This paper studies social optimal control of mean field LQG (linear-quadratic-Gaussian) models with uncertainty. Specially, the uncertainty is represented by a uncertain drift which is common for all agents. A robust optimization approach is applied by assuming all agents treat the uncertain drift as an adversarial player. In our model, both dynamics and costs of agents are coupled by mean field terms, and both finite- and infinite-time horizon cases are considered. By examining social functional variation and exploiting person-by-person optimality principle, we construct an auxiliary control problem for the generic agent via a class of forward-backward stochastic differential equation system. By solving the auxiliary problem and constructing consistent mean field approximation, a set of decentralized control strategies is designed and shown to be asymptotically optimal.
Continuing the first part of the paper, we consider scalar decentralized average-cost infinite-horizon LQG problems with two controllers. This paper focuses on the slow dynamics case when the eigenvalue of the system is small and prove that the single-controller optimal strategies ---linear strategies--- are constant ratio optimal among all distributed control strategies.
We consider scalar decentralized average-cost infinite-horizon LQG problems with two controllers, focusing on the fast dynamics case when the (scalar) eigenvalue of the system is large. It is shown that the best linear controllers' performance can be an arbitrary factor worse than the optimal performance. We propose a set of finite-dimensional nonlinear controllers, and prove that the proposed set contains an easy-to-find approximately optimal solution that achieves within a constant ratio of the optimal quadratic cost. The insight for nonlinear strategies comes from revealing the relationship…
Here we will provide the proofs of the results stated in the Infinite Horizon LQG Control section of [1] by focusing on the control law and the related MARE. The analysis of the observation case can be achieved in a dual way and it is partially covered by [2].
Distributed control problems under some specific information constraints can be formulated as (possibly infinite dimensional) convex optimization problems. The underlying motivation of this work is to develop an understanding of the optimal decision making architecture for such problems. In this paper, we particularly focus on the N-player triangular LQG problems and show that the optimal output feedback controllers have attractive state space realizations. The optimal controller can be synthesized using a set of stabilizing solutions to 2N linearly coupled algebraic Riccati equations, which t…
In this paper, our goal is to study fundamental foundations of linear quadratic Gaussian (LQG) control problems for stochastic linear time-invariant systems via Lagrangian duality of semidefinite programming (SDP) problems. In particular, we derive an SDP formulation of the finite-horizon LQG problem, and its Lagrangian duality. Moreover, we prove that Riccati equation for LQG can be derived the KKT optimality condition of the corresponding SDP problem. Besides, the proposed primal problem efficiently decouples the system matrices and the gain matrix. This allows us to develop new convex relax…
Optimal control theory and machine learning techniques are combined to formulate and solve in closed form an optimal control formulation of online learning from supervised examples with regularization of the updates. The connections with the classical Linear Quadratic Gaussian (LQG) optimal control problem, of which the proposed learning paradigm is a non-trivial variation as it involves random matrices, are investigated. The obtained optimal solutions are compared with the Kalman-filter estimate of the parameter vector to be learned. It is shown that the proposed algorithm is less sensitive t…
We consider the problem of finding an event-based sampling scheme that optimizes the trade-off between average sampling rate and control performance in a linear-quadratic-Gaussian (LQG) control problem setting with output feedback. Our analysis is based on a recently presented sampled-data controller structure, which remains LQG-optimal for any choice of sampling scheme. We show that optimization of the sampling scheme is related to an elliptic convection-diffusion type partial differential equation over a domain with free boundary, a so called Stefan problem. A numerical method is presented t…
