DOCSLIB.ORG

Stochastic Model Predictive Control with Driver Behavior Learning for Improved Powertrain Control

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stochastic Model Predictive Control with Driver Behavior Learning for Improved Powertrain Control 49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Stochastic Model Predictive Control with Driver Behavior Learning for Improved Powertrain Control M. Bichi, G. Ripaccioli, S. Di Cairano, D. Bernardini, A. Bemporad, I.V. Kolmanovsky Abstract— In this paper we advocate the use of stochastic applications to tackle the uncertainty arising from the vehi- model predictive control (SMPC) for improving the perfor- cle environment. Concrete examples are stochastic dynamic mance of powertrain control algorithms, by optimally control- programming applications to hybrid electric vehicles (HEV) ling the complex system composed of driver and vehicle. While the powertrain is modeled as the deterministic component of energy management [3], [4] and emission control [5], and the dynamics, the driver behavior is represented as a stochastic linear stochastic optimal control for chassis control [6]. system which affects the vehicle dynamics. Since stochastic However, stochastic linear control is often inadequate to MPC is based on online numerical optimization, the driver capture the variety of driver behaviors, while stochastic model can be learned online, hence allowing the control algo- dynamic programming is numerically intensive and cannot rithm to adapt to different drivers and drivers’ behaviors. The proposed technique is evaluated in two applications: adaptive be easily updated if the underlying statistical model changes. cruise control, where the driver behavioral model is used to In recent years various stochastic model predictive control predict the leading vehicle dynamics, and series hybrid electric algorithms (SMPC) have been proposed, based on different vehicle (SHEV) energy management, where the driver model is prediction models and stochastic optimal control problems, used to predict the future power requests. see [7]–[9] and the references therein. In this paper, we I. INTRODUCTION model the deterministic dynamics by a linear system and the stochastic driver model by a Markov chain. The choice of Modern automotive vehicles are complex systems where Markov chains to represent driver behaviors is motivated by the mechanical components interact with the control elec- previous literature [3], [5], and by the approximation prop- tronics and with the human driver. With the constant increase erties of Markov chains [10]. The overall model is used in a in powertrain complexity, tightening of emissions standards, finite horizon stochastic optimal control problem, where the increase in gas price, and the increased number of vehicle expected performance objective is optimized, subject to the functionalities, advanced control algorithms are needed to constraints on states and inputs. Concurrently, the Markov meet the specification requirements while keeping sensor and chain modeling the driver is updated online by applying a actuator costs limited. simple learning algorithm, based on linear filtering of the Model predictive control (MPC)[1] is an appealing can- transition frequencies. This allows the SMPC controller to didate for control of complex automotive systems, due to its adapt to changes in the driver behavior. capability of coordinating multiple constrained actuators and The paper is structured as follows. In Section II we discuss optimizing the system behavior with respect to a performance the driver model based on Markov chains and the correspond- objective. However, MPC requires a model of the dynamics, ing online learning algorithm, and in Section III we introduce and, in general, the more precisely the model represents the stochastic model predictive control algorithm used in this the real dynamics, the