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

Adaptive Control For Autonomous Navigation Of Mobile Robots Considering Time Delay And Uncertainty

Autonomous control of mobile robots has attracted considerable attention of researchers in the areas of robotics and autonomous systems during the past decades. One of the goals in the field of mobile robotics is development of platforms that robustly operate in given, partially unknown, or unpredictable environments and offer desired services to humans. Autonomous mobile robots need to be equipped with effective, robust and/or adaptive, navigation control systems. In spite of enormous reported work on autonomous navigation control systems for mobile robots, achieving the goal above is still an open problem. Robustness and reliability of the controlled system can always be improved. The fundamental issues affecting the stability of the control systems include the undesired nonlinear effects introduced by actuator saturation, time delay in the controlled system, and uncertainty in the model. This research work develops robustly stabilizing control systems by investigating and addressing such nonlinear effects through analytical, simulations, and experiments. The control systems are designed to meet specified transient and steady-state specifications. The systems used for this research are ground (Dr Robot X80SV) and aerial (Parrot AR.Drone 2.0) mobile robots. Firstly, an effective autonomous navigation control system is developed for X80SV using logic control by combining ‘go-to-goal’, ‘avoid-obstacle’, and ‘follow-wall’ controllers. A MATLAB robot simulator is developed to implement this control algorithm and experiments are conducted in a typical office environment. The next stage of the research develops an autonomous position (x, y, and z) and attitude (roll, pitch, and yaw) controllers for a quadrotor, and PD-feedback control is used to achieve stabilization. The quadrotor’s nonlinear dynamics and kinematics are implemented using MATLAB S-function to generate the state output. Secondly, the white-box and black-box approaches are used to obtain a linearized second-order altitude models for the quadrotor, AR.Drone 2.0. Proportional (P), pole placement or proportional plus velocity (PV), linear quadratic regulator (LQR), and model reference adaptive control (MRAC) controllers are designed and validated through simulations using MATLAB/Simulink. Control input saturation and time delay in the controlled systems are also studied. MATLAB graphical user interface (GUI) and Simulink programs are developed to implement the controllers on the drone. Thirdly, the time delay in the drone’s control system is estimated using analytical and experimental methods. In the experimental approach, the transient properties of the experimental altitude responses are compared to those of simulated responses. The analytical approach makes use of the Lambert W function to obtain analytical solutions of scalar first-order delay differential equations (DDEs). A time-delayed P-feedback control system (retarded type) is used in estimating the time delay. Then an improved system performance is obtained by incorporating the estimated time delay in the design of the PV control system (neutral type) and PV-MRAC control system. Furthermore, the stability of a parametric perturbed linear time-invariant (LTI) retarded type system is studied. This is done by analytically calculating the stability radius of the system. Simulation of the control system is conducted to confirm the stability. This robust control design and uncertainty analysis are conducted for first-order and second-order quadrotor models. Lastly, the robustly designed PV and PV-MRAC control systems are used to autonomously track multiple waypoints. Also, the robustness of the PV-MRAC controller is tested against a baseline PV controller using the payload capability of the drone. It is shown that the PV-MRAC offers several benefits over the fixed-gain approach of the PV controller. The adaptive control is found to offer enhanced robustness to the payload fluctuations

Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments

Exact eigenvalue assignment of linear scalar systems with single delay using Lambert W function

Eigenvalue assignment problem of a linear scalar system with a single discrete delay is analytically and exactly solved. The existence condition of the desired eigenvalue is established when the current and delay states are present in the feedback loop. Design of the feedback controller is then followed. Furthermore, eigenvalue assignment for the input-delay system is also obtained as well. Numerical examples illustrate the procedure of assigning the desired eigenvalue

On the Positive Effect of Delay on the Rate of Convergence of a Class of Linear Time-Delayed Systems

This paper is a comprehensive study of a long observed phenomenon of increase in the stability margin and so the rate of convergence of a class of linear systems due to time delay. We use Lambert W function to determine (a) in what systems the delay can lead to increase in the rate of convergence, (b) the exact range of time delay for which the rate of convergence is greater than that of the delay free system, and (c) an estimate on the value of the delay that leads to the maximum rate of convergence. For the special case when the system matrix eigenvalues are all negative real numbers, we expand our results to show that the rate of convergence in the presence of delay depends only on the eigenvalues with minimum and maximum real parts. Moreover, we determine the exact value of the maximum rate of convergence and the corresponding maximizing time delay. We demonstrate our results through a numerical example on the practical application in accelerating an agreement algorithm for networked~systems by use of a delayed feedback

Delay induced Turing-like waves for one species reaction-diffusion model on a network

A one species time-delay reaction-diffusion system defined on a complex networks is studied. Travelling waves are predicted to occur as follows a symmetry breaking instability of an homogenous stationary stable solution, subject to an external non homogenous perturbation. These are generalized Turing-like waves that materialize in a single species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time delayed differential equation. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz networks and with the inclusion of the delay