Application of D-Decomposition Technique to Selection of Controller Parameters for a Two-Mass Drive System

: In this work, issues related to the application of the D-decomposition technique to selection of the controller parameters for a drive system with ﬂexibility are presented. In the introduction the commonly used control structures dedicated to two-mass drive systems are described. Then the mathematical model as well as control structure are introduced. The considered structure has only basic feedbacks from the motor speed and PI type controller. Due to the order of the closed-loop system, the free location of the system’s poles is not possible. Large oscillations can be expected in responses of the plant. In order to improve the characteristics of the drive, the tuning methodology based on the D-decomposition technique is proposed. The initial working point is selected using an analytical formula. Then the value of controller proportional gain is decreasing, until the required value of overshoot is obtained. In the paper di ﬀ erent advantages of the D-decomposition technique are presented, for instance calculation of global stability area for the selected gain and phase margin, the impact of parameter changes, and additional delay evident in the system. Theoretical considerations are conﬁrmed by simulation and experimental results.


Introduction
At present, high precision control is required in most modern industrial processes [1][2][3][4][5]. Industrial automation is mainly based on the electrical servo systems with different types of driving motors. In order to ensure accurate control of servo, all factors which can affect the characteristic of the drive should be taken into account [1][2][3][4][5]. The mechanical characteristics of the system determine the performance of the speed/position control. The commonly-used approach, in which the mechanical shaft has an infinite stiffness coefficient, is not valid in many applications. As traditional examples rolling-mill drives or conveyer drives can be quoted [6][7][8][9][10][11]. In these applications large inertias and mechanical connections are evident. During transients the speed/position of the driving motor is different from the speed/position of the load side. The existing torsional vibrations influence the control performance significantly. Due to the progress of microprocessor technique and power electronics, which allows to generate the electromagnetic torque with a very small delay, nowadays this problem is also evident in different drive applications [7][8][9][10][11]. Torsional vibrations are recognised in servo systems, robot arm drives, CNC machines, hard disc drives, MEMS (micro electromechanical systems) [12][13][14][15][16][17][18][19], as well as in different industrial branches [20][21][22][23].
Different control strategies are proposed in literature in order to damp torsional vibrations [1][2][3][4][5][6][7]. The use of designated control strategies is one of the most effective and commonly used. The classical the drive system with a flexible connection is introduced briefly. The commonly used control structure for an electrical drive is described in details. The application of poles-placement methodologies to the selection of controller parameters is shown. The limitations of the classical control design methodology are emphasized. In the third section the D-decomposition technique is presented. Next, the application of the D-decomposition technique to selection of controller parameters is discussed. Different design factors, such as: phase margin, gain margin, delay times, changes of the system parameters are considered. Additionally, the stability issues are considered. Next, the selected experimental results are presented and discussed. The concluding remarks are formulated at the end of the paper.

Mathematical Model of the Plant and Control Structures
In order to analyze the plant there is a need of selection of a suitable mathematical model. In general, two groups of modeling can be distinguished in mechatronic systems. The first one is based on the finite element method. This approach allows to analyse the whole system very thoroughly, yet computational complexity is very high. This modeling is very popular, for example in designing electrical machines. The second modeling approach is based directly on differential equations. It allows to determine the behavior of the system vary fast, so it is popular in control engineering. However, the accuracy of obtained results is limited. The selection of one of the above-mentioned models usually depends on the application. In implementations for which time is crucial (real time control), only the second modeling approach is possible.
In the work, a commonly applied model of the drive system with flexible connection (two-mass system) is used. It is based on three differential equations, which describe the behavior of the three states of the system. This state equation representing the model is presented below [6,8]: where: ω 1 , ω 2 -the speeds of driving motor and load machine respectively, m e , m s , m L -the electromagnetic (driving), shaft (coupling), and load torques, T 1, T 2 -mechanical time constant of the driving motor and load machine, T c -the parameter which represents the elasticity of the coupling. The block diagram of the considered plant is shown in Figure 1.
Energies 2020, 13, x 4 of 21 control design methodology are emphasized. In the third section the D-decomposition technique is presented. Next, the application of the D-decomposition technique to selection of controller parameters is discussed. Different design factors, such as: phase margin, gain margin, delay times, changes of the system parameters are considered. Additionally, the stability issues are considered. Next, the selected experimental results are presented and discussed. The concluding remarks are formulated at the end of the paper.

Mathematical Model of the Plant and Control Structures
In order to analyze the plant there is a need of selection of a suitable mathematical model. In general, two groups of modeling can be distinguished in mechatronic systems. The first one is based on the finite element method. This approach allows to analyse the whole system very thoroughly, yet computational complexity is very high. This modeling is very popular, for example in designing electrical machines. The second modeling approach is based directly on differential equations. It allows to determine the behavior of the system vary fast, so it is popular in control engineering. However, the accuracy of obtained results is limited. The selection of one of the above-mentioned models usually depends on the application. In implementations for which time is crucial (real time control), only the second modeling approach is possible.