better the closed-loop performance paper. In Sections IV and V we present the applications of is. While suitable models are available for the dynamics of the proposed approach to the adaptive cruise control (ACC) most vehicle components [2], the overall automotive vehicle and to energy management in series hybrid electric vehicle behavior strongly depends on what the driver is doing. As (SHEV), respectively. The conclusions are summarized in such, including a prediction model of possible future driver’s Section VI. actions may increase the closed-loop performance of MPC. R Z Z In this paper we propose to model the driver as a stochas- Notation: , , 0+ denote the set of real, integer, and nonnegative integer numbers, respectively. For a set A, tic process whose output affects a deterministic model of th |A| denotes the cardinality. For a vector a, [a]i is the i the vehicle. Thus, the obtained model of the vehicle and th component, and for a matrix A, [A]ij is the ij element the driver dynamics is a stochastic dynamical system that th requires appropriate stochastic control algorithms. Stochastic and [A]i is the i row. We denote a square matrix of size control algorithms have been already proposed in automotive s × s entirely composed of zeros by 0s, and the identity by Is. Subscripts are dropped when clear from the context. M. Bichi, G. Ripaccioli, and D. Bernardini are with Dept. In- formation Engineering, University of Siena, Italy, ripaccioli, II. STOCHASTIC DRIVER MODEL AND LEARNING [email protected], [email protected] ALGORITHM S. Di Cairano is with Powertrain Control R&A, Ford Research and Adv. Engineering, [email protected] We model the actions of the driver on the vehicle by A. Bemporad is with Dept. Mechanical and Structural Engineering, a stochastic process w(·) where w(k) ∈ W, for all k ∈ University of Trento, Italy [email protected] Z I.V. Kolmanovsky is with the Dept. Aerospace Engineering, the Univer- 0+ . We assume that at time k, the value w(k) can be sity of Michigan, Ann Arbor, MI, [email protected] measured, and, with a little abuse of notation, we denote 978-1-4244-7744-9/10/$26.00 ©2010 IEEE 6077 by w(k) the measured realization of the disturbance at Algorithm 1 On-line driver’s model learning procedure k ∈ Z0+. Depending on the specific application, vector w(k) 1: Given T , at k = 0 set: may represent different quantities, such as power request in 2: N = 0s, τ = 0; an HEV, acceleration, velocity, angular rate applied to the 3: i = argmin |w(0) − wh|; steering wheel, or any combination of the above. h∈{1,...,s} 4: for all k ≥ 1 do For prediction purposes, the random process generat- 5: τ = τ + 1; ing w is modelled as a Markov chain with states W = 6: j = argmin |w(k) − wh|; {w1, w2,..., ws}, where obviously wi ∈ W, for all h∈{1,...,s} i ∈ {1,..., s}. The cardinality |W | defines the trade off 7: [N]ij = [N]ij + 1; between stochastic model complexity and its precision. The 8: if τ = τmax then 9: for all h ∈ {1, 2,..., s} do Markov chain is defined by a transition probability matrix s s s −1 T ∈ R × , such that 10: [T ]h = ([N]h + λ[T ]h)(λ + l=1[N]hl) ; 11: end for P [T ]ij = Pr[w(k + 1) = wj|w(k) = wi], (1) 12: N = 0, τ = 0; 13: end if for all i, j ∈ {1,..., s}, where w(k) is the state of the Markov chain at time k. By using the Markov chain model, 14: i = j; 15: end for given w(k) = wi, the probability distribution of w(k + ℓ) is computed as ℓ ′ Pr[w(k + ℓ) = wj|w(k) = wi] = T ǫi j. (2) III. STOCHASTIC MODEL PREDICTIVE CONTROL th £¡ ¢ ¤ where the ǫi is the i unitary vector, i.e., [ǫ]i = 1, [ǫ]j = 0, In this paper we apply the SMPC formulation based on for all j 6= i. scenario enumeration and multi-stage stochastic optimization A straightforward extension of this model is to model the introduced in [12]. Consider a process whose discrete-time dynamics are modelled by the linear system stochastic process w(·) by a set of Markov chains