Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers

We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the $H_2$ norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsity-promoting penalty functions into the optimal control problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize feedback gains subject to structural constraints determined by the identified sparsity patterns. In the first step, the sparsity structure of feedback gains is identified using the alternating direction method of multipliers, which is a powerful algorithm well-suited to large optimization problems. This method alternates between promoting the sparsity of the controller and optimizing the closed-loop performance, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of the sparsity-promoting penalty functions to decompose the minimization problem into sub-problems that can be solved analytically. Several examples are provided to illustrate the effectiveness of the developed approach.Comment: To appear in IEEE Trans. Automat. Contro

A Fast Algorithm for Sparse Controller Design

We consider the task of designing sparse control laws for large-scale systems by directly minimizing an infinite horizon quadratic cost with an $\ell_1$ penalty on the feedback controller gains. Our focus is on an improved algorithm that allows us to scale to large systems (i.e. those where sparsity is most useful) with convergence times that are several orders of magnitude faster than existing algorithms. In particular, we develop an efficient proximal Newton method which minimizes per-iteration cost with a coordinate descent active set approach and fast numerical solutions to the Lyapunov equations. Experimentally we demonstrate the appeal of this approach on synthetic examples and real power networks significantly larger than those previously considered in the literature

L1-induced Performance Analysis and Sparse Controller Synthesis for Interval Positive Systems

The Lecture Notes in Engineering and Computer Science(LNECS) vol. entitled: World Congress on Engineering: WCE 2013This paper is concerned with the design of L1- induced sparse controller for continuous-time positive systems with interval uncertainties. A necessary and sufficient condition for stability and L1-induced performance of positive linear systems is proposed in terms of linear inequalities. Based on this, conditions for the existence of robust state-feedback controllers are established. Moreover, the total number of all the nonzero elements of the controller gain is to be minimized, while satisfying a guaranteed level of L1-induced performance. Then, we propose an ℓ1-minimization problem to relax the ℓ0 objective function for optimization and an iterative convex optimization approach is developed to solve the conditions. Finally, an illustrative example is provided to show the effectiveness and applicability of the theoretical results.published_or_final_versio

Óptimo control y operación de redes de distribución eléctrica a través del algoritmo beta - radialidad.

In the following academic article, it is proposed a model that determined the optimal control and operation of a distribution network; the model is based on the Beta Radial algorithm. The objective function of the proposed model finds the maximum loss of power due to failure or damage of distribution lines, tracking the system load constant The model first obtains a matrix of radial beta combinations, for each row of the obtained matrix a simulation of the power flow was performed, obtaining, as a result, the matrix of power flows. In conclusion, two important cases were found; in the first case, the distribution lines that are faulty produce the greatest power loss in the system and in the second case, the lines fault that produce the greatest overload to the system. The model pursues an alternative to the decision making the process to design a preventive maintenance plan for an electrical power system in order to increase the useful life and reliability of the electric service.En el siguiente artículo académico se plantea un modelo que permite determinar la estrategia de control óptimo y de operación de una red de distribución. El modelo está basado en el algoritmo Beta Radial. La función objetivo del modelo propuesto encuentra la máxima pérdida de potencia por falla o avería en las líneas manteniendo la carga del sistema constante. Para el modelo propuesto se obtiene una matriz de combinaciones beta radial, por cada fila de la matriz obtenida se realizó una simulación del flujo de potencia, obteniéndose como consecuencia la matriz de flujos de potencia. Como resultado, se encontraron dos casos importantes, en el primer caso se presentan las líneas que al estar en falla producen la mayor pérdida de potencia del sistema y en el segundo caso se encontró las líneas que al estar en falla producen la mayor sobrecarga al mismo. El modelo persigue una alternativa a la toma de decisiones para diseñar un plan de mantenimiento preventivo de cualquier sistema eléctrico de potencia, de esta forma aumentar la vida útil y confiabilidad del servicio eléctrico

H2 Optimal Sensing Architecture with Model Uncertainty

In this thesis, I shall present an integrated approach to control and sensing design. The framework assumes sensor noise as a design variable along with the controller and determines l1 regularized optimal sensing precision. This design satisfies a given closed-loop performance in the presence of model uncertainty. Two methods will be proposed to achieve this. The first method designs a controller for an open loop uncertain system, which is scaled in order to have a finite H2 norm. Within this, two approaches have been pursued. In the first approach, uncertainty has been represented as polytopic and, in the second formulation, modelled using integral quadratic constraints (IQC). These two approaches have been applied to an active suspension control and sensing design problem and demonstrate that the IQC based approach provides better results and is able to incorporate larger system uncertainty. The second method finds an appropriate scaling to bound the H2 norm of an uncertain controlled system. The sensor precision is found as the minimal solution to an optimization problem. The design is tested for stability and robustness on a tensegrity robot arm model

Analysis & Synthesis of Distributed Control Systems with Sparse Interconnection Topologies

This dissertation is about control, identification, and analysis of systems with sparse interconnection topologies. We address two main research objectives relating to sparsity in control systems and networks. The first problem is optimal sparse controller synthesis, and the second one is the identification of sparse network. The first part of this dissertation starts with the chapter focusing on developing theoretical frameworks for the synthesis of optimal sparse output feedback controllers under pre-specified structural constraints. This is achieved by establishing a balance between the stability of the controller and the systems quadratic performance. Our approach is mainly based on converting the problem into rank constrained optimizations.We then propose a new approach in the syntheses of sparse controllers by em- ploying the concept of Hp approximations. Considering the trade-off between the controller sparsity and the performance deterioration due to the sparsification pro- cess, we propose solving methodologies in order to obtain robust sparse controllers when the system is subject to parametric uncertainties.Next, we pivot our attention to a less-studied notion of sparsity, namely row sparsity, in our optimal controller design. Combining the concepts from the majorization theory and our proposed rank constrained formulation, we propose an exact reformulation of the optimal state feedback controllers with strict row sparsity constraint, which can be sub-optimally solved by our proposed iterative optimization techniques. The second part of this dissertation focuses on developing a theoretical framework and algorithms to derive linear ordinary differential equation models of gene regulatory networks using literature curated data and micro-array data. We propose several algorithms to derive stable sparse network matrices. A thorough comparison of our algorithms with the existing methods are also presented by applying them to both synthetic and experimental data-sets