# Introduction

ecently the concept of time scales(or measure chains) is growing very rapidly into many areas of research. One reason being it includes both continuous and discrete systems as special cases. Hence many researchers [1] - [4] and [8]- [11] explored different concepts on time scales. Stephan Barnett [12] studied the control theory for both the continuous and discrete cases for simple dynamical systems. Control theory has developed rapidly over the past two decades and is now established as an important area of contemporary Applied Mathematics. Many problems of great importance in the contemporary world require a quite different approach, the aim being to compel or control a system to behave in some desired fashion. Basically control theory has involved the study of analysis and control of any dynamical system. This theory has been successfully applied in a variety of branches in the disciplines of engineering and particularly it is receiving great impetus from Aerospace engineering. A fascinating fact is that all the widely different disciplines of applications depend on a common core of mathematical techniques of the modern control system theory.

In this paper we establish the concept of controllability and observability of dynamical systems on measure chains. The results presented in this chapter generalises the existing results on controllability and observability for continuous and discrete cases and includes them as a particular case. This paper is organised as follows; In section 2, we outline the salient features of time scales. In section 3, we briefly mention the necessary and sufficient conditions for the controllability and observability of vector dynamical systems on time scales.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) S x t A t x t B t u t x t x y t C t x t 1 0 0 ? = + = = ? ? ?

# , ( )

Author: Department of Mathematics, University of Dodoma Dodoma, Tanzania. e-mail: gvrlsarma@rediffmail.com where A is n × n, B is n × m and C is r × n matrices whose elements are rd-continuous on a time scale T = [t 0 , t N ], the control u is (mx1) and x is an (nx1)vector. Section 4 presents the controllability and observability criteria of the most general matrix Lyapunov differential system on measure chains
S2 ) ( , ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 X t X t U t K t B t X t X t C t X t A t Y t X = + + ? ? ? = = ? ?
where A(t), B(t), K(t) and C(t) are square matrices of orders n×n, nxn, nxs and rxn respectively. The control U(t) is an sxn matrix and the out put Y(t) is an rxn matrix whose elements are rd-continuous on a measure chain T = [t 0 , t N ]. We firmly believe that these results will have a significant impact on control engineering problems.


# II.


# Salient Features of Time Scales

A measure chain (or time scale) is an arbitrary closed subset of real numbers R and it is denoted by T throughout the paper. Time scales are not necessarily connected and this topological handicap is eliminated by introducing the notion of jump operators ? and ? as follows: Definition 2.1: Let T be a measure chain. For t ? T define the forward jump operator ? : T ? T by ? (t) = Inf {s ? T : s > t } and the backward jump operator ? : T ? T by ? (t) = Sup {s ? T : s < t } .

A point t ? T is said to be right dense, right scattered, left dense and left scattered according as ? (t) = t, ? (t) > t, ? (t) = t and ? (t) < t respectively. The grainness
µ : T ? [o, ?) is defined by µ (t) = ? (t) -t .
The set T k which is derived from the measure chain T as follows:

If T has a left scattered maximum m, then
T k = T -{m}, otherwise T k = T. Definition 2.2: Let f : T ? R , t ? T k . Then define f ? (t) to
be the number (provided it exists ) with the property that given any ?>0, there exists a neighbourhood ? of t such that Year 2014
I (1.1) Global Journal of Researches in Engineering ( ) Volume XIV Issue II Version I Year 2014 6 I |[f(?(t)) -f(s)] -f ? (t) [?(t) -s] | ? ? |?(t) -s | for all s ? ? Then f ? (t) is called the delta derivative of f at t.
If T = R, the delta derivative is same as that of ordinary derivative and for T = Z, f ? (t) = f(t+1) -f(t) = ?f(t), which is the forward difference operator. Definition 2.3 : We say that f is delta differentiable on T k , if f ? (t) exists for all t ? T k . Result 2.1 : Assume f, g : T?R are delta differentiable functions at t?T k , then
? f+g : T ?R is delta differentiable at t with (f+g) ? (t) = f ? (t) +g ? (t). ? For any constant k, kf : T? R is delta differentiable at t with (kf) ? (t) = k f ? (t). ? : T x T? R is delta differentiable at t with (fg) ? (t) = f ? (t) g(t) + f(?(t)) g ? (t) = f(t) g ? (t) + f ? (t) g(?(t)) . Definition 2.4: A function F : T k ? R is called an antiderivative of f : T k ? R, provided F ? (t) = f(t) holds for all t ? T k .
Then the delta integral of f is defined = F (t) -F (a)) ? t ?T.

Definition 2.5: Let f : T?T be a function. We say that f is rd -continuous if it is continuous in right dense points and if limit f (s) exists as s ? t -for all left-dense points t ? T.

Result 2.2: Rd -Continuous functions possess an anti derivative.

