Linear Optimal Control Theory Research Articles

Optimal Control Strategy of Electro-hydraulic Position Servo System Using Genetic Algorithm

Background: The optimal control strategy has been widely used in electro-hydraulic position servo systems to achieve high-precision position tracking. However, the difficulty of selecting the weighted matrices in optimal control often leads to poor tracking accuracy. Objective: This patent proposes an optimal control strategy using a genetic algorithm to improve the tracking accuracy of the electro-hydraulic servo system. Methods: The patent first established the system state equation of the valve-controlled asymmetric cylinder. Secondly, based on linear quadratic optimal control theory and genetic algorithm, an optimal control strategy using a genetic algorithm was proposed. Finally, the simulation and experimental results showed that the designed controller has high position tracking accuracy . Results: The optimal controller using a genetic algorithm was designed using Matlab/Simulink, and the effectiveness of the controller was verified through simulation. Additionally, experimental results showed that the proposed optimal control controller using a genetic algorithm had higher tracking accuracy than the proportional-integral-derivative controller and traditional backstepping controller for a given reference signal. Conclusion: The control technology of the optimal controller using a genetic algorithm was found to be superior to proportional-integral-derivative and traditional backstepping controllers, and the tracking error of the linear quadratic regulator controller was reported to be relatively small. This demonstrated the effectiveness of the optimal control strategy using a genetic algorithm in this patent.

Maximum power tracking for wind energy conversion systems via a high-order optimal disturbance observer-based LQR without a wind speed sensor

A high-order optimal disturbance observer (HOODO) is introduced to precisely estimate the aerodynamic torque and the variable wind speed; and consequently calculate the optimal speed of the generator without measuring the wind speed. As the immeasurable wind speed is varying fast, the aerodynamic torque, which is considered as a disturbance, is also changed fast, then the conventional assumption that the disturbance is slowly-varying (i.e., its first-time derivative is zero), is not applicable. The proposed HOODO design considers the fast and stochastic characteristics of the wind speed by relaxing aforementioned conventional assumption. Moreover, via the linear optimal control theory, the parameters of the HOODO are tuned systematically by proper selection of the elements of the diagonal weighting matrices. In this study, a compromise between the observer’s convergence rate and the ability to Gaussian noise suppression is also considered. This helps to solve the difficulties of many existing observers regarding the gain selection algorithms. Also in the article, for the first time, a control scheme is considered in combination with the proposed HOODO and the LQ controller. To maximize a power captured from the wind, a linear-quadratic regulator (LQR) is utilized to maintain the angular speed of the generator at the optimal speed reference. A stability analysis of the designed control scheme is also discussed. Comparative results of the HOODO under different orders as well as other observers are given to prove the effectiveness of the introduced estimation method. The obtained simulation results reveal that the HOODO has a superior estimation of disturbance. All the simulations are carried out in the MATLAB/Simulink software.

A Multiobjective Feedback Linearization Control of a Cuk Converter

Cuk converter is one of the most common power electronic converters, but its controller design has always been a challenging problem due to its fourth-order nonlinear and nonminimum phase characteristics. In order to solve this problem, this paper proposes a multi-objective feedback linearization control method, which selecting the variables of interest to rebuild the output function to linearize the nonlinear system into a combination of a linear subsystem and a nonlinear subsystem. According to the structural characteristics of these two subsystems, the linear optimal control theory is adopted for the control design of the linear subsystem to make the original nonlinear system have a good output performance, while for the nonlinear subsystem, the coefficient of the output function is adjusted to ensure the stability of the original nonlinear system. The results of simulation and experiment show the feasibility and superiority of using the multi- objective feedback linearization control method to design the nonlinear control law of the Cuk converter.

Lyapunov equation in open quantum systems and non-Hermitian physics

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

Optimal Unmanned Ground Vehicle—Unmanned Aerial Vehicle Formation-Maintenance Control for Air-Ground Cooperation

This paper investigates the air–ground cooperative time-varying formation-tracking control problem of a heterogeneous cluster system composed of an unmanned ground vehicle (UGV) and an unmanned aerial vehicle (UAV). Initially, the structure of the UAV–UGV formation-control system is analyzed from the perspective of a cooperative combat system. Next, based on the motion relationship between the UAV–UGV in a relative coordinate system, the relative motion model between them is established, which can clearly reveal the physical meaning of the relative motion process in the UAV–UGV system. Then, under the premise that the control system of the UAG is closed-loop stable, the motion state of the UGV is modeled as an input perturbation. Finally, using a linear quadratic optimal control theory, a UAV–UGV formation-maintenance controller is designed to track the reference trajectory of the UGV based on the UAV–UGV relative motion model. The simulation results demonstrate that the proposed controller can overcome input perturbations, model-constant perturbations, and linearization biases. Moreover, it can achieve fast and stable adjustment and maintenance control of the desired UAV–UGV formation proposed by the cooperative combat mission planning system.

