Stability analysis and observer design for one-sided Lipschitz descriptor systems with time-varying delay

This paper investigates the problem of stability analysis and observer design for nonlinear descriptor systems with time-varying delay. In the systems, the nonlinear function satisfies the one-sided Lipschitz condition and the quadratic internal boundary condition. The disturbance is considered in both the state and the output equation. Using one-sided Lipschitz condition, the quadratic internal boundary condition, and the generalized Lyapunov method, we establish the non-strict bilinear matrix inequality (BMI)-based condition. We change the condition into strict bilinear matrix inequality (BMI) condition. Furthermore, we give the linear matrix inequality-based condition to ensure the gradual convergence of state estimation error and to accomplish robustness against   L2  norm bounded disturbances by utilizing change of variables for straight forward computation of the observer gain matrix. Finally, a numerical example is given to verify the effectiveness of the observer design scheme

Time-Delay Systems: Analysis and Control using the Lambert W Function.

Time-delay systems can arise due to inherent time-delays in the system or a deliberate introduction of time-delays into the system for control purposes. Such systems frequently occur in engineering and science. Time-delays can cause significant problems (e.g., instability and inaccuracy) and, thus, limit and degrade achievable performance. Time-delay terms lead to an infinite number of roots of the characteristic equation, and make analysis difficult using classical methods, especially, in determining stability and designing stabilizing controllers. Thus, such problems have been addressed mainly by using approximate, numerical, and graphical methods. However, such approaches constitute limitations, for example, on accuracy and robustness. The objective of this research is to develop an effective approach to analyze and control time-delay systems. Using the LambertWfunction, free and forced analytical solutions to delay differential equations are derived. The main advantage of this solution approach lies in the fact that the solution has an analytical form expressed in terms of system parameters and, thus, one can explicitly determine how each parameter affects each eigenvalue and the solution. Also, each eigenvalue in the infinite eigenspectrum is associated individually with a branch of the LambertWfunction. Solutions are obtained, for the first time, for systems of delay differential equations using the matrix Lambert W function. The obtained solutions are used to analyze essential system properties, such as stability, controllability and observability, and to design controllers for stabilizing systems, improving robustness and/or meeting time-domain specifications. Then, these methods are applied to biological systems to analyze the immune system via eigenvalue sensitivity analysis, to automotive powertrain systems to design feedback control with observers for improvements in fuel economy and emissions, and to manufacturing processes to improve productivity via stability analysis. The newly developed approach based on the matrix Lambert W function provides a tool for analysis and control, which is accurate (i.e., no approximation of timedelay terms), robust (i.e., no prediction of responses from models), and easy to implement (i.e., no need for complex nonlinear controllers).Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64756/1/syjo_1.pd

Stability of Uncertainty Piecewise Affine Time-Delay Systems with Application to All Wheel Drive Clutch Control.

Piecewise affine (PWA) systems provide good flexibility and traceability for modeling a variety of nonlinear systems. The stability of PWA systems is an important but challenging problem since the stability of the sub-systems does not directly imply the stability of the global system. Meanwhile, time delays and uncertainty exist in many practical systems in engineering and introduce various complex behaviors such as oscillation, instability and poor performance. To ensure the stability of the practical control systems developed via the PWA system framework, the stability of uncertain PWA time-delay systems is investigated. In addition, a quantitative description of asymptotic behavior for time-delay systems is also studied. First, the stability problem for uncertain piecewise affine time-delay systems is investigated. It is assumed that there exists a constant time delay in the system and the uncertainly is norm-bounded. Sufficient conditions for the stability of nominal systems and the stability of systems subject to uncertainty are derived using the Lyapunov-Krasovskii functional with a triple integration term. This approach handles switching based on the delayed states (in addition to the states) for a PWA time-delay system, considers structured as well as unstructured uncertainty, and reduces the conservativeness of previous approaches. Second, an application of the PWA system framework to the modeling and control of an automotive all wheel drive clutch system is presented. The open-loop system is modeled as a PWA system, followed by the design of a piecewise linear feedback controller. The stability of the closed-loop system is examined using the proposed stability method. Finally, a new Lambert W function based approach for estimation of the decay function for time-delay systems is presented. Using this solution form, a decay function estimate, which is less conservative than existing methods, is obtained.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86297/1/duansm_1.pd