Tm, m = 1,..., µ, where, in automotive applications, the currently x(k + 1) = Ax(k) + B1u(k) + B2w(k) (4a) active Markov chain is chosen at every instant depending y(k) = Cx(k) + D u(k) + D w(k), (4b) on current conditions such as velocity, temperature, or road 1 2 surface. where x(k) ∈ Rnx is the state, u(k) ∈ Rnu is the input, The Markov chain transition matrix T can be updated w(k) ∈ W is an additive stochastic disturbance. The state, online by different learning algorithms, with varying com- input, and output vectors may be subject to the following L constraints plexity [10], [11]. From a batch of measurement {w(k)}k=0, the Markov chain transition matrix T is estimated by x(k) ∈ X, u(k) ∈ U, y(k) ∈ Y, ∀k ∈ Z0+. (5) nij [T ]ij = , i, j ∈ {1,..., s}, (3) To simplify the exposition, we consider hereafter a scalar dis- n i turbance w. However, the approach described below is easily where nij = |Kij|, extended to multi-dimensional disturbances. For predicting k w k w i, the evolution of the disturbance w, a time-varying probabil- Kij = { | argmin | ( ) − h| = ′ h∈{1,...,s} ity vector p(k) = [p1(k), p2(k),..., ps(k)] is introduced, argmin |w(k + 1) − wh| = j}, which defines the probability of disturbance realization at h∈{1,...,s} time k, 1 s and ni = j=1 nij, for all i ∈ {1,..., s}. When new pj(k) = Pr[w(k) = wj], j = 1, 2,..., s, (6) values of w(·) are collected, the Markov chain is updated. P s Z The learning procedure used here is detailed in Algorithm 1, with j=1 pj(k) = 1, for all k ∈ 0+. We model the where N ∈ Zs×s is used to store the measured data. evolutiPon of p(k) by the Markov chain (1). As a consequence 0+ the complete model of plant and disturbance is Algorithm 1 is a linear filtering algorithm that estimates T and, if w(·) is generated by a Markov chain, it can be x(k + 1) = Ax(k) + B1u(k) + B2w(k) (7a) shown to converge to the correct value, see, e.g., [10], [11].
Load more

Recommended publications
  • STOCHASTIC CONTROL Contents 1. Introduction 1 2. Mathematical Probability 1 3. Stochastic Calculus 3 4. Diffusions 7 5. the Stoc
    STOCHASTIC CONTROL NAFTALI HARRIS Abstract. In this paper we discuss the Stochastic Control Problem, which deals with the best way to control a stochastic system and has applications in fields as diverse as robotics, finance, and industry. After quickly reviewing mathematical probability and stochastic integration, we prove the Hamilton- Jacobi-Bellman equations for stochastic control, which transform the Stochas- tic Control Problem into a partial differential equation. Contents 1. Introduction 1 2. Mathematical Probability 1 3. Stochastic Calculus 3 4. Diffusions 7 5. The Stochastic Control Problem 11 6. The Hamilton-Jacobi-Bellman Equation 12 7. Acknowledgements 18 References 18 1. Introduction The Stochastic Control Problem arises when we wish to control a continuous- time random system optimally. For example, we wish to fly an airplane through turbulence, or manage a stock portfolio, or precisely maintain the temperature and pressure of a system in the presence of random air currents and temperatures. To build up to a formal definition of this problem, we will first review mathematical probability and stochastic calculus. These will give us the tools to talk about It^o processes and diffusions, which will allow us to state the Stochastic Control Problem formally. Finally, we will spend the latter half of this paper proving necessary conditions and sufficient conditions for its solution. 2. Mathematical Probability In this section we briskly define the mathematical model of probability required in the rest of this paper. We define probability spaces, random variables, and conditional expectation. For more detail, see Williams [4]. Definition 2.1 (Probability Space). A probability space is a triple (Ω; F;P ), where Ω is an arbitrary set, F is a sigma algebra over Ω, and P is a measure on sets in F with P (Ω) = 1.
