Optimal adaptive control and consistent parameter estimate for deterministic systems with quadratic cost.

*(English)*Zbl 0649.93038This well-written paper describes results on deterministic adaptive control. The key results rely heavily on a theorem of H. Chen and L. Guo [ibid. 44, 1459-1476 (1986; Zbl 0604.93061); ibid. 45, 2183-2202 (1987; Zbl 0624.93036); SIAM J. Control Optimization 25, 845-867 (1987; Zbl 0632.93045)], which concludes consistency of parameter estimation under assumptions of growth conditions on the observations (system inputs and outputs) used in the parameter estimation algorithm. The author applies these results to get convergence rates and optimality of the applied control using certainty equivalence.

The problems that this reader has with the paper (aside from occasional typographical errors), are, firstly, that the use of a state space model for control of an input / output system is artificial, and, as stated, the states are not even available when the parameters are known (they must be constructed). Secondly, when the adaptive filter is constructed, (eq (20)), it takes an unusual form, in that the only way the observed outputs are used to update the state estimates is through feedback in the controls. Thus, even though eq’s (21) - (25) define an optimal control for (20), it is not clear that (20) gives a correct formulation for an estimate of the state equation in (7), so that it is not clear that x in (20) converges to x in (7), which is necessary for conclusion of optimality of the control applied to the system (1) - (5).

The problems that this reader has with the paper (aside from occasional typographical errors), are, firstly, that the use of a state space model for control of an input / output system is artificial, and, as stated, the states are not even available when the parameters are known (they must be constructed). Secondly, when the adaptive filter is constructed, (eq (20)), it takes an unusual form, in that the only way the observed outputs are used to update the state estimates is through feedback in the controls. Thus, even though eq’s (21) - (25) define an optimal control for (20), it is not clear that (20) gives a correct formulation for an estimate of the state equation in (7), so that it is not clear that x in (20) converges to x in (7), which is necessary for conclusion of optimality of the control applied to the system (1) - (5).

Reviewer: C.Schwartz

##### MSC:

93C40 | Adaptive control/observation systems |

62F12 | Asymptotic properties of parametric estimators |

93E10 | Estimation and detection in stochastic control theory |

93E12 | Identification in stochastic control theory |

##### Keywords:

deterministic adaptive control; consistency of parameter estimation; convergence rates; certainty equivalence
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DOI

##### References:

[1] | ANDERSON B. D. O., Stability of Adaptive Systems: Passivity and Averaging Analysis (1986) |

[2] | DOI: 10.1016/0005-1098(82)90021-8 · Zbl 0474.93080 |

[3] | ANDERSON B. D. O., Linear Optimal Control (1971) · Zbl 0321.49001 |

[4] | DOI: 10.1109/TIT.1984.1056898 · Zbl 0542.93063 |

[5] | CHEN H. F., Scientia sin. 25 pp 777– (1982) |

[6] | DOI: 10.1080/00207178608933679 · Zbl 0604.93061 |

[7] | DOI: 10.1109/TAC.1979.1102127 · Zbl 0412.93047 |

[8] | GOODWIN G. C., Adaptive Filtering, Prediction and Control (1984) · Zbl 0653.93001 |

[9] | DOI: 10.1080/0020718508961206 · Zbl 0568.93071 |

[10] | PETERKA , V. , 1986 , Algorithms for LQG self-tuning control based on input output data models. Preprints of 2nd IFAC Workshop on Adaptive Systems in Control and Signal Processing , Lund Institute of Technology , Lund , Sweden . |

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