SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1857-1883, June 2024. Abstract. In this paper, we are concerned with the tracking performance and antidisturbance ability of the widely used proportional-integral-derivative (PID) controllers in practice. Towards this end, we consider a basic class of second-order nonlinear stochastic control systems subject to model uncertainties and external disturbances, and focus on the ability of the classical PID controller to track time-varying reference signals. First, under some suitable conditions on the system nonlinear functions, reference signals, and external disturbances, we show that such control systems can be stabilized in the mean square sense, provided that the three PID gains are selected from a stability region constructed in the paper. Besides, it is shown that the steady-state tracking error has an upper bound proportional to the sum of the varying rates of reference signals, the varying rates of external disturbances, and the intensity of random noises. Meanwhile, its proportional coefficient depends on the selection of PID gains, which can be made arbitrarily small by choosing suitably large PID gains. Finally, by introducing a desired transient process which is shaped from the reference signal, a new PID tuning rule is presented, which can guarantee both the expected steady state and transient tracking performance.
{"title":"On Tracking and Antidisturbance Ability of PID Controllers","authors":"Cheng Zhao, Shuo Yuan","doi":"10.1137/22m1522498","DOIUrl":"https://doi.org/10.1137/22m1522498","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1857-1883, June 2024. <br/> Abstract. In this paper, we are concerned with the tracking performance and antidisturbance ability of the widely used proportional-integral-derivative (PID) controllers in practice. Towards this end, we consider a basic class of second-order nonlinear stochastic control systems subject to model uncertainties and external disturbances, and focus on the ability of the classical PID controller to track time-varying reference signals. First, under some suitable conditions on the system nonlinear functions, reference signals, and external disturbances, we show that such control systems can be stabilized in the mean square sense, provided that the three PID gains are selected from a stability region constructed in the paper. Besides, it is shown that the steady-state tracking error has an upper bound proportional to the sum of the varying rates of reference signals, the varying rates of external disturbances, and the intensity of random noises. Meanwhile, its proportional coefficient depends on the selection of PID gains, which can be made arbitrarily small by choosing suitably large PID gains. Finally, by introducing a desired transient process which is shaped from the reference signal, a new PID tuning rule is presented, which can guarantee both the expected steady state and transient tracking performance.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1832-1856, June 2024. Abstract. Consider the motion of a Brownian particle in [math] dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly [math] of the coordinate processes get a (known) nonzero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which the [math] coordinate processes get the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimizes the probability of a false early detection and the expected delay of a missed late detection. The elliptic case [math] has been settled in [] where the hypoelliptic case [math] resolved in the present paper was left open (the case [math] reduces to the classic case [math] having a known solution). We also show that the methodology developed solves the problem in the general case where exactly [math] is relaxed to any number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multidimensional hypoelliptic problem has been solved exactly in the literature.
{"title":"Quickest Real-Time Detection of Multiple Brownian Drifts","authors":"P. A. Ernst, H. Mei, G. Peskir","doi":"10.1137/23m1587750","DOIUrl":"https://doi.org/10.1137/23m1587750","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1832-1856, June 2024. <br/> Abstract. Consider the motion of a Brownian particle in [math] dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly [math] of the coordinate processes get a (known) nonzero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which the [math] coordinate processes get the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimizes the probability of a false early detection and the expected delay of a missed late detection. The elliptic case [math] has been settled in [] where the hypoelliptic case [math] resolved in the present paper was left open (the case [math] reduces to the classic case [math] having a known solution). We also show that the methodology developed solves the problem in the general case where exactly [math] is relaxed to any number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multidimensional hypoelliptic problem has been solved exactly in the literature.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1809-1831, June 2024. Abstract. In this paper, we consider the problem of placing a minimal number of controls to achieve controllability for a class of networked control systems that are based on the original Turing reaction-diffusion model, which is governed by a set of ordinary differential equations with interactions defined by a ring graph. Turing model considers two morphogens reacting and diffusing over the spatial domain and has been widely accepted as one of the most fundamental models to explain pattern formation in a developing embryo. It is of great importance to understand the mechanism behind the various reaction kinetics that generate such a wide range of patterns. As a first step towards this goal, in this paper we study controllability of Turing model for the case of cells connected as a square grid in which controls can be applied to the boundary cells. We first investigate the minimal control placement problem for the diffusion only system. The eigenvalues of the diffusion matrix are classified by their geometric multiplicity, and the properties of the corresponding eigenspaces are studied. The symmetric control sets are designed to categorize control candidates by symmetry of the network topology. Then the necessary and sufficient condition is provided for placing the minimal control to guarantee controllability for the diffusion system. Furthermore, we show that the necessary condition can be extended to Turing model by a natural expansion of the symmetric control sets. Under certain circumstances, we prove that it is also sufficient to ensure controllability of Turing model.
