TABLE OF CONTENTS The concept of the transfer matrix is an extension of that of the transfer function. Here we shall first obtain transfer function of a linear single-input-single-output control system and then transfer the matrix of a linear multiple-input-multiple-output control system using respective state and output equations.
Consider the state model of a linear single-input-single-output system
Taking the Laplace transform of Equations (1.43) we have
Consider Equation (1.44 A)
Assume zero initial conditions, i.e., x(0) = 0, we get
Put X(s) from Equation (1.45) into (1.44 B) and we get
Equation (1.46) gives the expression of transfer function of a linear single-input-single-output system.
Consider the state model of a liner multi-input-multi-output system
Take the Laplace transform of Equations (1.47) and we have
Consider Equation (1.48 A) assuming zero initial conditions, i.e., x(0) = 0, we get
Put X( s) from Equation (1.49) into (1.48 B) and we get
Equation (1.50) gives the expression of the transfer matrix of a linear multi-input-multi-output system, where transfer matrix G( s) relates the output Y( s) to the input U( s) as
or, in an expanded form
The (i, j) th element G ij(s);i= 1, 2, , p; j = 1, 2, m of G (s) is the transfer function relating the it i th ouput to the j th input.
It is...
