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Modern Control System Theory and Design, 2nd Edition by Stanley M. Shinners

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2.26.  STATE TRANSITION MATRIX

The state transition matrix relates the state of a system at t = t0 to its state at a subsequent time t, when the input u(t) = 0. In order to define the state transition matrix of a system, let us consider the general form of the state equation [see Eq. 2.197]:

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The Laplace transform of Eq. (2.252) is given by

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where X(s) is the Laplace transform of x(t) and U(s) is the Laplace transform of u(t). Solving for X(s), we obtain

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The inverse Laplace transform of Eq. (2.254) gives the state transition equation

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Figure 2.37   Unit step response of system shown in Figure 2.29 obtained using the MATLAB Program in Table 2.12.

where the state transition matrix is defined by

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The first term on the right-hand side of Eq. (2.255) is known as the homogeneous solution and is due only to the initial conditions; the second term on the right-hand side of Eq. (2.255), the convolution ...

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