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Essentials of control techniques and Keyword Stats_10

Tham khảo tài liệu 'essentials of control techniques and keyword stats_10', kỹ thuật - công nghệ, điện - điện tử phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 256 Essentials of Control Techniques and Theory and we are back in the world of eigenvectors and eigenvalues. If we can find an eigenvector x with its eigenvalue X then it will have the property that next x kx representing stable decay or unstable growth according to whether the magnitude of X is less than or greater than one. Note carefully that it is now the magnitude of X not its real part which defines stability. The stable region is not the left half-plane as before but is the interior of a circle of unit radius centered on the origin. Only if all the eigenvalues of the system lie within this circle will it be stable. We might just stretch a point and allow the value 1 j0 to be called stable. With this as an eigenvalue the system could allow a variable to stand constant without decay. 18.6 Initial and Final Value Theorems In the case of the Laplace transform we found the initial and final value theorems. The value of the time function at t 0 is given by the limit of sF s as s tends to infinity. As s tends to zero sF s tends to the value of the function at infinity. For the z-transform the initial value is easy. By letting z tend to infinity all the terms in the summation vanish except for x 0 . Finding the final value theory is not quite so simple. We can certainly not let z tend to zero otherwise every term but the first will become infinite. First we must stipulate that F z has no poles on or outside the unit circle otherwise it will not represent a function which settles to a steady value. If G z is the z-transform of g n then if we let z tend to 1 the value of G z will tend to the sum of all the elements of g n . If we can construct g n to be the difference between consecutive elements off n g n f n - f n - 1 then the sum of the terms of g n N E g n 0 N f n - f n - 1 0 f N - f -1 Digital Control in More Detail 257 Since fl l is zero as is every negative sample the limit of this when Nis taken to infinity is the final value off n . So what is the .