Plenary Lecture 2
"Invariant Subspaces and Structural Properties of 2-D Control Systems"

                          by Professor Valeriu Prepelita
                      University Politehnica of Bucharest
                    Department of Mathematics Informatics

Abstract: Since its birth about sixty years ago, Systems Theory has developed into a scientific
and engineering discipline connected with all aspects of modern society. At the beginning it
was studied as Control Theory by mathematicians and engineers, but soon Systems Theory
extended to the study and the applications of various domains such as economics, business,
political science, sociology, medicine, biology, psychology, ecology etc.
The past three decades have seen a continually growing interest of many researchers
in the theory of two-dimensional (2D), which became a distinct and important branch of the
Systems Theory. Two-dimensional models were developed in a series of papers by Roesser
[15], Fornasini and Marchesini [4], Attasi [1], Eising [3] and others [6], [7], [12], [13] . The
reasons for the interest in this domain are on one side the richness in potential application
fields and on the other side the richness and significance of the theoretical approaches. The
application fields include circuits, control and signal processing, image processing (which is
the core of this approach), computer tomography, seismology, control of multipass processes
[16], [17], iterative learning control [8] etc.

The invariant subspaces with respect to linear transformations represent the
fundamentals of the Geometric Approach, which is one of the main trends in Systems and
Control Theory.

Geometric Approach provides a very clear concept of minimality and explicit
geometric conditions for controllability, observability, constructibility, pole assignability,
tracking or regulation etc. These concepts are used to obtain efficient and elegant solutions of
important problems of controller synthesis such as decoupling and pole-assignment problems
or the compensator and regulator synthesis for linear time-invariant multivariable systems.
The history of the Geometric Approach starts in 1969 when Basile and Marro [2]
introduced and studied the basic geometric tools named by them controlled and conditioned
invariant subspaces. They applied these techniques to disturbance rejection or unknown-
input observability. In 1970 Wonham and Morse [18] applied a maximal controlled invariant
method to decoupling and noninteracting control problems and later on Wonham's book [19]
imposed the name of "(A,B)-invariant" instead of "(A,B)-controlled invariant". Basile and
Marro opened new prospects to many applications (disturbance rejection, noninteraction etc.)
by the robust controlled invariant and the emphasis of the duality [2]. The Maro’s monograph
[11] presents the various aspects of the Geometric Approach, from the fundamental concepts,
the structure of the linear systems, invariant subspaces, up to applications to the regulator
problem, noninteraction, feedback and robustness etc. The LQ problem was studied in a
geometric framework by Silverman, Hautus and Willems. An important series of researchers
among which Anderson, Akashi, Bhattacharyya, Kucera, Malabre, Molinari, Pearson,
Francis and Schumacher developed the theory and applications of the Geometric Approach.
The range of the applications of the Geometric Approach was extended to various areas,
including, for instance, Markovian representations (Lindquist, Picci and Ruckebusch [10]) or
modeling and estimation of linear stochastic systems [9]). All these books and papers refers to
the „classical” 1D systems.

In the present lecture some aspects of the Geometric Approach are extended to a class
of 2D systems described by a partial differential state equation. The state space representation
of these systems is characterized by five matrices: two drift matrices and , an input-state
matrix , a state-output matrix and a input-output matrix . These systems represent the
continuous counterpart of Attasi's 2D discrete-time model.
The behavior of these 2D systems is described and their general response formula is
obtained. The concepts of complete controllability and complete observability are introduced
and they are characterized by means of two suitable 2D controllability and observability
Gramians. In the case of time-invariant 2D some controllability and observability criteria are
derived. The controllability and observability matrices are constructed (by extending the usual
1D ones). The first is used to characterize the space of the controlable states as the minimal -
invariant subspace which contains the columns of the matrix B and to obtain necessary and
sufficient conditions of controlability for 2D systems.
An algorithm is presented which compute the minimal -invariant subspace included
in , (i.e. the subspace of the controllable states of the system ) and which extends the 1D
algorithm from [9].
The observability Gramian and the observability matrix are employed to obtain the
description of the space of non-observable states as the maximal -invariant subspace
contained in and to derive some observability criteria. An algorithm is proposed which
compute this invariant subspace. These invariant subspaces can be used to obtain the Kalman
canonical decomposition of the state space and to reduce the system to a minimal realization.


