**
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

Romania

vprepelita@mathem.pub.ro

**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.

**REFERENCES**

[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),

866-868.

[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),

301-311.

[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,

1999.

[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.