Optimal Linear Feedback Control Research Articles

Resolvent-based approach for pmb {H_2}-optimal estimation and control: an application to the cylinder flow

We consider estimation and control of the cylinder wake at low Reynolds numbers. A particular focus is on the development of efficient numerical algorithms to design optimal linear feedback controllers when there are many inputs (disturbances applied everywhere) and many outputs (perturbations measured everywhere). We propose a resolvent-based iterative algorithm to perform (i) optimal estimation of the flow using a limited number of sensors, and (ii) optimal control of the flow when the entire flow is known but only a limited number of actuators are available for control. The method takes advantage of the low-rank characteristics of the cylinder wake and provides full-dimensional solutions by implementing a terminal reduction technique based on resolvent analysis. Optimal feedback controllers are also obtained by combining the solutions of the estimation and control problems. We show that the performance of the estimators and controllers converges to the true global optima, indicating that the important physical mechanisms for estimation and control are of low rank.Graphic abstract

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.

Mittag–Leffler synchronization for impulsive fractional-order bidirectional associative memory neural networks via optimal linear feedback control

In this paper, we are concerned with the synchronization scheme for fractional-order bidirectional associative memory (BAM) neural networks, where both synaptic transmission delay and impulsive effect are considered. By constructing Lyapunov functional, sufficient conditions are established to ensure the Mittag–Leffler synchronization. Based on Pontryagin’s maximum principle with delay, time-dependent control gains are obtained, which minimize the accumulative errors within the limitation of actuator saturation during the Mittag–Leffler synchronization. Numerical simulations are carried out to illustrate the feasibility and effectiveness of theoretical results with the help of the modified predictor-corrector algorithm and the forward-backward sweep method.

Optimal Linear Controller for Minimizing DC Voltage Oscillations in MMC-Based Offshore Multiterminal HVDC Grids

The paper aims at minimizing DC voltage oscillations in offshore multiterminal high-voltage direct current (HVDC) grids based on modular multilevel converters (MMCs). The DC voltage stability is a crucial factor in multiterminal HVDC networks since it is associated with the grid power balance. Furthermore, DC voltage oscillations can cause the propagation of significant disturbances to the interconnected AC grids. This paper proposes an optimal control technique based on semidefinite programming to improve the DC voltage stability margins under the worst-case perturbation scenario. A centralized optimal linear feedback controller is introduced to achieve this goal while ensuring compliance with the control inputs’ and state variables’ constraints. Furthermore, the methodology is adapted to develop a decentralized optimal linear feedback controller with naturally decoupled constraints on the control inputs and state variables. It is shown that the proposed centralized and decentralized optimal linear controllers can minimize the DC voltage oscillations under the worst-case perturbation scenario in the presence or absence of the droop control gain. The performance of these controllers is verified via eigenvalue stability analysis and time-domain simulations of the MMC-based four-terminal HVDC test grid. Finally, a DC voltage oscillation index is introduced as a potential decision-support criterion. Its applicability is exemplified by identifying, among several options, the HVDC link that gives minimum DC voltage oscillations between independent point-to-point networks while considering the wind intermittency effect.

H∞ controller synthesis for AGV trajectory tracking using a linearized kinematic model

This paper proposes a design procedure for H∞ optimal linear feedback controllers for trajectory tracking of an automated guided vehicle (AGV) based on a linearized kinematic model. The design specifications are expressed as a constrained H∞ norm optimization problem to assess the potential of the H∞ framework for the considered application. This problem is then solved for a linear parameter-varying (LPV), a full-block and a decoupled linear time-invariant (LTI) controller. The performance of all controllers is compared through numerical simulations and experiments on an AGV prototype targeted at agricultural applications.

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.

Stochastic Linear-Quadratic Control with State Dependent Fractional Brownian Noise and Stochastic Coefficients

Stochastic linear-quadratic control problems for bilinear equations with some stochastic coefficients and driven by fractional Brownian motions with the Hurst parameters H ϵ (½,1) are formulated and solved. The family of admissible controls is the collection of linear feedback controls. An optimal control in this family is not optimal in the admissible family of all adapted controls. However an optimal linear feedback control obtained in this paper can be relatively easily implemented whereas an optimal control in the family of adapted controls has a term that predicts the future noise so it is more difficult to implement. An optimal control is determined from the solution of a Riccati equation which differs from the Riccati equation for the linear-quadratic control problem with a Brownian motion replacing the noise term. Since the Riccati equation has some stochastic coefficients it is solved as a backward stochastic differential equation. Given the solution of the stochastic Riccati equation, a direct method is used to determine the optimal linear feedback control.

