FEEDBACK LINEARIZATION DESIGNS FOR UNCERTAIN NONLINEAR TIME-DELAY PROCESSES: A COMPARATIVE STUDY

This work concerns the phenomena in which the feedback linearization control is applied to uncertain nonlinear time-delay processes. Under the I/O linearization algorithm, both nonlinear controllers are used to stabilize the closed-loop system with transformed delay inputs. When the effect of input perturbations can converge to zero or asymptotically vanish, these nonlinear feedback designs with only an adjustable parameter can directly improve the tracking performance. The simple linearizing controller can directly regulate the system output at unstable operating point. Combined with deadtime compensation the nonlinear predictive controller with the aid of appropriate state prediction is valid for the real process in the presence of large time delay. Finally, via computer simulation and test of control ability of both feedback control designs the useful comparative results are presented.     

GRAPHICS-BASED CONTROL DESIGNS FOR NONMINIMUM-PHASE NONLINEAR SYSTEMS SUBJECT TO INPUT AND OUTPUT CONSTRAINTS

This article deals with the output regulation of nonminimum-phase systems subject to input and output constraints. Through the off-line static data reconciliation algorithm for a class of stable nonlinear bioreactors, the static feedforward control can ensure constant disturbance attenuation in spite of input multiplicities and actuators constraints. Under the pseudo-steady-state error diagnoses and a graphics-based mechanism for disturbance estimation, the proposed error feedback control scheme will induce the piecewise output regulation. Based on the “intelligent” algorithm for tuning improvement, the closed-loop simulation shows that the piecewise control strategy turns out to be robust against the unknown disturbances.     

EXACT LEAST SQUARES MULTI-STEP PREDICTION FROM NONLINEAR AUTOREGRESSIVE MODELS

Abstract. We obtain an integral equation recurrence relation for the optimal least squares predictor of a nonlinear autoregressive time series model. Numerical solutions are given for a first order threshold model up to three steps ahead.     

DISTURBANCE COMPENSATION FOR NONLINEAR PROCESSES

This paper focuses on the feedback linearization of nonlinear processes with external measurable and immeasurable disturbances. The proposed disturbance compensator, based on a set of the adjustable parameters, is used as a technique to robustify the cancellation of nonlinear terms under a suitable tuning framework. The bound for the adjustable parameters is given to ensure the stability of the closed-loop system. The proposed methodology is applied to the composition control of a CSTR, such that the output regulation and system robustness are achieved. Computer simulation shows results in satisfactory control.     

NONLINEAR MODEL PREDICTIVE CONTROL

Nonlinear Model Predictive Control (NMPC), a strategy for constrained, feedback control of nonlinear processes, has been developed. The algorithm uses a simultaneous solution and optimization approach to determine the open-loop optimal manipulated variable trajectory at each sampling instant. Feedback is incorporated via an estimator, which uses process measurements to infer unmeasured state and disturbance values. These are used by the controller to determine the future optimal control policy. This scheme can be used to control processes described by different kinds of models, such as nonlinear ordinary differential/algebraic equations, partial differential/algebraic equations, integra-differential equations and delay equations. The advantages of the proposed NMPC scheme are demonstrated with the start-up of a non-isothermal, non-adiabatic CSTR with an irreversible, first-order reaction. The set-point corresponds to an open-loop unstable steady state. Comparisons have been made with controllers designed using (1) nonlinear variable transformations, (2) a linear controller tuned using the internal model control approach, and (3) open-loop optimal control. NMPC was able to bring the controlled variable to its set-point quickly and smoothly from a wide variety of initial conditions. Unlike the other controllers, NMPC dealt with constraints in an explicit manner without any degradation in the quality of control. NMPC also demonstrated superior performance in the presence of a moderate amount of error in the model parameters, and the process was brought to its set-point without steady-state offset.     

