Linear Optimal Control Law Research Articles

A data-driven LQG method for linear time-varying systems to monitor the controller performance in the batch process

PurposeThe flexibility of batch process enables its wide application in fine-chemical, pharmaceutical and semi-conductor industries, whilst its complexity necessitates control performance monitoring to ensure high operation efficiency. This paper proposes a data-driven approach to carry out controller performance monitoring within batch based on linear quadratic Gaussian (LQG) method.Design/methodology/approachA linear time-varying LQG method is proposed to obtain the joint covariance benchmark for the stochastic part of batch process input/output. From historical golden operation batch, linear time-varying (LTV) system and noise models are identified based on generalized observer Markov parameters realization.FindingsOpen/closed loop input and output data are applied to identify the process model as well as the disturbance model, both in Markov parameter form. Then the optimal covariance of joint input and output can be obtained by the LQG method. The Hotelling's Tˆ2 control chart can be established to monitor the controller.Originality/value(1) An observer Markov parameter approach to identify the time-varying process and noise models from both open and closed loop data, (2) a linear time-varying LQG optimal control law to obtain the optimal benchmark covariance of joint input and output and (3) a joint input and output multivariate control chart based on Hotelling's T2 statistic for controller performance monitoring.

Resolvent-based tools for optimal estimation and control via the Wiener–Hopf formalism

The application of control tools to complex flows frequently requires approximations, such as reduced-order models and/or simplified forcing assumptions, where these may be considered low rank or defined in terms of simplified statistics (e.g. white noise). In this work we propose a resolvent-based control methodology with causality imposed via a Wiener–Hopf formalism. Linear optimal causal estimation and control laws are obtained directly from full-rank, globally stable systems with arbitrary disturbance statistics, circumventing many drawbacks of alternative methods. We use efficient, matrix-free methods to construct the matrix Wiener–Hopf problem, and we implement a tailored method to solve the problem numerically. The approach naturally handles forcing terms with space–time colour; it allows inexpensive parametric investigation of sensor/actuator placement in scenarios where disturbances/targets are low rank; it is directly applicable to complex flows disturbed by high-rank forcing; it has lower cost in comparison to standard methods; it can be used in scenarios where an adjoint solver is not available; or it can be based exclusively on experimental data. The method is particularly well suited for the control of amplifier flows, for which optimal control approaches are typically robust. Validation of the approach is performed using the linearized Ginzburg–Landau equation. Flow over a backward-facing step perturbed by high-rank forcing is then considered. Sensor and actuator placement are investigated for this case, and we show that while the flow response downstream of the step is dominated by the Kelvin–Helmholtz mechanism, it has a complex, high-rank receptivity to incoming upstream perturbations, requiring multiple sensors for control.

Linear Quadratic Optimal Control Design: A Novel Approach Based on Krotov Conditions

This paper revisits the problem of synthesizing the optimal control law for linear systems with a quadratic cost. For this problem, traditionally, the state feedback gain matrix of the optimal controller is computed by solving the Riccati equation, which is primarily obtained using calculus of variations- (CoV-) based and Hamilton–Jacobi–Bellman (HJB) equation-based approaches. To obtain the Riccati equation, these approaches require some assumptions in the solution procedure; that is, the former approach requires the notion of costates and then their relationship with states is exploited to obtain the closed-form expression of the optimal control law, while the latter requires a priori knowledge regarding the optimal cost function. In this paper, we propose a novel method for computing linear quadratic optimal control laws by using the global optimal control framework introduced by V. F. Krotov. As shall be illustrated in this article, this framework does not require the notion of costates and any a priori information regarding the optimal cost function. Nevertheless, using this framework, the optimal control problem gets translated to a nonconvex optimization problem. The novelty of the proposed method lies in transforming this nonconvex optimization problem into a convex problem. The convexity imposition results in a linear matrix inequality (LMI), whose analysis is reported in this work. Furthermore, this LMI reduces to the Riccati equation upon imposing optimality requirements. The insights along with the future directions of the work are presented and gathered at appropriate locations in this article. Finally, numerical results are provided to demonstrate the proposed methodology.

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.

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.

Synthesis and Implementation of Different Control Law Strategies for a Winding Process

This article presents the synthesis and the real tests of three control law strategies to a winding process. The principal no linearity of this type of system is due to the variation, during operation, of the radius of the winding reels. A linear time varying model can thus describe the dynamic behavior of this system. A linear optimal control law, a gain scheduled controller and a non linear control law (input-output linearizing and decoupling approach) are selected for the control law synthesis. These control laws are then applied to the process with a DSP dSpace board interfaced with Matlab/Simulink. The experimental results illustrate the decoupling advantage of the nonlinear approach.

