The book covers both state-space methods and those based on the polynomial approach. Our approach By utilizing the Dirichlet process, we model the unknown distribution of the underlying stochastic process as a random probability measure and achieve online learning in a Bayesian manner. The results of this section are proved in Appendix C. Let us report from Subbaraman and Teel (2013) the basic definitions. In NRMPC, an optimal control sequence is obtained by solving an optimization problem based on the current state, and then the first portion of this sequence is applied to the real system in an open-loop manner during each sampling period. Stochastic MPC and robust MPC are two main approaches to deal with uncertainty (Mayne, 2016).In stochastic MPC, it usually “soften” the state and terminal constraints to obtain a meaningful optimal control problem (see Dai, Xia, Gao, Kouvaritakis, & Cannon, 2015; Grammatico, Subbaraman, & Teel, 2013; Hokayem, Cinquemani, Chatterjee, Ramponi, & Lygeros, 2012; Zhang, Georghiou, & Lygeros, 2015).This paper focuses on robust MPC and will present two robust MPC schemes for a classical unicycle robot tracking problem. In tube-MPC, the control signal consists of a control action and a nonlinear feedback law based on the deviation of the actual states from the states of a nominal system. It is known that there exist stabilizable deterministic discrete-time nonlinear control systems that cannot be stabilized by continuous state feedback (Rawlings & Mayne, 2009, Example 2.7) even though they admit a continuous control-Lyapunov function (Grimm, Messina, Tuna, & Teel, 2005, Example 1) and thus can be robustly stabilized by discontinuous state feedback (Kellett & Teel, 2004). Anantharaman Subbaraman received the B.Tech. Publication:1996, 330 pages, softcover. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. His research contributions are primarily in control and system theory, in particular in the subareas of stochastic control, filtering, stochastic realization, control of discrete-event systems and of hybrid systems, and control and system theory of rational systems. Applications of the theory in the book include the control of ships, shock absorbers, traffic and communications networks, and power systems with fluctuating power flows. • Infinite Time Horizon Control: Positive, Discounted and Nega-tive Programming. This fact motivates our investigations. An example shows that without strict causality we may have no robustness even to arbitrarily small perturbations. Purchase Techniques in Discrete-Time Stochastic Control Systems, Volume 73 - 1st Edition. Correspondingly, based on this definition, some sufficient conditions are provided for nSSNL systems and SSNL systems. ...you'll find more products in the shopping cart. In this section we present our main results, proved in Appendix A, on robustness of Lyapunov conditions to sufficiently small, state-dependent, strictly causal, worst-case perturbations. A stable weighted multiple model adaptive control system for uncertain linear, discrete‐time stochastic plant is presented in the paper. Published by Elsevier Ltd. All rights reserved. In this paper we propose a new methodology for solving a discrete time stochastic Markovian control problem under model uncertainty. The aim of this paper is to study the stability of discrete stochastic time-delayed systems with multiplicative noise, where the coefficients are assumed to be time-varying with a general time-varying rate or a small time-varying rate. In this paper, we consider discrete-time stochastic systems with basic regularity properties and we investigate robustness of asymptotic stability in probability and of recurrence. The chapters include treatments of optimal stopping problems. Similarities and differences between these approaches are highlighted. x+=(x1x2)+=(x1+vux2+vu3)=f(x,u,v) where x=(x1,x2)⊤∈X=R2,u∈U=R,v∈V={−1,1} with μ({−1})=p and μ({1})=1−p,p∈[0,1]. Abstract: This paper is concerned with the event-based security control problem for a class of discrete-time stochastic systems with multiplicative noises subject to both randomly occurring Denial-of-Service (DoS) attacks and randomly occurring deception attacks. Since we deal with discontinuous systems, we introduce generalized random solutions to generate enough random solutions which provide an accurate picture of robustness with respect to strictly causal perturbations. Firstly, by the Kronecker algebra theory and H-representation technique, the exponential stability of the stochastic system with common time-varying coefficients is investigated by the spectral approach. Sergio Grammatico received the B.Sc., M.Sc. The extension to the continuous-time setting is highly non-trivial as one needs to continuously randomize actions, and there has been little understanding (if any) of how to appropriately incorporate stochastic policies … At each time period new