Fundamental Results on Determining Matrices for a Certain Class of Hereditary Systems

Three major tools are required to investigate the controllability of control systems, namely, determining matrices, index of control systems and controllability Grammian. Determining matrices are the preferred choice for autonomous control systems due to the fact that they are devoid of integral operators in their computations. This article developed the structure of certain parameter-ordered determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for Euclidean controllability of the system. Expressions for the relevant determining matrices were formulated and it was established that the determining matrices for double time-delay linear autonomous functional differential control systems do not exist if one of the time-delays is not an integer multiple of the other paving the way for the investigation of the Euclidean controllability of generic double time-delay control systems.

Global relative controllability of fractional stochastic dynamical systems with distributed delays in control

This paper is concerned with the global relative controllability of linear and nonlinear fractional stochastic dynamical systems with distributed delays in control for finite dimensional spaces. Sufficient conditions for controllability results are obtained using the Banach fixed point theorem and the controllability Grammian matrix which is dened by the Mittag-Leffler matrix function. An example is provided to illustrate the theory.

On the controllability of fractional dynamical systems

This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

Structural Damage Identification and Location Using Grammian Matrices

In this paper, an approach using observability and controllability grammian matrices is proposed to determine if structural damage has occurred together with an estimate of its location. The theory is outlined and simulations are carried out on a simple structure to demonstrate the method. Experimental tests were also carried out to demonstrate the validity of the approach using real signals. The dynamic properties of the structure are identified using the eigensystem realization algorithm (ERA) and a reduced order state-space model of the system subsequently constructed. Either the observability or controllability grammians can then be used depending on the number of sensors available. It is shown that these are sensitive to both the degree and location of the damage and offer promise for structural health monitoring applications.

Adaptive Vibration Control of Flexible Plate Structures With Actuator Placement Optimization

This paper investigates piezoelectric actuator placement optimization on a rectangular plate structure and the vibration control of the structure. In the first part, an actuator placement optimization method is developed based on maximizing the controllability grammian. It is then implemented using Structuring Analysis in ANSYS Finite Element Analysis Package and Genetic Algorithm. In the second part, a filtered-x LMS-based multi-channel adaptive control is used to suppress vibration response of the plate. Numerical simulations are performed in suppressing tri-sinusoidal response at three points of the plates and the results show the control algorithm is efficient and robust in reducing the plate’s vibration. The results also demonstrate that the developed actuator placement optimization method is effective to reduce the energy required for achieving significant vibration reduction.