Linear Optimal Control Research Articles

Feedback control of limit cycle oscillations and transonic buzz, using the nonlinear transonic small disturbance aerodynamics

In this paper, a systematic method to suppress transonic buzz with feedback is presented. A trailing edge control surface in the form of part-span flap was used only to modify and control the unsteady aerodynamic loading on the wing. The flap rotation was used to provide feedback, which consisted of a weighted linear combination of the amplitudes of the principal modes of the structure, referred to as the control law. A linear, optimal feedback control law, that is synthesized systematically based on pseudo-spectral time domain analysis, may be used in principle, to assess its capacity to actively suppress the buzz in the transonic flow domain by using a servo-controlled control surface to modify the unsteady, nonlinear aerodynamic loads on the wing. Thus, it is essential that a set of feasible control laws are first constructed. In this paper, this is done by applying the doublet-lattice method. Restrictions, such as near-zero structural damping in the flap mode, were imposed on the aeroelastic model to facilitate the occurrence of transonic buzz. The feasible set of control laws were then assessed using the nonlinear transonic small disturbance theory and an optimum control is selected to suppress the buzz. The essential differences of the behaviour of the closed-loop system in nonlinear transonic flow, when compared to the applications of linear optimal control in linear potential flow, are presented and discussed.

The true potential of nonlinear stiffness for point absorbing wave energy converters

A spherical point absorbing wave energy converter was simulated in floating and submerged conditions for regular and irregular waves. A nonlinear stiffness was included to augment a linear stiffness and damper control system. The linear control parameters were optimised for each regular and irregular wave condition. The optimal linear control stiffness for the floating system was negative to counteract the hydrostatic stiffness. The nonlinear stiffness was shown to increase the converted power if no linear control stiffness was used, and reduce the converted power if optimal linear control parameters were used for regular waves. Similarly, for irregular conditions, nonlinear stiffness degrades the performance of the system when compared to optimised parameters, but enhances the robustness of the system to changing sea states and stochastic phases. Therefore, a nonlinear stiffness mechanism may improve the power output for poorly tuned control systems arising from plant uncertainty for point absorbing wave energy converters.

Tank testing experiment of the Mocean M100 wave energy converter: linear non-causal optimal control and wave prediction

This paper presents tank testing experimental results of applying a non-causal optimal control strategy to a hinged-raft wave energy converter (WEC). The linear non-causal optimal control (LNOC) algorithm is designed and implemented in real-time to calculate the PTO torque signal based on the WEC response feedback information and the incoming wave feed-forward information. A controllable DC motor is used to provide the required torque to the WEC hinge. Tank testing results show that, compared with a well-tuned passive damper, the LNOC can improve the absorbed mechanical power in most of the tested sea conditions by up to 126%.

On optimal stochastic linear quadratic control with inversely proportional time-weighting in the cost

Рассматривается задача оптимального стохастического линейного регулятора для системы с абсолютно интегрируемыми на бесконечности матрицами при состоянии в уравнении процесса и в функционале потерь. При этом множители, входящие в матрицы интегрального квадратичного целевого функционала, имеют взаимно обратно пропорциональную динамику во времени. Показывается, что известный линейный установившийся закон управления является оптимальным по критериям из класса обобщенных долговременных средних. Результаты применяются при анализе системы управления с динамическим масштабированием параметров.

Deep Reinforcement Learning Based Adaptation of Pure-Pursuit Path-Tracking Control for Skid-Steered Vehicles

