Riccati Differential Equation Research Articles

Optimal Control of Temporal Networks With Variable Input and Node-Source Connection.

Many networked systems built upon real-life physical or social interactions have time-varying connections among individual units, where the temporal changes in connectivity and/or interaction strength lead to complicated dynamics. The temporal network model was proposed in the form of controlled linear dynamical systems acting in an ordered sequence of time intervals. One of the core challenges in network science is the control of networks and the optimization of the control strategy. However, most canonical frameworks for solving optimal control problems were established for static networks featuring constant topology. New theories and techniques are yet to be developed for the temporal networks, with an important case being that the input and the source-node connection are both variables. In this work, by formulating a quadratic energy cost without solving the Riccati differential equation, we show that the control effort can be reduced substantially by improving either the system trajectories or the input matrices. The two approaches are further combined in a coordinate descent framework, integrating linearly constrained quadratic programming, and a projected gradient descent method. Taken together, the results underline the potential of temporal networks as energy-efficient control systems and present strategies to improve the control input. Moreover, the proposed algorithms can serve as a starting point for future engineering of real-world temporal networks.

Solving Fractional Riccati Differential Equation with Caputo-Fabrizio Fractional Derivative

This article offers an analytical solution for the fractional Riccati differential equation in three distinct cases. These cases are determined by the discriminant and the analytical solution based on the properties of the Caputo-Fabrizio fractional derivative and integral. Several examples were tested using this analytical solution. It is noteworthy that various methods have yielded related results as indicated in the literature.

Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs

Abstract We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${\mathcal O}(h^{2}+\tau )$ for its solution, and conclude from it a rate of almost ${\mathcal O}(h^{2}+\tau )$ resp. ${\mathcal O}(h^{2}+\tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.

Open-loop and closed-loop Nash equilibria for the LQ stochastic difference game

This paper investigates the conditions for the existence of open-loop and closed-loop Nash equilibria for the linear quadratic (LQ) two-player stochastic difference game. It is shown that the former is characterized by the solvability of a forward–backward stochastic difference equation (FBSΔE), while the latter is described by the solvability of a class of coupled symmetric difference Riccati equations (ΔREs). Different from the existing works, the relationship between the closed-loop representation of open-loop Nash equilibrium and the closed-loop Nash equilibrium has been studied. It is found that the closed-loop representation of open-loop Nash equilibrium is different from the outcome of closed-loop Nash equilibrium for the general case. However, for the LQ zero-sum stochastic difference game, they are consistent.

Optimal tracking control for underactuated airship

A non-linear mathematical model of underactuated airship is derived in this paper based on Euler-Newton approach. The model is linearized with small disturbance theory, producing a linear time varying (LTV) model. The LTV model is verified by comparing its output response with the result of the nonlinear model for a given input signal. The verified LTV model is used in designing the LQT controller. The controller is designed to minimize the error between the output and required states response with acceptable control signals using a weighted cost function. Two LQT controllers are presented in this work based on two different costates transformations used in solving the differential Riccati equation (DRE). The first proposed assumption of costates transformation has a good tracking performance, but it is sensitive to the change of trajectory profile, whereas the second one overcomes this problem due to considering the trajectory dynamics. Therefore, the first assumption is performed across the whole trajectory tracking except for parts of trajectory profile changes where the second assumption is applied. The hybrid LQT controller is used and tested on circular, helical, and bowed trajectories. The simulation assured that the introduced hybrid controller results in improving airship performance.

Finite-Horizon H∞ State Estimation for Complex Networks With Uncertain Couplings and Packet Losses: Handling Amplify-and-Forward Relays.

This article is concerned with the state estimation problem for a class of complex networks (CNs) with uncertain inner couplings and packet losses over communication networks. The inner couplings are allowed to be uncertain and varying in a specific interval. The amplify-and-forward (AaF) relay protocols are introduced to improve the communication quality and enhance the propagation distance. The Bernoulli random variables are used to characterize the randomly occurring packet losses encountered in communication channels. The focus of this article is on the design of a state estimator for each node of CNs such that a prescribed H∞ performance constraint is satisfied for the dynamical error system over a finite horizon. A sufficient condition is first provided to verify the existence of the desired H∞ state estimator, and the estimator gain is then determined by solving two coupled backward Riccati difference equations (RDEs). Subsequently, a recursive state estimation algorithm is put forward that is suitable for online computation. Finally, a numerical example is given to demonstrate the effectiveness of the proposed estimation method.