observations are made, and the control variables are to be adjusted optimally. For any closed set C⊆Rn and x∈Rn,|x|C≔infy∈C|x−y| is the Euclidean distance to the set C. B(B∘) denotes the closed (open) unit ball in Rn. For the study of GASiP, the definition which we considered is not the usual notion of asymptotic stability in probability (stability in probability plus attractivity in probability); it can depict the properties of the system quantitatively. The condition graph(z)⊂(Z≥0×(Ā+εB∘)) is equivalent to z(ω)∈Ā+εB∘ for all i∈{0,…,Jz(ω)−1}. This paper addresses a version of the linear quadratic control problem for mean-field stochastic differential equations with deterministic coefficients on time scales, which includes the discrete time and continuous time as special cases. By continuing you agree to the use of cookies. Professor Jan H. van Schuppen gained his PhD from the Department of Electrical Engineering and Computer Science of the University of California at Berkeley in 1973. This research monograph, first published in 1978 by Academic Press, remains the authoritative and comprehensive treatment of the mathematical foundations of stochastic optimalcontrol of discrete … Stochastic Optimal Control: The Discrete-Time Case Dimitri P. Bertsekas and Steven E. Shreve This book was originally published by Academic Press in 1978, and republished by … 2569-2576, Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations, Dynamic Stability of Passive Bipedal Walking on Rough Terrain: A Preliminary Simulation Study, Lyapunov-based model predictive control of stochastic nonlinear systems, Economic model predictive control without terminal constraints for optimal periodic behavior, Lyapunov conditions certifying stability and recurrence for a class of stochastic hybrid systems, Stochastic input-to-state stability of switched stochastic nonlinear systems. In Teel (in press) the notion of random solutions to set-valued discrete-time stochastic systems is introduced. Finally, some examples are provided to demonstrate the applicability of our results. He also received a M.Sc. It was found that the average maximum Floquet multiplier increases with surface roughness in a non-linear form. Tel. 1970 edition. By introducing a robust state constraint and tightening the terminal region, recursive feasibility and input-to-state stability are guaranteed. The main results are shown in Section  4. (gross), ca. The application of the proposed LMPC method is illustrated using a nonlinear chemical process system example. Section  2 contains the basic notation and definitions. Our results show that the passive walker can walk on rough surfaces subject to surface roughness up to approximately 0.1% of its leg length. This book provides a comprehensive introduction to stochastic control problems in discrete and continuous time. degree in Engineering Sciences from Dartmouth College in Hanover, New Hampshire, in 1987, and his M.S. We have a dedicated site for France. stochastic optimal control problem for discrete-time Markovian switching systems. Properties of the value function and the mode-dependent optimal policy are derived under a variety of … An event-triggered 952-961, Automatica, Volume 48, Issue 9, 2012, pp. Research supported in part by the National Science Foundation grant number NSF ECCS-1232035 and the Air Force Office of Scientific Research grant number AFOSR FA9550-12-1-0127. The material is presented logically, beginning with the discrete-time case before proceeding to the stochastic continuous-time models. chapters 8-11 (5.353Mb) chapters 5 - 7 (7.261Mb) Chap 1 - 4 (4.900Mb) Table of Contents (151.9Kb) Metadata Show full … (submitted for publication). degree in Engineering from the Sant’Anna School of Advanced Studies, Pisa, Italy, in 2011. In this paper, we introduce a Newton-based approach to stochastic extremum seeking and prove local stability of Newton-based stochastic extremum seeking algorithm in the sense of both almost sure convergence and convergence in probability. R≥0(R>0) denotes the set of non-negative (positive) real numbers, and Z≥0(Z>0) denotes the set of non-negative (positive) integers. By using the stochastic comparison principle, the Itô formula, and the Borel- Cantelli lemma, we obtain two sufficient criteria for stochastic intermittent stabilization. