Computation of Gram Matrix and Its Partial Derivative Using Precise Integration Method for Linear Time-Invariant Systems

Gram matrix is an important tool in system analysis and design as it provides a description of the input-output behavior for system; its partial derivative matrix is often required in some numerical algorithms. It is essential to study computation of these matrices. Analytical methods only work in some special circumstances; for example, the system matrix is diagonal matrix or Jordan matrix. In most cases, numerical integration method is needed, but there are two problems when compute using traditional numerical integration method. One is low accuracy: as high accuracy requires extremely small integration step, it will result in large amount of computation; and another is stability and stiffness issues caused by the dependence on the property of system matrix. In order to overcome these problems, this paper proposes an efficient numerical method based on the key idea of precise integration method (PIM) for the Gram matrix and its partial derivative of linear time-invariant systems. Since matrix inverse operation is not required in this method, it can be used with high precision no matter the system is normal or singular. The specific calculation algorithm and block diagram are also given. Finally, numerical examples are given to demonstrate the correctness and validity of this method.