    [Show full text]
  • MARKOV DECISION PROCESSES Abstract: the Theory of Markov
    July 06, 2010, 15:00 Draft, don’t distribute MARKOV DECISION PROCESSES NICOLE BAUERLE¨ ∗ AND ULRICH RIEDER‡ Abstract: The theory of Markov Decision Processes is the theory of controlled Markov chains. Its origins can be traced back to R. Bellman and L. Shapley in the 1950’s. During the decades of the last century this theory has grown dramatically. It has found applications in various areas like e.g. computer science, engineering, operations research, biology and economics. In this article we give a short intro- duction to parts of this theory. We treat Markov Decision Processes with finite and infinite time horizon where we will restrict the presentation to the so-called (gen- eralized) negative case. Solution algorithms like Howard’s policy improvement and linear programming are also explained. Various examples show the application of the theory. We treat stochastic linear-quadratic control problems, bandit problems and dividend pay-out problems. AMS 2010 Classification: 90C40, 60J05, 93E20 Keywords and Phrases: Markov Decision Process, Markov Chain, Bell- man Equation, Policy Improvement, Linear Programming 1. Introduction Do you want to play a card game? Yes? Then I will tell you how it works. We have a well-shuffled standard 32-card deck which is also known as a piquet deck. 16 cards are red and 16 cards are black. Initially the card deck lies on the table face down. Then I start to remove the cards and you are able to see its faces. Once you have to say ”stop”. If the next card is black you win 10 Euro, if it is red you loose 10 Euro.
    [Show full text]
  • On Some Recent Aspects of Stochastic Control and Their Applications 507
    Probability Surveys Vol. 2 (2005) 506–549 ISSN: 1549-5787 DOI: 10.1214/154957805100000195 On some recent aspects of stochastic control and their applications∗ Huyˆen Pham† Laboratoire de Probabilit´es et Mod`eles Al´eatoires, CNRS, UMR 7599 Universit´eParis 7, e-mail: [email protected], and CREST Abstract: This paper is a survey on some recent aspects and developments in stochastic control. We discuss the two main historical approaches, Bell- man’s optimality principle and Pontryagin’s maximum principle, and their modern exposition with viscosity solutions and backward stochastic differ- ential equations. Some original proofs are presented in a unifying context including degenerate singular control problems. We emphasize key results on characterization of optimal control for diffusion processes, with a view towards applications. Some examples in finance are detailed with their ex- plicit solutions. We also discuss numerical issues and open questions. AMS 2000 subject classifications: 93E20, 49J20, 49L20, 60H30. Keywords and phrases: Controlled diffusions, dynamic programming, maximum principle, viscosity solutions, backward stochastic differential equations, finance. Received December 2005. arXiv:math/0509711v3 [math.PR] 12 Jan 2006 ∗This is an original survey paper †I would like to thank Philip Protter for his encouragement to write this paper. 506 H. Pham/On some recent aspects of stochastic control and their applications 507 Contents 1 Introduction 508 2 Theproblemanddiscussiononmethodology 508 2.1 Problemformulation . 508 2.2 Bellman’s optimality principle . 509 2.2.1 The Hamilton-Jacobi-Bellman (in)equation . 511 2.2.2 Theclassicalverificationapproach . 512 2.3 Pontryagin’s maximum principle . 514 2.4 Othercontrolproblems . 515 2.4.1 Randomhorizon .
    [Show full text]
  • Stochastic Control
    1 Stochastic Control Roger Brockett April 3, 2009 2 PREFACE Teaching stochastic processes to students whose primary interests are in applications has long been a problem. On one hand, the subject can quickly become highly technical and if mathematical concerns are allowed to dominate there may be no time available for exploring the many interesting areas of applications. On the other hand, the treatment of stochastic calculus in a cavalier fashion leaves the student with a feeling of great uncertainty when it comes to exploring new material. Moreover, the problem has become more acute as the power of the differential equation point of view has become more widely appreciated. In these notes, an attempt is made to resolve this dilemma with the needs of those interested in building models and designing algorithms for estimation and control in mind. The approach is to start with Poisson counters and to identify the Wiener process with a certain limiting form. We do not attempt to define the Wiener process per se. Instead, everything is done in terms of limits of jump processes. The Poisson counter and differential equations whose right-hand sides include the differential of Poisson counters are developed first. This leads to the construction of a sample path representations of a continuous time jump process using Poisson counters. This point of view leads to an efficient problem solving technique and permits a unified treatment of time varying and nonlinear problems. More importantly, it provides sound intuition for stochastic differential equations and their uses without allowing the technicalities to dominate. In treating estimation theory, the conditional density equa- tion is given a central role.
