Feedback control design by Lyapunov's direct method

6 - Design and Performance of Feedback Controllers

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method.

A self-tuning vibration control of a rotational flexible arm using neural networks is presented. To the self-tuning control system, the control scheme consists of gain tuning neural networks and a variable-gain feedback controller. The neural networks are trained so as to make the root moment zero. In the process, the neural networks learn the optimal gain of the feedback controller. The feedback controller is designed based on Lyapunov's direct method. The feedback control of the vibration of the flexible system is derived by considering the time rate of change of the total energy of the system. This approach has the advantage over the conventional methods in the respect that it allows one to deal directly with the system's partial differential equations without resorting to approximations. Numerical and experimental results for the vibration control of a rotational flexible arm are discussed. It verifies that the proposed control system is effective at controlling flexible dynamical systems.

A design procedure for simultaneous position and force control is developed, using Lyapunov's direct method, for manipulators in contact with a rigid environment that can be described by holonomic constraints. Many manipulators that interact with their environment require taking into account the effects of these constraints in the control design. The forces of constraint play a critical role in constrained motion and are, along with displacements and velocities, to be regulated at specified values. Lyapunov's direct method is used to develop a class of position and force feedback controllers. The conditions for gain selection demonstrate the importance of the constraints. Force feedback has been shown not to be mandatory for closed-loop stabilization, but it is useful in improving certain closed-loop robustness properties.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper describes vibration control and position control of a micro-actuator for hard disk drives using self-sensing actuator. A stabilizing feedback control low is designed by using Lyapunov's direct method. The feedback control law depends only on the time derivative of tip slope in the micro-actuator. The self-sensing actuation using the electrical RC bridge circuit can measure strain rate in the micro-actuator. This sensing signals correspond to the time rate of tip slope in the micro-actuator. The feedback controller derived from Lyapunov's direct method can realize by using the self-sensing signal. A two-degree-of-freedom control system consists of a feedforward and the feedback controller. The feedforward controller is designed by using inverse dynamic system, while the stabilized feedback controller is designed by Lyapunov's direct method. Numerical simulation and experimental results confirm the feasibility of the proposed control system.

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.

In this paper, a family of hybrid control algorithms is presented; where it is merged a free camera-calibration image-based control scheme and a direct force controller, both with the same priority level. The aim of this generalised hybrid controller is to regulate the robot-environment interaction into a two-dimensional task-space. The design of the proposed control structure takes into account most of the dynamic effects present in robot manipulators whose inputs are torque signals. As examples of this generalised structure of hybrid force/vision controllers, a linear proportional-derivative structure and a nonlinear proportional-derivative one (based on the hyperbolic tangent function) are presented. The corresponding stability analysis, using Lyapunov's direct method and invariance theory, is performed to proof the asymptotic stability of the equilibrium vector of the closed-loop system. Experimental tests of the control scheme are presented and a suitable performance is observed in all the cases. Unlike most of the previously presented hybrid schemes, the control structure proposed herein achieves soft contact forces without overshoots, fast convergence of force and position error signals, robustness of the controller in the face of some uncertainties (such as camera rotation), and safe operation of the robot actuators when saturating functions (non-linear case) are used in the mathematical structure. This is one of the first works to propose a generalized structure of hybrid force/vision control that includes a closed loop stability analysis for torque-driven robot manipulators.

It is well-known that phase-shifting controllers used for active combustion control must be manually adjusted in order to maintain control over a broad range of operating combustor operating conditions. If one assumes that the thermoacoustic instabilities are linearly stabilizable, then what is needed is a method to determine, and ultimately predict, the frequency response of the plant for any range of operating conditions, so the controller design can be automatically updated to track the changing plant gain/phase relationships that are observed with changing heat release. A unique test-based, design process has been proposed to predict the gain/phase characteristics required of a proportional, phase-shifting controller that can stabilize the thermoacoustic instabilities. In this paper, that process is used to automate the design of a fixed-gain feedback controller that limits the amplitudes of any feedback induced instabilities (to some pre-specified level) while providing the best control of the targeted limit cycling pressure oscillations. The paper describes how a neural network was trained, using the suggested design process, to predict the frequency response of the thermoacoustics in a tube combustor at frequencies adjacent to the limit cycle frequency using certain operating conditions that included a sparsely-sampled temperature profile, total air/fuel flow rate, and equivalence ratio. The neural net training was performed using complex valued, open-loop frequency response function data as the desired signal with the previously mentioned operating conditions as the input signals. (The open loop data was collected for a narrow frequency range surrounding the limit cycle instability by performing a sine dwell at discrete frequencies). Once the neural network was trained, it was used to predict the approximate phase and gain margins as a function of temperature and flow conditions. The margins were then used to automatically update and design a fixed shape feedback controller having the proper phase and magnitude to ensure stability and control in the face of changing operating conditions. A companion paper describes the methodology that underlies the automated design of the feedback controller gain and phase delay.