In the work, a commonly applied model of the drive system with flexible connection (two-mass system) is used. It is based on three differential equations, which describe the behavior of the three states of the system. This state equation representing the model is presented below [6,8]: where: ω1, ω2-the speeds of driving motor and load machine respectively, me, ms, mL-the electromagnetic (driving), shaft (coupling), and load torques, T1,T2-mechanical time constant of the driving motor and load machine, Tc-the parameter which represents the elasticity of the coupling.
The block diagram of the considered plant is shown in Figure 1. The electromagnetic torque is an input variable in that system. It is compared with the shaft torque (which can be treated as a load torque here and in normal condition has zero value at start) and the difference is accelerating the first mass. The mechanical time constant is proportional to the inertia od the driving motor. The changes of motor speed result in changing of its position, which causes the twist of the shaft. This, in turn, generates a non-zero value of shaft torque which with some delay is acting on the second inertia. The second speed is calculated as integration of the difference between the shaft and the load torque. During transients both speeds can have different values.
In industry, a standard control structure for an electrical drive has the following form. It is based on the two main control loops [8]. The inner loop provides the control of the electromagnetic torque. It can have different forms resulting from the type of the driving motor. In the case of a DC motor it is based on an armature current controller. For induction or PMSM motor the DFOC (direct field oriented control) or DTC (direct torque control) is a standard solution. The task for the inner loop is The electromagnetic torque is an input variable in that system. It is compared with the shaft torque (which can be treated as a load torque here and in normal condition has zero value at start) and the difference is accelerating the first mass. The mechanical time constant is proportional to the inertia od the driving motor. The changes of motor speed result in changing of its position, which causes the twist of the shaft. This, in turn, generates a non-zero value of shaft torque which with some delay is acting on the second inertia. The second speed is calculated as integration of the difference between the shaft and the load torque. During transients both speeds can have different values.
In industry, a standard control structure for an electrical drive has the following form. It is based on the two main control loops [8]. The inner loop provides the control of the electromagnetic torque. It can have different forms resulting from the type of the driving motor. In the case of a DC motor it is based on an armature current controller. For induction or PMSM motor the DFOC Energies 2020, 13, 6614 5 of 21 (direct field oriented control) or DTC (direct torque control) is a standard solution. The task for the inner loop is to follow the reference value of torque relatively fast. The outer loop in the electrical drive encompasses the mechanical part of the drive. The main task for this loop is to control the system speed. The standard drive has only one feedback from the motor speed, and this signal is used for control. The speed PI controller evident here is tuned usually by symmetry criterion. In the case of a stiff mechanical connection, this feedback can ensure accurate control. However, in the case of a flexible connection there is a possibility to introduce torsional vibrations into the system. As it is mentioned in the introduction, there are many different control structures which can be used to supress torsional vibrations. The most advanced ones are based on the additional feedbacks from all state variables of the plant (motor speed, shaft torque, load speed and in some cases the load torque). This allows to damp the torsional vibrations successfully. The drawback of these approaches is a complicated control algorithms-this limits the application of such structures in industry. One of the possible solutions, which does not influence the form of the control structure, is changing the methodology used for selection of the parameters of the speed controller.
In the paper the classical control structure with basic feedback from the motor speed is considered ( Figure 2). encompasses the mechanical part of the drive. The main task for this loop is to control the system speed. The standard drive has only one feedback from the motor speed, and this signal is used for control. The speed PI controller evident here is tuned usually by symmetry criterion. In the case of a stiff mechanical connection, this feedback can ensure accurate control. However, in the case of a flexible connection there is a possibility to introduce torsional vibrations into the system. As it is mentioned in the introduction, there are many different control structures which can be used to supress torsional vibrations. The most advanced ones are based on the additional feedbacks from all state variables of the plant (motor speed, shaft torque, load speed and in some cases the load torque). This allows to damp the torsional vibrations successfully. The drawback of these approaches is a complicated control algorithms-this limits the application of such structures in industry. One of the possible solutions, which does not influence the form of the control structure, is changing the methodology used for selection of the parameters of the speed controller.
In the paper the classical control structure with basic feedback from the motor speed is considered ( Figure 2).
In the block diagram presented in Figure 2, two control loops are evident. The inner, torque control loop, is approximated by first order term with small time constant ( ). Because this time is much smaller than time constants evident in the mechanical part of the drive, it is usually neglected in analysis. The outer loop encompasses the following elements: the mechanical part of the driving motor, shaft, load machine, speed sensor and speed controller (C ω (s)). Additional delays caused by the speed sensor, sampling time in processor are also often neglected (represented in Figure 2 by following element − 2 ). This element represents delays caused by the sensor itself (which usually is small) and by sampling time of speed loop. In the D-decomposition technique all these delays can be taken into account in a natural way. In order to show the advantages of the D-decomposition technique, design process with the help of classical poles-placement is introduced [6,7]. It is based on the following procedure.
