A Note on Commensurate-Order Characteristic Root Equivalency Class of Linear Time Invariant Systems

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.


I. INTRODUCTION
HERE has been a growing interest for utilization of Fractional Calculus (FC) in engineering and science problems because of its promises of better describing real world objects and phenomenon [1][2][3][4][5]. It was suggested that real world objects can be more accurately modeled and analyzed by using FC because real world objects do not have to precisely comply with integer order system models: Majority of them may exhibit fractionalty even at a low degree [1,2,[5][6][7][8][9][10]. The fractional order derivative and integration were shown to act upon solutions of many problems in physics [6,7,8], thermodynamics [9], electrical circuits theory and fractances [10,11,12], mechatronics systems [13], chaos theory [14], control systems [15,16,17] etc.
Linear Time Invariant (LTI) systems have been used as a fundamental and substantial modeling tool for theoretical analyses of real systems [18]. Extensive researches on LTI systems offered well established mathematical background, simplified solutions, experimentally proven methods for the characterization and analysis of real systems. In these analyses, LTI system models can be expressed in many forms such as differential equations, state space models and transfer function forms [18]. LTI system modeling techniques, system stability, observability and controllability issues have been studied extensively and applied in engineering problems.
Commensurate order LTI systems are indeed a class of fractional order systems, which provides simplification for analysis and design efforts because it has a proper expression of fractional orders in the form of  , the system turns into conventional integer order LTI systems. As known, system behavior and model structure strongly depends on root locus of LTI system.
Investigating effects of fractional order  on root placement of characteristic equations on complex plane can be helpful to explain connections between fractional order LTI system model and integer order LTI system model.
This study investigates properties of a special class of commeasure order systems. This class of system models has the same characteristic root set for different commeasure order (

B. Effects of fractional order root equivalency
In this section, we study root equivalency of the first order LTI systems. In further sections, we extent our analyses on root equivalency to second order systems. The first order LTI systems establish the most basic form of root equivalent fractional order systems.
Let consider a commensurate order LTI system, defined in form of are input and state of the system, respectively. The system output was assumed as ) ( and a zero initial state ( ) is used for the sake of computational simplicity. The transfer function of the first order LTI system for 1   is expressed in general form as, In equation (1)   Effects of fractional order on system stability and the first Riemann sheet were discussed in detail in Ref [1,[21][22][23][24]. By applying m u s  transformation to characteristic polynomial of fractional order systems, the stability analysis of a fractional order system based on the root locus was confined in the first Riemann sheet that is defined as the portion of complex plane with angle range of m m / / is already known that root magnitude takes effect on the response time of a stable system. As magnitude decreases, system response is getting slower.
in equation (2), one can express the root equivalent family of first order LTI systems in the form of The  order root equivalency mostly generates complex coefficient system models from real coefficient first order LTI system models due to the fact that first order system has a single real root without any conjugate pair with respect to real axis and this root moves in a round trajectory without conjugate trajectory as illustrated in Fig. 1.
Here, 0234 The magnitude of root is changed from 2 to 1.7411 ( and the angle of root changes from  to  8 . 0 radian. For 1   , magnitude and angle of the root decrease. Fig. 3 shows this effect, graphically. Fig. 3 illustrates possible root equivalency trajectories of the first order systems The root trajectories in the figure demonstrate that the system model has single complex root without a complex conjugate. Practically, it is difficult to implement transfer functions containing complex roots without its conjugate pairs. One see that, for 0   , all trajectories converge to one. For 0   , type of root equivalency trajectory of the system (narrowing, circling or expending trajectories) depends on the magnitude of the root ( M ).

