Design of a control system using linear matrix inequalities for the active vibration control of a plate

The study of algorithms for active vibration control in smart structures is an area of interest, mainly due to the demand for better performance of mechanical systems, such as aircraft and aerospace structures. Smart structures, formed using actuators and sensors, can improve the dynamic performance with the application of several kinds of controllers. This article describes the application of a technique based on linear matrix inequalities (LMI) to design an active control system. The positioning of the actuators, the design of a robust state feedback controller and the design of an observer are all achieved using LMI. The following are considered in the controller design: limited actuator input, bounded output (energy) and robustness to parametric uncertainties. Active vibration control of a flat plate is chosen as an application example. The model is identified using experimental data by an eigensystem realization algorithm (ERA) and the placement of the two piezoelectric actuators and single sensor is determined using a finite element model (FEM) and an optimization procedure. A robust controller for active damping is designed using an LMI framework, and a reduced model with observation and control spillover effects is implemented using a computer. The simulation results demonstrate the efficacy of the approach, and show that the control system increases the damping in some of the modes.

Optimal and robust modal control of a flexible structure using an active dynamic vibration absorber

This paper is concerned with feedback vibration control of a lightly damped flexible structure that has a large number of well-separated modes. A single active electrical dynamic absorber is used to reduce a particular single vibration mode selectively or multiple modes simultaneously. The absorber is realized electrically by feeding back the structural acceleration at one position to a collocated piezoceramic patch actuator via a controller consisting of one or several second order lowpass filters. A simple analytical method is presented to design a modal control filter that is optimal in that it maximally flattens the mobility frequency response of the target mode, as well as robust in that it works within a prescribed maximum control spillover of 2 dB at all frequencies. Experiments are conducted with a free-free beam to demonstrate its ability to control any single mode optimally and robustly. It is also shown that an active absorber with multiple such filters can effectively control multiple modes simultaneously.

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the control of a non-ideal engineering system: A friction-driven oscillating system with limited power supply

In this work a switching feedback controller for stick-slip compensation of a 2-DOF mass-spring-belt system which interacts with an energy source of limited power supply (non-ideal case) is developed. The system presents an oscillatory behavior due to the stick-slip friction. As the system equilibrium for a conventional feedback controller is not the origin, a switching control law combining a state feedback term and a discontinuous term is proposed to regulate the position of the mass. The problem of tracking a desired periodic trajectory is also considered. The feedback system is robust with respect to the friction force that is assumed to be within known upper and lower bounds.

Phase-Locked loops lock-in range in Frequency Modulated-Atomic Force Microscope nonlinear control system

Since the mid 1980s the Atomic Force Microscope is one the most powerful tools to perform surface investigation, and since 1995 Non-Contact AFM achieved true atomic resolution. The Frequency-Modulated Atomic Force Microscope (FM-AFM) operates in the dynamic mode, which means that the control system of the FM-AFM must force the micro-cantilever to oscillate with constant amplitude and frequency. However, tip-sample interaction forces cause modulations in the microcantilever motion. A Phase-Locked loop (PLL) is used to demodulate the tip-sample interaction forces from the microcantilever motion. The demodulated signal is used as the feedback signal to the control system, and to generate both topographic and dissipation images. As a consequence, a proper design of the PLL is vital to the FM-AFM performance. In this work, using bifurcation analysis, the lock-in range of the PLL is determined as a function of the frequency shift (Q) of the microcantilever and of the other design parameters, providing a technique to properly design the PLL in the FM-AFM system. (C) 2011 Elsevier B.V. All rights reserved.

On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin-Huxley (HH) mathematical model

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Robust state-derivative feedback LMI-based designs for multivariable linear systems

In some practical problems, for instance in the control systems for the suppression of vibration in mechanical systems, the state-derivative signals are easier to obtain than the state signals. New necessary and sufficient linear matrix inequalities (LMI) conditions for the design of state-derivative feedback for multi-input (MI) linear systems are proposed. For multi-input/multi-output (MIMO) linear time-invariant or time-varying plants, with or without uncertainties in their parameters, the proposed methods can include in the LMI-based control designs the specifications of the decay rate, bounds on the output peak, and bounds on the state-derivative feedback matrix K. These design procedures allow new specifications and also, they consider a broader class of plants than the related results available in the literature. The LMIs, when feasible, can be efficiently solved using convex programming techniques. Practical applications illustrate the efficiency of the proposed methods.

CONTROL OF UNCERTAIN DYNAMIC-SYSTEMS USING STRICTLY POSITIVE REAL SYSTEMS

This paper presents necessary and sufficient conditions to turn a linear time-invariant system with p outputs, m inputs, p greater-than-or-equal-to m and using only inputs and outputs measurements into a Strictly Positive Real (SPR).Two results are presented. In the first, the system compensation is made by two static compensators, one of which forward feeds the outputs and the second back feeds the outputs of the nominal system.The second result presents conditions for the Walcott and Zak variable structure observer-controller synthesis. In this problem, if the nominal system is given by {A,B,C}, then the compensated system is given by {A+GC,B,FC} where F and G are the constant compensation matrices. These results are useful in the control system with uncertainties.