Firstly, closed-loop transfer functions from the reference signal to the speed of load machine are calculated (assuming that all delays in the system are neglected): where Gr-PI controller transfer function.
Then the characteristic equation of the system given by: is compared to the desired polynomial which has following form: where: ξ-damping coefficient, ω0-resonant frequency of the closed-loop system.
After multiplication, the Equation (10) can be rewritten as follows:  In the block diagram presented in Figure 2, two control loops are evident. The inner, torque control loop, is approximated by first order term with small time constant (T m e ). Because this time is much smaller than time constants evident in the mechanical part of the drive, it is usually neglected in analysis. The outer loop encompasses the following elements: the mechanical part of the driving motor, shaft, load machine, speed sensor and speed controller (C ω (s)). Additional delays caused by the speed sensor, sampling time in processor are also often neglected (represented in Figure 2 by following element e −sτ A2D ). This element represents delays caused by the sensor itself (which usually is small) and by sampling time of speed loop. In the D-decomposition technique all these delays can be taken into account in a natural way.
In order to show the advantages of the D-decomposition technique, design process with the help of classical poles-placement is introduced [6,7]. It is based on the following procedure.
Firstly, closed-loop transfer functions from the reference signal to the speed of load machine are calculated (assuming that all delays in the system are neglected): where G r -PI controller transfer function.
Then the characteristic equation of the system given by: is compared to the desired polynomial which has following form: where: ξ-damping coefficient, ω 0 -resonant frequency of the closed-loop system. After multiplication, the Equation (10) can be rewritten as follows: Comparing Equations (3) with (5), the set of four equations is created. Finally the equations which allow to set the controller parameter are obtained.
The following remarks can be formulated through the analysis of Equation (6). The gains of the controller depend on the particular parameter of the plant. It means that the system closed-loop poles cannot be located freely in an imaginary plane. It results from the fact that there are four poles of the closed-loop system (three poles from the plant and one from the controller) and there are only two design parameters. Equation (6) ensures double location of the closed-loop poles.
In order to demonstrate the properties of the system with proposed approach the simulation study are performed. The system with two value of inertia ratio R = T 2 /T 1 is considered, namely R = 1, R = 0.5. The system is working with the following cycle. At the time t 1 = 0 s the reference value of the speed is set to 0.2. Then at the time t 2 = 0.4 s the nominal torque is applied to the system. Finally, the load torque is switched off at t 3 = 0.6 s. The obtained transients of the system speeds and torques are presented in Figure 3.
The following remarks can be formulated through the analysis of Equation (6). The gains of the controller depend on the particular parameter of the plant. It means that the system closed-loop poles cannot be located freely in an imaginary plane. It results from the fact that there are four poles of the closed-loop system (three poles from the plant and one from the controller) and there are only two design parameters. Equation (6) ensures double location of the closed-loop poles.
In order to demonstrate the properties of the system with proposed approach the simulation study are performed. The system with two value of inertia ratio R = T2/T1 is considered, namely R = 1, R = 0.5. The system is working with the following cycle. At the time t1 = 0 s the reference value of the speed is set to 0.2. Then at the time t2 = 0.4 s the nominal torque is applied to the system. Finally, the load torque is switched off at t3 = 0.6 s. The obtained transients of the system speeds and torques are presented in Figure 3. The following remarks can be formulated from the transients presented in Figure 3. The overshoot in the system speeds depends strictly on inertia ratio. For a small value of R, in the system speeds there are large overshoots and oscillations. These oscillations decrease as R increase. However, the raising time of the system in case of bigger value of R also increased. Those oscillations are also visible in transients of driving and shaft torques. Because oscillations in system state transients are not acceptable in modern drives, different tuning methodologies of the PI controller are looking for. In the [6] possibility of location of system closed-loop poles on the circle, line with identical damping coefficient and line with equal real part has been proposed. The more detailed studies concerning the influence of alternative closed-loop The following remarks can be formulated from the transients presented in Figure 3. The overshoot in the system speeds depends strictly on inertia ratio. For a small value of R, in the system speeds there are large overshoots and oscillations. These oscillations decrease as R increase. However, the raising Energies 2020, 13, 6614 7 of 21 time of the system in case of bigger value of R also increased. Those oscillations are also visible in transients of driving and shaft torques. Because oscillations in system state transients are not acceptable in modern drives, different tuning methodologies of the PI controller are looking for. In the [6] possibility of location of system closed-loop poles on the circle, line with identical damping coefficient and line with equal real part has been proposed. The more detailed studies concerning the influence of alternative closed-loop poles location to drive performances are presented in [7]. This alternative closed-loop poles location improve system performances, yet only in some limited range. Additionally, the effect of additional delays and parameter changes are not included in classical poles-placement methodology. Therefore, other tuning criteria are sought after. In this paper the D-decomposition methodology, presented in the next section, is proposed.

D-Decomposition Technique
Relationship between frequency domain criterion (phase and gain) and of the n th -order characteristic equation, and a space of permissible parameters for which the stability condition is met can be established by means of the D-decomposition technique.
Boundary of the stable region can be calculated by substituting s = jω in the characteristic equation, where the pulsation ω is to be understood as a real number in range −∞ < ω < +∞. As next, the real and imaginary parts of obtained expression must be equated to zero. By solving such equations one can reach dependencies describing parametric hypersurface which designates the stability boundary in the so called D surface. The D surface, as a parametric surface D(l, r = n − l), relies on l and r standing for number of the characteristic equation roots in the left and right half-plane, respectively. There are no roots in the right half-plane, r = 0 if l = n. In such condition the designated surface D(l = n, 0) indicates a stable region [34]. Once the stability boundary is known, it is necessary to indicate the stable side. For a single parameter, the left-hand side of the boundary is the stable region, where left-hand side is referred to drawing the boundary by changing frequency in direction from −∞ to +∞. For two tuneable parameters an additional boundary is needed. It is so called ∆D 0 hyperplane [34]. It is designated by comparison of the characteristic equation to 0 at the origin of the s-plane, s = 0. Solution leads to a complementary criterion for the second parameter region.
By introducing additional Gain Margin (GM), and Phase Margin (PM), criteria an internal boundary must be found within the asymptotic stability region. This can be realized by equating the characteristic equation to a complex number instead of zero. The complex number represents the required GM and PM.
A well-known expression for the closed-loop transfer function of a standard control structure can be written as: where: C(s)-the regulator transfer function; P(s)-controlled plant transfer function; R(s)-reference signal; Y(s)-the plant output. One should notice, that the 1 + C(s)P(s) = 0 represents the characteristic equation. The C(s)P(s) part stands for the open loop transfer function, G OL (s). For being more specific, assume a 3rd-order plant transfer function as shown in Figure 2: where: T 1 is the mechanical time constant of the motor; T 2 is the mechanical time constant of the load; T C is the mechanical time constant of the shaft. In such cases Equation (7), with assumed IP regulator, can be rewritten: where: K I is the integral gain; K P is the proportional gain. The denominator of (9) is the characteristic equation with the −1 comprised inside.
In general, the characteristic equation in the frequency domain, s = jω, can be written as following: The equation can be rewritten as: Solving the (11) and (12) for the K P and K I leads to: Additionally, the ∆D 0 hyperplane boundary is: The Equations (13) and (14) are describing the parametric hypersurface designating the stability boundary on the D surface. Nevertheless, they do not take into account the reality in form of the system delays, e.g., the signal conversion delay, τ A2D , as shown in (3).
When τ A2D time delay is taken into account, the block diagram shown in Figure 2 leads to the closed loop transfer function written as: where, for equation simplicity reasons: Additionally, the T me represents time constant of an electromagnetic torque control loop and C ω (s) represents transfer function of the speed compensator.
Following the same procedure as above, the K P and K I can be found as functions of ω: Energies 2020, 13, 6614 9 of 21 Now, for certain gain and phase margins, GM and PM, the open loop transfer function must be evaluated as: where the a + jb stands for coordinates of an arbitrary point in the polar plane. In such conditions the K P and K I gains for required GM in dB can be written as: Similarly, for the PM in degrees the K P and K I are: In terms of qualitative visualization, the Equations (25)-(28) together with (22) and (23), and the ∆D 0 = 0 lead to trajectories as shown in Figure 4. The intersection point between GM and PM lines stands for our desired gains fulfilling the GM and PM conditions.
In terms of qualitative visualization, the Equations (25)-(28) together with (22) and (23), and the ∆ 0 = 0 lead to trajectories as shown in Figure 4. The intersection point between GM and PM lines stands for our desired gains fulfilling the GM and PM conditions.

Results
In this section the simulation and selected experimental results concerning the application of the D-decomposition technique are presented. According to the description included in the previous section, the proposed methodology allows to determinate the controller parameters taking into account different factors, such as gain margin, phase margin, additional delay. Additionally, the effect of the changes of the plant parameters can be analysed using the D-decomposition technique. All these cases are presented below. The system with two inertia ratio value is considered: R = 1 and R = 0.25.
Firstly, the general properties of the D-decomposition technique are shown. The algorithm allows to calculate the stability region of the controller for hypothetical parameters of the plant (T1 = 812 ms, T2 = 203 ms, Tc = 2.6 ms for R = 0.25 and T1 = T2 = 203 ms, Tc = 2.6 ms for R = 1 and additional delays m e = 0.1 ms, A2D = 0.5 ms). In Figure 5 the graphical illustration of the stability region of the control structure for different value of the gain margin and phase margin are presented.
