Page 259 - Modern Control Systems

Terms and Concepts 233

Jordan canonical form A block diagonal canonical form State differential equation The differential equation for
for systems that do not possess distinct system poles. the state vector: x = Ax + Bu.
Matrix exponential function An important matrix func- State of a system A set of numbers such that the knowl-
2
tion, defined as e Af = I + At + (A/) /2! + • • • + edge of these numbers and the input function will,
k
(A() /kl + • • •, that plays a role in the solution of lin- with the equations describing the dynamics, provide
ear constant coefficient differential equations. the future state of the system.
Output equation The algebraic equation that relates the State-space representation A time-domain model com-
state vector x and the inputs u to the outputs y prising the state differential equation x = Ax + Bu
through the relationship y = Cx + Du. and the output equation, y = Cx + Du.
Phase variable canonical form A canonical form described State variables The set of variables that describe the system.
by n feedback loops involving the a n coefficients of the State vector The vector containing all n state variables,
nth order denominator polynomial of the transfer func- Xi, X2, . . . , X n.
tion and m feedforward loops involving the b,„ coeffi- Time domain The mathematical domain that incorpo-
cients of the /nth order numerator polynomial of the rates the time response and the description of a sys-
transfer function. tem in terms of time t.
Phase variables The state variables associated with the Time-varying system A system for which one or more pa-
phase variable canonical form. rameters may vary with time.
Physical variables The state variables representing the Transition matrix ¢(/) The matrix exponential function
physical variables of the system. that describes the unforced response of the system.