Riccati Differential Equation Research Articles

Linear Exponential Quadratic Control for Mean Field Stochastic Systems

In this technical note, we consider linear exponential quadratic (LEQ) control for mean field stochastic differential equations (MFSDEs). The MFSDE includes the expectation value of state and control, and the objective functional is exponential of a quadratic functional in state, control, and their expectations. We obtain the explicit optimal solution as well as the optimal cost. The corresponding optimal solution is linear in state and its expectation, which is characterized by the Riccati differential equations (RDEs). The results are obtained by showing that after applying the completion of squares method, the remaining exponentiated stochastic integral and additional RDE terms can be eliminated together by taking expectation since they constitute the associated Radon-Nikodym derivative. As an extension of the problem, the LEQ zero-sum differential game is considered, for which we obtain the explicit optimal solution (saddle-point equilibrium) as well as the optimal cost.

Solution of Scalar Riccati Differential Equation of Fractional Order

The main objective of this paper is to generalize the scalar Riccati differential equation for factional order derivatives using Caputo definition, and then to find its approximate solution using the variational iteration method. The present work consists of the statement and the proof of the variational iteration formula and then prove of its convergent to the exact solution. Some illustrative examples are considered and simulated using Mathcad 15 computer program.

APPROXIMATE SOLUTION TO FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

Robust Protocol-Based ℋ∞ Consensus Control of Time-Varying Uncertain Multiagent Systems Subject to Missing Measurements

In this paper, we consider the design problem of a robust ℋ∞ consensus controller for discrete time-varying uncertain multiagent systems (DTVUMASs) with stochastic communication protocol (SCP) and missing measurements. The SCP is described by a set of random variables with a known probability to arrange signal transmission of addressed multiagent systems. Moreover, we depict the missing measurement phenomenon by a sequence of Bernoulli-distributed random variables having known probabilities. The controller parameters are designed to ensure that the closed-loop DTVUMASs satisfy the ℋ∞ performance with the satisfactory consensus criterion. Together with the completing squares approach and the stochastic analysis methodology, some sufficient conditions are proposed by solving coupled backward recursive Riccati difference equations (BRRDEs) to guarantee the ℋ∞ consensus performance. Finally, we present a numerical simulation example to illustrate the effectiveness of the designed controller design scheme.

Guidance strategies for interceptor against active defense spacecraft in two-on-two engagement

This paper investigates the guidance problem for interceptors against spacecraft with active defense in a two-on-two engagement. There are four adversaries in the engagement which are interceptor, protector, spacecraft and active defender. The interceptor is required to capture the spacecraft and evade the defender with the assist of the protector. Two classes of guidance laws are proposed based on norm differential game strategy and linear quadratic differential game strategy, respectively. Interceptor using norm differential game guidance laws is able to pursue the spacecraft without being intercepted by defender, even though the maneuverability of both interceptor and protector is lower than defender. Additionally, linear quadratic differential game guidance laws derived by numerical solution of Riccati differential equation take into account the control saturation, fuel cost and chattering phenomenon simultaneously. Finally, the effectiveness and performance of the developed guidance approaches are demonstrated by nonlinear numerical simulations.

Reduction of the optimal control problem for a magnetoelectric power drive

A linear-quadratic optimal control problem is considered for a singularly perturbed differential system, which describes the dynamics of a magnetoelectric. The optimal control law is constructed as feedback. Using the method of integral manifolds allows us to reduce the order of the matrix differential Riccati equation.

RETRACTED ARTICLE: Finite-Horizon Optimal Nonlinear Regulation and Tracking Using Differential State Dependent Riccati Equation

A number of computational techniques have been proposed for synthesizing nonlinear control based on state dependent Riccati equation (SDRE). Most of these techniques focusing on infinite-horizon regulator problems. This paper presents a novel and computationally efficient online technique used for finite-horizon nonlinear regulation and tracking problems. In the finite-horizon SDRE problems, the solution is time dependent and a differential equation, rather than an algebraic equation in infinite-horizon SDRE. This technique is based on change of variables that converts the differential Riccati equation to a linear Lyapunov differential equation. Simulation results are provided to illustrate the effectiveness of the proposed technique.

