Algebraic Riccati Research Articles

Yaw Angle Tracking Control Design of Underactuated AUV By Using State Dependent Riccati Equations (SDRE)-LQT

In realizing yaw angle control tracking on AUV, the use of the State Dependent Riccati Equations method based on Linear Quadratic Tracking (SDRE-LQT) is realized. This algorithm calculates changes in yaw angle tracking problems through calculation of parameter changes from online AUV with Algebraic Riccati Equations.So that the control signal given to the plant can follow the changing conditions of the plant itself.

Cooperative output feedback tracking control for multi-agent consensus with time-varying delays and switching topology

This paper investigates the consensus tracking problem for leader-following multi-agent systems with general dynamics. The agents that are at most critically unstable and with time-varying delays and arbitrary switching topology are considered in our work. A novel framework for cooperative tracking control with time-varying input delay and communication delay under switching topology is constructed. The distributed observer-based tracking protocol is proposed for desired tracking based on the truncated predictor feedback method, algebraic graph theory and Riccati equation. Moreover, we design the observer gain and feedback gain independent of each other. A simulation example is provided to demonstrate the effectiveness of our proposed method.

Event-Triggered Control Over Noisy Feedback Channels

In this paper, the stability of a discrete-time event-triggered control system over noisy feedback channels is analyzed. The transmission between the controller and the actuator is triggered by an event involving estimated control, desirable control, and a trigger threshold. It is revealed that if the trigger threshold is less than a bound determined by the system and controller matrices, then the closed-loop system is mean square stable. In addition, two equivalent conditions based on algebraic Riccati inequality/equation (ARI/ARE) are proposed to facilitate further analysis.

Solving Matrix Equations on Multi-Core and Many-Core Architectures

We address the numerical solution of Lyapunov, algebraic and differential Riccati equations, via the matrix sign function, on platforms equipped with general-purpose multicore processors and, optionally, one or more graphics processing units (GPUs). In particular, we review the solvers for these equations, as well as the underlying methods, analyze their concurrency and scalability and provide details on their parallel implementation. Our experimental results show that this class of hardware provides sufficient computational power to tackle large-scale problems, which only a few years ago would have required a cluster of computers.

Adaptive Consensus of High Order Multi-Agent Systems with Unknown Nonlinear Dynamics

In this paper, high order consensus problem of multi-agent systems with non-identical unknown nonlinear dynamics following an unknown and nonlinear leader is investigated in networks with fixed and switching topologies. By parameterizations of unknown nonlinear dynamics of agents in the system, neighbor-based adaptive consensus algorithms are proposed by incorporating consensus errors in addition to relative position feedback. Base on algebraic graph theory, Lyapunov theory, Riccati inequalities and PE condition, analysis of stability and parameter convergence of the proposed algorithm are conducted. Finally, a simulation in switching networks is worked out to illustrate the effectiveness of the theoretical results.

Leader-Following Consensus of Linear Multiagent Systems with State Observer under Switching Topologies

The leader-following consensus problem of higher order multiagent systems is considered. The dynamics of each agent are given in general form of linear system, and the communication topology among the agents is assumed to be directed and switching. To track the leader, two kinds of distributed observer-based consensus protocols are proposed for each following agent, whose distributed observers are used to estimate the leader’s state and tracking error based on the relative outputs of the neighboring agents, respectively. Some sufficient consensus conditions are established by using parameter-dependent Lyapunov function method under a class of directed interaction topologies. As special cases, the consensus conditions for balanced and undirected interconnection topology cases can be obtained directly. The protocol design technique is based on algebraic graph theory, Riccati equation, and Sylvester equation. Finally, a simulation example is given to illustrate our obtained result.

The shift techniques for a nonsymmetric algebraic Riccati equation

In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure-preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.

Stochastic optimal control of flexible aircraft taxiing at constant or variable velocity

Active control landing gears are used to alleviate vibration during aircraft taxiing. A nonlinear stochastic dynamics model is established, considering the aircraft body pitch movement and elastic vibration excited by the random runway. The equivalent linearization method is adopted to ensure the model linearity near the balance point, and the Gaussian random process of the runway is generated from the Gaussian white noise using a shape filter. Based on the stochastic optimal control theory, the LQG controller is designed along with weighted quadratic performance index for a better ride comfort, shock absorption, road holding and least energy expenditure. The algebraic Riccati and Lyapunov equations are solved to obtain stationary response while taxiing aircraft at a constant velocity and the differential Riccati and Lyapunov equations are solved to obtain the nonstationary response while taxiing aircraft at a variable velocity. The aircraft dynamic responses are obtained through the runway random process modeled by Monte Carlo method. Simulation results show that active control landing gear can give a better ride comfort, shock absorption and road holding performance no matter whether taxiing is at constant or variable velocity.