[1] S. Attasi, Introduction d'une classe de systèmes linéaires reccurents à deux indices,
Comptes Rendus Acad. Sc. Paris, 277, série A (1973), 1135.
[2] G. Basile and G. Marro, L'invarianza rispetto ai disturbi studiata nello spazio degli stati,
Rendiconti della LXX Riunione Annuale AEI, paper 1.4.01 (1997).
[3] R. Eising, Low Order Realization for 2-D Transfer Functions. Proc.IEEE, 67, 5 (1997),
[4] E. Fornasini and G. Marchesini, State space realization theory of two-dimensional filters,
IEEE Trans. Aut. Control, AC-21 (1976), 484-492.
[5] K. Gałkovski, E. Rogers and D.H. Owens, New 2D models and a transition matrix for
discrete linear repetitive processes, Int. J. Control, 72, 15 (1999), 1365-1380.
[6] T. Kaczorek, Controllability and minimum energy control of 2D continuous-discrete
linear systems, Appl. Math. and Comp. Sci., 5, 1 (1995), 5-21.
[7] S.-Y. Kung, B.C. Lévy, M. Morf, T. Kailath, New results in 2D systems theory. Part II:
2D state-space models realization and the notions of controllability, observability and
minimality, Proceedings of the IEEE, 65, 6 (1977), 945-978.
[8] J. Kurek and M.B. Zaremba, Iterative learning control synthesis on 2D system theory.
IEEE Trans. Aut. Control, AC-38, 1 (1993), 121-125.
[9] A. Lindquist and G. Picci, A geometric approach to modeling and
estimation of linear stochastic systems, Journal of Math. Systems, Estimation
and Control, 1 (1991), 241–333.
[10] A. Lindquist, G. Picci and G. Ruckebusch, On minimal splitting sub-spaces and
Markovian representation, Mathematical Systems Theory, 12, (1979), 271-279.
[11] G. Marro, Teoria dei sistemi e del controlo, Zanichelli, Bologna, 1989.
[12] V. Prepeliţă, Systèmes linéaires à N indices, Comptes Rendus Acad. Sci. Paris, 279,
Série A (1974), 387.
[13] V. Prepeliţă, A dynamical system described by partial differential equations, Rev.
Roumaine Sci. Techn., Électrotechn. et Énerg., 1 (1979), 117-126.
[14] V. Prepelita, Controllability Criteria for a Class of Multidimensional Hybrid Systems,
Proceedings of the Sixth Congress of Romanian Mathematicians, Bucharest, 1 (2007),
[15] R. P. Roesser, A discrete state-space model for linear image processing, IEEE Trans.
Aut. Control, AC-20, (1975), 1-10.
[16] E. Rogers, D.H. Owens, Stability Analysis for Linear Repetitive Processes, Lecture Notes
in Control and Information Sciences, 175, Ed. Thoma H, Wyner W., Springer Verlag Berlin,
[17] K. Smyth, Computer aided analysis for linear repetitive Processes, PhD Thesis,
University of Strathclyde, Glasgow, UK, 1992.
[18] W. M. Wonham and A. S. Morse, Decoupling and pole assignment in linear
multivariable systems: A geometric approach, SIAM J. Control, 8 (1970), 1-18.
[19] W. M. Wonham, Linear Multivariable Control, a Geometric Approach, 2nd ed., Springer
Verlag, New York, 1979.

Brief Biography: Prof. Valeriu Prepelita graduated from the Faculty of Mathematics-Mechanics of the
University of Bucharest in 1964. He obtained Ph.D. in Mathematics at the University
of Bucharest in 1974. He is currently Professor at the Faculty of Applied Sciences,
the University Politehnica of Bucharest, Director of the Department Mathematics-
Informatics. His research and teaching activities have covered a large area of domains
such as Systems Theory and Control, Multidimensional Systems, Functions of a Complex
Variables, Linear and Multilinear Algebra, Special Functions, Ordinary Differential
Equations, Partial Differential Equations, Operational Calculus, Probability Theory and
Stochastic Processes, Operational Research, Mathematical Programming, Mathematics of
Finance. Professor Valeriu Prepelita is author of more than 110 published papers in
refereed journals or conference proceedings and author or co-author of 15 books. He has
participated in many national and international Grants. He is member of the Editorial
Board of some journals, member in the Organizing Committee and the Scientific
Committee of several international conferences, keynote lecturer or chairman of some
sections of these conferences. He is a reviewer for five international journals. He received
the Award for Distinguished Didactic and Scientific Activity of the Ministry of Education
and Instruction of Romania.