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APPENDIX F State variable models and transient analysis  303





                   Example F.1
                    Consider a system with two inputs (f) and two outputs (y). Then Eq. (F.15) may be written as

                                          y 1 s ðÞ  G 11 G 12  f 1 s ðÞ
                                               ¼                              (F.16)
                                          y 2 sðÞ  G 21 G 22  f 2 sðÞ
                       Thus, the transfer function between output y i and input f j is given by the matrix element.
                                          y i s ðÞ
                                             ¼ G ij sðÞ; i ¼ 1,2; j ¼ 1,2     (F.17)
                                          f j s ðÞ



                  F.3.3 Transient response of MIMO systems
                  The response x(t) for a given initial condition x(0) and forcing term f(t) is determined
                  by taking the inverse Laplace transform of Eq. (F.9). Thus,

                                        h          i    h            i
                                                1
                                                                1
                                 xtðÞ ¼ L  1  ð sI  AÞ x 0ðÞ + L  1  ð sI  AÞ BFsðÞ  (F.18)
                                                     h       i
                                        Define ϕ tðÞ   L  1  ð sI  AÞ  1        (F.19)
                  The inverse transform in Eq. (F.18) can be accomplished by using the convolution
                  integral (see Appendix D). If Y(s) ¼G(s)X(s), the inverse transform is achieved by
                  the following convolution integral.
                                                      ð ∞
                                    ytðÞ ¼ L  1  ½ GsðÞXsðފ ¼  gt τÞx τðÞ dτ
                                                        ð
                                                      0
                  Using the convolution integral and the definition (F.19), x(t) becomes.
                                                   ð t
                                       xtðÞ ¼ φ tðÞx 0ðÞ + φ t τÞBf τðÞdτ       (F.20)
                                                     ð
                                                   0
                  (The convolution between two time functions p(t) and q(t) is defined by the following
                  integral:    t
                               Ð
                                 ð
                     ptðÞ∗qtðÞ ¼  pt τÞ q τðÞ dτ, p(t) and q(t) are causal functions).
                               0
                     Eq. (F.20) is similar the solution of a linear first order system with initial condi-
                  tion. Furthermore, the solution of a first order linear time invariant system.
                                              dx
                                                ¼ ax + ftðÞ                     (F.21)
                                               dt
                  is given by
                                                    ð t
                                              at
                                                       ð
                                        xtðÞ ¼ e x 0ðÞ + e at τÞ f τðÞdτ        (F.22)
                                                    0
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