Solutions and local stability of the Jacobsthal system of difference equations

<abstract><p>We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.</p></abstract>

Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. We propose a continuous-discrete McKean–Vlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.

Expected power utility maximization with delay for insurers under the 4/2 stochastic volatility model

In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer with delay under the $ 4/2 $ stochastic volatility model. Indeed, the objective of the insurer is to maximize the expected power utility of the terminal wealth and the average wealth on finite time horizon. The other objective is to maximize the growth rate of expected power utility per unit time on infinite time horizon. The wealth of the insurer is described by an approximation of the classical Cramér–Lundberg process. Then, these problems can be formulated as stochastic control problems with delay. A pair of forward-backward stochastic differential equations that are derived via the stochastic maximum principle has an explicit solution obtained by solving a Riccati differential equation. So, the optimal strategy of the finite time horizon problem can be constructed explicitly. And, by investigating asymptotics of the Riccati equation, the infinite time horizon problem can be solved explicitly. Finally, we present some numerical results to illustrate our model, optimal strategies and sensitivities of some parameters.

A SHORT SOLUTION OF THE LOCAL FRACTIONAL (2+1)-DIMENSIONAL DISPERSIVE LONG WATER WAVE SYSTEM

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional [Formula: see text]-dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported.

Novel closed-loop controllers for fractional linear quadratic time-varying systems

A novel method for finding the closed-loop optimal controllers of the finite-time fractional linear quadratic regulator systems is introduced. The necessary optimality conditions for the fractional optimal control problems are obtained. The matrix fractional differential Riccati equation is presented. A new method for solving the fractional differential Riccati equation is given for the first time which can also be applied to the integer order Riccati equation. Several illustrative examples (including a high order system) are presented to show the applicability and capabilities of the method.

Local well-posedness of the mortensen observer

The analytical background of nonlinear observers based on minimal energy estimation is discussed. It is shown that locally the derivation of the observer equation based on a trajectory with pointwise minimal energy can be done rigorously. The result is obtained by a local sensitivity analysis of the value function based on Pontryagin’s maximum principle and the Hamilton-Jacobi-Bellman equation. The consideration of a differential Riccati equation reveals that locally the second derivative of the value function is a positive definite matrix. The local convexity ensures existence of a trajectory minimizing the energy, which is then shown to satisfy the observer equation.

Stochastic optimal control and piecewise parameterization and optimization method for inventory control system improvement

In recent decades, stochastic control systems have been widely used in industrial production, biomedicine, aerospace, military strategy and so on. In this paper, an approximate optimal strategy derived from a stochastic linear quadratic (SLQ) optimal control problem is considered, and a piecewise parameterization and optimization (PPAO) method is proposed. Firstly, using the principle of dynamic programming, the control form of SLQ optimal control problem is relevant to a Riccati differential equation. It is well known that the Riccati differential equation is difficult to be solved analytically. Thus, we present a PPAO method for finding an approximate optimal strategy for stochastic control problems. Here, a piecewise parameter control can be obtained by solving first-order differential equations rather than Riccati differential equations. Finally, the inventory control problems with different dimensions are used to justify the feasibility of PPAO method, and the results compared with the original optimal control are given. The results show that parametric control greatly simplifies the control form, and a small model error is ensured.

Linear-quadratic optimal control of second-order parabolic systems

This paper considers the linear-quadratic (LQ) optimal control problem for systems governed by a class of second-order parabolic partial differential equations (PDEs). This problem is significant in many areas of mathematics, control, engineering and so on, and thus has received great attention in the past decades. Different from the previous articles where the operator is applied to present the controller, the main contribution of this paper is to propose the discretisation-then-continuousization method, which is explicit and implementable. The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations, and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.

