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It has been previously revealed that the stability and the monotonicity of step response in an integer-order LTI system are two specifications, which are preserved in its fractional counterparts possessing commensurate orders between zero and one. In this paper, it is shown that the monotonicity of magnitude-frequency is another specification which is inherited from integer-order systems to their commensurate order counterparts. This finding completes the trilogy of stability, extrema-freeness of the step response, and monotonicity of magnitude-frequency, as the specifications whose holding in integer-order systems yields in meeting them in their fractional-order counterparts with... 

In this paper, an effective way is proposed for reduction of oscillations in the response of dynamical systems. In this regard, it is analytically shown that the undesirable oscillations in the response of dynamical systems can be reduced using a simple input shaping fractional order filter. The effectiveness of the proposed fractional calculus based technique is numerically verified in reduction of oscillations in a mass-spring-damper system, a low-pass Chebyshev filter, and a PI-controlled two-mass drive system  

Finding the oscillatory region in the order space is one of the most challenging problems in nonlinear fractional-order systems. This paper proposes a method to find the possible oscillatory region in the order space for a nonlinear fractional-order system. The effectiveness of the proposed method in finding the oscillatory region and special order sets placed in its boundary is confirmed by presenting some examples  

The aim of this paper is to provide a survey on the recently obtained results which are useful in time response analysis of fractional-order control systems. In this survey, at first some results on error signal analysis in fractional-order control systems are presented. Then, some previously obtained results which are helpful for system output analysis in fractional-order control systems are summarized. In addition, some results on the analysis of the control signal and the system response to the load disturbances in fractional-order control systems are reviewed  

This paper deals with irrational transfer functions having monotonic nondecreasing step responses. Firstly, some results on the monotonicity of step responses in irrational transfer functions describing fractional- or distributed-order systems are presented. Then, some conditions guaranteeing the existence of monotonic nondecreasing step responses in more general forms of irrational transfer functions are found. Various examples are brought to show the usefulness of the obtained results in time response analysis of fractional/distributed-order systems. The achievements of the paper can be used in the design of control systems having monotonic step responses  

This study deals with the investigation of the magnitude-frequency responses of all-pole fractional order systems in the viewpoint of extrema existence in these responses. In this investigation, a sufficient algebraic condition, two necessary algebraic conditions, and a necessary and sufficient algebraic condition are obtained to guarantee the non-existence of extrema in the magnitude-frequency response of all-pole fractional order systems. Some examples are presented to show the effectiveness of the results of the paper  

In this paper, a sufficient condition for existence of an overshoot in the step response of fractional-order systems is presented. Based on this condition, it can be shown that the existence of an overshoot in the step responses of some classes of fractional-order systems (for example, the class of fractional-order systems having commensurate orders between 1 and 2) is unavoidable. To show the usefulness of the obtained condition, this condition is applied to prove some results on the time response analysis of fractional-order control systems  

Proportional-integral (PI) controllers are the most common form of feedback used in industrial applications today [1][3]. The use of proportional and integral feedback also has a long history of practical applications [4]. For example, in the middle of the 18th century, centrifugal governors as the proportional feedback were applied to regulate the speed of windmills [5]. By the 19th century, it was known that using integral feedback could remove the offsets appearing in working with governors [6]. At present, PI control, still a very basic form of feedback, is also one of the first solutions often considered in the control of industrial systems [7]. On the other hand, in some applications,... 

In this letter, it is shown that some of the equalities that were used in the proof of the main theorem of the paper given by Lin and Lee are not consistent with fractional calculus principles. Simple counterexamples are provided to confirm this point. Moreover, correct versions of equations that were derived in the mentioned theorem are presented. Based on these corrections, the synchronization scheme proposed in the mentioned paper is investigated  

This paper investigates the step responses of fractional order systems in the viewpoint of extrema existence in such responses. It is proven that a fractional order system with a commensurate order between zero and one has an extrema-free step response if its integer counterpart has such a step response. In addition, it is shown that the step response of a stable fractional order system with a commensurate order between one and two cannot be monotonic. Based on these achievements, some further results on the step response of different classes of fractional order systems are presented  

This paper deals with the problem of stabilizing the unstable fixed points of a class of fractional-order chaotic systems via using static output feedback. At first, a static output feedback controller designed to stabilize a fixed point of a fractional-order chaotic system is considered. Then, the maximal allowable perturbation bound around the nominal value of the output feedback gain of the designed controller, such that the stability of the intended fixed point in the closed-loop system is guaranteed, is analytically determined. Also, some numerical examples are presented to confirm the validity of the analytical results of the paper  

In this note, the type number concept is defined for fractional-order systems. Based on this definition, it is shown that fractional-order systems having type numbers more than 1 can not track some classes of reference inputs without any overshoot  

In approximation of fractional order systems, a significant objective is to preserve the important properties of the original system. The monotonicity of time frequency responses is one of these properties whose preservation is of great importance in approximation process. Considering this importance, the issues of monotonicity preservation of the step response and monotonicity preservation of the magnitude-frequency response are independently investigated in this paper. In these investigations, some conditions on approximating filters of fractional operators are found to guarantee the preservation of step magnitude-frequency response monotonicity in approximation process. These conditions... 

This study deals with the properties of magnitude-frequency responses in fractional order systems. Using Phragmén-Lindelöf theorem in complex analysis, it is shown that the supremum of the magnitude-frequency response of a fractional system with a commensurate order less than one cannot be greater than that of its integer order bounded-input, bounded-output stable counterpart. Further results are also obtained on magnitude-frequency response of stable/unstable fractional order systems. Moreover, it is found that the supremum (infimum) of the magnitude-scaling frequency of the family of fractional order systems having a fixed structure and different orders in the range (0, 2) is a piecewise... 

In this note, it is shown that some of the results reported in the above titled paper on the behavior analysis of a fractional order system are not consistent with reality. To support this claim, theoretical justifications and numerical results are presented. Also, it is analytically explained that why such inconsistencies have been occurred in the aforementioned paper  

In this paper, an effective way is proposed for reduction of oscillations in the response of dynamical systems. In this regard, it is analytically shown that the undesirable oscillations in the response of dynamical systems can be reduced using a simple input shaping fractional order filter. The effectiveness of the proposed fractional calculus based technique is numerically verified in reduction of oscillations in a mass-spring-damper system, a low-pass Chebyshev filter, and a PI-controlled two-mass drive system  

In this paper, it is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the... 

Integral performance indices as quantitative measures of the performance of a system are commonly used to evaluate the performance of designed control systems. In this paper, it is pointed out that due to existence of non-exponential modes in the step response of a fractional-order control system having zero steady state error, integral performance indices of such a system may be infinite. According to this point, some simple conditions are derived to guarantee the finiteness of different integral performance indices in a class of fractional-order control systems. Finally, some numerical examples are presented to show the applicability of the analytical achievements of the paper  

It has been pointed out that the numerical simulation results presented in the above paper are not consistent with reality. The reason for this inconsistency has analytically been clarified in this note. © 2009 IEEE  

In this paper, a simple fractional calculus-based control law is proposed for asymptotic tracking of ramp reference inputs in dynamical systems. Without need to add any zero to the loop transfer function, the proposed technique can guarantee asymptotic ramp tracking in plants having nonminimum phase zeros. The appropriate range for determining the parameters of the proposed control law is also specified. Moreover, the performance of the designed control system in tracking ramp reference inputs is illustrated by different numerical examples. © 2016 by ASME