By R. Balasubramanian

Kingdom house research of platforms. Modal regulate. Quadratic optimum keep watch over. layout of observers. different chosen layout tools within the nation area. Frequency area research of multivariable platforms. The Inverse Nyquist array technique. The attribute locus technique. Frequency area layout via factorisation equipment. chosen stochastic difficulties

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17) have n rows linearly independent for t0 ^ / ^ oo. Define W(t09 h) = I [e~-AtBBTe~ATt] dt We maintain that W(t0, tx) is nonsingular. For, if it were singular there exists a n a / 0 such that which in turn implies or equivalently that the rows of e AtB are linearly dependent. This is a contradiction. Hence W(t0, tx) is nonsingular. Now choose u(t) =BTe-AtW~l(t0, tx)[e~Atix{tx)~-e-At«x0} Substituting this in eqn. 17, the equation is satisfied. This means that we have explicitly constructed one n(t), t0 ^ x ^ tx to transfer any initial state x0 to any final state x(t{).

5 Evaluation of eAt In eqn. 4 eAt was defined via an infinite series. Often a truncated form of this infinite series, based on some error criterion, is used in machine computation of eAt. A number of 'methods' have been suggested, for evaluating eAt as a closed form expression. We consider below two of the most frequently used methods for this purpose. Method 1 Find (si — A) ~l and then evaluate its inverse Laplace transform. ;•,] _\(s+2)(s Hence 2e-2t- Method 2 We defined eAt as However, according to the Cayley-Hamilton theorem An+ 1 n- X ajAJ = 0 o The state space analysis of systems 19 Thus the terms involving A\ i^n in the series expansion for eAt can be replaced in terms of A\ i < n.

N — 1. If a particular eigenvalue, Xj say, is repeated (M + 1) times, one uses j =0 f o r / ? = 0 , 1 , . ,Af Now that an adequate number of independent equations are available, a,(0 and hence eAt can be calculated. 6 Let •[-] The eigenvalues are readily found to be 5 and 2. 7 Let r -2 3i L-l/3 -4J The eigenvalues are —3, —3. One now gets e~3t =