A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rossler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.

On controlling the vibrations and energy transfer in MEMS gyroscope system with simultaneous resonance

In this paper, we study the dynamics, transfer of energy and control of the vibrations of micro-electromechanical system (MEMS) gyroscope with linear and nonlinear parametric excitations. This leads to two-degree-of-freedom system. The coupling of the system equations is responsible for energy transfers of the two vibration modes (drive mode and sense mode) and for the resonance in MEMS gyroscope. The resulting governing equation is in the form of a cubic Mathieu equation coupled to a Duffing equation. The averaging method is applied to obtain the frequency response equations near simultaneous sub-harmonic and internal resonance. The stability of the steady-state solution near the worst resonance case is studied and discussed. The effects of the different parameters on the two modes of system behavior are studied numerically. Poincare maps are used to determine stability and plot bifurcation diagrams. Comparison between optimal linear feedback control and active control via negative nonlinear cubic velocity feedback in the presence of parameter uncertainties and noise measurement are done. The numerical results of stability, phase planes and time history are achieved using MATLAB and MAPLE programs.

Distributionally Robust Control of Constrained Stochastic Systems

We investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process, i.e., when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. In the first case, we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems. We prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust CVaR and chance constrained problems. The proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives.

An online optimal linear state feedback controller based on MLS approximations and a novel straightforward PSO algorithm

Selection of control parameters is an important issue in the field of control design. This selection depends on the initial condition of the systems. On the other hand, controller parameters should be adjusted under all initial conditions to reach the optimal performance. To overcome this problem, in this paper, an online optimal linear state feedback control is introduced. Firstly, a straightforward particle swarm optimization is used to optimize the state feedback control parameters under some certain initial conditions. Then, in order to adapt the optimal controller to different initial conditions, the moving least squares approximation is used. The proposed technique is applied to an inverted pendulum system. The feasibility and efficiency of the proposed controller are assessed in simulation results.

Nonlinear control in an electromechanical transducer with chaotic behaviour

The electromechanical transducer considered in this work is composed of a mechanical oscillator linked to an electronic circuit. Simulations results have determined that for some combination of parameters the electromechanical system is subject to chaotic motion with resonant transient behavior, and after the resonant transient the mechanical system (MS) synchronizes with the electrical system (ES). In order to improve the transient response, avoiding both the chaotic and resonant behaviors, a nonlinear control system is designed, a feedback control strategy is used to drive the system into the desired periodic orbit, and a nonlinear feedforward strategy is used to keep the system into the periodic orbit, obtained by the Fourier series. Two control techniques are used and compared, namely: the state dependent Ricatti equation and the optimal linear feedback control. Numerical simulations results are shown in order to compare the results, considering parametric uncertainties. Additionally, the energy transfer “pumping” between the ES and the MS is also analysed.

On nonlinear dynamics and control of a robotic arm with chaos

In this paper a robotic arm is modelled by a double pendulum excited in its base by a DC motor of limited power via crank mechanism and elastic connector. In the mathematical model, a chaotic motion was identified, for a wide range of parameters. Controlling of the chaotic behaviour of the system, were implemented using, two control techniques, the nonlinear saturation control (NSC) and the optimal linear feedback control (OLFC). The actuator and sensor of the device are allowed in the pivot and joints of the double pendulum. The nonlinear saturation control (NSC) is based in the order second differential equations and its action in the pivot/joint of the robotic arm is through of quadratic nonlinearities feedback signals. The optimal linear feedback control (OLFC) involves the application of two control signals, a nonlinear feedforward control to maintain the controlled system to a desired periodic orbit, and control a feedback control to bring the trajectory of the system to the desired orbit. Simulation results, including of uncertainties show the feasibility of the both methods, for chaos control of the considered system.

A global sliding mode controller for missile electromechanical actuator servo system

A global sliding mode controller (GSMC) is proposed for the missile electromechanical actuator (EA) servo system, where exists high uncertainties, such as parameter variations and external disturbances. By the design of an optimal integral switching function based on optimal linear quadratic regulator (LQR) theory, the initial state of system is set on the switching surface, and the optimal sliding mode motion is produced. The proposed GSMC is composed of an optimal linear state feedback controller (OLSFC), and a fuzzy nonlinear robust controller (FNRC), which can be designed respectively. The OLSFC, generated by the designed switching function, intends to minimise a quadratic performance index, and then improves the dynamic performance of system. Meanwhile, the FNRC employs a fuzzy decision maker (FDM), which estimates the upper bound of uncertainties as FNRC’s gain adaptively, and then makes GSMC robust and control input smooth. With the computer simulations on an EA experiment plant, it presents that the proposed scheme possesses good tracking precision, effective suppression against chattering at control input, and strong robustness against system uncertainties.

