Periodic Riccati Differential Equation Research Articles

Symmetry preserving nonlinear observer for attitude estimation with magnetometer only

This paper addresses designing a nonlinear symmetry observer for nano-satellite attitude determination with only Three-Axis Magnetometer (TAM). Hence, two nested filters are delineated to compensate for the lack of data and to solve the observability problem. In the first filter, the magnetic field derivative is estimated by the extended Kalman filter. Subsequently, the magnetic field and its derivative are applied in the second filter. The proposed methodology relies on invariant observers under the action of a Lie group to estimate the attitude and angular velocity simultaneously. Later, the invariant error dynamic equations are utilized in designing the observer. Consequently, the desired performance of this proposed observer tuned by periodic Riccati differential equation is validated through both simulation and exponential stability proof.

A periodic linear–quadratic controller for suppressing rotor-blade vibration

This paper presents an active control strategy, based on a time-varying linear–quadratic optimal control problem, to attenuate the tip vibration of a two-dimensional coupled rotor-blade system whose dynamics is periodic. First, a periodic full-state feedback controller based on the linear–quadratic regulator (LQR) problem is designed. If all the states are not available for feedback, then an optimal periodic time-varying estimator, using the Kalman–Bucy filter, is computed. Both the Kalman filter gain and the LQR gain are obtained as the solution of a periodic Riccati differential equation (PRDE). Together, these gains provide the observer-based linear–quadratic–Gaussian (LQG) controller. An algorithm to solve the PRDE is also presented. Both controller designs ensure closed-loop stability and performance for the linear time-varying rotor-blade equation of motion. Numerical simulations show that the LQR and the LQG controllers are able to significantly attenuate the rotor-blade tip vibration.

Quasi-Potential Calculation and Minimum Action Method for Limit Cycle

We study the noise-induced escape from a stable limit cycle of a non-gradient dynamical system driven by a small additive noise. The fact that the optimal transition path in this case is infinitely long imposes a severe numerical challenge to resolve it in the minimum action method. We first consider the landscape of the quasi-potential near the limit cycle, which characterizes the minimal cost of the noise to drive the system far away form the limit cycle. We derive and compute the quadratic approximation of this quasi-potential near the limit cycle in the form of a positive definite solution to a matrix-valued periodic Riccati differential equation on the limit cycle. We then combine this local approximation in the neighborhood of the limit cycle with the minimum action method applied outside of the neighborhood. The neighborhood size is selected to be compatible with the path discretization error. By several numerical examples, we show that this strategy effectively improves the minimum action method to compute the spiral optimal escape path from limit cycles in various systems.

On the Stabilizing Solution of Periodic Riccati Differential Equations Related to a Class of Stochastic Linear Quadratic Differential Game

We consider a two players linear quadratic differential game associated to a positive θ-periodic dynamical system with jump Markov perturbations. We introduce the concept of stabilizing solution and strong stabilizing solution of the associated system of coupled matrix Riccati differential equations and we will point out the relation between these kinds of solutions. Moreover, we provide a set of conditions which guarantee the existence of strong stabilizing and θ-periodic solution (stabilizing and θ-periodic solution, respectively) of the considered set of coupled matrix Riccati differential equations.

A novel extended precise integration method based on Fourier series expansion for periodic Riccati differential equations

SummaryA new, reliable algorithm for nonnegative, stabilizing solutions for the periodic Riccati differential equation (PRDE) is proposed based on Fourier series expansion and the precise integration method (PIM). Taking full advantages of periodicity, we expand coefficient matrices of the underlying linear time‐varying periodic Hamiltonian system associated with the PRDE in Fourier series, and a novel extended PIM for the transition matrix of linear time‐varying periodic systems is developed by combining the doubling algorithm with the increment‐storage technique. This method needs to compute the matrix exponential and its related integrals only once for all evenly divided subintervals, which greatly improves the computational efficiency. Further, by introducing the Riccati transformation, a fast recursive formula for the PRDE is derived based on the block form of the transition matrix computed by the extended PIM. Finally, two numerical examples are presented to verify the numerical accuracy and efficiency of the proposed algorithm with compared results.

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

ABSTRACTNumerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.

Formula omitted] optimal filtering for continuous-time periodic linear stochastic systems with state-dependent noise

This note addresses the problem of optimal H2 filtering for a class of continuous-time time-varying stochastic systems. The time-variation character is considered to be periodic. The state space representation of the optimal filter is designed based on the unique periodic solution of a suitable Lyapunov differential equation and the periodic and stabilizing solution of a suitable generalized periodic Riccati differential equation. Some numerical experiments are also added to show the effectiveness of the proposed method.

Robust [formula omitted] control of spacecraft rendezvous on elliptical orbit

This paper investigates the robust H∞ control problem of spacecraft rendezvous on elliptical orbit. Based on Lyapunov theory, a sufficient condition for the existence of the robust H∞ controller is given in terms of the periodic Riccati differential equation (RDE). Then, a stabilizable controller design method based on a fast multi-shot algorithm is proposed. With the controller obtained, the spacecraft rendezvous mission can be accomplished. An illustrative example is provided to show the effectiveness of the proposed control design method.

