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.Comment: Provisionally accepted to appear in IEEE Transactions on Automatic Contro

A structure exploiting SDP solver for robust controller synthesis

In this paper, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP Lemma and they play a crucial role in e.g. robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require $O(n^6)$ arithmetic operations per Newton step to solve this class of problems, where $n$ is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to $O(n^3)$. However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this paper, we show how to include such constraints in structure exploiting KYP-SDP solvers.Comment: Submitted to Conference on Decision and Control, copyright owned by iee

Bundle-based pruning in the max-plus curse of dimensionality free method

Recently a new class of techniques termed the max-plus curse of dimensionality-free methods have been developed to solve nonlinear optimal control problems. In these methods the discretization in state space is avoided by using a max-plus basis expansion of the value function. This requires storing only the coefficients of the basis functions used for representation. However, the number of basis functions grows exponentially with respect to the number of time steps of propagation to the time horizon of the control problem. This so called "curse of complexity" can be managed by applying a pruning procedure which selects the subset of basis functions that contribute most to the approximation of the value function. The pruning procedures described thus far in the literature rely on the solution of a sequence of high dimensional optimization problems which can become computationally expensive. In this paper we show that if the max-plus basis functions are linear and the region of interest in state space is convex, the pruning problem can be efficiently solved by the bundle method. This approach combining the bundle method and semidefinite formulations is applied to the quantum gate synthesis problem, in which the state space is the special unitary group (which is non-convex). This is based on the observation that the convexification of the unitary group leads to an exact relaxation. The results are studied and validated via examples