MP
M.G. Potters
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Estimation of physical parameters in dynamical systems driven by linear partial differential equations is an important problem. In this paper, we introduce the least costly experiment design framework for these systems. It enables parameter estimation with an accuracy that is specified by the experimenter prior to the identification experiment, while at the same time minimising the cost of the experiment. We show how to adapt the classical framework for these systems and take into account scaling and stability issues. We also introduce a progressive subdivision algorithm that further generalises the experiment design framework in the sense that it returns the lowest cost by finding the optimal input signal, and optimal sensor and actuator locations. Our methodology is then applied to a relevant problem in heat transfer studies: estimation of conductivity and diffusivity parameters in front-face experiments. We find good correspondence between numerical and theoretical results.
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Estimation of physical parameters in dynamical systems driven by linear partial differential equations is an important problem. In this paper, we introduce the least costly experiment design framework for these systems. It enables parameter estimation with an accuracy that is specified by the experimenter prior to the identification experiment, while at the same time minimising the cost of the experiment. We show how to adapt the classical framework for these systems and take into account scaling and stability issues. We also introduce a progressive subdivision algorithm that further generalises the experiment design framework in the sense that it returns the lowest cost by finding the optimal input signal, and optimal sensor and actuator locations. Our methodology is then applied to a relevant problem in heat transfer studies: estimation of conductivity and diffusivity parameters in front-face experiments. We find good correspondence between numerical and theoretical results.
We consider the input design problem of finding the minimal required experiment time such that accuracy constraints on the parameter estimate of an identification experiment are satisfied, while also respecting signal amplitude bounds. The input signal is parameterized as a multi-sine. We first show how multiple linear matrix inequalities from the least-costly and applications-oriented experiment design frameworks can be transformed into a generalised E-optimality constraint. Then, the solution to our problem is found by: (i) designing a multi-sine of one period with the Guillaume-Manchester algorithm [12], [10] that minimises the generalised E-optimality criterion under signal amplitude bounds, and (ii) utilising periodicity and an optimality condition to scale the experiment time such that the imposed accuracy constraints are also respected. An example shows an experiment time reduction of 50% compared with a traditional least-costly experiment design approach.
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We consider the input design problem of finding the minimal required experiment time such that accuracy constraints on the parameter estimate of an identification experiment are satisfied, while also respecting signal amplitude bounds. The input signal is parameterized as a multi-sine. We first show how multiple linear matrix inequalities from the least-costly and applications-oriented experiment design frameworks can be transformed into a generalised E-optimality constraint. Then, the solution to our problem is found by: (i) designing a multi-sine of one period with the Guillaume-Manchester algorithm [12], [10] that minimises the generalised E-optimality criterion under signal amplitude bounds, and (ii) utilising periodicity and an optimality condition to scale the experiment time such that the imposed accuracy constraints are also respected. An example shows an experiment time reduction of 50% compared with a traditional least-costly experiment design approach.
Control configurations in distillation columns
A comparative study
Model-plant mismatch in model predictive controlled industrial processes is a phenomenon that may lead to relevant performance losses. Research in finding a suitable procedure to effectively detect and revert model plant mismatch often uses distillation columns as benchmark testing models. However, it was found that sub optimal distillation columns and control configurations were used in these studies. This paper aims to provide answers for the issues as just discussed. At first a benchmark distillation column is designed using the work of Sigurd Skogestad. Secondly, two common control configurations are compared, in which it is shown that in high purity regions the a more accurate model can be identified in the (L/D)(V/B)-configuration. Further, in the case of disturbances on the feed rate and feed composition channels of the distillation column, it is shown that the (L/D)(V/B)-configuration out performs the LV-configuration in terms of disturbance rejection and handling model-plant mismatch.
...
Model-plant mismatch in model predictive controlled industrial processes is a phenomenon that may lead to relevant performance losses. Research in finding a suitable procedure to effectively detect and revert model plant mismatch often uses distillation columns as benchmark testing models. However, it was found that sub optimal distillation columns and control configurations were used in these studies. This paper aims to provide answers for the issues as just discussed. At first a benchmark distillation column is designed using the work of Sigurd Skogestad. Secondly, two common control configurations are compared, in which it is shown that in high purity regions the a more accurate model can be identified in the (L/D)(V/B)-configuration. Further, in the case of disturbances on the feed rate and feed composition channels of the distillation column, it is shown that the (L/D)(V/B)-configuration out performs the LV-configuration in terms of disturbance rejection and handling model-plant mismatch.
Experiment Design for system identification involves the design of an optimal input signal with the purpose of accurately estimating unknown parameters in a system. Specifically, in the Least-Costly Experiment Design (LCED) framework, the optimal input signal results from an optimisation problem in which a weighted input power (the cost) is minimised subject to parameter accuracy constraints. In this particular formulation, the problem is convex and can be solved with efficient numerical tools. The LCED framework, however, has the following limitations: (i) no interpretation follows from its numerical solutions, (ii) it can not be applied to systems with unknown or nonlinear controllers, (iii) it cannot be applied in full generality to structured (physical) systems, and (iv) the problem formulation has so far mainly considered input power as the cost, whereas other possibilities exist. In this thesis these four limitations are addressed.
Firstly, we calculate analytical solutions for a class of LCED problems for models with one or two parameters. For uni-parametric models we have proven that the solution is always a single sinusoid, whereas for bi-parametric models we have provided arguments that a single sinusoid is often the solution. From our theoretical analysis we also, at a formal level, classify the LCED problems as generalised and weighted dual E-optimality problems.
