This website contains more than 250 free tutorials! Every tutorial is accompanied by a YouTube video. By applying the inverse Laplace transform to every entry of the matrix ( 20), we obtain the desired matrix exponential.Īnd consequently, our state trajectory is determined by The inverse Laplace transform of a matrix, is obtained by taking inverse Laplace transforms of every entry of the matrix. Next, we need to compute the inverse Laplace transform of this matrix. The inverse of this matrix is the resolvent matrix, and this inverse is given by First, we need to compute the resolvent matrix. To compute the matrix exponential, we will use the formula ( 15). We consider the following problem: Compute the state trajectory of a linear dynamical system Computation of Matrix Exponential – Application to State Trajectory Computation These approaches will be explained in Part 2 of this tutorial. Of course, there are also other approaches for computing the matrix exponential. Instead, we can use the inverse Laplace transform to compute the matrix exponential. Direct computation of this expansion might be a tedious task. That is, we do not need to directly compute the series expansion given by ( 4). This formula is important since it tells us that we can compute the matrix exponential by computing the inverse Laplace transform of the resolvent matrix. By using this simple result, we can express the state of the system at the time instant as follows Also, this simple proof explains the main motivation for defining the matrix exponential as a series. That is, the state transition matrix is equal to the matrix exponential of. We haveīy taking the inverse Laplace transform, we obtain Let us use this formula to represent the resolvent matrix as a series. We assume that the series on the right-hand side of this equation converges. Next, we will show that the state transition matrix is actually equal to the matrix exponential of. That is the state of the system at the time instant is calculated by multiplying the state transition matrix by the initial state. On the other hand, the matrixĬonsequently, the equation ( 8) can be written as follows The matrix is called the resolvent matrix of the matrix. Where is the notation for the inverse Laplace transform. Let us apply the inverse Laplace transform to the last equation. Let us apply the Laplace transform to the system ( 16). Let us assume that the initial condition of the system is given by. Where is the state vector and is the system matrix. Let us consider a linear dynamical system Matrix exponential, dynamical systems, and control theory We answer these questions in the next section.
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