On Nonlinear Control Systems: State Observation And Orbital Stabilization

The dissertation is devoted to two more-practically and theoretically-challenging constructive problems for nonlinear control systems,i.e.,state observers design and orbital stabilization,which receive a great deal of attention from both academia and industry particularly for some modern control tasks.Many high-performance controller design techniques for nonlinear systems rely on the availability of the systems state.On the other hand,in many practical applications,the installation of sensors is stymied by cost and technological considerations.Observer design can then be invoked to reconstruct the state via input and output measurements.Although great progress in the systematic development of observers for linear systems has been made ever since Luenberger's cornerstone work,there are still numerous unsolved challenges in the nonlinear counterpart,which are usually related to some practically important engineering problems.Motivated by the fact,we propose a new state observer design technique and a unifying framework for general nonlinear systems,called [KKL+PEB] observer.It is clear,however,that the system must satisfy some observability properties for the success of any kind of observer design techniques.Another challenge of observers design,hence,lies in being to guarantee observability.In the case when the latter is not satisfied,it is still possible to extract additional information from the system via probing signal injection-that is a well-known technique widely used in several applications.We then pursue the virtual output observation,via signal injection,from both regression and frequency domain perspectives,showing how the new filtering technique can be applied to estimate the electrical coordinates of a general class of electromechanical systems assuming that only the current and voltage are measurable.The other topic of the dissertation is concerned with the orbital stabilization of nonlinear systems.In many practical tasks the system under control is required to operate along periodic motions.We show that a slight modification to the widely popular interconnection and damping assignment passivity-based control method allows us to provide a solution to the more challenging orbital stabilization problem.Two different,though related,ways how this procedure can be applied are proposed.The main contributions of the dissertation are threefold.1.The dissertation proposes the second-order averaging analysis of the signal injection method to multi-input-multi-output systems,thus proposing a new filtering technique to identify high-frequency components of the output induced by signal injection,which is applicable to general nonlinear systems.The filter design technique proposed in the dissertation relies on the following two key observations.First,that the task of reconstructing the virtual output can be recast as a problem of estimation of(slowly time-varying)parameters in a linear regression model.Second,the observation that the particular form of the regressor can be exploited to apply the dynamic regressor extension and mixing estimator with some suitable operators that guarantee the excitation condition needed for exponential parameter convergence.Two significant advantages of the proposed filter are,on one hand,that the exponential stability property makes the filter robust to unavoidable measurement noise.On the other hand,since it relies on the use of linear time-varying filters,it has very simple practical implementation.The dissertation shows how the new filtering technique can be applied to estimate the electrical coordinates of a general class of electromechanical systems for sensorless control.We then provide a theoretical framework for the analysis to the conventional linear time invariant filtering from the frequency domain perspective,which is indeed the de facto standard in industrial practice.It can be proved that the new design admits a high-pass filtering and low-pass filtering interpretation.This is an important issue,since it shows the connection-and downwards-compatibility-of the new method with standard industrial practice.The above-mentioned results are experimentally verified on the interior permanent magnet synchronous motors and the 1-dof magnetic levitation system.2.The dissertation proposes a new observer design technique,called [KKL+PEB] observer that combines-in a seamless way-the Kazantzis-Kravaris-Luenberger(KKL)observer and the parameter estimation-based(PEB)observer designs,yielding a new design applicable to a broader class of systems.It follows by the proof that KKL,PEB and[KKL+PEB] observers can be recast as particular cases of generalized immersion and invariance observers-providing in this way a unified framework for their analysis and design.We then present simulation results of the well-known DC-DC (?)uk converter,which illustrate the superior performance of the proposed observer compared to other existing observer designs.Another contribution of this part is to propose a technique to solve the partial differential equation(PDE)using mappings pseudo-inverses and Poincaré Lemma.This new procedure,which translates the problem of solution of the PDE into the solvability of a set of(nonlinear)algebraic equations and an,easily verifiable,integrability condition-both steps some free mappings-is more systematic and provides additional degrees of freedom to solve the problem.3.The last aim of the dissertation is to show that the widely popular interconnection and damping assignment passivity based control(IDA-PBC),originally proposed for stabilization of equilibria,can be easily adapted to address the problem of orbital stabilization of general nonlinear systems.This leads to two constructive solutions for this problem that-as usual in PBC-have clear interpretation from the energy viewpoint.First,the assignment of an energy function that has connected equilibria in the closed curve,i.e.,with the shape of a Mexican sombrero.Second,the use of a damping matrix that changes “sign” according to the position of the state trajectory relative to the desired orbit,that is,pumping or dissipating energy.As usual in all constructive nonlinear controller designs,the success of the proposed methods hinges upon our ability to solve a partial differential equation.At the end,we use the proposed energy pumping-anddamping technique to regulate nonholonomic systems described by kinematic models.It provides a suitable framework for the solution of the problem: find a globally defined,smooth,time-invariant state-feedback that achieves the objective to drive the state of nonholonomic systems to zero.