QUALITATIVE AND QUANTITATIVE IDENTIFIABILITY ANALYSIS OF NONLINEAR CHEMICAL KINETIC MODELS

Identiriability analysis deals with the problem of uniqueness of the parameters when fitting a model to a set of observations. If the model is not qualitatively identifiable, then several or infinitely many parameter sets generate identical predictions of the observed quantities. Three rigorous approaches are evaluated to study qualitative identiriability of nonlinear dynamic models, with emphasis on chemical kinetic modelling. Analysis of a large variety of systems of higher-order reactions shows that under reasonable experimental conditions such models are rarely unidentifiable in the qualitative sense, although there exist well-known examples of unidentifiable models for monomolecular reaction systems. Kinetic models are, however, frequently unidentifiable in a quantitative sense, when a particular sel of error-corrupted data does not allow for obtaining reliable estimates of the parameters. In such cases the goodness-of-fit depends only on some combinations of the parameters. Performing a logarithmic transformation, the well-known principal component analysis is shown to offer an efficient method for detecting and identifying nonlinear dependences among the parameters, thereby suggesting simpler models leading to meaningful estimates     

GLOBALLY OPTIMAL NONLINEAR FEEDBACK: APPLICATION TO NONISOTHERMAL CSTR CONTROL

Approximations of the globally optimal nonlinear quadratic regulator (NQR) strategy are employed to control the unstable steady state of a continuous stirred tank reactor. It is shown that the proposed approximate NQR controllers can be uniquely determined through solution of Riccati and linear matrix equations and can deliver better dynamic performance than the linear quadratic regulator (LQR) if the process model is proper. Stability, optimality, and constraint satisfaction regions for these controllers are identified.     

MODELING AND PREDICTIVE CONTROL OF MIMO NONLINEAR SYSTEMS USING WIENER-LAGUERRE MODELS

In this work, a Weiner-type nonlinear black box model was developed for capturing dynamics of open loop stable MIMO nonlinear systems with deterministic inputs. The linear dynamic component of the model was parameterized using orthogonal Laguerre filters while the nonlinear state output map was constructed either using quadratic polynomial functions or artificial neural networks. The properties of the resulting model, such as open loop stability and steady-state behavior, are discussed in detail. The identified Weiner-Laguerre model was further used to formulate a nonlinear model predictive control (NMPC) scheme. The efficacy of the proposed modeling and control scheme was demonstrated using two benchmark control problems: (a) a simulation study involving control of a continuously operated fermenter at its optimum (singular) operating point and (b) experimental verification involving control of pH at the critical point of a neutralization process. It was observed that the proposed Weiner-Laguerre model is able to capture both the dynamic and steady-state characteristics of the continuous fermenter as well as the neutralization process reasonably accurately over wide operating ranges. The proposed NMPC scheme achieved a smooth transition from a suboptimal operating point to the optimum (singular) operating point of the fermenter without causing large variation in manipulated inputs. The proposed NMPC scheme was also found to be robust in the face of moderate perturbation in the unmeasured disturbances. In the case of experimental verification using the neutralization process, the proposed control scheme was found to achieve much faster transition to a set point close to the critical point when compared to a conventional gain-scheduled PID controller.     

MODELING AND PREDICTIVE CONTROL OF MIMO NONLINEAR SYSTEMS USING WIENER-LAGUERRE MODELS

In this work, a Weiner-type nonlinear black box model was developed for capturing dynamics of open loop stable MIMO nonlinear systems with deterministic inputs. The linear dynamic component of the model was parameterized using orthogonal Laguerre filters while the nonlinear state output map was constructed either using quadratic polynomial functions or artificial neural networks. The properties of the resulting model, such as open loop stability and steady-state behavior, are discussed in detail. The identified Weiner-Laguerre model was further used to formulate a nonlinear model predictive control (NMPC) scheme. The efficacy of the proposed modeling and control scheme was demonstrated using two benchmark control problems: (a) a simulation study involving control of a continuously operated fermenter at its optimum (singular) operating point and (b) experimental verification involving control of pH at the critical point of a neutralization process. It was observed that the proposed Weiner-Laguerre model is able to capture both the dynamic and steady-state characteristics of the continuous fermenter as well as the neutralization process reasonably accurately over wide operating ranges. The proposed NMPC scheme achieved a smooth transition from a suboptimal operating point to the optimum (singular) operating point of the fermenter without causing large variation in manipulated inputs. The proposed NMPC scheme was also found to be robust in the face of moderate perturbation in the unmeasured disturbances. In the case of experimental verification using the neutralization process, the proposed control scheme was found to achieve much faster transition to a set point close to the critical point when compared to a conventional gain-scheduled PID controller.     

