Gabriela Iuliana Bara

Observer Design for Nonlinear Systems: An Approach Based on the Differential Mean Value Theorem.

In this note, observers design for a class of non linear dynamical systems has been investigated. The main contribution lies in the use of the differential mean value theorem (DMVT) to transform the nonlinear error dynamics into a LPV system. The stability analysis is, therefore, performed using a standard Lyapunov function that leads to the solvability of a set of Linear Matrix Inequalities (LMIs), easily tractable. Numerical examples are provided to show high performances of the proposed approach and the large class of nonlinear dynamical systems that are concerned.

Robust analysis and control of parameter-dependent uncertain descriptor systems

In this paper, we address the robust admissibility analysis and control of descriptor systems with parameter-dependent, possibly time-varying, uncertainties. First, we propose a new necessary and sufficient condition for the admissibility of linear time-invariant (LTI) descriptor systems. This condition uses a strict linear matrix inequality (LMI) formulation and involves two additional matrix variables usually called slack variables. Then, we present an extension of this condition to robust admissibility of time-varying parameter-dependent descriptor systems. Using parameter-dependent Lyapunov functions and employing two slack variables, this extension states robust admissibility LMI conditions that are less restrictive, as supported by our numerical examples, than the existing ones. Our robust admissibility analysis result, which can be directly used for state estimation, cannot be so for state-feedback control design. This is due to the absence of duality between state estimation and state-feedback control in the case of parameter-dependent systems with uncertain parameters as recently reported in the literature. In order to overcome this problem, we also present in this paper a new robust admissibility analysis result, involving only one slack variable, which we apply to state-feedback control design.

Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-time polytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary and sufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant (LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article, we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptor systems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependent Lyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMI conditions which are easily tractable numerically compared to the non-strict ones often encountered when dealing with descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directly exploitable for state estimation and the other for state-feedback control. The need for different admissibility analysis conditions for each synthesis problem is motivated by the fact that the duality between state estimation and state-feedback control, which hold in the case of measurable parameters, does not hold when dealing with uncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on the dynamical part of the descriptor system. The application of our analysis result to both robust state-feedback control and polytopic state-feedback control are also presented in this article. Our analysis results extend to descriptor systems, some existing results developed for regular systems.

Flexible-link Robot Control Using a Linear Parameter Varying Systems Methodology

This paper addresses the issues of the Linear Parameter Varying (LPV) modelling and control of flexible-link robot manipulators. The LPV formalism allows the synthesis of nonlinear control laws and the assessment of their closed-loop stability and performances in a simple and effective manner, based on the use of Linear Matrix Inequalities (LMI). Following the quasi-LPV modelling approach, an LPV model of a flexible manipulator is obtained, starting from the nonlinear dynamic model stemming from Euler-Lagrange equations. Based on this LPV model, which has a rational dependence in terms of the varying parameters, two different methods for the synthesis of LPV controllers are explored. They guarantee the asymptotic stability and some level of closed-loop ℒ2-gain performance on a bounded parametric set. The first method exploits a descriptor representation that simplifies the rational dependence of the LPV model, whereas the second one manages the troublesome rational dependence by using dilated LMI conditions and taking the particular structure of the model into account. The resulting controllers involve the measured state variables only, namely the joint positions and velocities. Simulation results are presented that illustrate the validity of the proposed control methodology. Comparisons with an inversion-based nonlinear control method are performed in the presence of velocity measurement noise, model uncertainties and high-frequency inputs.

Observer-based controller synthesis for LPV descriptor systems using dilated LMI conditions

This paper deals with the stabilization and the H2 control synthesis of Linear Parameter Varying (LPV) descriptor systems. The compensator design problem is examined through a parameter-dependent observer-based control structure. The proposed approach uses parameter-dependent Lyapunov functions and involves slack variables in order to achieve less restrictive synthesis conditions. These synthesis conditions are expressed as a Linear Matrix Inequality (LMI) solvability problem. The admissibility and the H2 performance are ensured by computing appropriate gains of the state-feedback controller and the observer. Simulation results obtained with a numerical example, show the effectiveness of the proposed approach.

