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A Guide to Arithmetic [Lecture notes] by Robin Chapman

By Robin Chapman

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C. W ILLEMS, Introduction to Mathematical Systems Theory: A Behavioural Approach, Springer-Verlag, New York, 1998. [27] R. E. ROBERSON AND R. S CHWERTASSEK, Dynamics of Multibody Systems, Springer-Verlag, Heidelberg, 1988. [28] W. S CHIEHLEN, Multibody Systems Handbook, Springer-Verlag, Heidelberg, 1990. [29] W. S CHIEHLEN, Advanced Multibody System Dynamics, Kluwer Academic Publishers, Stuttgart, Germany, 1993. [30] K. S CHLACHER AND A. K UGI, Automatic control of mechatronic systems, Int. J. Math.

Since G ∗1 has full row rank k, G + ∗1 is continuous and G ∗1 G ∗1 = Ik . 14) with arbitrary functions w such that G ∗1 w = 0. 14) into + − (Dx) − K Dx + DG + ∗1 B∗ D Dx − DG ∗1 q = Dw, Q0 x − + Q0 G+ ∗1 B∗ D Dx − Q0 G+ ∗1 q = Q 0 w. 16) ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 42 Chapter 3. Linearization of DAEs Set w = 0. 13) is uniquely solvable and provides a continuously differentiable Dx. 16). 13), which proves the first assertion. If m = k, then G ∗1 is also injective, and thus the unique solvability is evident and so is the second assertion.

In a general behavior setting this does not matter, since the variables are not distinguished and one can make a decision which variables one wants to consider as inputs and states at this point; see [12, 13, 26]. If, however, the application clearly defines which variables are input or state variables, then a reinterpretation of these variables is necessary. If d + a > n, then we just introduce a new vector x˜ by attaching d + a − n of the input variables in u to the vector x and considering a new input vector u˜ which contains the remaining variables.