Distributed Algebraic Riccati Equations in Multi-Agent Systems

In the present paper we obtain a closed-form solution for the class of continuous-time algebraic Riccati equations (AREs) with vanishing state weight. The AREs in such a class solve a minimum energy control problem. The present work extends on previous preliminary results by considering the more challenging case of repeated unstable eigenvalues. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove comparable in terms of numerical precision over current solving algorithms.

We present a unified theory of matrix pencil techniques for solving both continuous and discrete-time algebraic Riccati equations (AREs) under fairly general conditions on the coefficient matrices. The theory applies to a large class of AREs and Riccati-like equations arising from the singular H/sup /spl infin//- and H/sup 2/-control problems, singular linear quadratic control, the 4-block Nehari problem, or from singular J-spectral factorizations. The underlying concept is the so-called proper deflating subspace of a (possibly singular) matrix pencil in terms of which necessary and sufficient conditions for the solvability of Riccati equations are given. It is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm.

Rotational Motion Control by Feedback with Minimum L1-Norm

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This paper proposes a multi-rate state estimation method using a Kalman filter and an H-infinity filter. First, we derive periodic steady state Kalman/H-infinity gains for multi-rate systems since a state estimation problem with multi-rate sensors reduces to that for periodic time-varying systems. This result enables one to compute the optimal Kalman/H-infinity gains off-line using a class of algebraic Riccati equations. Furthermore, the effectiveness of the proposed design method is demonstrated with numerical simulations.

The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration.

: Numerical issues related to the computational solution of the algebraic matrix Riccati equation were studied. The approach used the generalized eigenproblem formulation for the solution of general forms of algebraic Riccati equations arising in both continuous- and discrete-time applications. These general forms result from control and filtering problems for systems in generalized state space form. A Newton-type iterative refinement procedure for the generalized Riccati solution was derived. Balancing to improve numerical condition was studied. A Fortran package called RICPACK was developed. Numerical experiments with RICPACK were performed to investigate a number of proposed condition numbers. Experience with RICPACK to date indicates that it is the most powerful software yet developed to solve general classes of Riccati equations reliably for a few hundred or less problems. The special structure of models of physical systems given in linear second-order form was also examined. Exploiting that structure in solving associated Riccati equations was studied. Tests for controllability and observability were derived in terms of the original second-order-model matrices. Originator-supplied keywords include: Riccati equations, Generalized matrix Riccati equations, Discrete Riccati equation, Kalman filtering, Numerical condition, Balancing, Generalized eigneproblem, Numerical software, Second-order models.

This paper studies leaderless state synchronization of continuous-time homogeneous multi-agent systems (MAS) with identical general linear agents. The communication network is with full-state coupling, or partial-state coupling but with additional information exchange. We assume that the communication topology is completely unknown and undirected. An adaptive nonlinear dynamic protocol design is presented to achieve leaderless state synchronization for a MAS. For the full-state coupling case, the protocol design is based on two methods. One is algebraic Riccati equation (ARE) method, and the other is asymptotic time-scale eigenstructure assignment (ATEA) method. We show that the adaptive nonlinear dynamic protocol is universal and does not depend on any information of communication topology such as lower bound of non-zero eigenvalue for the Laplacian matrix associated with the communication topology and the number of agents.

We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.

In this paper, we propose a method for estimating the algebraic Riccati equation (ARE) with respect to an unknown discrete-time system from the system state and input observation. The inverse optimal control (IOC) problem asks, “What objective function is optimized by a given control system?” The inverse linear quadratic regulator (ILQR) problem is an IOC problem that assumes a linear system and quadratic objective function. The ILQR problem can be solved by solving a linear matrix inequality that contains the ARE. However, the system model is required to obtain the ARE, and it is often unknown in fields in which the IOC problem occurs, for example, biological system analysis. Our method directly estimates the ARE from the observation data without identifying the system. This feature enables us to economize the observation data using prior information about the objective function. We provide a data condition that is sufficient for our method to estimate the ARE. We conducted a numerical experiment to demonstrate that our method can estimate the ARE with less data than system identification if the prior information is sufficient.

Algebraic Riccati Equation (ARE) solutions play an important role in many optimal/suboptimal control problems. However, a key assumption in formulation and solution of the ARE is a certain ‘regularity condition’ on the feedthrough term D of the system. For example, formulation of the ARE requires nonsingularity of \(D+D^T\) in positive real dissipative systems and, in the case of bounded real dissipative systems, one requires nonsingularity of \(I-D^T D\). Note that for lossless systems \(D+D^T=0\), while for all-pass systems \(I-D^TD=0\); this rules out the formulation of the ARE. Noting that the ARE solutions are also extremal “storage functions” for dissipative systems, one can speak of storage function for the lossless case too. This contributed chapter formulates new properties of the ARE solution; we then show that this property is satisfied by the storage function for the lossless case too. The formulation of this property is via the set of trajectories of minimal dissipation. We show that the states in a first-order representation of this set satisfy static relations that are closely linked to ARE solutions; this property holds for the lossless case too. Using this property, we propose an algorithm to compute the storage function for the lossless case.

In this paper, a time-varying leader-following consensus of multi-agent systems under fixed, connected and undirected communication topology is presented. In the proposed method, the dynamics of each agent including the followers and their corresponding leader is a linear nth order system. Moreover, the communication topology between the leader and its neighbours depends upon bounded and time-varying functions, assumed to remain connected as time passes. To tackle this problem, a set of time-varying distributed control laws for each follower agent is designed, based on algebraic graph theory, algebraic Riccati equation and Lyapunov direct method. Simulation results indicate the effectiveness of the proposed method and display convergence to consensus is achieved in a finite time via time-varying distributed control laws.

This article investigates the consensus problem of nonlinear leader-following multi-agent systems (MASs) via adaptive impulsive control. A distributed impulsive controller is employed, in which an adaptive updating law in the discrete-time setting is designed for the impulsive gain. Moreover, in order to reduce the communication cost, a distributed self-triggered strategy is proposed to determine when the impulsive instant occurs. Some criteria are derived to guarantee consensus of leader-following MASs based on Lyapunov stability theory and algebraic Riccati equation. It is proved that the self-triggered impulsive sequence does not exhibit Zeno behaviour. Finally, an illustrative example is presented to empirically validate the effectiveness of the theoretical results.

The secure consensus problem of multiagent systems with fixed topology and external disturbances is investigated in this paper. Along with external disturbances, the studied multiagent system is also subjected to denial-of-service (DoS) attacks. First, some graph-based Lyapunov functions are presented for the robust consensus analysis. Second, the duration and attack frequency are introduced to quantitatively analyze the DoS attacks’ effects on the consensus. Third, the gain of the controller is acquired by solving an algebraic Riccati equation (ARE), which can always be guaranteed compared with solving linear matrix inequalities (LMIs). Finally, a numerical simulation of a microgrid test system is provided to demonstrate the effectiveness of the proposed strategy.

Consensusability of multi-agent systems with time-varying communication delay