Abstract In this paper, we consider the robust error feedback output regulation problem for a linear $ 2 times 2 $ hyperbolic system with system uncertainties and disturbance signals. Using the backstepping method, the state feedback controller for the system without the system uncertainties and the external disturbance signals (nominal system) is designed. In terms of the measurable regulation error, an observer system is designed to recover the state of the nominal system, and an observer-based error feedback controller is obtained to solve the robust output regulation problem. Moreover, we prove that the controller is robust to the system uncertainties and the disturbance signals, and show that the state of the closed-loop system is bounded. The numerical simulations are presented to illustrate the effectiveness of the theoretical results.
{"title":"Robust error feedback output regulation of a 2 × 2 hyperbolic system","authors":"Wei-Wei Liu, Jun-Min Wang, Xiang-Dong Liu","doi":"10.1093/imamci/dnad015","DOIUrl":"https://doi.org/10.1093/imamci/dnad015","url":null,"abstract":"Abstract In this paper, we consider the robust error feedback output regulation problem for a linear $ 2 times 2 $ hyperbolic system with system uncertainties and disturbance signals. Using the backstepping method, the state feedback controller for the system without the system uncertainties and the external disturbance signals (nominal system) is designed. In terms of the measurable regulation error, an observer system is designed to recover the state of the nominal system, and an observer-based error feedback controller is obtained to solve the robust output regulation problem. Moreover, we prove that the controller is robust to the system uncertainties and the disturbance signals, and show that the state of the closed-loop system is bounded. The numerical simulations are presented to illustrate the effectiveness of the theoretical results.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"513 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135642750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.
{"title":"Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay","authors":"Habib Ayadi, Mariem Jlassi","doi":"10.1093/imamci/dnad016","DOIUrl":"https://doi.org/10.1093/imamci/dnad016","url":null,"abstract":"Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135643522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper designs a Dirichlet boundary controller to stabilize a heat equation with boundary disturbance within a prescribed finite time independent of initial conditions. We first use boundary measurements and time-varying gain to construct a disturbance estimator that estimates the disturbance itself and the system state within a prescribed time. We then design the estimation-based prescribed time boundary controller by the backstepping transformation with a time-varying kernel. The control gain proposed here diverges as the time approaches the prescribed time. Nevertheless, we can demonstrate the controller’s boundedness and the system’s prescribed time stability. A simulation example illustrates the theoretical result.
{"title":"Prescribed-time stabilization of uncertain heat equation with Dirichlet boundary control","authors":"Chengzhou Wei, Junmi Li","doi":"10.1093/imamci/dnad017","DOIUrl":"https://doi.org/10.1093/imamci/dnad017","url":null,"abstract":"\u0000 This paper designs a Dirichlet boundary controller to stabilize a heat equation with boundary disturbance within a prescribed finite time independent of initial conditions. We first use boundary measurements and time-varying gain to construct a disturbance estimator that estimates the disturbance itself and the system state within a prescribed time. We then design the estimation-based prescribed time boundary controller by the backstepping transformation with a time-varying kernel. The control gain proposed here diverges as the time approaches the prescribed time. Nevertheless, we can demonstrate the controller’s boundedness and the system’s prescribed time stability. A simulation example illustrates the theoretical result.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44008555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider parameter estimation and velocity signal extraction from a disturbed boundary velocity signal for an unstable wave equation. Firstly, an adaptive observer is designed based on the boundary displacement and the corrupted boundary velocity. Then the design of the feedback law adopts the backstepping method of infinite dimensional system. Finally, as time approaches infinity, the estimated parameters converge to the unknown parameters, the initial value disturbance can be obtained, and the velocity signal can be asymptotically recovered. Meanwhile, the asymptotic stability of the closed-loop system can be proved by $C_{0}$-semigroup theory and Lyapunov method. Numerical simulation shows that the proposed scheme is reasonable.
{"title":"Parameter estimation and velocity signal extraction for one-dimensional wave equation with harmonic corrupted boundary observation","authors":"Shuang-yun Huang, Feng-Fei Jin","doi":"10.1093/imamci/dnad018","DOIUrl":"https://doi.org/10.1093/imamci/dnad018","url":null,"abstract":"\u0000 In this paper, we consider parameter estimation and velocity signal extraction from a disturbed boundary velocity signal for an unstable wave equation. Firstly, an adaptive observer is designed based on the boundary displacement and the corrupted boundary velocity. Then the design of the feedback law adopts the backstepping method of infinite dimensional system. Finally, as time approaches infinity, the estimated parameters converge to the unknown parameters, the initial value disturbance can be obtained, and the velocity signal can be asymptotically recovered. Meanwhile, the asymptotic stability of the closed-loop system can be proved by $C_{0}$-semigroup theory and Lyapunov method. Numerical simulation shows that the proposed scheme is reasonable.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"12 1","pages":"385-402"},"PeriodicalIF":1.5,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76289622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with adaptive stabilization for a wave equation subject to boundary control matched harmonic disturbance. We use the adaptive and Lyapunov approach to estimate unknown disturbance and construct an adaptive boundary feedback controller. By the semigroup theory and Lasalle‘s invariance theorem, the well-posedness and asymptotic stability of the closed-loop system is proved, respectively. At the same time, it is shown that the parameter estimates involved in the constructed controller converge to their own real values as time goes to infinity. Some numerical simulations are offered at the end of the paper to illustrate the effectiveness of theoretical results.
