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In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: [1] A X + X B = C . {\displaystyle AX+XB=C.} It is named after English mathematician James Joseph Sylvester .
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations.It is a popular method for solving the large matrix equations that arise in systems theory and control, [1] and can be formulated to construct solutions in a memory-efficient, factored form.
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
which is known as the discrete-time dynamic Riccati equation of this problem. The steady-state characterization of P , relevant for the infinite-horizon problem in which T goes to infinity, can be found by iterating the dynamic equation repeatedly until it converges; then P is characterized by removing the time subscripts from the dynamic equation.
Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable. Thus a non-time variable jumps from one value to another as time moves from one time period to the next.
The state of a linear, time-invariant discrete-time system is assumed to satisfy (+) = + () = + where, at time , () is the plant's state; () is its inputs; and () is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current states and the current inputs.
Again, as well as in the continuous time case and in the discrete time case, one may be interested in discovering if the system given by the pair ((), ()) is controllable or not. This can be done in a very similar way of the preceding cases.
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form ˙ = () + (), =, where () are the states of the system, () is the input signal, () and () are matrix functions, and is the initial condition at .