Active control of ALP with improved performance function

Conventional optimal control theories usually deal with performance function that involves only structural displacement and velocity. This paper presents an algorithm for the linear active control by considering also the structural acceleration in the performance function without bringing in the actuator dynamics. The linear optimal control law, thus achieved, is implemented to control the peak displacement response of a 180.5 m tall articulated leg platform (ALP) under the random sea state. Both controlled and uncontrolled responses of the ALP are obtained by an iterative frequency domain technique which duly considers all the nonlinearities present in the system. The controlled responses are compared with those obtained by the conventional linear feedback control strategy in order to investigate the efficiency of the proposed algorithm. Numerical results indicate that the proposed algorithm performs better than the conventional control strategy.

CONTROL OF BUILDING VIBRATIONS WITH ACTIVE/PASSIVE DEVICES

Two linear optimal control laws and a non-linear control strategy are critically evaluated. They are implemented in a ten-story frame structure. For the linear control laws, both an active bracing system and a hybrid mass damper are considered as control devices, while the non-linear control law can be implemented with either an active or semi-active bracing system. The active and semi-active systems are compared to a passive bracing system with linear viscous dampers and to a hybrid system consisting of a passive bracing and a hybrid mass damper. Dimensionless indices based on the reduction of the maximum story drift and on the maximum control force required are introduced to compare the efficiencies of different control strategies. While the linear optimal control laws exhibit an excellent performance, the non-linear control law, in addition to its simplicity and robustness, appears to be more efficient when the allowable control force is within a certain limit. Furthermore, one attractive feature of the latter is that it can be implemented with semi-active devices to minimize the power requirement.

Helicopter Station-Keeping Using Fuzzy Logic Control and Neural Networks

Station-keeping is an important, but difficult, flight control mode for helicopters, since they have very little forward speed and are dynamically unstable in that mode. In this paper the designs for a fuzzy logic controller and two neural network (NN) controllers to provide satisfactory station-keeping performance for a single rotor helicopter are presented. The effectiveness of the proposed designs can be judged from several simulation results of the helicopter's controlled response. The necessary training and recall data for the NN were obtained from the response of the same helicopter type controlled in a station-keeping mode by a linear optimal feedback control law obtained by solving a linear quadratic regulator (LQR) problem.

Active control of non-stationary response of a two-degree of freedom vehicle model with nonlinear suspension

Active control of non-stationary response of a two-degree of freedom vehicle model with nonlinear passive suspension elements is considered in this paper. The method of equivalent linearization is used to derive an equivalent linear model and optimal control laws are obtained by using stochastic optimal control theory based on full state information. Velocity squared quadratic damping and hysteretic type of stiffness nonlinearities are considered. The effect of the nonlinearities on the active system performance is studied. The performance of active suspensions with nonlinear passive elements is found to be superior to the corresponding passive suspension systems.

Vibration control of flexible linkage mechanisms using piezoelectric films

A control methodology for suppressing the elastodynamic responses of high-speed flexible linkage mechanism is presented by adopting the concept of smart structures featuring piezoelectric films. The dynamics of piezoelectric actuators and sensors bonded on the flexible link are developed for a one-dimensional beam model. The linear optimal feedback control law with a Luenberger observer is employed. The instability caused by the combined effect of control and observation spillover is investigated and carefully prevented. Numerical simulation is performed to evaluate the improvement of elastodynamic responses.

Robust Linear-Optimal Control Laws for Active Suspension Systems

The stability robustness of linear-optimal control laws for quarter-car active suspension systems is evaluated using stochastic robustness analysis. Simultaneous parameters variations and neglected actuator and sensor dynamics are considered for LQ active suspension systems and for a single-measurement LQG system to determine the effects of uncertainty on system stability. The results indicate that neglected actuator and sensor dynamics have a small effect on stability robustness, while parameter uncertainty, particularly that of the “sprung mass” is of great concern. The effectiveness of Loop Transfer Recovery on active suspension systems with both parameter uncertainty and higher-order uncertainty is discerned. The analysis shows that when Loop Transfer Recovery is applied arbitrarily to uncertain systems, both estimator performance and system robustness can decrease. Nevertheless, it is concluded that the impact of the robustness recovery method is determined by stochastic stability robustness analysis, and the recovery design parameter that provides sufficient robustness with minimal performance degradation is readily identified. The effect of LQ design parameters on robustness also is considered. The paper presents robustness analysis and synthesis methods for a quarter-car model that can be applied to higher-order active suspension systems.