A SOLUTION METHOD FOR A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMS

An approximation formula for finite Sturm-Liouville integral transforms of nonlinear functions allows determination of approximate analytical solutions to a certain class of nonlinear boundary value problems. The proposed method is extremely powerful in analysis of a variety of chemical engineering models. The influence of the degree of nonlinearity of the nonlinear term on results obtained using the approximate transform formula is investigated numerically. An iterative procedure in transform space is described for improving solution accuracy, and fast convergence is observed in three illustrative examples. The technique yields solutions which can be related to bounds on the exact solution derived by Kalaba. The asymptotic solution for the mode error, which is the error in approximating the nonlinear function in transform space, is obtained analytically.     

MODELING LONG-MEMORY PROCESSES FOR OPTIMAL LONG-RANGE PREDICTION

Abstract. We look at the implications of modeling observations from a fractionally differenced noise process using an approximating AR ( p ) model. The approximation is used because of computational difficulties in the estimation of the differencing parameter of the fractional noise model. Because the fractional noise process is long-range dependent, we assess the applicability of the approximating autoregressive (AR) model based on its long-range forecasting accuracy compared with that of the fractional noise model. We derive the asymptotic k -step-ahead prediction error for a fractional noise process modeled as an AR( p ) process and compare it with the k -step-ahead prediction error obtained when the model for the observed series is correctly specified as a fractional noise process and the fractional differencing parameter d is either known or estimated. We also assess the validity of the asymptotic results for a finite sample size via simulation. We see that AR models can be useful for long-range forecasting of long-memory data, provided that consideration is given to the forecast horizon when choosing an approximating model.     

LEVINSON-TYPE RECURSIVE ALGORITHMS FOR LEAST-SQUARES AUTOREGRESSION

Abstract. A set of formulae for calculating the least-squares autoregressive coefficients is given. It can be used for stable, unstable and explosive models. The calculations needed by this algorithm are less than those of the Burg and Marple algorithms. The method used to deduce these formulae has general significance. For example, it is also used to improve the Marple algorithm. Simulation shows that the estimates given by this algorithm are better than those given by the Levinson and Burg recursions.     

BATCH-TO-BATCH OPTIMAL CONTROL OF BATCH PROCESSES BASED ON RECURSIVELY UPDATED NONLINEAR PARTIAL LEAST SQUARES MODELS

A batch-to-batch optimal control approach for batch processes based on batch-wise updated nonlinear partial least squares (NLPLS) models is presented in this article. To overcome the difficulty in developing mechanistic models for batch/semi-batch processes, a NLPLS model is developed to predict the final product quality from the batch control profile. Mismatch between the NLPLS model and the actual plant often exists due to low-quality training data or variations in process operating conditions. Thus, the optimal control profile calculated from a fixed NLPLS model may not be optimal when applied to the actual plant. To address this problem, a recursive nonlinear PLS (RNPLS) algorithm is proposed to update the NLPLS model using the information newly obtained after each batch run. The proposed algorithm is computationally efficient in that it updates the model using the current model parameters and data from the current batch. Then the new optimal control profile is recalculated from the updated model and implemented on the next batch. The procedure is repeated from batch to batch and, usually after several batches, the control profile will converge to the optimal one. The effectiveness of this method is demonstrated on a simulated batch polymerization process. Simulation results show that the proposed method achieves good performance, and the optimization with the proposed NLPLS model is more effective and stable than that with a batch-wise updated linear PLS model.     