As can be concluded from the presented graphs, the D-decomposition technique allows to determine the stable regions (relationship between the gains of the controller) of the system. The bigger area is calculated for the zero value of the GM and PM. Increasing these values decreases the

Results
In this section the simulation and selected experimental results concerning the application of the D-decomposition technique are presented. According to the description included in the previous section, the proposed methodology allows to determinate the controller parameters taking into account different factors, such as gain margin, phase margin, additional delay. Additionally, the effect of the changes of the plant parameters can be analysed using the D-decomposition technique. All these cases are presented below. The system with two inertia ratio value is considered: R = 1 and R = 0.25.
Firstly, the general properties of the D-decomposition technique are shown. The algorithm allows to calculate the stability region of the controller for hypothetical parameters of the plant (T 1 = 812 ms, T 2 = 203 ms, T c = 2.6 ms for R = 0.25 and T 1 = T 2 = 203 ms, T c = 2.6 ms for R = 1 and additional delays T m e = 0.1 ms, τ A2D = 0.5 ms). In Figure 5 the graphical illustration of the stability region of the control structure for different value of the gain margin and phase margin are presented. Energies 2020, 13, x 11 of 23 As can be concluded from the presented graphs, the D-decomposition technique allows to determine the stable regions (relationship between the gains of the controller) of the system. The bigger area is calculated for the zero value of the GM and PM. Increasing these values decreases the stable area. For example, setting value PM = 80 results in a very small area of the stable work. Usually, these two conditions are specified during design of the control structure. In Figure 5c,f, the solution is presented for GM > 20 dB and PM > 70 • . All pairs of the controller coefficients laying into the area limited by two curves fulfilled the condition. The crossing point indicates the exact value of the conditions GM = 20 dB and PM = 70 • . The characteristics PM and GM as a function of total area of region A are similar for two considered inertia ratios. They differ by values, not by shapes. Although the presented methodology simplifies the design of the control structure, by indicating of the stable region of the work and limiting the hypothetical gains of the controller for setting condition, the selection of the particular values required expertise in this field. Therefore, in the next paragraph the procedure is developed to simplify the tuning process.
The drive with following value of the inertia ratio R = 0.25 is considered at first. The results of application of D-decomposition technique is a plane representing the stability area as the function of the controller parameters (proportional and integration gains). The enlarged plot is shown in Figure 6a. The following remarks can be formulated based on the following figure. The stable region of the system work is relatively wide. There is a problem to specify optimal working point (pairs of K I and K P ) at start because there is a lot of possible solutions. This is a problem for industrial engineers which are asking for a simple and reliable tuning procedure. Therefore, the starting point is selected with the Equation (6) coming from poles-placement methodology. This initial working point P 1 is indicated in Figure 6a by red circle. Then this point is shifted down keeping the constant value of the gain margin and increasing value of the phase margin until the transition point P 2 . The final point is marked with P 3 . In the considered case, the proportional gain is keeping at constant value when the integral gain is reduced from 750 to 450. In order to demonstrate the effectiveness of the proposed procedure transients of the state variables of the system, for different selected points (P 1 , P 2 , and P 3 ) are plotted in Figure 6d-f.
The system is working under the following cycle. At the time t 1 = 0 s the reference value of the speed is changing from 0 to 0.2 (the low value of the speed is used in order to avoid the electromagnetic torque limit). Then at the time t 2 = 0.3 s the nominal load torque is applied to the system. As can be concluded from the Figure 6, the drive system with the gains of the controller selected using classical methodologies (6) has big oscillations and slowly damped overshoots in all state variables. Changing the working point from P 1 , through P 2 and finally to P 3 results in decreasing oscillations in all state variables. The overshoot in load speed is reduced from almost 90% (P 1 ) through 40% (P 2 ) and in the final point to 8% (P 3 ). It should be stressed that the settling time of the system is the shortest for point P 3 . The changing of working point requires to decrease only K I controller coefficient. This makes this procedure easy to perform without advanced knowledge of D-decomposition. However, it should be stressed that this methodology offers more possibilities which is going to be shown later.
In order to confirm the effectiveness of the proposed solutions, additional characteristics of the system are generated. The relationship between the overshoot of load speed and controller parameters (Figure 6b) as well as the values of ITAE (integral of time-weighted absolute error- Figure 6c) are shown. The considered working points of the system are also marked in both diagrams. This figure is generated in a program which is not a part of the D-decomposition technique. Using this additional figures the effectiveness of the proposed tuning methodology can be confirmed. Additionally, for this particular system different values of plant responses can be obtained. The following remarks can be formulated on the basis of the presented characteristic. The application of changing of operation point allows to improve the system performance significantly. The overshoot is reduced to a small value. The further increase of the phase margin can eliminate the overshoot completely, yet the settling time of the system will be longer. This is clearly visible in Figure 6c, where the value of ITAE is presented. The point P 3 ensures quasi optimal solutions. The global optimum can be obtained by also decreasing slightly the integrational gain of the controller.
