##
**Impulse observability and impulse controllability of linear time-varying singular systems.**
*(English)*
Zbl 1058.93011

Linear, finite-dimensional, continuous-time singular control systems are considered. It is generally assumed that the system coefficients depend on time and the controls and observations are unconstrained. Using algebraic methods and the theory of singular differential equations, necessary and sufficient conditions for impulsive controllability and impulsive observability are formulated and proved. In the proofs of the main results canonical decomposition of singular control systems is extensively used. It should be pointed out that these conditions require verification of the images and kernels of linear operators. Moreover, a simple numerical example is given which illustrates the theoretical considerations. Finally, it should be mentioned that the paper contains also several remarks and comments on controllability and observability problems for singular control systems. Moreover, similar considerations can be found in the paper [J. D. Cobb, “Controllability, observability and duality in singular systems”, IEEE Trans. Autom. Control AC-29, 1076–1082 (1984)].

Reviewer: Jerzy Klamka (Gliwice)

PDFBibTeX
XMLCite

\textit{C.-J. Wang} and \textit{H.-E. Liao}, Automatica 37, No. 11, 1867--1872 (2001; Zbl 1058.93011)

Full Text:
DOI

### References:

[1] | Byers, R.; Kunkel, P.; Mehrmann, V., Regularization of linear descriptor systems with variable coefficients, SIAM Journal on Control and Optimization, 35, 117-133 (1997) · Zbl 0895.93026 |

[2] | Campbell, S. L., A general form for solvable linear time varying singular systems of differential equations, SIAM Journal on Mathematical Analysis, 18, 1101-1115 (1987) · Zbl 0623.34005 |

[3] | Campbell, S. L.; Nichols, N. K.; Terrell, W. J., Duality, observability and controllability for linear time-varying descriptor systems, Circuits, Systems, and Signal Processing, 10, 455-470 (1991) · Zbl 0752.93009 |

[4] | Campbell, S. L.; Petzold, L. R., Canonical forms and solvable singular systems of differential equations, SIAM Journal on Algebraic and Discrete Methods, 4, 517-521 (1983) · Zbl 0524.34003 |

[5] | Campbell, S. L.; Terrell, W. J., Observability of linear time varying descriptor systems, SIAM Journal on Matrix Analysis and Applications, 12, 484-496 (1991) · Zbl 0736.93007 |

[6] | Cobb, J. D., Controllability, observability, and duality in singular systems, IEEE Transactions on Automatic Control, 29, 1076-1082 (1984) |

[7] | Gelf́and, I. M.; Shilov, G. E., Generalized functions, vol. I (1964), Academic Press: Academic Press New York · Zbl 0115.33101 |

[8] | Kunkel, P.; Mehrmann, V., The linear quadratic optimal control problem for linear descriptor systems with variable coefficients, Mathematics of Control, Signals and Systems, 10, 247-264 (1997) · Zbl 0894.49020 |

[9] | Rabier, P. J.; Rheinbolt, W. C., Classical and generalized solutions of time-dependent linear differential-algebraic equations, Linear Algebra and its Applications, 245, 259-293 (1996) · Zbl 0857.34005 |

[10] | Rabier, P. J.; Rheinbolt, W. C., Time-dependent linear DAEs with discontinuous inputs, Linear Algebra and its Applications, 247, 1-29 (1996) · Zbl 0864.65044 |

[11] | Rath, W., Derivative and proportional state feedback for linear descriptor systems with variable coefficients, Linear Algebra and its Applications, 260, 273-310 (1997) · Zbl 0941.93030 |

[12] | Terrell, W. J., The output-nulling space, projected dynamics, and system decomposition for linear time-varying singular systems, SIAM Journal on Control and Optimization, 32, 876-889 (1994) · Zbl 0798.34002 |

[13] | Verghese, G. C.; Levy, B. C.; Kailath, T., A generalized state-space for singular systems, IEEE Transaction on Automatics Control, 26, 811-831 (1981) · Zbl 0541.34040 |

[14] | Wang, C.-J., State feedback impulse elimination of linear time-varying singular systems, Automatica, 32, 133-136 (1996) · Zbl 0857.93017 |

[15] | Wang, C.-J., Controllability and Observability of linear time-varying singular systems, IEEE Transactions on Automatic Control, 44, 1901-1905 (1999) · Zbl 0956.93003 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.