Matrix Riccati Research Articles

Gain margin and Lyapunov analysis of discrete‐time network synchronization via Riccati design

SummaryThis paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete‐time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single‐input controllable system is analyzed. The solution matrix entries are represented using unknown closed‐loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi‐input and single‐input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete‐time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.

On the solution of riccati matrix differential equations by an improved variational iteration scheme

On the solution of riccati matrix differential equations by an improved variational iteration scheme

Dynamic trading with Markov liquidity switching

In this paper, we solve a continuous-time portfolio choice problem of an investor under a Markov jump linear system that effectively captures stochasticity in asset returns, price impacts, and market resilience. Specifically, the investor chooses his portfolio to maximize the expected excess returns netting risks and trading costs due to the price impacts in a Markov switching market. We show that the value function and the optimal feedback trading strategy can be expressed in terms of the solution of a coupled matrix Riccati differential system. The presence of the market resilience and the Markov switching results in the coupled Riccati system being non-canonical. We manage to establish a set of transparent sufficient conditions to ensure the well-posedness of the coupled system. Our numerical results indicate that the investor trades more (less) aggressively in a liquidity regime where the temporary price impact is low (high) and permanent price impact is high (low) with low (high) market resilience.

Network optimality conditions

Abstract Optimality conditions for optimal control problems arising in network modeling are discussed. We confine ourselves to the steady state network models. Therefore, we consider only control systems described by ordinary differential equations. First, we derive optimality conditions for the nonlinear problem for a single beam. These conditions are formulated in terms of the local Pontryagin maximum principle and the matrix Riccati equation. Then, the optimality conditions for the control problem for networks posed on an arbitrary planar graph are discussed. This problem has a set of independent variables x i varying within their intervals [0, l i], associated with the corresponding beams at network edges. The lengths l i of intervals are not specified and must be determined. So, the optimization problem is non-standard, it is a combination of control and design of networks. However, using a linear change of the independent variables, it can be reduced to a standard one, and we show this. Two simple numerical examples for the single-beam problem are considered.

Model following adaptive control for the complex dynamic network with the dynamic matrix‐relation links

SummaryThis article formulates and solves a problem of model following adaptive control (MFAC) of nodes in the complex dynamic network (CDN) with the dynamic matrix‐relation links. From the perspective of composite systems, the CDN with the dynamic matrix‐relation links is considered as an interconnected system composed of intercoupling node subsystem (NS) and link subsystem (LS). Based on this idea, employing the Riccati matrix differential equation as the dynamic equation of LS, and the vector differential equation with the second derivative term as the dynamic equation of NS. Then, to remove the restriction that the state of LS is unavailable, an adaptive control scheme is proposed for NS with the help of constructed dynamic auxiliary tracking target for LS, such that the problem of MFAC of nodes is solved. Besides, it is proved mathematically that the state of NS is asymptotically tracking the state of the given reference model when the state of LS is asymptotically tracking the constructed dynamic auxiliary tracking target. Finally, the numerical simulation is given to verify the effectiveness of the theoretical results in this article.

Stabilizability of Riccati Matrix Fractional Delay Differential Equation

In this article, the backstepping control scheme is proposed to stabilize the fractional order Riccati matrix differential equation with retarded arguments in which the fractional derivative is presented using Caputo's definition of fractional derivative. The results are established using Mittag-Leffler stability. The fractional Lyapunov function is defined at each stage and the negativity of an overall fractional Lyapunov function is ensured by the proper selection of the control law. Numerical simulation has been used to demonstrate the effectiveness of the proposed control scheme for stabilizing such type of Riccati matrix differential equations.

THUẬT TOÁN ĐIỀU KHIỂN THEO ĐẦU RA HỆ THỐNG TUYẾN TÍNH KHÔNG DỪNG, SỬ DỤNG CÁC PHƯƠNG PHÁP NHẬN DẠNG THAM SỐ

The problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. For synthesis of the observer of state variables, a new approach is proposed, providing the possibility of obtaining monotonous estimates of convergence with the regulation of transition time. The main idea of the synthesis of observers is based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. Then the DREM algorithm is used to estimate the state variables. Some simulations on Matlab/Simulink prove the correctness of the theory built, ensuring the good quality of the response processes.

Dynamic asset-liability management with frictions

This paper studies a dynamic asset-liability management problem of a company with market frictions. Specifically, the asset prices are modeled by a multivariate geometric Brownian motion with their excess returns driven by some correlated stochastic signals; and the liability process is modeled by another geometric Brownian motion correlated to the asset price dynamics. The company trades dynamically to offset the risks from its liability and each trade induces both temporary and persistent price impacts. We characterize the optimal trading strategies in terms of the solutions to the coupled matrix Riccati differential systems. Due to the price impacts, the company should adopt a target-chasing strategy in which the dynamic target portfolio is expressed in terms of the return-predicting signals and realized liability. We also derive some sufficient conditions, based on the model parameters alone, to ensure the well-posedness of the coupled Riccati systems. Our numerical results indicate that the temporary and persistent price impacts have opposite implications on the company's trading behavior. While the temporary price impact slows down the company's trading speed toward the target portfolio, the persistent price impact may encourage the company to trade more aggressively to enhance the expected returns.

