Standard Algebraic Riccati Equation Research Articles

Distributed optimal consensus protocol for high-order integrator-type multi-agent systems

Optimal consensus control of high-order multi-agent systems (MASs) modeled by multiple integrator-type dynamics is studied. A fully distributed optimal control protocol that achieves the specific consensus behavior is designed for MASs with linear dynamics, where topology-dependent conditions are removed. Further, a distributed consensus protocol for high-order nonlinear MASs with one-sided Lipschitz continuity is presented using the optimization approach, and the optimal solution can be obtained by solving a standard algebraic Riccati equation. Some numerical examples are finally provided to illustrate the effectiveness of the presented approaches.

Optimal control and stabilization for linear mean‐field system with indefinite quadratic cost functional

AbstractIn this paper, we consider an optimal control and stabilization problem of linear mean‐field (MF) system, where the quadratic cost functional is allowed to be indefinite. Inspired by the equivalent cost functional method, we introduce a subset, which helps us to investigate the convergence property of generalized differential Riccati equations arising in indefinite mean‐field linear quadratic (MF‐LQ) problems with finite horizon. A coupled generalized algebraic Riccati equation (GARE) is thus obtained. More importantly, the solution pair of corresponding GARE can be decomposed into a solution pair of a coupled standard algebraic Riccati equation (SARE) and a matrix pair in the subset we introduce. Thus, an equivalent relationship is established between GARE and SARE. With this equivalence, we derive necessary and sufficient conditions to stabilize linear MF‐system with indefinite weighting matrices in mean‐square sense.

Distributed Fault-Tolerant Bipartite Output Synchronization of Discrete-Time Linear Multiagent Systems.

This article studies the distributed fault-tolerant bipartite output synchronization problem of discrete-time linear multiagent systems (MASs) with process faults under a general directed signed graph. The reference signal is generated by an autonomous exosystem, which can also be seen as a leader. All followers are divided into two subgroups with antagonistic interactions, and the followers in each subgroup are cooperative. We aim to solve the bipartite fault-tolerant control (FTC) problem via the output regulation theory such that bipartite output synchronization can be achieved in the presence of process faults, that is, the outputs of followers with different subgroups can approach the output of exosystem with the same magnitude and the opposite sign regardless of process faults. To estimate the states and the faults of each follower, a simultaneous state and fault estimator based on the neighboring signed output estimation error and the standard discrete-time algebraic Riccati equation (ARE) is designed. Besides, a new exosystem observer with two classes of convergence conditions relying on the respective solutions of standard and modified AREs is provided. All eigenvalues of the exosystem matrix can lie completely outside the unit circle. Based on these estimations, we present a distributed fault-tolerant output feedback controller, which can overcome the no-loops constraint. Finally, simulation results are given to demonstrate the analytic results.

Formula omitted]-learning algorithm in solving consensusability problem of discrete-time multi-agent systems

This paper solves the consensusability problem for the single-input discrete-time multi-agent system (MAS) over directed graphs by the linear quadratic regulator (LQR) design method. It is proved that the maximum consensus region is exactly the largest gain margin (GM) of LQR. Based on this, the necessary and sufficient condition on consensusability is derived by solving a standard algebraic Riccati equation (ARE). The developed framework permits that the consensusability problem can be solved when the agents’ models are completely unavailable. Q-learning algorithm is employed to compute the maximum consensus region and implement the consensus protocol design. The algorithm runs only on a single agent rather than the intercommunicating MAS hence the unattainable initial admissible protocols are not required. A numerical example is given to illustrate the effectiveness of the developed methods.

On the Existence of a Stabilizing Solution of Modified Algebraic Riccati Equations in Terms of Standard Algebraic Riccati Equations and Linear Matrix Inequalities

In this letter, we study conditions for the existence of stabilizing solutions of two classes of modified discrete algebraic Riccati equations (MAREs) emerging in stochastic control problems. In order to do so, we first rewrite each MARE in terms of a standard ARE subject to specific constraints, which allows us to connect their solution with a set of linear-control constrained problems. With this result we also determine, for each MARE, a linear matrix inequality problem whose feasibility is guaranteed if and only if a stabilizing solution of the original MARE exists.