Higher-order terms in asymptotic expansion for information loss in quantized random processes

Uniform quantization of random vectors onto e-grids eℤ n is considered. Higherorder terms in asymptotic expansions for the entropy of the e-quantized random vector and for the loss of the mutual information between two random vectors under such quantization as e→0+are obtained. The coefficients in these asymptotics are explicitly calculated for Gaussian distributed vectors. Taken for initial segments of stationary Gaussian sequences, these factors have limit average values per unit of time. For such sequences governed by state-space equations, computation of these average values is reduced to solutions of algebraic matrix Riccati and Lyapunov equations.

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.

A Nonrecursive Solution Method for the Linear-Quadratic Optimal ControlProblem with a Singular Transition Matrix

In the optimal linear regulator problem the control vector is usually determined by solving the algebraic matrix Riccati equation using successive substitutions. This, however, can be rather inefficient from a computational point of view. A nonrecursive method which requires that the transition matrix is nonsingular has been proposed by Vaughan (1970). In the present paper we present a nonrecursive solution to the matrix Riccati equation for the case that the transition matrix may be singular. We show that this procedure leads to the same numerical results as the standard iteration of the matrix Riccati equation.

A practical implementation for solutions to the algebraic matrix Riccati equation in a LQCM setting

Abstract The linear-quadratic control model (LQCM) is very popular due to its applicability to many economic situations. An infinite-horizon equilibrium solution to such a model requires that we solve the algebraic matrix Riccati Equation. Numerical methods based on a modified Newton–Raphson iteration converge quadratically to this solution. We show that the computational complexity of each iteration can be substantially reduced from O(n6) to O(n4) by applying Broyden’s method of rank-1 updates. This method achieves superlinear converges. Moreover, it maintains the generality of the original method in that it does not impose constraints on matrix structure.

Eigenvalue upper bounds of the solution of the continuous Riccati equation

New upper bounds for the solution eigenvalues of the continuous algebraic matrix Riccati equation are developed. They include bounds of the extreme eigenvalues, the summation and product of eigenvalues, the trace, and the determinant. It is shown that the majority of the present eigenvalue bounds, expressed in concise forms, are less restrictive and sharper than existing results.

Optimal entire eigenstructure assignment of discrete-time linear systems

An improved method which assigns the closed-loop eigenvalues of a discrete-time linear system in desired preselected stable locations and which simultaneously selects those eigenvectors which satisfy a quadratic cost criterion with suitable weighting matrices is presented. The optimal feedback gain-matrix is determined without solving the algebraic matrix Riccati equation. The proposed explicit solution of the Riccati equation is feasible for both real and complex closed-loop stable eigenvalues.

Spectral norm and trace bounds of algebraic matrix Riccati equations

Upper bounds on spectral norm and trace for the solution of continuous algebraic matrix Riccati equations are proposed. The proposed bounds are based on a form of the general analytic solution. It is shown that, in many cases, the spectral norm and the trace bounds are better than other results.

Eigenvalue upper and lower bounds of the solution for the continuous algebraic matrix Riccati equation

A lower bound and several upper bounds of the solution eigenvalues for the continuous algebraic matrix Riccati equation are developed. It is shown that the lower bound is new and tighter than the previous reported one and the upper bounds are concise and less restrictive than the majority of those appeared in the literature. Furthermore, by these results, some bounds for the extreme eigenvalues of the solution matrix of the continuous Lyapunov equation are also defined.

半正定の2次および定数項をもつ行列Riccati代数方程式の半正定解

半正定の2次および定数項をもつ行列Riccati代数方程式の半正定解

The linear-Quadratic Control Problem Revisited

A long-run, average-cost, stochastic, linear-quadratic control problem that incorporates different time scales is considered. The system dynamics and the cost functional are modeled with the help of a locally square-integrable semimartingale process with independent increments and the corresponding predictable quadratic variation process. The solution of the control problem is given in terms of the solution of certain system of algebraic and differential Riccati equations. The model considered here embodies as particular cases the "traditional" continuous-time and discrete-time linear quadratic control problems, and is applicable, for example, to certain hybrid control problems that cannot be treated using existing control methods.

Neural Network Implementation for Realrtime Closed-Loop Motion Control of Redundant Robots

Abstract This paper describes an implementation of closed-loop motion control of redundant robots using a multi-layer neural network. The control scheme is developed based on the well known Hamilton-Jacobi-Bellman algorithm of optimal control. The algorithm requires solving the algebraic matrix Riccati equation, which is very computationally time consuming, and not practical for real time control applications. This paper illustrates that the computing time can be drastically reduced by replacing solving the Riccati equation with a neural network. The effectiveness of the neural controller is demonstrated by applying it to a PUMA 650 robot with a simulated prismatic joint inserted between the elbow and the wrist to obtain an extra degree of freedom.

Convergence of an iterative technique for algebraic Matrix Riccati equations and applications to transport theory

Abstract A monotone iterative technique is applied to a class of algebraic matrix Riccati equations that is satisfied by the reflection matrix for linear particle transport in a half-space. The convergence of the iteration as well as the rate of convergence are addressed. Moreover, we show theoretically and numerically that our iterative procedure, in general, is faster than that of Shimizu and Aoki.