    [Show full text]
  • Stochastic Bridges of Linear Systems Yongxin Chen and Tryphon Georgiou
    1 Stochastic bridges of linear systems Yongxin Chen and Tryphon Georgiou Abstract We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed to have known states at the boundary. Thus, the movement of the particles can be modeled as an Ornstein-Uhlenbeck process conditioned on position and velocity measurements at the two end-points. It is shown that optimal stochastic control provides a stochastic differential equation (SDE) that generates such a bridge as a degenerate diffusion process. Generalizations to higher order linear diffusions are considered. I. INTRODUCTION The theoretical foundations on how molecular dynamics affect large scale properties of ensembles were layed down more than a hundred years ago. A most prominent place among mathematical models has been occupied by the Brownian motion which provides a basis for studying diffusion and noise [1], [2], [3], [4]. The Brownian motion is captured by the mathematical model of a Wiener process, herein denoted by w(t). It represents the random motion of particles suspended in a fluid where their inertia is negligible compared to viscous forces. Taking into account inertial effects under a “delta-correlated” stationary Gaussian force field η(t) (that is, white noise, loosely thought of as dw=dt [1, p. 46]) d2x(t) dx(t) m = −λ + η(t) dt2 dt represents the Langevin dynamics ; x represents position, m mass, t time, and λ viscous friction parameter. The corresponding SDE dx(t) 0 1 x(t) 0 = dt + dw(t); dv(t) 0 −λ/m v(t) 1=m where w is a Wiener process and v the velocity, is a degenerate diffusion in that the stochastic term does not affect all degrees of freedom.
    [Show full text]
  • Numerical Methods for Mean Field Games and Mean Field Type Control
    Numerical Methods for Mean Field Games and Mean Field Type Control Mathieu Lauri`ere Abstract. Mean Field Games (MFG) have been introduced to tackle games with a large number of competing players. Considering the limit when the number of players is infinite, Nash equilibria are studied by considering the interaction of a typical player with the population's distribution. The situa- tion in which the players cooperate corresponds to Mean Field Control (MFC) problems, which can also be viewed as optimal control problems driven by a McKean-Vlasov dynamics. These two types of problems have found a wide range of potential applications, for which numerical methods play a key role since most models do not have analytical solutions. In these notes, we review several aspects of numerical methods for MFG and MFC. We start by pre- senting some heuristics in a basic linear-quadratic setting. We then discuss numerical schemes for forward-backward systems of partial differential equa- tions (PDEs), optimization techniques for variational problems driven by a Kolmogorov-Fokker-Planck PDE, an approach based on a monotone operator viewpoint, and stochastic methods relying on machine learning tools. Contents 1. Introduction 2 1.1. Outline 3 1.2. Definition of the problems and notation 3 2. Warm-up: Linear-Quadratic problems 5 2.1. Problem definition and MFG solution 5 2.2. Time discretization 6 2.3. Picard (fixed point) iterations 6 arXiv:2106.06231v1 [math.OC] 11 Jun 2021 2.4. Fictitious play 9 2.5. Newton iterations 10 2.6. Mean Field Type Control and the Price of Anarchy 10 3.