Discrete-time [formula omitted] optimal control under intermittent and lossy communications

The paper studies the H2 optimal control for discrete-time systems under the constraint that the information exchange between the sensor- and actuator-side parts of the controller is intermittent, a priori unknown, and independent of the process. A closed-form analytic solution to the problem is derived. The solution is based on two standard algebraic Riccati equations and is readily implementable. An implementation maintaining optimality of the controller under packet losses, which can be modeled by the intermittent communication framework, is also proposed. On the technical side, the paper reveals an issue related to the need for extending the time-varying error system to the whole discrete time axis Z in order to solve the corresponding H2 model-matching problem and proposes a procedure to resolve it.

On Discrete-Time H Optimization under Intermittent Communications

Abstract The paper studies the H2 optimal control in discrete-time systems under the constraint that the information exchange between the sensor- and actuator-side parts of the controller is intermittent, a priori unknown, and independent of the process. A closed-form analytic solution to the problem is derived. The solution is based on two standard algebraic Riccati equations and is readily implementable.

Static analysis of the complex multilayered soil field using the modified scaled boundary finite element method

This paper extends the modified scaled boundary finite element method (SBFEM) to analyse the static behaviors of the complex model with rigid bedrock. In this method, the original scaling center is replaced by scaling line, which makes the proposed method be very suitable for analyzing the horizontal multilayered soil model. The displacement governing equation is obtained by introducing the dual variables in the Hamilton system. This new derivation approach for the modified SBFEM is firstly investigated. Meanwhile, the derivation process is greatly concise and straightforward by introducing the dual vectors. The displacement equation is transformed into an equivalent system of nonlinear first order differential equation. The static stiffness equation is a standard algebraic Riccati equation which can be solved directly. By coupling with the nearfield, the global static governing equation of the horizontal layered unbounded domain is obtained. By that, the accurate global displacement of the modified SBFEM can be solved. In order to model the inclined model, the sub-structure method is presented. Numerical examples demonstrate the accuracy and wide applicability of the proposed method for the complex multilayered model.

On the Existence of a Mean-Square Stabilizing Solution to a Modified Algebraic Riccati Equation

In this paper, we investigate a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE), which arises in our previous work on the linear quadratic optimal control for linear time-invariant discrete systems with random input gains. An explicit necessary and sufficient condition ensuring the existence of a mean-square stabilizing solution is given directly in terms of the system parameters with the help of theory of positive operators and the assumption of observability or detectability of certain stochastic systems is no longer needed. Such a necessary and sufficient condition is compatible with that for the existence of a stabilizing solution to the standard definite algebraic Riccati equation (ARE).

Formula omitted] optimization for systems with adobe input delays: A loop shifting approach

This paper studies the H2 optimization problem for systems with adobe input delay. These are systems having only two (possibly vector) input channels: one is delay free and the other is delayed. We present a solution based on the reduction of the problem to an equivalent delay-free problem via simple loop shifting arguments. This results in a solution based on two standard algebraic Riccati equations, which are associated with the delay-free version of the problem. The optimal controller is in the dead-time compensator form. We also derive an explicit and transparent expression for the cost of delay and (in the case when the problem is solvable without the delayed channel) a numerically stable form of the optimal solution, which includes exponentials of Hurwitz matrices only. The approach is readily extendible to more general multiple input/output delay cases.

New results in the Lyapunov-type algorithm for algebraic MCV Riccati equations

In this paper, we give a new proof for the convergence of the Lyapunov-type algorithm which can be used to compute the iterative solutions of the coupled algebraic Riccati equations appearing in the minimal cost variance control problem. This proof uses existence argument of a positive definite solution of a standard algebraic Riccati equation. Moreover, we will show that this algorithm is well defined and converges to a unique pair of stabilizing solutions of the coupled algebraic equations under study. A mean result of this work is that the existence of this stabilizing solution does not depend on the detectability assumption.

