Tore K. Gustafsson

Dynamic modeling and reaction invariant control of pH

Abstract A systematic treatment of the dynamics and control of acid-base reactions is presented. A state model consisting of mass balances for the species present is formed. Through a linear transformation of the concentration vector a new model is set up. The state vector of this model is partitioned into a reaction invariant and a reaction variant part. The model is thus split into two parts, one describing the physical properties of the reactor system independent of chemical reactions, the other describing the chemical reactions. For fast acid-base reactions the reaction invariant part of the model is sufficient to define the thermodynamic state of the system. The reaction rate vector is thus eliminated from the model. In this case the reaction variant part of the model consists of a static equation, relating pH to the reaction invariant state variables. The model obtained provides a sound basis for design of control loops for feedback and feedforward control of pH. As a direct application a new scheme for adaptive control of pH is proposed.

An experimental study of a class of algorithms for adaptive pH control

Abstract We start from an earlier presented method of formulating models for fast acid-base reaction processes, where the reaction invariant and the reaction variant parts of the dynamic model are separated. Models suited for the design of controllers of pH in fast acid-base reaction processes with varying buffer concentrations are set up. The set of models includes a nonlinear model resulting in a controller with linearized feedback of a reaction invariant state vector and linearized models resulting in adaptive linear feedback of pH. Experimental results with some different implementations of these controllers are presented.

Multimodel robust control of nonlinear plants: a case study

Abstract A multimodel controller design procedure is proposed and evaluated on a strongly nonlinear chemical process. The objective is to design a controller which achieves good quadratic (LQ) performance and robustness for an uncertain nonlinear plant. In order to capture both parametric and unstructured plant uncertainties, a multimodel process description is introduced, in which the plant is represented by a discrete set of linear models and associated norm-bounded uncertainties. The controller design problem is then stated as a multimodel mixed H 2 / H ∞ problem for optimal quadratic ( H 2 ) performance subject to ( H ∞ -type) robustness bounds for the multimodel plant description. The problem of finding a linear controller with the specified optimality and robustness properties lacks a closed solution, and it is therefore solved with parametric controller optimization techniques. The proposed controller design procedure is demonstrated on a simulated pH neutralization process, and is shown to give a linear controller with good performance and robustness characteristics. The achievable global performance is further improved by combining the proposed design procedure with controller scheduling. Two implementations of controller scheduling which satisfy a set of linearization conditions are presented and successfully tested on the neutralization process.

Multimodel robust control applied to a pH neutralization process

A multimodel controller design procedure based on robust LQ design is evaluated for a highly nonlinear chemical process. The procedure is combined with two gain-scheduling methods: a fixed robust controller with scheduled output gain, and complete scheduling of locally optimal controllers. The performances obtained with the fixed controller, the gain scheduled controllers and conventional PID control are compared and tested on a simulated pH neutralization process.

Modelling of uncertain systems via linear programming

Abstract Real plants are, in general, time-varying and uncertain. Yet most industrial control loops are designed using linear time-invariant (LTI) plant models. The present work formalizes the problem of best approximate LTI modelling of BIBO-stable linear time-varying (LTV) systems in a way that is compatible with the notion of induced l ∞ norm used in robust l 1 control. This setup is closely associated with certain identification methods and controlmotivated model validation/invalidation procedures that can be efficiently implemented with special-purpose linear programming (LP) techniques. Results are given for approximate modelling of BIBO-stable LTV and LTI systems using LTI models, especially BIBO-stable fixed-pole state-space model parametrizations. It is shown that such parametrizations are satisfactory from an approximate modelling point of view, and can be used in output-error-type LP identification techniques and in LP model validation procedures. Various useful constraints on model parameters etc. can be included, as long as they are linear in the unknown parameters, and both insensitive (statistically robust) and sensitive criteria can be used in identification and model validation. Simulation examples are included to illustrate that such techniques indeed give good results and can be used to solve problems of realistic size.

Modelling of uncertain systems with application to robust process control

Abstract A method for black-box identification of uncertain systems is presented. The method identifies a nominal model and an uncertainty model set, consisting of unfalsified uncertainty models. Minimisation of a Chebyshev criterion leads to computationally favourable linear programming problems and allows the possibility to include a priori information in the form of linear constraints without making the computations more complex. Using data compression via correlation computations solves the computation problem associated with identifying unfalsified uncertainty models. The application of set-valued uncertainty models to robust process control is illustrated in a simulation study of robust model predictive control of a distillation column.

Modelling of Uncertain Systems via Linear Programming

Abstract Linear programming methods are useful in problems of identification and model validation for uncertain systems. Several types of priors as well as uncertainty models are easily included. Simulation examples with smoothness priors, Laguerre-model identification and unmodelled dynamics are presented.

On system identification and model validation via linear programming

Linear programming methods for discrete l/sup 1/ approximation are used to provide solutions to problems of approximate identification with state space models and to problems of model validation for stable uncertain systems. Choice of model structure is studied via Kolmogorov n-width concept and a related n-width concept for state space models. Several results are given for FIR, Laguerre and Kautz models concerning their approximation properties in the space of bounded-input bounded-output (BIBO) stable systems. A robust convergence result is given for a modified least sum of absolute deviations identification algorithm for BIBO stable linear discrete-time systems. A simulation example with identification of Kautz models and subsequent model validation is given.<<ETX>>

Output-error criteria in multi-model adaptive control with experimental application to pH control

The use of output-error criteria in a multi-model adaptive PID controller is developed and evaluated in an experimental study using a nonlinear pH neutralization process. Variable hysteresis depending on model distance is included in the model switching. The performance and robustness characteristics of the multi-model controller are compared to those of a conventional PID controller, conventional self-tuning controller, and a multi-model adaptive PID controller using a prediction-error criterion. Both the output-error criterion and especially the variable hysteresis were found to be significant factors in the performance of the multi-model adaptive controller.

An approximative method for discrete optimal control problems and its application to a special class of transportation

We consider N-stage decision processes with state- and control constraints and with time dependend parameters. The optimal control problem for such discrete processes Is formulated in the policy space. Our aim is, to compute optimal or suboptimal controls for special technological processes, for instance the process “road surface construction”. Therefore, we obtain a mathematical model of this process, which is easy to extend to a more general class of transportation control problems. As optimization method we use the iterative Podyp algorithm given by E. Von Schalien and K. Waller. We extend this algorithm to make it available for problems with state-depending control constraints and derive error estimations. Examples show that our method is applicable to the considered class of problems.