Journal Publications

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Most of my articles/preprints are freely downloadable from this page on arXiv.

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Submitted

  • S. Das, A. Ghosh, and D. Chatterjee
    Algorithmic detection of sensor attacks in linear cyberphysical systems.
    Submitted.

    • Abstract

      Details
      This article introduces a sequential algorithm to detect the presence of attacks falling within the scope of data deception/integrity attacks in cyber-physical systems (CPSs) modeled as linear time-invariant stochastic systems. The core idea of this data-driven algorithm is based on the fact that an honest state (that is not infected by adversaries) generated by the CPS should concentrate near its weighted empirical mean of the immediate past samples. As main results, we provide strong guarantees on the false positive error and the false negative error under certain mild assumptions. Several remarks highlighting various technical aspects of the established algorithm and the main results are included. A numerical experiment conducted on the benchmark inverted pendulum model is included to demonstrate the effectiveness of the algorithm.

  • S. Ganguly and D. Chatterjee
    Exact solutions to minmax control problems for constrained noisy linear systems.
    Submitted.

    • Abstract

      Details
      We present a numerical technique for exact solutions to a class of robust minmax optimal control problems (OCPs) under constraints. The underlying dynamical system is assumed to be linear and subjected to bounded disturbances and the objective function is a sum of quadratic stage and quadratic terminal costs. We impose convex constraints on the states and the control; the control minimizes and the disturbance maximizes the objective. The resulting constrained minmax OCP admits a reformulation in the language of convex semi-infinite programs (CSIPs) and we employ recently developed numerical tools in CSIP to solve the ensuing optimization problem. A simple benchmark numerical example is provided to illustrate our results.

  • S. Ganguly, R. A. D’Silva, and D. Chatterjee
    QuITO v.2: Numerical solutions with uniform error guarantees to nonlinear optimal control problems under path constraints.
    Submitted.

    • arXiv version

    • The public github respository of QuITO v.2 is here.

    • Abstract

      Details
      This article introduces a new transcription, change point localization, and mesh refinement scheme for direct optimization-based solutions and for uniform approximation of optimal control trajectories associated with a class of nonlinear constrained optimal control problems. The base transcription algorithm for which we establish the refinement algorithm is a direct multiple shooting technique — QuITO v.2 (Quasi-Interpolation based Trajectory Optimization). The mesh refinement technique consists of two steps — localization of certain irregular regions in an optimal control trajectory via wavelets, followed by a targeted h-refinement approach around such regions of irregularity. Theoretical approximation guarantees on uniform grids are presented for optimal controls with certain regularity properties along with guarantees of localization of change points by wavelet transform. Numerical illustrations are provided for control profiles involving discontinuities to show the effectiveness of the localization and refinement strategy. We also announce and make freely available a new software package developed based on QuITO v.2 along with its functionalities to make the article complete.

  • M. Oza and D. Chatterjee
    Near-optimal solutions to geodesically convex semi-infinite programs.
    Submitted.

    • Abstract

      Details
      We establish a numerically tractable algorithm for the determination of the optimal value of a semi-infinite program with geodesically convex data on a uniquely geodesic Riemannian manifold. The uncertainty set of the semi-infinite program is assumed to be compact, and geodesic convexity is imposed on the objective function and the constraint function corresponding to each uncertainty element. This algorithm extends the results of DACC22 in the context of convex semi-infinite programs to the setting of geodesically convex data on uniquely geodesic Riemannian manifolds.

  • S. Ganguly, S. Gupta, and D. Chatterjee
    Robust model predictive control: explicit solutions for low- through moderate-dimensional systems.
    Submitted.

    • Abstract

      Details
      We present an approximate feedback synthesis algorithm to explicitly solve a class of robust minmax model predictive control (MPC) problems. The chief ingredients of the optimal control problem consists of a linear noisy dynamical system, a quadratic instantaneous and quadratic terminal cost, and convex constraints on the state, control and disturbance sequences. We proceed via two steps: first we reformulate the given baseline minmax problem into a convex semi-infinite program and employ some recently developed tools in semi-infinite convex programming theory to solve a newly recast problem online in a exact fashion. The second stage involves laying an uniform grid over the state-space and solving the semi-infinite program on the uniform grid to generate the point values of a implicit feedback policy. Finally we employ an approximation scheme that can furnish tight and uniform error bounds on a given domain to generate the approximate feedback policy. To this end, we employ (a) a quasi-interpolation based approximation engine for low-dimensional systems and (b) a ReLU neural network based approximation scheme for moderate- through high-dimensional systems. We show that the closed-loop system is stable and recursively feasible under the approximated feedback policy (generated via (a) and (b) respectively), which is guaranteed to be within an \(\varepsilon\)-ball of the original feedback policy. Numerical examples are provided to illustrate the results.

  • R. Gupta, S. Chattopadhyay, P. Paruchuri, and D. Chatterjee
    Algorithmic construction of Lyapunov functions for continuous vector fields.
    Submitted.

    • arXiv version

    • Abstract

      Details
      This article presents a novel numerically tractable technique for synthesizing Lyapunov functions for equilibria of nonlinear vector fields. In broad strokes, corresponding to an isolated equilibrium point of a given vector field, a selection is made of a compact neighborhood of the equilibrium and a dictionary of functions in which a Lyapunov function is expected to lie. Then an algorithmic procedure based on the recent work DACC22 is deployed on the preceding neighborhood-dictionary pair and charged with the task of finding a function satisfying a compact family of inequalities that defines the behavior of a Lyapunov function on the selected neighborhood. The technique applies to continuous nonlinear vector fields without special algebraic structures and does not even require their analytical expressions to proceed. Several numerical examples are presented to illustrate our results.

  • S. Ganguly and D. Chatterjee
    Explicit feedback synthesis driven by quasi-interpolation for nonlinear constrained model predictive control.
    Submitted.

    • arXiv version

    • Abstract

      Details
      We present QuIFS (Quasi-Interpolation driven Feedback Synthesis): an offline feedback synthesis algorithm for explicit nonlinear robust minmax model predictive control (MPC) problems with guaranteed quality of approximation. The underlying technique is driven by a particular type of grid-based quasi-interpolation scheme. The QuIFS algorithm departs drastically from conventional approximation algorithms that are employed in the MPC industry (in particular, it is neither based on multi-parametric programming tools and nor does it involve kernel methods), and the essence of its point of departure is encoded in the following challenge-answer approach: Given an error margin \(\varepsilon > 0\), compute in a single stroke a feasible feedback policy that is uniformly \(\varepsilon\)-close to the optimal MPC feedback policy for a given nonlinear system subjected to constraints and bounded uncertainties. Closed-loop stability and recursive feasibility under the approximate feedback policy are also established. We provide a library of numerical examples to illustrate our results.

  • S. Ganguly, S. Das, D. Chatterjee, and R. Banavar
    On optimal control problems under rate constraints.
    Submitted.

