The discretized Lyapunov functional method is extended to linear systems with both, discrete and distributed delays, and to H∞ control. The coefficients associated with the distributed delay are assumed to be piecewise constant. A new Bounded Real Lemma (BRL) is derived in terms of Linear Matrix Inequalities (LMIs) via descriptor approach. In three numerical examples considered for retarded type systems, the resulting values of H∞-norm converge to the exact ones. The analysis results are applied to state-feedback H∞ control of linear neutral systems with discrete and distributed delays, where the controller may be either instantaneous or may contain discrete or distributed delay terms. A numerical example illustrates the efficiency of the design method and the advantage of using distributed delay term in the feedback for H∞ control of systems with state delay.
{"title":"H∞ Control of Distributed and Discrete Delay Systems via Discretized Lyapunov Functional","authors":"E. Fridman, G. Tsodik","doi":"10.3166/ejc.15.84-94","DOIUrl":"https://doi.org/10.3166/ejc.15.84-94","url":null,"abstract":"The discretized Lyapunov functional method is extended to linear systems with both, discrete and distributed delays, and to H∞ control. The coefficients associated with the distributed delay are assumed to be piecewise constant. A new Bounded Real Lemma (BRL) is derived in terms of Linear Matrix Inequalities (LMIs) via descriptor approach. In three numerical examples considered for retarded type systems, the resulting values of H∞-norm converge to the exact ones. The analysis results are applied to state-feedback H∞ control of linear neutral systems with discrete and distributed delays, where the controller may be either instantaneous or may contain discrete or distributed delay terms. A numerical example illustrates the efficiency of the design method and the advantage of using distributed delay term in the feedback for H∞ control of systems with state delay.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"19 1","pages":"84-94"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90843843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The role of map invariance is examined within the context of the dynamic state reconstruction problem for nonlinear discrete-time systems. In particular, the key notion of invariant manifold for maps in nonlinear discrete-time dynamics is shown to be conceptually insightful and technically quite effective to address important issues related to the deterministic observerbased nonlinear state estimation problem in the discrete-time domain. As a necessary first methodological step, the problem of quantitatively characterizing the asymptotic long-term behavior of nonlinear discrete-time systems with a skew-product structure using the notion of map invariance is revisited. The formulation of this problem can be naturally realized through a system of invariance functional equations (FEs), for which a set of existence and uniqueness conditions of a solution is provided. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories/orbits, and therefore, the asymptotic long-term dynamic behavior of the system is determined through the restriction of the discrete-time system dynamics on the invariant manifold. Within the above analytical framework, the nonlinear full-order observer design problem in the discrete-time domain is considered, appropriately formulated and an interpretation of previous work on the problem is attempted through the notion of invariant manifolds for maps. Furthermore, this framework allows the development of a new approach to the nonlinear reduced-order observer design problem for multiple-output systems in the discrete-time domain, which is also presented in the present work. Finally, the performance of the proposed nonlinear reduced-order discrete-time observer is assessed in an illustrative bioreactor example through simulations.
{"title":"Map Invariance and the State Reconstruction Problem for Nonlinear Discrete-time Systems","authors":"N. Kazantzis","doi":"10.3166/ejc.15.105119","DOIUrl":"https://doi.org/10.3166/ejc.15.105119","url":null,"abstract":"The role of map invariance is examined within the context of the dynamic state reconstruction problem for nonlinear discrete-time systems. In particular, the key notion of invariant manifold for maps in nonlinear discrete-time dynamics is shown to be conceptually insightful and technically quite effective to address important issues related to the deterministic observerbased nonlinear state estimation problem in the discrete-time domain. As a necessary first methodological step, the problem of quantitatively characterizing the asymptotic long-term behavior of nonlinear discrete-time systems with a skew-product structure using the notion of map invariance is revisited. The formulation of this problem can be naturally realized through a system of invariance functional equations (FEs), for which a set of existence and uniqueness conditions of a solution is provided. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories/orbits, and therefore, the asymptotic long-term dynamic behavior of the system is determined through the restriction of the discrete-time system dynamics on the invariant manifold. Within the above analytical framework, the nonlinear full-order observer design problem in the discrete-time domain is considered, appropriately formulated and an interpretation of previous work on the problem is attempted through the notion of invariant manifolds for maps. Furthermore, this framework allows the development of a new approach to the nonlinear reduced-order observer design problem for multiple-output systems in the discrete-time domain, which is also presented in the present work. Finally, the performance of the proposed nonlinear reduced-order discrete-time observer is assessed in an illustrative bioreactor example through simulations.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"28 1","pages":"105-119"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84870621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Discussion on: ''Structural Analysis of the Partial State and Input Obervability for Structured Linear Systems. Application to Distributed Systems","authors":"M. Hou","doi":"10.3166/ejc.15.517-522","DOIUrl":"https://doi.org/10.3166/ejc.15.517-522","url":null,"abstract":"","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"144 1","pages":"517-522"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75584218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the design of quadratic and higher order normal forms for the left invertibility problem. The linearly observable case and one-dimensional linearly unobservable case are investigated. The interest of such a study in the design of a delayed discrete-time observer is examined. The example of the Burgers map with unknown input is treated and a delayed discrete-time observer is designed. Finally, some simulated results are commented.
