Stochastic Feedback Control Research Articles

Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE: Design and Application to a Sputtering Process

In this work, we develop a method for nonlinear feedback control of the roughness of a one-dimensional surface whose evolution is described by the stochastic Kuramoto-Sivashinsky equation (KSE), a fourthorder nonlinear stochastic partial differential equation. We initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations by using Galerkin’s method. A finite-dimensional approximation of the stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A nonlinear feedback controller is then designed based on the finite-dimensional approximation to control the surface roughness. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the nonlinear feedback controller in the closed-loop infinite-dimensional system. The effectiveness of the proposed nonlinear controller and the advantages of the nonlinear controller over a linear controller resulting from the linearization of the nonlinear controller around the zero solution are demonstrated through numerical simulations. Finally, a successful application of a stochastic KSE-based nonlinear feedback controller to the kinetic Monte Carlo model of a sputtering process is also demonstrated.

Adaptive stochastic output feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic output feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The system dynamics is experimentally determined via the extended least square method with an exponential forgetting factor and covariance resetting. The optimal output feedback controller is redesigned online periodically based on the system identification. The output measurements and past control inputs are used to construct new control inputs. The adaptive output controller can stabilize the time dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly three times the inverse of the growth rate. The design procedure is simpler and the computation time is shorter than the state feedback method reported earlier in Sun, Sen, and Longman [Phys. Plasmas13, 012512 (2006)].

Adaptive optimal stochastic state feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic state feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The extended least-square method with exponential forgetting factor and covariance resetting is used to identify (experimentally determine) the time-varying stochastic system model. A Kalman filter is used to estimate the system states. The estimated system states are passed on to an optimal state feedback controller to construct control inputs. The Kalman filter and the optimal state feedback controller are periodically redesigned online based on the identified system model. This adaptive controller can stabilize the time-dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly four times the inverse of the growth rate for the time-invariant model used.

Vibrations of Stiffened Composite Panels With Smart Materials

A numerical study of the use of electrorheological (ER) fluids and piezoelectric (PZT) actuators to control random vibrations of stiffened composite panels is presented. Active control of stiffness and damping is provided by the ER fluids and direct feedback control is provided by the PZT’s. New forms of transfer matrices are developed to include the effects of these smart materials. The modal equations of an equivalent uniform panel are converted into state-space form and digital stochastic feedback control is implemented. PZT direct feedback control is compared with digital stochastic feedback control. Parametric studies quantify the effect of actuator size and number, and ER fluid action.

Effective Information for Offline Stochastic Feedback and Optimal Control of Dynamic Systems

The impact of uncertain future events on decision making in a stochastic environment is modeled in this paper. Such modeling is presented for both feedback and optimal control problems. This research overcomes the difficulties of forecasting that arise when considering future information. In this paper, we seek to find the minimum amount of information (effective information) necessary to evaluating system performance offline or to optimally control a system. The existence of effective information is proved and a methodology for determining it is developed. It is also shown that ignoring information beyond the planning horizon leads to significant performance loss and may even lead to violating the constraints of a control problem.

On-line order estimation and parameter identification for linear stochastic feedback control systems (CARMA model)

This paper presents a new recursive estimate method for orders and coefficients of linear stochastic feedback control systems (CARMA model) under the assumption that the upper bounds of system orders are known. The strong consistency of the estimates for orders and coefficients is proved and the convergence rate of coefficient estimates to their true values is also obtained. The estimate algorithm is applied to adaptive tracking of the systems with unknown orders and unknown coefficients. The resulting closed-loop systems are then globally stable and the tracking sample mean square error is minimized as well. Simultaneously, the estimates of the adaptive tracking for orders and coefficients are also strongly consistent. The simulation results given here show that the new developed algorithms of both system identification and adaptive tracking are effective.

Discussion and analysis of stochastic feedback control adjustment

Process control techniques from engineering and statistical process control overlap at the interface of the two process control methodologies. In recent years, process control practitioners have focused their attention on statistical process monitoring and feedback control adjustment. It is quite common to encounter problems that are related to feedback (closed-loop) stability, controller limitations and dead-time compensation in order to obtain minimum variance (mean square error) at the output. A stochastic feedback control algorithm for input adjustment can be derived by use of techniques from the two process control methodologies. This paper briefly reviews statistical feedback control adjustment and describes a method to model stochastic feedback control adjustment in order to control the quality of products at the output of a continuous process. The method takes into consideration situations under which the product quality mean is on target/set point and situations where it is not on target by assigning suitable probabilities for the respective situations. A test of significance is conducted for such situations when a feedback control adjustment is required to bring the product quality mean on target and a suitable probability model is developed for such a situation.

Optimal feedback control as a theory of motor coordination.

A central problem in motor control is understanding how the many biomechanical degrees of freedom are coordinated to achieve a common goal. An especially puzzling aspect of coordination is that behavioral goals are achieved reliably and repeatedly with movements rarely reproducible in their detail. Existing theoretical frameworks emphasize either goal achievement or the richness of motor variability, but fail to reconcile the two. Here we propose an alternative theory based on stochastic optimal feedback control. We show that the optimal strategy in the face of uncertainty is to allow variability in redundant (task-irrelevant) dimensions. This strategy does not enforce a desired trajectory, but uses feedback more intelligently, correcting only those deviations that interfere with task goals. From this framework, task-constrained variability, goal-directed corrections, motor synergies, controlled parameters, simplifying rules and discrete coordination modes emerge naturally. We present experimental results from a range of motor tasks to support this theory.

Analysis of an up/down power control algorithm for the CDMA reverse link under fading

We analyze the performance of a code division multiple access (CDMA) reverse link a with an up/down power control algorithm in the presence of fading. We derive a stochastic nonlinear feedback control system model for the power controlled reverse link, and study the power control performance based on the nonlinear model using the technique of statistical linearization. We provide a general analysis framework that allows us to study the effect of mobile speed, power control step size, and fading channel parameters such as correlation coefficient and rate of fading on power control errors. Numerical results show excellent accuracy of our analysis, which can be used to design and optimize the system parameters without going through lengthy simulations. For example, in the presence of shadow fading, it is seen that a power control step size in the range of 0.5-1.3 dB is sufficient to keep power control errors near a minimum. In the case of Rayleigh fading, the standard deviation of power control errors grows quickly even at moderate mobile speeds.

Calibration and Control for Estuary System under Uncertainty

A comprehensive stochastic management methodology is developed for a distributed-parameter estuary system (DPES) which is described by partial differential equations (PDEs). The proposed stochastic management methodology consists of uncertainty-based calibration (i.e. parameter estimation under uncertainty) and uncertainty-based control (i.e. determination of optimal freshwater inflows into the estuary under uncertainty). The technique for uncertainty-based calibration is based on Gauss–Newton minimization method and uncertainty analysis method such as Rosenblueth's point estimate method or Harr's point estimate method. The uncertainty-based calibration technique is used to estimate the optimal parameters such as Manning's roughness coefficient and dispersion coefficient in the PDEs while considering the uncertainties in the boundary conditions or initial conditions for the PDEs. The technique for uncertainty-based control is based on a real-time control method which is discrete-time stochastic linear quadratic feedback control. Real-time control is achieved by feedback control which uses salinities at test stations in the estuary as the state vector and controlled freshwater inflows as the control vector. The feedback control law determines the real-time control vector using observed states in a manner that best approximates desired states and controls (freshwater inflows). This new methodology for real-time control has been applied to the Lavaca–Tres Palacios Estuary in Texas for purposes of testing and illustration.

Stochastic Verification Theorems within the Framework of Viscosity Solutions

This paper studies controlled systems governed by Ito's stochastic differential equations in which control variables are allowed to enter both drift and diffusion terms. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. This theorem is shown to have wider applicability than the restrictive classical verification theorems, which require the associated dynamic programming equations to have smooth solutions. Based on the new verification result, optimal stochastic feedback controls are obtained by maximizing the generalized Hamiltonians over both the control regions and the superdifferentials of the value functions.

A New Hierarchical Optimal Control Technique of Stochastic Large-Scale Systems

The stochastic optimal feedback control problem of the linear large-scale systems is considered. A new two-level hierarchical control algorithm and the structure of the regulator is given, the optimal control solution is reduced to a two-point boundary-value problem.

