A robust direct adaptive controller.

*(English)*Zbl 0628.93040A robust direct continuous-time adaptive control algorithm is presented in this paper. The plant is represented by
\[
G(s)=G_ 0(s)(1+\mu \Delta_ 2(s))+\mu \Delta_ 1(s)
\]
where \(G_ 0(s)\) is a modelled part and \(\mu \Delta_ 1(s)\), \(\mu \Delta_ 2(s)\) is an additive and a multiplicative plant perturbation, respectively. The algorithm is designed for the modelled part \(G_ 0(s)\) which is assumed to be minimum phase and of known relative degree and order, but applied to the full order plant G(s) which, due to the unmodelled dynamics, may be nonminimum phase and of unknown order. It is assumed that a positive constant \(M_ 0\) such that \(\| \theta^*\| <M_ 0\) is known where \(\theta^*\) is the desired control parameter vector, and that a lower bound \(p_ 0>0\) on the stability margin \(p>0\), for which the poles of \(\Delta_ 1(s-p)\), \(\Delta_ 2(s-p)\) are stable, is known. For the robustness, the proposed adaptive law to adjust the parameter vector consists of the normalization of the conventional algorithm part and the improved \(\sigma\)-modification part. It is shown that by using this algorithm, there exists a \(\mu^*>0\) such that for each \(\mu \in [0,\mu^*)\) all the signals of the total adaptive system for the full order plant are bounded for any bounded initial conditions. It is also shown that this algorithm guarantees small residual tracking error and that if there is no modelling error the tracking error converges to 0.

Reviewer: Y.Mutoh

##### MSC:

93C40 | Adaptive control/observation systems |

93B35 | Sensitivity (robustness) |

93B40 | Computational methods in systems theory (MSC2010) |

93D99 | Stability of control systems |