A Multi-Index Feedback Linearization Control for a Buck-Boost Converter

Due to the nonlinear and nonminimum phase characteristics of the buck-boost converter, the design of its controller has always been a challenging problem. In this paper, a multi-index feedback linearization control strategy is proposed to design the controller of the buck-boost converter. Firstly, by constructing an appropriate output function, the original nonlinear system is mapped into a combination of a linear subsystem and a nonlinear subsystem. Then, according to the structural characteristics of these two subsystems, the linear optimal control theory is adopted for the control design of the linear subsystem to make it have a good output performance, while for the nonlinear subsystem, the coefficient of the output function is adjusted to ensure its stability. Finally, based on the Hartman–Grobman theorem, the internal mechanism and coefficient adjustment basis of the proposed method are revealed; that is, by adjusting the coefficient of the output function and the feedback coefficient of the linear control law, the poles of the system are configured to achieve the purpose of adjusting the static and dynamic performance of the system. The simulation results show the feasibility and superiority of using the multi-index feedback linearization control strategy to design the nonlinear control law of the buck-boost converter.

Optimization design of active suspension of vehicle based on LQR control

In this article, we take the 1/4 vehicle model as an example, according to the requirements of the driving performance of vehicles. We establishes the dynamic model of the vehicle, and use the linear quadratic optimal control theory to design the LQG controller of the active suspension, and use MATLAB/simulink to simulate the vehicle dynamic model. The results show that the active suspension with LQG controller has a good effect on the improvement of vehicle running stability and riding comfort.

Feedback control of chaotic systems using multiple shooting shadowing and application to Kuramoto-Sivashinsky equation.

We propose an iterative method to evaluate the feedback control kernel of a chaotic system directly from the system's attractor. Such kernels are currently computed using standard linear optimal control theory, known as linear quadratic regulator theory. This is however applicable only to linear systems, which are obtained by linearizing the system governing equations around a target state. In the present paper, we employ the preconditioned multiple shooting shadowing (PMSS) algorithm to compute the kernel directly from the nonlinear dynamics, thereby bypassing the linear approximation. Using the adjoint version of the PMSS algorithm, we show that we can compute the kernel at any point of the domain in a single computation. The algorithm replaces the standard adjoint equation (that is ill-conditioned for chaotic systems) with a well-conditioned adjoint, producing reliable sensitivities which are used to evaluate the feedback matrix elements. We apply the idea to the Kuramoto-Sivashinsky equation. We compare the computed kernel with that produced by the standard linear quadratic regulator algorithm and note similarities and differences. Both kernels are stabilizing, have compact support and similar shape. We explain the shape using two-point spatial correlations that capture the streaky structure of the solution of the uncontrolled system.

Research on Maneuvering Trajectory Tracking Accuracy of Target Missile Based on Feedback Linearization

Aiming at the trajectory tracking accuracy of missile target, a trajectory tracking guidance law is designed based on feedback linearization theory and linear quadratic optimal control theory, and the tracking accuracy of the designed trajectory tracking guidance law under different maneuvering trajectories is tested. The motion equations of missiles are typical non-linear motion models, so feedback linearization is introduced as an exact linearization method, and the definition and necessary and sufficient conditions of feedback linearization theory are listed. Firstly, the non-linear equations of the longitudinal motion of target missile are established, and the feedback linearization method is used to linearize the non-linear model accurately. Then, the linearized model is designed by using the linear quadratic optimal control theory, and a target trajectory tracking guidance law is obtained. Finally, given gust disturbance and different maneuvering trajectories, the designed trajectory tracking guidance law is simulated, and the tracking accuracy of the guidance law is studied by changing the control coefficient. The results show that the trajectory tracking guidance law based on feedback linearization has excellent trajectory tracking ability, good tracking accuracy and strong anti-jamming ability under different maneuvering trajectories.