{"title":"Minimal Control Placement of Networked Reaction-Diffusion Systems Based on Turing Model","authors":"Yuexin Cao, Yibei Li, Lirong Zheng, Xiaoming Hu","doi":"10.1137/23m1616856","DOIUrl":"https://doi.org/10.1137/23m1616856","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1809-1831, June 2024. <br/> Abstract. In this paper, we consider the problem of placing a minimal number of controls to achieve controllability for a class of networked control systems that are based on the original Turing reaction-diffusion model, which is governed by a set of ordinary differential equations with interactions defined by a ring graph. Turing model considers two morphogens reacting and diffusing over the spatial domain and has been widely accepted as one of the most fundamental models to explain pattern formation in a developing embryo. It is of great importance to understand the mechanism behind the various reaction kinetics that generate such a wide range of patterns. As a first step towards this goal, in this paper we study controllability of Turing model for the case of cells connected as a square grid in which controls can be applied to the boundary cells. We first investigate the minimal control placement problem for the diffusion only system. The eigenvalues of the diffusion matrix are classified by their geometric multiplicity, and the properties of the corresponding eigenspaces are studied. The symmetric control sets are designed to categorize control candidates by symmetry of the network topology. Then the necessary and sufficient condition is provided for placing the minimal control to guarantee controllability for the diffusion system. Furthermore, we show that the necessary condition can be extended to Turing model by a natural expansion of the symmetric control sets. Under certain circumstances, we prove that it is also sufficient to ensure controllability of Turing model.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1766-1782, June 2024. Abstract. We study the boundary controllability of [math] system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be neither null-controllable nor null-controllable for any [math], where [math]. Here we recover this result in the case that [math] by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.
{"title":"Flatness Approach for the Boundary Controllability of a System of Heat Equations","authors":"Blaise Colle, Jérôme Lohéac, Takéo Takahashi","doi":"10.1137/23m1577833","DOIUrl":"https://doi.org/10.1137/23m1577833","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1766-1782, June 2024. <br/> Abstract. We study the boundary controllability of [math] system of heat equations by using a flatness approach. According to the relation between the diffusion coefficients of the heat equation, it is known that the system can be neither null-controllable nor null-controllable for any [math], where [math]. Here we recover this result in the case that [math] by using the flatness method, and we obtain an explicit formula for the control and for the corresponding solutions. In particular, the state and the control have Gevrey regularity in time and in space.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1783-1808, June 2024. Abstract. This paper proposes criteria for establishing the asymptotic moment stability of semi-Markovian impulsive switched systems. Under some mild assumptions, we formulate an auxiliary linear time-delayed system based on the Lyapunov characterizations of the subsystems and impulses, as well as the properties of the underlying semi-Markovian impulsive switching signal. Our main result provides an upper bound on the moment, which is directly related to a solution of the aforementioned linear time-delayed system. Specifically, the semi-Markovian impulsive switched system is asymptotically moment stable if the auxiliary linear time-delayed system is asymptotically stable. In situations where the mode-dependent sojourn time distributions of the underlying impulsive switching signals are all exponential, uniform, or trigonometric, we deduce explicit formulae for the auxiliary linear time-delayed systems. To prove the main result, we compute the expected gain function, which requires formulating a generalized renewal equation. Finally, we test our stability criteria on a numerical example in different scenarios and show that our stability results are nonconservative compared to the statistically obtained average of state-norms and state-norm-squares.
{"title":"Nonconservative Stability Criteria for Semi-Markovian Impulsive Switched Systems","authors":"Shenyu Liu, Penghui Wen","doi":"10.1137/23m1564833","DOIUrl":"https://doi.org/10.1137/23m1564833","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1783-1808, June 2024. <br/> Abstract. This paper proposes criteria for establishing the asymptotic moment stability of semi-Markovian impulsive switched systems. Under some mild assumptions, we formulate an auxiliary linear time-delayed system based on the Lyapunov characterizations of the subsystems and impulses, as well as the properties of the underlying semi-Markovian impulsive switching signal. Our main result provides an upper bound on the moment, which is directly related to a solution of the aforementioned linear time-delayed system. Specifically, the semi-Markovian impulsive switched system is asymptotically moment stable if the auxiliary linear time-delayed system is asymptotically stable. In situations where the mode-dependent sojourn time distributions of the underlying impulsive switching signals are all exponential, uniform, or trigonometric, we deduce explicit formulae for the auxiliary linear time-delayed systems. To prove the main result, we compute the expected gain function, which requires formulating a generalized renewal equation. Finally, we test our stability criteria on a numerical example in different scenarios and show that our stability results are nonconservative compared to the statistically obtained average of state-norms and state-norm-squares.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Philippe Bergault, Pierre Cardaliaguet, Catherine Rainer
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1737-1765, June 2024. Abstract. We investigate a stochastic differential game in which a major player has private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash mean field game equilibrium. The major player’s cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players.
