Neil K. Dhingra

An ADMM algorithm for optimal sensor and actuator selection

We consider the problem of the optimal selection of a subset of available sensors or actuators in large-scale dynamical systems. By replacing a combinatorial penalty on the number of sensors or actuators with a convex sparsity-promoting term, we cast this problem as a semidefinite program. The solution of the resulting convex optimization problem is used to select sensors (actuators) in order to gracefully degrade performance relative to the optimal Kalman filter (Linear Quadratic Regulator) that uses all available sensing (actuating) capabilities. We employ the alternating direction method of multipliers to develop a customized algorithm that is well-suited for large-scale problems. Our algorithm scales better than standard SDP solvers with respect to both the state dimension and the number of available sensors or actuators.

On identifying sparse representations of consensus networks

Abstract We consider the problem of identifying optimal sparse graph representations of dense consensus networks. The performance of the sparse representation is characterized by the global performance measure which quantifies the difference between the output of the sparse graph and the output of the original graph. By minimizing the sum of this performance measure and a sparsity-promoting penalty function, the alternating direction method of multipliers identifies sparsity structures that strike a balance between the performance measure and the number of edges in the graph. We then optimize the edge weights of sparse graphs over the identified topologies. Two examples are provided to illustrate the utility of the developed approach.

Sparsity-promoting optimal control of spatially-invariant systems

We study the optimal design of sparse and block sparse feedback gains for spatially-invariant systems on a circle. For this class of systems, the state-space matrices are jointly diagonalizable via the discrete Fourier transform. We exploit this structure to develop an ADMM-based algorithm that significantly reduces the computational complexity relative to standard approaches. Specifically, the complexity of the developed algorithm scales linearly with the number of subsystems. This is in contrast to a cubic scaling when circulant structure is not exploited. Two examples are provided to illustrate the effectiveness of the developed approach.

Controller architectures: Tradeoffs between performance and structure

Abstract This review article describes the design of static controllers that achieve an optimal tradeoff between closed-loop performance and controller structure. Our methodology consists of two steps. First, we identify controller structure by incorporating regularization functions into the optimal control problem and, second, we optimize the controller over the identified structure. For large-scale networks of dynamical systems, the desired structural property is captured by limited information exchange between physical and controller layers and the regularization term penalizes the number of communication links. Although structured optimal control problems are, in general, nonconvex, we identify classes of convex problems that arise in the design of symmetric systems, undirected consensus and synchronization networks, optimal selection of sensors and actuators, and decentralized control of positive systems. Examples of consensus networks, drug therapy design, sensor selection in flexible wing aircrafts, and optimal wide-area control of power systems are provided to demonstrate the effectiveness of the framework.

On the convexity of a class of structured optimal control problems for positive systems

We study a class of structured optimal control problems for positive systems in which the design variable modifies the main diagonal of the dynamic matrix. For this class of systems, we establish convexity of both the H2 and H∞ optimal control formulations. In contrast to previous approaches, our formulation allows for arbitrary convex constraints and regularization of the design parameter. We provide expressions for the gradient and subgradient of the H2 and norms and establish graph-theoretic conditions under which the H∞ norm is continuously differentiable. Finally, we develop a customized proximal algorithm for computing the solution to the regularized optimal control problems and apply our results for HIV combination drug therapy design.

A method of multipliers algorithm for sparsity-promoting optimal control

We develop a customized method of multipliers algorithm to efficiently solve a class of regularized optimal control problems. By exploiting the problem structure, we transform the augmented Lagrangian into a form which can be efficiently minimized using proximal methods. We apply our algorithm to an ℓ1-regularized state-feedback optimal control problem and compare its performance with a proximal gradient algorithm and an alternating direction method of multipliers algorithm. In contrast to other methods, our algorithm has both a theoretical guarantee of convergence and fast computation speed in practice.

Convex synthesis of symmetric modifications to linear systems

We develop a method for designing symmetric modifications to linear dynamical systems for the purpose of optimizing ℋ2 performance. For systems with symmetric dynamic matrices this problem is convex. While in the absence of symmetry the design problem is not convex in general, we show that the ℋ2 norm of the symmetric part of the system provides an upper bound on the ℋ2 norm of the original system. We then study the particular case where the modifications are given by a weighted sum of diagonal matrices and develop an efficient customized algorithm for computing the optimal solution. Finally, we illustrate the efficacy of our approach on a combination drug therapy example for HIV treatment.

Topology identification of undirected consensus networks via sparse inverse covariance estimation

We study the problem of identifying sparse interaction topology using sample covariance matrix of the state of the network. Specifically, we assume that the statistics are generated by a stochastically-forced undirected first-order consensus network with unknown topology. We propose a method for identifying the topology using a regularized Gaussian maximum likelihood framework where the ℓ1 regularizer is introduced as a means for inducing sparse network topology. The proposed algorithm employs a sequential quadratic approximation in which the Newtons direction is obtained using coordinate descent method. We provide several examples to demonstrate good practical performance of the method.

Convex reformulation of a robust optimal control problem for a class of positive systems

In this paper we consider the robust optimal control problem for a class of positive systems with an application to design of optimal drug dosage for HIV therapy. We consider uncertainty modeled as a Linear Fractional Transformation (LFT) and we show that, with a suitable change of variables, the structured singular value, μ, is a convex function of the control parameters. We provide graph theoretical conditions that guarantee μ to be a continuously differentiable function of the controller parameters and an expression of its gradient or subgradient. We illustrate the result with a numerical example where we compute the optimal drug dosages for HIV treatment in the presence of model uncertainty.