A new approach is presented to improve power swings damping of nonlinear power system model based on direct Lyapunov method. The approach combines superconducting magnetic energy storage (SMES) system with static series synchronous compensator (SSSC). Considering the energy absorption/injection ability of SMES, in transient states the combination exchanges both active and reactive powers with power system. Since direct Lyapunov method implies time derivative of both active and reactive powers, prepares an effective control strategy for the integrated SSSC-SMES system to decrease power swings damping time. Using direct Lyapunov method for structure preserving model (SPM) of power system, dq components of SSSC injecting voltage are calculated. Decentralizing method is used to locally control the SSSC-SMES independent of state variables of synchronous generators which require wide area measurements. PSCAD/EMTDC simulations verify the effectiveness of proposed method.

Circle Condition-Based Robust Feedback Control Against Plant Perturbation

This paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

Lyapunov's direct method has been adapted to the study of an electrostatic instability of a plasma by using Zubov's extension of the method to distributed parameter systems. Specifically, the linear stability of two inhomogeneous interpenetrating electron streams in a finite one-dimensional system is considered. Proper Lyapunov functionals are constructed for both stability and instability. It is seen that, as in the homogeneous case, the stability or instability depends on whether or not the thermal speed is greater or less than the streaming speed. In contrast to the methods of energy principle, the complete generality of Lyapunov's direct method is discussed.

Growing environmental concerns and shrinking water resources require methods beyond conventional wastewater treatment. Membrane bioreactor (MBR) is a technology that has become a ubiquitous choice for high quality treatment and reuse of wastewater. One of the key challenges in wastewater treatment is the high energy cost associated with aeration. MBR systems use feedback control to regulate the measured dissolved oxygen level at a predetermined set point by manipulating the blower throughputs. However, for high dynamic loads, feedback control may not result in the best performance and energy efficiency. Any attempt to optimize performance and power consumption beyond a simple controls strategy requires a proper trade‐off analysis between investments on additional sensors and the long‐term benefits. This article proposes a joint state and parameter estimation methodology, which measures and controls the MBR system using available measurements. Thus, limitations of feedback strategies can be overcome by predicting the impact of time varying disturbances on the outputs. The novelty in this approach is the ability to reconstruct the unknown states and parameters with available measurements. The unknown parameters are adaptively estimated online. Lyapunov's direct method is employed to show boundedness of state and parameter estimation errors. Simulation results illustrate the efficacy of the approach. Copyright © 2011 Curtin University of Technology and John Wiley &amp; Sons, Ltd.

So far, a considerable amount of research has been devoted to flexible arms, especially to their modelling, position control and vibration control. Meanwhile very few research has addressed the problem of force control, since it is one of the most important functions of the robot tasks. In this paper, we present a two-degree-of-freedom control system for the force control of flexible cartesian robots. Use of a two-degree-of freedom control system guarantees the independent design of the feedforward and feedback control characteristics. The feedforward controller is designed using inverse dynamics, while the design of the stabilized feedback controller is achieved by Lyapunov's direct method. Numerical simulation and experimental results confirm that faster precise force control can be realized with the proposed control system.

The dynamics of a web transport system are coupled and interactive among spans of web and rolls/rollers. As the process speed or variation of the speed is high, system friction and inertia of rotation of rolls could cause problems in implementation of observer techniques for tension estimation and control. In the author's previous study (see IEEE International Conference on Industrial Electronics, Control and Instrumentation, Denver, Colorado, USA (2001)), an observer-based tension feedback controller with friction and inertia compensation was proposed. However, the gains of observer and controller were determined based on the trial-and-error approach. This paper continues the author's previous study and presents an analytical approach for the gain design of observers in the time domain but design of controllers in the frequency domain. The linearization and decentralization techniques are implemented. Design of tension observers and feedback controllers with friction and inertia compensation is studied. Approaches for the gain design are discussed, and summarized procedures are given. The proposed analytical approach for design of tension observers and feedback controllers has been proven to be adequate by experiment.