Coupled Riccati Equations Research Articles

Online adaptive learning for team strategies in multi-agent systems

During mission execution in military applications, the TRADOC Pamphlet 525-66 Battle Command and Battle Space Awareness capabilities prescribe expectations that networked teams will perform in a reliable manner under changing mission requirements and changing team and individual objectives. In this paper we first present an overall view for dynamical decision-making in teams, both cooperative and competitive. Strategies for team decision problems, including optimal control, N-player games ( H∞ control, non-zero sum) and so on are normally solved offline by solving associated matrix equations such as the coupled Riccati equations or coupled Hamilton–Jacobi equations. However, using that approach, players cannot change their objectives online in real time without calling for a completely new offline solution for the new strategies. Therefore, in this paper we give a method for learning optimal team strategies online in real time as team dynamical play unfolds. In the linear quadratic regulator case, for instance, the method learns the coupled Riccati equations solution online without ever solving the coupled Riccati equations. This allows for truly dynamical team decisions where objective functions can change in real time and the system dynamics can be time-varying.

How to discretize differential systems in a systematic way

We present a systematic approach to the construction of discrete analogues for differential systems. Our method is tailored to first-order differential equations and relies on a formal linearization, followed by a Padé-like rational approximation of an exponential evolution operator. We apply our method to a host of systems for which there exist discretization results obtained by what we call the ‘intuitive’ method and compare the discretizations obtained. A discussion of our method as compared to one of the Mickens is also presented. Finally we apply our method to a system of coupled Riccati equations with emphasis on the preservation of the integrable character of the differential system.

A new sub-equation method applied to obtain exact travelling wave solutions of some complex nonlinear equations

By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein–Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.

Novel Stochastic Modeling and Filtering of the Attitude Quaternion

A novel continuous-time stochastic differential equation (SDE) for spacecraft attitude quaternion kinematics with state-multiplicative noise and a novel continuous-time exact optimal quaternion filter are developed in the framework of Ito (mean-square) calculus. The quaternion Ito SDE contains dissipative terms that ensure the mean-square stability of the process. Closed-form deterministic propagation equations are developed for the second-order moment of the quaternion. The quaternion filter produces the linear minimum-variance unbiased estimate of the quaternion from continuous observations with additive white noise. The filter gain computations include coupled Riccati equations of the estimation error matrix and of the quaternion second-order moment. These computations are not estimate-dependent and can therefore be performed off-line. The special case of gyro error white noise with independent identically distributed components is considered. The case of correlated components can be addressed straightforwardly. Additive gyro biases are easily handled via state augmentation. Extensive Monte-Carlo simulations are performed in order to validate the Ito model and to illustrate the proposed filter accuracy. Comparative simulations of the novel filter and of a standard additive extended Kaiman filter are performed both for attitude-only estimation and for attitude-bias estimation. For practical purposes, the novel modeling and filtering approach shows similar results as the standard approach when the process noise level is low. For high signal-to-noise ratio however, the numerical study suggests that the novel filter can increase the accuracy of a conventional Kaiman by orders of magnitudes.

Mixed ℋ︁2/ℋ︁∞ nonlinear filtering

AbstractIn this paper, we consider the mixed ℋ︁2/ℋ︁∞ filtering problem for affine nonlinear systems. Sufficient conditions for the solvability of this problem with a finite‐dimensional filter are given in terms of a pair of coupled Hamilton–Jacobi–Isaacs equations (HJIEs). For linear systems, it is shown that these conditions reduce to a pair of coupled Riccati equations similar to the ones for the control case. Both the finite‐horizon and the infinite‐horizon problems are discussed. Simulation results are presented to show the usefulness of the scheme, and the results are generalized to include other classes of nonlinear systems. Copyright © 2008 John Wiley & Sons, Ltd.

Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations

In this paper, we propose upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous coupled algebraic Riccati equation (CCARE) and the discrete coupled algebraic Riccati equation (DCARE), which are then used to infer upper bounds for the maximal eigenvalues of the solutions of each Riccati equation. By utilizing the upper bounds for the maximal eigenvalues of each equation, we then derive upper matrix bounds for the solutions of the CCARE and DCARE. Following the development of each bound, an iterative algorithm is proposed which can be used to derive tighter upper matrix bounds. Finally, we give numerical examples to demonstrate the effectiveness of the proposed results, making comparisons with existing results.

