Farhad Farokhi

Distributed MPC Via Dual Decomposition and Alternative Direction Method of Multipliers

A conventional way to handle model predictive control (MPC) problems distributedly is to solve them via dual decomposition and gradient ascent. However, at each time-step, it might not be feasible to wait for the dual algorithm to converge. As a result, the algorithm might be needed to be terminated prematurely. One is then interested to see if the solution at the point of termination is close to the optimal solution and when one should terminate the algorithm if a certain distance to optimality is to be guaranteed. In this chapter, we look at this problem for distributed systems under general dynamical and performance couplings, then, we make a statement on validity of similar results where the problem is solved using alternating direction method of multipliers.

Optimal structured static state-feedback control design with limited model information for fully-actuated systems

We introduce the family of limited model information control design methods, which construct controllers by accessing the plants model in a constrained way, according to a given design graph. We investigate the closed-loop performance achievable by such control design methods for fully-actuated discrete-time linear time-invariant systems, under a separable quadratic cost. We restrict our study to control design methods which produce structured static state feedback controllers, where each subcontroller can at least access the state measurements of those subsystems that affect its corresponding subsystem. We compute the optimal control design strategy (in terms of the competitive ratio and domination metrics) when the control designer has access to the local model information and the global interconnection structure of the plant-to-be-controlled. Finally, we study the trade-off between the amount of model information exploited by a control design method and the best closed-loop performance (in terms of the competitive ratio) of controllers it can produce.

Stochastic Sensor Scheduling for Networked Control Systems

Optimal sensor scheduling with applications to networked estimation and control systems is considered. We model sensor measurement and transmission instances using jumps between states of a continuous-time Markov chain. We introduce a cost function for this Markov chain as the summation of terms depending on the average sampling frequencies of the subsystems and the effort needed for changing the parameters of the underlying Markov chain. By minimizing this cost function through extending Brocketts recent approach to optimal control of Markov chains, we extract an optimal scheduling policy to fairly allocate the network resources among the control loops. We study the statistical properties of this scheduling policy in order to compute upper bounds for the closed-loop performance of the networked system, where several decoupled scalar subsystems are connected to their corresponding estimator or controller through a shared communication medium. We generalize the estimation results to observable subsystems of arbitrary order. Finally, we illustrate the developed results numerically on a networked system composed of several decoupled water tanks.

Estimation With Strategic Sensors

We introduce a model of estimation in the presence of strategic, self-interested sensors. We employ a game-theoretic setup to model the interaction between the sensors and the receiver. The cost function of the receiver is equal to the estimation error variance while the cost function of the sensor contains an extra term which is determined by its private information. We start by the single sensor case in which the receiver has access to a noisy but honest side information in addition to the message transmitted by a strategic sensor. We study both static and dynamic estimation problems. For both these problems, we characterize a family of equilibria in which the sensor and the receiver employ simple strategies. Interestingly, for the dynamic estimation problem, we find an equilibrium for which the strategic sensor uses a memory-less policy. We generalize the static estimation setup to multiple sensors with synchronous communication structure (i.e., all the sensors transmit their messages simultaneously). We prove the maybe surprising fact that, for the constructed equilibrium in affine strategies, the estimation quality degrades as the number of sensors increases. However, if the sensors are herding (i.e., copying each other policies), the quality of the receiver’s estimation improves as the number of sensors increases. Finally, we consider the asynchronous communication structure (i.e., the sensors transmit their messages sequentially).

A game-theoretic framework for studying truck platooning incentives

An atomic congestion game with two types of agents, cars and trucks, is used to model the traffic flow on a road over certain time intervals. In this game, the drivers make a trade-off between the time they choose to use the road, the average velocity of the flow at that time, and the dynamic congestion tax that they are paying to use the road. The trucks have platooning capabilities and therefore, have an incentive for using the road at the same time as their peers. The dynamics and equilibria of this game-theoretic model for the interaction between car traffic and truck platooning incentives are investigated. We use traffic data from Stockholm to validate the modeling assumptions and extract reasonable parameters for the simulations. We perform a comprehensive simulation study to understand the influence of various factors, such as the percentage of the trucks that are equipped with platooning devices on the properties of the pure strategy Nash equilibrium that is learned using a joint strategy fictitious play.