Proof: For the proof we refer [2]. IV.
Definition

# Controllability and Observability of Lyapunov Type Matrix Dynamical Systems on Measure Chains

In this section we consider Lyapunov type matrix dynamical system on measure chains
X t ? ( ) = A (t)X(t) X t t K t t X t X ( ( ))B( ) ( )U( ) , ( ) ? + = 0 0
and obtain necessary and sufficient conditions for the controllability and observability of S 2 . Throughout this section ? 1 (t) and ? 2 (t) represent fundamental matrix solutions of A(t)X(t) and B*(t) X(? (t))

Theorem 4.1 : The solution of the equation (4.1) is given by X t ( ) =?1(t) ? ? 
1 1 0 0 2 1 0 ? ? ( ) ( ) * t X t ? 2 * ( ) t +?1(t) ? ? ? ? 1 1 2 1 2 0 ? ? ? ? ? ? ? ? ? ? ? ( ( ))K( )U( ) ( ) ( ) * * s s s s s t t t ? .= ?1(t,t0) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , )( , )+ (4.1)
x ? (t) = respectively.

x ? (t) = Year 2014 I A New Look into the Controllability and Observability of Lyapunov type Matrix Dynamical Systems on Measure Chains X t ( ) =?1(t) ? ?
1 1 0 0 2 1 0 ? ? ( ) ( ) * t X t ? 2 * ( ) t +?1(t) ? ? ? ? 1 1 2 1 2 0 ? ? ? ? ? ? ? ? ? ? ? ( ( ))K( )U( ) ( ) ( ) * * s s s s s t t t ? .= ?1(t,t0) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , ) ( , ) * * ? where ? i (t,t 0 ) = ? i (t)? i -1 (t 0 ) (i=1,2)
Proof : Please refer [6] Theorem 4.2 : The Lyapunov type matrix dynamical system on measure chains S 2 is completely controllable if and only if the symmetric controllability matrix
W 1 (t 0 ,t N )= ? ? N t t o s s t s K s K s t 0 )) ( , ( ) ( ) ( )) ( , ( * 1 * 0 1 ? ? ? ? is non singular.
Then the control U (t) defined by
U(t= -K * (t) ( ) [ ] ? ? ? ? ? 1 0 1 1 0 0 1 0 2 0 2 1 0 * * * , ( ) ( , ) ( , ) ( , ) ( , ) t t W t t X t t X t t t t N N f N ? ? ? defined for t 0 < t < t N transfers X(t 0 ) = X 0 to X(t N ) = Xf.
Proof : First we suppose that W 1 (t 0 ,t N ) is non -singular.

Then U (t) defined as above exists. We know that any solution of (4.1) has the form
X t ( ) =? 1 (t,t 0 ) X t s s s t s s t t t t 0 1 0 2 0 2 0 0 + ? ? ? ? ? ? ? ? ? ? ? ? ? ( , ( ))K( )U( ) ( , )( , ) * *
? put t = t N and substitute U(t) defined as above we will get X t N ( ) = Xf and hence S 2 is controllable.

Conversely, suppose that S 2 is controllable. We have to show that W 1 (t 0 , t N ) is non-singular. Since W 1 (t 0 , t N ) is symmetric, clearly it is positive semi definite. Now suppose that there exists some column