Retraction: Finite-Horizon Optimal Nonlinear Regulation and Tracking Using Differential State Dependent Riccati Equation

Retraction: Finite-Horizon Optimal Nonlinear Regulation and Tracking Using Differential State Dependent Riccati Equation

Differential Riccati equation‐based adaptive stabilization for switched linear systems

SummaryIn this paper, a novel adaptive control design is proposed to stabilize a class of switched linear systems with parametric uncertainties and disturbances. In particular, a family of differential Riccati equations that holds for the finite switching intervals are established, of which the closed‐form solutions are exploited to develop the Lyapunov function for the closed‐loop system. This Lyapunov function is imposed to be decreasing at and between two arbitrary switching instants by solving a group of matrix inequalities, based on which the adaptive controllers for switched linear system with and without disturbances are proposed, respectively. In the nondisturbed setting, asymptotic stability of the adaptive switched linear system is achieved, while the system is bounded in a mean square sense in the disturbed setting. A numerical example of flight control for F4E fighter aircraft is used to illustrate the proposed adaptive control methodology.

Pareto optimal strategy for linear stochastic systems with H∞ constraint in finite horizon

In this paper, we derive conditions for the existence of Pareto efficient strategy and Pareto solution under H∞ constraint for the linear quadratic (LQ) finite horizon cooperative differential game of stochastic systems with state-, control- and disturbance-multiplicative noise. Firstly, we present a stochastic bounded real lemma (SBRL) with any initial condition for the considered stochastic system. Next, a necessary and sufficient condition for Pareto optimal strategy under the H∞ constraint is researched by two cross-coupled generalized differential Riccati equations (GDREs), which is the key contribution of this paper. Another contribution is that the Pareto solution for every controller is given under the Pareto efficient strategy and worst-case external disturbance. Finally, a practical example is given to illustrate the effectiveness of our results.

Application of Generalized Differential Transform Method to Special Cases of Riccati Differential Equation of Fractional Order

In this paper, we acquire the inexact solutions of Special cases of Riccati Differential equation of Fractional order using Generalized Differential Transform Method (GDTM). The fractional derivatives are described in the Caputo sense. Accuracy and competence of the proposed method is verified through numerical solution of some special cases of Riccati Differential equation of fractional order. The obtained results reveal that the performance of the proposed method is specific and predictable.

Finite-Horizon Robust Suboptimal Control-Based Impact Angle Guidance

This paper presents impact angle guidance based on finite-horizon robust optimal control. This approach, a fusion of newly proposed finite-time state-dependent Riccati equation and integral sliding-mode-control, solves finite-horizon tracking problem for input-affine nonlinear systems with specified terminal conditions. It also ensures robustness against external disturbances and uncertainties from beginning. The technique for ensuring specified final-time involves deriving closed-form control expression in terms of approximate solution of differential Riccati equation, which further reduces to solutions of algebraic Riccati and Lyapunov equations. This yields computationally efficient algorithm. To alleviate chattering, the supertwisting algorithm is incorporated with the proposed technique. Guidance strategy is derived after converting impact angle problem to tracking one using the proposed algorithm. Efficacy of the proposed guidance is vindicated for various engagement geometries, and also compared with an existing guidance strategy.

Finite Horizon Optimal Nonlinear Spacecraft Attitude Control

This research is devoted to present spacecraft attitude control via a finite-horizon nonlinear optimal control technique. The spacecraft kinematics are represented using the modified Rodrigues parameters, which possess singularities for eigenaxis rotations greater than 180 degree. The proposed technique is based on State Dependent Riccati Equation (SDRE). In this technique, the differential Riccati equation is converted to a linear Lyapunov differential equation. This technique can be applied for finite-horizon nonlinear regulation and tracking problems. The proposed technique is effective for a wide range of operating points. Simulation results are given to illustrate the effectiveness of the finite horizon technique.