A complex ansatz method applied to nonlinear equations of Schrödinger type

By using the general solutions of a coupled Riccati equations and the relations between them a direct algebra approach is described to construct several types of closed-form travelling wave solutions for the coupled Schrödinger–KdV equations. Several new solutions are explicitly obtained with the aid of symbolic computation.

Constructing exact solutions to differential-difference equations via the coupled Riccati equations

An algebraic method to construct the exact solutions of nonlinear differential-difference equations is presented by introducing the coupled Riccati equations. As an example, we apply this method to the general lattice equation, the relativistic Toda lattice equations and the (2+1) dimensional Toda lattice equation. Some kink solitary wave solutions and complexiton solutions are obtained with the help of symbolic system Mathematica. Our method can also be applied to other nonlinear differential-difference equations.

Generalized Coupled Algebraic Riccati Equations for Discrete-time Markov Jump with Multiplicative Noise Systems

In this paper we consider the existence of the maximal and mean square stabilizing solutions for a set of generalized coupled algebraic Riccati equations (GCARE for short) associated to the infinite-horizon stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a sufficient condition, based only on some positive semidefinite and kernel restrictions on some matrices, under which there exists the maximal solution and a necessary and sufficient condition under which there exists the mean square stabilizing solution for the GCARE. We also present a solution for the discounted and long run average cost problems when the performance criterion is assumed to be composed by a linear combination of an indefinite quadratic part and a linear part in the state and control variables. The paper is concluded with a numerical example for pension fund with regime switching.

Generalized Coupled Algebraic Riccati Equations for Discrete-Time Markov Jump with Multiplicative Noise Systems

In this paper we consider the existence of the maximal and mean square stabilizing solutions for a set of generalized coupled algebraic Riccati equations (GCARE for short) associated to the infinite-horizon stochastic quadratic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a sufficient condition under which there exists the maximal solution and a necessary and sufficient condition under which there exists the mean square stabilizing solution for the GCARE.

Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface

Travelling wave solution for Ibragimov-Shabat equation, is obtained by using an improved sine-cosine method and the Wu's elimination method. An infinite number of conserved quantities for the above equation are also obtained by solving a set of coupled Riccati equations.

Optimal Linear Quadratic Regulator for Markovian Jump Linear Systems, in the presence of one time-step delayed mode observations

In this paper, we provide the solution to the optimal Linear Quadratic Regulator (LQR) paradigm for Markovian Jump linear Systems, when the continuous state is available at the controller instantaneously, but the mode is available only after a delay of one time step. This paper is the first to investigate the LQR paradigm in the presence of such mismatch between the delay in observing the mode and the continuous state at the controller. We show that the optimal LQR policy is a time-varying matrix gain multiplied by the continuous component of the state, where the gain is indexed in time by the one-step delayed mode. The solution to the LQR is expressed as a collection of coupled Riccati iterations and equations, for the finite and the infinite horizon cases respectively. In the infinite horizon case the solution of the coupled Riccati equations or a certificate of infeasibility is obtained by solving a set of linear matrix inequalities. We also explain the difficulties of solving the LQR problem when the mode is observed by the controller with a delay of more than one step. We show that, with delays of more than one time-step, the optimal control will be a time-varying nonlinear function of the continuous state and of the control input, without presenting an exact solution.

Policy iteration based feedback control

It is well known that stochastic control systems can be viewed as Markov decision processes (MDPs) with continuous state spaces. In this paper, we propose to apply the policy iteration approach in MDPs to the optimal control problem of stochastic systems. We first provide an optimality equation based on performance potentials and develop a policy iteration procedure. Then we apply policy iteration to the jump linear quadratic problem and obtain the coupled Riccati equations for their optimal solutions. The approach is applicable to linear as well as nonlinear systems and can be implemented on-line on real world systems without identifying all the system structure and parameters.