    • arXiv version

    • Abstract

      Details
      Limited bandwidth and limited saturation in actuators are practical concerns in control systems. Mathematically, these limitations manifest as constraints being imposed on the control actions, their rates of change, and more generally, the global behavior of their paths. While the problem of actuator saturation has been studied extensively, little attention has been devoted to the problem of actuators having limited bandwidth. While attempts have been made in the direction of incorporating frequency constraints on state-action trajectories before, rate constraints on the control at the design stage have not been studied extensively in the discrete-time regime. This article contributes toward filling this lacuna. In particular, we establish a new discrete-time Pontryagin maximum principle with rate constraints being imposed on the control trajectories, and derive first-order necessary conditions for optimality. A brief discussion on the existence of optimal control is included, and numerical examples are provided to illustrate the results.

Journal publications

2024

  • S. Das, P. Dey, and D. Chatterjee
    Almost sure detection of the presence of malicious components in cyber-physical systems.
    To appear in Automatica, 2024.

    • arXiv version

    • Abstract

      Details
      This article studies a fundamental problem of security of cyber-physical systems (CPSs): that of detecting, almost surely, the presence of malicious components in the CPS. We assume that some of the actuators may be malicious while all sensors are honest. We introduce a novel idea of separability of state trajectories generated by CPSs in two situations: those under the nominal no-attack situation and those under the influence of an attacker. We establish its connection to security of CPSs in the context of detecting the presence of malicious actuators (if any) in them. As primary contributions we establish necessary and sufficient conditions for the aforementioned detection in CPSs modeled as Markov decision processes (MDPs). Moreover, we focus on the mechanism of perturbing the pre-determined control policies of the honest agents in CPSs modeled as stochastic linear systems, by injecting a certain class of random process called private excitation; sufficient conditions for detectability and non-detectability of the presence of malicious actuators assuming that the policies are randomized history dependent and randomized Markovian, are established. Several technical aspects of our results are discussed extensively.

    • Informal description of this work

      Details
      Most of the system-theoretic studies on security of cyber-physical systems assume certain types of adversarial attack models. This essentially means that the designers/defenders already know the plans of the attackers, which does not reflect realistic situations since no reasonable attacker would be inclined to announce their attack strategies to the world. Game-theoretic approaches to studying adversarial attacks are also not quite applicable directly and widely because the designers/defenders cannot possibly know the attackers' objectives (unless the dividing line between a designer and an attacker is blurred). Consequently, an approach based purely on the analysis of ‘signals’ — measurements of the systems under consideration — is, we believe, the most natural approach to tackling the problem of detecting adversarial attacks. This work contains some fundamental theoretical results in this direction; generalizations and algorithm development are on the anvil.

  • A. Ganguly and D. Chatterjee
    Moment stability of stochastic processes with applications to control systems.
    Mathematical Control and Related Fields, Vol. 14, No. 1, pp. 386—​412, 2024.

    • doi, pdf

    • arXiv version

    • Abstract

      Details
      We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the sizes of the one-step jumps of the processes. The state-dependent feature of the results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of Markovian property, new result on ergodicity has also been proved. There are several applications to iterative systems, control systems, and other dynamical systems with state-dependent multiplicative noise, and we include illustrative examples to demonstrate applicability of our results.

2023

  • S. Ganguly, N. Randad, R. A. D’Silva, M. Raj, and D. Chatterjee
    QuITO: Numerical software for constrained nonlinear optimal control problems.
    SoftwareX, Vol. 24, 2023.

    • doi

    • Abstract

      Details
      We introduce the MATLAB-based software QuITO (Quasi-Interpolation based Trajectory Optimization) to numerically solve a wide class of constrained nonlinear optimal control problems. The solver is based on the QuITO (the same abbreviation) algorithm, which is a direct multiple shooting technique that leverages a particular type of quasi-interpolation scheme for control trajectory parameterization. The software is equipped with several options for numerical integration, and optimization solvers along with a Graphical User Interface to make the process of designing and solving the OCPs smooth and seamless for users with minimum coding experience. We demonstrate with two benchmark numerical examples the procedure to generate constrained state and control trajectories using QuITO.

  • P. Paruchuri and D. Chatterjee
    Attaining the Chebyshev bound for optimal learning: a numerical algorithm.
    Systems & Control Letters, Vol. 181, 2023.

    • doi

    • arXiv version

    • Abstract

      Details
      Given a compact subset of a Banach space, the Chebyshev center problem consists of finding a minimal circumscribing ball containing the set. In this article we establish a numerically tractable algorithm for solving the Chebyshev center problem in the context of optimal learning from a finite set of data points. For a hypothesis space realized as a compact but not necessarily convex subset of a finite-dimensional subspace of some underlying Banach space, this algorithm computes the Chebyshev radius and the Chebyshev center of the hypothesis space, thereby solving the problem of optimal recovery of functions from data. The algorithm itself is based on, and significantly extends, recent results for near-optimal solutions of convex semi-infinite problems by means of targeted sampling, and it is of independent interest. Several examples of numerical computations of Chebyshev centers are included in order to illustrate the effectiveness of the algorithm.

    • Informal blurb

      Details
      The Chebyshev center problem (also known as the circumcenter problem) is known to be NP-hard — a stimulating discussion on this topic may be found in AT2021, Chapter 15. However, there is no democracy in the world of NP-hard problems — some of them may admit easy (i.e., numerically viable) approximations while others may appear to be intractable. This work contains the first theoretically exact and to date the only numerically viable approach to solve the Chebyshev center problem. It solves an open problem on optimal recovery posed in MR1977 in the signal processing literature.

  • A. Jindal, D. Chatterjee, and R. Banavar
    Estimates of the size of the domain of the implicit function theorem: a mapping degree based approach.
    Mathematics of Control, Signals, and Systems, 2023.

    • doi, pdf

    • arXiv version

    • Abstract

      Details
      In this article, we present explicit estimates of the size of the domain on which the Implicit Function Theorem and the Inverse Function Theorem are valid. For maps that are twice continuously differentiable, these estimates depend upon the magnitude of the first-order derivatives evaluated at the point of interest, and a bound on the second-order derivatives over a region of interest. One of the key contributions of this article is that the estimates presented require minimal numerical computation. In particular, these estimates are arrived at without any intermediate optimization procedures. We then present three applications in optimization and systems and control theory where the computation of such bounds turns out to be important. First, in electrical networks, the power flow operations can be written as Quadratically Constrained Quadratic Programs (QCQPs), and we utilize our bounds to compute the size of permissible power variations to ensure stable operations of the power system network. Second, the robustness margin of positive definite solutions to the Algebraic Riccati Equation (frequently encountered in control problems) subject to perturbations in the system matrices are computed with the aid of our bounds. Finally, we employ these bounds to provide quantitative estimates of the size of the domains for feedback linearization of discrete-time control systems.

    • A few words about our experience with the review process

      Details
      This submission received two of the most thorough and supportive reviews my students have received so far. If the anonymous reviewers happen to look at these lines, then please accept our sincere thanks for your supportive and detailed comments.

  • A. Aravind, D. Chatterjee, and A. Cherukuri
    Cross apprenticeship learning: properties and solution approaches.
    IEEE Open Journal of Control Systems, Vol. 2, pp. 36—​48, 2023.