Optimal Robust Fault Detection for Linear Discrete Time Systems

This paper considers robust fault‐detection problems for linear discrete time systems. It is shown that the optimal robust detection filters for several well‐recognized robust fault‐detection problems, such as ℋ−/ℋ∞, ℋ2/ℋ∞, and ℋ∞/ℋ∞ problems, are the same and can be obtained by solving a standard algebraic Riccati equatio n. Optimal filters are also derived for many other optimization criteria and it is shown that some well‐studied and seeming‐sensible optimization criteria for fault‐detection filter design could lead to (optimal) but useless fault‐detection filters.

Computation of general inner-outer and spectral factorizations

In this paper, we solve two problems in linear systems theory: the computation of the inner-outer and spectral factorizations of a continuous-time system considered in the most general setting. We show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic Riccati equation of order usually much smaller than the McMillan degree of the transfer function matrix of the system. The proposed procedures are completely general, being applicable for a polynomial/proper/improper system whose transfer function matrix could be rank deficient and could have poles/zeros on the imaginary axis or at infinity. As an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix. Numerical refinements and examples illustrating the proposed approach, are discussed in detail.

An efficient algorithm for the discrete-time algebraic Riccati equation

The authors develop an algorithm to solve the standard discrete-time algebraic Riccati equation by using a skew-Hamiltonian transformation and the square-root method. The algorithm is structure-preserving and efficient because the Hamiltonian structure is fully exploited and only orthogonal transformations are used. The efficiency and stability of the algorithm are analyzed, numerical examples are included.

Improved perturbation bounds for general quadratic matrix equations

Perturbation analysis of general algebraic quadratic matrix equations is presented, which include as particular cases the standard and descriptor algebraic matrix Riccati equations, arising in the optimal control and filtering of continuous, time-invariant, linear systems. The perturbation bounds derived are superior to the bounds, already available in the literature. They are applicable to the error analysis of general quadratic matrix equations. The results are based on the use of Lyapunov majorants, fixed point principles and detailed estimates of the norms of tensor products of matrices.

Perturbation bounds and characterisation of the solution of the associated algebraic Riccati equation

The paper deals with the associated algebraic matrix Riccati equation (AAMRE), closely related to the standard algebraic matrix Riccati equation arising in the theory of linear-quadratic optimisation and filtering. The sensitivity of the AAMRE relative to perturbations in its coefficients is studied. Both linear local (norm-wise and component-wise) and non-linear non-local perturbation bounds are obtained. The conditioning of the AAMRE is determined in particular. A full characterisation of the solution of AAMRE in terms of neutral subspaces of certain Hermitian matrix is given which is a counterpart of the characterisation of the solutions to the standard Riccati equation in terms of the invariant subspaces of the corresponding Hamiltonian matrix. A reliable method to obtain all solutions to AAMRE is briefly outlined.

Hermitian solutions of the equation [formula omitted

We consider the matrix equation X = Q + NX −1N∗ . Its Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil. We show that the equation admits both a largest and a smallest solution. The largest solution corresponds to the unique positive definite solution. The smallest solution is the unique negative definite solution if and only if N is nonsingular. If N is singular, no negative definite solution exists. An interesting relation between the given equation and a standard algebraic Riccati equation of Kalman filtering theory is also obtained. Finally, we present an algorithm which converges to the positive definite solution for a wide range of initial conditions.

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ( A,B,S −1/2 B *, D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H ∞ for positive real transfer functions of the form D+ S −1/2 B *( author− A) −1, B.

Generalized continuous-time Riccati theory

A Riccati-like equation which incorporates as special cases the standard continuous-time algebraic Riccati equation and the constrained continuous-time algebraic Riccati equation is introduced and studied. A characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil.

On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control

Existence of maximal solution is proved for a generalized version of the well-known standard algebraic Riccati equations which arise in certain stochastic optimal control problems.