Quadratic Optimal Control Research Articles

The Numerical Solution for Quadratic Optimal Control Problems by Using Chebyshev and Legendre Polynomials

الغرض من هذا البحث هو حل مشاكل التحكم الأمثل من الدرجة الثانية (QOCP) عدديًا بمساعدة متعددة حدود Chebyshev و Legendre كوظائف أساسية لإيجاد حل للتحكم الأمثل (QOC) تقريبًا. سنشرح خوارزميات الحل بالأمثلة ونستخدم برنامج Mathcad للوصول إلى النتيجة الدقيقة.مقدمةتتمثل مشكلة التحكم المثلى في العثور على عنصر تحكم u ^ * (t) يقلل من مؤشر أداء معين مع تلبية معادلات حالة النظام والقيود. [1]نستخدم طرق التقريب لحل مشكلة التحكم الأمثل اعتمادًا على متعدد حدود Chebyshev في المرة الأولى ومتعدد حدود Legendre ، وبعد ذلك سنقترب من هذه الحلول ذات الوقت الخطي المستمر. للوصول إلى الحلول التقريبية ، نستخدم المعادلات التفاضلية الخطية متعددة المدى لـ u (t) و x (t) لكل من متعدد حدود Chebyshev و Legendre ونجعل شروط هذه المعادلات كمصفوفة مربعة للعثور على هذه القيم بواسطة نظام المصفوفات.عندما نستخدم هذه كثيرات الحدود في حلول تقريبية ، تم تقييم النتائج باستخدام الفهرس مع n = 5.سنشرح هذه الخوارزميات من خلال أخذ بعض الأمثلة لمشاكل التحكم التربيعي.تم تحديد المشكلة التربيعية الخطية على النحو التالي ؛قلل من الوقت المستمر التربيعيدالة التكلفة J = ∫_ (t_0) ^ (t_f) ▒ 〖(x ^ T Qx + u ^ T Ru) □ (24 & dt)〗 ... (1)مع مراعاة معادلات حالة النظام الخطي ؛س ̇ (t) = Dx (t) + Eu (t) ، ... (2)حيث الشرط الأولي x (0) = x_0 والمصفوفات (D و E و Q و R) هي

Observer-based dynamic optimal recoil controller design for deepwater drilling riser systems

This paper deals with the observer-based dynamic optimal recoil controller design of a deepwater drilling riser system subject to friction force of fluid discharge and platform heave motion. First, by using exponential and polynomial functions conjunctively, a novel nonlinear computational model of the friction force is presented. Then, a linear system model is developed to describe dynamic characteristics of the friction force. Third, by using feedforward mechanisms and internal model principle jointly, a dynamic optimal recoil control scheme is proposed for the riser, where the feedforward mechanism is designed to compensate the friction force of fluid discharge, and the internal model principle is introduced to reject platform heave motion. The conditions of existence and uniqueness of a dynamic optimal recoil controller of the riser are developed. Fourth, a friction force observer is designed to solve the physically realizable problem of the recoil controller. Simulation results show that under the observer-based dynamic optimal recoil controller, the recoil of the riser can be attenuated significantly, and zero steady state errors of the riser can be guaranteed. In addition, from the perspective of achieving steady performance of the riser, the designed dynamic optimal recoil control scheme outperforms existing linear quadratic optimal control schemes.

Open-Loop Decentralized LQ Control Problem With Multiplicative Noise

This article considers the decentralized linear–quadratic (LQ) optimal control problem for the multiplicative-noise stochastic systems in discrete time with asymmetric information. There exist two controllers in the system, and each controller has its own information in general structure. The adopted information structure is adapted open loop for both controllers. The main contribution is to derive the necessary and sufficient condition for the unique solvability of the LQ control problem with a general asymmetric information structure and present the explicitly optimal controllers in terms of the proposed Riccati equations. The key is to explicitly solve the forward and backward stochastic difference equations, which are derived from the maximum principle.

Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps

The stochastic linear–quadratic optimal control problem with Poisson jumps is addressed in this paper. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed to be indefinite. The notion of closed-loop strategies is introduced, and the sufficient and necessary conditions for the closed-loop solvability are given. The optimal closed-loop strategy is characterized by a Riccati integral–differential equation and a backward stochastic differential equation with Poisson jumps. A simple example is given to demonstrate the effectiveness of the main result.

Preconditioners for robust optimal control problems under uncertainty

AbstractThe discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle‐point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriously challenging. In this manuscript, we study efficient preconditioners for all‐at‐once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle‐point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi‐iterative method are proposed to solve this reduced system. We consider a random elliptic partial differential equation whose diffusion coefficient is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates of the dependence of the spectrum of the preconditioned system matrix on the statistical properties of the random field and on the discretization of the probability space. Such estimates involve either the first or second moment of the random variables and , where is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.

Linear Quadratic Regulator Optimal Control of Two-Rotor Wind Turbine Mounted on Spar-Type Floating Platform

Abstract Interest is steadily growing for multi-rotor wind turbine concepts. This type of wind turbine offers a practical solution for scaling issues of large wind turbine components and for the reduction of costs associated with manufacturing, logistics, and maintenance. However, the literature lacks thorough knowledge of the dynamic performance of multi-rotor wind turbine concepts installed on floating platforms. Previous research studied the dynamic response of a two-rotor wind turbine concept mounted on a spar-type floating platform (2WT). Platform yaw motion is a significant dynamic factor directly caused by differential turbulence intensity experienced by the two hubs coupled with the distribution of thrust loads on the tower structure. Blade-pitch control analysis also showed how the 2WT yaw response is extremely sensitive to the control strategy employed. In this work, a linear quadratic regulator (LQR) is used to design an optimal controller for the 2WT prototype. Three LQR gain schedules corresponding to three operation regions are considered. An in-house tool for the dynamic analysis of multi-rotor floating wind turbines is used for linear state-space extraction and dynamic analysis. The control performance in different load conditions is assessed against the baseline OC3 proportional-integral (PI) control strategy and a PI-P control strategy in a previous article presented by the authors.

Research on the Control Method of Force Servo System of Liquid Hydrostatic Guide Oil Film Thickness Test Bench for Machine Tools

The test bench was designed to solve the problem of measuring the oil film thickness of the liquid hydrostatic guideway of machine tools. And based on the fast overshoot of the classical PID control, introducing the self-antidisturbance control algorithm, this study established a mathematical model of the test bench electro-hydraulic servo control system. At the same time, this paper also designed a specific structure of the self-turbulence controller, which adopts optimal quadratic control, pole configuration, PID control, and self-turbulence control to study the control methods of the test bench electro-hydraulic servo system, respectively, and uses the SIMULINK module to establish the system simulation model of the above-mentioned four control methods of the test bench. Simulation results of the test bench control system show that the response speed of the self-turbulence control is 0.4 seconds, 0.2 seconds, and 4.4 seconds faster than PID control, pole configuration, and optimal quadratic control, respectively, and the overshoot is significantly reduced. Finally, the test bench was built to complete the experiments, and the experimental results showed that the followability of the loading force and thickness detection accuracy of the test bench were better after adding the self-turbulence control algorithm and the stability of the oil film thickness was significantly improved by the test bench electro-hydraulic servo control system.

Online Learning Human Behavior for a Class of Human-in-the-Loop Systems via Adaptive Inverse Optimal Control

To enhance the machines’ intelligence, it is important for them to learn how humans perform tasks. In this article, the issue of online adaptive learning human behavior is addressed for a class of human-in-the-loop (HiTL) systems using the state measurement only. The hypothesis underlying our study is that human behavior can be described by a linear quadratic optimal control model with an unknown weighting matrix for the quadratic cost function. In this model, the weighting matrix depicts the human tradeoff of various objectives. Our aim is thus to only use the system state measurement for learning the weighting matrix under the condition that human feedback gain matrix is unknown. A novel adaptive inverse optimal control approach to online learning human behavior is proposed for the HiTL system, which integrates adaptive estimation and linear matrix inequality (LMI) optimization techniques. Our approach consists of two steps: First, an adaptive law is developed to learn the human feedback gain matrix online using the system state measurement only, and second, the weighting matrix of human cost function is retrieved by solving an LMI optimization problem with the learned feedback gain matrix. Finally, simulation and experiment results on a steering assist system of intelligent vehicles are presented to illustrate the effectiveness of the proposed method.

