^{1}

The Riemann-Liouville-, Caputo-, and Grünwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point, and secondly to distinguish initial conditions for particular operator.

The main topic of the paper concerns systems with fractional difference operators. Fractional sums and operators are generalizations of

In this contribution, the Riemann-Liouville-, Caputo-, and Grünwald-Letnikov-type fractional order difference operators are discussed and used to state and solve the controllability and observability problems of linear fractional order discrete-time control systems with multiorder and multistep. Our goal is to state basic properties of fractional difference multiorder and multistep linear control systems and to compare using some particular examples if there are some varieties between operators and numbers of steps that we need to have a system being controllable or observable or achieve some exact target. It is shown that the obtained results do not depend on the type of fractional operators and steps. The comparison of systems is made under the number of steps needed, firstly to achieve a final point and secondly to distinguish initial conditions for particular operator. As our results are formulated for multistep systems, they remain also true for systems with the same step along coordinates of solutions and particularly for the common step

In the paper definitions of operators for general step and orders are presented. In our systems we use

The paper is organized as follows. In Section

Now we are listing the necessary definitions and technical propositions that we use in the sequel of the paper. Let

For a function

For

Let

In [

Let

In our consideration the crucial role is played by the power rule formula; see [

As the tool in the present consideration we can define the next family of functions. Let

The properties of two-indexed functions

Let for

Family of functions

for all

For the family of functions

Let for

Let

Similarly as in [

The next step is to present three main fractional difference operators: Caputo-type, Riemann-Liouville-type, and Grünwald-Letnikov-type

Let

Note that

For the Caputo-type fractional difference operator there exists the inverse operator that is the tool in recurrence and direct solving fractional difference equations.

Let

We can also state, similarly as in Lemma

Let

From this moment for the case

The next presented operator is called fractional

Let

For

The next propositions give useful formula for transforming fractional difference equations into fractional summations.

Let

The next Lemma gives the transition formula for the Riemann-Liouville-type operator between the cases for any

Let

For the case

The third type of the operators that we take into our consideration is the fractional

Let

We can easily see the following comparison.

Let

From Proposition

The operators presented in this section can be extended to operators acting on vector valued functions in a componentwise manner.

In this section we consider initial value problems of fractional order systems of multiorder and multistep difference equations with three types of operators. In the next section we state as the conclusion the solution of initial value problem for control systems with the Caputo-, the Riemann-Liouville-, and the Grünwald-Letnikov-type difference operators and formulate common results for the controllability as well for observability. We consider systems with operators written in the general form

The solution of system (

The most important step in the proof is to use the properties of fractional summations with different orders of functions from families

Let

In this section we construct the solution of initial value problems and consider the controllability problem of multistep and multiorder difference linear control systems for different forms of operators.

We consider systems with operators written in the general form

Let

The steps in the proof are similar to the case with one value for the step

We can consider also the value

System (

Note that controllability means that the final state can be reached in

Let us define the controllability matrix for the system (

The next propositions give the formula for the values of steering control and also rank condition for complete controllability. The results have the same meaning like in the classical theory. For the fractional difference systems with the Caputo-type operator with one step

If the matrix

Control system (

Let us consider multiorder and multistep difference linear control system of the form (

System (

Let us define a real matrix

System (

At the end we give the example in which we show that for a given system with the Riemann-Liouville-type operator we have faster recognition of the initial state than for the system with the Caputo-type operator.

System (

columns of the matrix

The matrix

Let us consider linear control system with two fractional orders

Firstly we consider the situation with the Caputo-type difference and

Taking into account the Caputo-type operator and

Now we take the Riemann-Liouville-type operator; then we need values of matrix function

From our consideration it follows that we could not distinguish which model/type of the system is better according to the controllability or the observability properties. In fact it depends on the matrices inside the systems. Moreover parameters that one can include in the description of some process involve the model and the type operator with ramifications of the used model. We cannot downplay the role of the used orders and steps. However this paper does not discuss any particular process. This is rather theoretical consideration of possible properties.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is grateful to anonymous referees for valuable suggestions and comments, which improved the quality of the paper. The project was supported by the funds of National Science Centre granted on the bases of the decision number DEC-2011/03/B/ST7/03476. The work was supported by Bialystok University of Technology Grant G/WM/3/2012.