Linear Quadratic Systems Research Articles

Limited Attention Allocation in a Stochastic Linear Quadratic System With Multiplicative Noise

Limited Attention Allocation in a Stochastic Linear Quadratic System With Multiplicative Noise

Statistically consistent inverse optimal control for discrete-time indefinite linear–quadratic systems

The Inverse Optimal Control (IOC) problem is a structured system identification problem that aims to identify the underlying objective function based on observed optimal trajectories. This provides a data-driven way to model experts’ behavior. In this paper, we consider the case of discrete-time finite-horizon linear–quadratic problems where: the quadratic cost term in the objective is not necessarily positive semi-definite; the planning horizon is a random variable; we have both process noise and observation noise; the dynamics can have a drift term; and where we can have a linear cost term in the objective. In this setting, we first formulate the necessary and sufficient conditions for when the forward optimal control problem is solvable. Next, we show that the corresponding IOC problem is identifiable. Using the conditions for existence of an optimum of the forward problem, we then formulate an estimator for the parameters in the objective function of the forward problem as the globally optimal solution to a convex optimization problem, and prove that the estimator is statistical consistent. Finally, the performance of the algorithm is demonstrated on two numerical examples.

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

Novel closed-loop controllers for fractional linear quadratic time-varying systems

A novel method for finding the closed-loop optimal controllers of the finite-time fractional linear quadratic regulator systems is introduced. The necessary optimality conditions for the fractional optimal control problems are obtained. The matrix fractional differential Riccati equation is presented. A new method for solving the fractional differential Riccati equation is given for the first time which can also be applied to the integer order Riccati equation. Several illustrative examples (including a high order system) are presented to show the applicability and capabilities of the method.

Asymptotically Optimal Distributed Control for Linear Quadratic Mean Field Social Systems With Heterogeneous Agents

Asymptotically Optimal Distributed Control for Linear Quadratic Mean Field Social Systems With Heterogeneous Agents

ϵ-Nash mean-field games for stochastic linear-quadratic systems with delay and applications

ϵ-Nash mean-field games for stochastic linear-quadratic systems with delay and applications

Linear quadratic Gaussian-based clock steering system for GNSS real-time precise point positioning timing receiver

Linear quadratic Gaussian-based clock steering system for GNSS real-time precise point positioning timing receiver

Decentralized Linear Quadratic Systems With Major and Minor Agents and Non-Gaussian Noise

A decentralized linear quadratic system with a major agent and a collection of minor agents is considered. The major agent affects the minor agents, but not vice versa. The state of the major agent is observed by all agents. In addition, the minor agents have a noisy observation of their local state. The noise processes is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">not</i> assumed to be Gaussian. The structures of the optimal strategy and the best linear strategy are characterized. It is shown that major agent's optimal control action is a linear function of the major agent's MMSE (minimum mean squared error) estimate of the system state while the minor agent's optimal control action is a linear function of the major agent's MMSE estimate of the system state and a “correction term” which depends on the difference of the minor agent's MMSE estimate of its local state and the major agent's MMSE estimate of the minor agent's local state. Since the noise is non-Gaussian, the minor agent's MMSE estimate is a non-linear function of its observation. It is shown that replacing the minor agent's MMSE estimate by its LLMS (linear least mean square) estimate gives the best linear control strategy. The results are proved using a direct method based on conditional independence, common-information-based splitting of state and control actions, and simplifying the per-step cost based on conditional independence, orthogonality principle, and completion of squares.

Suboptimal control law for a multi fractional high order linear quadratic regulator system in the presence of disturbance

A new method for finding a suboptimal control law for a multi fractional spring–mass system as a fractional high order linear quadratic regulator system in the presence of known disturbance is introduced. The method is presented in a general form for solving practical multi fractional linear quadratic regulator systems with disturbances. After defining a closed-loop controller, formulations for finding the suboptimal controller are derived. In the proposed method, there is no need to solve the fractional vector differential equation. Simulation studies demonstrate the applicability of the method for practical purposes. Some new types of a spring–mass–damper system are considered.

