control system with two quadratic forms in its dynamical equations:
E(a(2)( ),Q (2)( )) E(a1 ,Q 1)
X(t0+ )
We can βi2 (i = s + 1, . . . , n) are positive, otherwise instead of related zeros we may add small positive terms in the same way as before. So instead of the system (22) we have (1) fλ (z)
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0.2 0.15 0.1 0.05 0 2.5
2 1.5 1 0.5
E(a,k12(B(1))-1) 2
1.5
1
0
E(a,k 22(B(2)) )
X0 0.5
x2
0
-0.5
x1
-1
-1.5 -0.5
-1 -1
-2 -1.5
-2
-2.5 -2.5
Fig. 2. Steps of the Algorithm of ellipsoidal estimating process for the trajectory tube X(t). x˙ 1 = 1.6x1 + x21 + 2x22 + u1 , x˙ 2 = 1.5x2 + 2.1x21 + x22 + u2 , (25) x0 ∈ X0 , t0 ≤ t ≤ T. Here t0 = 0, T = 0.3, X0 = B(0, 1) and U = B(0, 0.5). Several steps of the algorithm of external ellipsoidal estimating the reachable set X(t) with the resulting ellipsoidal tube E(a(t), Q(t)) are shown in Fig. 2. 5. CONCLUSIONS The paper deals with the problems of state estimation for uncertain dynamical control systems for which we assume that the initial system state is unknown but bounded with given constraints. Basing on the results of ellipsoidal calculus developed earlier for linear uncertain systems we present the modified state estimation approaches which use the special quadratic structure of nonlinearity of the control system and allow to construct the external ellipsoidal estimates of reachable sets. The special case when the quadratic form in the equations of dynamics of the controlled system may be not positive definite is studied. Examples and numerical results related to procedures of set-valued approximations of trajectory tubes and reachable sets are also presented. The applications of the problems studied in this paper are in guaranteed state estimation for nonlinear systems with unknown but bounded errors and in nonlinear control theory. REFERENCES Apreutesei, N. C. (2009). An optimal control problem for a prey-predator system with a general functional response. Appl. Math. Lett., 22(7), 1062-1065. Asselborn, L., Groß, D. and Stursberg, O. (2013). Control of uncertain nonlinear systems using ellipsoidal reachability calculus. In: Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems, Toulouse, France, September 4-6, 2013. IFAC, 50–55. August, E., Lu, J. and Koeppl, H.(2012). Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters. In: Proceedings of the 2012 American Control Conference, 27-29 June 2012, Montreal, QC. IEEE Computer Soc., 1488-1493.
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