Adaptive Control of Robot Manipulators: A Unified by An-chyau Huang
By An-chyau Huang
This e-book introduces an unified functionality approximation method of the keep an eye on of doubtful robotic manipulators containing basic uncertainties. it really works at no cost house monitoring regulate in addition to compliant movement regulate. it truly is appropriate to the inflexible robotic and the versatile joint robotic. inspite of actuator dynamics, the unified procedure continues to be possible. a majority of these beneficial properties make the publication stand proud of different present guides.
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Extra info for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach
Example text
Since the Lyapunov function in (28) is not radially unbounded, the global behavior cannot be concluded. If the reference input is PE, convergence of the estimated parameters can further be proved. 3 Persistent excitation We have introduced MRAC laws for linear time-invariant systems to have asymptotic tracking error convergence performance. 10 Model Reference Adaptive Control (MRAC) 51 obtain uniformly boundedness of parameter errors in those systems. In this section, we are going to investigate the problem of persistency of excitation of signals in the closed loop system, which relates to the convergence of the parameter vector.
From the computation n ∫ ∞ 0 x T Qxdt = − ∫ ∞ 0 p ɺ = V0 − V ∞ < ∞ Vdt we have x ∈ L2 . Boundedness of xɺ can be obtained by observing (30a). Therefore, by Barbalat’s lemma, we have proved x → 0 as t → ∞ . n 52 Chapter 2 Preliminaries To prove asymptotic convergence of z, we need to prove that ∀ε > 0 , ∃Tε > 0 such that z (t ) < ε , ∀t ≥ Tε . 6-3), inequality (33) becomes λmax (P ) x(Tε ) + z (Tε ) ≥ z (t ) 2 z (t ) ≤ λmax (P ) x(Tε ) + z (Tε ) 2 2 2 This further implies 2 (34) Since we have proved x → 0 as t → ∞ , this implies that ∀ε > 0 , ∃tε > 0 such that ∀t ≥ tε , x(t ) ≤ ε 2λmax (P ) .
Consider a special linear system (Marino and Tomei 1996) xɺ = Ax + ΩT (t )z (30a) zɺ = −Ω(t )Px (30b) where x ∈ℜ n and z ∈ℜ p . A ∈ℜ n × n is a Hurwitz matrix, and P ∈ℜ n × n is a T positive definite matrix satisfying A P + PA = −Q for some positive definite n × n matrix Q. The p × n real matrix Ω has the property that Ω(t ) and ɺ (t ) are uniformly bounded. System (30) is frequently encountered in the Ω parameter convergence analysis of adaptive systems. Equation (30a) usually corresponds to the tracking error dynamics and (30b) is the update law.