We consider a joint sensor and controller design problem for linear Gaussian stochastic systems in which a weighted sum of quadratic control cost and the amount of information acquired by the sensor is minimized. This problem formulation is motivated by situations where a control law must be designed in the presence of sensing, communication, and privacy constraints. We show that the optimal joint sensor-controller design is relatively easy when the sensing policy is restricted to be linear. Namely, an explicit form of the optimal linear sensor equation, the Kalman filter, and the certainty eq…
A decentralized control system with linear dynamics, quadratic cost, and Gaussian disturbances is considered. The system consists of a finite number of subsystems whose dynamics and per-step cost function are coupled through their mean-field (empirical average). The system has mean-field sharing information structure, i.e., each controller observes the state of its local subsystem (either perfectly or with noise) and the mean-field. It is shown that the optimal control law is unique, linear, and identical across all subsystems. Moreover, the optimal gains are computed by solving two decoupled …
In this work, we revisit the Linear Quadratic Gaussian (LQG) optimal control problem from a behavioral perspective. Motivated by the suitability of behavioral models for data-driven control, we begin with a reformulation of the LQG problem in the space of input-output behaviors and obtain a complete characterization of the optimal solutions. In particular, we show that the optimal LQG controller can be expressed as a static behavioral-feedback gain, thereby eliminating the need for dynamic state estimation characteristic of state space methods. The static form of the optimal LQG gain also make…
This paper investigates the linear-quadratic-Gaussian (LQG) mean-field game (MFG) for a class of stochastic delay systems. We consider a large population system in which the dynamics of each player satisfies some forward stochastic differential delay equation (SDDE). The consistency condition or Nash certainty equivalence (NCE) principle is established through an auxiliary mean-field system of anticipated forward-backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such consistency condition system can be further established by some continuation method instead t…
In this paper LQG control over unreliable communication links is derived. That is to say, the communication channels between the controller and the actuators and between the sensors and the controller are unreliable. Previous solutions to finite horizon discrete time hold-input LQG control for this case do not fully utilize the available information. Here a new solution is presented which resolves this limitation. The focus is to derive and present a full mathematical proof to derive the optimal control sequence.
We consider the problem of controlling a set of dynamically decoupled plants where the plants' subcontrollers communicate with each other according to a fixed and known network topology. We assume the communication to be instantaneous but there is a fixed processing delay associated with incoming transmissions. We provide explicit closed-form expressions for the optimal decentralized controller under these communication constraints and using standard LQG assumptions for the plants and cost function. Although this problem is convex, it is challenging due to the irrationality of continuous-time …
Inverse Problems for Costs and Controls in LQG MFGs via Mean Field Trajectories
Center for Data Science, New York University（纽约大学数据科学中心）；NYU Shanghai（纽约大学上海分校）；Shanghai Center for Data Science（上海数据科学中心）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学上海分校纽约-华东师范大学数学科学研究所）
In this work, we study a class of mean-field linear quadratic Gaussian (LQG) problems. Under suitable conditions, explicit solutions of the distribution-dependent optimal control problems are obtained. Riccati systems are derived by directly solving the associated master equations. Some extensions on controls with partial observations are also considered.
