Balanced truncation for time-delay systems via approximate gramians

In circuit simulation, when a large RLC network is connected with delay elements, such as transmission lines, the resulting system is a time-delay system (TDS). This paper presents a new model order reduction (MOR) scheme for TDSs with state time delays. It is the first time to reduce a TDS using balanced truncation. The Lyapunov-type equations for TDSs are derived, and an analysis of their computational complexity is presented. To reduce the computational cost, we approximate the controllability and observability Gramians in the frequency domain. The reduced-order models (ROMs) are then obtained by balancing and truncating the approximate Gramians. Numerical examples are presented to verify the accuracy and efficiency of the proposed algorithm. ©2011 IEEE.published_or_final_versionThe 16th Asia and South Pacific Design Automation Conference (ASP-DAC 2011), Yokohama, Japan, 25-28 January 2011. In Proceedings of the 16th ASP-DAC, 2011, p. 55-60, paper 1C-

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

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

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

Controllability of nonlinear fractional Langevin delay systems

In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α,β ≤ 1. Necessary and sufficient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufficient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders fixed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory

Stability and stabilization of fractional order time delay systems

U ovom radu predstavljeni su neki osnovni rezultati koji se odnose na kriterijume stabilnosti sistema necelobrojnog reda sa kašnjenjem kao i za sisteme necelobrojnog reda bez kašnjenja.Takođe, dobijeni su i predstavljeni dovoljni uslovi za konačnom vremenskom stabilnost i stabilizacija za (ne)linearne (ne)homogene kao i za perturbovane sisteme necelobrojnog reda sa vremenskim kašnjenjem. Nekoliko kriterijuma stabilnosti za ovu klasu sistema necelobrojnog reda je predloženo korišćenjem nedavno dobijene generalizovane Gronval nejednakosti, kao i 'klasične' Belman-Gronval nejednakosti. Neki zaključci koji se odnose na stabilnost sistema necelobrojnog reda su slični onima koji se odnose na klasične sisteme celobrojnog reda. Na kraju, numerički primer je dat u cilju ilustracije značaja predloženog postupka.In this paper, some basic results of the stability criteria of fractional order system with time delay as well as free delay are presented. Also, we obtained and presented sufficient conditions for finite time stability and stabilization for (non)linear (non)homogeneous as well as perturbed fractional order time delay systems. Several stability criteria for this class of fractional order systems are proposed using a recently suggested generalized Gronwall inequality as well as 'classical' Bellman-Gronwall inequality. Some conclusions for stability are similar to those of classical integerorder differential equations. Finally, a numerical example is given to illustrate the validity of the proposed procedure

Structural Controllability of Discrete-Time Linear Control Systems with Time-Delay: A Delay Node Inserting Approach

This paper is concerned with the structural controllability analysis for discrete-time linear control systems with time-delay. By adding virtual delay nodes, the linear systems with time-delay are transformed into corresponding linear systems without time-delay, and the structural controllability of them is equivalent. That is to say, the time-delay does not affect or change the controllability of the systems. Several examples are also presented to illustrate the theoretical results

Solution of a System of Linear Delay Differential Equations

An approach for the analytical solution to systems of delay differential equations (DDEs) has been developed using the matrix Lambert function. To generalize the Lambert function method for scalar DDEs, we introduce a new matrix, Q when the coefficient matrices in a system of DDEs do not commute. The solution has the form of an infinite series of modes written in terms of the matrix Lambert functions. The essential advantage of this approach is the similarity with the concept of the state transition matrix in linear ordinary differential equations (ODEs), enabling its use for general classes of linear delay differential equations. Examples are presented to illustrate by comparison to numerical methods.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106423/1/ACC_FinalDraft_Submitted.pd

Stability of fractional order time delay systems

In this paper, some basic results of the stability criteria of fractional order system with time delay as well as free delay are presented. Also, they are obtained and presented sufficient conditions for finite time stability for (non)linear (non)homogeneous as well as perturbed fractional order time delay systems. Several stability criteria for this class of fractional order systems are proposed using a recently suggested generalized Gronwall inequality as well as “classical” Bellman-Gronwall inequality. Some conclusions for stability are similar to that of classical integer-order differential equations. Last, a numerical example is given to illustrate the validity of the proposed procedure