#### Abstract

This study proposes an approach to synthesize a three-impulse sequence input shaper with a negative impulse, known as Unity Magnitude (UM) shaper. The corresponding analytic model has been already achieved for undamped and low-damped systems. In this paper, the analytic design of UM shaper is demonstrated for the generalized case of damped systems for both types: integer and fractional orders. Hence, the UM shaper model has been designed for second-order systems with damped dynamics, associating a graphical fitting and an analytical procedure; then, it has been extended to explicit fractional derivative systems. Moreover, the feasibility and the effectiveness of the proposed on-off profile prefilter applied on a second-generation controller have been substantiated by experimental results on an instrumented DC motor bench.

#### 1. Introduction

Industrial motion systems require the suppression of the vibration to ensure mostly the stability and also the accuracy. Researchers and engineers have worked on command generation approaches to ensure such a purpose. Input shaping strategy constitutes a key issue for reducing residual vibrations in motion control systems, based on its robustness and effectiveness [1, 2]. Singer and Seering are the pioneers by proposing a practical input shaping scheme for flexible structures [2, 3]. It is an open-loop control approach, implemented by convolving a train of impulses with any desired command profile, to generate the shaped reference command used to drive the system [1]. However, open-loop controllers are subject of external disturbances; thus, input shaping is used in conjunction with feedback control to ensure good efficiency [3]. The design of shapers is achieved by determining timing instants and amplitudes of the impulses, depending on the natural frequency and the damping ratio of the vibratory system, and by solving a set of constraint equations [2, 4].

Several works dealt with input shaping since its originality and simplicity to be implemented. By using a two-impulse set, the Zero Vibration (ZV) shaper cancels perfectly the output vibrations and eliminates the overshoot in case of linear systems [5, 6]. Robustness is strengthened by adding derivative constraints, inducing a new impulse for each order: Zero Vibration Derivative (ZVD) and Zero Vibration Derivative Derivative (ZVDD) [6]. These different options have been already implemented on industrial systems [7] such as flexible robot manipulators, rotary cranes [3], and flexible beams.

Further, input shaping scheme has been extended to fit fractional order systems, notably for the ZV and ZVD shapers [8–12]. It has been associated with CRONE controllers for both real and complex orders, as well as fractional order PID controllers [13].

To overcome the delay induced by added impulses, new shapers have been designed by employing negative amplitudes that permit improving the rise time, shortening the shaper duration, and cancelling undesired oscillations [14–16]. However, unmodeled high modes can be subject of excitation [15]. Unity Magnitude (UM) shaper is one of these shapers as it allows negative impulses to be generated by switching impulses amplitudes between positive and negative unity [17–19]. Analytic definition of impulses timing is complex to solve for damped systems [14, 15, 17]. In [17], a restricted solution of UM shaper impulses times has been presented, only for second-order undamped systems and low-damped ones, with a damping ratio value limited between 0 and 0.35.

The present paper deals basically with the design of the UM shaper for explicit fractional derivative order systems. For this purpose, the first challenge is to overcome the restriction on second-order damped systems. Thus an, in-depth work has been undertaken to write UM shaper impulse time location models [18]. A graphical approach has been associated with such development. In a second step, the elaborated approach has been extended for the case of explicit fractional derivative systems, with the purpose to associate it with the CRONE control scheme. Thus, the developed analytic model of the UM shaper has been validated by the implementation on a DC motor test bench in association with a second-generation CRONE controller.

This paper is organized as follows. Section 2 is reserved to a background on the CRONE control and explicit fractional order systems. Section 3 introduces a background on the Unity Magnitude shaper. In Section 4, the problem description and the main contribution of this work are presented. Section 5 details the analytical synthesis of the UM shaper for second-order damped systems. Section 6 deals with the extension of the new analytic solution to explicit fractional derivative systems. In Section 7, real experimentation on a test bench with a CRONE controller is run in order to validate the effectiveness of the proposed approach. Finally, this paper ends with a conclusion.

#### 2. Background on Second-Generation Crone Control and Explicit Fractional Order Systems

Fractional order robust control is named CRONE control by reference to the French nomination “Commande Robuste d’Ordre Non Entier.” The CRONE Control System Design (CSD) methodology has been introduced since the by Oustaloup [20, 21] and is basically considered as a frequency-domain method. It relies on the unity-feedback control theory. Three CRONE CSD methods have been established and applied in several application fields. The main idea among the synthesis of these three approaches is to set the controller or the open-loop transfer function with reduced number of parameters and to define it by a fractional order integrodifferentiation [22, 23]. Thus, the major advantage of the CRONE controller, mainly its robustness, is due to its few number of parameters.

