*(English)*Zbl 0614.93039

In this paper, the discrete model reference adaptive control system (DMRACS) using a new discrete-time operator, called the delta operator, is concerned. This approach resolves the problem of the non-minimum phase zeros of the DMRACS which is caused by the rapid sampling.

Let ${A}^{\text{'}}\left(D\right)y\left(t\right)={B}^{\text{'}}\left(D\right)u\left(t\right)$ be the stably invertible continuous model of the single-input single-output linear plant to be controlled, where $D$ is a differential operator. Assuming that the plant is sampled at interval ${\Delta}$ and the input is applied via a zero order hold, then the discrete model of the plant becomes $\overline{A}\left(q\right)y\left(k{\Delta}\right)=\overline{B}\left(q\right)u\left(k{\Delta}\right),$ where $q$ is the usual forward shift operator. If the continuous model of the plant has relative degree greater than one, the rapid sampling gives rise to the discrete model having unstable zeros, which implies that usual DMRACS cannot be applied to this plant. To avoid this difficulty, an operator $\delta $ defined by $\delta =(q-1)/{\Delta}$ is introduced. Then the alternative discrete model becomes $A\left(\delta \right)y\left(k{\Delta}\right)=B\left(\delta \right)u\left(k{\Delta}\right)$. Here, $B\left(\delta \right)$ can be expressed by ${B}_{R}\left(\delta \right)+B\in \left(\delta \right)$, where ${B}_{R}\left(\delta \right)$ converges to ${B}^{\text{'}}\left(\delta \right)$ and $B\in \left(\delta \right)$ converges to zero as ${\Delta}$ tends to zero. Since the polynomial in $\delta $ is stable if its zeros lie in a circle of radius $1/{\Delta}$ centered at $(-1/{\Delta},0)$, there exists a constant ${{\Delta}}_{i}$ such that ${B}_{R}\left(\delta \right)$ is exponentially stable for all ${\Delta}\le {{\Delta}}_{1}$ provided that the continuous model of the plant is stably invertible.

The model reference adaptive control system is designed based on this $\delta $-model, $A\left(\delta \right)y\left(k{\Delta}\right)={B}_{R}\left(\delta \right)u\left(k{\Delta}\right)+B\in \left(\delta \right)u\left(k{\Delta}\right)$, where $B\in \left(\delta \right)u\left(k{\Delta}\right)$ is ignored as a modelling error. The usual parameter estimation techniques for the system having the modelling error are applied to this $\delta $-model with a slight modification. Among them, the projection algorithm and the least squares algorithm of a dead zone type are presented here. The plant input is generated using the estimated parameters. The global stability of the total adaptive system is also established.

##### MSC:

93C40 | Adaptive control systems |

93B35 | Sensitivity (robustness) of control systems |

93C55 | Discrete-time control systems |

68U20 | Simulation |

93C57 | Sampled-data control systems |

93D99 | Stability of control systems |