Optimal periodic controller for formation flying on libration point orbits

An optimal periodic controller based on continuous low-thrust is proposed for the stabilization missions of spacecraft station-keeping and formation-keeping along periodic Libration point orbits of the Sun–Earth system. Additionally, a new numerical algorithm is proposed for solving the periodic Riccati differential equation in the design of the optimal periodic controller. Practical missions show that the optimal periodic controller (which is designed with the linear periodic time-varying equation of the relative dynamical model) overcomes the problems and limitations of the time-variant LQR controller. Furthermore, nonlinear numerical simulations are presented for the missions of a leader spacecraft station-keeping and three follower spacecraft formation-keeping. Numerical simulations show that the velocity increments for spacecraft control and relative position errors vary little with changes in the altitude of periodic orbits. In addition, the actual trajectories of the leader and the follower spacecraft track the periodic reference orbit with high accuracy under the perturbation of the eccentric nature of the Earth's orbit and the initial injection errors. In particular, the relative position errors obtained by the optimal periodic controller for spacecraft formation-keeping are all in the range of millimeters.

A New Iterative Algorithm to Solve Periodic Riccati Differential Equations With Sign Indefinite Quadratic Terms

An iterative algorithm to solve periodic Riccati differential equations (PRDE) with an indefinite quadratic term is proposed. In our algorithm, we replace the problem of solving a PRDE with an indefinite quadratic term by the problem of solving a sequence of PRDEs with a negative semidefinite quadratic term which can be solved by existing methods. The global convergence and the local quadratic rate of convergence are both established. A numerical example is given to illustrate our algorithm.

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.

A numerical evaluation of solvers for the periodic Riccati differential equation

Efficient and accurate structure exploiting numerical methods for solving the periodic Riccati differential equation (PRDE) are addressed. Such methods are essential, for example, to design periodic feedback controllers for periodic control systems. Three recently proposed methods for solving the PRDE are presented and evaluated on challenging periodic linear artificial systems with known solutions and applied to the stabilization of periodic motions of mechanical systems. The first two methods are of the type multiple shooting and rely on computing the stable invariant subspace of an associated Hamiltonian system. The stable subspace is determined using either algorithms for computing an ordered periodic real Schur form of a cyclic matrix sequence, or a recently proposed method which implicitly constructs a stable deflating subspace from an associated lifted pencil. The third method reformulates the PRDE as a convex optimization problem where the stabilizing solution is approximated by its truncated Fourier series. As known, this reformulation leads to a semidefinite programming problem with linear matrix inequality constraints admitting an effective numerical realization. The numerical evaluation of the PRDE methods, with focus on the number of states (n) and the length of the period (T) of the periodic systems considered, includes both quantitative and qualitative results.

COMPARING ONE-SHOT AND MULTI-SHOT METHODS FOR SOLVING PERIODIC RICCATI DIFFERENTIAL EQUATIONS

One-shot methods and recently proposed multi-shot methods for computing stabilizing solutions of continuous-time periodic Riccati differential equations are examined and evaluated on two test problems: (i) a stabilization problem for an artificially constructed time-varying linear system for which the exact solution is known; (ii) a nonlinear stabilization problem for a devil stick juggling model along a periodic trajectory. The numerical comparisons are performed using both general purpose and symplectic integration methods for solving the associated Hamiltonian differential systems. In the multi-shot method a stable subspace is determined using recent algorithms for computing a reordered periodic real Schur form. The results show the increased accuracy achievable by combining multi-shot methods with structure preserving (symplectic) integration techniques.

Convergence and existence results for continuous- and discrete-time Riccati equations

We give a new proof for the convergence of the solution of a terminal value problem for the periodic Riccati differential equation towards its strong solution as t → −∞. The proof is mainly based on well-known comparison results and also on an explicit representation formula for the solution that reflects precisely the dependence on the terminal value. Moreover, we give sufficient conditions for the existence of a periodic solution of the differential equation. Similar results are derived for the discrete-time Riccati equation.

Comparison and uniqueness theorems for periodic Riccati differential equations

This paper presents some comparison and uniqueness theorems for periodic Riccati differential equations (PRDEs). Precisely, it is first shown that for a PRDE with a positive semidefinite coefficient matrix in the quadratic term under the assumption of stabilizability, the solubility of the corresponding Riccati differential inequality implies that the PRDE admits a maximal solution. Furthermore, for the class of PRDEs mentioned above a monotonicity property is given. It is also proven that each stabilizing solution (if any) for this class of PRDE is just the maximal solution and thus is unique. Finally, for a PRDE with a positive definite coefficient matrix in the constant term, it is concluded that the stabilizing solution (if any) is also unique and using this uniqueness result another uniqueness theorem of the stabilizing solution for the general PRDE is derived.

Maximal solution of a certain class of periodic Riccati differential equations

This paper deals with a generalized version of the well-known periodic Riccati differential equation which arises in certain optimal filtering problems for linear periodic systems. A monotonicity property for symmetric periodic equilibrium solutions is established, and it is proved that there exists a maximal symmetric periodic equilibrium solution.

A note on the maximal solution of the periodic Riccati equation

The periodic symmetric solutions of the periodic Riccati differential equation associated with the filtering problem are considered by the authors. It is proven that, under the sole assumption of detectability, there exists a maximal solution. Moreover, such a solution turns out to be strong, i.e. the characteristic multipliers of the associated closed-loop system belong to the closed unit disk. The proof relies on an iterative linearization technique, which calls for a sequence of periodic Lyapunov equations. Similar results are given for the minimal solution. >

Riccati differential equation in optimal filtering of periodic non-stabilizable systems

This paper discusses the periodic solutions of the matrix Riccati differential equation in the optimal filtering of periodic systems. Special emphasis is given to non-stabilizable systems and the question addressed is the existence and uniqueness of a steady-state periodic non-negative definite solution of the periodic Riccati differential equation which gives rise to an asymptotically stable steady-state filter. The results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution. The convergence of the general solution of the periodic Riccati differential equation to a periodic equilibrium solution is also investigated. The results are extensions of existing time-invariant systems results to the case of periodic systems