Extremum seeking control and its application to process and reaction systems: A survey

The objective of this paper is to present a survey on extremum seeking control methods and their applications to process and reaction systems. Two important classes of extremum seeking control approaches are considered: perturbation-based and model-based methods.     

Extremum seeking control for soft landing of an electromechanical valve actuator

Many electromagnetic actuators suffer from high velocity impacts. One such actuator is the electromechanical valve actuator, recently receiving attention for enabling variable valve timing in internal combustion engines. Impacts experienced by the actuator are excessively loud and create unnecessary wear. This paper presents an extremum seeking controller designed to reduce the magnitude of these impacts. Based on a measure of the sound intensity at impact, the controller tunes a nonlinear feedback to achieve impact velocities of less than while maintaining transition times of less than . The control strategy is implemented with an eddy current sensor, to measure the valve position, and a microphone.     

Lyapunov functions for quadratic differential equations with applications to adaptive control

Conditions are derived for the existence of quadratic Lyapunov functions for vector differential equations with quadratic terms. These conditions are used to establish a set of nonlinear equations for which the null solution is asymptotically stable in the whole. Many of the recent results in the stability of model reference adaptive systems are particular cases of these general conditions.     

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.     

Extremum seeking under stochastic noise and applications to mobile sensors

In this paper the extremum seeking algorithm with sinusoidal perturbations has been extended and modified in two ways: (a) the output of the system is corrupted with measurement noise; (b) the amplitudes of the perturbation signals, as well as the gain of the integrator block, are time varying and tend to zero at a pre-specified rate. Convergence to the extremal point, with probability one, has been proved. Also, as a consequence of being able to cope with a stochastic environment, it has been shown how the proposed algorithm can be applied to mobile sensors as a tool for achieving the optimal observation positions. The proposed algorithm has been illustrated through several simulations.     

Adaptive nuclear reactor control for integral quadratic cost functions

The problem of optimally controlling the power level changes of a nuclear reactor is considered. The model of an existing power plant is used, which is a ninth-order nonlinear system, having time-varying parameters. A closed form solution of the optimal feedback controller is not known for such a system, and numerical solutions require prohibitive amounts of computer time and/or storage for on-line applications. The approach in the present work is, therefore, to approximate the response of the reactor by that of a low-order linear model in a piece-wise manner. The model parameters are chosen to minimize the deviations, in any desired sense, between the system and model responses. The optimal feedback parameters are readily computed for the model, and the same controller results in suboptimal system performance. The difference, however, has been found negligible in the examples considered. Since the model parameters are updated to reflect the plant nonlinearities as well as changes in the plant parameters, the resultant control system is adaptive. It has been found, that using a second-order model, optimal performance could be approached to within practical limits, and the necessary computer time is realizable for on-line applications.     

Extremum seeking with sliding mode gradient estimation and asymptotic regulation for a class of nonlinear systems

This paper presents a model-based extremum seeking approach for a class of single-input–single-output nonlinear systems, with the analytic form of the performance function unknown a priori. We focus on a practically implementable design with robustness to model uncertainties and disturbances. A discrete-time sliding mode gradient estimator is developed for estimating the gradient of the performance profile. Based on the estimate, a variable structure output feedback regulator is proposed to enforce the system states toward the optimal trajectory. We analyze convergence conditions of the switching system toward a neighborhood of the optimal trajectory, and establish an ultimate bound on the size of the neighborhood. The robustness of the proposed controller is discussed with respect to measurement noise.     

