A. Desages

High-level canonical piecewise linear representation using a simplicial partition

In this work, we propose a set of high-level canonical piecewise linear (HL-CPWL) functions to form a representation basis for the set of piecewise linear functions f: D/spl rarr/R/sup 1/ defined over a simplicial partition of a rectangular compact set D in R/sup n/. In consequence, the representation proposed uses the minimum number of parameters. The basis functions are obtained recursively by multiple compositions of a unique generating function /spl gamma/, resulting in several types of nested absolute-value functions. It is shown that the representation in a domain in R/sup n/ requires functions up to nesting level n. As a consequence of the choice of the basis functions, an efficient numerical method for the resolution of the parameters of the high-level (HL) canonical representation results. Finally, an application to the approximation of continuous functions is shown.

Orthonormal high-level canonical PWL functions with applications to model reduction

An inner product is defined for the linear vector space PWL/sub H/[S] of all the piecewise linear (PWL) continuous mappings defined over a rectangular compact set S, using a simplicial partition H. This permits us to assure that PWL/sub H/[S] is a Hilbert space. Then, the notion of orthogonality is introduced and orthonormal bases of PWL functions are formulated. A relevant consequence of the approach is that the problem of function approximation can be translated to the more studied field of approximation in Hilbert spaces of finite dimension. As will be shown, this powerful theoretical framework can be used to generate an optimal scheme for model reduction.

Canonical piecewise-linear approximation of smooth functions

This paper deals with the approximation of smooth functions using canonical piecewise-linear functions. The developing of tools in the field of analysis and control of nonlinear systems based on this kind of functions, as well as its efficiency in the representation of electronic devices, motivates the development of useful methods to obtain accurate approximations. A recursive method is proposed to obtain simultaneously all the parameters required and its convergence is studied. In addition, an iterative method to introduce new partitions on the domain, when the error obtained is not satisfactory, is described. This method takes advantage of the partitions already found to reduce the total number of parameters that the algorithm has to handle.

Robust characteristic polynomial assignment

In this paper a computational method for designing controllers which attempt to place the characteristic polynomial of an uncertain system inside some prescribed region is presented. An objective function consisting on two terms is proposed, penalizing both the distance to a given controller and the size of the uncertainty region developed to solve the robust assignment problem. The algorithm is based on semi-infinite programming theory, and the special structure of the problem is exploited to get both simpler methods and convergence proof. The way in which the uncertain parameters affect the plant coefficients is realistic, since it includes multinomial dependence. An example of application is given to illustrate the approach.

A parametrization of piecewise linear Lyapunov functions via linear programming

In this paper, we present a parametrization of piecewise linear (PWL) Lyapunov functions. To this end, we consider the class of all continuous PWL functions defined over a simplicial partition. We take advantage of a recently developed high level canonical PWL (HL CPWL) representation, which expresses the PWL function in a compact and closed form. Once the parametrization problem is properly stated, we focus on its application to the stabiilty analysis of dynamic systems. We consider uncertain non-linear systems and extend the sector condition obtained by Ohta et al. In addition, we propose a method of selecting an optimal candidate. One of the main advantages of this approach is that the parametrization and choice of the Lyapunov candidate, as well as the stability analysis, result in linear programming problems.

BIFURCATIONS AND HOPF DEGENERACIES IN NONLINEAR FEEDBACK SYSTEMS WITH TIME DELAY

An application of the well-developed frequency-domain approach to detect oscillations in nonlinear feedback systems with time delay is presented. The method depends on an early proof of the Hopf bifurcation theorem known as the Graphical Hopf Theorem (GHT). Several nondegeneracy conditions are included to apply the GHT in nonlinear systems with time delay. The singular conditions corresponding to degeneracies, which include static and dynamic bifurcations, as well as some special cases of degenerate Hopf bifurcations and multiple crossings, are also discussed. Two Single-Input Single-Output (SISO) feedback systems with odd nonlinearities are presented as examples to show that the proposed technique and a standard simulation method have very good agreement in the results, yet the GHT is much simpler in calculation. The first one shows an application of the GHT under classical Hopf conditions while the second emphasizes the presence of degenerate Hopf bifurcations and multiple crossings. For both examples, and others which have appeared recently in the literature, a considerable simplification of the formulas for recovering periodic solutions is also provided in this paper.

Paper: Variable structure control with a second-order sliding condition: Application to a steam generator

A method for the design of second-order sliding mode controllers is developed for a class of nonlinear systems. The key idea is the necessity of nullifying both the auxiliary output as well as its time derivative when the system is sliding. It is found that if the auxiliary output has a strong vector relative degree one, it is possible to design a continuous control input. This allows the application of the sliding mode control to systems that do not allow discontinuous control signals generated by the classical variable structure control. Robustness of the controllers is analyzed and dynamic bounds for the uncertainty are found for a class of nonlinear systems. The design technique could be extended to nth-order sliding controllers, though the complexity of the calculations increases and the robustness bounds become more restrictive. The application of the technique to a drum-type steam generating unit to solve a robust tracking problem is presented.

Degenerate Hopf bifurcations via feedback system theory: Higher-order harmonic balance

Abstract In this work we present some higher-order Hopf bifurcation formulas to analyze the rich dynamic behavior of chemical systems under the failure of some hypotheses of the classical Hopf theorem. The treatment is done entirely in an equivalent formulation in the so-called “frequency domain”. This approach gives a natural interpretation about some Hopf degeneracies involving limit points on the periodic branch and multiple periodic solutions. Moreover, we calculate the expression for the second curvature coefficient, extending previous results using this formulation. This coefficient is equivalent to one of the higher-order coefficients in the Hopf bifurcation normal form used previously by other researchers.

A multivariable delay compensator scheme

Abstract A simple multivariable predictor scheme is developed based on a previous methodology presented by the authors. In the proposed strategy the delays of compensation may be used as design parameters. The band of negligible unstable contribution (BNUC) of a complex function with respect to any proper or strictly signal is properly defined. The BNUC measure is used to develop an algorithm to evaluate the delays of compensation, which allows to achieve a given bandwidth for the closed-loop system. The stability of the nominal and perturbed feedback systems is discussed. Finally, several control examples show the utility of the proposed technique.

Robust controller design methodology for multivariable chemical processes

Abstract In this work, a methodology for a controller design of multivariable linear time invariant systems is presented. The singular values of the return difference operator and the time constant for each control loop can be easily specified through a set of rational functions in the Laplace domain. The frequency response of an ideal controller is directly obtained, allowing the implementation of different controller structures (PI, PID etc.). First, the discussion is focused on some new results of the perturbation theory which constitute the basis of the proposed methodology. Later, a complete analysis of the resulting control scheme is presented. Its robustness with respect to stability, performance and interactions as well as the normality of the closed-loop system is demonstrated. Some implementation aspects for the formulation of the ideal controller matrix are discussed. Approximation of this ideal controller to different realizable structures is then discussed. Finally, a case study of the dual-composition control of two typical distillation columns currently found in the literature is presented.