Harmonic-Copuled Riccati Equations and its Applications in Distributed Filtering

The coupled Riccati equations are cosisted of multiple Riccati-like equations with solutions coupled with each other, which can be applied to depict the properties of more complex systems such as markovian systems or multi-agent systems. This paper manages to formulate and investigate a new kind of coupled Riccati equations, called harmonic-coupled Riccati equations (HCRE), from the matrix iterative law of the consensus on information-based distributed filtering (CIDF) algortihm proposed in [1], where the solutions of the equations are coupled with harmonic means. Firstly, mild conditions of the existence and uniqueness of the solution to HCRE are induced with collective observability and primitiviness of weighting matrix. Then, it is proved that the matrix iterative law of CIDF will converge to the unique solution of the corresponding HCRE, hence can be used to obtain the solution to HCRE. Moreover, through applying the novel theory of HCRE, it is pointed out that the real estimation error covariance of CIDF will also become steady-state and the convergent value is simplified as the solution to a discrete time Lyapunov equation (DLE). Altogether, these new results develop the theory of the coupled Riccati equations, and provide a novel perspective on the performance analysis of CIDF algorithm, which sufficiently reduces the conservativeness of the evaluation techniques in the literature. Finally, the theoretical results are verified with numerical experiments.Comment: 14 pages, 4 figure

A numerical analysis of the Nash strategy for weakly coupled large-scale systems

his note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm; it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation

Controle amostrado ótimo de sistemas lineares com saltos markovianos através de realimentação de estados

Orientador: José Cláudio GeromelTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Este trabalho é inteiramente dedicado ao desenvolvimento de uma lei de controle ótimo amostrado aplicada a sistemas lineares com saltos markovianos, cujo principal uso são os sistemas controlados através da rede (NCS - Networked Control System). Neste contexto, duas características da rede são consideradas simultaneamente: a limitação da largura de banda, tratada através da existência de sinais amostrados no sistema, e a perda de pacotes, modelada através de uma cadeia de Markov a tempo contínuo. A fim de alcançar este objetivo, a abordagem geral adotada é dividida em quatro etapas: análise de estabilidade e cálculo de norma no contexto da norma H2; análise de estabilidade e cálculo de norma no contexto da norma Hoo; projeto de controle amostrado ótimo que minimiza o índice de desempenho J2 baseado na norma H2, o qual pode ser expresso em uma formulação convexa baseada em LMIs; projeto de controle amostrado ótimo que minimiza um certo índice de desempenho Joo baseado na norma Hoo, o qual também admite uma formulação convexa baseada em LMI, embora uma análise matemática mais aprofundada seja necessária. Cada uma destas etapas possui a mesma estrutura descrita a seguir. Primeiro, os resultados teóricos são matematicamente desenvolvidos e provados. Segundo, alguns casos particulares são derivados a partir destes resultados teóricos. Terceiro, um algoritmo convergente é proposto para resolver cada um dos casos mencionados. As convergências também são provadas. Finalmente, um exemplo teórico ilustra os principais desenvolvimentos em cada caso. A teoria aqui desenvolvida é nova, não havendo resultado similar na literatura atual. Para uma visão prática dos resultados desta dissertação, três exemplos são considerados e adaptados de trabalhos disponíveis: dois deles correspondem a sistemas físicos controlados através de uma rede sendo um originalmente estável e o outro instável, e o terceiro corresponde a um sistema econômico cujas políticas de controle são aplicadas a tempo discretoAbstract: This work is entirely devoted to develop an optimal sampled-data control law applied to Markov jump linear systems, whose main usage is Networked Control Systems (NCS). In this context, two network characteristics are simultaneously considered: the bandwidth limitation addressed by the existence of sampled-data signals in the system, and the packet dropouts modeled by a continuous-time Markov chain. In order to accomplish this goal, the general adopted approach is broken in four steps: stability analysis and norm evaluation based on the H2 norm; stability analysis and norm evaluation in the Hoo context; the optimal sampled-data control design that minimizes a J2 performance index based on the H2 norm, which can be expressed in a convex formulation based on LMIs; the optimal sampled-data control design that minimizes a certain Joo performance index based on the Hoo norm, which also admits a convex formulation based on LMIs, even though a deeper mathematical analysis is required. Each step has the same structure described in the sequel. First, the theoretical results are mathematically developed and proved. Second, some particular cases are derived from these theoretical results. Third, a convergent algorithm is proposed to solve each of the mentioned cases. The convergence of the algorithms are also proved. Finally, a numerical example illustrates the main developments in each step. The theory developed here is new and there is no similar result in the current literature. For a practical view of the outcomes, three practical examples are borrowed and adapted from available works: two of them are physical systems controlled through an NCS, where one is originally stable and the other unstable, and the third one is an economical system whose policy is applied in a discrete-time basisDoutoradoAutomaçãoDoutora em Engenharia Elétrica2012/23634-2FAPES

