Optimizing Networked Systems and Inverse Optimal Control

This thesis is concerned with the problems of optimizing networked systems, including designing a distributed energy optimal consensus controller for homogeneous networked linear systems, maximizing the algebraic connectivity of a network by projected saddle point dynamics. In addition, the inverse optimal control problems for discrete-time finite time-horizon Linear Quadratic Regulators (LQRs) are considered. The goal is to infer the Q matrix in the quadratic cost function using the observations (possibly noisy) either on the optimal state trajectories, optimal control input or the system output. In Paper A, an optimal energy cost controller design for identical networked linear systems asymptotic consensus is considered. It is assumed that the topology of the network is given and the controller can only depend on relative information of the agents. Since finding the control gain for such a controller is hard, we focus on finding an optimal controller among a classical family of controllers which is based on the Algebraic Riccati Equation (ARE) and guarantees asymptotic consensus. We find that the energy cost is bounded by an interval and hence we minimize the upper bound. Further, the minimization for the upper bound boils down to optimizing the control gain and the edge weights of the graph separately. A suboptimal control gain is obtained by choosing Q=0 in the ARE. Negative edge weights are allowed, meaning that "competitions" between the agents are allowed. The edge weight optimization problem is formulated as a Semi-Definite Programming (SDP) problem. We show that the lowest control energy cost is reached when the graph is complete and with equal edge weights. Furthermore, two sufficient conditions for the existence of negative optimal edge weights realization are given. In addition, we provide a distributed way of solving the SDP problem when the graph topology is regular. In Paper B, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization problems that are not necessarily strictly convex. The optimization variables are restricted by a convex set with computable projection operation on its tangent cone as well as equality constraints. We show that the projected dynamical system converges to one of the saddle points and hence finding an optimal solution. Moreover, the problem of distributedly maximizing the algebraic connectivity of an undirected network by optimizing the "port gains" of each nodes is considered. The original SDP problem is relaxed into a nonlinear programming (NP) problem that will be solved by the aforementioned projected dynamical system. Numerical examples show the convergence of the aforementioned algorithm to one of the optimal solutions. The effect of the relaxation is illustrated empirically with numerical examples. A methodology is presented so that the number of iterations needed to converge is reduced. Complexity per iteration of the algorithm is illustrated with numerical examples. In Paper C and D, the inverse optimal control problems over finite-time horizon for discrete-time LQRs are considered. The well-posedness of the inverse optimal control problem is first justified. In the noiseless case, when these observations of the optimal state trajectories or the optimal control input are exact, we analyze the identifiability of the problem and provide sufficient conditions for uniqueness of the solution. In the noisy case, when the observations are corrupted by additive zero-mean noise, we formulate the problem as an optimization problem and prove that the solution to this problem is statistically consistent. The following two scenarios are further considered: 1) the distributions of the initial state and the observation noise are unknown, yet the exact observations on the initial states and the noisy observations on the system output are available; 2) the exact observations on the initial states are not available, yet the observation noises are known to be white Gaussian and the distribution of the initial state is also Gaussian (with unknown mean and covariance). For the first scenario, we show statistical consistency for the estimation. For the second scenario, we fit the problem into the framework of maximum-likelihood and Expectation Maximization (EM) algorithm is used to solve this problem. The performance of the proposed method is illustrated through numerical examples.QC 20190121