{"title":"Adaptive stabilization for a wave equation subject to boundary control matched harmonic disturbance","authors":"Jun-Jun Liu, Yanxiao Zhao","doi":"10.1093/imamci/dnad010","DOIUrl":"https://doi.org/10.1093/imamci/dnad010","url":null,"abstract":"\u0000 In this paper, we are concerned with adaptive stabilization for a wave equation subject to boundary control matched harmonic disturbance. We use the adaptive and Lyapunov approach to estimate unknown disturbance and construct an adaptive boundary feedback controller. By the semigroup theory and Lasalle‘s invariance theorem, the well-posedness and asymptotic stability of the closed-loop system is proved, respectively. At the same time, it is shown that the parameter estimates involved in the constructed controller converge to their own real values as time goes to infinity. Some numerical simulations are offered at the end of the paper to illustrate the effectiveness of theoretical results.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42528303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Baroun, Hind El Baggari, Ilham Ouled Driss, S. Boulite
In this paper, we investigate the null approximate impulse controllability of the heat equation with an inverse square potential subject to dynamic boundary conditions in the ball $B(0, R_{0})$ of radius $R_{0}=left (frac{4}{3}right )^{frac{3}{2}}$. To that purpose, we use the Carleman commutator approach to show a logarithmic convexity estimate traducing an observability inequality at one instant of time.
{"title":"Impulse controllability for the heat equation with inverse square potential and dynamic boundary conditions","authors":"M. Baroun, Hind El Baggari, Ilham Ouled Driss, S. Boulite","doi":"10.1093/imamci/dnad012","DOIUrl":"https://doi.org/10.1093/imamci/dnad012","url":null,"abstract":"\u0000 In this paper, we investigate the null approximate impulse controllability of the heat equation with an inverse square potential subject to dynamic boundary conditions in the ball $B(0, R_{0})$ of radius $R_{0}=left (frac{4}{3}right )^{frac{3}{2}}$. To that purpose, we use the Carleman commutator approach to show a logarithmic convexity estimate traducing an observability inequality at one instant of time.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"34 1","pages":"353-384"},"PeriodicalIF":1.5,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79389228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns the stabilization problem for an underactuated robot called the Pendubot. Relying on a computational algorithm which is based on various results of the ‘polynomial matrix approach’, we propose an output-feedback-based internally stabilizing controller to stabilize the Pendubot at the unstable vertical upright position. The algorithm utilizes results for the solution of polynomial matrix Diophantine equations required for the computation and parameterization of proper ‘denominator assigning’ and internally stabilizing controllers for linear time invariant multivariable systems and reduces the problem to that of the solution of a set of numerical linear equations. The controller presented uses only the measured output which consists of the angles of the two links and does not require knowledge of the angular velocities which are usually not directly measurable. Comparative simulations are carried out to verify the good performance of the proposed controller. Finally, experimental results are provided to demonstrate the validity and feasibility of the proposed method.
{"title":"Stabilization of the Pendubot: a polynomial matrix approach","authors":"Cui Wei, A. Vardulakis, Tianyou Chai","doi":"10.1093/imamci/dnad011","DOIUrl":"https://doi.org/10.1093/imamci/dnad011","url":null,"abstract":"\u0000 This paper concerns the stabilization problem for an underactuated robot called the Pendubot. Relying on a computational algorithm which is based on various results of the ‘polynomial matrix approach’, we propose an output-feedback-based internally stabilizing controller to stabilize the Pendubot at the unstable vertical upright position. The algorithm utilizes results for the solution of polynomial matrix Diophantine equations required for the computation and parameterization of proper ‘denominator assigning’ and internally stabilizing controllers for linear time invariant multivariable systems and reduces the problem to that of the solution of a set of numerical linear equations. The controller presented uses only the measured output which consists of the angles of the two links and does not require knowledge of the angular velocities which are usually not directly measurable. Comparative simulations are carried out to verify the good performance of the proposed controller. Finally, experimental results are provided to demonstrate the validity and feasibility of the proposed method.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"11 1","pages":"332-352"},"PeriodicalIF":1.5,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88752112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jiehua Feng, Dongya Zhao, Xing-gang Yan, S. Spurgeon
In this paper, a class of non-linear systems in normal form is considered, which is composed of internal and external dynamics. An adaptive finite time sliding mode observer is first designed so that the system states, unmatched uncertain parameters and matched uncertainties can all be observed in finite time. Then, the systematic backstepping design procedure is employed to develop a novel output feedback backstepping control (OFBC). The proposed OFBC method can stabilize the considered non-linear systems despite the presence of non-linear internal dynamics and unmatched uncertainties. A Lyapunov method is used to ensure that the closed-loop system is asymptotically stable. Two MATLAB simulation examples are used to demonstrate the method.