A systematic review of uncertainty theory with the use of scientometrical method

Uncertainty theory is an area in axiomatic mathematics recently proposed by Professor Baoding Liu and aiming to deal with belief degrees. Retrieving 1004 journal articles from the Web of Science database between 2008 and 2019, and utilizing CiteSpace and Pajek software, we analyze the publications per year and by geographical distribution, productive scholars and their cooperation, key journals, highly cited articles and main paths of the field. In this way, seven key sub-fields of uncertainty theory and their research potential are derived. The results show the following: (1) The literature on uncertainty theory follows a linear growth trend, involves an extensive network of 1000 scholars worldwide and is published in 300 journals, indicating thus that uncertainty theory has become increasingly attractive, and its academic influence is gradually expanding. (2) Seven key sub-fields of uncertainty theory have clearly been identified, including the axiomatic system, uncertain programming, uncertain sets, uncertain logic, uncertain differential equations, uncertain risk analysis, and uncertain processes. Among them, uncertain differential equations and programming are the two main sub-fields with the largest numbers of published papers. Furthermore, for evaluating the research potential of sub-fields, maturity and recent attention indicators are calculated using the citations, total number of publications, quantity of most cited literature and half-life. Based on these indicators, uncertain processes shows the greatest development potential, and has remained a hot topic in recent years, being mainly concentrated on the uncertain renewal reward process, optimal control of discrete-time uncertain systems, and uncertain linear quadratic optimal control. Additionally, uncertain risk analysis is ranked second, and focuses on the analysis of expected losses, investment risk, and structural reliability of uncertain systems.

A study on cancer-obesity-treatment model with quadratic optimal control approach for better outcomes

In this study, we propose a cancer-obesity-treatment model under the control parameters: IL-2 therapy, ACI therapy, and nutritional diet. We analyze the proposed model and examine the existence and stability of the equilibrium points for the cases: with and without treatment cases. Further, to reduce the number of cancer cells and minimize the toxicity effect of the drug dose on other healthy cells, we consider an optimal control problem over a finite time interval under the treatment control parameters. To understand the treatment effect, we present simulation results for our proposed model considering different treatment strategies: no treatment; only IL-2 therapy; a combination of IL-2 therapy and ACI therapy; and a combination of IL-2 therapy, ACI therapy, and nutritional diet. Our results demonstrate that we could achieve an optimal treatment schedule for cancer management by controlling all three treatment parameters.

Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces

A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems.

Linear–Quadratic Optimal Control for Discrete-Time Mean-Field Systems With Input Delay

The linear–quadratic (LQ) optimal control and stabilization problems for mean-field systems with input delay (MFSID) are investigated in this article. The necessary and sufficient solvability conditions for LQ control of MFSID are first given in terms of a convexity condition and the solvability of equilibrium conditions. Consequently, by solving the associated mean-field forward and backward stochastic difference equations, the optimal control is derived in terms of the solution of a modified Riccati equation. Furthermore, for the infinite-horizon case, the stabilization problem for MFSID is studied, and the necessary and sufficient stabilizability conditions are obtained. We show that MFSID can be mean square stabilizable if and only if a modified algebraic Riccati equation admits a unique positive-definite solution.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

Slow Decay and Turnpike for Infinite-Horizon Hyperbolic Linear Quadratic Problems

This paper is devoted to analyzing the explicit slow decay rate and turnpike in infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Under suitable weak observability or controllability conditions, lower and upper bounds of the corresponding algebraic Riccati operator are proved. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to LQ optimal control problems constrained to networks of one-dimensional wave equations and also some multidimensional ones with local controls which lack a geometric control condition.

