Sina Khoshfetrat Pakazad

Robust Stability Analysis of Sparsely Interconnected Uncertain Systems

In this paper, we consider robust stability analysis of large-scale sparsely interconnected uncertain systems. By modeling the interconnections among the subsystems with integral quadratic constraints, we show that robust stability analysis of such systems can be performed by solving a set of sparse linear matrix inequalities. We also show that a sparse formulation of the analysis problem is equivalent to the classical formulation of the robustness analysis problem and hence does not introduce any additional conservativeness. The sparse formulation of the analysis problem allows us to apply methods that rely on efficient sparse factorization techniques, and our numerical results illustrate the effectiveness of this approach compared to methods that are based on the standard formulation of the analysis problem.

Distributed robust stability analysis of interconnected uncertain systems

This paper considers robust stability analysis of a large network of interconnected uncertain systems. To avoid analyzing the entire network as a single large, lumped system, we model the network interconnections with integral quadratic constraints. This approach yields a sparse linear matrix inequality which can be decomposed into a set of smaller, coupled linear matrix inequalities. This allows us to solve the analysis problem efficiently and in a distributed manner. We also show that the decomposed problem is equivalent to the original robustness analysis problem, and hence our method does not introduce additional conservativeness.

Sparse control using sum-of-norms regularized model predictive control

Some control applications require the use of piecewise constant or impulse-type control signals, with as few changes as possible. So as to achieve this type of control, we consider the use of regularized model predictive control (MPC), which allows us to impose this structure through the use of regularization. It is then possible to regulate the trade-off between control performance and control signal characteristics by tuning the so-called regularization parameter. However, since the mentioned trade-off is only indirectly affected by this parameter, its tuning is often unintuitive and time-consuming. In this paper, we propose an equivalent reformulation of the regularized MPC, which enables us to configure the desired trade-off in a more intuitive and computationally efficient manner. This reformulation is inspired by the so-called ε-constraint formulation of multi-objective optimization problems and enables us to quantify the trade-off, by explicitly assigning bounds over the control performance.

Distributed Robustness Analysis of Interconnected Uncertain Systems Using Chordal Decomposition

Abstract Large-scale interconnected uncertain systems commonly have large state and uncertainty dimensions. Aside from the heavy computational cost of performing robust stability analysis in a centralized manner, privacy requirements in the network can also introduce further issues. In this paper, we utilize IQC analysis for analyzing large-scale interconnected uncertain systems and we evade these issues by describing a decomposition scheme that is based on the interconnection structure of the system. This scheme is based on the so-called chordal decomposition and does not add any conservativeness to the analysis approach. The decomposed problem can be solved using distributed computational algorithms without the need for a centralized computational unit. We further discuss the merits of the proposed analysis approach using a numerical experiment.

Scalable anomaly detection in large homogeneous populations

Anomaly detection in large populations is a challenging but highly relevant problem. The problem is essentially a multi-hypothesis problem, with a hypothesis for every division of the systems into normal and anomal systems. The number of hypothesis grows rapidly with the number of systems and approximate solutions become a necessity for any problems of practical interests. In the current paper we take an optimization approach to this multi-hypothesis problem. We first observe that the problem is equivalent to a non-convex combinatorial optimization problem. We then relax the problem to a convex problem that can be solved distributively on the systems and that stays computationally tractable as the number of systems increase. An interesting property of the proposed method is that it can under certain conditions be shown to give exactly the same result as the combinatorial multi-hypothesis problem and the relaxation is hence tight.

Distributed Semidefinite Programming With Application to Large-Scale System Analysis

Distributed algorithms for solving coupled semidefinite programs commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper, we show that in case the coupled problem has an inherent tree structure, it is possible to devise an efficient distributed algorithm for solving such problems. The proposed algorithm relies on predictor–corrector primal-dual interior-point methods, where we use a message-passing algorithm to compute the search directions distributedly. Message passing here is closely related to dynamic programming over trees. This allows us to compute the exact search directions in a finite number of steps. This is because computing the search directions requires a recursion over the tree structure and, hence, terminates after an upward and downward pass through the tree. Furthermore, this number can be computed a priori and only depends on the coupling structure of the problem. We use the proposed algorithm for analyzing robustness of large-scale uncertain systems distributedly. We test the performance of this algorithm using numerical examples.

Distributed primal–dual interior-point methods for solving tree-structured coupled convex problems using message-passing

In this paper, we propose a distributed algorithm for solving coupled problems with chordal sparsity or an inherent tree structure which relies on primal–dual interior-point methods. We achieve this by distributing the computations at each iteration, using message-passing. In comparison to existing distributed algorithms for solving such problems, this algorithm requires far fewer iterations to converge to a solution with high accuracy. Furthermore, it is possible to compute an upper-bound for the number of required iterations which, unlike existing methods, only depends on the coupling structure in the problem. We illustrate the performance of our proposed method using a set of numerical examples.

Distributed Interior-point Method for Loosely Coupled Problems

Abstract In this paper, we put forth distributed algorithms for solving loosely coupled unconstrained and constrained optimization problems. Such problems are usually solved using algorithms that are based on a combination of decomposition and first order methods. These algorithms are commonly very slow and require many iterations to converge. In order to alleviate this issue, we propose algorithms that combine the Newton and interior-point methods with proximal splitting methods for solving such problems. Particularly, the algorithm for solving unconstrained loosely coupled problems, is based on Newtons method and utilizes proximal splitting to distribute the computations for calculating the Newton step at each iteration. A combination of this algorithm and the interior-point method is then used to introduce a distributed algorithm for solving constrained loosely coupled problems. We also provide guidelines on how to implement the proposed methods efficiently, and briefly discuss the properties of the resulting solutions.

Distributed solutions for loosely coupled feasibility problems using proximal splitting methods

In this paper, we consider convex feasibility problems (CFPs) where the underlying sets are loosely coupled, and we propose several algorithms to solve such problems in a distributed manner. These algorithms are obtained by applying proximal splitting methods to convex minimization reformulations of CFPs. We also put forth distributed convergence tests which enable us to establish feasibility or infeasibility of the problem distributedly, and we provide convergence rate results. Under the assumption that the problem is feasible and boundedly linearly regular, these convergence results are given in terms of the distance of the iterates to the feasible set, which are similar to those of classical projection methods. In case the feasibility problem is infeasible, we provide convergence rate results that concern the convergence of certain error bounds.

Applications of IQC-Based Analysis Techniques for Clearance

Results for stability analysis of the nonlinear rigid aircraft model and comfort and loads analysis of the integral aircraft model are presented in this chapter. The analysis is based on the theory for integral quadratic constraints and relies on linear fractional representations (LFRs) of the underlying closed-loop aircraft models. To alleviate the high computational demands associated with the usage of IQC based analysis to large order LFRs, two approaches have been employed aiming a trade-off between computational complexity and conservatism. First, the partitioning of the flight envelope in several smaller regions allows to use lower order LFRs in the analysis, and second, IQCs with lower computational demands have been used whenever possible. The obtained results illustrate the applicability of the IQCs based analysis techniques to solve highly complex analysis problems with an acceptable level of conservativeness.