Energies 2020, 13, x 11 of 21 40% (P2) and in the final point to 8% (P3). It should be stressed that the settling time of the system is the shortest for point P3. The changing of working point requires to decrease only KI controller coefficient. This makes this procedure easy to perform without advanced knowledge of Ddecomposition. However, it should be stressed that this methodology offers more possibilities which is going to be shown later. In order to confirm the effectiveness of the proposed solutions, additional characteristics of the system are generated. The relationship between the overshoot of load speed and controller parameters (Figure 6b) as well as the values of ITAE (integral of time-weighted absolute error- Figure 6. Characteristics of the system: stable region of work (a), overshoot in ω 2 (b), value of ITAE for ω 2 (c) transients of motor speed ω 1 (d), load speed ω 2 , (e) and shaft torque m s (f).
Then the system with R = 1 is considered. The analogical characteristic and transients are plotted in Figure 7. The tuning procedure of the controller parameters is repeated once again. The D-decomposition technique is used to determine the plane where the stable region of work. Next, using (8) the initial working point is specified. Then the value of phase margin is reduced until the point P 3 is reached. It should be noted that for the system with R = 1 the reduction of K I coefficient is smaller than in the previously considered case. The overshoot in load speed decreased from 28 to 2%.
Then the system with R = 1 is considered. The analogical characteristic and transients are plotted in Figure 7. The tuning procedure of the controller parameters is repeated once again. The D-decomposition technique is used to determine the plane where the stable region of work. Next, using (8) the initial working point is specified. Then the value of phase margin is reduced until the point P3 is reached. It should be noted that for the system with R = 1 the reduction of KI coefficient is smaller than in the previously considered case. The overshoot in load speed decreased from 28 to 2%. The application of the above-described procedure result is improving characteristics of the system. The overshoot of both speeds are reduced as well as the settling time is shorten. The additional characteristics plotted in Figure 8b,c show the similar properties as in the previously considered case. The point P3 is a quasi-optimal solution taking into account values of ITAE ( Figure  7c).
In the case of the two-mass system with different parameters of the plant, two types of characteristics are commonly used. The simplest solution is based on the presentation of influence of particular parameters of the drive to selected property (here stability). The more advanced approach relies on presentation of the changes of inertia ratio R (or resonant frequency) to properties of the system [6,7]. Although the second framework is more generalised, it is less visible for wide group of engineers not specialist in considered problem. Taking into account these considerations, the first solution is implemented in this paper. The application of the above-described procedure result is improving characteristics of the system. The overshoot of both speeds are reduced as well as the settling time is shorten. The additional characteristics plotted in Figure 8b,c show the similar properties as in the previously considered case. The point P 3 is a quasi-optimal solution taking into account values of ITAE (Figure 7c). characteristics are commonly used. The simplest solution is based on the presentation of influence of particular parameters of the drive to selected property (here stability). The more advanced approach relies on presentation of the changes of inertia ratio R (or resonant frequency) to properties of the system [6,7]. Although the second framework is more generalised, it is less visible for wide group of engineers not specialist in considered problem. Taking into account these considerations, the first solution is implemented in this paper. The D-decomposition technique can be also use to determinate the stable region of the control structure for different parameters of the plant. In Figure 9   In the case of the two-mass system with different parameters of the plant, two types of characteristics are commonly used. The simplest solution is based on the presentation of influence of particular parameters of the drive to selected property (here stability). The more advanced approach relies on presentation of the changes of inertia ratio R (or resonant frequency) to properties of the system [6,7]. Although the second framework is more generalised, it is less visible for wide group of engineers not specialist in considered problem. Taking into account these considerations, the first solution is implemented in this paper.
The D-decomposition technique can be also use to determinate the stable region of the control structure for different parameters of the plant. In Figure 9 the plots of the stable regions for different parameters of the plant are presented. The changes of the mechanical time constant of the motor T 1 (Figure 8a), changes of mechanical time constant of load machine T 2 (Figure 8b) and changes of elasticity time constant T c (Figure 8c) are considered.
The following assertions can be specified of the basis on the Figure 8. The changes of time constant of the driving motor influence the stability region of the control structure significantly. The increase of T 1 enlarge stability region significantly. The value of proportional gains rise from 550 (for T 1 = 203 ms) to 2200 (for T 1 = 812 ms). Similarly, the maximal values of integral gains change from 3000 to 12,500. The changes of the two resting parameters T 2 and T c have a much smaller impact to the stability regions (Figure 8b,c). Increasing these values hardly enlarges stability area of the controller.
The decomposition technique can be also used for analysing the impact of the delays evident in the control structure for the stability of the system. This is next investigated topic. First, the system with following parameters is tested: GM = 20 dB, PM = 70 • , R = 1 and two values of delay in torque control loop T m e = 0.1 ms (Figure 9a) and T m e = 2 ms (Figure 9b). The influence of delay τ A2D to stability regions are presented in Figure 9b. Then the following case is considered. The delay in the converter is kept constant τ A2D = 0.5 ms (Figure 9c) and τ A2D = 50 µs (Figure 9d) and the delay in the torque control loop T m e is variable (Figure 9c,d). The similar tests are repeated for the system with R = 0.25. The obtained results are presented in Figure 10.
