Stability analysis and controller synthesis for a class of piecewise smooth systems

This thesis deals with the analysis and synthesis of piecewise smooth (PWS) systems. In general, PWS systems are nonsmooth systems, which means their vector fields are discontinuous functions of the state vector. Dynamic behavior of nonsmooth systems is richer than smooth systems. For example, there are phenomena such as sliding modes that occur only in nonsmooth systems. In this thesis, a Lyapunov stability theorem is proved to provide the theoretical framework for the stability analysis of PWS systems. Piecewise affine (PWA) and piecewise polynomial (PWP) systems are then introduced as important subclasses of PWS systems. The objective of this thesis is to propose efficient computational controller synthesis methods for PWA and PWP systems. Three synthesis methods are presented in this thesis. The first method extends linear controllers for uncertain nonlinear systems to PWA controllers. The result is a PWA controller that maintains the performance of the linear controller while extending its region of convergence. However, the synthesis problem for the first method is formulated as a set of bilinear matrix inequalities (BMIs), which are not easy to solve. Two controller synthesis methods are then presented to formulate PWA and PWP controller synthesis as convex problems, which are numerically tractable. Finally, to address practical implementation issues, a time-delay approach to stability analysis of sampled-data PWA systems is presented. The proposed method calculates the maximum sampling time for a sampled-data PWA system consisting of a continuous-time plant and a discrete-time emulation of a continuous-time PWA state feedback controller

A duality-based convex optimization approach to L2 -gain control of piecewise affine slab differential inclusions

This paper introduces the dual parameter set of a piecewise affine (PWA) system. This is a key concept to enable a convex formulation of PWA controller synthesis for PWA slab differential inclusions using a new convex relaxation. Another important contribution of the paper is to present PWA L2-gain analysis and synthesis results for PWA systems whose output is also a PWA function of the state (as opposed to a piecewise-linear (PWL) function). Unlike other results existing in the literature, the sufficient LMI conditions in this paper are valid for synthesis, even when the PWA systems include sliding modes. A numerical example with sliding modes illustrates the new approac

Stability of sampled-data piecewise affine systems: A time-delay approach

This paper addresses the stability analysis of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant in feedback connection with a discrete-time emulation of a continuous-time state feedback controller. The sampled-data system is first considered as a continuous-time system with a variable time delay. Conditions under which the trajectories of the sampled-data closed-loop system will converge to an attracting invariant set are then presented. It is also shown that when the sampling period converges to zero, these conditions coincide with sufficient conditions for non-fragility of the stabilizing continuous-time PWA state feedback controller. The results are successfully applied to a helicopter examp

A unified dissipativity approach for stability analysis of piecewise smooth systems

The main objective of this paper is to present aunifieddissipativityapproach for stabilityanalysis of piecewisesmooth (PWS) systems with continuous and discontinuous vector fields. The Filippov definition is considered for the solution of these systems. Using the concept of generalized gradients for nonsmooth functions, sufficient conditions for the stability of a PWS system are formulated based on Lyapunov theory. The importance of the proposed approach is that it does not need any a-priori information about attractive sliding modes on switching surfaces, which is in general difficult to obtain. A section on application of the main results to piecewise affine (PWA) systems followed by a section with extensive examples clearly show the usefulness of the proposed unified methodology. In particular, we present an example with a stable sliding mode where the proposed method works and previously suggested methods fail

Extension of local linear controllers to global piecewise affine controllers for uncertain non-linear systems

A two-step controller synthesis method is proposed in this article for a class of uncertain non-linear systems described by piecewise affine differential inclusions. In the first step, a robust linear controller is designed for the linear differential inclusion that describes the dynamics of the non-linear system close to the equilibrium point. In the second step, a stabilising piecewise affine controller is designed that coincides with the linear controller in a region around the equilibrium point. The proposed method has two objectives: global stability and local performance. It thus enables us to use well-known techniques in linear control design for local stability and performance while delivering a global piecewise affine controller that is guaranteed to stabilise the non-linear system. To construct the required theoretical framework, a stability theorem for non-smooth Lyapunov functions is presented and proved. The new method will be applied to two examples

Investigation and Sensitivity analysis of experimental methods for estimating evaporation from free surface of Choghakhor lake

Evaporation losses from free surface water in reservoir dams are one of the important processes in meteorology and hydrology. Each experimental methods estimates the evaporation based on the climatic conditions of each region. For this reason, calibration of the relationship between evaporation estimates in different regions are required (Vanzyl et al., 1989). experimental methods are common in engineering sciences. Most scholars, by modifying experimental methods and discovering new relationships, try to find a simple and high-level relationship to replace with the field methods. More than 50 experimental relationships have been proposed for estimating evapotranspiration by various researchers (Naorem and Devi, 2014). Despite the importance of evaporation in hydrology, less attention has been paid to it. However, with the correct estimation, it can be applied in water resource planning with fewer errors (Saadatkhah et al., 2002).In this research, the experimental methods of Mayer, Marciano, Shahthin, Henfer, Ivanov and USBR have been used to study the evaporation from the free surface of the Choghakhor lake, located in Chaharmahal and Bakhtiari province of Iran. The results have been compared with actual evaporation values from the free surface

Development of a CuCl/phosphine system to catalyze phenylation and methylation of NN-tosyl aldimines with phenylboronic and methylboronic acids

The addition of phenylboronic and methylboronic acids to activated aromatic aldimines was demonstrated in the presence of copper(I)–phosphine complexes. The desired products were obtained using copper chloride/phosphine, and potassium fluoride in under toluene reflux, in moderate-to-good yield and a suitable reaction time

Assessment of the ratio of cross-matched to transfused blood amounts in pregnant patients in Alzahra Hospital in Tabriz before and after the implementation of the health reform program in 2013-2015

Background: Postpartum hemorrhage is the most common cause of mortality in women with vaginal or cesarean delivery. WHO statistics shows that about 500 thousands women have died of complications related to pregnancy or during childbirth in 2013. haemorrhage probability is the main reason to order blood requests in delivery and cesarean units. The purpose of this study is to evaluate the ratio of cross-matched to transfused blood in pregnant women during one year before and after the implementation of health reform program.Materials and Methods: In this retrospective descriptive study, the requests of blood reserves for pregnant patients with the gravid of 34±7 weeks and the age of 30±16 years old in two periods, before and after the implementation of health reform program were being collected and compared. Blood group antiserums and anti-human globulins with bovine albumins were purchased from LORN company and Baharafshan Company respectively.Results: The total number of requests for reserved cross-matched blood during the year before the health reform program were 2837 units which 277 of them had been transfused and a year later the total number of requests was for 3029 units, which 270 of them had been transfused. This study showed 6.76 percent increase in requests and 2.53 percent decrease in transfusion in comparison with its past year.Conclusion: The implementation of health reform program have accompanied by relative reduction of cesarean but considerable increase in vaginal delivery, and to prevent blood loss complications, the requests for reserved cross-matched blood have been increased without noticeable change in the proportion of necessarily transfused amount of them