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Linear Switching (Local Theory)

  • Mike R. Jeffrey
Chapter

Abstract

In this section we take a look at systems that depend only linearly on the switching multiplier s \(\boldsymbol{\lambda } = (\lambda _{1},\ldots,\lambda _{m})\) and are therefore expressible in the form
$$\displaystyle{ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x};\boldsymbol{\lambda }) = \mathbf{a}(\mathbf{x}) + \underline{\underline{B}}(\mathbf{x})\boldsymbol{\lambda }\;, }$$
where \(\underline{\underline{B}}\) is an n × m matrix. In relation to ( 5.3), the quantities in (8.1) are
$$\displaystyle{\mathbf{a} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,\qquad \mathbf{B}_{j} = \mbox{ $\frac{1} {2}$}\sum _{j=1}^{m}\sum _{ \kappa _{j}=\pm }\kappa _{j}\mathbf{f}^{\kappa _{1}\ldots \kappa _{j}\ldots \kappa _{m} }\;,}$$
where Bj is the jth column of \(\underline{\underline{B}}\).

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mike R. Jeffrey
    • 1
  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK

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