The following remarks can be formulated on the basis on presented graphs (Figures 9 and 10). As can be expected the biggest regions are obtained for the system with minimal values of additional delays (Figures 9a,d and 10a,d). The increase of both delays narrow the stability region. The systems with minimal value of τ A2D have the biggest stability area (Figures 9d and 10d), but when parameter T m e increases, the stability area is reduced significantly. In Figures 9b,c and 10b,c the characteristics for the system with parameters of used in the paper experimental set-up are presented. It is visible in total area A that the delay in torque control loop T m e has less impact on narrow the stability regions than delay τ A2D . Comparing the shape of characteristics for different inertia ratio R, it can be concluded that they have similar features, only the scale of controller parameters are different.
with minimal value of A2D have the biggest stability area (Figures 9d and 10d), but when parameter m e increases, the stability area is reduced significantly. In Figures 9b,c and 10b,c the characteristics for the system with parameters of used in the paper experimental set-up are presented. It is visible in total area A that the delay in torque control loop m e has less impact on narrow the stability regions than delay A2D . Comparing the shape of characteristics for different inertia ratio R, it can be concluded that they have similar features, only the scale of controller parameters are different.  The simulation results have been confirmed using a laboratory set-up. The main part of the laboratory stand consists of two DC motors connected by a long elastic shaft (made of steel with length 600 mm and 5 mm diameter). There are additional flywheels which allow to change the moment of inertia of motor and load machine. Both motors have a nominal power equal to 500 W. The driving motor is supplied by power converter (H bridge structure) which allows the flow of the armature current in both directions. The control of the load torque is obtained by switching on and off to armature winding to resistor. The speed of the motor and load machine is measured using incremental encoders (36,000 pulses per revolution). The control algorithm is implemented in real-time using dSpace control desk. The obtained sampling time is equal to 0.5 ms (including speed and torque loops). The picture of experimental ring is shown in Figure 11.
In the experimental tests the theoretical results have been confirmed. The system is working under the following cycle. At the time t 0 = 0 s the reference value is changing from 0 to 0.2 s. The initial value of load torque is set to zero. After start-up, at the time t 2 = 0.4 s and t 3 = 0.6 s nominal load torque is switched on and off to the drive system. Finally, at the time t 4 = 1 s the reference value changes to −0.2. At the beginning, the system with R = 1 is tested. armature current in both directions. The control of the load torque is obtained by switching on and off to armature winding to resistor. The speed of the motor and load machine is measured using incremental encoders (36,000 pulses per revolution). The control algorithm is implemented in realtime using dSpace control desk. The obtained sampling time is equal to 0.5 ms (including speed and torque loops). The picture of experimental ring is shown in Figure 11.   In the experimental tests the theoretical results have been confirmed. The system is working under the following cycle. At the time t0=0s the reference value is changing from 0 to 0.2 s. The initial value of load torque is set to zero. After start-up, at the time t2 = 0.4 s and t3 = 0.6 s nominal load torque is switched on and off to the drive system. Finally, at the time t4 = 1 s the reference value changes to -0.2. At the beginning, the system with R = 1 is tested. Firstly, the system with an IP controller with the coefficients set with the use of the classical method (6) is tested (point P1 in Figure 7). Next, the system with increased value of the PM is Firstly, the system with an IP controller with the coefficients set with the use of the classical method (6) is tested (point P 1 in Figure 7). Next, the system with increased value of the PM is considered (related to point P 2 ). The obtained transients are recorded. Finally, the control structure with final parameters of the speed controller (point P 3 ) is considered. In order to compare the characteristics saved for different operation points, transients of particular variable for different operation points are placed in one graph ( Figure 12). 0.2. At the beginning, the system with R = 1 is tested. Firstly, the system with an IP controller with the coefficients set with the use of the classical method (6) is tested (point P1 in Figure 7). Next, the system with increased value of the PM is considered (related to point P2). The obtained transients are recorded. Finally, the control structure with final parameters of the speed controller (point P3) is considered. In order to compare the characteristics saved for different operation points, transients of particular variable for different operation points are placed in one graph (Figure 12). The saved experimental transients are presented in Figure 12 in following order. As first transients of driving motor speed ( Figure 12a) and load speed (Figure 12b) are placed. Then the electromagnetic (Figure 12c) as well as shaft torque are presented (Figure 12d). The following remarks can be formulated with the help of the mentioned graph. The decrease of the PM lower the overshoot and settling time of the system speeds. The overshoot in load speed is reduced from 28 to 6%. Additionally, the oscillations into torque transients are the smallest for the final point of the works. However, it should be stated that the improvement of the system characteristic for the inertia ratio is not so visible.
Then the system with R = 0.25 is tested experimentally. In order to obtain such value of inertia coefficient, an additional flywheel has been added to the motor site. The cycle of work is similar as in the previous considered case. The obtained transients are shown in Figure 13.