C. Root addition for complex root conjugating of commensurate order equivalence of first order LTI Systems
System designers may question whether it is possible to derive a real coefficient commensurate order system from the first order LTI systems by using  -order root equivalency so that realization of complex coefficient systems is a complicated problem for practice. Previously, to carry out implementation of complex-order systems, several studies have addressed the implementation problems of complex coefficient transfer functions. To obtain real valued outputs from complex-order systems, conjugated-order system concept were introduced [34,35].
In this section, we benefit from a similar perspective, which involves complex conjugate root addition to system model in order to transform a complex coefficient root equivalent system into a real coefficient commensurate order systems. This method increases root counts of characteristic equation from one to two because of ) . Thus, this additional conjugate root allows real valued outputs from the system model. With the complex root conjugating, equation (3) expand to the model of real coefficient  commensurate order system as follows.
)) sin( ) cos( When this equation is arranged, one obtains the real coefficient commensurate order root equivalent system with complex conjugate as follows, Here, the root angle  can take values of    , 0 so that a real coefficient first order LTI system ) (s T has a characteristic root only on the real axis. If ) (s T is a stable system, the root resides on the left half plane and    . The real coefficient ) (s T  function expresses a special class for the second order commensurate order system that is based on root equivalency of first order LTI systems. Previously, some analyses addressing the stability of second order conjugate systems were addressed in detail by Radwan et al [23].
Let's investigate effects of  order on the system behavior.
A discussion on system behavior depending on system pole location was previously given by Monje et al. [32]. We performed simulations and observed the following remarks for root equivalencies of the first order stable LTI systems given by equation (7):  (7). In order . This property allows a straightforward solution for testing of the stability of a special class of commensurate order systems expressed in the form of equation (7). A stability check procedure for a given commensurate order system complying with the equation (7) can be summarized as follows: Step 1: By considering characteristic polynomial in the form of Step 2: By considering the term 0 a , calculate M by Step

D. An extension of root equivalency analysis for the second order LTI systems
In this section, we extent our investigation for root equivalence of the second order LTI systems. The transfer function of the second order LTI system can be written in general form as, ) )( ( The root equivalent characteristic polynomial of the equation (8) Table 2.
Step responses of these systems are shown in Fig. 4. The step responses in the Step responses obtained for commensurate order root equivalent systems, which are listed in Table 2 ALAGOZ: A NOTE ON COMMENSURATE-ORDER CHRACTERISTIC ROOT EQUIVALENCY CLASS OF LINEAR TIME Copyright © BAJECE ISSN: 2147-284X http://www.bajece.com 92 equivalent systems is faster than the first order integer order LTI system. For 5 . 0   , the step response of commensurate order root equivalent system is slower than the first order integer order LTI system. Another noteworthy point should be emphasized that, .. 33 . 1   yields oscillations due to root placement on the stability boundary. More comprehensive discussions on sinusoidal oscillation condition of fractional and integer order systems was given in [22,23]. Step 4: In this case, since 3 In order to validate this result, we demonstrate the step response of ) (s G in Fig. 5(a) and root locus in the in the first Riemann sheet in Fig. 5(b) to apply Radwan stability test procedure [23]. Radwan stability test procedure is essentially based on Matignon's theorem [21]. This procedure applies m u s  transformation to characteristic polynomial and investigates root angle of expanded degree integer order polynomial on the first Riemann sheet. . Accordingly, amount of digits in fractional part increases computational complexity, significantly. One of the advantages of root equivalency analyses may appear in such cases. If commensurate order LTI system is a member of root equivalency class, it can significantly reduce computational complexity in stability analyses.
We used step function proposed in [33] to numerically calculate step responses in this study. Fig. 6 shows oscillating behavior of the system. As ...     Fig.  7(a) indicates step response of an unstable system and Fig. 7(b) indicates the step response of a stable system.

IV. CONCLUSIONS
This study shows that one can establish analytical relations between fractional order LTI systems and integer order LTI systems via characteristic root equivalency. These relations can facilitate design and analysis efforts of commensurate order systems on bases of root equivalency of integer order LTI systems and help comprehension of effects of fractional orders on the system behavior.
In summary,  -order commensurate root equivalence classes of the first and second order stable LTI systems are investigated in the paper. The first order LTI system produces complex coefficient commensurate order root equivalent