Brief paperA necessary and sufficient condition for parameter insensitive disturbance-rejection problem with state feedback☆
Introduction
Since the equivalent concepts of controlled invariance (Basile & Marro, 1969) and -invariance (Wonham & Morse, 1970) were discovered independently, many ideas (e.g., the concepts of conditioned invariance (Basile & Marro, 1969) which was renamed as -invariance by Wonham (1985) and -pair (Schumacher, 1980)) have been extensively developed as the so-called geometric control theory for linear multivariable time-invariant systems. As basic control applications of the above concepts, disturbance-rejection problems and decoupling control problems have been studied in the framework of the geometric approach. And the above concepts and their applications have been stated in many books (e.g., Basile and Marro, 1992, Trentelman et al., 2001, Wonham, 1985).
On the other hand, geometric theories for various classes of systems with uncertain parameter variations have been studied from the practical viewpoint. Firstly, a disturbance-rejection problem with state feedback was considered by Bhattacharyya (1983) in the case that the coefficient linear maps of the systems depend linearly on uncertain parameters using the notion of generalized -invariance which is an extension of -invariance, and a sufficient condition for the problem to be solvable was given. However, its necessity was pointed out as an open problem, and necessary and sufficient conditions for the solvability have not been solved. After that Basile and Marro (1987) investigated the robust controlled invariant subspace and robust self-bounded controlled invariance which are straightforward extensions of the previous studies for unstructured uncertain systems. Further, simultaneously -invariance and simultaneously invariance for a finite number of linear systems were investigated by Ghosh (1986), and disturbance-rejection problems for uncertain linear systems in the sense that the coefficient linear maps of the systems are represented as convex combinations of two given maps were studied. Moreover, a disturbance-rejection problem with state feedback was studied by Conte and Perdon (1993) in the case that the elements of the coefficient matrices depend polynomially on uncertain parameters, and necessary and sufficient conditions for the problem to be solvable which depend on uncertain parameters were given for linear left-invertible systems. Further, a problem with static output feedback and measurable disturbances was studied by Koumboulis and Tzierakis (1999) in the case that the elements of the coefficient matrices of the systems have uncertain nonlinear forms, and necessary and sufficient conditions for the problem to be solvable which depend on uncertain parameters were given for linear left-invertible systems. And the rejection problem of atmospheric disturbances to the longitudinal motion of an aircraft with angle sensor uncertainties was studied as practical applications. Recently, Otsuka, 1999, Otsuka, 2000 investigated the two concepts of generalized -invariance and generalized -pair which are the dual concepts of generalized -invariance and an extension of -pair investigated by Schumacher (1980), respectively.
In this paper, a necessary and sufficient condition for a parameter insensitive disturbance-rejection problem with state feedback which was investigated by Bhattacharyya (1983) to be solvable is proved.
In Section 2, simultaneously -invariant subspaces for a family of linear systems are introduced and a key lemma to prove the main result is given. In Section 3, a relationship between simultaneously -invariant subspaces and generalized -invariant subspaces for uncertain linear systems is discussed. In Section 4 the main result is proved by using some important results in the previous sections. Finally, concluding remarks are given in Section 5.
Section snippets
Simultaneously -invariant subspace
At first, some notations used throughout this investigation are given. For a linear map from a vector space into a vector space and a subspace of the image, the kernel, the dimension and the inverse image are denoted by , , and , respectively.
Next, consider a family of linear systems defined in : where is the state, is the input and , . For a set of subspaces of the
Generalized -invariant subspace
In this section some important properties of generalized -invariant subspaces which are used to prove the main result are investigated.
Consider the following linear system defined in : where is the state and is the input. And coefficient linear maps and have uncertain parameters in the sense thatwhere and . Further, , and
Main result
This section gives a solution for an open problem which was pointed out in the paper (Bhattacharyya, 1983).
Consider the following uncertain systems : where is the state, is the input, is the controlled output and is the disturbance. And coefficient linear maps have the following uncertain parameters:
Concluding remarks
In this paper, a relationship between simultaneously -invariant subspaces and generalized -invariant subspaces was firstly investigated. And then, a necessary and sufficient condition for a parameter insensitive disturbance-rejection problem with state feedback to be solvable which was pointed out as an open problem by Bhattacharyya (1983) was proved by using the results on the above two invariant subspaces for linearly dependent structured uncertain systems.
As future important
Acknowledgements
This work was supported in part by the Grant-in-Aid for the 21st Century COE (Center of Excellence) Research by the Japanese Ministry of Education, Culture, Sports, Science and Technology. The author also would like to thank Professor Bijoy K. Ghosh for useful discussions with him. Finally, the author wishes to thank the reviewers for their useful comments to revise this paper
Naohisa Otsuka was born in Niigata, Japan. He received the B.S. and M.S. degrees, both in Mathematical Sciences, from Tokyo Denki University (TDU) in 1984 and 1986, respectively. In 1992, he received the Doctor of Science in Mathematical Sciences from Graduate School of Science and Engineering of TDU. From 1986 to 1992 he served as a Research Associate in the Department of Information Sciences, TDU. In April 1992, he moved to the Institute of Information Sciences and Electronics, University of
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Naohisa Otsuka was born in Niigata, Japan. He received the B.S. and M.S. degrees, both in Mathematical Sciences, from Tokyo Denki University (TDU) in 1984 and 1986, respectively. In 1992, he received the Doctor of Science in Mathematical Sciences from Graduate School of Science and Engineering of TDU. From 1986 to 1992 he served as a Research Associate in the Department of Information Sciences, TDU. In April 1992, he moved to the Institute of Information Sciences and Electronics, University of Tsukuba as a Research Associate and served as an Assistant Professor of the same University from 1993 to September 2000. From October 2000 to September 2002 he served as an Associate Professor in the Department of Information Sciences, TDU. Since October 2002 he has been with the same department in TDU as a Professor. His main research interests are in mathematical systems theory, geometric control theory, robust stability of parametric approach, infinite-dimensional linear systems and periodic systems. Dr. Otsuka is a member of the Society of Instrument and Control Engineers of Japan (SICE), IEEE and SIAM.
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This paper was partly presented at IFAC World Congress 2005. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo.