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**Rapprochement between continuous and discrete model reference adaptive control.**
*(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 \(A'(D)y(t)=B'(D)u(t)\) 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 \(\bar A(q)y(k\Delta) = \bar B(q)u(k\Delta),\) 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 \(A(\delta)y(k\Delta) = B(\delta)u(k\Delta)\). Here, \(B(\delta)\) can be expressed by \(B_ R(\delta)+B\in (\delta)\), where \(B_ R(\delta)\) converges to \(B'(\delta)\) and \(B\in (\delta)\) 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 \(B_ R(\delta)\) is exponentially stable for all \(\Delta \leq \Delta_ 1\) provided that the continuous model of the plant is stably invertible.

The model reference adaptive control system is designed based on this \(\delta\)-model, \(A(\delta)y(k\Delta) = B_ R(\delta)u(k\Delta) + B\in (\delta)u(k\Delta)\), where \(B\in (\delta)u(k\Delta)\) 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.

Let \(A'(D)y(t)=B'(D)u(t)\) 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 \(\bar A(q)y(k\Delta) = \bar B(q)u(k\Delta),\) 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 \(A(\delta)y(k\Delta) = B(\delta)u(k\Delta)\). Here, \(B(\delta)\) can be expressed by \(B_ R(\delta)+B\in (\delta)\), where \(B_ R(\delta)\) converges to \(B'(\delta)\) and \(B\in (\delta)\) 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 \(B_ R(\delta)\) is exponentially stable for all \(\Delta \leq \Delta_ 1\) provided that the continuous model of the plant is stably invertible.

The model reference adaptive control system is designed based on this \(\delta\)-model, \(A(\delta)y(k\Delta) = B_ R(\delta)u(k\Delta) + B\in (\delta)u(k\Delta)\), where \(B\in (\delta)u(k\Delta)\) 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.

Reviewer: Y.Mutoh

### MSC:

93C40 | Adaptive control/observation systems |

93B35 | Sensitivity (robustness) |

93C55 | Discrete-time control/observation systems |

68U20 | Simulation (MSC2010) |

93C57 | Sampled-data control/observation systems |

93D99 | Stability of control systems |

### Keywords:

discrete model reference adaptive control system; non-minimum phase zeros; rapid sampling; modelling error; projection algorithm; least squares algorithm
Full Text:
DOI

### References:

[1] | Agarwal, R. C.; Burrows, C. S., New recursive digital filter structures having low sensitivity and roundoff noise, IEEE Trans. Ccts Syst., CAS22, 12 (1975) |

[2] | Ǎström, K. J.; Hagander, P.; Sternby, J., Zeros of sampled systems, Automatica, 20, 31-39 (1984) · Zbl 0542.93047 |

[3] | Åström, K. J.; Wittenmark, B., Computer Controlled Systems (1984), Prentice-Hall: Prentice-Hall New Jersey |

[4] | Clarke, D. W.; Gawthrop, P. J., Hybrid self tuning control and its interpretation, (Proc. 3rd IMA Conf. Control Theory (1981)) · Zbl 0507.93048 |

[5] | Desoer, C. A., Slowly varying discrete system \(x_{i+l} = A_{i\) |

[6] | Edmunds, J. M., Identifying sampled data systems using difference operator models, Control Systems Centre UMIST Technical Report No. 601 (1985) |

[7] | Egardt, B., Unification of some continuous-time adaptive control schemes, IEEE Trans. Aut. Control, AC-24, 588-592 (1979) · Zbl 0417.93046 |

[8] | Gawthrop, P. J., Hybrid self-tuning control, (Proc. IEE, 127 (1980)), 5, Pt.D. · Zbl 0507.93048 |

[9] | Goodwin, G. C.; Hill, D. J.; Palaniswami, M., Towards an adaptive robust controller, (7th IFAC Symp. Ident. Syst. Param. Estim.. 7th IFAC Symp. Ident. Syst. Param. Estim., York (1985)) · Zbl 0544.93044 |

[10] | Goodwin, G. C.; Middleton, R. H., (Leondes, C. T., Continuous and Discrete Adaptive Control, Vol. XXXIV. Control and Dynamics Systems (1985)) · Zbl 0672.93046 |

[11] | Goodwin, G. C.; Ramadge, P. J.; Caines, P. E., Discrete time multivariable adaptive control, IEEE Trans. Aut. Control, AC-25, 449-456 (1980) · Zbl 0429.93034 |

[12] | Goodwin, G. C.; Sin, K. S., Adaptive Filtering Prediction and Control (1984), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0653.93001 |

[13] | Karwoski, R. J., Introduction to the Z-transform and its derivation, (Tutorial paper TRW Products, El Segundo, California (1979)) |

[14] | Kreisselmeier, G.; Anderson, B. D.O., Robust model reference adaptive control, DFVLR Tech. Report (1985), Also IEEE Trans Aut. Control, to appear |

[15] | Middleton, R.; Goodwin, G. C., Improved finite word length characteristics in digital control using Delta operators, (Technical Report (1985), Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308: Department of Electrical and Computer Engineering, University of Newcastle, NSW 2308 Australia) · Zbl 0672.93046 |

[16] | Morse, A. S., Global stability of parameter-adaptive control systems, IEEE Trans. Aut. Control, AC-25, 433-439 (1980) · Zbl 0438.93042 |

[17] | M’Saad, M.; Ortega, R.; Landau, I. D., Adaptive controllers for discrete time systems with arbitrary zeros—an overview, Automatica, 21, 413-425 (1985) · Zbl 0571.93041 |

[18] | Narendra, K. S.; Lin, Y. H.; Valavani, L. S., Stable adaptive controller design, Part II: Proof of stability, IEEE Trans. Aut. Control, AC-25, 440-449 (1980) · Zbl 0467.93049 |

[19] | Orlandi, G.; Martinelli, G., Low sensitivity recursive digital filters obtained via the delay replacement, IEEE Trans. Ccts Syst., CAS-31, 7 (1984) |

[20] | Tschauner, J., Introduction à la théorie des systèms èchantillonnés, ((1963), Dunod: Dunod Paris), 42 · Zbl 0126.30501 |

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