A Linear Quadratic Regression-Based Synchronised Health Monitoring System (SHMS) for IoT Applications

In recent days, the IoT along with wireless sensor networks (WSNs), have been widely deployed for various healthcare applications. Nowadays, healthcare industries use electronic sensors to reduce human errors while analysing illness more accurately and effectively. This paper proposes an IoT-based health monitoring system to investigate body weight, temperature, blood pressure, respiration and heart rate, room temperature, humidity, and ambient light along with the synchronised clock model. The system is divided into two phases. In the first phase, the system compares the observed parameters. It generates advisory to parents or guardians through SMS or e-mails. This cost-effective and easy-to-deploy system provides timely intimation to the associated medical practitioner about the patient’s health and reduces the effort of the medical practitioner. The data collected using the proposed system were accurate. In the second phase, the proposed system was also synchronised using a linear quadratic regression clock synchronisation technique to maintain a high synchronisation between sensors and an alarm system. The observation made in this paper is that the synchronised technology improved the performance of the proposed health monitoring system by reducing the root mean square error to 0.379% and the R-square error by 0.71%.

Optimal stealthy false data injection attacks for networked linear quadratic Gaussian control systems with two channels

SummaryIn this paper, a security problem in networked control systems which are modeled as linear quadratic Gaussian control systems is investigated from the perspective of attackers. We consider the scenario that the attackers can inject false data into two channels: sensor‐to‐controller and controller‐to‐actuator channels. The goal of attackers is to design the optimal stealthy false data injection attacks to maximize the linear quadratic Gaussian control cost function, where Kullback‐Leiber divergence is used as a stealthiness metric. Firstly, the cost function is divided into a fixed part and a part which is related to a remote estimation error covariance. Then, when the two channels are attacked by attackers, the recursion of remote estimation error covariance is derived. In the presence of attacks, the optimization problem with multiple decision variables, which may not be a convex problem, is transformed into a convex optimization problem and the corresponding optimal solution is obtained. In addition, an upper bound of benefit loss when using the optimal solution of the convex optimization problem is presented. Finally, a simulation example is presented to show the effectiveness of the obtained results.

Optimization Under Rational Expectations: A Framework of Fully Coupled Forward-Backward Stochastic Linear Quadratic Systems

In this paper, we propose a general modeling framework for optimal control of stochastic fully coupled forward-backward linear quadratic (FBLQ) problems with indefinite control weight costs that stem from rational expectations models. We propose a new decoupling technique to obtain the optimal feedback control, which is accompanied by one kind of non-Riccati-type ordinary differential equation (ODE). By applying the completion-of-squares method, we prove the existence of the solutions for the obtained ODEs. The obtained results make it possible to compute the control and value function. For this FBLQ problem, the optimal control should depend on the entire trajectory of the state process. Several examples are given to illustrate our results. Funding: M. Hu’s research was supported by the National Science Foundation (NSF) [Grant 11671231] and the Young Scholars Program of Shandong University [Grant 2016WLJH10]. S. Ji’s research was supported by the NSF [Grant 11571203]. X. Xue’s research was supported by “The Fundamental Research Funds of Shandong University,” the NSF [Grants 12001316 and 61907022], and the Natural Science Foundation of Shandong Province [Grant ZR2019BF015].

Robust optimal sliding mode control for the deployment of Coulomb spacecraft formation flying

Inter-spacecraft electrostatic force (Coulomb force) is desirable for close formation flying control because of its propellant-less and free contaminate characteristics attributed to the propellant exhaust emission. This paper presents robust optimal sliding mode control to deal with the problem of thruster saturation in tracking the formation trajectory for Coulomb spacecraft formation flying. The robust controller design is based on optimal control theory as a linear quadratic system, and it is augmented with an integral sliding mode control technique. The stability of the closed-loop system is guaranteed using the second Lyapunov method. The developed controller outperforms the existing ones, because it has a higher degree of fine-tuning to cope with the uncertainty. Numerical simulations are employed to confirm the efficiency of the developed controller.

Balancing control of a motorcycle

Abstract This article deals with balancing an autonomous motorcycle model along a straight line and curve lines. The dynamic model of the motorcycle balancing is described with an inverted physical pendulum loaded with torque. The torque is provided by the inertia of a rotor driven by a direct current motor. The lean angle of the motorcycle is measured by a smart sensor, which is the feedback signal for the linear quadratic regulator control system. The main purpose of this study is to compensate the error of the smart sensor. Controlling the necessary lean angle of the motorcycle during cornering is also addressed.

