Costas Kravaris

Identification of Parameters in Distributed Parameter Systems by Regularization

Identification of spatially varying parameters in distributed parameter systems from noisy data is an ill-posed problem. The concept of regularization, widely used in solving linear Fredholm integral equations, is developed for the identification of parameters in distributed parameter systems. A general regularization identification theory is first presented and then applied to a parabolic identification problem. Methods for the numerical implementation of the regularization identification approach are also presented.

From Continuous-Time Design to Sampled-Data Design of Observers

In this work, a sampled-data nonlinear observer is designed using a continuous-time design coupled with an inter-sample output predictor. The proposed sampled-data observer is a hybrid system. It is shown that under certain conditions, the robustness properties of the continuous-time design are inherited by the sampled-data design, as long as the sampling period is not too large. The approach is applied to linear systems and to triangular globally Lipschitz systems.

Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems

The present work addresses the problem of synthesizing nonlinear state feedback controllers for second-order nonminimum-phase nonlinear systems. The concept of a first-order nonlinear all-pass is first introduced. A class of static state feedback control laws is then developed that makes the closed-loop system equivalent, under an appropriate coordinate transformation, to a nonlinear first-order all-pass in series with a linear first-order lag. A particular control law from this class is calculated that results in I.%!?-optimal response. The performance of the proposed methodology in set point tracking is evaluated through numerical simulations in a CSTR example. Major contributions in the area of linear process control in the last decade have established the idea that a controller must explicitly or implicitly generate a process inverse (Garcia and Morari, 1982). When dealing with minimum-phase linear systems, such an inverse is stable and can be used for controller synthesis. When dealing with nonminimum-phase linear systems, an appropriate decomposition of the process model into a part with stable inverse and a part with unstable inverse is necessary and the con- troller must invert only the part with stable inverse. In the context of linear state feedback, the same idea arises in placing closed-loop poles at the left-half plane zeros and at the reflection of the right-half plane zeros. Such a control strategy has been shown to be ZSE-optimal for step changes. In the field of nonlinear process control, the idea of a controller that generates a process inverse is central in general synthesis methods for minimum-phase sys- tems, like the nonlinear IMC structure (Economou et al., 1986; Parrish and Brosilow, 1988), which explic- itly generates a process inverse on-line and the input/ output linearization method (Kravaris and Chung, 1987) which implicitly generates a process inverse. The latter has been shown to lead to ZSE-optimal responses for step changes (Kravaris, 1988). How- ever, the control of nonminimum-phase nonlinear systems in this vein remains a major unmet challenge. In this work, a control law for second-order non- minimum-phase nonlinear systems will be developed, that leads to an ZSE-optimal closed-loop response for changes in the set point. The proposed methodology hopes to serve as a starting point for the development of a more general framework for the control of nonminimum-phase nonlinear systems and to moti- vate further research effort in this area. In Section 2 the characterization of nonminimum- phase behavior for second-order systems will be reviewed following the approach of Bymes and Isidori (1985). In Section 3, a nonlinear analog of the linear first-order all-pass will be introduced. Section 4 will review a standard result for ZSE-optimal state feedback control of linear second-order nonmini- mum-phase systems and will motivate the develop- ment that follows. In Section 5, a class of control laws will be developed that lead to a closed-loop response of a nonlinear first-order all-pass in series with a linear first-order lag. In Section 6, a particular control law from this class will be calculated, that results in ZSE-optimal closed-loop response in the limit as the time constant of the linear lag tends to zero. Finally, Section 7 will illustrate the application of the pro- posed control methodology and evaluate its perform- ance in a CSTR example.

Time-discretization of nonlinear control systems via Taylor methods

Abstract A new discretization method for the calculation of a sampled-data representation of a nonlinear continuous-time system is proposed. It is based upon the well-known Taylor method and the zero-order hold (ZOH) assumption. The mathematical structure of the new discretization scheme is analyzed and characterized as being particularly useful in establishing concrete connections between numerical properties and system-theoretic properties. In particular, the effect of the Taylor discretization procedure on key properties of nonlinear systems, such as equilibrium properties and asymptotic stability, is examined. Within a control context, numerical aspects of Taylor discretization are also discussed, and ‘hybrid’ discretization schemes, that result from a combination of the ‘scaling and squaring’ technique with the Taylor method, are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method’s parameters to meet CPU time and accuracy requirements, are examined as well. Finally, the performance of the proposed discretization procedure is evaluated in a chemical reactor example, that exhibits nonlinear behavior and is subject to various sampling rates.