    • doi

    • arXiv version

    • Abstract

      Details
      Apprenticeship learning is a framework in which an agent learns a policy to perform a given task in an environment using example trajectories provided by an expert. In the real world, one might have access to expert trajectories in different environments where the system dynamics is different while the learning task is the same. For such scenarios, two types of learning objectives can be defined. One where the learned policy performs very well in one specific environment and another when it performs well across all environments. To balance these two objectives in a principled way, our work presents the cross apprenticeship learning (CAL) framework. This consists of an optimization problem where an optimal policy for each environment is sought while ensuring that all policies remain close to each other. This nearness is facilitated by one tuning parameter in the optimization problem. We derive properties of the optimizers of the problem as the tuning parameter varies. Since the problem is nonconvex, we provide a convex outer approximation. Finally, we demonstrate the attributes of our framework in the context of a navigation task in a windy gridworld environment.

2022

  • S. Das, A. Aravind, A. Cherukuri, and D. Chatterjee
    Near-optimal solutions of convex semi-infinite programs via targeted sampling.
    Annals of Operations Research, Vol. 318, pp. 129—​146, 2022.

    • doi, pdf

    • Abstract

      Details
      We propose an approach to find the optimal value of a convex semi-infinite program (SIP) that involves identifying a finite set of relevant constraints by solving a finite-dimensional global maximization problem. One of the major advantages of our approach is that it admits a plug-and-play module where any suitable global optimization algorithm can be employed to obtain the optimal value of the SIP. As an example, we propose a simulated annealing based algorithm which is useful especially when the constraint index set is high-dimensional. A proof of convergence of the algorithm is included, and the performance and accuracy of the algorithm itself are illustrated on several benchmark SIPs lifted from the literature.

    • Informal blurb

      Details
      This work establishes a theoretically exact and (to date the only) numerically viable algorithm to solve convex semi-infinite programs exactly. We claimed ‘near-exact’ in its title to stay conservative against numerical errors in the algorithms available today. Convex semi-infinie programs constitute some of the most important problems in robust optimization; in particular, minmax problems with continuous objective functions that are convex in the minimizing variable fall within this class of problems.

  • M. Rayyan Sheriff and D. Chatterjee
    Novel min-max reformulations of linear inverse problems.
    Journal of Machine Learning Research, Vol. 23, Paper No. 28, pp. 1—​46, 2022.

    • pdf

    • arXiv version

    • Abstract

      Details
      In this article, we dwell into the class of so-called ill-posed Linear Inverse Problems (LIP) which simply refer to the task of recovering the entire signal from its relatively few random linear measurements. Such problems arise in a variety of settings with applications ranging from medical image processing, recommender systems, etc. We propose a slightly generalized version of the error constrained linear inverse problem and obtain a novel and equivalent convex-concave min-max reformulation by providing an exposition to its convex geometry. Saddle points of the min-max problem are completely characterized in terms of a solution to the LIP, and vice versa. Applying simple saddle point seeking ascend-descent type algorithms to solve the min-max problems provides novel and simple algorithms to find a solution to the LIP. Moreover, the reformulation of an LIP as the min-max problem provided in this article is crucial in developing methods to solve the dictionary learning problem with almost sure recovery constraints.

    • Informal blurb

      Details
      This is an unusual take on linear inverse problems, and Rayyan’s later studies on the numerical side of the results reported here indicate superior performance relative to most other techniques out there.

  • Y. Kumar, S. Srikant, D. Chatterjee, and M. Nagahara
    Sparse optimal control problems with intermediate constraints: necessary conditions.
    Optimal Control, Applications and Methods, Vol. 43, No. 2, pp. 369—​385, 2022.

    • doi

    • arXiv version

    • Abstract

      Details
      This article treats optimal sparse control problems with multiple constraints defined at intermediate points of the time domain. For such problems with intermediate constraints, we first establish a new Pontryagin maximum principle that provides first order necessary conditions for optimality in such problems. Then we announce and employ a new numerical algorithm to arrive at, in a computationally tractable fashion, optimal state-action trajectories from the necessary conditions given by our maximum principle. Several detailed illustrative examples are included.

  • A. A. Joshi, D. Chatterjee, and R. Banavar
    Robust discrete-time Pontryagin maximum principle on matrix Lie groups.
    IEEE Transactions on Automatic Control, Vol. 67, No. 7, pp. 3545—​3552, 2022.

    • doi

    • arXiv version

    • Abstract

      Details
      This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min–max optimal control wherein the disturbance is the adversary and tries to maximize a cost that the control tries to minimize. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This PMP features a saddle point condition on the Hamiltonian and a set of backward difference equations for the adjoint dynamics. We also present a special case of our result on Euclidean spaces. We conclude with applying the PMP to robust version of single axis rotation of a rigid body.

  • T. Ikeda, M. Nagahara, D. Chatterjee, and S. Srikant
    Constrained smoothing splines by optimal control.
    IEEE Control Systems Letters, Vol. 6, pp. 1298—​1303, 2022.

    • doi

    • Abstract

      Details
      In this letter, we consider the problem of constructing constrained smoothing splines, which are of great importance in data science. The novelty of this work is to formulate the problem as an optimal control problem, and we mathematically analyze the optimal smoothing spline with intermediate constraints using first-order optimality condition from a nonstandard version of the Pontryagin maximum principle. We also propose a novel algorithm to compute the optimal constrained splines based on Newton-Raphson iterative scheme combined with stochastic approximation. A numerical example is shown to illustrate the advantages of the proposed method.

  • S. Kadam, K. S. Phogat, R. Banavar, and D. Chatterjee
    Exact isoholonomic motion of the planar Purcell’s swimmer.
    IEEE Transactions on Automatic Control, Vol. 67, No. 1, pp. 429—​435, 2022.

    • doi

    • arXiv preprint

    • Abstract

      Details
      In this article, we present the discrete-time isoholonomic problem of the planar Purcell’s swimmer and solve it using the discrete-time Pontryagin’s maximum principle. The three-link Purcell’s swimmer is a locomotion system moving in a low Reynolds number environment. The kinematics of the system evolves on a principal fiber bundle. A structure-preserving discrete-time kinematic model of the system is obtained in terms of the local form of a discrete connection. An adapted version of the discrete maximum principle on matrix Lie groups is then employed to come up with the necessary optimality conditions for an optimal transfer from a given initial state while minimizing the mechanical energy expended in the presence of constraints on the controls. These necessary conditions appear as a two-point boundary value problem and are solved using a numerical technique. Results from numerical experiments are presented to illustrate the algorithm and compared with the existing results for a similar case in the literature.

2021

  • C. Maheshwari, S. Srikant, and D. Chatterjee
    Stabilization under round-robin scheduling of control inputs in nonlinear systems.
    Automatica, Vol. 134, 2021.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We study stability of multivariable control-affine nonlinear systems under sparsification of feedback controllers. Sparsification in our context refers to the scheduling of the individual control inputs one at a time in rapid periodic sweeps over the set of control inputs, which corresponds to round-robin scheduling. We prove that if a locally asymptotically stabilizing feedback controller is sparsified via the round-robin scheme and each control action is scaled appropriately, then the corresponding equilibrium of the resulting system is stabilized when the scheduling is sufficiently fast; under mild additional conditions, local asymptotic stabilization of the corresponding equilibrium can also be guaranteed. Moreover, the basin of attraction for the equilibrium of scheduled system also remains same as the original system under sufficiently fast switching. Our technical tools are derived from optimal control theory, and our results also contribute to the literature on the stability of switched systems in the fast switching regime. Illustrative numerical examples depicting several subtle features of our results are included.