Secondly, we introduce Stealth and Sensitivity methods that enable the applicability of the LCED framework to structured and unstructured systems regulated by unknown or nonlinear controllers. The requirement of an explicit and known expression of the sensitivity function, necessary to solve the LCED problems, is circumvented with the above two novel methods. The Stealth method adapts the classical Direct Identification scheme such that the controller does not sense the excitation signal, reducing the sensitivity function to unity. The Sensitivity method, instead, relies on the usual Direct Identification scheme and finds an approximation of the sensitivity function. Three numerical studies show the strength of both methods.
Thirdly, we generalise the LCED framework such that it can be applied to structured systems governed by linear partial differential equations with constant coefficients. We use a systematic approach to simulate such systems using harmonic signals, which in turn are designed by the LCED framework. Issues such as stability and scaling will be formally addressed. Since structured systems are concomitant with degrees of freedom in the experiment set-up, we also develop a progressive subdivision algorithm that can efficiently solve the corresponding LCED problems.
Fourthly, we introduce the novel Minimum Experiment Time (MET) algorithm that, by designing an optimal harmonic input signal, solves an optimisation problem formulated by Ebadat et al. (2014b): minimise the experiment time subject to parameter accuracy constraints and amplitude bounds on the input and output signal. The MET algorithm is applicable to systems that are in open or closed loop, and is relevant for many industrial processes. It can also deal with multiple accuracy constraints, in contrast to traditional methods. We show with several examples that optimal experiment times can be achieved that are up to 50% shorter compared with the solutions of the classical LCED framework.
Finally, using the above methods, we address an important problem in petrophysics: the estimation of permeability and porosity values of a porous rock sample using Pressure Oscillation experiments. We show how to design the optimal input spectrum and inlet and outlet volumes of the experiment set-up such that the experiment time is minimised, while respecting parameter accuracy and actuator constraints. Furthermore, we design such signals for the Direct and Indirect identification methods. We show that identifiability issues can arise with the former method. The latter method has no such issues. Consequently, the Indirect method delivers optimal experiment times that are a factor fourteen shorter compared with those of the Direct Method.
...
Experiment Design for system identification involves the design of an optimal input signal with the purpose of accurately estimating unknown parameters in a system. Specifically, in the Least-Costly Experiment Design (LCED) framework, the optimal input signal results from an optimisation problem in which a weighted input power (the cost) is minimised subject to parameter accuracy constraints. In this particular formulation, the problem is convex and can be solved with efficient numerical tools. The LCED framework, however, has the following limitations: (i) no interpretation follows from its numerical solutions, (ii) it can not be applied to systems with unknown or nonlinear controllers, (iii) it cannot be applied in full generality to structured (physical) systems, and (iv) the problem formulation has so far mainly considered input power as the cost, whereas other possibilities exist. In this thesis these four limitations are addressed.
Firstly, we calculate analytical solutions for a class of LCED problems for models with one or two parameters. For uni-parametric models we have proven that the solution is always a single sinusoid, whereas for bi-parametric models we have provided arguments that a single sinusoid is often the solution. From our theoretical analysis we also, at a formal level, classify the LCED problems as generalised and weighted dual E-optimality problems.
Secondly, we introduce Stealth and Sensitivity methods that enable the applicability of the LCED framework to structured and unstructured systems regulated by unknown or nonlinear controllers. The requirement of an explicit and known expression of the sensitivity function, necessary to solve the LCED problems, is circumvented with the above two novel methods. The Stealth method adapts the classical Direct Identification scheme such that the controller does not sense the excitation signal, reducing the sensitivity function to unity. The Sensitivity method, instead, relies on the usual Direct Identification scheme and finds an approximation of the sensitivity function. Three numerical studies show the strength of both methods.
Thirdly, we generalise the LCED framework such that it can be applied to structured systems governed by linear partial differential equations with constant coefficients. We use a systematic approach to simulate such systems using harmonic signals, which in turn are designed by the LCED framework. Issues such as stability and scaling will be formally addressed. Since structured systems are concomitant with degrees of freedom in the experiment set-up, we also develop a progressive subdivision algorithm that can efficiently solve the corresponding LCED problems.
Fourthly, we introduce the novel Minimum Experiment Time (MET) algorithm that, by designing an optimal harmonic input signal, solves an optimisation problem formulated by Ebadat et al. (2014b): minimise the experiment time subject to parameter accuracy constraints and amplitude bounds on the input and output signal. The MET algorithm is applicable to systems that are in open or closed loop, and is relevant for many industrial processes. It can also deal with multiple accuracy constraints, in contrast to traditional methods. We show with several examples that optimal experiment times can be achieved that are up to 50% shorter compared with the solutions of the classical LCED framework.
Finally, using the above methods, we address an important problem in petrophysics: the estimation of permeability and porosity values of a porous rock sample using Pressure Oscillation experiments. We show how to design the optimal input spectrum and inlet and outlet volumes of the experiment set-up such that the experiment time is minimised, while respecting parameter accuracy and actuator constraints. Furthermore, we design such signals for the Direct and Indirect identification methods. We show that identifiability issues can arise with the former method. The latter method has no such issues. Consequently, the Indirect method delivers optimal experiment times that are a factor fourteen shorter compared with those of the Direct Method.