Nonlinear Control System Applied to Atomic Force Microscope Including Parametric Errors

The performance of the optimal linear feedback control and of the state-dependent Riccati equation control techniques applied to control and to suppress the chaotic motion in the atomic force microscope are analyzed. In addition, the sensitivity of each control technique regarding to parametric uncertainties are considered. Simulation results show the advantages and disadvantages of each technique.

On two kinds of intermediate orbits for asteroid explorations

In the paper, two kinds of intermediate orbits for asteroid explorations are proposed. One is around the collinear libration points of the Sun-asteroid restricted three-body problem. The other is around the asteroid itself. The first kind of intermediate orbit is applicable to asteroids with known masses, while the second is suitable for asteroids with unknown or negligible masses. Analytical solutions of these two intermediate orbits in the simplified models are introduced first, and then numerical algorithms are used to refine them to obtain the true orbits in the real force model. At last, the problem of station-keeping is addressed. The linear optimal feedback control law is used, and numerical simulations are made to both kinds of intermediate orbits. The results show that both kinds of orbits are feasible. The cost is reasonable and mainly depends on the initial insertion error.

Nonlinear State Estimation and Control for Chaos Suppression in MEMS Resonator

During the last decade the chaotic behavior in MEMS resonators have been reported in a number of works. Here, the chaotic behavior of a micro-mechanical resonator is suppressed. The aim is to control the system forcing it to an orbit of the analytical solution obtained by the multiple scales method. The State Dependent Riccati Equation (SDRE) and the Optimal Linear Feedback Control (OLFC) strategies are used for controlling the trajectory of the system. Additionally, the SDRE technique is used in the state estimator design. The state estimation and the control techniques proved to be effective in controlling the trajectory of the system. Additionally, the robustness of the control strategies are tested considering parametric errors and measurement noise in the control loop.

Nonlinear State Estimation and Control for Chaos Suppression in MEMS Resonator

During the last decade the chaotic behavior in MEMS resonators have been reported in a number of works. Here, the chaotic behavior of a micro-mechanical resonator is suppressed. The aim is to control the system forcing it to an orbit of the analytical solution obtained by the multiple scales method. The State Dependent Riccati Equation (SDRE) and the Optimal Linear Feedback Control (OLFC) strategies are used for controlling the trajectory of the system. Additionally, the SDRE technique is used in the state estimator design. The state estimation and the control techniques proved to be effective in controlling the trajectory of the system. Additionally, the robustness of the control strategies are tested considering parametric errors and measurement noise in the control loop.

Microcantilever chaotic motion suppression in tapping mode atomic force microscope

The tapping mode is one of the mostly employed techniques in atomic force microscopy due to its accurate imaging quality for a wide variety of surfaces. However, chaotic microcantilever motion impairs the obtention of accurate images from the sample surfaces. In order to investigate the problem the tapping mode atomic force microscope is modeled and chaotic motion is identified for a wide range of the parameter's values. Additionally, attempting to prevent the chaotic motion, two control techniques are implemented: the optimal linear feedback control and the time-delayed feedback control. The simulation results show the feasibility of the techniques for chaos control in the atomic force microscopy.

Control of diesel engines using NOx-emission feedback

Variations of engine-out emissions due to ageing, component drift or production tolerances pose serious problems to meet legislative restrictions on exhaust tailpipe pollutant emissions. This paper addresses feedback of the raw emissions for improved control of diesel engines. A discussion of issues regarding the inclusion of raw-emission feedback into the engine control structure is provided, and a novel control structure for combined feedback control of the air-path variables boost pressure and exhaust gas recirculation rate and of the NOx emissions is presented. The proposed control structure basically consists of an optimal linear output feedback controller and a setpoint-adaption loop on the exhaust gas recirculation rate. With this approach, a simple control structure is available requiring a marginal calibration effort to meet desired NOx-emission values. Unfavourable injection timing in connection with NOx control is minimized by adapting the exhaust gas recirculation rate setpoint. The performance of the proposed approach is demonstrated by experimental results.