The growing need for autonomous vehicles in the offroad space raises certain complexities that need to be considered more rigorously in comparison to onroad vehicle automation. Popular path control frameworks in onroad autonomy deployments such as the pure-pursuit controller use geometric and kinematic motion models to generate reference trajectories. However in the offroad settings these controllers, despite their merits (low design and computation requirements), could compute dynamically infeasible trajectories as several of the nominal assumptions made by these models don't hold true when operating in a 2.5D terrain. Outside of the notable challenges such as uncertainties and non-linearities/disturbances introduced by the unknown/unmapped 2.5D terrains, additional complexities arise from the use of vehicle architectures such as the skid-steer that experience lateral skidding for achieving simple curvilinear motion. Additionally, linear models of skid-steer vehicles often consist of high modeling uncertainty which renders traditional linear optimal and robust control techniques inadequate given their sensitivity to modeling errors. Nonlinear MPC has emerged as an upgrade, but needs to overcome realtime deployment challenges (including slow sampling time, design complexity, and limited computational resources). This provides an unique opportunity to utilize data-driven adaptive control methods in tailored application spaces to implicitly learn and hence compensate for the unmodeled aspects of the robot operation. In this study, we build an adaptive control framework called Deep Reinforcement Learning based Adaptive Pure Pursuit (DRAPP) where the base structure is that of a geometric Pure-Pursuit (PP) algorithm which is adapted through a policy learned using Deep Reinforcement Learning (DRL). An additional law that enforces a mechanism to account for the rough terrain is added to the DRL policy to prioritize smoother reference-trajectory generation (and thereby more feasible trajectories for lower-level controllers). The adaptive framework converges quickly and generates smoother references relative to a pure 2D-kinematic path tracking controller. This work includes extensive simulations and a bench marking of the DRAPP framework against Nonlinear Model Predictive Control (NMPC) that is an alternate popular choice in literature for this application.

Control strategies with multiple closing instants for linear optimal control problems with disturbances*

This paper deals with an optimal control problem for a linear discrete system subject to input and state constraints and unknown bounded disturbances, where the control goal is to minimize a cost function used in linear explicit model predictive control. Since open-loop worst-case solution is conservative or may suffer feasibility problems and dynamic programming solution is computationally demanding, we propose a compromise between those solutions. This compromise is an optimal control strategy that takes into account the state measurements of the system at several future time instants (closing instants). We define control strategies with one and multiple closing instants and propose efficient numerical methods for their construction.

A Few Methods of Solving Optimal Control Problem in Hamilton-Jacobi-Bellman Equation Form

Non-linear optimal control problem arises in many different areas, for example, engineering, medical sciences, economics, industries, etc. The solution of Hamilton-Jacobi-Bellman equation is connected with the non -linear optimal control problem. In this paper, we formulate the Hamilton-Jacobi-Bellman equation using nonlinear optimal control problem. We also discuss its solutions using Adomian decomposition method, Laplace transform-Homotopy perturbation method and variational iteration method.

Optimality conditions in optimal control problems for a Gursa-Darboux system with a multipoint objective functional

We consider one linear optimal control problem described by a system of hyperbolic equations with the Goursat boundary condition and a multipoint linear objective functional. A necessary and sufficient optimality condition of the Pontryagin maximum principle type is proved.

Data-Driven Model-Free Adaptive Control of Z-Source Inverters.

The universal paradigm shift towards green energy has accelerated the development of modern algorithms and technologies, among them converters such as Z-Source Inverters (ZSI) are playing an important role. ZSIs are single-stage inverters which are capable of performing both buck and boost operations through an impedance network that enables the shoot-through state. Despite all advantages, these inverters are associated with the non-minimum phase feature imposing heavy restrictions on their closed-loop response. Moreover, uncertainties such as parameter perturbation, unmodeled dynamics, and load disturbances may degrade their performance or even lead to instability, especially when model-based controllers are applied. To tackle these issues, a data-driven model-free adaptive controller is proposed in this paper which guarantees stability and the desired performance of the inverter in the presence of uncertainties. It performs the control action in two steps: First, a model of the system is updated using the current input and output signals of the system. Based on this updated model, the control action is re-tuned to achieve the desired performance. The convergence and stability of the proposed control system are proved in the Lyapunov sense. Experiments corroborate the effectiveness and superiority of the presented method over model-based controllers including PI, state feedback, and optimal robust linear quadratic integral controllers in terms of various metrics.