{"title":"Output feedback backstepping control for non-linear systems using an adaptive finite time sliding mode observer","authors":"Jiehua Feng, Dongya Zhao, Xing-gang Yan, S. Spurgeon","doi":"10.1093/imamci/dnad014","DOIUrl":"https://doi.org/10.1093/imamci/dnad014","url":null,"abstract":"\u0000 In this paper, a class of non-linear systems in normal form is considered, which is composed of internal and external dynamics. An adaptive finite time sliding mode observer is first designed so that the system states, unmatched uncertain parameters and matched uncertainties can all be observed in finite time. Then, the systematic backstepping design procedure is employed to develop a novel output feedback backstepping control (OFBC). The proposed OFBC method can stabilize the considered non-linear systems despite the presence of non-linear internal dynamics and unmatched uncertainties. A Lyapunov method is used to ensure that the closed-loop system is asymptotically stable. Two MATLAB simulation examples are used to demonstrate the method.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"118 1","pages":"306-331"},"PeriodicalIF":1.5,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89355275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is addressed to the study of the null controllability for integer order integro-differential equations. Unlike the known results for partial differential equations, we need to consider the equation involving a $beta -$power of the Laplace operator $(-varDelta )^beta $ and an integral term. The key point is to construct a suitable state space of the controlled system at the final time. We first discuss a class of hyperbolic integro-differential equation. We prove that the controlled system is null controllable by an Ingham-type estimate. Also, the controllability time is given. On the other hand, by reduction to absurdity, we deduce that the null controllability property fails for a class of parabolic integro-differential equation with $beta in mathbb{N}^+$.
本文研究了整阶积分微分方程的零可控性问题。与偏微分方程的已知结果不同，我们需要考虑涉及拉普拉斯算子$(-varDelta)^beta $幂和一个积分项的方程。关键是在最终时刻构造一个合适的被控系统状态空间。首先讨论了一类双曲型积分微分方程。利用ingham型估计证明了被控系统是零可控的。并给出了控制时间。另一方面，通过谬论化简，我们推导出了一类具有$beta in mathbb{N}^+$的抛物型积分微分方程的零可控性不成立。
{"title":"On the null controllability of integer order integro-differential equations","authors":"Xiuxiang Zhou, Li Cheng, Xin Wang","doi":"10.1093/imamci/dnad013","DOIUrl":"https://doi.org/10.1093/imamci/dnad013","url":null,"abstract":"\u0000 This paper is addressed to the study of the null controllability for integer order integro-differential equations. Unlike the known results for partial differential equations, we need to consider the equation involving a $beta -$power of the Laplace operator $(-varDelta )^beta $ and an integral term. The key point is to construct a suitable state space of the controlled system at the final time. We first discuss a class of hyperbolic integro-differential equation. We prove that the controlled system is null controllable by an Ingham-type estimate. Also, the controllability time is given. On the other hand, by reduction to absurdity, we deduce that the null controllability property fails for a class of parabolic integro-differential equation with $beta in mathbb{N}^+$.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"18 1","pages":"285-305"},"PeriodicalIF":1.5,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87344725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $dot{x}left ( tright )=A(t)x(t)+B(t)u(t)$ defined on an interval $I subset mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.
{"title":"Topological equivalence of linear time-varying control systems","authors":"Jing Li, Zhixiong Zhang","doi":"10.1093/imamci/dnad009","DOIUrl":"https://doi.org/10.1093/imamci/dnad009","url":null,"abstract":"\u0000 In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $dot{x}left ( tright )=A(t)x(t)+B(t)u(t)$ defined on an interval $I subset mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.","PeriodicalId":56128,"journal":{"name":"IMA Journal of Mathematical Control and Information","volume":"1 1","pages":"253-284"},"PeriodicalIF":1.5,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86908846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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