arXiv subjects

Debasish Chatterjee

Publications and source records attributed to Debasish Chatterjee.

At least 37 records · Page 2Linked to original sources

arXiv2023

A discrete-time Pontryagin maximum principle under rate constraints

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.

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arXiv2022

Cross apprenticeship learning framework: Properties and solution approaches

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.

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arXiv2022

Moment stability of stochastic processes with applications to control systems

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.

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arXiv2021

Efficient constrained sensor placement for observability of linear systems

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.

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arXiv2021

Maintaining Ferment: On Opinion Control Over Social Networks

We consider the design of external inputs to achieve a control objective on the opinions, represented by scalars, in a social network. The opinion dynamics follow a variant of the discrete-time Friedkin-Johnsen model. We first consider two minimum cost optimal control problems over a finite interval $(T_0,T),$ $T_0 >0$ -- (1) TF where opinions at all nodes should exceed a given $τ,$ and (2) GF where a scalar function of the opinion vector should exceed a given $τ.$ For both problems we first provide a Pontryagin maximum principle (PMP) based control function when the controllable nodes are specified. We then show that both these problems exhibit the turnpike property where both the control function and the state vectors stay near their equilibrium for a large fraction of the time. This property is then used to choose the optimum set of controllable nodes. We then consider a third system, MF, which is a cost-constrained optimal control problem where we maximize the minimum value of a scalar function of the opinion vector over $(T_0,T).$ We provide a numerical algorithm to derive the control function for this problem using non-smooth PMP based techniques. Extensive numerical studies illustrate the three models, control techniques and corresponding outcomes.

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arXiv2020

Reference tracking stochastic model predictive control over unreliable channels and bounded control actions

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 that 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.

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arXiv2020

Necessary condition for sparse optimal control problem with intermediate constraints

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.

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arXiv2020

Stabilization under round robin scheduling of control inputs in nonlinear systems

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.

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arXiv2020

Robust Discrete-Time Pontryagin Maximum Principle on Matrix Lie Groups

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 maximise a cost that the control tries to minimise. 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.

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arXiv2020

Dictionary Learning with Almost Sure Error Constraints

A dictionary is a database of standard vectors, so that other vectors / signals are expressed as linear combinations of dictionary vectors, and the task of learning a dictionary for a given data is to find a good dictionary so that the representation of data points has desirable features. Dictionary learning and the related matrix factorization methods have gained significant prominence recently due to their applications in Wide variety of fields like machine learning, signal processing, statistics etc. In this article we study the dictionary learning problem for achieving desirable features in the representation of a given data with almost sure recovery constraints. We impose the constraint that every sample is reconstructed properly to within a predefined threshold. This problem formulation is more challenging than the conventional dictionary learning, which is done by minimizing a regularised cost function. We make use of the duality results for linear inverse problems to obtain an equivalent reformulation in the form of a convex-concave min-max problem. The resulting min-max problem is then solved using gradient descent-ascent like algorithms.

cs.LG↗

arXiv2020

Novel min-max reformulations of Linear Inverse Problems

In this article, we dwell into the class of so-called ill-posed Linear Inverse Problems (LIP) which simply refers 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.

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arXiv2020

Stochastic Predictive Control under Intermittent Observations and Unreliable Actions

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.

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arXiv2020

On Convex Duality in Linear Inverse Problems

In this article we dwell into the class of so called ill posed Linear Inverse Problems (LIP) in machine learning, which has become almost a classic in recent times. The fundamental task in an LIP is to recover the entire signal / data from its relatively few random linear measurements. Such problems arise in variety of settings with applications ranging from medical image processing, recommender systems etc. We provide an exposition to the convex duality of the linear inverse problems, and obtain a novel and equivalent convex-concave min-max reformulation that gives rise to simple ascend-descent type algorithms to solve an LIP. Moreover, such a reformulation is crucial in developing methods to solve the dictionary learning problem with almost sure recovery constraints.

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arXiv2019

On optimal multiplexing of an ensemble of discrete-time constrained control systems on matrix Lie groups

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.

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arXiv2019

Structure-Preserving Constrained Optimal Trajectory Planning of a Wheeled Inverted Pendulum

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.

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arXiv2019

On Minimum Cost Sparsest Input-Connectivity for Controllability of Linear Systems

We deal with algorithmic techniques for minimal cost input-connectivity while maintaining controllability of linear systems. The input matrix is assumed to be constrained in the sense that the set of states that each input (if present) can influence is known a priori, and that each interconnection between an input and a state is associated with a certain cost. In this setting we determine a set of input-connections that lead to the minimum cost and ensures that the resulting system is structurally controllable. We also identify a sparsest set of input-connections with minimum cost while maintaining structural controllability of the system. A large class of systems are identified for which these problems are solvable in polynomial time using efficient algorithms. A 2-approximation solution is presented for the general case. Graph-theoretic tools are employed to tackle the above class of constrained design problems. Illustrative examples are included to demonstrate the efficacy of the techniques developed here.

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arXiv2019

Robust matrix commutator conditions for stability of switched linear systems under restricted switching

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.

eess.SY↗

arXiv2019

Scenario approach for minmax optimization with emphasis on the nonconvex case: positive results and caveats

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.

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