New optimization methods in predictive control

This thesis is mainly concerned with the efficient solution of a linear discrete-time finite horizon optimal control problem (FHOCP) with quadratic cost and linear constraints on the states and inputs. In predictive control, such a FHOCP needs to be solved online at each sampling instant. In order to solve such a FHOCP, it is necessary to solve a quadratic programming (QP) problem. Interior point methods (IPMs) have proven to be an efficient way of solving quadratic programming problems. A linear system of equations needs to be solved in each iteration of an IPM. The ill-conditioning of this linear system in the later iterations of the IPM prevents the use of an iterative method in solving the linear system due to a very slow rate of convergence; in some cases the solution never reaches the desired accuracy. A new well-conditioned IPM, which increases the rate of convergence of the iterative method is proposed. The computational advantage is obtained by the use of an inexact Newton method along with the use of novel preconditioners. A new warm-start strategy is also presented to solve a QP with an interior-point method whose data is slightly perturbed from the previous QP. The effectiveness of this warm-start strategy is demonstrated on a number of available online benchmark problems. Numerical results indicate that the proposed technique depends upon the size of perturbation and it leads to a reduction of 30-74% in floating point operations compared to a cold-start interior point method. Following the main theme of this thesis, which is to improve the computational efficiency of an algorithm, an efficient algorithm for solving the coupled Sylvester equation that arises in converting a system of linear differential-algebraic equations (DAEs) to ordinary differential equations is also presented. A significant computational advantage is obtained by exploiting the structure of the involved matrices. The proposed algorithm removes the need to solve a standard Sylvester equation or to invert a matrix. The improved performance of this new method over existing techniques is demonstrated by comparing the number of floating-point operations and via numerical examples

Static output feedback: a survey

This paper reviews the static output feedback problem in the control of linear, time-invariant (LTI) systems. It includes analytical and computational methods and presents in a unified fashion, the knowledge gained in the decades of research into this most important problem

Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

In this paper, we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct non-uniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions

Differential Games For Multi-agent Systems Under Distributed Information

In this dissertation, we consider differential games for multi-agent systems under distributed information where every agent is only able to acquire information about the others according to a directed information graph of local communication/sensor networks. Such games arise naturally from many applications including mobile robot coordination, power system optimization, multiplayer pursuit-evasion games, etc. Since the admissible strategy of each agent has to conform to the information graph constraint, the conventional game strategy design approaches based upon Riccati equation(s) are not applicable because all the agents are required to have the information of the entire system. Accordingly, the game strategy design under distributed information is commonly known to be challenging. Toward this end, we propose novel open-loop and feedback game strategy design approaches for Nash equilibrium and noninferior solutions with a focus on linear quadratic differential games. For the open-loop design, approximate Nash/noninferior game strategies are proposed by integrating distributed state estimation into the open-loop global-information Nash/noninferior strategies such that, without global information, the distributed game strategies can be made arbitrarily close to and asymptotically converge over time to the global-information strategies. For the feedback design, we propose the best achievable performance indices based approach under which the distributed strategies form a Nash equilibrium or noninferior solution with respect to a set of performance indices that are the closest to the original indices. This approach overcomes two issues in the classical optimal output feedback approach: the simultaneous optimization and initial state dependence. The proposed open-loop and feedback design approaches are applied to an unmanned aerial vehicle formation control problem and a multi-pursuer single-evader differential game problem, respectively. Simulation results of several scenarios are presented for illustration