Backward Riccati Equation Research Articles

A necessary and sufficient condition for the existence of the stabilizing solution of a large class of discrete-time Riccati type equations with periodic coefficients

This paper is devoted to the study of a large class of discrete-time backward Riccati equations arising in several linear quadratic (LQ) type control problems both in the deterministic and in the stochastic frameworks. The periodic time-varying case is considered. We propose existence and uniqueness conditions for a global special solution named stabilizing solution for such equations. Beside the stabilizability condition, the criterion derived in this paper is expressed based on some suitable properties of the characteristic multipliers of a discrete-time, periodic linear equation adequately constructed using the coefficients of the given equation. Our result englobes, as particular cases, several existing results in the literature.

Decentralized Output-Feedback Control With Asymmetric One-Step Delayed Information

This paper considers the decentralized LQG control problem for discrete-time decentralized system controlled by two players. In this scenario, player 1 shares a unit delayed observations and control inputs with the controller of player 2, whereas due to the limiting capacity, the controller of player 1 cannot obtain the observations and control inputs of player 2 which leads to the asymmetric one-step delay information. It should be emphasized that this structure makes the classical separation principle fail. Under the assumption of linear control strategies, we derive the optimal estimators based on asymmetric information. In virtue of the Pontryagin's maximum principle, the optimal explicit decentralized controllers are obtained by solving the forward and backward stochastic difference equations (FBSDEs). It is noted that the control gains are coupled with the estimation gains. Moreover, the estimation gains satisfy the forward Riccati equations and the control gains follow the backward Riccati equations which cannot be solved simultaneously. To this end, we present iterative solutions to the coupled forward and backward Riccati equations. Finally, a sufficient condition for the stabilization problem is given in terms of the coupled algebraic Riccati equations. Numerical examples are illustrated to show the effectiveness of the proposed algorithm.

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

AbstractThe decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.

Suboptimal Linear Output Feedback Control of Discrete-Time Systems With Multiplicative Noises

This brief considers the fundamental problem of output feedback control for a class of discrete-time systems with multiplicative noises. It is well-known that the classical separation principle fails due to the existence of multiplicative noises in state and control variables. Thus, the traditional optimal solutions for the controller and estimator can not be obtained. Different from the previous methods of enforced separation principle, the explicit expression of the suboptimal linear controller is derived by using the solution of the forward and backward stochastic difference equations (FBSDEs) and the linear optimal estimator. However, it leads to the other mathematical difficult problem that the control gain and the estimator gain are coupled with each other. Specifically, the control gain obeys the backward Riccati equation and the estimator gain satisfies the forward Riccati equation such that the solutions can not be acquired simultaneously. A novel method of iterative solutions to the coupled backward and forward Riccati equations (CBFREs) is proposed. Numerical examples are illustrated to show that the proposed algorithms have better performance than the previous methods.

Decentralized LQG Control With $d$-Step Delayed Information Sharing Pattern

In this article, we study a class of discrete-time decentralized linear-quadratic-Gaussian (LQG) control problems with two controllers and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -step delayed information sharing pattern. An explicit form of the pair of optimal linear controllers is obtained in terms of backward Riccati equations and forward equations of estimation error covariance. To overcome the difficulty that arises due to the coupling of the backward–forward equations, we propose a forward iteration approach to derive a steady-state solution. This accordingly leads to an asymptotically optimal linear controller where the feedback gains are designed by using the convergent steady-state solutions.

