Riccati Differential Equation Research Articles

Linear-Quadratic Stochastic Stackelberg Differential Games for Jump-Diffusion Systems

This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem of the leader using the variational method. However, to obtain the state-feedback representation of the open-loop solution for the leader, there is a technical challenge due to the jump process. To overcome this limitation, we consider two different cases, in which the state-feedback type optimal control for the leader in terms of the ISRDE can be characterized by generalizing the Four-Step Scheme. Finally, in these two cases, we show that the state-feedback representation of the open-loop optimal solutions for the leader and the follower constitutes the Stackelberg equilibrium. Note that the (indefinite) LQ control problem of the leader is new and nontrivial due to the coupled FBSDE constraint induced by the rational behavior of the follower.

Explicit Symplectic-Precise Iteration Algorithms for Linear Quadratic Regulator and Matrix Differential Riccati Equation

Efficient, robust and precise algorithms for linear quadratic regulator (LQR) and matrix differential Riccati equation (MDRE) are essential in optimal control. However, there are lack of good algorithms for time-varying LQR problem because of the difficulty of solving the nonlinear time-varying MDRE. In this paper, we proved that the n-th order LQR problem is equivalent to n parallel 1-dim Hamiltonian systems and proposed the explicit symplectic-precise iteration method (SPIM) for solving LQR and MDRE. The explicit symplectic-precise iteration algorithms (ESPIA) designed with SPIM have three typical merits: firstly, there are no accumulative errors in the sense of long-term time which inherits from symplectic difference scheme; secondly the stiffness problem due to the inverse of matrix is avoided by the precise iteration method; and finally the algorithmic structure of ESPIA is simple and no extra assumptions are required. Systematic analysis shows that the time complexity of the symplectic algorithms for the n-th order LQR and MDRE is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) where k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> is the iteration times specified by the time duration. Numerical examples and simulations are provided to validate the performance of the ESPIA.

Unified Riccati Theory for Optimal Permanent and Sampled-Data Control Problems in Finite and Infinite Time Horizons

We revisit and extend the Riccati theory, unifying continuous-time linear-quadratic optimal permanent and sampled-data control problems, in finite and infinite time horizons. In a nutshell, we prove that: -- when the time horizon T tends to $+\infty$, one passes from the Sampled-Data Difference Riccati Equation (SD-DRE) to the Sampled-Data Algebraic Riccati Equation (SD-ARE), and from the Permanent Differential Riccati Equation (P-DRE) to the Permanent Algebraic Riccati Equation (P-ARE); -- when the maximal step of the time partition $\Delta$ tends to $0$, one passes from (SD-DRE) to (P-DRE), and from (SD-ARE) to (P-ARE). Our notations and analysis provide a unified framework in order to settle all corresponding results.

Exponential integrators for large-scale stiff Riccati differential equations

Riccati differential equations arise in many different areas and are particularly important within the field of control theory. In this paper we consider numerical integration for large-scale systems of stiff Riccati differential equations. We show how to apply exponential Rosenbrock-type integrators to get approximate solutions. Two typical exponential integration schemes are considered. The implementation issues are addressed and some low-rank approximations are exploited based on high quality numerical algebra codes. Numerical comparisons demonstrate that the exponential integrators can obtain high accuracy and efficiency for solving large-scale systems of stiff Riccati differential equations.

Formula omitted] and [formula omitted] optimal sampled-data controller synthesis: A hybrid systems approach with mixed discrete/continuous specifications

In this paper, a tractable sampled-data controller synthesis method is proposed for linear time-invariant plants that gives guarantees for stability and performance of the closed-loop system in terms of the H∞- and H2-norm. This is done by taking a so-called jump/hybrid systems approach, which allows formulating linear matrix inequalities using an explicit solution to the Riccati differential equation. Furthermore, the sampled-data problem is formulated such that continuous-time design techniques like H∞ loop-shaping can be used in a sampled-data context. To do so, it is essential to consider mixed discrete/continuous specifications, in which both discrete and continuous signals are weighted using weighting filters. This leads to a generalised plant that has both continuous-time and discrete-time dynamics. The controller design method is demonstrated on a benchmark example, in which we compare to existing results in the literature, and on a design example of reference tracking of a two-mass–spring–damper system.

