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**Guaranteed properties of gain scheduled control for linear parameter- varying plants.**
*(English)*
Zbl 0754.93022

Summary: Gain scheduling has proven to be a successful design methodology in many engineering applications. However in the absence of a sound theoretical analysis, these designs come with no guarantees on the robustness, performance, or even nominal stability of the overall gain scheduled design.

This paper presents such an analysis for one type of gain schedule system, namely, a linear parameter-varying plant scheduling on its exogenous parameters. Conditions are given which guarantee that the stability, robustness, and performance properties of the fixed operating point designs carry over to the global gain schedule design. These conditions confirm and formalize popular notions regarding gain scheduled design, such as the scheduling variable should “vary slowly”.

This paper presents such an analysis for one type of gain schedule system, namely, a linear parameter-varying plant scheduling on its exogenous parameters. Conditions are given which guarantee that the stability, robustness, and performance properties of the fixed operating point designs carry over to the global gain schedule design. These conditions confirm and formalize popular notions regarding gain scheduled design, such as the scheduling variable should “vary slowly”.

### MSC:

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93C15 | Control/observation systems governed by ordinary differential equations |

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\textit{J. S. Shamma} and \textit{M. Athans}, Automatica 27, No. 3, 559--564 (1991; Zbl 0754.93022)

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### References:

[1] | Burton, T. A., (Volterra Integral and Differential Equations (1983), Academic Press: Academic Press New York) · Zbl 0515.45001 |

[2] | Callier, F. M.; Desoer, C. A., An algebra of transfer functions of distributed linear time-invariant systems, IEEE Trans. Circ. Syst., CAS-25, 651-662 (1978) · Zbl 0418.93037 |

[3] | Chen, M. J.; Desoer, C. A., Necessary and sufficient condition for robust stability of linear distributed feedback systems, Int. J. Control, 35, 255-267 (1982) · Zbl 0489.93041 |

[4] | Corduneanu, C., Some differential equations with delay, (Proc. Equadiff 3 (Czechoslovak Conference on Differential Equations and their Applications). Proc. Equadiff 3 (Czechoslovak Conference on Differential Equations and their Applications), Brno (1973)), 105-114 · Zbl 0331.34066 |

[5] | Corduneanu, C.; Lakshmikantham, V., Equations with unbounded delay: A survey, Nonlinear Anal. Theory Methods Appl., 4, 831-877 (1980) · Zbl 0449.34048 |

[6] | Corduneanu, C.; Luca, N., The stability of some feedback systems with delay, J. Math. Anal. Applic., 51, 377-393 (1975) · Zbl 0312.34051 |

[7] | Desoer, C. A., Slowly varying system \(ẋ = A(t) x\), IEEE Trans. Aut. Control, AC-14, 780-781 (1969) |

[8] | Desoer, C. A.; Vidyasagar, M., (Feedback Systems: Input-Output Properties (1975), Academic Press: Academic Press New York) · Zbl 0327.93009 |

[9] | Doyle, J., Analysis of feedback systems with structural uncertainties, (Proc. IEE, 129 (1982)), 242-250, Part D |

[10] | Doyle, J. C.; Stein, G., Multivariable feedback design: concepts for a classical/modern synthesis, IEEE Trans. Aut. Control, AC-26, 4-16 (1981) · Zbl 0462.93027 |

[11] | Doyle, J. C.; Wall, J. E.; Stein, G., Performance and robustness analysis for structural uncertainty, (Proc. 21st Conf. Decision and Control (1982)), 629-636 |

[12] | Driver, R. D., (Ordinary and Delay Differential Equations (1977), Springer: Springer New York) · Zbl 0374.34001 |

[13] | Hale, J., (Theory of Functional Differential Equations (1977), Springer: Springer New York) |

[14] | Luca, N., The stability of the solutions of a class of integrodifferential systems with infinite delays, J. Math. Anal. Applics., 67, 323-339 (1979) · Zbl 0415.45008 |

[15] | Miller, R. K., (Nonlinear Volterra Integral Equations (1971), Benjamin: Benjamin Menlo Park, CA) · Zbl 0448.45004 |

[16] | Shamma, J. S.; Athans, M., Guaranteed properties of nonlinear gain scheduled control systems, (Proc. 27th IEEE Conf. on Decision and Control (1988)) · Zbl 0754.93022 |

[17] | Shamma, J. S.; Athans, M., Analysis of nonlinear gain scheduled control systems, IEEE Trans. Aut. Control, 35, 898-907 (1990) · Zbl 0723.93022 |

[18] | Stein, G.; Hartmann, G. L.; Hendrick, R. C., Adaptive control laws for F-8 Flight Test, IEEE Trans. Aut. Control, AC-22, 758-767 (1977) |

[19] | Stein, G., Adaptive flight control—a pragmatic view, (Narendra, K. S.; Monopoli, R. V., Applications of Adaptive Control (1980), Academic Press: Academic Press New York) |

[20] | Willems, J. C., (The Analysis of Feedback Systems (1971), MIT Press: MIT Press Cambridge, MA) · Zbl 0244.93048 |

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