Simulation and Experimental Study on the Active Stability of High-Speed Trains

An active hunting stability scheme is proposed for the high-speed trains based on frame lateral vibration control. The stability of the vehicle is improved by exerting active control force on the front and rear frames of the bogie. First, a simplified lateral vibration model of a single bogie is established to the control system design. The feedback gain matrix is obtained according to the linear quadratic optimal control theory. Then, the multibody dynamics model of the vehicle is built using SIMPACK, and the linear stability and straight running performance are analyzed under different working configurations. Finally, the active control effects are verified by a scaled roller rig. The results show that this active control scheme to improve the stability performances of vehicles through the feedback of frame vibration state is feasible.

Kalman 1960: The birth of modern system theory

ABSTRACTRudolph E. Kalman is mainly known for the Kalman filter, first published in 1960. In this year, he published two equally important contributions, one about linear state space system theory and the other about linear quadratic optimal control theory. These three domains are intertwined in the later theory of linear quadratic Gaussian control. An extended version of linear quadratic optimal control is put into practice in an example of cooperation in population ecology.

Cooperative Guidance and Collision Avoidance for Multiple Pursuers

This paper presents a cooperative guidance law for an -on- engagement scenario that ensures target interception and collision avoidance between the pursuers along with minimizing their team effort. The guidance law is derived using linear quadratic optimal control theory for a linearized engagement model. The pursuers cooperate with each other to accommodate differences in maneuver capabilities of different team members. A special case of 2-on-2 engagement is also analyzed, for which analytical closed-form formulas for the pursuer’s guidance law are obtained. Various simulation results and experimental validation results, exemplifying the cooperation in different engagements, are also presented.

Time-Coordinated Control for Unmanned Aerial Vehicle Swarm Cooperative Attack on Ground-Moving Target

In this paper, cooperative attack control law analysis and design problems for unmanned aerial vehicle (UAV) swarm attack on ground-moving target with time-coordinated strategies are investigated. First, the time-coordinated control problem for a single UAV is formulated, which is the foundation of solving the problem of a single UAV arriving at the desired attack position relative to ground-moving target at a specific terminal time. Then, relative motion between each UAV and ground-moving target is considered as a finite-time time-varying tracking system problem, and the difference between expected output and system output is defined as tracking error vector. The control law is obtained by linear quadratic optimal control theory to minimize the energy cost in the whole process and the tracking error at the terminal time. Besides, time-coordinated function, which is critical to coordinate terminal time among all UAVs in the UAV swarm, is proposed to model time-coordinated strategies. Finally, numerical simulations show that the proposed control law can steer UAV swarm to arrive at the desired attack positions and achieve the time-coordinated strategies effectively.

Adaptive Optimal Control With Guaranteed Convergence Rate for Continuous-Time Linear Systems With Completely Unknown Dynamics

The traditional linear quadratic optimal control can be summarized as finding the state feedback controller so that the closed-loop system is stable and the performance index is minimum. And, it is well-known that the solution of the linear quadratic optimal control problem can be solved by algebraic Riccati equation. However, the feature of the traditional linear quadratic optimal control theory is that the convergence rate is not specified. This may result in the phenomena of slow convergence. In this paper, we mainly consider the linear quadratic optimal control with guaranteed convergence rate (LQOCGCR) and propose a policy iteration-based adaptive dynamic programming algorithm that includes offline and online versions for finding the solution of the LQOCGCR. Finally, a numerical example is worked out to show the effectiveness of the proposed approach.

Assessment of shunted piezoelectric devices for simultaneous noise and vibration reduction: comparison of passive, active and hybrid networks

Structural vibration and noise control of a cavity-backed three-layered smart piezo-coupled rectangular panel system under harmonic or transient loads is achieved by using purely active, passive, and hybrid active/passive piezoelectric shunt networks. Problem formulation is based on the classical lamination plate theory, Maxwell’s equation for piezoelectric materials, linear circuit theory, and wave equation for the enclosed acoustic domain. The orthogonal mode expansions along with the modal coupling theory are employed to obtain the coupled differential equations of the electro-mechanical-acoustic system, which are then put into the convenient state-space form, and subsequently solved numerically in both frequency and time domains. A triple-mode hybrid RLC shunt circuit, in series with an external active voltage source and connected to a single electroded piezoelectric segment, is tuned to the dominant resonance frequencies of the composite structure. The linear quadratic optimal control (LQR) theory is adopted for obtaining the active control gains. The frequency and time domain performances of the passive, active and hybrid multi-modal piezoelectric systems are calculated and discussed in terms of sensor output voltage, local sound pressure, and control effort. It is found that the hybrid control methodology with properly tuned circuit parameters can be an excellent candidate for simultaneous vibration and structure-borne noise control of the cavity-coupled smart panel with decreased control effort. Also, the active control strategy integrated in the hybrid control system is demonstrated to enhance the overall system damping characteristics and improve the control authority at frequencies where the passive shunt network performs weakly. Limiting cases are considered and correctness of the mathematical model is verified by using a commercial finite element software as well as by comparisons with the literature.

