Nikolaos Kazantzis

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.

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.

Brief A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of energy dissipation of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubovs partial differential equation based on the systems drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.

Singular PDEs and the single-step formulation of feedback linearization with pole placement

The present work proposes a new formulation and approach to the problem of feedback linearization with pole placement. The problem under consideration is not treated within the context of geometric exact feedback linearization, where restrictive conditions arise from a two-step design method (transformation of the original nonlinear system into a linear one in controllable canonical form with an external reference input, and the subsequent employment of linear pole-placement techniques). In the present work, the problem is formulated in a single step, using tools from singular PDE theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived, by using Lyapunovs auxiliary theorem. The solution to the system of singular PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law computed through the solution of the system of singular PDEs, both feedback linearization and pole-placement design objectives may be accomplished in a single step, effectively overcoming the restrictions of the other approaches by bypassing the intermediate step of transforming the original system into a linear controllable one with an external reference input.

Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems

Abstract The present research work proposes a new approach to the problem of finding invariant manifolds for nonlinear real analytic dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear partial differential equations (PDEs) and a rather general set of conditions for solvability is derived using Lyapunovs auxiliary theorem. The solution of the aforementioned system of PDEs is proven to be a locally analytic invariant manifold that under certain conditions coincides with the stable or unstable manifold, and which can be easily computed with the aid of a symbolic software package.

Synthesis of state feedback regulators for nonlinear processes

Abstract The present work proposes a new approach to the state feedback regulator synthesis problem for multiple-input nonlinear processes. The problem under consideration is not treated within the context of exact feedback linearization, where restrictive conditions arise, but is conveniently formulated in the context of singular partial differential equations (PDE) theory. In particular, the mathematical formulation of the problem is realized via a system of first-order quasi-linear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of singular PDEs 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 such as MAPLE. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback control law that is computed through the solution of the above system of singular PDEs, both feedback linearization and pole-placement design objectives can be accomplished in a single step. Finally, the proposed nonlinear state feedback regulator synthesis method is applied to a continuous stirred tank reactor (CSTR) in non-isothermal operation that exhibits steady-state multiplicity. The control objective is to regulate the reactor at the middle unstable steady state by manipulating the dilution rate. Simulation studies have been conducted to evaluate the performance of the proposed nonlinear state feedback regulator, as well as to illustrate the main design aspects of the proposed approach. It is shown that the nonlinear state feedback regulator clearly outperforms the standard linear one, especially in the presence of adverse conditions under which linear regulation at the unstable steady state is not always feasible.

Energy-predictive control: a new synthesis approach for nonlinear process control

The present work proposes a new controller synthesis framework, that is based on the methodological principles of Lyapunov design. A notion of short-horizon energy-predictive control is introduced, that predicts and optimizes the rate of energy dissipation of the system and leads to the derivation of a continuous state feedback control law. The control law possesses two adjustable parameters that directly influence the size of the closed-loop stability region and the performance characteristics. The derived state feedback law may be combined with an open-loop state observer, leading to a dynamic output feedback controller with integral action. Finally, the performance of the proposed controller design methodology is evaluated in a chemical reactor example, that exhibits nonminimum-phase characteristics.

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

Abstract The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problems formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.

Nonlinear observer design using Lyapunov's auxiliary theorem

The work proposes an approach to the nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer theory, the proposed approach develops a nonlinear analogue. The formulation of the observer design problem is realized via a system of first-order linear singular PDEs, and a rather general set of necessary and sufficient conditions for solvability is derived by using Lyapunovs auxiliary theorem. The solution to the above system of PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package. Within the proposed design framework, both full-order and reduced-order observers are studied.

Design of discrete-time nonlinear observers

Proposes an 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 functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived using results from linear functional equation theory. The solution to the above system of linear 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.