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**A Riccati equation approach to the stabilization of uncertain linear systems.**
*(English)*
Zbl 0602.93055

The paper deals with the problem of designing a controller when no accurate model is available for the process to be controlled. Specifically, the problem of stabilizing an uncertain system using state feedback control is considered. The unknown parameters are assumed to be bounded and to vary within time.

The procedure presented can be regarded as being an extension of the linear-quadratic regulator design procedure, since a candidate quadratic Lyapunov function which assures the stability of the closed-loop uncertain system can be obtained by solving an augmented matrix Riccati equation. The augmentation specifically accounts for the uncertainty in the system allowed in matrices A and B of a state-space description. For systems without uncertainties the augmented matrix Riccati equation reduces to the ordinary Riccati equation which arises in the linear- quadratic regulator problem.

Uncertain linear systems can be stabilized if the uncertainty satisfies so-called matching conditions. These constitute sufficient conditions and are known to be duly restrictive. The main aim of the present paper is to enlarge the class of uncertain linear systems for which a stabilizing feedback control law can be constructed. This problem may be reduced to the construction of suitable quadratic Lyapunov function for the system. To this end a computationally feasible algorithm is developed for the construction of such a Lyapunov function. Classes of uncertain systems are identified for which the success of the algorithm becomes necessary and sufficient for the existence of a suitable quadratic Lyapunov function.

In an illustrative example the method is applied to ascertain the longitudinal stability of the A4D aircraft.

The procedure presented can be regarded as being an extension of the linear-quadratic regulator design procedure, since a candidate quadratic Lyapunov function which assures the stability of the closed-loop uncertain system can be obtained by solving an augmented matrix Riccati equation. The augmentation specifically accounts for the uncertainty in the system allowed in matrices A and B of a state-space description. For systems without uncertainties the augmented matrix Riccati equation reduces to the ordinary Riccati equation which arises in the linear- quadratic regulator problem.

Uncertain linear systems can be stabilized if the uncertainty satisfies so-called matching conditions. These constitute sufficient conditions and are known to be duly restrictive. The main aim of the present paper is to enlarge the class of uncertain linear systems for which a stabilizing feedback control law can be constructed. This problem may be reduced to the construction of suitable quadratic Lyapunov function for the system. To this end a computationally feasible algorithm is developed for the construction of such a Lyapunov function. Classes of uncertain systems are identified for which the success of the algorithm becomes necessary and sufficient for the existence of a suitable quadratic Lyapunov function.

In an illustrative example the method is applied to ascertain the longitudinal stability of the A4D aircraft.

Reviewer: H.D.Fischer

### MSC:

93D15 | Stabilization of systems by feedback |

15A24 | Matrix equations and identities |

93C05 | Linear systems in control theory |

34D20 | Stability of solutions to ordinary differential equations |

93B40 | Computational methods in systems theory (MSC2010) |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

### Keywords:

time-dependent
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\textit{I. R. Petersen} and \textit{C. V. Hollot}, Automatica 22, 397--411 (1986; Zbl 0602.93055)

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### References:

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