  • P. Dey, N. Balachandran, and D. Chatterjee
    Complexity of constrained sensor placement problems for optimal observability.
    Automatica, Vol. 131, 2021.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations. (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed \(k > 2\), the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most \(k\), is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of sensors in order to maximize the number of states of the system that are structurally observable is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a \((1-\frac{1}{e})\)-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly \(400\) states in its linearized model.

  • Mishal Assif P.K., M. Rayyan Sheriff, and D. Chatterjee
    Measure of quality of finite-dimensional linear systems: a frame-theoretic view.
    Systems & Control Letters, Vol. 151, 2021.

    • doi

    • arXiv preprint

    • Abstract

      Details
      A measure of quality of a control system is a quantitative extension of the classical binary notion of controllability. In this article we study the quality of linear control systems from a frame-theoretic perspective. We demonstrate that all LTI systems naturally generate a frame on their state space, and that three standard measures of quality involving the trace, minimum eigenvalue, and the determinant of the controllability Gramian achieve their optimum values when this generated frame is tight. Motivated by this, and in view of some recent developments in frame-theoretic signal processing, we propose a natural measure of quality for continuous time LTI systems based on a measure of tightness of the frame generated by it and then discuss some properties of this frame-theoretic measure of quality.

  • P. K. Mishra, S. Diwale, C. Jones, and D. Chatterjee
    Reference tracking stochastic model predictive control over unreliable channels and bounded control actions.
    Automatica, Vol. 127, 2021.

    • doi

    • Abstract

      Details
      A stochastic model predictive control framework over unreliable Bernoulli communication channels, in the presence of unbounded process noise and under bounded control inputs, is presented for tracking a reference signal. The data losses in the control channel are compensated by a carefully designed transmission protocol, and those of the sensor channel by a dropout compensator. A class of saturated, disturbance feedback policies is proposed for control in the presence of noisy dropout compensation. A reference governor is employed to generate trackable reference trajectories and stability constraints are employed to ensure mean-square boundedness of the reference tracking error. The overall approach yields a computationally tractable quadratic program, which can be iteratively solved online.

  • C. Maheshwari, S. Srikant, and D. Chatterjee
    Optimal multiplexing of discrete time constrained control systems on matrix Lie groups.
    IEEE Transactions on Automatic Control, Vol. 66, No. 4, pp. 1895—​1901, 2021.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We study a constrained optimal control problem for an ensemble of control systems. Each sub-system (or plant) evolves on a matrix Lie group, and must satisfy given state and control action constraints pointwise in time. In addition, certain multiplexing requirement is imposed: the controller must be shared between the plants in the sense that at any time instant the control signal may be sent to only one plant. We provide first-order necessary conditions for optimality in the form of suitable Pontryagin maximum principle in this problem. Detailed numerical experiments are presented for a system of two satellites performing energy optimal maneuvers under the preceding family of constraints.

2020

  • P. Paruchuri, S. Kotpalliwar, K. S. Phogat, D. Chatterjee, and R. Banavar
    A frequency-constrained geometric Pontryagin maximum principle on matrix Lie groups.
    International Journal of Robust and Nonlinear Control, Vol. 30, pp. 6281—​6297, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      In this article we present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We validate our theoretical results with a numerical experiment on the attitude control of a spacecraft on the Lie group \(SO(3)\).

  • K. Albert, K. S. Phogat, F. Anhalt, R. Banavar, D. Chatterjee, and B. Lohmann
    Structure-preserving constrained optimal trajectory planning of a wheeled inverted pendulum.
    IEEE Transactions on Robotics, Vol. 36, No. 3, pp. 910—​923, 2020.

    • doi

    • arXiv preprint

    • Experiment

    • Abstract

      Details
      The Wheeled Inverted Pendulum (WIP) is an underactuated, nonholonomic mechatronic system, and has been popularized commercially as the Segway. Designing a control law for motion planning, that incorporates the state and control constraints, while respecting the configuration manifold, is a challenging problem. In this article we derive a discrete-time model of the WIP system using discrete mechanics and generate optimal trajectories for the WIP system by solving a discrete-time constrained optimal control problem. Further, we describe a nonlinear continuous-time model with parameters for designing a closed loop LQ-controller. A dual control architecture is implemented in which the designed optimal trajectory is then provided as a reference to the robot with the optimal control trajectory as a feedforward control action, and an LQ-controller in the feedback mode is employed to mitigate noise and disturbances for ensuing stable motion of the WIP system. While performing experiments on the WIP system involving aggressive maneuvers with fairly sharp turns, we found a high degree of congruence in the designed optimal trajectories and the path traced by the robot while tracking these trajectories. This corroborates the validity of the nonlinear model and the control scheme. Finally, these experiments demonstrate the highly nonlinear nature of the WIP system and robustness of the control scheme.

  • M. Rayyan Sheriff and D. Chatterjee
    A complete characterization of optimal dictionaries for least squares representation.
    Linear Algebra and Its Applications, Vol. 601, pp. 219—​264, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      Dictionaries are collections of vectors used for representations of elements in Euclidean spaces. While recent research on optimal dictionaries is focussed on providing sparse (i.e., \(\ell_0\)-optimal,) representations, here we consider the problem of finding optimal dictionaries such that representations of samples of a random vector are optimal in an \(\ell_2\)-sense. For us, optimality of representation is equivalent to minimization of the average \(\ell_2\)-norm of the coefficients used to represent the random vector, with the lengths of the dictionary vectors being specified a priori. With the help of recent results on rank-1 decompositions of symmetric positive semidefinite matrices and the theory of majorization, we provide a complete characterization of \(\ell_2\)-optimal dictionaries. Our results are accompanied by polynomial time algorithms that construct \(\ell_2\)-optimal dictionaries from given data.

  • P. K. Mishra, D. Chatterjee, and D. Quevedo
    Stochastic predictive control under intermittent observations and unreliable actions.
    Automatica, Vol. 118, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We propose a provably stabilizing and tractable approach for control of constrained linear systems under intermittent observations and unreliable transmissions of control commands. A smart sensor equipped with a Kalman filter is employed for the estimation of the states from incomplete and corrupt measurements, and an estimator at the controller side optimally feeds the intermittently received sensor data to the controller. The remote controller iteratively solves constrained stochastic optimal control problems and transmits the control commands according to a carefully designed transmission protocol through an unreliable channel. We present a (globally) recursively feasible quadratic program, which is solved online to yield a stabilizing controller for Lyapunov stable linear time invariant systems under any positive bound on control values and any non-zero transmission probabilities of Bernoulli channels.