Optimal pulse-position pulsewidth-modulated control of sampled data position servosystems

The paper considers the optimisation of pulse-position pulsewidth-modulated (PPPWM) control of a position servosystem. The optimisation is carried out via an equivalent PPPWM transformation of the linear optimal control law. The technique developed here is applicable to both tracking and regulator problems, and the equivalence assumes the existence of at least one pole at the origin for the plant transfer function, which is normally true for any position servosystems. The control signal is optimised with respect to the discrete quadratic performance index and a closed-loop analytical solution can be obtained. The solution for a second-order system is simple, as shown, and readily applicable to real-time sampled-data position servosystems.

A Microcomputer Based Stochastic Controller of a Feeding Process for Bulk Solids

A microcomputer-based implementation of an optimal stochastic regulator for a feeding process is presented. It is shown that the regulator design can be interpreted as a problem of linear optimal signal estimation. The theory of the stationary Kalman - or Wiener - Filter is applied to the problem to obtain the - in the sense of minimum mean square feeding error variance - linear optimal control law. A parametric model of the noise processes is derived.It is shown how to obtain the paraméters of the model out of measurements at the physical process during the normal feeding operation using a linear Least Squares Estimator.

On the Modelling of Planning and Supervisory Behaviour of the Navigator: An Example of the Application of Linear Optimal Control Theory

The development of a control theoretic model of the behaviour of a navigator, requires the solution of a control problem that is characterized by the optimal solution not being a priori known. An algorithm which features iterative application of a linear optimal control law is presented. On the basis of this technique a model of the navigator’s short term planning and supervisory behaviour has been successfully developed.

A slewing control experiment for flexible structures

A hardware setup has been developed to study slewing control for flexible structures including a steel beam and a solar panel. The linear optimal terminal control law is used to design active controllers that are implemented in an analog computer. The objective of this experiment is to demonstrate and verify the dynamics and optimal terminal control laws as applied to flexible structures for large-angle maneuver. Actuation is provided by an electric motor while sensing is given by strain gages and angle potentiometers. Experimental measurements are compared with analytical predictions in terms of modal parameters of the system stability matrix, and sufficient agreement is achieved to validate the theory.

Automatic Inventory Control in a Biological Wastewater Treatment System

To treat wastewater from a chemical plant by a bacteriological reactor, it is necessary to store the stream in an equalization tank prior to feeding to the reactor. The tank is one of the major capital investments in the treatment system. For a given magnitude and frequency of the wastewater load changes, a smaller tank will require a more advanced automatic system for tank inventory control. To determine the minimum required volume, an automatic control system has been designed. The plant waste stream data, the Kalman filter, and a linear optimal control law have been employed in designing the control system. By the assumptions and approximations which may be justified to use in this particular case study, mathematical formulation of the process has been derived to which the filter and the control law are applied.

Positive definiteness of a quadratic functional

In a 1975 paper, Molinari [1] proved that under certain continuity and controllability hypotheses, the infinum of a quadratic linear functional subject to linear differential equations constraints and a linear terminal constraint, is a quadratic function of the initial state. We show here how to constructively find this quadratic form under the addition of a positivity assumption. We also show that if a strengthened generalized Legendre-Clebsch condition holds then there is a linear optimal feedback control law.

The use of model following methods to generate suboptimal feedback control†

Implementation of linear optimal control laws oftentimes requires that all system state variables be physically available for measurement and feedback. The well known impracticality of this requirement motivates the work described in this paper which is addressed to the use of model following methods in the synthesis of sub-optimal feedback controllers which do not require sensing all of the system state variables. The approach presented here consists of first computing a complete set of optimal feedback gains and thus generating optimum system response which is then used as a model toward which a suboptimal design can strive. A suboptimal control is postulated with zero feedback gains as desired, the remaining gains being as yet arbitrary. The average integrated difference between the model closed loop response and the suboptimal closed loop response is then minimized by selection of these remaining feedback gains. For a given feedback configuration, the method thus determines the suboptimal closed loop control yielding a response which is as close as possible in a given well-defined sense to the model response. Defining equations are derived and parameter optimisation methods are utilized to compute the free feedback gains. The method is particularly well suited to digital computation requiring only the use of standard parameter optimization algorithms as well as an algorithm to solve a set of bilinear matrix equations. An example involving regulation of the longitudinal modes of a VTOL flight vehicle is presented to illustrate the model following capabilities of the synthesis procedure.