Analysis of Nonlinear Active Noise Behavior of Fuzzy Controller Using Non-Perturbation Methods

In this paper, a Fuzzy controller model has been converted into a time-dependent nonlinear model and then quadratic Riccati differential equation was analyzed to satisfy the solution of the nonlinear active noise behavior of Fuzzy controller. Further, the approximate solutions of this equation using non-perturbation methods i.e., adomian decomposition method (ADM), variational iterational method (VIM) and homotopy perturbation method (HPM) were investigated. A comparison of these methods has also been given with tabular and graphical presentations. Our results reveal that VIM provides the closest approximate solution and fast convergence for the proposed model as compare to ADM and HPM.

EFFECT OF A BARRIER ON ENERGY EIGENVALUE OF THE NONRELATIVISTIC (SCHRÖDINGER EQUATION) AND SOME EXPECTATION VALUES OF COMBINED POTENTIALS

Schrödinger equation was proposed in theoretical physics to provide information and the behavior of a system of particles. The Schrödinger equation was recently solved for a combination of different potentials using different traditional methodologies. The studies do not consider the effect of a barrier between the potentials. In reality, the barrier between the potentials has an effect on the energy eigenvalues. In the present work, a Kratzer-Mie-type potential, a combination of Kratzer and Mie-type-constant potentials, was proposed as the interacting potential. The solutions of the radial Schrödinger equation was obtained in the presence of the combined potentials by considering a barrier between the potentials. The energy equation and the corresponding wave functions were explicitly calculated. The calculations were performed by using the methodology of supersymmetric quantum mechanics. This method involves the proposition of superpotential function as a solution to its Riccati differential equation. Numerical results were generated for some diatomic molecules using the energy equation and the diatomic model parameters. Some expectation values for the combined potential were calculated using Hellmann Feynman Theorem, and the effect of the barrier on the expectation values were numerically studied. It was observed that when a particle moves from the lower end to a higher end of a barrier, it absorbs energy from the system for some time. This same behavior was also noted when the particle penetrates through a barrier. Thus, its vibration will only depend on the initial energy, which is absorbed. It was equally seen that as the width of the barrier becomes larger than the height, the energy of the system decreases drastically.

Asymptotic dynamics of hermitian Riccati difference equations

In this paper, we consider the hermitian Riccati difference equations. Analogous to a Riccati differential equation, there is a connection between a Riccati difference equation and its associated linear difference equation. Based on the linear difference equation, we can obtain an explicit representation for the solution of the Riccati difference equation and define the extended solution. Further, we can characterize the asymptotic state of the extended solution and the rate of convergence. Constant equilibrium solutions, periodic solutions and closed limit cycles are exhibited in the investigation of asymptotic behavior of the hermitian Riccati difference equations.

Parallel Computing for Rolling Mill Process with a Numerical Treatment of the LQR Problem

The considerable increase in computation of the optimal control problems has in many cases overflowed the computing capacity available to handle complex systems in real time. For this reason, alternatives such as parallel computing are studied in this article, where the problem is worked out by distributing the tasks among several processors in order to accelerate the computation and to analyze and investigate the reduction of the total time of calculation the incremental gradually the processors used in it. We explore the use of these methods with a case study represented in a rolling mill process, and in turn making use of the strategy of updating the Phase Finals values for the construction of the final penalty matrix for the solution of the differential Riccati Equation. In addition, the order of the problem studied is increasing gradually for compare the improvements achieved in the models with major dimension. Parallel computing alternatives are also studied through multiple processing elements within a single machine or in a cluster via OpenMP, which is an application programming interface (API) that allows the creation of shared memory programs.

Approximation of functions with bounded derivative and solution of Riccati differential equations by Jacobi wavelet operational matrix

In this paper, functions with bounded derivative is considered. Two new estimators E2k,0(f) and E2k,M(f) of functions with bounded derivative have been obtained. These estimators are sharper and best possible. Also, this paper aims to construct a Jacobi wavelet operational matrix based on Jacobi polynomials. The main aim is to solve Riccati differential equation which appears in various fields of science such as Physics and Engineering. This approach is generalization of other wavelet operational matrix methods,e.g., Chebyshev wavelets of first kind, Chebyshev wavelets of second kind, Legendre wavelets, Gegenbauer wavelets, etc, which are special cases of Jacobi wavelets. The obtained estimators, the solutions of Riccati differential equation and real world problem by Jacobi wavelet operational matrix and its comparison with the exact solution are the significant achievement of this research paper.

