Linear Optimal Control Research Articles

Recursive approximate solution to time-varying matrix differential Riccati equation: linear and nonlinear systems

ABSTRACTAn approximate solution is proposed for linear-time-varying (LTV) systems based on Taylor series expansion in a recursive manner. The intention is to present a fast numerical solution with reduced sampling time in computation. The proposed procedure is implemented on finite-horizon linear and nonlinear optimal control problem. Backward integration (BI) is a well known method to give a solution to finite-horizon optimal control problem. The BI performs a two-round solution: first one elicits an optimal gain and the second one completes the answer. It is very important to finish the backward solution promptly lest in practical work, system should not wait for any action. The proposed recursive solution was applied for mathematical examples as well as a manipulator as a representative of complex nonlinear systems, since path planning is a critical subject solved by optimal control in robotics.

On an Optimal Control Applied in MEMS Oscillator with Chaotic Behavior including Fractional Order

The dynamical analysis and control of a nonlinear MEMS resonator system is considered. Phase diagram, power spectral density (FFT), bifurcation diagram, and the 0-1 test were applied to analyze the influence of the nonlinear stiffness term related to the dynamics of the system. In addition, the dynamical behavior of the system is considered in fractional order. Numerical results showed that the nonlinear stiffness parameter and the order of the fractional order were significant, indicating that the response can be either a chaotic or periodic behavior. In order to bring the system from a chaotic state to a periodic orbit, the optimal linear feedback control (OLFC) is considered. The robustness of the proposed control is tested by a sensitivity analysis to parametric uncertainties.

An LQR-Based Controller Design for an LCL-Filtered Grid-Connected Inverter in Discrete-Time State-Space under Distorted Grid Environment

In order to alleviate the negative impacts of harmonically distorted grid conditions on inverters, this paper presents a linear quadratic regulator (LQR)-based current control design for an inductive-capacitive-inductive (LCL)-filtered grid-connected inverter. The proposed control scheme is constructed based on the internal model (IM) principle in which a full-state feedback controller is used for the purpose of stabilization and the integral terms as well as resonant terms are augmented into a control structure for the reference tracking and harmonic compensation, respectively. Additionally, the proposed scheme is implemented in the synchronous reference frame (SRF) to take advantage of the simultaneous compensation for both the negative and positive sequence harmonics by one resonant term. Since this leads to the decrease of necessary resonant terms by half, the computation effort of the controller can be reduced. With regard to the full-state feedback control approach for the LCL-filtered grid connected inverter, additional sensing devices are normally required to measure all of the system state variables. However, this causes a complexity in hardware and high implementation cost for measurement devices. To overcome this challenge, this paper presents a discrete-time current full-state observer that uses only the information from the control input, grid-side current sensor, and grid voltage sensor to estimate all of the system state variables with a high precision. Finally, an optimal linear quadratic control approach is introduced for the purpose of choosing optimal feedback gains, systematically, for both the controller and full-state observer. The simulation and experimental results are presented to prove the effectiveness and validity of the proposed control scheme.

Active Vibration Control of a Doubly Curved Composite Shell Stiffened by Beams Bonded With Discrete Macro Fiber Composite Sensor/Actuator Pairs

Doubly curved stiffened shells are essential parts of many large-scale engineering structures, such as aerospace, automotive and marine structures. Optimization of active vibration reduction has not been properly investigated for this important group of structures. This study develops a placement methodology for such structures under motion base and external force excitations to optimize the locations of discrete piezoelectric sensor/actuator pairs and feedback gain using genetic algorithms for active vibration control. In this study, fitness and objective functions are proposed based on the maximization of sensor output voltage to optimize the locations of discrete sensors collected with actuators to attenuate several vibrations modes. The optimal control feedback gain is determined then based on the minimization of the linear quadratic index. A doubly curved composite shell stiffened by beams and bonded with discrete piezoelectric sensor/actuator pairs is modeled in this paper by first-order shear deformation theory using finite element method and Hamilton's principle. The proposed methodology is implemented first to investigate a cantilever composite shell to optimize four sensor/actuator pairs to attenuate the first six modes of vibration. The placement methodology is applied next to study a complex stiffened composite shell to optimize four sensor/actuator pairs to test the methodology effectiveness. The results of optimal sensor/actuator distribution are validated by convergence study in genetic algorithm program, ANSYS package and vibration reduction using optimal linear quadratic control scheme.

