Nader Motee

Optimal Control of Spatially Distributed Systems

In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over arbitrary graphs. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD) operators. We study the structural properties of infinite-horizon linear quadratic optimal controllers for such systems by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. We prove that the kernel of the optimal feedback of each subsystem decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the system.

Optimal partitioning in distributed model predictive control

In this paper we develop the algorithms for optimal partitioning of a distributed control system into subsystems of manageable size for which control actions are found using model predictive control (MPC) technology. We will first define a realization-invariant weighting matrix to represent the distributed system as a directed graph. We then develop a formulation in which an open loop performance metric is used to partition the distributed system into subsystem in which local MPC problems will be solved. This partitioning however is balanced against the closed loop cost of the control actions for the overall distributed system. Effective algorithms for the distributed control of the large-scale systems are then proposed. Future work will include the study of the effect of the constraints in the partitioning, and the development of efficient problem formulations aimed at improving numerical properties of the proposed control algorithms.

Fundamental Limits and Tradeoffs on Disturbance Propagation in Linear Dynamical Networks

We investigate performance deterioration in linear consensus networks subject to external stochastic disturbances. The expected value of the steady state dispersion of the states of the network is adopted as a performance measure. We develop a graph-theoretic methodology to relate structural specifications of the coupling graph of a linear consensus network to its performance measure. We explicitly quantify several inherent fundamental limits on the best achievable levels of performance and show that these limits of performance are emerged only due to the specific interconnection topology of the coupling graphs. Furthermore, we discover some of the inherent fundamental tradeoffs between notions of sparsity and performance in linear consensus networks.

Fundamental limits on robustness measures in networks of interconnected systems

We investigate robustness of interconnected dynamical networks with respect to external distributed stochastic disturbances. In this paper, we consider networks with linear time-invariant dynamics. The ℋ2 norm of the underlying system is considered as a robustness index to measure the expected steady-state dispersion of the state of the entire network. We present new tight bounds for the robustness measure for general linear dynamical networks. We, then, focus on two specific classes of networks: first- and second-order consensus in dynamical networks. A weighted version of the ℋ2 norm of the system, so called LQ-energy of the network, is introduced as a robustness measure. It turns out that when LQ is the Laplacian matrix of a complete graph, LQ-energy reduces to the expected steady-state dispersion of the state of the entire network. We quantify several graph-dependent and graph-independent fundamental limits on the LQ-energy of the networks. Our theoretical results have been applied to two application areas. First, we show that in power networks the concept of LQ-energy can be interpreted as the total resistive losses in the network and that it does not depend on specific structure of the underlying graph of the network. Second, we consider formation control with second-order dynamics and show that the LQ-energy of the network is graph-dependent and corresponds to the energy of the flock.

On decentralized optimal control and information structures

A canonical decentralized optimal control problem with quadratic cost criteria can be cast as an LQR problem in which the stabilizing controller is restricted to lie in a constraint set. We characterize a wide class of systems and constraint sets for which the canonical problem is tractable. We employ the notion of operator algebras to study the structural properties of the canonical problem. Examples of some widely used operator algebras in the context of distributed control include the subspace of infinite and finite dimensional spatially decaying operators, lower (or upper) triangular matrices, and circulant matrices. For a given operator algebra, we prove that if the trajectory of the solution of an operator differential equation starts inside the operator algebra, it will remain inside for all times. Using this result, we show that if the constraint set is an operator algebra, the canonical problem is solvable and equivalent to the standard LQR problem without the information constraint.

Approximation methods and spatial interpolation in distributed control systems

We propose an approximation method to solve large-scale optimal control problems for spatially distributed systems. The finite-section method is employed to construct finite-dimensional approximations to the large-scale optimal control problem. Then, we study the limit behavior of the approximation method and show that the solution of the approximate problems converge strongly to the solution of the large-scale problem. These techniques are applied to design finite-dimensional local optimal controllers. Finally, a spatial interpolation method is proposed that can patch all locally designed controllers to construct a parameterized family of stabilizing controller for the spatially distributed system. Furthermore, we characterize the class of stabilizing controllers which have finite supports.