Operational Absolutely Optimal Dynamic Control of the Stochastic Differential Plant’s State by Its Output

The problem of synthesizing the average-optimal control law for a dynamic plant subject to random disturbances, if its state variables are measured partially or with random errors, is considered. Using the method of a posteriori sufficient coordinates (SCs), the complexity of constructing the well-known interval-optimal Mortensen controller is described and a much simpler algorithm for finding its operational-optimal analog is obtained. The new controller does not require the solution of the corresponding Bellman equation in inverse time, since it is optimal in the sense of a time-varying criterion. This makes it possible to disregard information about the future behavior of the object and reduces the procedure for finding the dependence of a control on sufficient coordinates to direct-time integration of the Fokker–Planck–Kolmogorov equation and to solving a problem of parametric nonlinear programming. The application of the obtained algorithm is demonstrated by the example of a linear-quadratic-Gaussian problem, as a result of which a new operational version of the well-known separation theorem is formulated. It represents a stochastic control device as a combination of a linear Kalman–Bucy filter and a linear operational-optimal positional controller. The latter differs from the traditional interval-optimal controller by the well-known gain and does not require the solution of the corresponding matrix Riccati equation in inverse time

Adaptive guaranteed‐performance consensus for high‐order nonlinear multi‐agent systems with switching topologies

AbstractThis article investigates the adaptive guaranteed‐performance consensus problem for high‐order nonlinear multi‐agent systems (MASs) with switching topologies, and briefly analyzes the special leader‐following situation. To further handle the problem of adaptive guaranteed‐performance consensus for higher‐order nonlinear MASs with switching topologies, two novel distributed control protocols are introduced, which are applied to the leaderless and leader tracking cases, respectively. In addition, the methods based on the linear matrix inequality (LMI) and Riccati inequality provide the judgment basis for the stability of Lyapunov function theorem. Eventually, the theoretical results are efficiently and successfully validated by numerical simulations.

Learning by Population Genetics and Matrix Riccati Equation

A model of learning as a generalization of the Eigen's quasispecies model in population genetics is introduced. Eigen's model is considered as a matrix Riccati equation. The error catastrophe in the Eigen's model (when the purifying selection becomes ineffective) is discussed as the divergence of the Perron-Frobenius eigenvalue of the Riccati model in the limit of large matrices. A known estimate for the Perron-Frobenius eigenvalue provides an explanation for observed patterns of genomic evolution. We propose to consider the error catastrophe in Eigen's model as an analog of overfitting in learning theory; this gives a criterion for the presence of overfitting in learning.

Flow equation and fermion gap in the holographic superconductors

We reconsider the fermion spectral function in the presence of the Cooper pair condensation and identified the interaction type of complex scalar and fermion, which gives consistent results with the expected s-wave superconductor for the first time. We derive the matrix Riccati equation, which allows the precise calculation of the fermion spectral function. Apart from the gap structure, we studied the effect of the chemical potential and the density and compared it with the BCS theory. We found that two theories give similar results in small chemical potential but very different ones in the high-density case, which we attribute to the correlation effect.

Solvability conditions for nonlinear matrix equations

Conditions for the existence of solutions of nonlinear matrix equations have been found and a method of their construction has been proposed. As an example of the iterative scheme construction, approximations for the solutions of the nonlinear algebraic matrix Riccati equation and their accuracy errors were determined.

Symmetries and Zero Modes in Sample Path Large Deviations

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile.

Asymptotically Tracking Control of Structural Balance for Discrete-Time Links System Associated with External Stimulations and State Observer

Considering that the link states of some practical networks are not measurable, it is impossible to control the links system directly to achieve structural balance. In order to handle this problem, the state observer design and the structural balance tracking control for the links system with complex dynamical networks are investigated. The dynamical model of the links system is molded as the Riccati matrix difference equation, which contains a coupling term associated with external stimulations. For the discrete-time model, the pole assignment algorithm is adopted to calculate the gain matrix for the observer of the links system. It could guarantee that the estimation error of observer for links system is globally asymptotically stable. Further, combining the effective estimation information of observer and external stimulations, an asymptotically tracking controller is proposed, which could achieve the asymptotic tracking of the reference structure balance network to the links system. Finally, the validity of the proposed method is verified by the simulation example.

Riccati matrix differential equation and the discrete order preserving property

In this paper we recall discrete order preserving property related to the discrete Riccati matrix equation. We present results obtained by applying this property to the solutions of the Riccati matrix differential equation.

A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

<abstract><p>A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.</p></abstract>

Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.

FINITE APPROXIMATIONS OF THE PROBLEM OF OPTIMAL IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT

A method for constructing finite-dimensional approximations for the optimal impulse stabilizing control of a linear autonomous system of differential equations with aftereffect is proposed. Auxiliary finite-dimensional problems of optimal stabilization of linear autonomous systems of ordinary differential equations with non-impulse controls are used. A procedure for reducing the dimension of a system of algebraic matrix Riccati equations is described.

On stability of motion of unbalanced gyroscope

In this paper we consider the stability of motion of an unbalanced gyroscope with a flexible shaft in gimbal suspension.Gyroscopes are used to control the stability of boom tower cranes. First, we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems arise in transportation problems. We prove several properties of the Hamiltonian matrix. Next, we consider the normalization of Hamiltonian matrix. We solve the system of matrix equations to find the generating function of the canonical transformation. Further, we find the symmetric solution of nonlinear algebraic matrix Riccati equation in one case. We propose the analytical method for solving this equation. Further, we investigate the motion of gyroscope. We consider the system of equations of motion of the gyroscope rotor and frames in the first approximation. We consider eigenvalues of characteristic equation corresponding to the motion equations. We obtain stability criterion in several cases.