Optimal linear control of the output voltage of a grid-connected single-phase photovoltaic power source

The object of the study is an autonomous source of sinusoidal voltage, which consists of a bridge frequency converter with PWM, equipped with an output LC filter and a single-phase nonlinear output transformer with an additional output filter, which is connected to the non-autonomous grid. The nonlinearity of the magnetic system of the transformer is approximated by an odd function of the arctg type. The substitution of variables made possible to develop a mathematical description of the circuit in the form of a nonlinear singularly perturbed system of differential equations. It is assumed that there is an attractive invariant surface, which makes it possible to reduce the order of the original system of equations. An analytical form of the equations of this surface in the form of a series in powers of a small parameter is obtained. A system of lower-order differential equations is obtained that is equivalent in some region to the original nonlinear singularly perturbed system. The non-linear reduced system is transformed to a form that has made it possible to apply the linear feedback control strategies. The Lyapunov function is obtained as a quadratic form, the coefficients of which are solutions of the matrix algebraic Riccati equation. A condition is set out in which the use of these coefficients guarantees the local asymptotic stability of this nonlinear system. The results of digital simulation are presented. The simulation was performed taking into account the limitations inherent in the real object of power electronics and affecting the possibility of technical implementation of the obtained control strategy. Local equivalency and isomorphism in the control strategies of full-length and reduced systems avoids the need to measure all variables in the space of states, which facilitates practical implementation. The possibility of reducing the impact of load jumps on the form of the output voltage has been checked, taking into account the constraints specific to real systems, has been verified using digital simulation.Ref. 11, fig. 2.

Darrieus wind turbine prototype: Dynamic modeling parameter identification and control analysis

This paper details the process required to develop a linear dynamic model and control of a Darrieus wind turbine equipped with a Permanent Magnet Synchronous Generator. The turbine is to be integrated in a smart grid for future installation in urban context. An active control system is integrated in such a way that, even in the presence of non-ideal phenomena and disturbances, optimal operation is achieved. For the class of linear (or linearized) systems such optimal control solution corresponds to the Linear Quadratic Regulator (LQR). A set of tests were performed on a wind tunnel, data was acquired allowing the application of well rooted identification techniques, leading to the deduction and validation of several low order dynamic models. Optimal control solutions were implemented to guarantee that close to ideal operational conditions are maintained. As such, efficiency is improved without jeopardizing the integrity of the wind turbine in a broad set of operational conditions.

Optimal linear quadratic control for wireless sensor and actuator networks with random delays and packet dropouts

In this article, the optimal linear quadratic control problem is considered for the wireless sensor and actuator network with stochastic network-induced delays and packet dropouts. Considering the event-driven relay nodes, the optimal solution is obtained, which is a function of the current plant state and all past control signals. It is shown that the optimal control law is the same for all locations of the controller placement. Since the perfect plant state information is available at the sensor, the optimal controller should be collocated with the sensor. In addition, some issues such as the plant state noise and suboptimal solution are also discussed. The performance of the proposed scheme is investigated by an application of the load frequency control system in power grid.

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.

Optimal Strategy with One Closing Instant for a Linear Optimal Guaranteed Control Problem

We consider an optimal guaranteed control problem for a linear time-varying system that is subject to unknown bounded disturbances. A control strategy is defined that guarantees steering the system to a given terminal set for any realization of disturbances and takes into account that at one future time instant the control loop will be closed. An efficient method for constructing the optimal control strategy and an algorithm for optimal feedback control based on this type of strategies are proposed.

