Abstract
Parameter adaptation algorithms are the key step for building an adaptive control system. An extensive coverage of the subject is provided in this chapter. Both synthesis and analysis of the parameter adaptation algorithms in a deterministic environment will be considered. Stability and convergence issues will be emphasized.
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- 1.
A symmetric square matrix F is termed positive definite if x T Fx>0 for all x≠0, x∈ℝn. In addition: (i) all the terms of the main diagonal are positive, (ii) the determinants of all the principals minors are positive.
- 2.
It is assumed that the matrix \(\sum ^{t}_{i=1} \phi(i-1)\phi^{T}(i-1)\) is invertible. As it will be shown later this corresponds to an excitation condition.
- 3.
This condition corresponds to “persistent excitation” condition on the observation vector φ(t). See Sect. 3.4.
- 4.
In fact, taking advantage of the term \(\sum_{t=0}^{t_{1}}\phi^{T}(t)F(t)\phi(t)\nu^{2}(t+1)\) in (3.226) and the fact that the equivalent linear block is characterized by a unitary gain, one can conclude that: lim t→∞[1+φ T(t)F(t)φ(t)]ε 2(t+1)=0.
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Landau, I.D., Lozano, R., M’Saad, M., Karimi, A. (2011). Parameter Adaptation Algorithms—Deterministic Environment. In: Adaptive Control. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-664-1_3
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DOI: https://doi.org/10.1007/978-0-85729-664-1_3
Publisher Name: Springer, London
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