Riccati equation as topology-based model of computer worms and discrete SIR model with constant infectious period

We propose discrete and continuous infection models of computer worms via e-mail or social networking site (SNS) messengers that were previously classified as worms spreading through topological neighbors. The discrete model is made on the basis of a new classification of worms as “permanently” or “temporarily” infectious. A temporary infection means that only the most recently infected nodes are infectious according to a difference equation. The discrete model is reduced to a Riccati differential equation (the continuous model) at the limit of a zero difference interval for the difference equation. The discrete and continuous models well describe actual data and are superior to a linear model in terms of the Akaike information criterion (AIC). Both models overcome the overestimation that is generated by applying a scan-based model to topology-based infection, especially in the early stages. The discrete model gives a condition in which all nodes are infected because the vulnerable nodes of the Ricatti difference equation are finite and the solution of the Riccati difference equation plots discrete values on the exact solution of the Riccati differential equation. Also, the discrete model can also be understood as a model for the spread of infections of an epidemic virus with a constant infectious period and is described with a discrete susceptible–infected–recovered (SIR) model. The discrete SIR model has an exact solution. A control to reduce the infection is considered through the discrete SIR model.

Binomial series and complex difference equations

In this study, we consider the properties of binomial series ∑n=0∞anzn_, where zn_=z(z−1)⋯(z−n+1), and the convergence of binomial series in the complex domain. We discuss the order of growth for the entire and meromorphic solutions of some difference equations represented by binomial series. Some examples are provided. As an application, we construct a difference Riccati equation that possesses a transcendental meromorphic solution of order 1/2.

Linear Optimal Estimation for Discrete-time and Continuous-time Systems with Multiple Measurement Delays

In this paper, we investigate the linear optimal estimation problems of discrete-time and continuous-time systems with multiple state delays in measurements. For discrete-time systems, we obtain the linear optimal estimation of state by direct calculation of optimal gain in terms of the solution to a retarded Riccati-like difference equation instead of a group of Riccati difference equations. For continuous-time systems, we also obtain the analytical expression of linear optimal estimation without resorting to Riccati partial differential equations. All the Riccati equations are of the same dimension as the system to be estimated and the computational cost is much saved. Infinite horizon case is also studied by stability analysis. Kalman filter can be recovered from our result when delays disappear. A numerical example is provided to demonstrate the results.

An Efficient Algorithm to Compute Norms for Finite Horizon, Linear Time-Varying Systems

We present an efficient algorithm to compute the induced norms of finite-horizon Linear Time-Varying (LTV) systems. The formulation includes both induced L2 and terminal Euclidean norm penalties on outputs. Existing computational approaches include the power iteration and the bisection of a Riccati Differential Equation (RDE). The power iteration has low computation time per iteration but overall convergence can be slow. In contrast, the RDE condition provides guaranteed bounds on the induced gain but single RDE integration can be slow. The complementary features of these two algorithms are combined to develop a new algorithm that is both fast and provides provable upper and lower bounds on the induced norm within the desired tolerance. The algorithm also provides a worst-case disturbance input that achieves the lower bound on the norm. We also present a new proof which shows that the power iteration for this problem converges monotonically. Finally, we show a controllability Gramian based simpler computational method for induced L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -to-Euclidean norm. This can be used to compute the reachable set at any time on the horizon. Numerical examples are provided to demonstrate the proposed algorithm.

Intrinsical localization of both topological (anti-kink) envelope and gray (black) gap solitons of the condensed bosons in deep optical lattices.

By developing quasi-discrete multiple-scale method combined with tight-binding approximation, a novel quadratic Riccati differential equation is first derived for the soliton dynamics of the condensed bosons trapped in the optical lattices. For a lack of exact solutions, the trial solutions of the Riccati equation have been analytically explored for the condensed bosons with various scattering length as. When the lattice depth is rather shallow, the results of sub-fundamental gap solitons are in qualitative agreement with the experimental observation. For the deeper lattice potentials, we predict that in the case of as>0, some novel intrinsically localized modes of symmetrical envelope, topological (kink) envelope, and anti-kink envelope solitons can be observed within the bandgap in the system, of which the amplitude increases with the increasing lattice spacing and (or) depth. In the case of as<0, the bandgap brings out intrinsically localized gray or black soliton. This well provides experimental protocols to realize transformation between the gray and black solitons by reducing light intensity of the laser beams forming optical lattice.