Parametric variational procedure to solve index-3 semi explicit linear-regulator control problem with constrained robot illustration

The aim of this work is to solve 2-dimensional, 3-links linearized constrained robot problem. This problem is of index-three linear descriptor control problem. Since the aim is to regulate its output, a quadratic performance index is proposed to form an index-3 linear regulator optimal control problem. Firstly, the problem has been transformed into standard constrained optimal control problem over class of consistent initial condition with solvability assumptions. Secondly, the solution of differential-algebraic equation has been shown as equivalent to find the critical points of some variational formulation and vise- versa. Then aggregating all functional (performance one) with consistent initial conditions in one performance (weighted) index. Under standard topology of the solvability, a parametric approach has been proposed to approximate the state-space solutions with control over the class of constraints. Hence, an infinite dimensional problem is transformed into finite one, and then an approximate solutions is obtained by solving the class of linear algebraic equations. The proposed approach is easily implemented and accurate up to decision maker selections of the parameterization basis functions and their numbers.

Gravity compensation and optimal control of actuated multibody system dynamics

This work investigates the gravity compensation topic, from a control perspective. The gravity could be levelled by a compensating mechanical system or in the control law, such as proportional derivative (PD) plus gravity, sliding mode control, or computed torque method. The gravity compensation term is missing in linear and nonlinear optimal control, in both continuous- and discrete-time domains. The equilibrium point of the control system is usually zero and this makes it impossible to perform regulation when the desired condition is not set at origin or in other cases, where the gravity vector is not zero at the equilibrium point. The system needs a steady-state input signal to compensate for the gravity in those conditions. The stability proof of the gravity compensated control law based on nonlinear optimal control and the corresponding deviation from optimality, with proof, are introduced in this work. The same concept exists in discrete-time control since it uses analog to digital conversion of the system and that includes the gravity vector of the system. The simulation results highlight two important cases, a robotic manipulator and a tilted-rotor hexacopter, as an application to the claimed theoretical statements.

Methodology of determining of the transfer function of engagement kinematics of accelerations of an aircraft and its elliptic coordinates in alternative representation

In this work, simplified expressions have been obtained that describe the kinematics parameters of the aircraft movement and its accelerations. These expressions are needed to obtain an optimal linear control law in order to provide movement of the aircraft using the hyperbola guidance method with a time difference of arrival navigation system. The key feature of the hyperbola navigation method is the ability to reduce the number of navigation positions by one when the on-board navigation equipment operates in a passive mode, carrying out only the reception of navigation information, like consumers of satellite navigation information.

High-Capacity Wave Energy Conversion by Multi-Float, Multi-PTO, Control and Prediction: Generalized State-Space Modelling With Linear Optimal Control and Arbitrary Headings

Wave energy converters with capacity similar to, or greater than, wind turbines are desirable for the supply of electricity to the grid. It is shown that this may be provided by multiple floats in a hinged raft-type configuration with multi-mode forcing. The case analysed has 8 floats and 4 power take off (PTO) units. Analysis is based on linear diffraction-radiation modelling, validated in wave basin experiments with a smaller number of floats. Control is desirable to improve energy capture, mainly demonstrated for point absorbers, but this has not previously been applied to such a complex problem with many degrees of freedom. The linear hydrodynamic model in a state-space form makes it possible to implement advanced control algorithms in real time. Linear non-causal optimal control (LNOC) is applied with wave force prediction from auto-regression. For the design case with zero heading, as the configuration heads naturally into the wave direction, energy capture is improved by between 21% and 83%. The energy capture is about 62% the maximum possible from idealised analyses. Off-design, non-zero headings are also analysed to indicate how energy capture can be reduced; the contribution from different modes of forcing varies with heading and energy capture is always improved by control, by several times at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$90^{\circ }$</tex-math></inline-formula> heading.

Linear-quadratic control for a class of stochastic Volterra equations: Solvability and approximation

We provide an exhaustive treatment of linear-quadratic control problems for a class of stochastic Volterra equations of convolution type, whose kernels are Laplace transforms of certain signed matrix measures which are not necessarily finite. These equations are in general neither Markovian nor semimartingales, and include the fractional Brownian motion with Hurst index smaller than 1/2 as a special case. We establish the correspondence of the initial problem with a possibly infinite dimensional Markovian one in a Banach space, which allows us to identify the Markovian controlled state variables. Using a refined martingale verification argument combined with a squares completion technique, we prove that the value function is of linear quadratic form in these state variables with a linear optimal feedback control, depending on nonstandard Banach space valued Riccati equations. Furthermore, we show that the value function of the stochastic Volterra optimization problem can be approximated by that of conventional finite dimensional Markovian linear-quadratic problems, which is of crucial importance for numerical implementation.

