Near-optimum steady state regulators for stochastic linear weakly coupled systems.

*(English)*Zbl 0701.93104Summary: This paper presents an approach to the decomposition and approximation of the linear quadratic Gaussian estimation and control problems for weakly coupled systems. The global Kalman filter is decomposed into separate reduced-order local filters via the use of a decoupling transformation. A near-optimal control law is derived by approximating the coefficients of the truly optimal control law. The order of approximation of the optimal performance is \(O(\epsilon^ N)\), where N is the order of approximation of the coefficients. A real world power system example demonstrates the failure of \(O(\epsilon^ 2)\) and \(O(\epsilon^ 4)\) approximations and the necessity for the existence of the \(O(\epsilon^ N)\) theory. The proposed method produces the reduction in both off-line and on-line computational requirements and leads to convergence under mild assumptions. In addition, only low-order systems are involved in algebraic calculations and no analyticity requirement (a standard assumption for the power series method) is imposed on system coefficients.

##### MSC:

93E20 | Optimal stochastic control |

93E11 | Filtering in stochastic control theory |

93A15 | Large-scale systems |

##### Keywords:

decomposition; approximation; linear quadratic Gaussian estimation; weakly coupled systems; global Kalman filter; near-optimal control law
PDF
BibTeX
XML
Cite

\textit{X.-M. Shen} and \textit{Z. Gajic}, Automatica 26, No. 5, 919--923 (1990; Zbl 0701.93104)

Full Text:
DOI

##### References:

[1] | Chang, K., Singular perturbations of a general boundary value problem, SIAM J. math. anal., 3, 520-526, (1972) · Zbl 0247.34062 |

[2] | Delacour, J.D.; Darwish, M.; Fantin, J., Control strategies of large-scale power systems, Int. J. control, 27, 753-767, (1978) · Zbl 0373.93003 |

[3] | Gajic, Z., Numerical fixed point solution for near optimum regulators of linear quadratic Gaussian control problems for singularly perturbed systems, Int. J. control, 43, 373-387, (1986) · Zbl 0581.93073 |

[4] | Gajic, Z.; Petkovski, Dj.; Shen, X., () |

[5] | Gajic, Z.; Shen, X., Decoupling transformation for weakly coupled linear systems, Int. J. control, 50, 1517-1523, (1989) · Zbl 0686.93011 |

[6] | Geromel, J.; Peres, P., Decentralized load-frequency control, (), 225-230 · Zbl 0589.93004 |

[7] | Grodt, T.; Gajic, Z., The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation, IEEE trans. aut. control, AC-33, 751-754, (1988) · Zbl 0649.93023 |

[8] | Harkara, N.; Petkovski, Dj.; Gajic, Z., The recursive algorithm for optimal output feedback control problem of linear weakly coupled systems, Int. J. control, 50, 1-11, (1989) · Zbl 0683.49009 |

[9] | Ishimatsu, T.; Mohri, A.; Takata, M., Optimization of weakly coupled systems by a two-level method, Int. J. control, 22, 877-882, (1975) · Zbl 0315.49001 |

[10] | Khalil, H.; Gajic, Z., Near-optimum regulators for stochastic linear singularly perturbed systems, IEEE trans. aut. control, AC-29, 531-541, (1984) · Zbl 0535.62077 |

[11] | Khalil, H.; Kokotovic, P., Control strategies for decision makers using different models of the same system, IEEE trans. aut. control, AC-23, 289-298, (1978) · Zbl 0384.93007 |

[12] | Kokotovic, P.; Cruz, J., An approximation theorem for linear optimal regulators, J. math. anal., 27, 249-252, (1969) · Zbl 0192.51801 |

[13] | Kokotovic, P.; Khalil, H., () |

[14] | Kokotovic, P.; Perkins, W.; Cruz, J.; D’Ans, Ε-coupling for near-optimum design of large scale linear systems, (), 889-992 |

[15] | Kwakernaak, H.; Sivan, R., () |

[16] | Mahmoud, M., A quantitative comparison between two decentralized control approaches, Int. J. control, 28, 261-275, (1978) · Zbl 0379.93024 |

[17] | Petkovski, Dj.; Rakic, M., A series solution of feedback gains for output constrained regulators, Int. J. control, 29, 661-669, (1979) · Zbl 0498.93018 |

[18] | Petrovic, B.; Gajic, Z., Recursive solution of linear-quadratic Nash games for weakly interconnected systems, J. optimiz. theory applic., 56, 463-477, (1988) · Zbl 0622.93005 |

[19] | Sezer, M.; Siljak, D., Nested ε-decomposition and clustering of complex systems, Automatica, 22, 321-331, (1986) · Zbl 0594.93008 |

[20] | Shen, X., Near-optimum reduced-order stochastic control of linear discrete and continuous systems with small parameters, () |

[21] | Washburn, H.D.; Mendel, J., Multistage estimation of dynamical and weakly coupled systems in continuous-time linear systems, IEEE trans. aut. control, AC-25, 71-76, (1980) · Zbl 0429.93055 |

[22] | West, P.; Bingulac, S.; Perkins, W., L-A-S: A computer-aided control system design language, () |

[23] | Zangwill, W.; Garcia, C., () |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.