Differential Algebraic Approach Research Articles

Dynamical sliding mode control strategies in the regulation of nonlinear chemical processes

In this article, the use of variable structure feedback control strategies is proposed for the asymptotic stabilization of nonlinear dynamic systems describing chemical processes. The proposed discontinuous controller is dynamic in nature and it effectively eliminates the traditional bang-bang nature of the control input signals and the associated chattering responses. The controller is based on recent results of the differential algebraic approach to system dynamics; in particular Fliess's generalized observability canonical form is used in the derivation of the dynamical discontinuous controller. Some illustrative examples, including simulations, are provided.

Dynamical Discontinuous Feedback Control of Non-Linear Systems

Abstract Discontinuous feedback stabilization of nonlinear systems, expressed in Generalized State representation form, is accomplished by zeroing of a suitable input-dependent manifold. Pulse-Frequency-Modulation, Pulse Width-Modulation and Sampled Sliding Mode Control strategies are treated from a unified viewpoint which naturally arises from fundamental results of the Differential Algebraic approach to systems dynamics and control. The approach naturally leads to dynamical discontinuous feedback policies resulting in (chattering-free) smoothed constrained linearization and induced robust asymptotic output error stabilization. The results are applicable in a variety of nonlinear control problems, including stabilization, tracking, and model matching. Illustrative examples from non-traditional application areas, such as chemical process control and hydraulic systems control, are presented with simulations.

A Differential Algebraic Approach to the Classical Right Model Matching problem

The right model matching problem is considered in the differential algebraic setting. Simple rank conditions for the solvability of this problem are given. These are necessary and sufficient in the case of non degenerated compensation. Tangent linearized systems are shown to be most useful in this context: The problem is shown to be solvable if, and only if, it is solvable for the tangent linearized systems. The problem of existence of real compensators for real systems is considered and the conditions of the non degenerated compensator case are shown to be valid in the real case too.

The differential algebraic approach in nonlinear dynamical feedback controlled landing maneuvers

A differential algebraic approach is proposed for the synthesis of a dynamical feedback controller regulating a spacecraft smooth descent toward the surface of a planet which exhibits nonnegligible atmospheric resistance. An exact linearization-based controller is synthesized using Fliess' generalized observability canonical form of the controller system. The smooth controlled trajectory is regulated by means of assumed amplitude modulated thrusting capabilities of the spacecraft. The robustness of the regulator is tested in the presence of significant unmodelled spatial changes in the coefficient of atmospheric resistance. A simulation example is provided. >

Asymptotic output stabilization for nonlinear systems via dynamical variable-structure feedback control

In this article, the problem of asymptotic output stabilization in nonlinear controlled systems is approached from the perspective of dynamical sliding-mode control. The proposed controller is based on Fliess's Generalized Observability Canonical Form, recently derived from the differential algebraic approach to system dynamics.

Differential algebraic approach in non-linear dynamical compensator design for d.c.-to-d.c. power converters

Abstract A non-linear dynamical compensator design, based on Fliess’ generalized controller canonical form for non-linear dynamical systems, is proposed for the asymptotic stabilization of output load voltage, or current, in pulse-width-modulation controlled d.c.-to-d.c. power supplies. The technique rests on the recently introduced differential algebraic approach to the study of controlled dynamical systems.

Computational aspects of optics design and simulation: COSY INFINITY

Abstract The new differential algebraic (DA) techniques allow very efficient treatment and understanding of nonlinear motion in optical systems as well as circular accelerators. To utilize these techniques in their most general way, a powerful software environment is essential. A language with structure elements similar to Pascal was developed. It has object oriented features to allow for a direct utilization of the elementary operations of the DA package. The compiler of the language is written in Fortran 77 to guarantee wide portability. The language was used to write a very general beam optics code, COSY INFINITY. At its lowest level, it allows the computation of the maps of standard beam line elements including fringe fields and system parameters to arbitrary order. The power of the DA approach coupled with an adequate language environment reveals itself in the very limited length of COSY INFINITY of only a few hundred lines. Grouping of elements as well as structures for optimization and study are readily available through the features of the language. Because of the openness of the approach, it offers a lot of power for more advanced purposes. For example, it is very easy to construct new particle optical elements. There are also many ways to efficiently manipulate and analyze the maps.

Arbitrary order description of arbitrary particle optical systems

Abstract The differential algebraic approach for the design and analysis of particle optical systems and accelerators is presented. It allows the computation of transfer maps to arbitrary orders for arbitrary arrangements of electromagnetic fields, including the dependence on system parameters. The resulting maps can be cast into different forms. In the case of a Hamiltonian system, they can be used to determine the generating function or Eikonal representation. Also various factored Lie operator representations can be determined directly. These representations for Hamiltonian systems cannot be determined with any other method beyond relatively low orders. In the case of repetitive systems, a combination of the power series representation and the Lie operator representation allows a nonlinear change of variables such that the motion is very simple and its long term behaviour can be studied very efficiently. Furthermore, it is now possible to compute quantities relevant to the study of circular machines like tune shifts and chromaticities much more efficiently. Besides these aspects, the ability to compute maps depending on parameters provides analytical insight into the system. In addition, this approach allows very efficient optimization, to the extent that in many cases it is almost completely analytic.

Prime Differential Ideals in Nonlinear Rational Control Systems

The paper gives a direct proof of the fact that the differential ideal generated by the explicit state space equations {x.=A(x)+B(x)uy=C(x) is prime when the entries of A(x), B(x) and C(x) are rational functions of x. This differential ideal yields the definition of some differential fields which embody the so called differential algebraic approach for nonlinear system analysis. The work which is reponed here is introductory for a general result valid for a broader class of systems, e.g. the class of analytic or meromorphic systems.

Rank Invariants of Nonlinear Systems

A linear algebraic framework for the analysis of rank properties of nonlinear systems is introduced. This framework gives a high-level interpretation of several existing algorithms built around the recursive computation of certain algebraic ranks associated with right-invertibility, left-invertibility, and dynamic decoupling. Furthermore, it can be used to establish links between these algorithms and the differential algebraic approach, as well as to solve some static and dynamic noninteracting control problems.

A differential algebraic approach to mechanics with perfect nonholonomic constraints

Differential algebra permits the generalization of several features of symplectic formalism to mechanics with perfect nonholonomic constraints.