Decentralized Control for Networked Control Systems With Asymmetric Information

This article considers the decentralized control for networked control systems (NCSs) with asymmetric information. In this NCSs model, the controller 2 (C2) shares its observations and part of its historical control inputs with the controller 1 (C1), whereas C2 cannot obtain the information of C1 due to network constraints. Under the linear control strategies assumption, we present the optimal estimators for C1 and C2 respectively based on asymmetric observations. Since the information for C1 and C2 are asymmetric, the estimation error covariance (EEC) is coupled with the controller which means that the classical separation principle fails. By applying the Pontryagin’s maximum principle, we obtain a solution to the forward and backward stochastic difference equations. Based on this solution, we derive the optimal controllers to minimize a quadratic cost function. Combining the linear optimal controllers with the EEC, the controller C1 is decoupled from the EEC. It should be emphasized that the control gain is dependent on the estimation gain. What is more, the estimation gain satisfies the forward Riccati equation and the control gain satisfies the backward Riccati equation which makes the problem more challenging. We propose iterative solutions to the Riccati equations and give a suboptimal solution to the optimal decentralized control problem.

H∞ Pinning Control of Complex Dynamical Networks Under Dynamic Quantization Effects: A Coupled Backward Riccati Equation Approach.

In this article, a pinning control strategy is developed for the finite-horizon H∞ synchronization problem for a kind of discrete time-varying nonlinear complex dynamical network in a digital communication circumstance. For the sake of complying with the digitized data exchange, a feedback-type dynamic quantizer is introduced to reflect the transformation from the raw signals into the discrete-valued ones. Then, a quantized pinning control scheme takes place on a small fraction of the network nodes with the hope of cutting down the control expenses while achieving the expected global synchronization objective. Subsequently, by resorting to the completing-the-square technique, a sufficient condition is established to ensure the finite-horizon H∞ index of the synchronization error dynamics against both quantization errors and external noises. Moreover, a controller design algorithm is put forward via an auxiliary H2 -type criterion, and the desired controller gains are acquired in terms of two coupled backward Riccati equations. Finally, the validity of the presented results is verified via a simulation example.

Finite-Horizon H∞ State Estimation for Stochastic Coupled Networks With Random Inner Couplings Using Round-Robin Protocol.

This article is concerned with the problem of finite-horizon H∞ state estimation for time-varying coupled stochastic networks through the round-robin scheduling protocol. The inner coupling strengths of the considered coupled networks are governed by a random sequence with known expectations and variances. For the sake of mitigating the occurrence probability of the network-induced phenomena, the communication network is equipped with the round-robin protocol that schedules the signal transmissions of the sensors' measurement outputs. By using some dedicated approximation techniques, an uncertain auxiliary system with stochastic parameters is established where the multiplicative noises enter the coefficient matrix of the augmented disturbances. With the established auxiliary system, the desired finite-horizon H∞ state estimator is acquired by solving coupled backward Riccati equations, and the corresponding recursive estimator design algorithm is presented that is suitable for online application. The effectiveness of the proposed estimator design method is validated via a numerical example.

Fault‐tolerant cooperation for time‐varying multi‐vehicle systems with schemes

This study dealt with the fault-tolerant cooperation problem for a class of time-varying multi-vehicle system with actuator faults under the mixed structure. Through solving coupled backward Riccati equations, a distributed cooperation fault-tolerant controller was designed for satisfying the proposed constraints. Finally, a numerical simulation verified the effectiveness of the proposed method.

Stochastic linear quadratic control problem of switching systems with constraints

This paper is devoted to the optimal control problem for stochastic linear switching systems with a quadratic cost functional. A necessary and sufficient condition of optimality for mentioned linear control systems under endpoint constraints is obtained. A linear quadratic controller is simply constructed via a set of stochastic backward Riccati equations.

Duality of linear estimation for multiplicative noise systems with measurement delay

This study firstly investigates the estimation problem for systems with multiplicative noise and measurement delay. Based on the innovation analysis approach, the estimators are developed in terms of a Riccati equation and a Lyapunov equation. The equations are of the same dimension as the plant; therefore compared with the augmentation approach, the presented approach lessens the computational demand. Then the linear quadratic regulation (LQR) problem for input delay systems is discussed based on non-augmented approach, and the controller is given in terms of a backward Riccati equation and a backward Lyapunov equation. Finally, the authors establish a duality between the estimation problem for measurement delay systems with multiplicative noise and the LQR problem for deterministic input delay systems with constraint conditions.

