American Journal of Applied Mathematics
Volume 3, Issue 4, August 2015, Pages: 179-184

Higher Order Sylster’s Equation on Measure Chains-Controllability and Observability

Goteti V. R. L. Sarma

Department of Mathematics, University of Dodoma, Dodoma, Tanzania

Goteti V. R. L. Sarma. Higher Order Sylster’s Equation on Measure Chains-Controllability and Observability. American Journal of Applied Mathematics. Vol. 3, No. 4, 2015, pp. 179-184. doi: 10.11648/j.ajam.20150304.13

Abstract: In this article we derive the solution of higher order Sylster’s type differential equation on measure chains in terms of two fundamental matrices. Later by defining the controllability and observability on measure chains, necessary conditions for the controllability and observability of the higher order Sylster’s type differential system on measure chains is established.

Keywords: Measure Chains, Controllability Observability, Fundamental Matrix

1. Introduction

Control theory has emerged into the main core of applied mathematical studies as it sets the necessary and sufficient conditions for controllability and observability of the designed dynamical systems. Many of the present day real world problems arises in robotics, industrial engineering, automata theory, modeling biological systems and in space dynamics are mostly control theoretic in nature as the aim being to compel or control the system to behave in some desired fashion. These systems can be continuous or discrete nature and the researchers earlier used to study them separately.

At the progress and advancement of knowledge on time scales (or measure chains) which includes both continuous and discrete systems as special cases make it possible to study the complex dynamical systems rigorously. Stephan Barnett [1] studied the control theory for both the continuous and discrete cases for simple dynamical systems. 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. The results established in this article coincide with the findings of [1],[2],[5] and [11] and include them as a sub case of this article.

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. Section 3, deals with the method of solution of the initial value problem on higher order Sylster’s type equation on measure chains

In terms of two fundamental matrix solutions, finally in section 4, we obtain the necessary and sufficient conditions for the controllability and observability of higher order Sylster’s type matrix dynamical systems on measure chains

Where A and B are constant square matrices of order and  are all variable matrics whose elements are rd-continuous on a measure chain T = [t0, tN]. We firmly believe that these results will have a significant impact on robotic and control engineering applications.

2. 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 s and r as follows:

Definition 2.1: Let T be a measure chain. For t Î T define the forward jump operator s : T ® T by s (t) = Inf {s Î T : s > t } and the backward jump operator r : T ® T by r(t) = Sup {s Î T : s < t } .

A point t Î T is said to be right dense, right scattered, left dense and left scattered according as s(t) = t, s(t) > t, r(t) = t and r(t) < t respectively.

The grainness m : T ® [o,¥) is defined by m(t) = s (t) - t .

The set Tk which is derived from the measure chain T as follows:

If T has a left scattered maximum m, then Tk = T - {m}, otherwise Tk = T.

Definition 2.2: Let f : T ® R , t Î Tk. Then define fD(t) to be the number (provided it exists ) with the property that given any Î>0, there exists a neighbourhood È of t such that

|[f(s(t)) - f(s)] - fD(t) [s(t) - s] | £ Î |s(t) - s | for all s Î È

Then fD(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, fD(t) = f(t+1) - f(t) = Df(t), which is the forward difference operator.

Definition 2.3: We say that f is delta differentiable on Tk, if fD(t) exists for all t Î Tk.

Result 2.1: Assume f, g : T ® R are delta differentiable functions at t Î Tk, then

(i) f+g : T ® R is delta differentiable at t with (f+g)D(t) = fD(t) +gD(t).

(ii) for any constant k, kf : T® R is delta differentiable at t with (kf) D(t) = k fD(t).

(iii) fg : T x T® R is delta differentiable at t with

Result 2.2: Leibnitz like theorem on measure chains: If f(t) and g(t) are continuously  times delta differentiable functions on a measure chain T=[a,b] then

Definition 2.4 : A function F : Tk ® R is called an antiderivative of f : Tk ® R, provided FD(t) = f(t) holds for all t Î Tk . Then the delta integral of f is defined = F(t) - F(a) " t Î 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 Î T.

Result 2.3 : Rd - Continuous functions possess an anti derivative.

Proof: For the proof we refer [2].

3. Solving Higher order Syslster’s Equation

Throughout this article  denotes the fundamental matrix solution of  and Z(t) denotes the fundamental matrix solution of .

Theorem 3.1: If  is a fundamental matrix solution of  then  is also fundamental matrix solution of .

Proof:  is a fundamental matrix solution of

This implies

Delta differentiating on both sides gives

Similarly

Continuing like this we can conclude that

Hence  is also fundamental matrix solution of .

Theorem 3.2: Any solution of

(3.1)

is of the form where C is any square matrix of order .

