Matrix Riccati Research Articles

An extended block Golub–Kahan algorithm for large algebraic and differential matrix Riccati equations

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments.

Total reflection of near infrared range wave from subwavelength silicon 1D photonic crystal with small packing density

One-dimensional (1D) photonic crystals have been widely used in silicon photonics due to their simple structure and multiple working regimes. In this paper we present the computation of reflection and transmission spectra and spatial electric field patterns for 1D photonic crystal with packing density less than 10 percent with use of matrix Riccati equation technique in the theory of multiple electromagnetic wave scattering in inhomogeneous media. Narrow (Δλ ≲ 30 nm with maximum at λ = 1550 nm) reflectivity peaks reaching 100 percent maximum were observed for photonic crystals represented as periodically located silicon threads with rectangular or round cross section. The shape and dimensions of the threads were altered to investigate their influence on the peaks width.

On one method of finding optimal control in construction industry

In this paper we consider options for solving optimal control problems with a quadratic quality criterion. There are optimal management tasks in the construction industry also. For example, in optimizing the construction technology of large complexes, construction work cycles at the facilities. A quadratic quality criterion leads to Riccati matrix differential equations that are generally not solvable in quadratures. In this paper we consider solution options in the case of a stationary system. We find a symmetric solution of algebraic matrix Riccati equation. We offer some new analytical methods for solving this nonlinear equation in various cases.

Adaptive control for complex dynamical networks with structural balance via external stimulus signals

This paper investigates the adaptive structural balance control of complex dynamical networks by employing the controlled external stimulus signals which are coupled and transmitted to the dynamics of complex dynamical network. The control objective is to assure the asymptotical convergence of the dynamical links to the structural balance by the controlled external stimulus signals. The dynamical links of complex dynamical network are represented in this paper mathematically as the Riccati matrix differential equation with the controlled external stimulus signals which are coupled approximately in the form of Hebb rule. Compared with the existing results which are mainly concerned with the dynamical characteristics of nodes such as synchronization, this paper is mainly focused on the dynamical characteristic of links so named as the structural balance which is asymptotically obtained by the adaptive control scheme of external stimulus signals. Finally, a simulation example is given to show the validity of result proposed in this paper.

Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Sepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

Tracking control for the dynamic links of discrete-time complex dynamical network via state observer

In this paper, a class of discrete-time complex dynamical networks with time-varying links is regarded to be composed of the nodes subsystem and links subsystem which are mutually coupled. In the existing literatures on complex dynamical networks, the emphasizes is synchronization or stabilization of nodes, the dynamic behaviors of the links between nodes are always ignored. It is noted that the state of links is very important to the networks in practical, such as, the tension value of webs in web-winding system, the relationships between individuals in social network, therefore, discussing the tracking control problem for links subsystem is also of practical significance. Considering the state of links is difficult to be measured accurately in practice applications, this makes it difficult to design the controller. In view of this, a state observer is proposed for the links subsystem modeled mathematically by the Riccati matrix difference equation in this paper. And then the control scheme is proposed for the links subsystem by employing the state observer such that the links subsystem asymptotically converging to the given target. Finally, the simulations are used to show the validity of the method in this paper.

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

SummaryIn this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.

Linear Quadratic Regulator: II. Robust Formulations

The classical linear quadratic regulation problem is considered in the robust formulations where the matrices of the system and/or initial conditions are not know precisely. Several approaches are proposed where the quadratic cost is minimized against the worst-case uncertainties. Finding such controllers is performed via reducing the matrix Riccati equation with uncertainty to a single linear matrix inequality. The properties of the solutions are discussed and the comparison with previously known approaches is performed.

