# Digital Control Systems by Prof. Dr.-Ing. Rolf Isermann (auth.)

By Prof. Dr.-Ing. Rolf Isermann (auth.)

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**Extra resources for Digital Control Systems**

**Example text**

5-13) of a transfer function B (z) (3. 5-14) G(z) = A(z) leading to 1+w m-1 + ... 5-15) Multiplying by (1-w)m yields A (w) = ( 1+w) m + a 1 ( 1+w) ~1 ( 1-w) + ... +am ( 1-w) m (3. 5-16) Now the Hurwitz criterion can be applied for the polynomial :1\(w) = 0. 5-17) 38 3. Discrete-time Systems This criterion states that the coefficients have to exist and have to carry the same algebraic sign. Then the system is not monotonic unstable. To avoid oscillatory unstability the Hurwitz-determinants have to be positive for systems higher than second order (or the Routh criterion has to be falfilled).

6-27) are presented. A first possibility which corresponds to the recursive solution of difference equations for a given input signal u(k) and initial conditions ~ (0) is 3. ::_k~(O) + ~ (k) k L Ai- 1 b u(k-i). 6-44) homoge- particular solution (conneous solution volution sum) where Ak A•A ... A. --' k y(k) can finally be obtained from Eq. 6-27). If u(k) is given explicitely as a z-transform, a second possible solution can be used. The z-transform then furnishes }{~(k)} = ~(z) }{~(k+1)} = z[~(z)- ~(0)] (applying the theorem of shifting to the left) .

B) The exponential function: x(kT 0 ) = e z z-e-aTo -akT 0 (a is real) z c) The sine function: x(kT 0 ) With it follows, using the result of b) with a iw 1 , or with a that x(z) 2i [z_)w1To zsinw 1 T0 2 z +a 1 z+1 0 These examples have shown how the z-transforms of some simple functions can be obtained. In this way, a table of transforms of common functions can be assembled. A short table of corresponding continuous time functions, Laplace-transforms and z-transforms, is given in the Appendix. This table shows: --- a) There is a direct correspondence of the denominators of x(s) and x(z) (s-s 1 ) n (z-z 1 ) n with z 1 = T s e 0 1 b) There is no direct correspondence of the numerators of x(s) and x(z).