Computing the complex wave and dynamic behavior of one-dimensional phononic systems using a state-space formulation

Inspired by advances in photonics, band structures have been used to investigate the wave behavior along phononic structures as well as their singular properties such as band gaps, defects and topological modes. The most commonly used method for computing band structures is the plane wave expansion, but it cannot be used to compute the forced response of a finite structure with boundary conditions and applied loads. To obtain such dynamic response, it is usually necessary to employ a full finite element model of the whole structure or spectral models that take advantage of the periodicity. This paper proposes a new spectral approach, based on a Riccati differential equation with the impedance as variable, to compute the transfer matrix of one-dimensional phononic systems with arbitrary geometric and material profiles. With this formulation, not only the complex band structure can be computed, but also the forced response. Results for ducts, rods and beams are presented and validated with the extended plane wave expansion and scaled spectral element methods. The proposed approach can also be applied to any one-dimensional periodic structures such as Levy plates, cylindrical shells and photonic systems. It can also be used to investigate with high accuracy and efficient computational cost the physical behavior of generic structures with non-slowly varying properties, gradient-index systems and cloaking devices.

Nash Tracking Game with Preview by State Feedback for Linear Continuous-Time Systems

In this paper, we study Nash tracking game problems with preview for linear continuous-time systems over finite time intervals. We introduce cross-coupled differential Riccati equations (CDREs) to give solutions of the tracking problems with multi-players. Corresponding to the coupled properties of the CDREs, the forms of preview compensator dynamics are cross-coupled. In this paper, we focus on the derivation of the necessary conditions for the solvability of the tracking problems by variational approach.

On the Existence of the Stabilizing Solution of a Class of Periodic Stochastic Riccati Equations

This paper is devoted to the characterization of existence and uniqueness conditions for the stabilizing solution of a large class of Riccati equations arising in stochastic dynamic games. As an application of the obtained results, we consider in a second step the problem of a zero-sum two players linear quadratic difference game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such a control problem, a crucial role is played by the stabilizing solution of the considered class of Riccati difference equations.

An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

Abstract This manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): D μ u ( x ) + a ( x ) u 2 ( x ) + b ( x ) u ( x ) = g ( x ) , 0 ≤ μ ≤ 1 , 0 ≤ x ≤ t , u ( 0 ) = d , $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$ where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.

Finite‐horizon fault estimation for time‐varying systems with multiple fading measurements under torus‐event–based protocols

SummaryIn this paper, the issue of the finite‐horizon H∞ fault estimation is dealt with for a class of discrete time‐varying systems subject to randomly occurring faults and multiple fading measurements. The missing phenomena may occur in a random way from different sensors, which is represented by an individual stochastic variable meeting a certain probability distribution. Furthermore, in order to alleviate the communication burden, the torus‐event–based protocols are adopted to schedule the data transmissions only when some significant events occur. Our aim of the presented issue is to estimate the fault such that, with multiple fading measurements via the received information governed by torus‐event–based protocols, the H∞ index is satisfied over a given finite horizon. Sufficient conditions are obtained for the desired time‐varying estimator in terms of the technique of stochastic analysis and the methods of completing squares. The desired estimator gains are calculated by working out two backward recursive Riccati difference equations. Finally, a numerical simulation is given to verify the usefulness of our designed fault estimation approach.

Tracking Control for the Connection Relationships of Discrete-time Complex Dynamical Network Associated with the Controlled Nodes

In this paper, a general discrete-time complex dynamical network is regarded to be composed of the nodes subsystem and the links subsystem which are mutually coupled. Different from the existing researches on the stabilization and synchronization of nodes group, this paper mainly focuses on the dynamical behavior of connection relationships between the nodes, and the network nodes only play a helping and secondary role in the dynamics of connection relationships. Due to the state of the links subsystem is difficult to be measured accurately in practice appliances, the Riccati difference equation without any control input is employed as the dynamic model of the links subsystem, in which the dynamic coupling term is the first order polynomial about the state of nodes. With the helping of controlled nodes, the tracking problem is discussed for the links subsystem. By using the coupling algebraic relation between the reference tracking targets of the nodes subsystem and the given tracking target of the links subsystem, the tracking control scheme is proposed for the nodes subsystem to force the links subsystem converges asymptotically to the given target. Finally, the simulations are used to show the validity of the method in this paper.

Solving Riccati differential equation using Fourier polynomial basis

Solving Riccati differential equation using Fourier polynomial basis