Nonlinear dynamics and robust control of sloshing in a tank

Sloshing is a complex nonlinear dynamic phenomenon which has a significant influence on the stability of structure–fluid systems. In this study, the dynamic equation of sloshing based on Hamilton principle is established and linearized into a state space equation. Considering the uncertainty of the system, a robust H infinite guaranteed cost control method is proposed to mitigate the response of fluid wave height to horizontal acceleration of the tank body. Simulation results are given to demonstrate the closed-loop performance of the nonlinear dynamic modeling and linear optimal control method.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

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

Optimal Decentralized Control With Asymmetric One-Step Delayed Information Sharing

We consider optimal control of decentralized LQG problems for plants having nested subsystems controlled by two players with asymmetric information sharing patterns between them. In the main scenario, the players are assumed to have a unidirectional error-free, unlimited-rate communication channel with a unit delay in one direction and no communication in the other. A second model, presented for completeness, considers a channel with no delay in one direction and a unit delay in the other. Delayed information-sharing patterns do not, in general, admit linear optimal control laws and are thus difficult to control optimally. However, in these scenarios, we show that the problems have a partially nested information structure and, thus, linear optimal control laws exist. Summary statistics are identified and analytical solutions to the optimal control laws are derived. State and output feedback cases are solved for both scenarios.

Generalized Kleinman‐Newton method

SummaryThis paper addresses the general problem of optimal linear control design subject to convex gain constraints. Classical approaches based exclusively on Riccati equations or linear matrix inequalities are unable to treat problems that incorporate feedback gain constraints, for instance, the reduced‐order (including static) output feedback control design. In this paper, these two approaches are put together to obtain a genuine generalization of the celebrated Kleinman‐Newton method. The convergence to a local minimum is monotone. We believe that other control design problems can be also considered by the adoption of the same ideas and algebraic manipulations. Several examples borrowed from the literature are solved for illustration and comparison.

Renewable resource management in a seasonally fluctuating environment with restricted harvesting effort

This paper presents bio-economics of a renewable resources in a seasonally changing environment in which the resource exploitation is subjected to restrictions on harvesting effort. The dynamics of the resource is assumed to be governed by the logistic equation. Seasonality is incorporated into the system by choosing the coefficients in the growth equation to be periodic functions with the same period. A linear optimal control problem involving binding constraints on the control variable has been considered. As a result the concept of blocked interval plays a key role in the construction of optimal solution. In view of the periodicity associated with the considered problem, we first construct an optimal periodic solution. The optimal solution is established using the most rapid approach path to the said optimal periodic solution. The global asymptotic stability property of the optimal periodic solution enables construction of a suboptimal solution. The optimal solution is found to be periodic after some finite time and suboptimal solution approaches the optimal periodic solution asymptotically. Key results are illustrated through numerical simulation.

Discrete-Time Nonlinear Stochastic Optimal Control Problem Based on Stochastic Approximation Approach

In this paper, a computational approach is proposed for solving the discrete-time nonlinear optimal control problem, which is disturbed by a sequence of random noises. Because of the exact solution of such optimal control problem is impossible to be obtained, estimating the state dynamics is currently required. Here, it is assumed that the output can be measured from the real plant process. In our approach, the state mean propagation is applied in order to construct a linear model-based optimal control problem, where the model output is measureable. On this basis, an output error, which takes into account the differences between the real output and the model output, is defined. Then, this output error is minimized by applying the stochastic approximation approach. During the computation procedure, the stochastic gradient is established, so as the optimal solution of the model used can be updated iteratively. Once the convergence is achieved, the iterative solution approximates to the true optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, an example on a continuous stirred-tank reactor problem is studied, and the result obtained shows the applicability of the approach proposed. Hence, the efficiency of the approach proposed is highly recommended.