On some iterations for optimal control of jump linear equations

We consider different iterative methods for computing Hermitian solutions of the coupled Riccati equations of the optimal control problem for jump linear systems. We have constructed a sequence of perturbed Lyapunov algebraic equations whose solutions define matrix sequences with special properties proved under proper initial conditions. Several numerical examples are included to illustrate the effectiveness of the considered iterations.

Mixed [formula omitted] autopilot design of bank-to-turn missiles using fuzzy basis function networks

This paper presents a novel mixed H 2 / H ∞ autopilot design for bank-to-turn missiles. Based on sliding control theory, the proposed autopilot utilizes a fuzzy basis function network to approximate the unknown nonlinear functions of bank-to-.turn missile dynamics, enabling a mixed H 2 / H ∞ controller with weight updating laws to be applied to attenuate the effects of approximation errors and external disturbances. Moreover, the mixed H 2 / H ∞ autopilot design can minimize H 2 performance criteria under a desired H ∞ disturbance rejection constraint. Unlike general mixed H 2 / H ∞ control problems, the proposed H 2 / H ∞ autopilot with self-tuning fuzzy basis function network can avoid solving coupled Riccati equations by adequately selecting diagonal weighting matrices in the H 2 performance criterion and H ∞ constraint. Therefore, the proposed H 2 / H ∞ control approach can be implemented efficiently. Computer simulation results are presented to illustrate the effectiveness of the proposed mixed H 2 / H ∞ autopilot for BTT missiles.

An improved lower matrix bound of the solution of the unified coupled Riccati equation

A new lower matrix bound of the solution for the unified coupled algebraic Riccati equation (UCARE) is proposed. This bound improves the drawback of the results presented in a previous paper.

Properties of Coupled Riccati Equations in Stackelberg Games with Time Preference Rate

In this paper we deal with Stackelberg equilibrium in linear-quadratic games when a time preference rate is introduced in the players' cost functions with an open loop information structure. We apply general necessary conditions from (Simaan and Cruz, 1973b; Simaan and Cruz, 1973a) to our framework with a time preference rate. Such conditions lead to a set of coupled differential (or algebraic) Riccati equations. Necessary conditions for constant real solutions for the algebraic Riccati equations (ARE) are derived We give a bound for the number of real, constant and stabilizing solutions. When many solutions of (ARE) exist, a robustness study gives informations to choose the most robust control. A numerical example illustrates these results.

Mixed [formula omitted] control: the discrete-time case

A stochastic H ∞ and a mixed stochastic H 2/ H ∞ control problem for discrete-time systems are considered and solved. Conditions for existence of a solution are derived, based on the solvability of an equivalent minimax problem. In this framework, it is possible to consider at the same time both stochastic and deterministic disturbances highlighting their mutual effects. The controller can be designed by solving a system of three coupled Riccati equations. Such equations display how the H 2 -type minimization problem and the H ∞ -type constraint influence each other.

Extended Hyperbola Function Method and Its Application to Nonlinear Equations**The project supported by National Natural Science Foundation of China under Grant No. 10072013, and the State Key Basic Research Program under Grant No. G1998030600

An extended hyperbola function method is proposed to construct exact solitary wave solutions to nonlinear wave equation based upon a coupled Riccati equation. It is shown that more new solitary wave solutions can be found by this new method, which include kink-shaped soliton solutions, bell-shaped soliton solutions and new solitary wave. The new method can be applied to other nonlinear equations in mathematical physics.

Min–max Kalman filtering

The problem of H ∞-optimal state estimation of linear continuous-time systems that are measured with an additive white noise is addressed. The relevant cost function is the expected value of the standard H ∞ performance index, with respect to the measurement noise statistics. The solution is obtained by applying the matrix version of the maximum principle to the solution of the min–max problem in which the estimator tries to minimize the mean square estimation error and the exogenous disturbance tries to maximize it while being penalized for its energy. The solution is given in terms of two coupled Riccati difference equations from which the filter gains are derived. In the case where an infinite penalty is imposed on the energy of the exogenous disturbance, the celebrated Kalman filter is recovered. In the stationary case, where all the signals are stationary, an upper-bound on the solutions of the coupled Riccati equations is obtained via a solution of coupled linear matrix inequalities. The resulting filter then guarantees a bound on the estimation error covariance matrix. An illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the position.