Recursive Filtering With Measurement Fading: A Multiple Description Coding Scheme

In this article, the recursive filtering problem is studied for a class of discrete-time nonlinear stochastic systems subject to fading measurements. In order to facilitate the data transmission in a resource-constrained communication network, the multiple description coding scheme is adopted to encode the fading measurements into two descriptions with the identical importance. Two independent Bernoulli distributed random variables are introduced to govern the occurrences of the packet dropouts in two channels from the encoders to the decoders. The channel fading phenomenon is characterized by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> th-order Rice fading model whose coefficients are mutually independent random variables obeying certain probability distributions. The purpose of the problem addressed is to design a recursive filter, such that in the simultaneous presence of the stochastic noises, the channel fading and the data coding–decoding mechanism, an upper bound of the filtering error variance is obtained and then minimized at each time step. In virtue of the Riccati difference equation technique and the stochastic analysis approach, the explicit form of the desired filter parameters is derived by solving a sequence of coupled algebraic Riccati-like difference equations. Finally, a simulation experiment is provided to show the applicability of the developed filtering scheme.

Finite-horizon H ∞ state estimation for time-varying complex networks based on the outputs of partial nodes

In this paper, the partial-nodes-based resilient filtering problem for a class of discrete time-varying complex networks is investigated. In order to reduce the effect of imprecision of filter parameters on estimation performance, a set of resilient filters is proposed. The measurement output from all network nodes may not be available in the actual system, but only from a fraction of nodes. The state estimators are designed for the time-varying complex network based on partial nodes to make the estimation error achieve the performance constraint over a finite horizon. By employing the completing-the-square technique and the backward recursive Riccati difference equations, the sufficient conditions for the existence of the estimator are derived. Then the gain of the estimator is calculated. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.

A geometric-analytic study of linear differential equations of order two

&lt;p style='text-indent:20px;'&gt;We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a &lt;i&gt;global holonomy group&lt;/i&gt;. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.&lt;/p&gt;

Pareto-Optimal Strategy for Linear Mean-Field Stochastic Systems With H∞ Constraint.

This article presents results on designing the Pareto-optimal strategy under H∞ constraint for the linear mean-field stochastic systems disturbed by external disturbances. First, combining the stochastic H∞ control theory with the stochastic mean-field theory, we derive the stochastic bounded real lemma (SBRL) of our considered linear mean-field stochastic systems with the stochastic initial condition. Second, we use the mean-field forward-backward stochastic differential equation to solve the mean-field linear quadratic Pareto-optimal problem with indefinite cost functionals. It is proved that the existence of a closed-loop Pareto-optimal strategy is equivalent to the solvability of the coupled generalized differential Riccati equations when some conditions are satisfied. Finally, a necessary and sufficient condition for the Pareto-optimal strategy under the H∞ constraint is researched by four-coupled matrix-valued equations. Besides, we also obtain the Pareto frontier for the mean-field stochastic system with only state-dependent noise. A practical example is presented to show the effectiveness of our main results.

Nonlinear control strategies for a UAV carrying a load with swing attenuation

Two novel nonlinear control schemes for Unmanned Aerial Vehicles (UAV) carrying a load and their comparative results are presented in this paper. The goal is to carry the load to a desired position, with oscillation attenuation along the trajectory. The proposed control structures are hierarchical schemes consisting of nonlinear controllers to stabilize the vehicle translational movements and the payload swing together with a well-known state-dependent differential Riccati equation controller to stabilize the rotational dynamics. We present new methodologies where two nonlinear controllers are proposed to obtain precise aerial vehicle positioning and efficient load oscillation reduction by exploiting the natural coupling between the horizontal quadcopter movement and the payload oscillation. It is shown that asymptotic stability can be guaranteed by the use of the Lyapunov approach and La Salle’s invariance principle. Numerical experiments were carried out to validate the nonlinear control behaviors where the results show improvements with respect to a strategy from the literature.