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. Now we study how Lyapunov conditions predict the stochastic stability properties for random solutions associated with the stochastic difference equation x+=f(x,κ(x),v)(4) when the random input v is generated by the random variables vi:Ω→V, for i∈Z≥0. Remark 7Any stabilizing feedback control law for the deterministic discrete cubic integrator, namely system (26) with v≡1, is necessarily discontinuous (Meadows et al.. Any stabilizing feedback control law for the deterministic discrete cubic integrator, namely system (26) with v≡1, is necessarily discontinuous (Meadows et al., For discrete-time stochastic systems allowing discontinuous control laws, the existence of a continuous stochastic Lyapunov function implies that asymptotic stability in probability of the attractor for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Recursive feasibility and input-to-state stability are established and the constraints are ensured by tightening the input domain and the terminal region. This indicates that bipedal walkers based on passive dynamics may possess some intrinsic stability to adapt to rough terrains although the maximum roughness they can tolerate is small. There is a growing need to tackle uncertainty in applications of optimization. : +41 44 632 3469; fax: +41 44 632 1211. Abstract. His research interests include robust Lyapunov-based control and stochastic control systems. Different kinds of methods have been adopted to find less conservative criteria of stability.It can be remarked that, in spite of time-invariant systems or time-varying systems, the Lyapunov function method serves as a main technique for most existing works about the stability analysis, but finding suitable Lyapunov functions is still a difficult task; see [2,24,35–37] and so on.Another method is to investigate special cases of time-varying systems by decomposing the system matrix of a linear time-varying system into two parts, one is a constant matrix and the other one is a time-varying derivation, which satisfies certain conditions, see [11,27]. An illustrative MPC example is provided in Section  8. The number of consecutive steps before falling was used to measure the walking stability after the passive walker started to fall over. and Ph.D. degrees in Automation Engineering from the University of Pisa, Italy, respectively in 2008, 2009, and 2013. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Section  5 introduces the notion of generalized random solutions. Springer is part of, Please be advised Covid-19 shipping restrictions apply. Print Book & E-Book. It was also found that shifting the phase angle of the surface profile has apparent affect on the system stability. Example 4x+=(x1x2)+=(x1+vux2+vu3)=f(x,u,v) where x=(x1,x2)⊤∈X=R2,u∈U=R,v∈V={−1,1} with μ({−1})=p and μ({1})=1−p,p∈[0,1]. The condition graph(z)⊂(Z≥0×(Ā+εB∘)) is equivalent to z(ω)∈Ā+εB∘ for all i∈{0,…,Jz(ω)−1}. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. The set-valued mappings studied here satisfy the basic regularity properties considered in Teel et al. It is shown that, if the product of the input/output coupling matrices is a full-column rank, then the input error covariance matrix converges uniformly to zero in the presence of … 2271-2276, Annual Reviews in Control, Volume 37, Issue 1, 2013, pp. Author(s) Bertsekas, Dimitir P.; Shreve, Steven. Concluding comments are presented in Section  9. at discrete time epochs, one at a time, for an MDP. In this paper, we study asymptotic properties of problems of control of stochastic discrete time systems with time averaging and time discounting optimality criteria, and we establish that the Cesa´ro and Abel limits of the optimal values in such problems can be evaluated with the help of a certain infinite- When the roughness magnitude approached to 0.73% of the walker's leg length, it fell down to the ground as soon as it entered into the uneven terrain. After receiving his Ph.D., Dr. Teel was a postdoctoral fellow at the Ecole des Mines de Paris in Fontainebleau, France. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas mole… The first step in determining an optimal control policy is to designate a set of control policies which are admissible in a particular application. and Ph.D. degrees in Electrical Engineering from the University of California, Berkeley, in 1989 and 1992, respectively. It is shown that the time-varying stochastic systems with state delays is exponentially stable in mean square sense if and only if its corresponding generalized spectral radius is less than one. The paper is organized as follows. Contents, Preface, Ordering. Recently, there has been interest regarding stochastic systems with non-unique solutions (Teel, 2009) due to the interaction between random inputs and worst-case behavior. The discrete-time stochastic multi-agent system with the undirected graph G and the event-triggered control law is ε-consensusable if there exist a matrix K, two positive definite matrices Q and P, and a positive scalar δ satisfying (16) Q = P − (1 + δ) (A + Ξ ⊗ (B K C)) T P (A + Ξ ⊗ (B K C)) − D T P D − (Ξ ⊗ (B K E)) T P (Ξ ⊗ (B K E)) − σ 2 D T P (Ξ ⊗ (B K E)) − σ 2 (Ξ ⊗ (B K E)) T P D and the … Copyright © 2013 Elsevier Ltd. This behavior is analyzed in detail, and we show that under suitable dissipativity and controllability conditions, desired closed-loop performance guarantees as well as convergence to the optimal periodic orbit can be established. Here, the constraints mustbesatis eduniformly,overalladmissibleswitching paths. Two coupled Riccati equations on time scales are given and the optimal control can be expressed as a linear state feedback. 