Edward P. Gatzke

Model based control of a four-tank system

Abstract A multi-disciplinary laboratory for control education has been developed at the University of Delaware to expose students to realistic process system applications and advanced control methods. One of the experiments is level control of a four-tank system. This paper describes two model-based methods students can implement for control of this interacting four-tank system. Sub-space identification is used to develop an empirical state space model of the experimental apparatus. This model is then used for model based control using internal model control (IMC). This represents an application of inner—outer factorization for non-minimum phase multivariable IMC design. Modeling is also performed using step tests and Aspen software for use with dynamic matrix control (DMC).

Identification of metabolic system parameters using global optimization methods

BackgroundThe problem of estimating the parameters of dynamic models of complex biological systems from time series data is becoming increasingly important.Methods and resultsParticular consideration is given to metabolic systems that are formulated as Generalized Mass Action (GMA) models. The estimation problem is posed as a global optimization task, for which novel techniques can be applied to determine the best set of parameter values given the measured responses of the biological system. The challenge is that this task is nonconvex. Nonetheless, deterministic optimization techniques can be used to find a global solution that best reconciles the model parameters and measurements. Specifically, the paper employs branch-and-bound principles to identify the best set of model parameters from observed time course data and illustrates this method with an existing model of the fermentation pathway in Saccharomyces cerevisiae. This is a relatively simple yet representative system with five dependent states and a total of 19 unknown parameters of which the values are to be determined.ConclusionThe efficacy of the branch-and-reduce algorithm is illustrated by the S. cerevisiae example. The method described in this paper is likely to be widely applicable in the dynamic modeling of metabolic networks.

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

Abstract.A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.

Model predictive control of a granulation system using soft output constraints and prioritized control objectives

Abstract A granulation system presented by Pottman et al. [J. Powder Technol., 108 (2) (2000) 192] is used to demonstrate two Model Predictive Control (MPC) control methods. The first method penalizes process output constraint violations using soft constraints in the objective function. It is found that the soft constraints must be much tighter than the actual constraints for effective control of the granulation system. The soft constraint formulation is presented as a variation of the asymmetric objective function formulation described by Parker et al. [Proc. American Control Conf. Chicago, IL (2000)]. The second control method is based on the prioritized objective formulation originally proposed by Tyler and Morari [Automatica 35 (1999) 565]. The prioritized objective method uses optimization constraints involving binary variables to explicitly represent and prioritize control objectives. The formulation presented in this article demonstrates a multi-level objective function which first maximizes the number of objectives satisfied in order of priority, then maximizes the number of total objectives, and finally minimizes the traditional MPC error tracking and move suppression terms. This prioritized objective formulation also allows for delayed implementation of output objective constraints, allowing for relaxation of control objectives.

Construction of Convex Relaxations Using Automated Code Generation Techniques

This paper describes how the automated code generation tool DAEPACK can be used to construct convex relaxations of codes implementing nonconvex functions. Modern deterministic global optimization algorithms involving continuous and/or integer variables often require such convex relaxations. Within the described framework, the user supplies a code implementing the objective and constraints of a nonconvex optimization problem. DAEPACK then analyzes this code and automatically generates a collection of subroutines based upon various symbolic transformations used by automatic convexification algorithms. The methods considered include the convex relaxations of McCormick, αBB of Floudas and coworkers, and the linearization strategy of Tawarmalani and Sahinidis. It should be noted that the user supplied code can be quite complex, including arbitrary nonlinear expressions, subroutines, and iterative loops.The code generation approach has the advantage that it can be applied to general, legacy models coded in programming languages such as FORTRAN. It also provides a generic symbolic transformation service for researchers interested in developing new global optimization algorithms. Numerical results are presented, including a study of how these techniques can be used to generate convex relaxations based on a hybridization of αBB and the method of McCormick.

Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis

Abstract An integrated fault detection, fault isolation, and parameter estimation technique is presented in this paper. Process model parameters are treated as disturbances that dynamically affect the process outputs. A moving horizon estimation technique minimizes the error between process and model measurements over a finite horizon by calculating model parameter values across the estimation horizon. To implement qualitative process knowledge, this minimization is constrained such that only a limited number of different faults (parameters) may change during a specific horizon window. Multiple linear models are used to capture nonlinear process characteristics such as asymmetric response, variable dynamics, and changing gains. Problems of solution multiplicity and computational time are addressed. Results from a nonlinear chemical reactor simulation are presented.

Multiple Model Approach for CSTR Control

Abstract A method for combining multiple local models to describe a nonlinear system. The local model weights are based on the linear interpolation of the current operating point from the closest local model operating points defined as the filtered value of the current process input. This local modeling method can be used to describe systems with changing system gains and dynamics, as well as input multiplicity. A case study for a benchmark CSTR reactor is presented. Local linear models are used to synthesize IMC based PID controllers.

Practical case studies for undergraduate process dynamics and control using process control modules

An effective environment for the incorporation of realistic case studies is introduced for a process dynamics and control course. Process Control Modules can be used in a computer laboratory under MATLAB 5.1 and Simulink 2.1 (Mathworks). Several industrial case studies are reviewed which exemplify the use of such a tool for control education.

Using Piecewise Polynomials to Model Open-Circuit Potential Data

Curve fitting is commonly used to determine a mathematical expression for sets of experimental data. While there are many instances when theory dictates a linear, quadratic, or other simple mathematical form for a relationship, there are often times when the theoretical form is either unknown or too unwieldy to be used effectively. A common example is the open-circuit-potential relationship of a Li-ion battery. One may use empirical correlations such as high-order polynomials (HOP) or various types of splines which are piecewise polynomials of low order. Despite their simplicity, HOPs are often undesirable given their numerical instability as well as their poor extrapolation performance. Certain splines generally produce acceptably smooth curves but use an unacceptably large number of fitting parameters. A method is presented for construction of continuous and smooth piecewise polynomials. Using significantly fewer polynomial segments than experimental data points, the number of parameters required to develop the calibration curves of a Li-ion battery is substantially reduced from that of splines while maintaining a similar level of fit quality. Mixed-integer programming techniques are employed to ensure that the knot (transition point) placement is optimal.

TYPE 1 DIABETIC PATIENT INSULIN DELIVERY USING ASYMMETRIC PI CONTROL

Type 1 diabetes is characterized by the destruction of the only insulin producing cells in the body. The typical course of action consists of daily insulin injections or an insulin pump. Assuming available methods for online monitoring of glucose concentrations, feedback control can be applied to this problem to improve regulation of glucose concentrations. A control algorithm is presented for feedback control of glucose levels in Type 1 patients. The control problem may be viewed as asymmetric, with negative variation from normal values treated with a more aggressive response than positive deviation. A simple asymmetric proportional-integral (PI) controller is presented where controller parameters vary depending on the sign of the current error value. Optimal closed-loop tuning parameters for the asymmetric control system are determined using local search methods. The asymmetric control system is then considered for robustness analysis using standard techniques from linear systems theory.