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# Point to point control of fractional differential linear control systems

- Andrzej Dzieliński
^{1}Email author and - Wiktor Malesza
^{1}

**2011**:13

https://doi.org/10.1186/1687-1847-2011-13

© Dzieliński and Malesza; licensee Springer. 2011

**Received:**9 December 2010**Accepted:**22 June 2011**Published:**22 June 2011

## Abstract

In the article, an alternative elementary method for steering a controllable fractional linear control system with open-loop control is presented. It takes a system from an initial point to a final point in a state space, in a given finite time interval.

## Keywords

- fractional control systems
- fractional calculus
- point to point control

## 1 Introduction

Fractional integration and differentiation are generalizations of the notions of integer-order integration and differentiation. It turns out that in many real-life cases, models described by fractional differential equations much more better reflect the behavior of a phenomena than models expressed by means of the classical calculus (see, e.g., [1, 2]). This idea was used successfully in various fields of science and engineering for modeling numerous processes [3]. Mathematical fundamentals of fractional calculus are given in the monographs [4–9]. Some fractional-order controllers were developed in, e.g., [10, 11]. It is also worth mentioning that there are interesting results in optimal control of fractional order systems, e.g., [12–14].

In this article, it will be shown how to steer a controllable single-input fractional linear control system from a given initial state to a given final point of state space, in a given time interval. There is also shown how to derive hypothetical open-loop control functions, and some of them are presented. This method of control is an alternative to, e.g., introduced in [15], in which a derived open-loop control is based on controllability Gramian matrix, defined in [16] that seems to be much more complex to calculate than in our approach.

The article is divided into two main parts: in Sect. 2 we study control systems described by the Riemann-Liouville derivatives and in Sect. 3--systems expressed by means of the Caputo derivatives. In each of these sections, we consider three cases of linear control systems: in the form of an integrator of fractional order *α*, in the form of sequential *nα*-integrator, and finally, in a general (controllable) vector state space form. In Sect. 3.3, an illustrative example is given. Conclusions are given in Sect. 4.

## 2 Fractional control systems with Riemann-Liouville derivative

*α*∈ ℂ, on a finite interval of the real line [4, 9]:

where *n* = [ℜ(*α*)] + 1, and [ℜ(*α*)] denotes the integer part of ℜ(*α*).

i.e., the limit taken in ]*t*
_{s}, *t*
_{s} + *ε* [for *ε* > 0.

The existence and uniqueness of solutions of (2.1) and (2.2) were considered by numerous authors, e.g., [4, 8].

### 2.1 Linear control system in the form of *α*-integrator

where 0 < *α* < 1, *z*(*t*) is a scalar solution of (2.3), and *v*(*t*) is a scalar control function.

*z*(

*t*), from the

*start point*

in a finite time interval *t*
_{f} - *t*
_{s}. In other words, we are looking for such an open-loop control function *v* = *v*(*t*), which will achieve it in a finite time interval *t*
_{f} - *t*
_{s}. The start and final points will be also called the *terminal points*.

*w*

_{1}corresponding to (2.4). To this end, initial condition (2.6) can be rewritten (see [4]) as

**Proposition 1**.

*A control v*(

*t*)

*that steers system*(2.3)

*from the start point*(2.4)

*to the final point*(2.5)

*is of the form*

which finally yields *z*(*t*) = *φ* (*t*). In particular, *z*(*t*
_{f}) = *φ* (*t*
_{f}) = *z*
_{f}. □

*Example*

**2**. We want to steer system (2.3) from the start point (2.4) to the final point (2.5) by means of the control given by (2.8), where

*a*

_{0}and

*a*

_{1}have to be chosen such that conditions (2.9) hold, i.e., from

*v*(

*t*) is right, we integrate (2.14) by means of , giving

*z*

_{s}is given by (2.6) and (2.7), and substituting already calculated coefficients

*a*

_{0}and

*a*

_{1}given by (2.13), we get

Since
for *α* < 1, evaluating (2.15) at *t* = *t*
_{s} yields *z*(*t*
_{s}) = *z*
_{s}, and for *t* = *t*
_{f} gives *z*(*t*
_{f} ) = *z*
_{f} .

### 2.2 Linear control system in the form of *nα*-integrator

*z*(

*t*) is a scalar solution of (2.16), (2.17), and

*v*(

*t*) is a scalar control function. By we mean

at time *t*
_{f} , in the finite time interval *t*
_{f} - *t*
_{s}.

**Proposition 3**.

*A control v*(

*t*)

*that steers system*(2.16)

*from the start point*(2.20)

*to the final point*(2.21)

*is of the form*

*n*integrations, yields

□

satisfying (2.22).

*i*= 0, ..., 2

*n*- 1 and

*α*> 0 (0 <

*α*< 1). It follows that for the function (given by (2.28)), we have

in the finite time interval *t*
_{f} - *t*
_{s}.

from which we can calculate coefficients *a*
_{
i
} , 0 ≤ *i* ≤ 3, assuming that *t*
_{f} > *t*
_{s}.

*Z*

_{s}to the final point

*Z*

_{f}, is

where *a*
_{
i
} , 0 ≤ *i* ≤ 3, are already calculated from (2.29).