423-433, Automatica, Volume 50, Issue 3, 2014, pp. First, since in Assumption 2 we have not assumed that the control law κ:X→U is a measurable function, there is no guarantee that the iterationxi+1(ω)≔. Please review prior to ordering, Motivates detailed theoretical work with relevant real-world problems, Broadens reader understanding of control and system theory, Provides comprehensive definitions of multiple related concepts, Offers in-depth treatment of stochastic control with partial observation, Equips readers with uniform treatment of various system probability distributions, ebooks can be used on all reading devices, Institutional customers should get in touch with their account manager, The final prices may differ from the prices shown due to specifics of VAT rules. An open, bounded set Ō⊂Rn̄ is Uniformly Globally Recurrent for (17) if for each ϱ∈R>0 and R∈R>0 there exists J∈Z≥0 such that z∈RB∩(Rn̄∖Ō),z∈Sr(z)⟹P[(graph(z)⊂(Z≤J×Rn̄))∨(graph(z)∩(Z≤J×Ō)≠∅)]≥1−ϱ, where ∨ is the logical “or”. The robust control problem for discrete-time stochastic interval system (DTSIS) with time delay is investigated in this paper. The material in this paper was not presented at any conference. In 1992 he joined the faculty of the Electrical Engineering Department at the University of Minnesota where he was an assistant professor. We first show by means of a counterexample, that a classical receding horizon control scheme does not necessarily result in an optimal closed-loop behavior. The results show that the number of steps before falling decreases exponentially with the increase in surface roughness. Regarding stochastic systems, different stability notions and Lyapunov conditions have been studied in the literature (Kolmanovskii and Shaikhet, 2002, Kozin, 1969, Kushner, 1967, Kushner, 1971, Meyn, 1989, Meyn and Tweedie, 1993). Also, the existence of a continuous stochastic Lyapunov function implies, Sergio Grammatico received the B.Sc., M.Sc. In terms of the average dwell-time of the switching laws, a sufficient SISS condition is obtained for SSNL systems. Definition 3 UGRAn open, bounded set Ō⊂Rn̄ is Uniformly Globally Recurrent for (17) if for each ϱ∈R>0 and R∈R>0 there exists J∈Z≥0 such that z∈RB∩(Rn̄∖Ō),z∈Sr(z)⟹P[(graph(z)⊂(Z≤J×Rn̄))∨(graph(z)∩(Z≤J×Ō)≠∅)]≥1−ϱ, where ∨ is the logical “or”. Instead, a multi-step MPC scheme may be needed in order to establish near optimal performance of the closed-loop system. In this work, we design a Lyapunov-based model predictive controller (LMPC) for nonlinear systems subject to stochastic uncertainty. enable JavaScript in your browser. In combining these two approaches, the state mean propagation is constructed, where the adjusted parameter is added into the model output used. This further allows us to also relate the existence of a continuous stochastic Lyapunov function for the nominal closed-loop system to certain stochastic stability properties of the perturbed closed-loop system, in view of the results in Teel et al. He has acted as research advisor of 12 post-doctoral researchers and of 19 Ph.D. students. Let us consider the attractor A={0}. The set {ω∈Ω∣graph(z(ω))⊂(Z≥0×(, A compact set Ā⊂Rn̄ is said to be uniformly stable in probability for (17) if for each ε∈R>0 and ϱ∈R>0 there exists δ∈R>0 such that z∈Ā+δB,z∈Sr(z)⟹P[graph(z)⊂(Z≥0×(Ā+εB∘))]≥1−ϱ. (submitted for publication). Regularity conditions are given that guarantee the existence of random solutions and robustness of the Lyapunov conditions. For any set S⊆Rn, we define the, Consider a function f:X×U×V→X, where X⊆Rn and U⊆Rm are closed sets, V⊆Rp is measurable, and a stochastic controlled difference equation x+=f(x,u,v) with state variable x∈X, control input u∈U, and random input v∈V, eventually specified as a random variable, that is a measurable function from a probability space (Ω,F,P) to V. From an infinite sequence of independent, identically distributed (i.i.d.) In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a family of random variables. The stochastic interval system is equivalently transformed into a kind of stochastic uncertain time-delay system firstly. He is. Central themes are dynamic programming in discrete time and HJB-equations in continuous time. His organizational activities include being Co-Editor of the journal Mathematics of Control, Signals, and Systems, Co-Editor-at-Large of the journal IEEE Transactions on Automatic Control, co-editor of two conference proceedings, co-editor of two edited books, coordinator of four projects which were financially supported by the European Commission, and being director of the Dutch Network Systems and Control for the organization of a course program of systems and control for Ph.D. students. A similar robustness result holds for the recurrence property, under a weaker Lyapunov condition. A similar result showing the equivalence between the existence of a smooth Lyapunov function and a weaker stochastic stability property called recurrence is presented in Subbaraman and Teel (2013). He visited the Department of Mathematics at the University of Hawai’i at Manoa in 2010 and 2011, and the Department of Electrical and Computer Engineering at the University of California Santa Barbara in 2012. Since the MPC feedback law may be discontinuous, having a continuous Lyapunov function for the closed-loop system is necessary to establish nominal robustness (Grimm et al., 2005, Kellett and Teel, 2004). Let us consider the attractor A={0}. Price:$34.50. 