  • Mishal Assif P.K., D. Chatterjee, and R. Banavar
    Scenario approach for minmax optimization in the nonconvex setting: positive results and caveats.
    SIAM Journal of Optimization, Vol. 30, No. 2, pp. 1119—​1143, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We treat the so-called scenario approach, a popular probabilistic approximation method for robust minmax optimization problems via independent and indentically distributed (i.i.d) sampling from the uncertainty set, from various perspectives. The scenario approach is well-studied in the important case of convex robust optimization problems, and here we examine how the phenomenon of concentration of measures affects the i.i.d sampling aspect of the scenario approach in high dimensions and its relation with the optimal values. Moreover, we perform a detailed study of both the asymptotic behaviour (consistency) and finite time behaviour of the scenario approach in the more general setting of nonconvex minmax optimization problems. In the direction of the asymptotic behaviour of the scenario approach, we present an obstruction to consistency that arises when the decision set is noncompact. In the direction of finite sample guarantees, we establish a general methodology for extracting 'probably approximately correct' type estimates for the finite sample behaviour of the scenario approach for a large class of nonconvex problems.

    • Informal discussion

      Details
      We submitted this article initially to one of our flagship journals in 2019, and there our experience with the reviews was the following: This submission received a desk-reject with the statement that the journal would not publish papers on the scenario approach that have little to do with control per se. Fair enough, we thought, and moved our submission to SIOPT, where it appeared after passing through a standard and supportive round of reviews. With considerable interest I note that several papers focussing on the scenario approach continue to appear in the same flagship journal since 2019.

  • A. Kundu and D. Chatterjee
    Robust matrix commutator conditions for stability of switched linear systems under restricted switching.
    Automatica, Vol. 115, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article treats global uniform exponential stability (GUES) of discrete-time switched linear systems under restricted switching. Given admissible minimum and maximum dwell times, we provide sufficient conditions on the subsystems under which they admit a set of switching signals that obeys the given restrictions on dwell times and preserves stability of the resulting switched system. Our analysis relies on combinatorial arguments applied to matrix commutators and avoids the employment of Lyapunov-like functions. The proposed set of stabilizing switching signals is characterized in terms of duration of activation of Schur stable subsystems and non-consecutive activation of distinct unstable subsystems.

  • Mishal Assif P.K., D. Chatterjee, and R. Banavar
    A simple proof of the discrete time geometric Pontryagin maximum principle.
    Automatica, Vol. 114, 2020.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in Cha11 in the context of continuous-time optimal control.

  • M. Nagahara, D. Chatterjee, N. Challapalli, and M. Vidyasagar
    CLOT norm minimization for continuous-time hands-off control.
    Automatica, Vol. 113, 2020.

    • doi

    • Abstract

      Details
      In this paper, we propose optimal control that is both sparse and continuous, unlike previously proposed alternatives to maximum hands-off control. The maximum hands-off control is the \(L_0\)-optimal (or sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. The proposed control is obtained via minimization of the CLOT (Combined \(L\)-One and Two) norm of the control input along with the constraints on the state variable. The constraints on the state variable ensures that the states are not blown up while achieving the optimal control. By using the non-smooth maximum principle, we prove that the CLOT-norm optimal control is unique, and it is continuous in the time variable. We show by numerical simulations that the CLOT control is continuous unlike \(L_0\)-optimal control (or maximum hands-off control) and much sparser (i.e. has longer time duration on which the control equals 0) than the conventional EN (elastic net) control, which is a convex combination of \(L_1\)- and squared \(L_2\)-norms.

2019

  • S. Kotpalliwar, P. Paruchuri, D. Chatterjee, and R. Banavar
    Discrete time optimal control with frequency constraints for nonsmooth systems.
    Automatica, Vol. 107, pp. 493—​501, 2019.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of the type (b), while less standard in the literature, effectively serve the purpose of describing important spectral properties of inertial actuators and systems. The conjunction of constraints of the type (a) and (b) is a relatively new phenomenon in optimal control but are important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability our theory, and we include several illustrations of our results on standard problems.

  • Y. Kumar, S. Srikant, and D. Chatterjee
    Optimal sparse multiplexing of linear control systems.
    Automatica, Vol. 106, pp. 134—​142, 2019.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article treats three problems of sparse and optimal multiplexing a finite ensemble of linear control systems. Given an ensemble of linear control systems, multiplexing of the controllers consists of an algorithm that selects, at each time \(t\), only one from the ensemble of linear systems is actively controlled whereas the other systems evolve in open-loop. The first problem treated here is a ballistic reachability problem where the control signals are required to be maximally sparse and multiplexed, the second concerns sparse and optimally multiplexed linear quadratic control, and the third is a sparse and optimally multiplexed Mayer problem. Numerical experiments are provided to demonstrate the efficacy of the techniques developed here.

  • P. Paruchuri and D. Chatterjee
    A discrete time Pontryagin maximum principle under state-action-frequency constraints.
    IEEE Transactions on Automatic Control, Vol. 64, No. 10, pp. 4202—​4208, 2019.

    • doi

    • arXiv preprint, poster

    • Abstract

      Details
      We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

  • N. Balachandran, A. Kundu, and D. Chatterjee
    A randomized algorithm for stabilizing switching signals.
    Mathematical Control and Related Fields, Vol. 9, No. 1, pp. 159—​174, 2019.

    • doi, pdf (differs a little from the published version)

    • Abstract

      Details
      Qualitative behaviour of switched systems has attracted considerable research attention in the recent past. In this article we study ‘how likely’ is it for a family of systems to admit stabilizing switching signals. A weighted digraph is associated to a switched system in a natural fashion, and the switching signal is expressed as an infinite walk on this digraph. We provide a linear time probabilistic algorithm to find cycles on this digraph that have a desirable property (we call it “contractivity”), and under mild statistical hypotheses on the connectivity and weights of the digraph, demonstrate that there exist uncountably many stabilizing switching signals derived from such cycles. Our algorithm does not require the vertex and edge weights to be stored in memory prior to its application, has a learning/exploratory character, and naturally fits very large scale systems.

2018

  • K. S. Phogat, D. Chatterjee, and R. Banavar
    A discrete-time Pontryagin maximum principle on matrix Lie groups.
    Automatica, Vol. 97, pp. 376—​391, 2018.

    • doi

    • arXiv preprint

    • Received an Automatica Paper Prize, 2020

    • Experiment based on this work

    • Abstract

      Details
      In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then solved to extract optimal control trajectories. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. In this article we bridge this lacuna and establish a discrete-time PMP on matrix Lie groups. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme,) leading to the preservation of the underlying manifold over time, thereby resulting in greater numerical accuracy of our technique. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences.

  • N. Roy Chowdhury, S. Srikant, and D. Chatterjee
    A new condition for asymptotic consensus over switching graphs.
    Automatica, Vol. 97, pp. 18—​26, 2018.

    • doi

    • Abstract

      Details
      We investigate asymptotic consensus of linear systems under a class of switching communication graphs. We significantly relax several reciprocity and connectivity assumptions prevalent in the consensus literature by employing switched-systems techniques to establish consensus. Our results rely solely on asymptotic properties of the switching communication graphs in contrast to classical average dwell-time conditions. A bound on the uniform rate of convergence to consensus is also established as part of this work.