We study the problem of adaptive control in partially observable linear quadratic Gaussian control systems, where the model dynamics are unknown a priori. We propose LqgOpt, a novel reinforcement learning algorithm based on the principle of optimism in the face of uncertainty, to effectively minimize the overall control cost. We employ the predictor state evolution representation of the system dynamics and deploy a recently proposed closed-loop system identification method, estimation, and confidence bound construction. LqgOpt efficiently explores the system dynamics, estimates the model param…
The goal of this paper is to study a multi-objective linear quadratic Gaussian (LQG) control problem. In particular, we consider an optimal control problem minimizing a quadratic cost over a finite time horizon for linear stochastic systems subject to control energy constraints. To solve the problem, we suggest an efficient bisection line search algorithm which is computationally efficient compared to other approaches such as the semidefinite programming. The main idea is to use the Lagrangian function and Karush-Kuhn-Tucker (KKT) optimality conditions to solve the constrained optimization pro…
This paper is concerned with linear-quadratic-Gaussian (LQG) control for a field-mediated feedback connection of a plant and a coherent (measurement-free) controller. Both the plant and the controller are multimode open quantum harmonic oscillators governed by linear quantum stochastic differential equations. The control objective is to make the closed-loop system internally stable and to minimize the infinite-horizon quadratic cost involving the plant variables and the controller output subject to quantum physical realizability (PR) constraints. This coherent quantum LQG (CQLQG) control probl…
We study a general class of entropy-regularized multi-variate LQG mean field games (MFGs) in continuous time with $K$ distinct sub-population of agents. We extend the notion of actions to action distributions (exploratory actions), and explicitly derive the optimal action distributions for individual agents in the limiting MFG. We demonstrate that the optimal set of action distributions yields an $ε$-Nash equilibrium for the finite-population entropy-regularized MFG. Furthermore, we compare the resulting solutions with those of classical LQG MFGs and establish the equivalence of their existenc…
Information communicated within cyber-physical systems (CPSs) is often used in determining the physical states of such systems, and malicious adversaries may intercept these communications in order to infer future states of a CPS or its components. Accordingly, there arises a need to protect the state values of a system. Recently, the notion of differential privacy has been used to protect state trajectories in dynamical systems, and it is this notion of privacy that we use here to protect the state trajectories of CPSs. We incorporate a cloud computer to coordinate the agents comprising the C…
University of Pennsylvania（宾夕法尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
Linear-Quadratic-Gaussian (LQG) control is concerned with the design of an optimal controller and estimator for linear Gaussian systems with imperfect state information. Standard LQG assumes the set of sensor measurements, to be fed to the estimator, to be given. However, in many problems, arising in networked systems and robotics, one may not be able to use all the available sensors, due to power or payload constraints, or may be interested in using the smallest subset of sensors that guarantees the attainment of a desired control goal. In this paper, we introduce the sensing-constrained LQG …
We consider a discrete-time Linear-Quadratic-Gaussian (LQG) control problem in which Massey's directed information from the observed output of the plant to the control input is minimized while required control performance is attainable. This problem arises in several different contexts, including joint encoder and controller design for data-rate minimization in networked control systems. We show that the optimal control law is a Linear-Gaussian randomized policy. We also identify the state space realization of the optimal policy, which can be synthesized by an efficient algorithm based on semi…
In this paper, we define and solve the Inverse Stochastic Optimal Control (ISOC) problem of the linear-quadratic Gaussian (LQG) and the linear-quadratic sensorimotor (LQS) control model. These Stochastic Optimal Control (SOC) models are state-of-the-art approaches describing human movements. The LQG ISOC problem consists of finding the unknown weighting matrices of the quadratic cost function and the covariance matrices of the additive Gaussian noise processes based on ground truth trajectories observed from the human in practice. The LQS ISOC problem aims at additionally finding the covarianc…
Control problems with incomplete information and memory limitation appear in many practical situations. Although partially observable stochastic control (POSC) is a conventional theoretical framework that considers the optimal control problem with incomplete information, it cannot consider memory limitation. Furthermore, POSC cannot be solved in practice except in the special cases. In order to address these issues, we propose an alternative theoretical framework, memory-limited POSC (ML-POSC). ML-POSC directly considers memory limitation as well as incomplete information, and it can be solved…
In the design of closed-loop networked control systems (NCSs), induced transmission delay between sensors and the control station is an often-present issue which compromises control performance and may even cause instability. A very relevant scenario in which network-induced delay needs to be investigated is costly usage of communication resources. More precisely, advanced communication technologies, e.g. 5G, are capable of offering latency-varying information exchange for different prices. Therefore, induced delay becomes a decision variable. It is then the matter of decision maker's willingn…