ADSORPTION AND DESORPTION KINETICS FOR NONLINEAR ADSORPTION ISOTHERMS

The theoretical calculations of the mass transfer rate are based on a model which considers a nonlinear adsorption isotherms and simultaneous resistance to mass transfer in The pore adsorbent and the convective mass transfer. Numerical solutions to the diffusive-convective-controlled adsorption and desorption processes are calculated for the Langmuir and rectangular adsorption isotherms. It is shown that for the rectangular adsorption isotherm the adsorption kinetics is governed by the diffusive-convective mass transfer over a wide range of times and the desorption kinetics is irreversible process. The equations are derived to calculate from the experimental kinetics data: (a) the coefficient diffusion in the pore adsorbent, (b) the relaxation times characterizing the adsorption and desorption processes, and (c) the times needed to reach the quasiequilibrium state for the adsorption and desorption processes.     

SPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON-GAUSSIAN TIME SERIES

Abstract. In the present paper we consider nonlinear wavelet estimators of the spectral density f of a zero mean, not necessarily Gaussian, stochastic process, which is stationary in the wide sense. It is known in the case of Gaussian regression that these estimators outperform traditional linear methods if the degree of smoothness of the regression function varies considerably over the interval of interest. Such methods are based on a nonlinear treatment of empirical coefficients that arise from an orthonormal series expansion according to a wavelet basis.
The main goal of this paper is to transfer these methods to spectral density estimation. This is done by showing the asymptotic normality of certain empirical coefficients based on the tapered periodogram. Using these results we can show the risk equivalence to the Gaussian case for monotone estimators based on such empirical coefficients. The resulting estimator of f keeps all interesting properties such as high spatial adaptivity that are already known for wavelet estimators in the case of Gaussian regression.
It turns out that appropriately tuned versions of this estimator attain the optimal uniform rate of convergence of their L ₂ risk in a wide variety of Besov smoothness classes, including classes where linear estimators (kernel, spline) are not able to attain this rate. Some simulations indicate the usefulness of the new method in cases of high spatial inhomogeneity.     

A NONLINEAR EXPLICIT ALGORITHM FOR EFFICIENT INTEGRATION OF STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

An efficient algorithm for the integration of stiff systems of ordinary differential equations based on the boundary value technique is derived. The automatic step-size control procedure can be easily adapted. Results for a very stiff problem are compared with those obtained by other methods and the high effectiveness of the method is shown.     

AUTOMATED GAIN SCHEDULE GENERATION FOR NONLINEAR ADAPTIVE CONTROL

The generalized minimum variance (GMV) control law includes a penalty function that permits control activity to become more or less vigorous as process character changes. This penalty, if cast as a function of steady state process gain, can enable control of nonlinear processes. To make such an approach easy to implement, a method is detailed for automatically generating and updating a gain schedule while the process is in closed loop. The method is demonstrated on a simulated process that possesses a very nonlinear character. The scope of this paper is limited to single-input single-output, open-loop stable processes     

A NONLINEAR EXPLICIT ALGORITHM FOR EFFICIENT INTEGRATION OF STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

An efficient algorithm for the integration of stiff systems of ordinary differential equations based on the boundary value technique is derived. The automatic step-size control procedure can be easily adapted. Results for a very stiff problem are compared with those obtained by other methods and the high effectiveness of the method is shown.     

DYNAMIC MODELING AND OPERATION OF A SEEDED BATCH COOLING CRYSTALLIZER

In this study, a dynamic model of a batch cooling crystallizer is developed. The seeded crystallization of potash alum from aqueous solutions. Four different cooling policies namely natural cooling, linear cooling, optimal cooling, and controlled cooling (nonlinear geometric control (NGC) cooling) are presented. The simulation results indicate that both optimal and controlled cooling improve the weight mean size of the final product significantly. The influence of seed loading policy on the end product quality is also studied for the NGC cooling. It is found that the appropriate seed loading is important to achieve a good final CSD, especially for a fixed batch time.