Proof: Using Lebnitz like theorem on measure chains for the  delta derivative of product of functions we have

+ . . . +

+ . . . .+

Hence  is a solution of (3.1)

To prove that every solution of (3.1) is of this form, let X(t) be a solution of defined by  where  is any variable matrix of order .

Then using Leibnitz like theorem we have

+ . . . +

This is equivalent to R.H.S. of (3.1) if

i.e.

i.e.  is a solution of

Since Z(t) is the fundamental matrix solution of  hence

Hence

Hence  . This completes the proof.

Theorem 3.3: Any solution of the initial value problem

is given by

where and

Proof: From theorem 3.2 any solution of (3.1) is of the form

where C is any square matrix of order .

Now substituting  gives

Hence

Hence the solution of the above initial value problem is  where and . Hence the proof.

Theorem 3.4: Let  be a fundamental matrix solution of  and Z(t) be a fundamental matrix solution of . Further suppose that the variable matrix  is such that

(3.2)

for each .

Then the particular solution of

(3.3)

is

Proof: Any solution of the corresponding homogeneous equation of (3.3) is in the form . Where C is any constant square matrix of order  Since such a solution can not be the solution of non homogeneous equation (3.3), by variation of parameters formula we can assume the C is a variable matrix and seek the particular solution of (3.3) in the form

Substituting this in the equation (3.3) and using the condition (3.2) we get

By delta integrating  times on both sides we get

Hence the particular solution of (3.2) is

Theorem 3.5: Any solution of the non homogeneous system (3.2) is of the form

X(t)

Proof: Follows from the theorems 3.2 and 3.4

Theorem 3.6: Any solution of the initial value problem

is given by

Proof : Follows from the theorem 3.3 and 3.4

4. Controllability and Observability

In this section we consider the higher order Sylster’s type matrix dynamical systems on measure chains

(4.1)

and obtain the necessary and sufficient conditions for the controllability and observability of the corresponding control engineering system

Definition 4.1 : The higher order Sylster’s type time varying dynamical system S on measure chain T=[t0 ,tN] is said to be completely controllable if "for any initial time t0 and any initial state x (to) = xo and any given final state xf, there exists a finite time tN > to and a control u(t), to £ t £ tN such that x (tN) = xf".

Definition 4.2: The higher order Sylster’s type time varying dynamical system defined by S on measure chains is said to be completely observable if and only if the knowledge of the control u(t) and the out put Q(t) suffice to determine x(t0) = x0 uniquely for a finite time tN ³ t0.

Theorem 4.1: The solution of the equation (4.1) is given by

Proof: Follows from theorem 3.6.

Theorem 4.2: The Lyapunov type matrix dynamical system on measure chains  is completely controllable if and only if the symmetric controllability matrix

is non singular.

Then the control U(t) defined by

defined for t0 < t < tN transfers X(t0) = X0 to X(tN) = Xf.

Proof: First we suppose that is non - singular. Then  defined as above exists. We know that any solution of (4.1) has the form

put  and substitute  defined as above we will get  and hence  is controllable.

Conversely, suppose that  is controllable. We have to show that  is non-singular.

Since is symmetric, clearly it is positive semi definite.

Now suppose that there exists some column vector such that

where

Hence  on .

By our assumption that  is completely controllable, there exists a control V(t) (say) making  if  where  is any non zero  matrix.

Now it can be easily shown that

Therefore  Hence  which is a contradiction.

Therefore  is a positive definite matrix hence it is non-singular.

Theorem 4.3: The system  is completely observable if and only if the symmetric observability matrix

is non singular.

Proof: Suppose  is nonsingular.

Without loss of generality suppose that U(t) º 0 " t Î [t0, tN]

Then

For this the output is

Delta integrating from  to  we get

Which gives

Therefore  is completely observable.

Conversely suppose  is completely observable.

We will show that  is non singular.

Since is symmetric, clearly it is positive semi definite.

If possible suppose that there exists a column vector  such that

. Then

.

If  where K is a matrix of order n, then the output is

i.e  can not determined with the knowledge of  in this case. This contradicts our assumption that  is completely observable.

Therefore is positive definite and hence  is non - singular.

Observation 4.1: From the theorem (4.2) it is observed that the controllability matrix is independent of fundamental matrix. The fundamental matrix  alone determines the controllability criterion of the dynamical system. We also observe that the controllability criterion can be determined using the fundamental matrix  alone through properly defined controllability matrix.

Observation 4.2: From the theorem (4.3) it is observed that the observability matrix is independent of . The fundamental matrix  alone determines the observability criterion of the dynamical system .

5. Summary and Conclusions

These results coincide with the findings [1],[2],[5] and [11] of and include them as a sub case of this article. Hence this article generalizes the findings of the article and can be applicable to wider range of control engineering problems. Using these results we can study further class of differential systems on measure chains for their controllability and observability.

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