Periodic Riccati Research Articles

Optimal consensus control for multi-satellite assembly in elliptic orbit with input saturation

This paper presents an optimal control design method for multi-satellite assembly in an elliptic orbit in close proximity operations considering input saturation constraints. Tschauner–Hempel (T–H) equations are utilized to model the relative motion dynamics of the satellites in the Local Vertical Local Horizontal (LVLH) frame of the target satellite, where every satellite in the assembly is modeled as a point mass. The communication topology between the satellites in the assembly is modeled using graph theory, where the graph has several nodes representing the satellites and several edges representing the data exchange direction. It is assumed that every satellite exchanges its states with a few neighbors in the team, where the neighbors of every satellite are prespecified and do not change during the mission. This graph is utilized to construct the global dynamic model of the assembly, in which every node is represented by the dynamic model of a single satellite. The local control law, which is applied to every satellite in the assembly, consists of a state feedback gain and a coupling gain. The state feedback gain is designed using the discrete-time periodic Riccati equation to guarantee the local stability of every satellite separately, while the coupling gain is selected to satisfy the Lyapunov condition to guarantee the global stability of the assembly. The invariant-set method is utilized to prove the stability of the global dynamic system that is subjected to symmetric input saturation. The precision and optimality of the proposed control law are successfully demonstrated by numerical nonlinear simulations in a MATLAB environment for a cubic formation satellite assembly.

Optimal Stabilization of Periodic Orbits

In this contribution the optimal stabilization problem of periodic orbits is studied in a symplectic geometry setting. For this, the stable manifold theory for the point stabilization case is generalized to the case of periodic orbit stabilization. Sufficient conditions for the existence of a normally hyperbolic invariant manifold (NHIM) of the Hamiltonian system are derived. It is shown that the optimal control problem has a solution if the related periodic Riccati equation has a unique periodic solution. For the analysis of the stable and unstable manifold a coordinate transformation is used which is moving along the orbit. As an example, an optimal control problem is considered for a spring-mass oscillator system, which should be stabilized at a certain energy level.

Discrete gain scheduling control approach to elliptical orbit rendezvous system with actuator saturation

AbstractThis paper studies the discrete gain scheduling control design problem of elliptical orbit spacecraft rendezvous system with actuator saturation. Due to the presence of actuator saturation, the dynamic performance of the spacecraft rendezvous system degrades significantly. In order to improve the dynamic performance of the system, a discrete gain scheduling control approach is adopted to construct a group of time‐invariant ellipsoidal invariant sets, which can be used to determine the switching points of the discrete gain scheduling control. By choosing some discrete parameter values, the discrete gain scheduling control is obtained from a solution of a periodic Riccati matrix differential equation. Under the control obtained, the dynamic performance of the system is much improved while accomplishing successfully the rendezvous mission of the spacecraft. Finally, a practical example is provided to show the effectiveness of the proposed control design approach.

Periodic and almost periodic solutions of the Riccati equations with linear reflecting function

The method of Mironenko’s reflecting function is used for investigation of Riccati equations. The class of Riccati equations with certain-type reflecting function has been preliminarily constructed. The necessary and sufficient conditions, under which the Riccati equation would have a reflecting function linear in phase variable, are proved. These conditions are constructive in nature, since on their basis the formula is obtained, which shows the linear in phase variable reflecting function in terms of the coefficients of the Riccati equation. Additionally, the relationship between the parity (oddness) property of the coefficients of the Riccati equation and the existence of a reflecting function linear in phase variable is investigated. The application of the method of Mironenko’s reflecting function to the constructed class of Riccati equations revealed sufficient conditions, under which all its solutions are periodic or almost periodic. A sign of no periodic solutions for almost periodic Riccati equations is obtained. An example of the quasi-periodic Riccati equation with quasi-periodic reflecting function, which has a periodic solution, is given.

On extended state estimation for nonlinear uncertain systems with round-robin protocol

This work studies the state estimation problem for nonlinear uncertain systems over a shared communication channel. We transform the original system into an extended state system to handle the nonlinear uncertainty. The round-robin protocol is used to overcome the communication capacity constraint by sending the measurements periodically. The measurements’ structure is changed by the scheduling protocol, which may cause the round-robin-induced periodic extended state system (RRIPESS) to be unobservable. A necessary and sufficient condition of a class of RRIPESS’s observability is established. An extended state estimator is designed. The upper bound of the priori estimation error covariance is derived via a set of periodic Riccati equations (PREs). The stability of these PREs is ensured by the detectability and the stabilizability of the original system’s parameters. In addition, we give an iterative algorithm to improve the upper bound by optimizing a set of scalars induced by Young’s inequality. Finally, a simulation example is given to illustrate the effectiveness of the developed scheme.