176,79 €, © 2020 Springer Nature Switzerland AG. For any set S⊆Rn, the notation cl(S) denotes the closure of S. For any closed set C and ε∈R>0,C+εB denotes the set {x∈Rn∣|x|C≤ε}. In this paper, we present stochastic intermittent stabilization based on the feedback of the discrete time or the delay time. Our results are related to stochastic stability properties respectively in Sections  6 Stochastic stability, 7 Lyapunov conditions for robust recurrence. degree in Control Engineering from the National Institute of Technology, Trichy, India, in 2010, and the M.S. Given a continuous stochastic Lyapunov function V relative to the compact attractor A for the nominal closed-loop system (4), we show that there exists a concave function Γ∈K∞ such that the function Γ(V) is a continuous stochastic Lyapunov function relative to A for a perturbed closed-loop system. Simulation shows the effectiveness and advantage of the proposed algorithm over gradient-based stochastic extremum seeking. The key idea is to use stochastic Lyapunov-based feedback controllers, with well characterized stabilization in probability, to design constraints in the LMPC that allow the inheritance of the stability properties by the LMPC. In this paper, global asymptotic stability in probability (GASiP) and stochastic input-to-state stability (SISS) for nonswitched stochastic nonlinear (nSSNL) systems and switched stochastic nonlinear (SSNL) systems are investigated. Finding the optimal solution for the present time may involve iterating a matrix Riccati equation backwards in time from the last period to the present period. Stochastic Optimal Control: The Discrete-TIme Case. The equivalence between the existence of a continuous Lyapunov function and global asymptotic stability in probability of a compact attractor for stochastic difference inclusions without control inputs is established in Teel, Hespanha, and Subbaraman (submitted for publication) under certain regularity assumptions. This is probably because point contact was used to simulate the heel strikes and the resulted variations in system states at heel strikes may have pronounced impact on the passive gaits, which have narrow basins of attraction. degree in Engineering from the Sant’Anna School of Advanced Studies, Pisa, Italy, in 2011. Discrete-Time Stochastic Sliding-Mode Control Using Functional Observation will interest all researchers working in sliding-mode control and will be of particular assistance to graduate students in understanding the changes in design philosophy that arise when changing from continuous- to discrete-time … He visited the Department of Mathematics at the University of Hawai’i at Manoa in 2010 and 2011, and the Department of Electrical and Computer Engineering at the University of California Santa Barbara in 2012. From previous studies, the IOCPE algorithm is for solving the discrete-time nonlinear stochastic optimal control problem, while the stochastic approximation is for the stochastic optimization. In the proof of the above results, to overcome the difficulties coming with the appearance of switching and the stochastic property at the same time, we generalize the past comparison principle and fully use the properties of the functions which we constructed. In Section  3 we present the class of discrete-time stochastic systems along with certain regularity and Lyapunov conditions. We could consider random solutions of system (4) directly, but there are the following two issues. The field of Preview Control is concerned with using advanced knowledge of disturbances or references in order to improve tracking quality or disturbance rejection. Discrete-time Stochastic Systems gives a comprehensive introduction to the estimation and control of dynamic stochastic systems and provides complete derivations of key results such as the basic relations for Wiener filtering. We use cookies to help provide and enhance our service and tailor content and ads. Discrete-Time Controlled Stochastic Hybrid Systems Alessandro D'Innocenzo, Alessandro Abate, and Maria D. Di Benedetto Abstract This work presents a procedure to construct a nite abstraction of a controlled discrete-time stochastic hy-brid system.