The second-generation CRONE controller defines the open-loop transfer function in a limited frequency interval around the cross-over frequency as *n* fractional order integrator (equation (1)), , and *n* is limited to to ensure stability:where .

In this interval range, the Nichols locus of is a vertical straight line, called frequency template, whose phase position is strictly related to the noninteger order . As the plant parameter varies, the template slides vertically keeping the same phase location. Thus, this vertical displacement allows the robustness in phase margin, resonant peak, modulus margin, and damping ratio of the closed-loop system [22, 23].

In closed loop, a second-generation CRONE controller is described by a fundamental transfer function, which refers to explicit fractional derivative systems and is given bywhere is called time constant and is the order of the system. Obviously, works and findings corresponding to explicit fractional derivative systems are valuable for second-generation controllers once synthesized according to desired performances.

#### 3. Background on Unity Magnitude Command Inputs

Classical shapers, called Zero Vibration (ZV), are based on convolving two positive impulses to any command signal so that output vibrations are cancelled [2] as shown in Figure 1.

Such a preshaping technique adds a delay to the plant response in comparison to its natural dynamic. Consequently, in order to improve the rise time and shorten the shaper duration, negative impulse amplitudes are introduced as in [15, 24]. The UM shaper is the case where amplitudes are required to switch between and as illustrated in Figure 2 [15]. Thus, the UM shaped command can be easily generated using on-off actuations differently from the ZV shaped input profile (Figure 1) [24].

In general, for a second-order system with a natural frequency and a damping ratio , an input shaper is determined by satisfying residual vibration constraint equation (3) [1, 2, 5, 17]. and are, respectively, the amplitude and the location time of the impulse and is the number of impulses of the shaper. Equation (4) is considered so that the shaped command produces the same rigid body motion as the unshaped input command.

Considering the following system (5) dealing with the on-off amplitude profile, UM shaper needs to be identified through the location times of its three impulses. The first one is intuitively made at to ensure optimal shaper duration.

Now, substituting into equation (3), it becomes

However, solving analytically equation (6) is likely to be difficult and needs linear and nonlinear approximations [17, 18]. Though, the proposed solution does not give exact results for all damping ratio values.

The solution for undamped systems is easy to be determined using the amplitude summation constraint, the zero vibration constraint, and a trigonometric identity, which lead to designing the UM shaper for undamped systems as follows [15]:

#### 4. Problem Description and Main Contribution

Robust control and vibration cancellation constitute two potential objectives for a large variety of real problems coming from different fields of engineering sciences and branches of industries. In several solutions, both objectives have been solved separately, and even in combined solutions, some drawbacks remain. In fact, the fractional order controllers (CRONE controllers) and the input shaping approach are well known in robust control and vibration cancellation, respectively. Even when combined with ZV and ZVD shapers [8–12], the CRONE control loop induces a delayed system response. Besides, the delay is increased if the robustness is strengthened by means of the ZVD shaper.

The main contribution of this work consists of designing a control loop that ensures both objectives, combining the robustness and the simplicity of the implementation of a CRONE controller and the vibration reduction using the UM shaper, with the advantage of no delay time generation. This challenge is achieved following two steps:(i)First, the analytic synthesis of the UM shaper for second-order damped systems(ii)Second, the UM shaper design for explicit fractional derivative systems

Results show that, with the UM shaper, the desired objective is ensured: actuators are less solicited due to the vibration cancellation, and the system dynamic is fastened as no delay time is introduced. Moreover, real experimentation on a DC motor test bench is run to validate the effectiveness of the proposed control approach.

#### 5. Analytic Synthesis of UM Shaper for Damped Systems

Unity Magnitude shaper has the key feature to be easily implemented and to provide reduced control duration that guarantees no response delay. Thus, it has been adopted in many application areas as material-handling, lifting, and positioning applications [3, 25, 26]. The big challenge is to succeed to prove an analytical solution for time locations of the UM shaper in case of damped systems, which is likely to be complex as mentioned in previous works [17]. However, Gürleyük in [17] has elaborated one solution which has the disadvantage to be limited to low values of damping ratios varying between 0 and 0.35. In the sequel, analytic synthesis of UM shaper for second-order systems is successfully elaborated without any restriction on the damping ratio, which can largely vary from 0 to 1. The work is based on graphical estimations that permit deducing a key relation between second and third impulses times; then, further analytical development is completed.

##### 5.1. Graphical Solution for UM Impulses Times

As a first step, a graphical solution for times and has been found by using a curve fitting technique [17]. In fact, the idea is to fix randomly and then search for graphically, satisfying constraints of equation (6). An application example is used and is defined by a damping ratio and a natural frequency that are, respectively, given as and . Time was minutely chosen equal to , and time value of satisfying the constraints is deduced graphically and . Consequently, the resulting UM shaper configuration is as follows:

Figure 3 depicts the effect of the above synthesized UM shaper on the studied system.