  • P. K. Mishra, D. Chatterjee, and D. Quevedo
    Stabilizing stochastic predictive control under Bernoulli dropouts.
    IEEE Transactions on Automatic Control, Vol. 63, No. 6, pp. 1489—​1500, 2018.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article presents tractable and recursively feasible optimization-based controllers for stochastic linear systems with bounded controls. The stochastic noise in the plant is assumed to be additive, zero mean and fourth moment bounded, and the control values transmitted over an erasure channel. Three different transmission protocols are proposed having different requirements on the storage and computational facilities available at the actuator. We optimize a suitable stochastic cost function accounting for the effects of both the stochastic noise and the packet dropouts over affine saturated disturbance feedback policies. The proposed controllers ensure mean square boundedness of the states in closed-loop for all positive values of control bounds and any non-zero probability of successful transmission over a noisy control channel.

  • K. S. Phogat, D. Chatterjee, and R. Banavar
    Discrete-time optimal attitude control of a spacecraft with momentum and control constraints.
    Journal of Guidance, Control, and Dynamics, Vol. 41, No. 1, pp. 199—​211, 2018.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article solves an optimal control problem arising in attitude control of a spacecraft under state and control constraints. We first derive the discrete-time attitude dynamics by employing discrete mechanics. The orientation transfer, with initial and final values of the orientation and momentum and the time duration being specified, is posed as an energy optimal control problem in discrete-time subject to momentum and control constraints. Using variational analysis directly on the Lie group SO(3), we derive first order necessary conditions for optimality that leads to a constrained two point boundary value problem. This two point boundary value problem is solved via a novel multiple shooting technique that employs a root finding Newton algorithm. Robustness of the multiple shooting technique is demonstrated through a few representative numerical experiments.

  • P. K. Mishra, D. Chatterjee, and D. Quevedo
    Sparse and constrained stochastic predictive control for networked systems.
    Automatica, Vol. 87, pp. 40—​51, 2018.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. The control channel is assumed to have i.i.d. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. We further consider a regularization term in a quadratic performance index to promote sparsity in control. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission.

2017

  • M. Rayyan Sheriff and D. Chatterjee
    Optimal dictionary for least squares representation.
    Journal of Machine Learning Research, Vol. 18, Paper No. 107, pp. 1—​28, 2017.

    • pdf

    • arXiv preprint

    • Abstract

      Details
      Dictionaries are collections of vectors used for the representation of a class of vectors in Euclidean spaces. Recent research on optimal dictionaries is focused on constructing dictionaries that offer sparse representations, i.e., \(\ell_0\)-optimal representations. Here we consider the problem of finding optimal dictionaries with which representations of a given class of vectors is optimal in an \(\ell_2\)-sense: optimality of representation is defined as attaining the minimal average \(\ell_2\)-norm of the coefficients used to represent the vectors in the given class. With the help of recent results on rank-1 decompositions of symmetric positive semidefinite matrices, we provide an explicit description of \(\ell_2\)-optimal dictionaries as well as their algorithmic constructions in polynomial time.

  • P. K. Mishra, D. Chatterjee, and D. Quevedo
    Output feedback stochastic predictive control with hard input constraints.
    IEEE Systems & Control Letters, Vol. 1, No. 2, pp. 382—​387, 2017.

    • doi

    • arXiv preprint (contains corrections to typos)

    • Abstract

      Details
      We present a stochastic predictive controller for discrete time linear time invariant systems under incomplete state information. Our approach is based on a suitable choice of control policies, stability constraints, and employment of a Kalman filter to estimate the states of the system from incomplete and corrupt observations. We demonstrate that this approach yields a computationally tractable problem that should be solved online periodically, and that the resulting closed loop system is mean-square bounded for any positive bound on the control actions. Our results allow one to tackle the largest class of linear time invariant systems known to be amenable to stochastic stabilization under bounded control actions via output feedback stochastic predictive control.

  • A. Kundu and D. Chatterjee
    Stabilizing switching signals: a transition from point-wise to asymptotic conditions.
    Systems & Control Letters, Vol. 106, pp. 16—​23, 2017.

    • doi

    • arXiv preprint

    • Abstract

      Details
      Characterization of classes of switching signals that ensure stability of switched systems occupies a significant portion of the switched systems literature. This article collects a multitude of stabilizing switching signals under an umbrella framework. We achieve this in two steps: Firstly, given a family of systems, possibly containing unstable dynamics, we propose a new and general class of stabilizing switching signals. Secondly, we demonstrate that prior results based on both point-wise and asymptotic characterizations follow our result. This is the first attempt in the switched systems literature where these switching signals are unified under one banner.

  • P. K. Mishra, D. Chatterjee, and D. Quevedo
    Resource efficient stochastic predictive control under packet dropouts.
    IET Control Theory & Applications, Vol. 11, No. 11, pp. 1666—​1673, 2017.

    • doi

    • Abstract

      Details
      This study presents a resource efficient framework for a class of stochastic control systems that utilises state dependent control strategies in order to reduce the online computational load. When the states are in a given neighbourhood of the desired operating point, the controller is switched off, and data from the feedback channel is not transmitted. Outside this neighbourhood, the authors pay close attention to the performance of the controller by adopting a stochastic predictive algorithm when the states are in a predefined comfort zone, and activate a recovery algorithm beyond the comfort zone that secures at least good qualitative properties. The authors demonstrate that the proposed controller leads to mean square boundedness of the closed loop states in the presence of stochastic noise, bounded control authority, and control channel erasures, while entailing a dramatic reduction in network traffic and computational resources.

  • A. Kundu and D. Chatterjee
    On stability of discrete-time switched systems.
    Nonlinear Analysis: Hybrid Systems, Vol. 23, pp. 191—​210, 2017.

    • doi

    • Abstract

      Details
      This article deals with stability of discrete-time switched systems. Given a family of nonlinear systems and the admissible switches among the systems in the family, we first propose a class of switching signals under which the resulting switched system is globally asymptotically stable. We allow unstable systems in the family and our stability condition depends solely on asymptotic behaviour of the switching signals. We then discuss algorithmic construction of the above class of switching signals, and show that in the presence of exogenous inputs and outputs, a switching signal so constructed also ensures input/output-to-state stability for discrete-time switched nonlinear systems. We finally show that our class of switching signals that ensures global asymptotic stability also extends to the continuous-time setting with minor modifications under standard assumptions. We employ multiple Lyapunov-like functions and graph theoretic tools as the main apparatuses for our analysis.

2016

  • S. Srikant and D. Chatterjee
    A jammer’s perspective of reachability and LQ optimal control.
    Automatica, Vol. 70, pp. 295—​302, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article treats two problems dealing with control of linear systems in the presence of a jammer that can sporadically turn off the control signal. The first problem treats the standard reachability problem, and the second treats the standard linear quadratic regulator problem under the above class of jamming signals. We provide necessary and sufficient conditions for optimality based on a nonsmooth Pontryagin maximum principle.

  • A. Kundu and D. Chatterjee
    A graph theoretic approach to input-to-state stability of switched systems.
    European Journal of Control, Vol. 29, No. 5, pp. 44—​50, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article deals with input-to-state stability (ISS) of discrete-time switched systems. Given a family of nonlinear systems with exogenous inputs, we present a class of switching signals under which the resulting switched system is ISS. We allow non-ISS systems in the family and our analysis involves graph-theoretic arguments. A weighted digraph is associated to the switched system, and a switching signal is expressed as an infinite walk on this digraph, both in a natural way. Our class of stabilizing switching signals (infinite walks) is periodic in nature and affords simple algorithmic construction.

