By Sivan, Kwaternaak

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E,,), A = diag (A,, A,, . . , AJ. 1-77a 1-77b 16 Elemen$ of Linenr System Theory Tlterz T is noruingr~larand A can be represented as A = TAT-'. 1-78 Here the notation 1-77a implies that the vectors e,, e,, . . , e,, are the columns of the matrix T , and 1-77b means that A is a diagonal matrix with A,, A,, . ,A, as diagonal elements. I t is said that T diagonalizes A. The following fact is easily verified. 6. 5. Then 1-79 This result makes it simple to compute exp ( A t ) once A is diagonalized.

A , and corresponding characteristic vectors e,, e,, . , e,,, the response of the system to any initial state . 28 Elements of Linenr System Theory can be represented as ,I x(t) = 2 piek'"ei, i=l where the scalars pi, i = 1,2, . . , ] I follow from the initial state x(0). 4). Clearly, the stability of the system in both cases is determined by the characteristic values A+ We have the following result. 13. Tlze tirne-illvariant linear sj~steii~ d(t) = Ax(t) is stable in tlre seiwe of Lj~ap~rnou ifarid aiily if (a) all of tlre cl~aracteristicualties of A haue i~orlpositiverealparts, and (h) to arg~clraracteristic value a11 the iiiiagiiiarj~axis with ~rrriltiplicity111 tlrere correspond exactly n1 characferistic uectors of the matrix A.

4 is quite useful for the computation of the transition matrix by a digital computer since the repeated multiplications and additions are easily programmed and performed. Such programs must include a stopping rule to truncate the infinite series after a finite number of terms. A usual stopping rule is to truncate when the addition of a new term changes each of the elements of the partial sum by less than a specified fraction. Numerical difficulties may occur when M is too large; this means that t - to in 1-68 cannot be chosen too large (see Kalman, 1966; Knlman and Englar, 1966).

### Linear optimal control system by Sivan, Kwaternaak

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