On normal forms and return maps for pseudo-focus points

For planar systems with a pseudo-equilibrium point of focus type within its discontinuity line, computable normal forms are introduced. Thus, the classical theory of normal forms is adapted for dealing with piecewise smooth systems having a common invisible tangency from each side. The methodology looks for removing unessential terms in the expression of the vector field, and is based upon expanding the vector field as a sum of quasi-homogeneous terms, next applying adequate changes of variables that preserve every point of the discontinuity line. From these normal forms, it is easier to compute the associated half-return maps and to determine the maximal number of periodic orbits than can bifurcate from a pseudo-focus. Furthermore, a recent conjecture in this journal on the behaviour of the displacement function around an invisible fold-fold singularity is shown to be true. We illustrate the obtained results by considering some relevant examples concerning piecewise linear and linear-quadratic systems, also revisiting a piecewise smooth vector field of quasi-homogeneity degree four studied by A.F. Filippov.

Robust mean field social control problems with applications in analysis of opinion dynamics

This paper investigates the social optimality of linear quadratic mean field control systems with unmodelled dynamics. The objective of agents is to optimise the social cost, which is the sum of costs of all agents. By variational analysis and direct decoupling methods, the social optimal control problem is analysed, and two equivalent auxiliary robust optimal control problems are obtained for a representative agent. By solving the auxiliary problem with consistent mean field approximations, a set of decentralised strategies is designed, and its asymptotic social optimality is further proved. Next, the results are applied into the study of opinion dynamics in social networks. The evolution of opinions is analysed over finite and infinite horizons, respectively. All opinions are shown to reach agreement with the average opinion in a probabilistic sense. Finally, local interactions among multiple sub-populations are examined via graphon theory.

A Novel Wavelet-based Optimal Linear Quadratic Tracker for Time-varying Systems with Multiple Delays

A novel method for solving optimal tracking control of linear quadratic time-varying systems with differentforms of time delays in state and input variables and with constraints is presented in this paper. Using the concepts of two powerful wavelets, Legendre and Chebyshev wavelets, we convert the optimal tracking problem to a static optimization one. The method is presented in a general from by which one can utilize it by other wavelets. The proposed method has the ability to solve the problems with systems of integer and fractional orders. After determining open-loop solutions of time-delay tracking systems, closed-loop suboptimal controller is designed. A highly successful wavelet-based suboptimal controller is introduced in this work. This alternative method is applied on some optimal tracking systems.

Inverse optimal control from incomplete trajectory observations

This article develops a methodology that enables learning an objective function of an optimal control system from incomplete trajectory observations. The objective function is assumed to be a weighted sum of features (or basis functions) with unknown weights, and the observed data is a segment of a trajectory of system states and inputs. The proposed technique introduces the concept of the recovery matrix to establish the relationship between any available segment of the trajectory and the weights of given candidate features. The rank of the recovery matrix indicates whether a subset of relevant features can be found among the candidate features and the corresponding weights can be learned from the segment data. The recovery matrix can be obtained iteratively and its rank non-decreasing property shows that additional observations may contribute to the objective learning. Based on the recovery matrix, a method for using incomplete trajectory observations to learn the weights of selected features is established, and an incremental inverse optimal control algorithm is developed by automatically finding the minimal required observation. The effectiveness of the proposed method is demonstrated on a linear quadratic regulator system and a simulated robot manipulator.

Semifoldover plans for three-level orthogonal arrays with quantitative factors

Although foldover designs can de-alias many effects, they involve at least twice the original number of runs. A semifoldover design, one kind of the partial foldover designs, is typically much more efficient since such a design adds only half of the new runs of a foldover design to the initial design. Semifoldover designs for two-level orthogonal arrays have been investigated in recent literatures. With the use of linear-quadratic system, this paper considers semifoldover designs for three-level orthogonal arrays with quantitative factors. We examine when the linear effects can be de-aliased from their aliased two-factor interactions for regular and nonregular designs, and obtain some good properties via semifolding over on partial factors or all factors. Theoretical properties and some examples are provided to illustrate the usefulness of the proposed designs.

The optimal control system of the ship based on the linear quadratic regular algorithm

In this paper, the authors propose an optimal controller for the ship motion. Firstly, the model and dynamic equations of the ship motion are presented. Based on the model of the ship motion, the authors build the linear quadratic regular algorithm-based control system of ship motion to minimize the error between the desired trajectory and the response trajectory. The task of the controller is to control the trajectory of the ship to coincide with the desired trajectory. The ship model and controller are built to investigate the system quality through Matlab-Simulink software. The results show that the quality of the hold control system is very high. The trajectory of a ship always follows the desired trajectory with very small errors.