  • D. Chatterjee, M. Nagahara, D. Quevedo, and K. S. Mallikarjuna Rao
    Characterization of maximum hands-off control.
    Systems & Control Letters, Vol. 94, pp. 31—​36, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      Maximum hands-off control aims to maximize the length of time over which zero actuator values are applied to a system when executing specified control tasks. To tackle such problems, recent literature has investigated optimal control problems which penalize the size of the support of the control function and thereby lead to desired sparsity properties. This article gives the exact set of necessary conditions for a maximum hands-off optimal control problem using an \(L_0\)-(semi)norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an \(L_0\) cost leads to a sparse control, whereas an \(L_1\)-relaxation in singular problems leads to a non-sparse solution.

  • P. Mohajerin Esfahani, D. Chatterjee, and J. Lygeros
    On a stochastic reach-avoid problem and set characterization for diffusions.
    Automatica, Vol. 70, pp. 43—​56, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost-sure stochastic requirements. In contrast, we propose a methodology to tackle problems with less stringent requirements than almost sure. To this end, we first establish a connection between two distinct stochastic reach-avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions. Subsequently, we focus on solutions of one of the classes of stochastic optimal control problems---the exit-time problem, which solves both the two reach-avoid problems mentioned above. We then derive a weak version of a dynamic programming principle (DPP) for the corresponding value function; in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions. Moreover, based on our DPP, we provide an alternative characterization of the value function as a solution of a partial differential equation in the sense of discontinuous viscosity solutions, along with boundary conditions both in Dirichlet and viscosity senses. Theoretical justifications are also discussed to pave the way for deployment of off-the-shelf PDE solvers for numerical computations. Finally, we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.

  • A. Kundu, D. Chatterjee, and D. Liberzon
    Generalized switching signals for input-to-state stability of switched systems.
    Automatica, Vol. 64, No. 2, pp. 270—​277, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article deals with input-to-state stability (ISS) of continuous-time switched nonlinear systems. Given a family of systems with exogenous inputs such that not all systems in the family are ISS, we characterize a new and general class of switching signals under which the resulting switched system is ISS. Our stabilizing switching signals allow the number of switches to grow faster than an affine function of the length of a time interval, unlike in the case of average dwell time switching. We also recast a subclass of average dwell time switching signals in our setting and establish analogs of two representative prior results.

  • P. Mohajerin Esfahani, D. Chatterjee, and J. Lygeros
    Motion planning via optimal control for stochastic processes.
    IEEE Transactions on Automatic Control, Vol. 61, No. 8, pp. 2155—​2170, 2016.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We study stochastic motion planning problems which involve a controlled process, with possibly discontinuous sample paths, visiting certain subsets of the state-space while avoiding others in a sequential fashion. For this purpose, we first introduce two basic notions of motion planning, and then establish a connection to a class of stochastic optimal control problems concerned with sequential stopping times. A weak dynamic programming principle (DPP) is then proposed, which characterizes the set of initial states that admit a control enabling the process to execute the desired maneuver with probability no less than some pre-specified value. The proposed DPP comprises auxiliary value functions defined in terms of discontinuous payoff functions. A concrete instance of the use of this novel DPP in the case of diffusion processes is also presented. In this case, we establish that the aforementioned set of initial states can be characterized as the level set of a discontinuous viscosity solution to a sequence of partial differential equations, for which the first one has a known boundary condition, while the boundary conditions of the subsequent ones are determined by the solutions to the preceding steps. Finally, the generality and flexibility of the theoretical results are illustrated on an example involving biological switches.

2015

  • A. Kundu and D. Chatterjee
    Stabilizing switching signals for switched systems.
    IEEE Transactions on Automatic Control, Vol. 60, No. 3, pp. 882—​888, 2015.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article deals with stability of continuous-time switched linear systems under constrained switching. Given a family of linear systems, possibly containing unstable dynamics, we characterize a new class of switching signals under which the switched linear system generated by it and the family of systems is globally asymptotically stable. Our characterization of such stabilizing switching signals involves the asymptotic frequency of switching, the asymptotic fraction of activation of the constituent systems, and the asymptotic densities of admissible transitions among them. Our techniques employ multiple Lyapunov-like functions, and extend preceding results both in scope and applicability.

  • D. Chatterjee and J. Lygeros
    Stability and performance of stochastic receding horizon control.
    IEEE Transactions on Automatic Control, Vol. 60, No. 2, pp. 509—​514, 2015.

    • doi

    • arXiv preprint (contains more material compared to the published version)

    • Abstract

      Details
      This article is concerned with stability and performance of controlled stochastic processes under receding horizon policies. We carry out a systematic study of methods to guarantee stability under receding horizon policies via appropriate selections of cost functions in the underlying finite-horizon optimal control problem. We also obtain quantitative bounds on the performance of the system under receding horizon policies as measured by the long-run expected average cost. The results are illustrated with the help of several simple examples.

2013

  • T. Sutter, D. Chatterjee, F. Ramponi, and J. Lygeros
    Isospectral flows on a class of finite-dimensional Jacobi matrices.
    Systems & Control Letters, Vol. 62, No. 5, pp. 388—​394, 2013.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We present a new matrix-valued isospectral ordinary differential equation that asymptotically block-diagonalizes \(n\times n\) zero-diagonal Jacobi matrices employed as its initial condition. This o.d.e.\ features a right-hand side with a nested commutator of matrices, and structurally resembles the double-bracket o.d.e.\ studied by R.W.\ Brockett in 1991. We prove that its solutions converge asymptotically, that the limit is block-diagonal, and above all, that the limit matrix is defined uniquely as follows: For \(n\) even, a block-diagonal matrix containing \(2\times 2\) blocks, such that the super-diagonal entries are sorted by strictly increasing absolute value. Furthermore, the off-diagonal entries in these \(2\times 2\) blocks have the same sign as the respective entries in the matrix employed as initial condition. For \(n\) odd, there is one additional \(1\times 1\) block containing a zero that is the top left entry of the limit matrix. The results presented here extend some early work of Kac and van Moerbeke.

  • D. Chatterjee, P. Hokayem, F. Ramponi, and J. Lygeros
    On mean-square boundedness of stochastic linear systems with quantized observations.
    IEEE Transactions on Automatic Control, Vol. 58, No. 8, pp. 2082—​2085, 2013.

    • doi

    • Abstract

      Details
      We propose a procedure to design a state-quantizer with a fixed finite alphabet for a Lyapunov stable stochastic linear system, and a bounded policy based on the resulting quantized state measurements to ensure bounded second moments of the states in closed-loop.

2012

  • P. Hokayem, D. Chatterjee, F. Ramponi, and J. Lygeros
    Stable networked control systems with bounded control authority.
    IEEE Transactions on Automatic Control, Vol. 57, No. 12, pp. 3153—​3157, 2012.

    • doi

    • Abstract

      Details
      We study the stability of a class of networked control systems with hard bounds on the control authority. The plant dynamics are discrete-time, linear, and time-invariant, with stochastic process noise and measurement noise. The controller is designed as a norm-bounded causal history-dependent function of the past outputs perturbed by bounded noise. The resulting control signals are assumed to be transmitted through a lossy channel with packet dropouts. We show that under mild assumptions on the system matrices, the statistics of the process and measurement noise sequences, and the probability of dropouts, it is possible to ensure bounded variance of the system in closed-loop.