Postural control in children: Coupling to dynamic somatosensory information

The purpose of this investigation was to determine whether the coupling between dynamic somatosensory information and body sway is similar in children and adults. Thirty children (4-, 6-, and 8-year-olds) and 10 adults stood upright, with feet parallel, and lightly contacting the fingertip to a rigid metal plate that moved rhythmically at 0.2, 0.5, and 0.8 Hz. Light touch to the moving contact surface induced postural sway in all participants. The somatosensory stimulus produced a broadband frequency response in children, while the adult response was primarily at the driving frequency. Gain, as a function of frequency, was qualitatively the same in children and adults. Phase decreased less in 4-year-olds than other age groups, suggesting a weaker coupling to position information in the sensory stimulus. Postural sway variability was larger in children than adults. These findings suggest that, even as young as age 6, children show well-developed coupling to the sensory stimulus. However, unlike adults, this coupling is not well focused at the frequency specified by the somatosensory signal. Children may be unable to uncouple from sensory information that is less relevant to the task, resulting in a broadband response in their frequency spectrum. Moreover, higher sway variability may not result from the sensory feedback process, but rather from the children's underdeveloped ability to estimate an internal model of body orientation.

The LQ control problem for Markovian jumps linear systems with horizon defined by stopping times

This paper deals with a stochastic optimal control problem involving discrete-time jump Markov linear systems. The jumps or changes between the system operation modes evolve according to an underlying Markov chain. In the model studied, the problem horizon is defined by a stopping time τ which represents either, the occurrence of a fix number N of failures or repairs (TN), or the occurrence of a crucial failure event (τΔ), after which the system is brought to a halt for maintenance. In addition, an intermediary mixed case for which T represents the minimum between TN and τΔ is also considered. These stopping times coincide with some of the jump times of the Markov state and the information available allows the reconfiguration of the control action at each jump time, in the form of a linear feedback gain. The solution for the linear quadratic problem with complete Markov state observation is presented. The solution is given in terms of recursions of a set of algebraic Riccati equations (ARE) or a coupled set of algebraic Riccati equation (CARE).

Robust control applications for smart truss structure

A finite element modeling of an intelligent truss structure with piezoelectric stack actuators for the purpose of active damping and structural vibration attenuation is presented. This paper concerns with the following issues aspects: the design of intelligent truss structure considering electro-mechanical coupling between the host structure and piezoelectric stack actuators; the H 2 norm approach to search for optimal placement of actuators and sensors; and finally some aspects in robust control techniques. The electro-mechanical behavior of piezoelectric elements is directly related to the successful application of the actuators in truss structures. In order to achieve the desired damping in the interested bandwidth frequency it is used the H ∞ output feedback solved by convex optimization. The constraints to be reached are written by linear matrix inequalities (LMI). The paper concludes with a numerical example, using Matlab and Simulink, in a cantilevered, 2-bay space truss structure. The results demonstrated the approach applicability.

An efficient modal control strategy for the active vibration control of a truss structure

In this paper an efficient modal control strategy is described for the active vibration control of a truss structure. In this approach, a feedback force is applied to each mode to be controlled according to a weighting factor that is determined by assessing how much each mode is excited by the primary source. The strategy is effective provided that the primary source is at a fixed position on the structure, and that the source is stationary in the statistical sense. To test the effectiveness of the control strategy it is compared with an alternative, established approach namely, Independent Modal Space Control (IMSC). Numerical simulations show that with the new strategy it is possible to significantly reduce the control effort required, with a minimal reduction in control performance. © 2007 - IOS Press and the authors. All rights reserved.

Variable structure control of switched systems based on Lyapunov-Metzler-SPR systems

This paper presents two Variable Structure Controllers (VSC) for continuous-time switched plants. It is assumed that the state vector is available for feedback. The proposed control system provides a switching rule and also the variable structure control input. The design is based on Lyapunov-Metzler (LM) inequalities and also on Strictly Positive Real (SPR) systems stability results. The definition of Lyapunov-Metzler-SPR (LMS) systems and its direct application in the design of VSC for switched systems are introduced in this paper. Two examples illustrate the design of the proposed VSC, considering a plant given by a switched system with a switched-state control law and two linear time-invariant systems, that are not controllable and also can not be stabilized with state feedback. ©2008 IEEE.

Comparative study of LMI-based output feedback spr synthesis for plants with different numbers of inputs and outputs

New Linear Matrix Inequalities (LMI) conditions are proposed for the following problem, called Strictly Positive Real (SPR) synthesis: given a linear time-invariant plant, find a constant output feedback matrix Ko and a constant output tandem matrix F for the controlled system to be SPR. It is assumed that the plant has the number of outputs greater than the number of inputs. Some sufficient conditions for the solution of the problem are presented and compared. These results can be directly applied in the LMI-based design of Variable Structure Control (VSC) of uncertain plants. ©2008 IEEE.

Structural real time control using fluid magnetorheologic damper

In the last decades there was a great development in the study of control systems to attenuate the harmful effect of natural events in great structures, as buildings and bridges. Magnetorheological fluid (MR), that is an intelligent material, has been considered in many proposals of project for these controllers. This work presents the controller design using feedback of states through LMI (Linear Matrix Inequalities) approach. The experimental test were carried out in a structure with two degrees of freedom with a connected shock absorber MR. Experimental tests were realized in order to specify the features of this semi-active controller. In this case, there exist states that are not measurable, so the feedback of the states involves the project of an estimator. The coupling of the MR damper causes a variation in dynamics properties, so an identification methods, based on experimental input/output signal was used to compare with the numerical application. The identification method of Prediction Error Methods - (PEM) was used to find the physical characteristics of the system through realization in modal space of states. This proposal allows the project of a semi-active control, where the main characteristic is the possibility of the variation of the damping coefficient.