Study of seismic-responses and semi-active control of Taizhou Bridge under multi-support excitation

PurposeIn order to make sure of the safety of a long-span suspension bridge under earthquake action, this paper aims to study the traveling wave effect of the bridge under multi-support excitation and optimize the semi-active control schemes based on magneto-rheological (MR) dampers considering reference index as well as economical efficiency.Design/methodology/approachThe finite element model of the long-span suspension bridge is established in MATLAB and ANSYS software, which includes different input currents and semi-active control conditions. Six apparent wave velocities are used to conduct non-linear time history analysis in order to consider the seismic response influence in primary members under traveling wave effect. The parameters α and β, which are key parameters of classical linear optimal control algorithm, are optimized and analyzed taking into account five different combinations to obtain the optimal control scheme.FindingsWhen the apparent wave velocity is relatively small, the influence on the structural response is oscillatory. Along with the increase of the apparent wave velocity, the structural response is gradually approaching the response under uniform excitation. Semi-active control strategy based on MR dampers not only restrains the top displacement of main towers and relative displacement between towers and girders, but also affects the control effect of internal forces. For classical linear optimal control algorithm, the values of two parameters (α and β) are 100 and 8 × 10–6 considering the optimal control effect and economical efficiency.Originality/valueThe emphasis of this study is the traveling wave effect of the triple-tower suspension bridge under multi-support excitation. Meanwhile, the optimized parameters of semi-active control schemes using MR dampers have been obtained, providing relevant references in improving the seismic performance of three-tower suspension bridge.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

Application of Intelligent Technology in Power System Automation

This paper describes the theory of intelligent control,and introduces the implementation scheme and measures of intelligent technology in power system,including expert system control technology,fuzzy control technology,artificial intelligence technology,linear optimal control technology,monitoring technology and integrated intelligent system,etc.

A Coordinated Optimal Strategy for Voltage and Reactive Power Control with Adaptive Amplitude Limiter Based on Flexible Excitation System

The flexible excitation system (FES) is a kind of novel excitation system with two channels for damping control. Besides the basic functions of traditional excitation systems, flexible excitation systems can provide reactive power support for the terminal voltage, and the large-capacity FES can improve the voltage stability and power-angle stability of synchronous generator units. However, with the increase in system capacity and the complication of control objectives, the difficulty of controller design will be increased. The randomness and fluctuation of new energy resources such as photovoltaic and wind turbines may cause disturbance and fault to the power system, which requires the coordinated control strategy for the FES to achieve stability in voltage and power angle. In this paper, the basic characteristics of FES are analyzed, and the mathematic model of the single machine infinite bus (SMIB) system based on FES is derived. The coordinated control strategy based on decoupling control of stator and rotor is proposed according to the optimal objectives of voltage stability and power-angle stability, and the linear optimal excitation control (LOEC) is adopted with the adaptive amplitude limiter (AAL) determined by fuzzy rules. The MATLAB/Simulink platform is established and the results verify the superiority of the proposed LOEC + AAL control strategy in large disturbance working conditions, which showed better robustness. The proposed coordinated control strategy provides an effective solution for industrial application and performance improvement of FES.

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

A modified Picard iteration method to solve fractional optimal control problems

An effective modified of the Picard iteration method ( PIM ) is presented for solving the linear and nonlinear fractional optimal control problems ( FOCP ) in the Caputo sense. Here, the control function is first approximated by a finite series with unknown coefficients. Then the modified PIM is utilized to simulate the resulting fractional equations. Finally, the unknown coefficients could be computed by applying an optimization procedure. Some test examples are given to show the accuracy and validity of the method.