Comparing both curves, the overshoot and the oscillations are perfectly cancelled for the shaped response regarding the unshaped one. Thus, the synthesized UM shaper by the curve fitting technique proves its effectiveness; nevertheless, advanced experimentation and studies should be achieved to formulate a generalized analytical solution. To ensure such a purpose, different simulations have been undertaken for varying damping ratio values. It derives an important characteristic: as the damping ratio values vary, impulses times and remain symmetrically positioned regarding the rise time . Thus, the following property is written:

Property of equation (9) will be used next to express impulses times *t*_{2} and *t*_{3}, for damped systems.

##### 5.2. Analytic UM Shaper Command Profile

Basically, the analytic equation of a second-order system step response is written as

Further, in the case of unity step command, the rise time is expressed by the following equation:

By simply equaling equation (9) and equation (11), it leads to write that

Even with equation (12), the resolution of equation (6) is still complex. Therefore, deeper graphical investigation is worth of conducting to reduce difficulties.

Let us consider that the interval of time separating the second from the third impulse is likely short; then, it can be assumed that the response of the system upon this range of time is perfectly fitted to the tangent at the middle time . The curve slope at the instant is deduced as follows:

Moreover, by means of graphical approximation, the slope of the tangent at the point is identified as follows [18]:

Bringing together equation (9) and equation (14), the second impulse time *t*_{2} can be expressed as follows:

The third impulse time expression *t*_{3} is also deduced as in equation (16):

Now, having expressions of equations (11), (13), (15), and (16), UM shaper is analytically designed as follows:

Validation of the graphical approximation used for the analytical development is proved through a set of systems with different damping ratios. A comparison between computed time values and estimated ones by curve fitting is established in Table 1. Relevant error values are also mentioned in Table 1. It shows that, for all the cases, the error is at least of order; therefore, it can be neglected.

To conclude, the analytical approach of the UM shaper design for damped systems is well approved in the case of second-order systems without any restriction on damping ratio values, and UM shaper model is given by equation (17).

#### 6. UM Shaper Design for Explicit Fractional Derivative Systems

In the previous section, the main contribution is the elaboration of an analytical model for UM shapers for second-order damped systems. Now, the work deals with the extension of such an approach on explicit fractional derivative systems [18, 27] so that it can be applied for the case of second-generation CRONE control.

##### 6.1. Response in Time Domain

In the operational field, a fundamental system is called explicit fractional derivative when its transfer function is described by equation (2) [21].

In the time domain, the step response of an explicit fractional system is written as mentioned in equation (18) [21, 28]:

For practical reasons, equation (18) is divided into two modes: the aperiodic mode as in equation (19) and the oscillatory mode as in equation (20).

As the aperiodic mode goes towards zero rapidly, the oscillatory mode is dominant and used to write constraints of residual vibrations, robustness, impulse times, and amplitudes [9, 18, 27]. For this mode, the damping ratio is set by the fractional order and expressed as in equation (21).

##### 6.2. Extension of UM Shaper for Explicit Fractional Derivative Systems

In previous works, Poty et al. [8, 9] and Jallouli-Khlif et al. [11, 12, 27] have already extended the ZV and ZVD shapers to explicit fractional derivative systems for real order and complex order, respectively. In the present work, the UM shaper is extended to real order explicit fractional derivative systems [18, 27].

Based on system (3), explicit fractional system constraints used for the time impulse calculation are deduced as in (22), using only the oscillatory mode [8, 11]. In the sequel, is referred to by , and is referred to by .

In the case of UM shaper (on-off profile), these constraints are upgraded as follows:

Advanced analytic development undertaken in [16, 18] proved that both constraints of system (6) devoted to second-order systems and (23) dealing with explicit fractional order systems are equivalent. Thus, the analytical approach design of UM shaper detailed in Section (5) is extended to explicit fractional derivative order systems, and the solution is perfectly written as follows:with

##### 6.3. Simulation Results

Figures 4–6 outline the UM shaper action hold on a set of explicit fractional derivative systems, for , , , and . Each figure presents both shaped and unshaped responses.

Results show a residual overshoot that still exists, unlike the case of a second-order system. This is related to the fact that only oscillatory mode is considered for calculations, which induces a nonperfect UM shaper action on the global response.

Nevertheless, comparing both responses of each figure (Figures 4–6) and dealing with performances values of Table 2, it can be depicted that the UM shaper enhances system performances in a considerable way as(i)It allows an evident reduction of the overshoot especially for high values of order (ii)It improves the settling time(iii)It respects the rising time for all the cases

Besides, oscillation cancellation is proved.