선형 이산 시변시스템을 위한 고정시간 이동구간 제어

In the case of a linear time-varying system, it is difficult to apply the conventional stability conditions of RHC (Receding Horizon Control) to real physical systems because of computational complexity comes from time-varying system and backward Riccati equation. Therefore, in this study, a frozen time RHC for a linear discrete time-varying system is proposed. Since the proposed control law is obtained by time-invariant Riccati equation solved by forward iterations at each control time, its stability can be ensured by matrix inequality condition and the stability condition based on horizon for a time-invariant system, and they can be applied to real physical systems effectively in comparison with the conventional RHC.

New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences

Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Volterra‐type recursions describing characteristics of the corresponding optimal filter. In the case of Gauss‐Markov sequences, where the previous equations reduce to ordinary forward recursive equations, an alternative approach prices another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transforms, various relationships between the corresponding entries are identified. In particular, relationships between the solutions of matched forward and backward Riccati equations are thus proved probabilistically; they are proved again directly. In various specific cases, a further analysis of the concerned equations lead to completely explicit formulas for the Laplace transform.

Nonsymmetric Riccati equations: Partitioned algorithms

Generalized partitioned solutions (GPS) of nonsymmetric matric Riccati equations are presented in terms of forward and backward time differential equations that are of theoretical interest and also are computationally powerful. The GPS are the natural framework for the effective change of initial conditions, and the transformation of backward Riccati equation to forward Riccati equation and vice versa. Based on the GPS, computationally effective algorithms are obtained for the numerical solution of Riccati equations. These partitioned numerical algorithms have a decomposed or “partitioned” structure. They are given exactly in terms of a set of elemental solutions which are completely decoupled, and as such computable in either a parallel or serial processing mode. The overall solution is given exactly in terms of a simple recursive operation on the elemental solutions. Except for a subinterval of the total computation interval, the partitioned numerical algorithms are integration-free for the Riccati equation with constant or periodic matrices. Most importantly based on the GPS, a computationally attractive numerical algorithm is obtained for the computation of the steady-state solution of time-invariant Riccati equations. By making use of the GPS and some simple iterative operations, the Riccati solution is obtained in an interval which is twice as long as the previous interval requiring integration only in the initial subinterval.

Partitioned Ricatti solutions and integration-free doubling algorithms

Generalized partitioned solutions (GPS) of Riccati equations (RE) are presented in terms of forward and backward time differential equations that are theoretically interesting, possibly computationally advantageous, as well as provide interesting interpretations of these resuits, e.g., in terms of generalized partial observability and controllability matrices. The GPS are the natural framework for the effective change of initial conditions, and the transformation of backward RE to forward RE and vice-versa. The GPS are given in terms of families of forward or backward RE, and constitute generalizations to time-varying RE of well-known solution algorithms such as the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X - Y</tex> or Chandrasekhar algorithms. Most importantly, based on the GPS, computationally effective algorithms are obtained for the numerical solution of RE. These partitioned numerical algorithms (PNA) have a decomposed or "partitioned" structure, namely, they are given exactly in terms of a set of elemental solutions which are completely decoupled, and as such computable in either a parallel or serial processing mode. Further, the overall solution is given exactly in terms of a simple recursive operation on the elemental solutions. Finally, the PNA for a large class of RE, namely those with periodic or constant matrices, are completely integration-free, other than for a subinterval of the total computation interval, whose length, moreover, can be chosen arbitrarily. Also based on the GPS, a computationally attractive numerical algorithm is obtained for the computation of the steady-state solution of time-invariant RE. This algorithm results from doubling the length of the partitioning interval, and straightforward use of the GPS. The resulting "doubling" PNA is fast and is also essentially, integration-free requiring integration only in an initial subinterval, whose length is arbitrary, and subsequently consisting of simple iterative operations at the end of each time-interval which is twice as long as the interval in the previous iterations, i.e., doubling.