As can be concluded from the presented transients the system is working correctly. For the controller tuned with the help of the classical pole-placement methodology (P 1 ) both speeds possesses large overshoot and settling time (Figure 13a,b-yellow line). Additionally, big oscillations are visible in the electromagnetic and shaft torques transients (Figure 13c,d-yellow line). It should be stressed that in this case, the electromagnetic torque is slightly limited (Figure 13c) which results of reducing overshoots. These features result from the classical poles-placement methodology which guarantee double location of the system closed-loop poles. Because these performances are not acceptable in many practical applications, then D-decomposition technique is proposed to improve the system features. The transients saved for operation point P 2 (red line) and point P 3 (blue line) are presented in Figure 13. that in this case, the electromagnetic torque is slightly limited (Figure 13c) which results of reducing overshoots. These features result from the classical poles-placement methodology which guarantee double location of the system closed-loop poles. Because these performances are not acceptable in many practical applications, then D-decomposition technique is proposed to improve the system features. The transients saved for operation point P2 (red line) and point P3 (blue line) are presented in Figure 13. As in the simulation study, the final point ensures the most optimal characteristics of the system. The overshoots of the driving motor and load velocities are reduced significantly (in load speed from 70 to 10%). Additionally, the oscillations of the electromagnetic and shaft torques are much smaller as compared to the previously considered cases. This confirms the effectiveness of the proposed methodology, especially for the system with a small inertia ratio R.
In order to analyze the obtained transients (Figures 12 and 13) more accurately the characteristic values such as rising time, time to first maximum, settling time, overshoot, and others for different working points (P1, P2, P3) and inertia ratio R are calculated. The results are presented in Table 1.  As in the simulation study, the final point ensures the most optimal characteristics of the system. The overshoots of the driving motor and load velocities are reduced significantly (in load speed from 70 to 10%). Additionally, the oscillations of the electromagnetic and shaft torques are much smaller as compared to the previously considered cases. This confirms the effectiveness of the proposed methodology, especially for the system with a small inertia ratio R.
In order to analyze the obtained transients (Figures 12 and 13) more accurately the characteristic values such as rising time, time to first maximum, settling time, overshoot, and others for different working points (P 1 , P 2 , P 3 ) and inertia ratio R are calculated. The results are presented in Table 1.
The values presented in Table 1 confirm the previously presented analysis. The systems with working point P 3 have oscillatory responses. The overshoots in both speeds are very large. Additionally, the maximal values of the electromagnetic and shaft torques are the biggest. The changing of working points improve the system performances. The overshoots are reduced significantly as well as the maximal values of torques. The values of settling times are decreases. Where: δ ωx -overshoot of speed, t m ωx -time to first maximum of speed, t s2% ωx , t s5% ωx -2% and 5% settling time of speed, t s2% ωx -rising time, m e max -maximum value of the electromagnetic torque. m s max -maximum value of the shaft torque.

Conclusions
The application of the D-decomposition methodology for determination of the control parameters for the drive system with a flexible connection is demonstrated in this work. The classical two-mass drive system with only basic feedback from the driving motor speed is a demanding plant to control. Especially for the small value of inertia ratio R, in motor and load speeds large oscillations are generated. In order to damp those oscillations different control concepts can be used. Usually, advanced control structures based on additional feedback(s) are proposed. However, in many industrial cases simple methodology is required which allows to improve the drive performance without modifying the existing structure. For these situations, the D-decomposition is proposed in this work. Regarding the theoretical considerations and results of the tests, the following concluding remarks can be formulated.
The stable regions of work can be determined with the help of the D-decomposition technique. The designer can assume the values of the phase and gain margins and D-decomposition technique will determine the parameters of the controller. The selection of optimal value of the gain and phase margins for two-mass drive system requires some expertise. Therefore the initial point is calculated with the help of a formula resulting from classical poles-placement methodology. The procedure can be presented in a few steps. First, the classical equations which ensure double location of the system closed-loop poles are used. The obtained results are not optimal-there is a large overshoot in a load speed (especially for a system with small value of R). Then, phase margin is increased until the overshoot in the load speed is reduced to an acceptable value. This also improves transients of the other state variables of the system, such as motor speed as well as electromagnetic and shaft torques. The D-decomposition technique can be used to determine the stable working region of the system taking into account the delay caused by the torque control loop. The obtained area is determined as a function of the controller parameters. Additionally, the influence of the delay in the speed measurement loop can be taken into account. The proposed method is especially effective in the system with a small value of the inertia ratio R. For instance, for the system with R = 0.25 the overshoot in the load machine speed is reduced from 70 to 10%. Additionally, the oscillations in other drive states are also suppressed successfully.
The future work will be devoted to formulate general guidelines based on the PM and GM for the two-mass drive system with different values of inertia ratio and resonant frequency as well as Energies 2020, 13, 6614 20 of 21 changeable parameters. Additionally, the analysis of the system with an additional feedback (from e.g., shaft torque) will be done.