Infinite Horizon H2/H∞ Control for Discrete-Time Mean-Field Stochastic Systems

In this paper, we deal with the H2/H∞ control problem in the infinite horizon for discrete-time mean-field stochastic systems with (x,u,v)-dependent noise. First of all, a stochastic-bounded real lemma, which is the core of H∞ analysis, is derived. Secondly, a sufficient condition in terms of the solution of coupled difference Riccati equations (CDREs) is obtained for solving the H2/H∞ control problem above. In addition, an iterative algorithm for solving CDREs is proposed and a numerical example is given for verification of the feasibility of the developed results.

Relationship between Nash Equilibrium Strategies and H2/H∞ Control of Mean-Field Stochastic Differential Equations with Multiplicative Noise

The relationship between finite-horizon mean-field stochastic H2/H∞ control and Nash equilibrium strategies is investigated in this technical note. First, the finite-horizon mean-field stochastic bounded real lemma (SBRL) is established, which is key to developing the H∞ theory. Second, for mean-field stochastic differential equations (MF-SDEs) with control- and state-dependent noises, it is revealed that the existence of Nash equilibrium strategies is equivalent to the solvability of generalized differential Riccati equations (GDREs). Furthermore, the existence of Nash equilibrium strategies is equivalent to the solvability of H2/H∞ control for MF-SDEs with control- and state-dependent noises. However, for mean-field stochastic systems with disturbance-dependent noises, these two problems are not equivalent. Finally, a sufficient and necessary condition is presented via coupled matrix-valued equations for the finite-horizon H2/H∞ control of mean-field stochastic differential equations with disturbance-dependent noises.

Finite Horizon Worst Case Analysis of Linear Time-Varying Systems Applied to Launch Vehicle

This article presents an efficient approach to compute the worst case gain of the interconnection of a finite time horizon linear time-varying system and a perturbation. The input/output behavior of the uncertainty is described by integral quadratic constraints (IQCs). A condition for the worst case gain of such an interconnection can be formulated using dissipation theory as a parameterized Riccati differential equation, which depends on the chosen IQC multiplier. A nonlinear optimization problem is formulated to minimize the upper bound of the worst case gains over a set of admissible IQC multipliers. This problem can be efficiently solved with a custom-tailored logarithmically scaled, adaptive differential evolution algorithm. It provides a fast alternative to similar approaches based on solving semidefinite programs. The algorithm is applied to the worst case aerodynamic load analysis of an expendable launch vehicle (ELV). The worst case load of the uncertain ELV is calculated under wind turbulence during the atmospheric ascend and compared to results from nonlinear simulation.

Fast SDDRE-Based Maneuvering-Target Interception at Prespecified Orientation

This study considers the 3-D guidance law based on target lead angle information and the state-dependent differential Riccati equation (SDDRE) scheme. In an application-oriented manner, it presents theories to significantly improve critical computational performance and thus aims at a fast implementation for impact-angle-constrained interception of agile maneuvering targets. More specifically, regarding the two major computational burdens using SDDRE, we have replaced the burden in numerical applicability checking by a simple, equivalent, and closed-form condition for the entire state space, which is actually the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dominant</i> burden as supported by complexity analysis and extensive validations. Notably, the proposed analysis not only complements the early findings of applicability guarantee in literature, but also promotes the efficiency of the proposed philosophy when compared to the classic method, where the latter has caused concerns/reservations due to its feasibility and difficulty. On the other hand, we have largely mitigated the second major burden of SDDRE by—after exhaustive trials—selecting the most efficient Riccati-equation solver until the latest benchmarks. Such evaluations are: 1) in favor of a much-less-known achievement, rather than the common QR-based benchmark and 2) subject to both numerical and hardware experiments including, notably, implementations on microcontrollers and field-programmable gate arrays.

Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.

Trajectory‐based robustness analysis for nonlinear systems

AbstractThis article considers the robustness of an uncertain nonlinear system along a finite‐horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time‐varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time‐domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst‐case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst‐case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati differential equation. This yields a computational method that avoids heuristics often used to solve DLMIs, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.