{"title":"Discrete-time Normal Form for Left Invertibility Problem","authors":"M. Djemai, J. Barbot, I. Belmouhoub","doi":"10.3166/ejc.15.194204","DOIUrl":"https://doi.org/10.3166/ejc.15.194204","url":null,"abstract":"This paper deals with the design of quadratic and higher order normal forms for the left invertibility problem. The linearly observable case and one-dimensional linearly unobservable case are investigated. The interest of such a study in the design of a delayed discrete-time observer is examined. The example of the Burgers map with unknown input is treated and a delayed discrete-time observer is designed. Finally, some simulated results are commented.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"50 1","pages":"194-204"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90552825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear control system into the observer form using the state and output transformations. The necessary and sufficient solvability conditions are formulated in terms of differential one-forms, associated with the input-output equation of the control system. These conditions simplify the existing conditions [10] and extend them for the systems with inputs. The procedures to find the state and output transformations are given.
{"title":"Transformation the Nonlinear System into the Observer Form: Simplification and Extension","authors":"T. Mullari, Ü. Kotta","doi":"10.3166/ejc.15.177183","DOIUrl":"https://doi.org/10.3166/ejc.15.177183","url":null,"abstract":"The paper addresses the problem of transforming the discrete-time single-input single-output nonlinear control system into the observer form using the state and output transformations. The necessary and sufficient solvability conditions are formulated in terms of differential one-forms, associated with the input-output equation of the control system. These conditions simplify the existing conditions [10] and extend them for the systems with inputs. The procedures to find the state and output transformations are given.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"93 1","pages":"177-183"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83835601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies sampled-data output feedback control of nonlinear systems using high-gain observers in the presence of measurement noise. The observer is designed in continuous time, then discretized. It is shown that the sampled-data output feedback controller recovers the performance of the continuous-time state feedback controller as the observer gain is increased. However, the effect of measurement noise dominates beyond a certain point, and the gap between the performance of the two controllers widens as the observer gain is increased further. The analysis quantifies the relationship between the high-gain observer parameter and the amplitude of the noise.
{"title":"Analysis of Sampled-data High-gain Observers in the Presence of Measurement Noise","authors":"H. Khalil","doi":"10.3166/ejc.15.166176","DOIUrl":"https://doi.org/10.3166/ejc.15.166176","url":null,"abstract":"This paper studies sampled-data output feedback control of nonlinear systems using high-gain observers in the presence of measurement noise. The observer is designed in continuous time, then discretized. It is shown that the sampled-data output feedback controller recovers the performance of the continuous-time state feedback controller as the observer gain is increased. However, the effect of measurement noise dominates beyond a certain point, and the gap between the performance of the two controllers widens as the observer gain is increased further. The analysis quantifies the relationship between the high-gain observer parameter and the amplitude of the noise.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"33 1","pages":"166-176"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84401776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
performance requirement, and design of state feed-backcontrol.Asisnowwellknown,inordertoobtainstability conditions using Lyapunov-Krasovskiifunctional (LKF) approach, it is necessary to use acomplete quadratic LKF, as is done in this article. Asa general quadratic functional involve infinite numberof parameters, some discretization process is neces-sary in order to render conditions in a computableform. A piece-wise linear parameterization such aswidely used in finite element methods seems to be themost natural. However, there is a fundamental dif-ference between these two discretizations: in finiteelement analysis, the objective is purely to approx-imate the solution; in the discretization of LKF, it isalso necessary to guarantee the satisfaction of quad-raticinequalities.Therefore,thediscretizationofLKFis a much more sophisticated process.
{"title":"Discussion on: \"H∞ Control of Distributed and Discrete Delay Systems via Discretized Lyapunov Functional\"","authors":"K. Gu","doi":"10.3166/ejc.15.95-96","DOIUrl":"https://doi.org/10.3166/ejc.15.95-96","url":null,"abstract":"performance requirement, and design of state feed-backcontrol.Asisnowwellknown,inordertoobtainstability conditions using Lyapunov-Krasovskiifunctional (LKF) approach, it is necessary to use acomplete quadratic LKF, as is done in this article. Asa general quadratic functional involve infinite numberof parameters, some discretization process is neces-sary in order to render conditions in a computableform. A piece-wise linear parameterization such aswidely used in finite element methods seems to be themost natural. However, there is a fundamental dif-ference between these two discretizations: in finiteelement analysis, the objective is purely to approx-imate the solution; in the discretization of LKF, it isalso necessary to guarantee the satisfaction of quad-raticinequalities.Therefore,thediscretizationofLKFis a much more sophisticated process.","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"2 1","pages":"95-96"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76316027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Discussion on: \"Analysis of Control Relevant Coupled Nonlinear Oscillatory Systems\"","authors":"R. Pušenjak, M. Oblak","doi":"10.3166/ejc.14.283-285","DOIUrl":"https://doi.org/10.3166/ejc.14.283-285","url":null,"abstract":"","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"24 1","pages":"283-285"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73874863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Discussion on: \"Optimality Properties and Driver Input Parameterization for Trail-braking Cornering\"","authors":"Emilio Frazzoli, G. Klančar, I. Škrjanc","doi":"10.3166/ejc.14.321-328","DOIUrl":"https://doi.org/10.3166/ejc.14.321-328","url":null,"abstract":"","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"3 1","pages":"321-328"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81976038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Discussion on: \"Reconfigurable Fault-tolerant Control : A Tutorial Introduction\". Reply by the Authors","authors":"J. Maciejowski","doi":"10.3166/ejc.14.387-390","DOIUrl":"https://doi.org/10.3166/ejc.14.387-390","url":null,"abstract":"","PeriodicalId":11813,"journal":{"name":"Eur. J. Control","volume":"62 1","pages":"387-390"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89318741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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