Systemic measures for performance and robustness of large-scale interconnected dynamical networks

In this paper, we develop a novel unified methodology for performance and robustness analysis of linear dynamical networks. We introduce the notion of systemic measures for the class of first-order linear consensus networks. We classify two important types of performance and robustness measures according to their functional properties: convex systemic measures and Schur-convex systemic measures. It is shown that a viable systemic measure should satisfy several fundamental properties such as homogeneity, monotonicity, convexity, and orthogonal invariance. In order to support our proposed unified framework, we verify functional properties of several existing performance and robustness measures from the literature and show that they all belong to the class of systemic measures. Moreover, we introduce new classes of systemic measures based on (a version of) the well-known Riemann zeta function, input-output system norms, and etc. Then, it is shown that for a given linear dynamical network one can take several different strategies to optimize a given performance and robustness systemic measure via convex optimization. Finally, we characterized an interesting fundamental limit on the best achievable value of a given systemic measure after adding some certain number of new weighted edges to the underlying graph of the network.

Sparsity measures for spatially decaying systems

We consider the omnipresent class of spatially decaying systems, where the sensing and controls is spatially distributed. This class of systems arises in various applications where there is a notion of spatial distance with respect to which couplings between the subsystems can be quantified using a class of coupling weight functions. We exploit spatial decay property of the dynamics of the underlying system in order to introduce system-oriented sparsity measures for spatially distributed systems. We develop a new mathematical framework, based on notions of quasi-Banach algebras of spatially decaying matrices, to relate spatial decay properties of spatially decaying systems to sparsity features of their underlying information structures. Moreover, we show that the inverse-closedness property of matrix algebras plays a central role in exploiting various structural properties of spatially decaying systems. We show that the quadratically optimal state feedback controllers for spatially decaying systems are sparse and spatially localized in the sense that they have near-optimal sparse information structures. Finally, our results are applied to quantify sparsity and spatial localization features of a class of randomly generated power networks.

Distributed receding horizon control of spatially invariant systems

We present a rigorous framework for the study of distributed spatially invariant systems with input and state constraints. The proposed approach is based on blending tools from operator theory and Fourier analysis of spatially invariant systems with receding horizon control and multi parametric quadratic programming (MPQP). Our contributions are twofold: on one hand, we extend the recent results of Bamieh et al. on infinite-horizon optimal control of spatially invariant systems to finite receding horizon control with input and state constraints. On the other hand, our results can be interpreted as an extension of the finite dimensional MPQP-based analysis of receding horizon control to distributed, spatially invariant systems. It is assumed that the dynamics of each subsystem is uncoupled to the others, but the coupling appears through the finite horizon cost function. Specifically, we prove that for spatially invariant systems with constraints, optimal receding horizon controllers are piecewise affine (represented as a convolution sum plus an offset). Moreover, the kernel of each convolution sum decays exponentially in the spatial domain mirroring the unconstrained infinite-horizon case. Simulation results are provided for a simple example with 5 identical systems coupled in a loop

Active magnetic bearing control with zero steady-state power loss

In this paper, we address the problem of controlling an active magnetic bearing with reduced ohmic power losses. A solution is proposed which introduces an exponentially decaying bias flux in the nonlinear control law. To avoid control singularities, the closed-loop system is exponentially stabilized at a faster rate than the bias decay. In the steady state, the proposed controller has zero bias flux; hence, there are no ohmic power losses. The control scheme is applied to two different bearing modes of operation all electromagnets active at all times and only one opposing electromagnet is active at any given time. A rigorous analysis of the latter mode of operation is presented along with an implementation strategy. Simulations are provided to illustrate the performance of the proposed controller for the two operating modes in comparison to a standard constant-bias controller.