    [Show full text]
  • Markov Chains
    Chapter 1 Markov Chains A sequence of random variables X0,X1,...with values in a countable set S is a Markov chain if at any time n, the future states (or values) Xn+1,Xn+2,... depend on the history X0,...,Xn only through the present state Xn.Markov chains are fundamental stochastic processes that have many diverse applica- tions. This is because a Markov chain represents any dynamical system whose states satisfy the recursion Xn = f(Xn−1,Yn), n ≥ 1, where Y1,Y2 ... are independent and identically distributed (i.i.d.) and f is a deterministic func- tion. That is, the new state Xn is simply a function of the last state and an auxiliary random variable. Such system dynamics are typical of those for queue lengths in call centers, stresses on materials, waiting times in produc- tion and service facilities, inventories in supply chains, parallel-processing software, water levels in dams, insurance funds, stock prices, etc. This chapter begins by describing the basic structure of a Markov chain and how its single-step transition probabilities determine its evolution. For in- stance, what is the probability of reaching a certain state, and how long does it take to reach it? The next and main part of the chapter characterizes the stationary or equilibrium distribution of Markov chains. These distributions are the basis of limiting averages of various cost and performance param- eters associated with Markov chains. Considerable discussion is devoted to branching phenomena, stochastic networks, and time-reversible chains. In- cluded are examples of Markov chains that represent queueing, production systems, inventory control, reliability, and Monte Carlo simulations.
    [Show full text]
  • Applications of Stochastic Optimal Control to Economics and Finance
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Federico, Salvatore (Ed.); Ferrari, Giorgio (Ed.); Regis, Luca (Ed.) Book — Published Version Applications of stochastic optimal control to economics and finance Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Federico, Salvatore (Ed.); Ferrari, Giorgio (Ed.); Regis, Luca (Ed.) (2020) : Applications of stochastic optimal control to economics and finance, ISBN 978-3-03936-059-8, MDPI, Basel, http://dx.doi.org/10.3390/books978-3-03936-059-8 This Version is available at: http://hdl.handle.net/10419/230540 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von
    [Show full text]
  • Machine Learning, Lecture 5 Kernel Machines
    Outline lecture 5 2(39) Machine Learning, Lecture 5 1. Summary of lecture 4 Kernel Machines 2. Introductory GP example 3. Stochastic processes Thomas Schön 4. Gaussian processes (GP) Division of Automatic Control Construct a GP from a Bayesian linear regression model • GP regression Linköping University • Examples where we have made use of GPs in recent research Linköping, Sweden. • 5. Support vector machines (SVM) Email: [email protected], Phone: 013 - 281373, Chapter 6.4 – 7.2 (Chapter 12 in HTF, GP not covered in HTF) Office: House B, Entrance 27. AUTOMATIC CONTROL Machine Learning REGLERTEKNIK Machine Learning T. Schön LINKÖPINGS UNIVERSITET T. Schön Summary of lecture 4 (I/II) 3(39) Summary of lecture 4 (II/II) 4(39) A kernel function k(x, z) is defined as an inner product A neural network is a nonlinear function (as a function expansion) from a set of input variables to a set of output variables controlled by k(x, z) = φ(x)Tφ(z), adjustable parameters w. where φ(x) is a fixed mapping. This function expansion is found by formulating the problem as usual, which results in a (non-convex) optimization problem. This problem is Introduced the kernel trick (a.k.a. kernel substitution). In an solved using numerical methods. algorithm where the input data x enters only in the form of scalar products we can replace this scalar product with another choice of Backpropagation refers to a way of computing the gradients by kernel. making use of the chain rule, combined with clever reuse of information that is needed for more than one gradient.
    [Show full text]
  • Mean Field Simulation for Monte Carlo Integration MONOGRAPHS on STATISTICS and APPLIED PROBABILITY
    Mean Field Simulation for Monte Carlo Integration MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F. Bunea, V. Isham, N. Keiding, T. Louis, R. L. Smith, and H. Tong 1. Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2. Queues D.R. Cox and W.L. Smith (1961) 3. Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4. The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5. Population Genetics W.J. Ewens (1969) 6. Probability, Statistics and Time M.S. Barlett (1975) 7. Statistical Inference S.D. Silvey (1975) 8. The Analysis of Contingency Tables B.S. Everitt (1977) 9. Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10. Stochastic Abundance Models S. Engen (1978) 11. Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12. Point Processes D.R. Cox and V. Isham (1980) 13. Identification of OutliersD.M. Hawkins (1980) 14. Optimal Design S.D. Silvey (1980) 15. Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16. ClassificationA.D. Gordon (1981) 17. Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18. Residuals and Influence in RegressionR.D. Cook and S. Weisberg (1982) 19. Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20. Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21. Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22. An Introduction to Latent Variable Models B.S. Everitt (1984) 23. Bandit Problems D.A. Berry and B.