Control design with limited model information

We introduce the family of limited model information control design methods, which construct controllers by accessing the plants model in a constrained way, according to a given design graph. This class generalizes the notion of communication-less control design methods recently introduced by one of the authors, which construct each sub-controller using only local plant model information. We study the trade off between the amount of model information exploited by a control design method and the quality of controllers it can produce. In particular, we quantify the benefit (in terms of the competitive ratio and domination metrics) of giving the control designer access to the global interconnection structure of the plant-to-be-controlled, in addition to local model information.

Dynamic control design based on limited model information

The design of optimal H2 dynamic controllers for interconnected linear systems using limited plant model information is considered. Control design strategies based on various degrees of model information are compared using the competitive ratio as a performance metric, that is, the worst case control performance for a given design strategy normalized with the optimal control performance based on full model information. An explicit minimizer of the competitive ratio is found. It is shown that this control design strategy is not dominated by any other strategy with the same amount of model information. The result applies to a class of system interconnections and design information characterized through given plant, control, and design graphs.

A faithful distributed implementation of dual decomposition and average consensus algorithms

We consider large scale cost allocation problems and consensus seeking problems for multiple agents in which agents are suggested to collaborate in a distributed algorithm to find a solution. If agents are strategic and minimize their own individual cost rather than the global social cost, they are endowed with an incentive not to follow the intended algorithm, unless the tax/subsidy mechanism is carefully designed. Inspired by the classical Vickrey-Clarke-Groves mechanism and more recent algorithmic mechanism design theory, we propose a tax mechanism that incentivises agents to faithfully implement the intended algorithm. In particular, a new notion of asymptotic incentive compatibility is introduced to characterize a desirable property of such class of mechanisms. The proposed class of tax mechanisms provides a sequence of mechanisms that gives agents a diminishing incentive to deviate from suggested algorithm.

Optimal Control Design Under Limited Model Information for Discrete-Time Linear Systems With Stochastically-Varying Parameters

The value of plant model information available in the control design process is discussed. We design optimal state-feedback controllers for interconnected discrete-time linear systems with stochastically-varying parameters. The parameters are assumed to be independently and identically distributed random variables in time. The design of each controller relies only on (i) exact local plant model information and (ii) statistical beliefs about the model of the rest of the system. We consider both finite-horizon and infinite-horizon quadratic cost functions. The optimal state-feedback controller is derived in both cases. The optimal controller is shown to be linear in the state and to depend on the model parameters and their statistics in a particular way. Furthermore, we study the value of model information in optimal control design using the performance degradation ratio which is defined as the supremum (over all possible initial conditions) of the ratio of the cost of the optimal controller with limited model information scaled by the cost of the optimal controller with full model information. An upper bound for the performance degradation ratio is presented for the case of fully-actuated subsystems. Comparisons are made between designs based on limited, statistical, and full model information. Throughout the paper, we use a power network example to illustrate concepts and results.

Gaussian cheap talk game with quadratic cost functions: When herding between strategic senders is a virtue

We consider a Gaussian cheap talk game with quadratic cost functions. The cost function of the receiver is equal to the estimation error variance, however, the cost function of each senders contains an extra term which is captured by its private information. Following the cheap talk literature, we model this problem as a game with asymmetric information. We start by the single sender case in which the receiver also has access to a noisy but honest side information in addition to the message transmitted by a strategic sender. We generalize this setup to multiple sender case. For the multiple sender case, we observe that if the senders are not herding (i.e., copying each other policies), the quality of the receivers estimation degrades rapidly as the number of senders increases.