#### 7. UM Shaper for CRONE Control Experimentation on a DC Motor Bench

##### 7.1. Description of the Plant

The association of UM shaper with second-generation CRONE control is applied to the speed control of a DC motor testing bench which rotates a disk with 14 equal weights. The inertia driven by the motor depends on the number of mounted weights (Figure 7). This motor can be coupled to a second DC motor to simulate the disturbance torque, but in the present case, the motor is uncoupled [29].

The transmittance of the DC motor is given as follows:and specifications are as mentioned below.

, , .

Therefore, the transmittance of the motor bench is expressed by the following equation [29]:

Parametric variation of the process depends on the number of used weights. Besides, the disc must be loaded symmetrically to have a regular distribution of the inertia. In the proposed application, three parametric states are considered:(i)With minimum load (2 flyweights): (ii)With of the maximum load (8 flyweights): (iii)At maximum load (14 flyweights):

##### 7.2. Control Problem

The objective is to control the motor in spite of inertia variation and to guarantee:(i)A static error equal to zero(ii)No saturation of the actuators (linear mode) for a step variation of (iii)A response time of approximately (iv)A total reject of a step type control disturbance

##### 7.3. Controller Synthesis

The synthesis of the second-generation controller is achieved by means of the module “CRONE Control System Design” from the Matlab Toolbox “Crone Toolbox” [30]. Details of the synthesis of the second-generation CRONE control are explained in [22, 23, 30].

The second-generation CRONE controller is synthesized and has a fractional order . It is approximated by a rational transmittance, obtained by fitting its frequency response.

##### 7.4. UM Shaper Design

In order to reduce the rate of oscillations and the overshoot of the response to a speed step, the motor is controlled through the UM shaper. The shaper synthesis is made for the three configurations of masses mentioned above and offers the following designs, respectively, for the minimum load, of the maximum load, and maximum load:

##### 7.5. Results and Performances

Figures 8–10 present experimental results in speed for the three cases of masses configuration of the motor bench (maximum weight, of maximum weight, and minimum weight). A comparison between the classical CRONE control step response and a CRONE control with UM shaper response is presented. Corresponding command signals are given in Figures 11–13.

The settling time for each experiment is as displayed in Table 3. Overshoot values are presented in Table 4.

According to Tables 3 and 4, control specifications are satisfied, and four advantages related to the use of the UM shaper are deduced:(i)The settling time is reduced by a factor of 3(ii)The maximum overshoot is significantly reduced by a factor of 3.5(iii)From a control point of view, the actuator is less stressed, which reduces the risk of saturation(iv)Overshoots in all shaped cases are very close which proves the robustness of the UM shaper design approach

#### 8. Conclusion

This work dealt with the synthesis of a three-impulse sequence input shaper known as Unity Magnitude (UM) shaper. It has the property to generate an on-off signal profile that permits cancelling undesired oscillations as well as improving the rise time and shortening the shaper duration in comparison with ZV and ZVD shapers. The main contribution has been achieved in two steps. First, the analytic design of the UM shaper was detailed for the case of damped second-order systems. Second, it has been generalized for explicit fractional order systems. The proposed approach associates graphical fitting and analytical procedure. Simulation results have proved good performances of the proposed model of the UM shaper. Feasibility and effectiveness are also supported by an experimental implementation on a DC motor bench commanded by a second-generation CRONE controller. It proves that the actuator is less stressed and oscillations are reduced.

As servicing has an important role in controlling robotic applications, this approach will be potentially applied on flexible robot arms with variable loads. In parallel, an in-depth research work can be held on the generalized explicit derivative fractional order system response. It incorporates both aperiodic and oscillatory dynamics in order to ensure global controller design by the synthesis of a global response UM shaper.

#### Data Availability

Theoretical analysis and findings and experimental implementation parameters are the work result of the authors.

#### Disclosure

A part of the theoretical analysis was presented in 2019 16^{th} International Multi-Conference on Systems Signals and Devices (SSD). The implementation on the instrumental DC motor bench has never been published yet.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was achieved within the context of a research thesis on “Robust path tracking by input shaping in association with fractional order systems,” in collaboration between Laboratory of Control & Energy Management (CEM Lab), University of Sfax, Tunisia, and Integration from Materials to Systems Laboratory (IMS-UMR 5218 CNRS), Université de Bordeaux--Bordeaux INP, Bordeaux, France. The instrumental DC motor bench belongs to the CRONE team of IMS Laboratory. The authors would like to acknowledge the IMS Laboratory and particularly the CRONE team for supporting this work.