    [Show full text]
  • Hidden Markov Model Multiarm Bandits: a Methodology for Beam Scheduling in Multitarget Tracking Vikram Krishnamurthy, Senior Member, IEEE, and Robin J
    IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 12, DECEMBER 2001 2893 Hidden Markov Model Multiarm Bandits: A Methodology for Beam Scheduling in Multitarget Tracking Vikram Krishnamurthy, Senior Member, IEEE, and Robin J. Evans Abstract—In this paper, we derive optimal and suboptimal beam scheduling algorithms for electronically scanned array tracking systems. We formulate the scheduling problem as a multiarm bandit problem involving hidden Markov models (HMMs). A finite-dimensional optimal solution to this multiarm bandit problem is presented. The key to solving any multiarm bandit problem is to compute the Gittins index. We present a finite-dimensional algorithm that computes the Gittins index. Suboptimal algorithms for computing the Gittins index are also presented. Numerical examples are presented to illustrate the algorithms. Index Terms—Dynamic programming, hidden Markov models, optimal beam steering, scheduling, sequential decision procedures. Fig. 1. Multitarget tracking with one intelligent sensor. The scheduling problem involves which single target to track with the steerable electronic scanned array (ESA). I. INTRODUCTION AND PROBLEM FORMULATION ONSIDER the following hidden Markov model (HMM) work considers one important part of a complete agile beam C target tracking problem: targets (e.g., aircraft) are tracking system. We show that the multiarm bandit formulation being tracked by an electronically scanned array (ESA) with can provide a rigorous and potentially useful approach to the only one steerable beam. The coordinates of each target problem of scheduling an agile beam among a finite number of evolve according to a finite state Markov established target tracks. chain (see [27]), and the conditional mean estimate target state The multiarm bandit problem is an example of a dynamic sto- is computed by an HMM tracker.
    [Show full text]
  • A Framework of Stochastic Power Management Using Hidden Markov Model
    A Framework of Stochastic Power Management Using Hidden Markov Model Ying Tan, Qinru Qiu Department of Electrical and Computer Engineering Binghamton University, State University of New York Binghamton, New York 13902, USA {ying, qqiu}@binghamton.edu Abstract - The effectiveness of stochastic power management correspondence because the workload is not only controlled by the relies on the accurate system and workload model and effective hardware and software but also affected by user and environment. policy optimization. Workload modeling is a machine learning For example, user working style and user mood have significant procedure that finds the intrinsic pattern of the incoming tasks impact of the workload distribution of a computer system. Consider based on the observed workload attributes. Markov Decision a power manager (PM) of a wireless adapter. The observed Process (MDP) based model has been widely adopted for workload attribute is the frequency and the size of the stochastic power management because it delivers provable incoming/outgoing TCP/IP packets. During a teleconference, the optimal policy. Given a sequence of observed workload user may choose to send a video image, send the audio message, attributes, the hidden Markov model (HMM) of the workload share files or share the image on whiteboard. Different operations is trained. If the observed workload attributes and states in the generate different communication requirements. An accurate workload model do not have one-to-one correspondence, the workload model of the wireless adapter must reflect how the user MDP becomes a Partially Observable Markov Decision Process switches from one operation to another. However, this information (POMDP). This paper presents a framework of modeling and is not observable by the PM.
    [Show full text]