{"title":"Mean Field Games in a Stackelberg Problem with an Informed Major Player","authors":"Philippe Bergault, Pierre Cardaliaguet, Catherine Rainer","doi":"10.1137/23m1615188","DOIUrl":"https://doi.org/10.1137/23m1615188","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1737-1765, June 2024. <br/> Abstract. We investigate a stochastic differential game in which a major player has private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash mean field game equilibrium. The major player’s cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xiaoling Wang, Juan Qian, Housheng Su, Xiujuan Lu, J. Lam
{"title":"Distributed Global Consensus of LTI Mass with Heterogeneous Actuator Saturation and Communication Noises","authors":"Xiaoling Wang, Juan Qian, Housheng Su, Xiujuan Lu, J. Lam","doi":"10.1137/23m1584678","DOIUrl":"https://doi.org/10.1137/23m1584678","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141353937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joel A. Rosenfeld, Benjamin P. Russo, Rushikesh Kamalapurkar, Taylor T. Johnson
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1643-1668, June 2024. Abstract. This manuscript presents a novel approach to nonlinear system identification leveraging densely defined Liouville operators and a new “kernel” function that represents an integration functional over a reproducing kernel Hilbert space (RKHS) dubbed an occupation kernel. The manuscript thoroughly explores the concept of occupation kernels in the contexts of RKHSs of continuous functions and establishes Liouville operators over RKHS, where several dense domains are found for specific examples of this unbounded operator. The combination of these two concepts allows for the embedding of a dynamical system into an RKHS, where function-theoretic tools may be leveraged for the examination of such systems. This framework allows for trajectories of a nonlinear dynamical system to be treated as a fundamental unit of data for a nonlinear system identification routine. The approach to nonlinear system identification is demonstrated to identify parameters of a dynamical system accurately while also exhibiting a certain robustness to noise.
{"title":"The Occupation Kernel Method for Nonlinear System Identification","authors":"Joel A. Rosenfeld, Benjamin P. Russo, Rushikesh Kamalapurkar, Taylor T. Johnson","doi":"10.1137/19m127029x","DOIUrl":"https://doi.org/10.1137/19m127029x","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1643-1668, June 2024. <br/> Abstract. This manuscript presents a novel approach to nonlinear system identification leveraging densely defined Liouville operators and a new “kernel” function that represents an integration functional over a reproducing kernel Hilbert space (RKHS) dubbed an occupation kernel. The manuscript thoroughly explores the concept of occupation kernels in the contexts of RKHSs of continuous functions and establishes Liouville operators over RKHS, where several dense domains are found for specific examples of this unbounded operator. The combination of these two concepts allows for the embedding of a dynamical system into an RKHS, where function-theoretic tools may be leveraged for the examination of such systems. This framework allows for trajectories of a nonlinear dynamical system to be treated as a fundamental unit of data for a nonlinear system identification routine. The approach to nonlinear system identification is demonstrated to identify parameters of a dynamical system accurately while also exhibiting a certain robustness to noise.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1669-1689, June 2024. Abstract. We analyze the role of the bang-bang property in affine optimal control problems. We show that many essential stability properties of affine problems are only satisfied when minimizers are bang-bang. We employ Stegall’s variational principle to prove that almost any linear perturbation leads to a bang-bang strict global minimizer. Examples are given to show the applicability of our results to specific optimal control problems.
{"title":"Stability and Genericity of Bang-Bang Controls in Affine Problems","authors":"Alberto Domínguez Corella, Gerd Wachsmuth","doi":"10.1137/23m1586446","DOIUrl":"https://doi.org/10.1137/23m1586446","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1669-1689, June 2024. <br/> Abstract. We analyze the role of the bang-bang property in affine optimal control problems. We show that many essential stability properties of affine problems are only satisfied when minimizers are bang-bang. We employ Stegall’s variational principle to prove that almost any linear perturbation leads to a bang-bang strict global minimizer. Examples are given to show the applicability of our results to specific optimal control problems.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号