On the dissipative analysis and control of state-space symmetric systems

This paper addresses the quadratic dissipativity analysis and static output feedback control of linear time-invariant systems which are state-space symmetric. By considering particular weighting matrices, we present a necessary and sufficient inequality condition for checking asymptotic stability and quadratic dissipativity for this class of systems. Our analysis condition involves only system state matrices and known weighting matrices. Therefore, this condition is easy to check numerically and particularly suitable in the case of large-scale symmetric systems. The application of our analysis result to symmetric static output feedback (SSOF) control design is also reported in this paper. An easily tractable numerically, necessary and sufficient condition for the existence of a SSOF control law and an explicit parametrization of all SSOF controllers guaranteeing the asymptotic stability and the quadratic dissipativity of the closed-loop system is given. Note that the results presented in this paper generalize some results already proposed in the literature to a more general case of quadratic dissipativity analysis and control.

Dilated LMI conditions for the robust analysis of uncertain parameter-dependent descriptor systems

This paper addresses the robust admissibility/ℌ2 performance analysis for continuous-time uncertain parameter-dependent descriptor systems. In order to achieve less conservative results, the proposed approach uses parameter-dependent Lyapunov functions and slack variables. Our main contribution consists in proposing new necessary and sufficient conditions for the admissibility and ℌ2 performance analysis of time-invariant singular systems. These conditions are formulated as a strict linear matrix inequality (LMI) solvability problem and represent a generalization to singular systems of some dilated LMI analysis results developed in the literature for state-space systems. Also, we have extended our results to the analysis of descriptor systems with time-varying parametric uncertainties. A numerical example shows the applicability of our approach.

Phase space identification method for modeling the viscosity of bone cement

A lot of studies have recently emphasized various parameters influencing the viscosity evolution of acrylic bone cement currently used in percutaneous vertebroplasty. Despite all these studies, only very few mathematical models have been proposed to describe the evolution of bone cement viscosity over time. These models are either unexploitable for viscosity control purpose or unable to simultaneously show the viscosity dependence over the most important parameters. In this paper, we propose a new mathematical model of bone cement viscosity obtained through developing a phase space identification method. This model is expressed as a differential equation involving the most important parameters that are temperature and shear rate. In addition to being more complete that the ones already proposed in the literature, it is also easily exploitable for control purpose.

Bone cement modeling for percutaneous vertebroplasty: Bone cement modeling for percutaneous vertebroplasty

Vertebroplasty procedures provide a significant benefit for patients suffering from vertebral fractures. In order to address current issues of vertebroplasty procedures, an injection device able to control the bone cement viscosity has been developed. In addition, this device allows to protect the practitioner by removing him/her from the X-rays area. In this context, a study is first proposed to quantify the bone cement viscosity during its polymerization reaction on a rotational rheometer. These experimental measurements have led to the identification of a complete behavior law that takes into account the simultaneous effects of shear rate, time, and temperature. Based on this preliminary study, this article finally aims to prove the ability of estimating the viscosity of the flowing bone cement on the developed injection system. A final set of experiments validates that the injection device dedicated to vertebroplasty procedures can control the flowing bone cement viscosity by acting on the temperature.

Design and control of a thermal device for bone cement injection

This paper presents an original robotized system that reduces potential bone cement leakage during vertebroplasty procedures while providing a longer injection time to radiologists. The system design meets the requirements of vertebroplasty procedures in terms of injection speed, consumables and medical environment. Our contribution relies on the design, dimensioning and control of a device regulating the injected cement temperature, that is one of the main factors affecting the evolution of the cement viscosity. To control the system, thermal exchanges are first modeled from the physical equations of heat and mass transfer. Then, a simple control method is presented and experimentally assessed to confirm the ability to influence the flowing cement temperature at the center of the stream within an acceptable time.