Design of an Optimal Preview Controller for Dual-Rate Linear Discrete-Time Systems

Abstract A dual-rate preview control strategy for a type of discrete-time system is proposed based on the theory of multirate control. First, by using the discrete lifting technique, the general dual-rate discrete-time system is converted into a single-rate augmented system. On this basis, the augmented error system is constructed by introducing a first-order difference operator and the previewable reference signal. Then the tracking problem is transformed into a regulator problem of the augmented error system. The optimal preview control law of the augmented error system is obtained by using standard linear quadratic optimal preview control theory, and then the optimal preview controller of the original system is derived. In addition, the necessary and sufficient conditions for the controller are given. Finally, simulation results show the effectiveness of the proposed method.

Optimisation and Control of Vehicle Suspension Using Linear Quadratic Gaussian Control

Abstract The paper deals with the optimal design and analysis of quarter car vehicle suspension system based on the theory of linear optimal control because Linear Quadratic Gaussian (LQG) offers the possibility to emphasize quantifiable issues like ride comfort or road holding very easily by altering the weighting factor of a quadratic criterion. The theory used assumes that the plant (vehicle model + road unevenness model) is excited by white noise with Gaussian distribution. The term quadratic is related to a quadratic goal function. The goal function is chosen to provide the possibility to emphasize three main objectives of vehicle suspensions; ride comfort, suspension travel and road holding. Minimization of this quadratic goal function results in a law of feedback control. For optimal designs are used the optimal parameters which have been derived by comparison of two optimisation algorithms: Sequential Quadratic Program (SQP) and Genetic Algorithms (GA's), for a five chosen design parameters. LQG control is considered to control active suspension for the optimal parameters derived by GA's, while the main focus is to minimise the vertical vehicle body acceleration

Cooperative global optimal preview tracking control of linear multi-agent systems: an internal model approach

ABSTRACTThis paper investigates the cooperative global optimal preview tracking problem of linear multi-agent systems under the assumption that the output of a leader is a previewable periodic signal and the topology graph contains a directed spanning tree. First, a type of distributed internal model is introduced, and the cooperative preview tracking problem is converted to a global optimal regulation problem of an augmented system. Second, an optimal controller, which can guarantee the asymptotic stability of the augmented system, is obtained by means of the standard linear quadratic optimal preview control theory. Third, on the basis of proving the existence conditions of the controller, sufficient conditions are given for the original problem to be solvable, meanwhile a cooperative global optimal controller with error integral and preview compensation is derived. Finally, the validity of theoretical results is demonstrated by a numerical simulation.

My Experiences with Rudolf Kalman [Historical Perspectives

The first time I saw Rudy Kalman was around 1964 at Purdue University while attending a lecture he was giving. I can't remember the subject. At that time I was a young, inexperienced graduate student, and I didn't know who he was. That changed quite fast with my exposure to courses on linear systems, estimation theory, optimal control, and stability theory, since these were all subjects upon which Rudy's research was having a huge impact. I soon realized that the person who I had seen give that lecture was actually a celebrity.

The Research of Discrete Mean - Variance Portfolio Problem with Time-Delay

Due to the financial sector complicated variety of events, each financial problems from changes to know its essence, the change rule, from the change of strategy to formulate relevant policy and policy into effect, etc., the process inevitably has a certain lag. Therefore, in order to better reflect the actual situation, we study the portfolio model with delays in this paper. By joining our delay control item, the optimization model was established, the goal is to maximize earnings expectations. In this paper, it studies the continuous time without delay the mean - variance portfolio problems on the basis of existing research. It established auxiliary problem using the stochastic linear quadratic optimal control theory. Using the maximum principle, the solution of the optimal investment strategy are given and it analysis the case, the conclusion is in conformity with the actual. It studies the existing time delay portfolio strategy problem in discrete time case. Based on the stochastic LQ (linear quadratic) optimal control theory, it established the discrete time model with time delay. The paper has carried on the solution and example analysis.