vector
? 1 ?0 such that ? ? 1 1 0 1 * ( , ) W t t N = 0 ? ? ? ? 1 1 0 1 0 1 0 * * * ( , ( ))K( ) ( ) ( , ( )) ? ? ? ? t s s k s t s s t t N ? = 0 ? ? ? 1 0 1 0 0 * ( , ) ( , ) s t s t s t t N ? ? = 0, where ?1(s, t0) = K s t s * * ( ) ( , ( )) ? ? 1 0 1 ? ? ? 1 2 0 0 ?s t t N ? = . Hence ?1 ? 0 on [t0, tN].
By our assumption since S 2 is completely controllable, there exists a control V (t) (say) making X (t N ) = 0 if X (t 0 ) ? 1 g where g is any non zero 1xn matrix. i.e X 0 can not determined with the knowledge of Y(t) in this case. This contradicts our assumption that S 3 is completely observable. Therefore V(t 0 , t N ) is positive definite and hence V(t 0 , t N ) is non -singular.
X (t N ) = 0 ? X (t0) + ? ? ? 1 0 2 0 0 ( , ( ))K( )V( ) ( , ) * t s s s t s s t t N ? ? = 0 ? ?1g = -? ? ? 1 0 2 0 0 ( , ( ))K( )V( ) ( , ) * t s s s t s s t t N ? ? Now ? 1 2 g = (?1g)* (?1g) = - ? ? ? 2 0 1 0 1 0 ( , )V ( )K ( ) ( ,(
Observation 4.1: From the theorem (4.2) it is observed that the controllability matrix is independent of ? 2 . The fundamental matrix ? 1 alone determines the controllability criterion of the dynamical system S 2 . We also observe that the controllability criterion can be determined using the fundamental matrix ? 2 alone. In this case the controllability matrix is given by W 2 (t 0 ,t N ) = ( ( , ( ))K( )K ( ) ( , ( )) that the observability matrix is independent of ? 2 . The fundamental matrix ? 1 alone determines the observability criterion of the dynamical system S 2 . We can also determine the observability criterion of S 2 using the fundamental matrix ? 2 alone.
Proof : Please refer [6]Theorem 3.2 : Suppose u -(t) is any other control takingX (t 0 ) = x 0 to x(t N ) = xf.Thent ? N( ) u s2s ?t ? N( ) u ss 2 ?,totowhere u(s) = -B*(s) ? * to, ? (s)) W -1 (t o ,t N ) [x o -? (t o ,t N )xf]provided u(s) ? u(s) ?s ? [t o , t N ] .Proof : Please refer [6]Theorem 3.3 : The time varying dynamical system S 1 onmeasure chain T=[ t 0 ,t N ] is completely observable ifand only if the symmetric observability matrix V 1 (t 0 , t N ) =tN?( ? * , s t0)toDefinition 2.7 (Observability) : The linear time varyingdynamical system defined by S 1 on measure chains iscompletely observable if and only if the knowledge ofthe control u(t) and the out put y(t) suffice to determinex(t 0 ) = x 0 uniquely for a finite time t N ? t 0 .III.Controllability and Observabilityof Vector Dynamical Systems onMeasure ChainsTheorem 3.1: The system S 1 is completelycontrollable if and only if the n × n symmetriccontrollability matrix W (t o , t N ) =tN?(( ?to,?( ) ) ( ) ( ) * ? s B s B s*(to,?( ) ) s)sto(Controllability) 2.6 : The linear time varying dynamical system S 1 on measure chain T=[t 0 ,t N ] is completely controllable if "for any initial time t0 and any initial state x (t o ) = x o and any given final state xf, there exists a finite time tN > to and a control u(t), to ? t ? t N such that x (t N ) = xf". C* (s) C(s) ? (s,t 0 ) ? s is non singular.Proof : Please refer[6] 
t N ??2t0?ss*s?* 2to?s s ?, andt0the control U(t) defined byU(t) = -K * (t)() ( ( )K ( ) K t t * ? ? 1 1 1 ?t0) [ t X , ( ) ?0??1t t X N ( , ) 0f?* 2] ( , ) t t N 0W t t 2 1 0 ? ( , ) ( , )K( )K ( ) t t t t N 2 0* ? Observation 4.2 : From the theorem (4.3) it is observed
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* 
	
		Stability of abstract dynamic equations on time scales
		
			AlaaEHamza
		
		
			KarimaMOraby
		
	
	
		Advances in Difference Equations
				
			2012. 2012
			143
		
	




* 
	
		
			ZBartosiewicz
		
		
			EPaw?uszewicz
		
		Realizations of linear control systems on time scales, Control & Cybernetics
				
			2006
			35
			
		
	




* 
	
		Linear control systems on time scale: unification of continuous and discrete
		
			JZBartosiewicz
		
		
			EPaw?uszewicz
		
	
	
		Proceedings of the 10th IEEE International Conference On Methods and Models in Automation and Robotics MMAR
				the 10th IEEE International Conference On Methods and Models in Automation and Robotics MMARMiedzyzdroje, Poland
		
			2004. 2004
		
	




* 
	
		Stability for time varying linear dynamic systems on time scales
		
			JJDa Cunha
		
	
	
		J. Comput. Appl. Math
		
			176
			
			2005
		
	




* 
	
		Instability results for slowly time varying linear dynamic systems on time scales
		
			JJDa Cunha
		
	
	
		J. Math. Anal. Appl
		
			328
			
			2007
		
	




* 
	
		Dynamic systems on measure chains
		
			VLakshmikantham
		
		
			SSivasundaram
		
		
			BKaymakclan
		
		
			1996
			Kluwer academic Publishers
		
	




* 
	
		Nonregressivity in switched linear circuits and mechanical systems
		
			RJMarks
		
		
			II
		
		
			IAGravagne
		
		
			JMDavis
		
		
			JJDa Cunha
		
	
	
		Mathematical and Computer Modelling
		
			43
			
			2006
		
	




* 
	
		S Two point boundary value problems inhomogeneous time scale linear dynamic process
		
			KNMurty
		
		
			YRao
		
	
	
		JMAA
		
			184
			
			1994
		
	




* 
	
		Non linear Lyapunov type matrix system on time scales-Existence, Uniqueness and Sensitivity analysis. Dynamic systems and applications
		
			KNMurty
		
		
			GV S RDeekshitulu
		
		
			Dec 1998
			7
			
		
	




* 
	
		On the exponential stability of dynamic equations on time scales
		
			HuuNguyen
		
		
			Le HuyDu
		
		
			Tien
		
	
	
		J. Math. Anal. Appl
		
			331
			
			2007
		
	




* 
	
		A spectral characterization of exponential stability for linear time-invariant systems on time scales
		
			SP?tzsche
		
		
			FSigmund
		
		
			Wirth
		
	
	
		Discrete and Continuous Dynamical Systems
		
			9
			5
			
			2003
		
	




* 
	
		Introduction to mathematical control
		
			StephenBarnet
		
		
			1975
			Clarendon Press-Oxford