An Exponential Linear Model Matching for a Closed Kinematics Chain
Carlos Aguilar. I1, Moisés Bonilla2 and Oscar. Chavoya3
1 Centro de Investigación en Computación del IPN. Laboratorio de Metrología y Control. Av. J. de Dios Batiz s/n; México D.F., C.P. 07738.; México. E–mail: caguilar@cic.ipn.mx
2 CINVESTAV–IPN. Departamento Control Automático.
3 Camelback High School, Phoenix, AZ85016, U.S.A.
]]> Abstract
In this paper we propose an implicit linear control law for a two degree freedom manipulator whose aim is to stabilize and match a linear model. We show that for any finite initial condition there exists a sufficient small control parameter, ε, such that the model matching is exponentially achieved.
Keywords: Parallel Robots, Lyapunov 2nd method, Stability, Implicit Systems, PD control law.
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Appendix
Proof of the Theorem 2: Let us first separate the space of e in two regions using negativeness of the function Z (.), as follows
We proceed to consider two interesting cases: First Case: Let us first consider that e (t) never leaves Si, then
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Second Case: Let us next consider that e (t) comes into Se for some T. Then by continuity of e we have
Integrating the last inequality for Δ t > 0, and using properties (15), (16) of V we get (recall a definition of 
(.) (18))
using now conditions (19) in the above inequality, we have that
Then e (T + Δ t) never lives Se for any arbitrary Δ t > 0. Let us finally analyze inequality (40). For this we need to consider the following to cases.
They by indication:
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where 
= min 
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