  • D. Chatterjee, F. Ramponi, P. Hokayem, and J. Lygeros
    On mean square boundedness of stochastic linear systems with bounded controls.
    Systems & Control Letters, Vol. 61, No. 2, pp. 375—​380, 2012.

    • doi

    • Abstract

      Details
      We start by summarizing the state of the art in stabilization of stochastic linear systems with bounded inputs and highlight remaining open problems. We then report two new results concerning mean-square boundedness of a linear system with additive stochastic noise. The first states that, given any nonzero bound on the controls, it is possible to construct a policy with bounded memory requirements that renders a marginally stable stabilizable system mean-square bounded in closed-loop. The second states that it is not possible to ensure mean-square boundedness in closed-loop with a bounded control policy for systems affected by unbounded noise and having at least one eigenvalue outside the unit circle.

  • P. Hokayem, E. Cinquemani, D. Chatterjee, F. Ramponi, and J. Lygeros
    Stochastic MPC with output feedback and bounded control inputs.
    Automatica, Vol. 48, No. 1, pp. 77—​88, 2012.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We provide a solution to the problem of receding horizon control for stochastic discrete-time systems with bounded control inputs and imperfect state measurements. For a suitable choice of control policies, we show that the finite-horizon optimization problem to be solved on-line is convex and successively feasible. Due to the inherent nonlinearity of the feedback loop, a slight extension of the Kalman filter is exploited to estimate the state optimally in mean-square sense. We show that the receding horizon implementation of the resulting control policies renders the state of the overall system mean-square bounded under mild assumptions. Finally, we discuss how some of the quantities required by the finite-horizon optimization problem can be computed off-line, reducing the on-line computation, and present some numerical examples.

2011

  • D. Chatterjee and S. Pal
    An excursion-theoretic view of stability of stochastic hybrid systems.
    Applied Mathematics and Optimization, Vol. 63, No. 2, pp. 217—​237, 2011.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for \(L_1\)-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in \(L_1\). The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.

  • D. Chatterjee, E. Cinquemani, and J. Lygeros
    Maximizing the probability of attaining a target before extinction.
    Nonlinear Analysis: Hybrid Systems, Vol. 5, No. 2, pp. 367—​381, 2011.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We present a dynamic programming-based solution to the problem of maximizing the probability of attaining a target set before hitting a cemetery set for a discrete-time Markov control process. Under mild hypotheses we establish that there exists a deterministic stationary policy that achieves the maximum value of this probability. We demonstrate how the maximization of this probability can be computed through the maximization of an expected total reward until the first hitting time to either the target or the cemetery set. Martingale characterizations of thrifty, equalizing, and optimal policies in the context of our problem are also established.

  • D. Chatterjee, P. Hokayem, and J. Lygeros
    Stochastic model predictive control with bounded control inputs: a vector space approach.
    IEEE Transactions on Automatic Control, Vol. 56, No. 11, pp. 2704—​2710, 2011.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We design receding horizon control strategies for stochastic discrete-time linear systems with additive (possibly) unbounded disturbances, while obeying hard bounds on the control inputs. We pose the problem of selecting an appropriate optimal controller on vector spaces of functions and show that the resulting optimization problem has a tractable convex solution. Under the assumption that the zero-input and zero-noise system is asymptotically stable, we show that the variance of the state is bounded when enforcing hard bounds on the control inputs, for any receding horizon implementation. Throughout the article we provide several examples that illustrate how quantities needed in the formulation of the resulting optimization problems can be calculated off-line, as well as comparative examples that illustrate the effectiveness of our control strategies.

  • E. Cinquemani, M. Agarwal, D. Chatterjee, and J. Lygeros
    Convexity and convex approximations of discrete-time stochastic control problems with constraints.
    Automatica, Vol. 47, No. 9, pp. 2082—​2087, 2011.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost over a finite horizon. Hard constraints are introduced first, and then reformulated in terms of probabilistic constraints. It is shown that, for a suitable parametrization of the control policy, a wide class of the resulting optimization problems are convex, or admit reasonable convex approximations.

  • D. Chatterjee and D. Liberzon
    Stabilizing randomly switched systems.
    SIAM Journal on Control and Optimization, Vol. 49, No. 5, pp. 2008—​2031, 2011.

    • doi

    • arXiv preprint

    • Abstract

      Details
      This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design.

2010

  • F. Ramponi, D. Chatterjee, A. Milias-Argeitis, P. Hokayem, and J. Lygeros
    Attaining mean-square boundedness of a marginally stable stochastic linear system with bounded inputs.
    IEEE Transactions on Automatic Control, Vol. 55, pp. 2414—​2418, 2010.

    • doi

    • arXiv preprint

    • Abstract

      Details
      We construct control policies that ensure bounded variance of a noisy marginally stable linear system in closed-loop. It is assumed that the noise sequence is a mutually independent sequence of random vectors, enters the dynamics affinely, and has bounded fourth moment. The magnitude of the control is required to be of the order of the first moment of the noise, and the policies we obtain are simple and computable.

2007

  • D. Chatterjee and D. Liberzon
    On stability of randomly switched systems.
    IEEE Transactions on Automatic Control, Vol. 52, No. 12, pp. 2390—​2394, 2007.

    • doi

    • Abstract

      Details
      This note is concerned with stability analysis and stabilization of randomly switched systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure stability and stability in the mean using Lyapunov-based methods when individual subsystems are stable and a certain ldquoslow switchingrdquo condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results, therefore, hold for Markovian jump systems in particular. For systems with control inputs, we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems.

  • L. Vu, D. Chatterjee, and D. Liberzon
    Input-to-state stability of switched systems and switching adaptive control.
    Automatica, Vol. 43, No. 4, pp. 639—​646, 2007.

    • doi

    • Abstract

      Details
      In this paper we prove that a switched nonlinear system has several useful input-to-state stable (ISS)-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.

2006

  • D. Chatterjee and D. Liberzon
    Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions.
    SIAM Journal on Control and Optimization, Vol. 45, No. 1, pp. 174—​206, 2006.

    • doi

    • Abstract

      Details
      This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching.

2002

  • D. Chatterjee, A. Patra, and H. K. Joglekar
    Swing-up and stabilization of a cart-pendulum system under restricted cart track length.
    Systems & Control Letters, Vol. 47, No. 4, pp. 353—​362, 2002.

    • doi

    • Abstract

      Details
      This paper describes the swing-up and stabilization of a cart–pendulum system with a restricted cart track length and restricted control force using generalized energy control methods. Starting from a pendant position, the pendulum is swung up to the upright unstable equilibrium configuration using energy control principles. An “energy well” is built within the cart track to prevent the cart from going outside the limited length. When sufficient energy is acquired by the pendulum, it goes into a “cruise” mode when the acquired energy is maintained. Finally, when the pendulum is close to the upright configuration, a stabilizing controller is activated around a linear zone about the upright configuration. The proposed scheme has worked well both in simulation and a practical setup and the conditions for stability have been derived using the multiple Lyapunov functions approach.