A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators. (English) Zbl 0632.34005

Paden, Brad E.; Sastry, Shankar S.

IEEE Trans. Circuits Syst. 34, 73-82 (1987).

A link is established between the definition of Filippov’s solution concept for ordinary differential equations with a discontinuous right- hand side [A. F. Filippov, Mat. Sb., N. Ser. 51(93), 99-128 (1960; Zbl 0138.322)] and Clarke’s generalized gradient [F. H. Clarke, Optimization and nonsmooth analysis (1983; Zbl 0582.49001)]. According to Filippov’s definition, solutions to \(\dot x=f(x)\) are those to the differential inclusion \(\dot x(t)\in K(f)(x(t))\), where K(f) is a suitably defined multifunction depending on f. The authors remark that if f is locally Lipschitz, then \(K(\nabla f)=\partial f\), where \(\partial f\) denotes Clarke’s generalized gradient. This relation is useful for computing K in various situations. Such a calculus is applied to the variable structure control of a robot manipulator.

Reviewer: T.Zolezzi

MSC:

34A34

Nonlinear ordinary differential equations and systems

Keywords:

discontinuous right-hand side; Clarke’s generalized gradient; differential inclusion

Citations:

Zbl 0138.322; Zbl 0582.49001