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**Extended \(H_2\) and \(H_\infty\) norm characterizations and controller parametrizations for discrete-time systems.**
*(English)*
Zbl 1029.93020

This paper presents some synthesis procedures for discrete-time linear systems. The paper is based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability conditions for systems with uncertain parameters. These new synthesis conditions have some nice properties.

(a) they can be expressed in terms of LMI (linear matrix inequalities) and

(b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic Lyapunov function used to test stability.

This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, a parameter dependent Lyapunov function can be considered with an impact on the design of the design of robust \(H_2\) and \(H_\infty\) control problems. Third, the design of a controller with mixed objectives (also gain-scheduled controllers) can be addressed without employing a unique Lyapunov matrix to test all objectives (scheduled operation points).

Several numerical examples have been solved in order to illustrate the improvement in performances obtained with the proposed techniques.

(a) they can be expressed in terms of LMI (linear matrix inequalities) and

(b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic Lyapunov function used to test stability.

This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, a parameter dependent Lyapunov function can be considered with an impact on the design of the design of robust \(H_2\) and \(H_\infty\) control problems. Third, the design of a controller with mixed objectives (also gain-scheduled controllers) can be addressed without employing a unique Lyapunov matrix to test all objectives (scheduled operation points).

Several numerical examples have been solved in order to illustrate the improvement in performances obtained with the proposed techniques.

Reviewer: V.Dragan (Bucureşti)

### MSC:

93B36 | \(H^\infty\)-control |

93B50 | Synthesis problems |

93C55 | Discrete-time control/observation systems |

15A39 | Linear inequalities of matrices |

### Keywords:

conservativeness issues; linear matrix inequalities; synthesis procedures; discrete-time linear systems; quadratic stability; parameter dependent Lyapunov function; design; mixed objectives; Lyapunov matrix
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\textit{M. C. de Oliveira} et al., Int. J. Control 75, No. 9, 666--679 (2002; Zbl 1029.93020)

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### References:

[1] | DOI: 10.1007/BF00939145 · Zbl 0549.93045 |

[2] | DOI: 10.1016/0167-6911(89)90022-4 · Zbl 0678.93042 |

[3] | COLANERI P., Control Theory and Design: An RH2-RH8 Viewpoint (1997) |

[4] | DOI: 10.1016/S0167-6911(99)00035-3 · Zbl 0948.93058 |

[5] | DE OLIVEIRA, M. C., GEROMEL, J. C. and BERNUSSOU, J. An LMI optimization approach to multi objective controller design for discrete-time systems. Proceedings of the 38th IEEE Conference on Decision and Control. Phoenix, AZ, USA. pp.3611–3616. |

[6] | DOI: 10.1080/002071700219551 · Zbl 1006.93503 |

[7] | DOI: 10.1109/9.508913 · Zbl 0857.93088 |

[8] | DOI: 10.1016/0005-1098(96)00033-7 · Zbl 0855.93025 |

[9] | DOI: 10.1109/9.774121 · Zbl 0954.93003 |

[10] | DOI: 10.1016/0005-1098(94)90096-5 · Zbl 0816.93036 |

[11] | GEROMEL, J. C., DE OLIVEIRA, M. C. and BERNUSSOU, J. Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions. Proceedings of the 38th IEEE Conference on Decisions and Control. Phoenix, AZ, USA. pp.570–575. · Zbl 1022.93048 |

[12] | DOI: 10.1137/0329021 · Zbl 0741.93020 |

[13] | DOI: 10.1137/S0363012992238230 · Zbl 0842.93017 |

[14] | DOI: 10.1109/9.508904 · Zbl 0857.93078 |

[15] | DOI: 10.1109/9.85062 · Zbl 0748.93031 |

[16] | DOI: 10.1002/(SICI)1099-1239(19980715)8:8<669::AID-RNC337>3.0.CO;2-W · Zbl 0921.93012 |

[17] | DOI: 10.1109/9.599969 · Zbl 0883.93024 |

[18] | SILJAK D. D., Large Scale Dynamic Systems: Stability and Structure (1978) · Zbl 0384.93002 |

[19] | DOI: 10.1109/9.119629 · Zbl 0745.93025 |

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