This paper studies convex stochastic dynamic team problems with finite and infinite time horizons under decentralized information structures. First, we introduce two notions called exchangeable teams and symmetric information structures. We show that in convex exchangeable team problems an optimal policy exhibits a symmetry structure. We give a characterization for such symmetrically optimal teams for a general class of convex dynamic team problems under a mild conditional independence condition. In addition, through concentration of measure arguments, we establish the convergence of optimal p…
This paper revisits the classical Linear Quadratic Gaussian (LQG) control from a modern optimization perspective. We analyze two aspects of the optimization landscape of the LQG problem: 1) connectivity of the set of stabilizing controllers $\mathcal{C}_n$; and 2) structure of stationary points. It is known that similarity transformations do not change the input-output behavior of a dynamical controller or LQG cost. This inherent symmetry by similarity transformations makes the landscape of LQG very rich. We show that 1) the set of stabilizing controllers $\mathcal{C}_n$ has at most two path-c…
First order policy optimization has been widely used in reinforcement learning. It guarantees to find the optimal policy for the state-feedback linear quadratic regulator (LQR). However, the performance of policy optimization remains unclear for the linear quadratic Gaussian (LQG) control where the LQG cost has spurious suboptimal stationary points. In this paper, we introduce a novel perturbed policy gradient (PGD) method to escape a large class of bad stationary points (including high-order saddles). In particular, based on the specific structure of LQG, we introduce a novel reparameterizati…
We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of di…
This paper is concerned with the coherent quantum linear-quadratic-Gaussian control problem of minimising an infinite-horizon mean square cost for a measurement-free field-mediated interconnection of a quantum plant with a stabilising quantum controller. The plant and the controller are multimode open quantum harmonic oscillators, governed by linear quantum stochastic differential equations and coupled to each other and the external multichannel bosonic fields in the vacuum state. We discuss an interplay between the quantum physical realizability conditions and the Luenberger structure associa…
In this paper, we propose and analyze an attack detection scheme for securing the physical layer of a networked control system against attacks where the adversary replaces the true observations with stationary false data. An independent and identically distributed watermarking signal is added to the optimal linear quadratic Gaussian (LQG) control inputs, and a cumulative sum (CUSUM) test is carried out using the joint distribution of the innovation signal and the watermarking signal for quickest attack detection. We derive the expressions of the supremum of the average detection delay (SADD) f…
Massachusetts Institute of Technology（麻省理工学院）；University of Pennsylvania（宾夕法尼亚大学）
We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynom…
This paper examines mean field linear-quadratic-Gaussian (LQG) social optimum control with volatility-uncertain common noise. The diffusion terms in the dynamics of agents contain an unknown volatility process driven by a common noise. We apply a robust optimization approach in which all agents view volatility uncertainty as an adversarial player. Based on the principle of person-by-person optimality and a two-step-duality technique for stochastic variational analysis, we construct an auxiliary optimal control problem for a representative agent. Through solving this problem combined with a con…
System level synthesis is a promising approach that formulates structured optimal controller synthesis problems as convex problems. This work solves the distributed linear-quadratic regulator problem under communication constraints directly in infinite-dimensional space, without the finite-impulse response relaxation common in related work. Our method can also be used to construct optimal distributed Kalman filters with limited information exchange. We combine the distributed Kalman filter with state-feedback control to perform localized LQG control with communication constraints. We provide a…
We study distributed output feedback control for a heterogeneous multi-agent system (MAS), consisting of N different continuous-time linear dynamical systems. For achieving output consensus, a virtual reference model is assumed to generate the desired trajectory that the MAS is required to track and synchronize. A distinct feature of our results lies in the local optimality and robustness achieved by our proposed consensus control algorithm. In addition our study is focused on the case when the available output measurements contain only relative information from the neighboring agents and refe…
In this paper, we revisit the sequential source coding framework to analyze fundamental performance limitations of discrete-time stochastic control systems subject to feedback data-rate constraints in finite-time horizon. The basis of our results is a new characterization of the lower bound on the minimum total-rate achieved by sequential codes subject to a total (across time) distortion constraint and a computational algorithm that allocates optimally the rate-distortion for any fixed finite-time horizon. This characterization facilitates the derivation of analytical, non-asymptotic, and fini…