Estimation of valve stiction in control loops using separable least-squares and global search algorithms

This contribution presents a new procedure for quantifying valve stiction in control loops based on global optimisation. Measurements of the controlled variable (PV) and controller output (OP) are used to estimate the parameters of a Hammerstein system, consisting of a connection of a two-parameter stiction model and a linear low-order process model. As the objective function is non-smooth, gradient-free optimisation algorithms, i.e., pattern search (PS) methods or genetic algorithms (GA), are used for fixing the global minimum of the parameters of the stiction model, subordinated with a least-squares estimator for identifying the linear model parameters. Some approaches for selecting the model structure of the linear model part are discussed. Results show that this novel optimisation-based technique recovers accurate and reliable estimates of the stiction model parameters, dead-band plus stick band (S) and slip jump (J), from normal (closed-loop) operating data for self-regulating and integrating processes. The robustness of the proposed approach was proven considering a range of test conditions including different process types, controller settings and measurement noise. Numerous simulation and industrial case studies are described to demonstrate the applicability of the presented techniques for different loops and for different amounts of stiction.     

Performance of unstable control loops with limits

Unstable control loops can, and often do, perform well, although not better than properly tuned linear loops. However, the associated limit cycling may cause unnecessary mechanical and thermal fatigue in the offending loops and elsewhere in the plant.     

Stability of steepest descent with momentum for quadratic functions

This paper analyzes the effect of momentum on steepest descent training for quadratic performance functions. We demonstrate that there always exists a momentum coefficient that will stabilize the steepest descent algorithm, regardless of the value of the learning rate. We also demonstrate how the value of the momentum coefficient changes the convergence properties of the algorithm.     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.     

Sequential quadratic programming and modified lagrange functions

Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 51–67, September–October, 1994.     

Extremum seeking for moderately unstable systems and for autonomous vehicle target tracking without position measurements

We remove the long standing restriction that plant dynamics in extremum seeking control must be stable and provide an extension that allows single integrators, double integrators, and moderately unstable single poles. An application of the result for single and double integrators is in control of autonomous vehicles. Extremum seeking is used for finding a source of a signal (chemical, electromagnetic, etc.) whose strength decays with the distance. This is achieved without the measurement of the position vector and using only the measurement of the scalar signal.     

Energy-shaping control of synchronous generators with exciter-governor dual control loops

In this paper we extend the energy-shaping controller design technique, which has been successfully used for single-loop excitation regulation of synchronous generators in power systems, to the exciter–governor dual configuration. As in the simpler excitation problem, the aim of the controller is to shape the total energy function via modification of the energy transfer between electrical and mechanical subsystems—but in this dual scenario we also have to take into account the energy contribution of the governor control loop dynamics. Motivated by practical considerations we also present a partial state feedback version of the controller that is derived applying thoe immersion and invariance technique recently reported in the literature. Representative simulations show how the tuning parameters effectively shape the domain of attraction of the desired stable equilibrium enhancing the transient stability of the power system.     

Linear quadratic control with stability degree constraint

The problem of static state feedback Linear Quadratic (LQ) optimal control subject to a prescribed degree of stability for the closed-loop system is considered in this paper. A necessary optimality condition is given via the Lagrange multiplier method. A globally convergent algorithm is provided to solve the optimization problem. It is shown that the algorithm recovers the standard LQ feedback gain provided the desired stability degree is small enough to be within the range by the standard LQ design. As for other cases the optimum occurs on the boundary of the α-region. A numerical example shows that the proposed algorithm provides a better design compared to the existing methods.     

Optimal stochastic adaptive control with quadratic index

The discrete-time stochastic control system with complete observation is considered with quadratic loss function when the constant coefficients of the system are unknown. The parameter estimates given by the recursive least-squares method are used to define the feedback gain, and the adaptive control is taken to be either a linear-state feedback disturbed by a sequence of random vectors with variances tending to zero or only the disturbance without the feedback term in accordance with stopping times defined on the trajectory of the system. It is proved that the parameter estimates are strongly consistent and the loss function reaches its minimum, i.e. the adaptive control is optimal.     

Linear quadratic differential games with cheap control

This paper considers a linear quadratic differential game in which the weighting on the minimizing control is allowed to approach zero. It is shown that if a certain minimum phase condition is satisfied then the value of the game will approach zero as the weighting on the minimizing control approaches zero.