### 2.3 Linear control system in the general state space form

*x*(

*t*) = (

*x*

_{1}(

*t*), ...,

*x*

_{ n }(

*t*))

^{ T }∈ ℝ

^{ n }is a state space vector,

*A*∈ ℝ

^{ n×n },

*u*(

*t*) ∈ ℝ,

*b*∈ ℝ

^{ n×1}and . The initial conditions are

*t*

_{f}-

*t*

_{s}. To this end, since Λ is assumed to be controllable [15, 16], i.e.,

*t*

_{1}satisfies (2.33), then, using the linearity of Riemann-Liouville derivative, we have

where
is the *n* th row of the matrix *R*
^{-1}(*A*, *b*).

In such a way, we have transformed the problem of finding a control u(t) for the system (2.30) steering from the start point (2.31) to the final point (2.32), into an equivalent problem of finding a control *v*(*t*) for system (2.35) steering from the start point (2.36) to the final point (2.37), which has already been explained in Sect. 2.2.

*C*

^{ n }-function

*φ*(

*t*) satisfying (2.22) for given (2.36) and (2.37). For such a function

*φ*(

*t*), the control is

## 3 Fractional control systems with Caputo derivative

*α*∈ ℂ and ℜ(α) ≥ 0. If

*α*∉ ℕ

_{0},

*n*= [ℜ(

*α*)] + 1, and then

It has been already shown, e.g., in [4] that for (3.1) and (3.2) a solution exists.

### 3.1 Linear control system in the form of *α*-integrator

where *z*(*t*) is a scalar solution and *v*(*t*) is a scalar control function.

*t*

_{f}-

*t*

_{s}. In contrast to the equation defined by means of Riemann-Liouville derivative, initial conditions (3.4) coincide with start point (3.5), i.e.,

**Proposition 5**.

*A control v*(

*t*)

*that steers system*(3.3)

*from the start point*(3.5)

*to the final point*(3.6)

*is of the form*

□

*Example* **6**. Consider control system (3.3), for 0 < *α* < 1, where *n* = [*α*] + 1 = 1. We want to find a control function *v*(*t*), which steers (3.3) from the given start point *z*(*t*
_{s}) = *z*
_{s0} to the given final point *z*(*t*
_{f} ) = *z*
_{f0}.

Evaluating (3.16) at *t* = *t*
_{s} gives *z*(*t*
_{s}) = *z*
_{s0} and for *t* = *t*
_{f} yields *z*(*t*
_{f} ) = *z*
_{f0}, which means that control (3.14) correctly steers the system from *z*
_{s0} to *z*
_{f0}.

**Remark 7**. For 0 <

*α*< 1, the problem of steering system (3.3) from start point (initial condition) (3.5) to final point (3.6) can be also solved using the known relation between Caputo and Riemanna-Liouville derivative, i.e.,

### 3.2 Linear control system in the form of *nα*-integrator

*nα*, where α ∈ ℝ, 0 <

*α*≤ 1, and

*n*∈ ℕ

_{+}, such that

*nα*< 1, given by

where *z*(*t*) is a scalar solution, *v*(*t*) is a scalar control function, and
is defined like in (2.18), but for the Caputo derivative.

**Proposition 8**.

*A control v*(

*t*)

*that steers system*(3.19)

*from start point*(3.20)

*to final point*(3.21)

*is of the form*

□

*t*-

*t*

_{s})

^{ iα }is

### 3.3 Linear control system in the general state space form

*x*(

*t*) = (

*x*

_{1}(

*t*), ...,

*x*

_{ n }(

*t*))

^{ T }∈ ℝ

^{ n }is the state space vector,

*A*∈ ℝ

^{ n×n },

*u*(

*t*) ∈ ℝ,

*b*∈ ℝ

^{ n×1}and . The initial conditions are

*t*

_{f}-

*t*

_{s}. Then, obviously, the initial conditions have to be set to

for which we apply the theory presented in Sect. 3.2.

*u*(

*t*) taking system Λ from the start (initial) point (at time

*t*

_{s}= 1 s)

in the finite time interval *t*
_{f} - *t*
_{s} = 4 s.

## Conclusions

In the article, a method for steering a control system from one point to another in a state space was presented. Both for the system described by Riemann-Liouville derivative and Caputo derivative three forms of control systems were studied. In both cases, the *nα*-integrator form was introduced as a scalar representation equation of a control system in a controllable state space form. Because of the specific nature of initial conditions for systems defined by means of Riemann-Liouville derivative, numerical example was given only for the systems with Caputo derivative. The choice of possible candidates for control functions presented in the article is not the only one possible. Other functions achieving the task can also be found. Since in our approach no restrictions are posed on the trajectory joining two given points, a family of such trajectories, and thereby "base-functions," can be relatively wide, and authors have proposed some selected examples of such functions (e.g., (2.27), (3.11)). If one would wish, additionally, to steer a system from a given point to another one in an optimal way, i.e., with minimizing some cost function, this implies a specific trajectory. In such a case it is still possible to look for the other type of functions (satisfying one of Propositions 1, 3, 5, and 8) restricted additionally by these optimality constraints. In other words, it can be possible to find some other type of functions, perhaps different from these selected by authors, achieving the desired task. Interesting results in optimal control of fractional systems can be found, e.g., in [12–14].

## Declarations

## Authors’ Affiliations

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