Synthesis of state feedback laws for end-point optimization in batch processes

nature of the feedback law (static or dynamic) are completely characterized in terms of the Lie bracket structure of the system dynamics. Explicit synthesis formulae for the state feedback laws are first obtained for time-invariant systems and then extended to time-varying systems. As illustrative examples of application of the proposed methodology, we consider several end-point optimization problems in batch chemical and biochemical reactors. INTRCJDUCTJON Batch and semi-batch processes are of great importance to the chemical industry. A wide variety of specialty chemicals such as antibiotics and polymers are produced in batch reactors; they are preferred due to their ease and flexibility of operation. Batch reactors are used when there are many processing steps in the chemical process, when isolation is required for reasons of sterility or safety and when the materials involved are hard to handle.

Advances and selected recent developments in state and parameter estimation

Abstract This paper deals with two topics from state and parameter estimation. The first contribution of this work provides an overview of techniques used for determining which parameters of a model should be estimated. This is a question that commonly arises when fundamental models are used as these models often contain more parameters than can be reliably estimated from data. The decision of which parameters to estimate is independent of the observer/estimator design, however, it is directly affected by the structure of the model as well as the available data. The second contribution is an overview of recent developments regarding the design of nonlinear Luenberger observers, with special emphasis on exact error linearization techniques, but also discussing more general issues, including observer discretization, sampled data observers and the use of delayed measurements.

Structural evaluation of control configurations for multivariable nonlinear processes

Abstract This paper addresses the problem of evaluation of alternative control configurations on the basis of structural characteristics of the process. Relative order is proposed as the main analysis tool for this purpose. Using tools from graph theory, it is shown that generic calculation of relative orders requires only structural information about the process. Relative order is interpreted as a structural measure of the initial sluggishness of the response, as well as a structural analog of dead time, which expresses fundamental structural limitations in the control quality. A matrix of relative orders of input/output pairs is introduced, which leads to a characterization of structural coupling among input and output process variables. On the basis of the above properties, general structural evaluation guidelines are proposed for alternative sets of manipulated inputs and alternative input/output pairs. The application of the theory is illustrated in the case of an evaporation unit, a chemical reactor and a network of heat exchangers.

Dynamic output feedback control of nimimum-phase nonlinear processes

Abstract This paper concerns the synthesis of dynamic output feedback controllers for minimum-phase nonlinear processes. The problem is addressed first for open-loop stable and then for general minimum-phase nonlinear processes, leading to one- and two-degree-of-freedom controllers, respectively. The synthesis of the controllers essentially involves combination of state feedback and state observers. An input/output interpretation of the resulting control structures illustrates the importance of alternative state—space realizations of the process inverse for the controller implementation. Internal stability conditions are derived for the closed-loop system. Simulation studies in a chemical reactor example illustrate the application of the control methodology developed.

System-theoretic properties of sampled-data representations of nonlinear systems obtained via Taylor-Lie series

It is possible to obtain a sampled-data representation of a nonlinear dynamic system under zero-order hold using Taylor-Lie series. Within the context of Taylor-Lie series theory, explicit formulae for the resulting discrete-time system are derived in terms of the system Lie derivatives. The effect of sampling on systemtheoretic properties, such as equilibrium properties, relative order, stability, zero dynamics and minimum-phase characteristics, is examined, revealing the natural and transparent way in which the use of Taylor-Lie series permeates the relevant theoretical aspects. The results derived are also illustrated in two chemical engineering examples.

Discrete-time nonlinear observer design using functional equations

Abstract The present research work proposes a new approach to the discrete-time nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer, the proposed approach develops a nonlinear analogue. The formulation of the discrete-time nonlinear observer design problem is realized via a system of first-order linear nonhomogeneous functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived using results from functional equations theory. The solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.