Indefinite linear quadratic optimal control for discrete time‐varying linear rectangular descriptor systems

AbstractIn this article, the indefinite linear quadratic (LQ) optimal control problem for discrete time‐varying rectangular descriptor systems is investigated, in which the indefinite weight matrices are considered in the quadratic cost function. With a suite of equivalent transformations, the indefinite LQ problem for discrete time‐varying rectangular descriptor systems can be substituted by an equivalent indefinite LQ problem for standard state space systems. Then, the transformed equivalent indefinite LQ problem is solved by establishing a dual Krein space model, and the optimal control and optimal cost value are obtained by invoking the Kalman filter of Krein space. The form of state feedback optimal control corresponding to the original system is given. Next, to guarantee the dynamic part for the resulting optimal closed‐loop system has a unique solution, some necessary and sufficient conditions are proposed. Finally, a numerical example is carried out to demonstrate the availability of established results.

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.

Recoil attenuation for deepwater drilling riser systems via delayed [formula omitted] control

This paper deals with the recoil suppression problem of a deepwater drilling riser system via active H∞ control using both current and delayed states. First, based on the three degrees of freedom spring–mass–damping model of the riser system, an incremental dynamic equation of the system subject to the platform heave motion and the friction force induced by drilling discharge mud and seawater is established. Then, to reject recoil movements of the riser, a delayed state feedback H∞ controller with delayed states as well as current states is designed. The existence conditions and the design method of the delayed H∞ recoil controllers are presented. Third, the effects of the introduced time-delays on the recoil control of the riser are analyzed, and the design of optimal artificial time-delays is formulated as the minimum value problem of a series of quintic algebraic polynomials, which are related to the weights of average response amplitudes, steady-state errors, and the control force. Lastly, simulation results are provided to demonstrate the effectiveness of delay-free and delayed H∞ recoil control schemes for the riser. It is shown that (i) under the delayed H∞ controllers, the recoil responses of the riser can be controlled significantly; (ii) the decay rate of the recoil response under the delay-free H∞ controller is slightly faster than the one under the delayed H∞ controllers. However, the former requires more control cost than the latter; (iii) compared with the delayed H∞ controller with the existing linear quadratic optimal controller, the control cost by the former is larger than that by the latter. However, the steady-state errors of the riser under the latter are slightly smaller than that under the former; (iv) the introduced time-delays with proper size play positive role of suppressing recoil response of the system, and the corresponding delayed H∞ controller series provide more options for recoil control of the riser.

Numerical Methods for Solving Linear Time Varying Quadratic Optimal Control Problems

In this article, we discussed linear time varying optimal control problems with quadratic performance index, and approximated control variable, state variable and performance index. There are many different numerical processes for approximating of linear time varying quadratic optimal control problem. Two of those are explained here the Legendre scaling function and Chebyshev scaling function methods using the MATLAB. The methods will be explained, and then applied to test problems to see how they perform. The results show that the Legendre scaling function was the best of the Chebyshev scaling function methods being a computation. Hence, the Legendre scaling function method is more suitable for solving the linear time varying quadratic optimal control systems.

Optimal control of fractional Sturm–Liouville wave equations on a star graph

ABSTRACT In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

Active Flutter Suppression of a Two-Dimensional Wing Using Linear Quadratic Gaussian Optimal Control

This paper aims to utilize the modern control theory in designing an active flutter suppression controller for a two-dimensional wing with a flap. In this study, the wing’s pitch–plunge linear aeroelastic response, in addition to the control flap motion, was modeled using Lagrange’s energy method, and Theodorsen’s classic unsteady aerodynamic theory. The derived mathematical model was represented in state-space so the modern control techniques can be implemented. The open-loop system was analyzed, and the flutter speed for a selected experimental-wing model was calculated. For model verification purposes, the open-loop simulation results were compared to an experimental study from the literature. Next, a linear quadratic Gaussian (LQG) compensator with integral action was designed and tuned using MATLAB®simulation environment to reach the best possible performance at a selected speed in the flutter region, where the results have proven excellent performance for this kind of controller in flutter suppression, as the stable flight region has successfully been extended by 29%.