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.

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.

On the Existence of the Stabilizing Solution of a Class of Periodic Stochastic Riccati Equations

This paper is devoted to the characterization of existence and uniqueness conditions for the stabilizing solution of a large class of Riccati equations arising in stochastic dynamic games. As an application of the obtained results, we consider in a second step the problem of a zero-sum two players linear quadratic difference game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such a control problem, a crucial role is played by the stabilizing solution of the considered class of Riccati difference equations.

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.

On periodic optimal solutions of persistent sensor planning for continuous-time linear systems

This work investigates informative planning of sensing agents over infinite time horizon when the system of interest is expressed as a continuous-time linear system. The objective of this planning problem, termed persistent monitoring problem, is to maintain the monitoring uncertainty at the minimum. The method reduces the persistent planning problem into a periodic planning problem; it is formulated as a periodic optimal control or optimization problem to determine the optimal periodic sensor plan as well as the period. The plan induces the periodic Riccati equation and is proven to lead an arbitrary initial uncertainty state to the optimized periodic trajectory. It is also proven that any infinite-horizon (non-periodic) sensor plan is able to be approximated arbitrary well by a periodic sensor plan. A suboptimal filtering mechanism is proposed by using the resulting optimal periodic solution. Two numerical examples on (a) a relaxed periodic sensor scheduling for a two dimensional linear system, and (b) persistent monitoring by a mobile sensor of two-dimensional diffusion dynamics show the validity of the proposed approach.

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.

ДИСКРЕТНАЯ ПРОЦЕДУРА ОПТИМАЛЬНОЙ СТАБИЛИЗАЦИИ ПЕРИОДИЧЕСКИХ ЛИНЕЙНЫХ СИСТЕМ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

We propose procedure to solve the optimal stabilization problem for linear periodic systems of differential equations. Stabilizing controls, formed as a feedback, are defined by the system states at the fixed instants of time. Equivalent discrete-time linear periodic problem of optimal stabilization is considered. We propose a special procedure for the solution of discrete periodic Riccati equation. We investigate the relation between continuous-time and discrete-time periodic optimal stabilization problems. The proposed method is used for stabilization of mechanical 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.

An efficient algorithm for periodic Riccati equation with periodically time-varying input matrix

This paper proposes an efficient algorithm for the periodic discrete-time Riccati equation arising from a linear periodic time-varying system (Ak,Bk). Although the algorithm can be used for general problem, it is particularly efficient when Ak is time-invariant and Bk is periodic, a special property associated with the problem of spacecraft attitude control using magnetic torques. We will compare the proposed algorithm, by using analysis and simulation, to some well-established algorithm to confirm our claim.

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.

The Maximal Solution and Comparison Theorems for the Periodic Discrete-Time Riccati Equation

AbstractProperties and comparison theorems for the maximal solution of the periodic discrete-time Riccati equation are supplemented by an extension of some earlier results and analysis, for the discrete-time Riccati equation to the periodic case.

Rapidly Exploring Random Cycles: Persistent Estimation of Spatiotemporal Fields With Multiple Sensing Robots

This paper considers the problem of planning trajectories for both single and multiple sensing robots to best estimate a spatiotemporal field in a dynamic environment. The robots use a Kalman filter to maintain an estimate of the field value and to compute the error covariance matrix of the estimate. Two new sampling-based path-planning algorithms (rapidly expanding random cycles and an improved variant of rapidly expanding random cycles) are proposed to find periodic trajectories for the sensing robots that minimize the largest eigenvalue of the error covariance matrix over an infinite horizon. The algorithms are proven to find the minimum infinite horizon cost cycle in a random graph, which grows by successively adding random points. The algorithms leverage recently developed methods for periodic Riccati recursions to efficiently compute the infinite horizon cost of the cycles, and they use the monotonicity property of the Riccati recursion to efficiently compare the costs of different cycles without explicitly computing their costs. The algorithms are demonstrated in a study using National Oceanic and Atmospheric Administration data to plan sensing trajectories in the Caribbean Sea. Our algorithms significantly outperform random, greedy, and receding horizon approaches in this environment.

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.

On computing the stabilizing solution of a class of discrete‐time periodic Riccati equations

SummaryThis paper addresses the problem of solving a class of periodic discrete‐time Riccati equation with an indefinite sign of its quadratic term. Such an equation is closely related to the so‐called full‐information H ∞ control of discrete‐time periodic systems. A globally convergent iterative algorithm with a local quadratic convergence rate is proposed for this purpose. An application to the problem of H ∞ filtering of discrete‐time periodic systems is also developed and illustrated via a numerical example. Copyright © 2013 John Wiley & Sons, Ltd.