Nonquadratic cost and nonlinear feedback control

AbstractNonlinear controllers offer significant advantages over linear controllers in a variety of circumstances. Hence there has been significant interest in extending linear‐quadratic synthesis to nonlinear‐nonquadratic problems. The purpose of this paper is to review the current status of such efforts and to present, in a simplified and tutorial manner, some of the basic ideas underlying these results. Our approach focuses on the role of the Lyapunov function in guaranteeing stability for autonomous systems on an infinite horizon. Sufficient conditions for optimality are given in a form that corresponds to a steady‐state version of the Hamilton‐Jacobi‐Bellman equation. These results are used to provide a simplified derivation of the nonlinear feedback controller obtained by Bass and Webber (1966)38 and to obtain a deterministic variation of the stochastic nonlinear feedback controller developed by Speyer (1976).45.

Variance formulae for feedback stochastic control systems

Abstract An approach to evaluating the performance of a feedback stochastic controller is presented. The controller is evaluated by the values of the autocovariances of the feedback output and input variable time series. To calculate the autocovariances, the feedback output and input variable time series are identified first, then using time series literature, the formula for the autocovariances is derived. The approach can be used for both univariable or multivariable systems and delayed or non-delayed processes.

A stochastic optimal feedback control problem with random-sized jumps

An optimal feedback control problem for a partially observed linear system with noise of fixed-sized jumps occurring at random times driven by a Poisson process is extended to include noise with random-sized jumps. The control structure is appropriately modified to compensate for the mean behavior of the system jumps via an additional deterministic term.

Identification of Coefficients, Orders and Time-delay for ARMAX Systems

Consistent estimates of coefficients, orders and time-delay for the stochastic feedback control system described by the ARMAX model are given in two forms: 1. The estimation algorithm is recursive when the upper bounds for unknown orders and the lower bound for time-delay are available and the strictly positive realness (SPR) condition is satisfied for the system noise; 2. The estimates are no longer recursive and the computation for th em is more time-consuming when the bounds for parameters are unavailable and the SPR condition is replaced by some stability condition.

Least-squares identification for ARMAX models without the positive real condition

Recursive identification problems of linear stochastic feedback control systems described by ARMAX models are studied without imposing the strictly positive real condition on the noise model. An increasing lag least squares is used in the first step to estimate the noise process, while the parameter estimate is formed in the second step by an extended least squares. In the algorithm, there is an increase in computational cost in comparison to traditional algorithms. However the results of this note do not need any a priori information or conditions on the noise model except that of stability, and they are applicable to the identification of general feedback control systems. >

Consistent order estimation for linear stochastic feedback control systems (CARMA model)

A new criterion CIC is introduced to estimate orders ( p 0, q 0, r 0) of the linear stochastic feedback control system with correlated noise described by a CARMA model. It is proved that the estimate is strongly consistent when the upper bounds for p 0, q 0 and r 0 are available, but neither the stability condition nor the ergodicity of the input and output are imposed on the system.

Theory of System Identification and Adaptive Control for Stochastic Systems

This paper presents a series of theoretical results on parameter identification and adaptive control for discrete-time stochastic feedback control systems. Without a use of persistent excitation condition the unknown coefficients of the system are consistently estimated by use of the extended least squares (ELS) and the stochastic gradient (SG) algorithms. Then a strong consistent estimate for orders of the system is also given by minimizing an information criterion designed for feedback control systems. In order to apply results of parameter estimation to adaptive control systems an attenuating excitation technique is introduced so that for strong consistency of the parameter estimate the conditions imposed on eigenvalues are replaced by the growth rate restriction on the input-output of the system. Further, optimal controls are designed respectively for tracking a given reference signal and for minimizing a quadratic loss function so that the parameter estimates are strongly consistent and, simultaneously, the performance indeces are minimized. When the system contains an unmodeled dynamics but the adaptive control is designed on the basis of the modeled part of the system a robust analysis is presented by estimating the error in parameter estimation and in control performances.

Polynomial Optimization of Stochastic Feedback Control for Stable Plants

Polynomial Optimization of Stochastic Feedback Control for Stable Plants