Solution Sets for Inverse Non-Cooperative Linear-Quadratic Differential Games

This letter addresses the inverse problem of differential games, where the aim is to compute cost functions which lead to observed Nash equilibrium trajectories. The solution of this problem allows the generation of a model for inferring the intent of several agents interacting with each other. We present a method to find all cost functions which lead to the same Nash equilibrium in an infinite-horizon linear-quadratic (LQ) differential game. The approach relies on a reformulation of the coupled matrix Riccati equations which arise out of necessary and sufficient conditions for Nash equilibria. Furthermore, based on our findings, we present an approach to compute a solution given a set of observed state and control trajectories. Our results highlight properties of feedback Nash equilibria in LQ differential games and provide an efficient approach for the estimation of cost function matrices in such a scenario.

Offshore Platform Actuator Fault Optimal Anti-Interference Self-Repair Fault-Tolerant Control

Li, F., 2019. Offshore platform actuator fault optimal anti-interference self-repair fault-tolerant control. In: Gong, D.; Zhu, H., and Liu, R. (eds.), Selected Topics in Coastal Research: Engineering, Industry, Economy, and Sustainable Development. Journal of Coastal Research, Special Issue No. 94, pp. 153–156. Coconut Creek (Florida), ISSN 0749-0208.A fault-tolerant control method based on extended fault diagnostics is proposed for the actuator failure of offshore platforms under the influence of ocean waves. Firstly, a self-repairing fault-tolerant control law is designed for actuator failure. Then an optimal anti-interference self-repairing fault-tolerant controller is designed according to Riccati and Sylvester matrix equations to ensure that the offshore platform system is optimal under fault and wave interference. In order to solve the physical unrealizable problem that the system state and faults in fault-tolerant control cannot be directly measured, an extended fault diagnostic device is designed. Finally, numerical simulation results verify the feasibility and effectiveness of this scheme.

A novel iterative algorithm for solving coupled Riccati equations

In this paper, a novel iterative algorithm is developed for solving the coupled algebraic Riccati equation arising from the quadratic optimal control problem for continuous-time Markovian jump linear systems. First, two existing iterative algorithms to solve the coupled Riccati matrix equation are reviewed. Next, based on analysis for these two algorithms, a new iterative algorithm that combines both the information in the current iterative step and the information in the last iterative step is proposed. It is shown that the proposed algorithm with proper initial conditions can monotonically converge to the unique positive definite solution of the coupled Riccati matrix equation if the associated Markovian jump system is stochastically stabilizable. Also, numerical examples show that the presented algorithm is faster than some previous algorithms when the weighted parameter is appropriately selected.

Iterative algorithms for discrete periodic Riccati matrix equations

In this paper, two iterative algorithms are constructed to obtain the positive definite solutions of the discrete periodic algebraic Riccati matrix equations. In these two algorithms, the estimation of the unknown matrices are updated by using the available estimation information at the current iteration step. The convergence properties of the proposed algorithms are also given. Finally, numerical examples are employed to illustrate the effectiveness of the proposed algorithms.

Systems of matrix Riccati equations, linear fractional transformations, partial integrability and synchronization

We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed case. Provided that certain unitarity conditions are satisfied, we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved. Special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions, with well-known synchronization behavior, and models with complex vector wavefunctions that describe synchronization in quantum systems, possibly infinite-dimensional.

Optimal control of second‐order and high‐order descriptor systems

SummaryThis study investigates the optimal control problem of second‐order descriptor systems. The optimal control is characterized by using a new second‐order generalized Riccati equation, which is directly derived in terms of the original coefficient matrices of the system. Under some assumption conditions, applying matrix's singular value decomposition and matrix transformation, a nonlinear generalized Riccati matrix equation is transformed into a linear matrix equation, and the optimal control gain can be determined by solving the linear matrix equation. Furthermore, relevant results are also extended to the optimal control problems for the high‐order descriptor system. Finally, several simulation examples and the comparison with the existing linearization method are provided to illustrate the effectiveness of the developed approach.