PID2018 Benchmark Challenge: Model Predictive Control With Conditional Integral Control Using A General Purpose Optimal Control Problem Solver – RIOTS.

This paper presents a multi-variable Model Predictive Control (MPC) based controller for the one-staged refrigeration cycle model described in the PID2018 Benchmark Challenge. This model represents a two-input, two-output system with strong nonlinearities and high coupling between its variables. A general purpose optimal control problem (OCP) solver Matlab toolbox called RIOTS is used as the OCP solver for the proposed MPC scheme which allows for straightforward implementation of the method and for solving a wide range of constrained linear and nonlinear optimal control problems. A conditional integral (CI) compensator is embedded in the controller to compensate for the small steady state errors. This method shows significant improvements in performance compared to both discrete decentralized control (C1) and multi-variable PID controller (C2) originally given in PID2018 Benchmark Challenge as a baseline. Our solution is introduced in detail in this paper and our final results using the overall relative index, J, are 0.2 over C1 and 0.3 over C2, respectively. In other words, we achieved 80% improvement over C1 and 70% improvement over C2. We expect to achieve further improvements when some optimized searching efforts are used for MPC and CI parameter tuning.

Filtering method for linear and non-linear stochastic optimal control of partially observable systems II

In this paper we studied stochastic optimal control problem based on partially observable systems (SOCPP) with a control factor on the diffusion term. A SOCPP has state and observation processes. This kind of problem has also a minimum payoff function. The payoff function should be minimized according to the partially observable systems consist of the state and observation processes. In this regard, the filtering method is used to evaluat this kind of problem and express full consideration of it. Finally, presented estimation methods are used to simulate the solution of a partially observable system corresponding to the control factor of this problem. These methods are numerically used to solve linear and nonlinear cases.

Adaptation of the control synthesized by the ACAR method to the control based on the implementation of the maximum principle

A special case of synthesizing the electromechanical control system by the maximum method and using the Analytical Construction method of Aggregate Regulators (ACAR) is considered in the article. For the basis the task of synthesizing the optimal for speed electromechanical positioning system was chosen, while the moment of resistance to movement linearly depended on the output coordinate of the system, that is, on the angle of the engine rotor rotation. Synthesis of the optimal system for speed makes it possible to increase the efficiency of the entire production process in many production tasks, and the synthesis of the optimal linear control system based on the maximum principle is a fairly well-formalized problem. Here it should be noted that the procedure for synergistic synthesis of the optimal control system has no such formalization. An approach that brings together the solutions obtained by these two methods, which makes it possible to increase the efficiency of the ACAR method by adding some features of the methodology for synthesizing optimal systems by introducing nonlinearity of the “saturation” type is proposed in the article. The results obtained made it possible to formulate the following basic scientific proposition: the synthesis of a control system based on the synergetic approach makes it possible to obtain a system close to optimal (quasi-optimal, but after the modification of the synergetic synthesis method itself.) Here we also formulate the hypothesis of a connection between the time constants, using the ACAR method, with the optimal control switching time determined in the maximum method.

Optimization of Lagrange problem with higher order differential inclusions and endpoint constraints

In the paper minimization of a Lagrange type cost functional over the feasible set of solutions of higher order differential inclusions with endpoint constraints is studied. Our aim is to derive sufficient conditions of optimality for m th-order convex and non-convex differential inclusions. The sufficient conditions of optimality containing the Euler-Lagrange and Hamiltonian type inclusions as a result of endpoint constraints are accompanied by so-called ?endpoint? conditions. Here the basic apparatus of locally adjoint mappings is suggested. An application from the calculus of variations is presented and the corresponding Euler-Poisson equation is derived. Moreover, some higher order linear optimal control problems with quadratic cost functional are considered and the corresponding Weierstrass-Pontryagin maximum principle is constructed. Also at the end of the paper some characteristic features of the obtained result are illustrated by example with second order linear differential inclusions.