Stability Analysis of Following Vehicles on a Highway for Safety of Automated Transportation Systems

The development and integration of advanced control technologies in vehicles and infrastructure can make intelligent (or automated) transportation systems much more safe and efficient. This paper focuses on a proper approach to stability and control of two adjacent vehicles following each other on the same traffic lane of a highway to ensure the functional safety of their automated driving systems. These vehicles are as entirely automated as they have the ability to follow a line path at a high speed and reduce the distance that separates them while adjusting their dynamic responses automatically without intervention from humans (i.e. drivers). To meet requirements for driving safety and stability, this paper explores a new approach to string stability of (any) two adjacent vehicles traveling in the same direction based on using optimal control and Lyapunov stability theories. Therefore, sufficient conditions for string stability of a vehicle traffic using continuous-time Lyapunov function and matrix Riccati equation are presented. The search for a quadratic Lyapunov function is formulated as a convex optimization problem in terms of a linear invariant system towards a quadratic optimal criterion. The analysis of these methods for minimizing the desired performance criteria is also given. Finally, the paper ends with a conclusion and perspective for future work.

Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting directed spanning trees.

Robust State Observer Design for Dynamic Connection Relationships in Complex Dynamical Networks

The complex dynamical network with the time-varying links may be regarded to be composed of the two mutually coupled subsystems, which are called the nodes subsystem and the connection relationships subsystem respectively. Designing the state observer for the nodes subsystem has been discussed in some existing researches by employing the known connection relationships. However, the state observer design for the connection relationships subsystem is not shown in the existing researches. In practical applications, the connection relationships subsystem possesses also the state variables such as the relationship strength in social network, the size of web tension in the web winding system. Therefore, designing the state observer for the connection relationships subsystem is also of practical significance. In this paper, a novel state observer is proposed for the connection relationships subsystem modeled mathematically by the Riccati matrix differential equation. The observer utilizes the output measurements of the connection relationships subsystem and the state of nodes subsystem. Compared with the state observer of nodes subsystems which employs the known connection relationships, the state observer for the connection relationships subsystem employs the state of nodes subsystem and is represented in form of the matrix differential equation. By using Lyapunov stability theory, it is proved in this paper that the state observer for the connection relationships subsystem is asymptotical under certain mathematic conditions. Finally, the illustrative simulation is given to show the efficiency and validity of the proposed method in this paper.

A periodic solution of the coupled matrix Riccati differential equations

A periodic solution of the coupled matrix Riccati differential equations

A Novel Iterative Learning Control Approach Based on Steady-State Kalman Filtering

This paper presents a novel off-line iterative learning control algorithm for multiple-input–multiple-output time-varying discrete stochastic systems. Using the steady-state Kalman filtering method, we provide a novel framework for the selection of optimal/sub-optimal fixed learning gain matrices in real applications, which is convenient for engineers. Meanwhile, this framework considerably decreases the calculation about the operations of inverting matrix by introducing a matrix Riccati equation at every iteration. It is strictly proved that the input error covariance converges to its steady-state value asymptotically in the mean square sense, and accordingly, the tracking error covariance also converges. The numerical simulations verify the theoretical results.

Absolute Instantaneous Optimal Control Performance Index for Active Vibration Control of Structures under Seismic Excitation

In this paper, an instantaneous optimal control performance index for active control of structures under seismic excitation is analytically proposed. Absolute velocity and absolute displacement terms are implemented to the conventional state vector terms and eventually to the resulting performance index expression. The seismic response reduction effectiveness of the proposed performance index is compared with the linear quadratic regulator control (LQR). For numerical verification of the performance index, an eight‐story shear building with a fully active tendon controller system under unidirectional earthquake is considered as the first example. For a more complex model, a three‐dimensional tier building under the effect of bidirectional earthquakes is selected as second numerical example. Unidirectional near fault and bidirectional near fault earthquakes are used in the simulations. The control energy demand of each control method is also considered in the comparison. It is obtained from numerical simulations that the proposed performance index is as effective as LQR in attenuating structural vibrations. However, the resulting performance index does not require a priori knowledge of the seismic excitation like the LQR. The nonlinear Riccati matrix equation solution of the LQR is not required in the proposed performance index as well.