Enhancing PD-controller efficiency via time-delays to suppress nonlinear system oscillations

W. A. EL-Ganaini

Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menouf, 32952, Menoufia University, Egypt.

*Corresponding author contact details (e-mail address): [email protected]

Abstract

Within this work, the conventional PD-controller is proposed to suppress the lateral oscillation of a nonlinear Jeffcott-rotor system. Time-delays in the control loop are included in the system model. The slow-flow modulating equations governing the whole system dynamics are obtained. The Influence of the controller gains and time-delays on response curves is explored. The analyses showed that the controller efficiency has been enhanced at specific values of time-delays; while the system may lose its stability at other values of time-delays. The phenomenon of enhancing the controller efficiency was explained and accordingly, the optimal values of the loop-delays have been reported.

Keywords Time-delays control, Limit point, Hopf-bifurcations, Whirling-orbit, multiple scales method

Nomenclature

x_1,x ''_1,x ''_1

Displacement, velocity and acceleration at X_1 direction.

x_2,x ''_2,x ''_2

Displacement, velocity and acceleration at X_2 direction.

''_1,''_2

Linear damping coefficients of both vibration modes.

'' Cubic nonlinearity coefficient.

''_1,''_2

The position and velocity gains of the position-velocity controller, respectively.

''_1,''_2

Time-delays in the control loop.

'' The disc spinning speed.

f The disc eccentricity.

a_1,a_2 Oscillation amplitudes at X_1 and X_2 directions, respectively.

'' An external detuning parameter that characterizes the closeness of the disc spinning speed '' to the normalized critical speed where ''=1+''.

2010 Mathematics Subject Classification 70K99, 34E13, and 34D10

Introduction

The dynamical interaction among the stator, rotating parts and the mass imbalance of the rotor are the main reasons of the rotating machinery vibrations. Due to the wide industrial applications of the rotating machines, hundreds of research articles are dedicated annually to study and control the nonlinear oscillations of those machines. Generally, the vibration control methods are classified into three main categories, which are active, semi-active, and passive control methods. Involving active controllers in the control loop raises the well-known problem of time-delays that may be the cause of controlled system instability. Many of the recently published papers concerning vibration control have discussed the influences of time-delays on the controlled system's stability and the connected controller performance, where Hu et al. [1] introduced time-delayed state feedback controller to suppress the nonlinear oscillations of the Duffing oscillator at primary and sub-harmonic resonance causes. The analyses illustrated that involving time-delays in the control loop can provide active damping resulting in the enhancement of the controller efficiency. Moreover, the authors concluded that the combination of positive position and negative velocity feedback control could mitigate the resonant peak efficiently. Maccari studied the nonlinear vibrations control of van der pol oscillator when excited parametrically [2] and externally [3], utilizing time-delayed state feedback controller. He concluded that the feedback gains and time-delays could reduce the vibration peaks and suppress the system quasi-periodic motions at both external and parametric excitations (excitement). Maccari [4] discussed the nonlinear vibrations of a harmonically excited cantilever beam model at the primary resonance case. Time-delayed position-velocity controller was employed to suppress the system oscillations, where the author concluded that the appropriate choice of the controller gains and loop delays might enhance the controller efficiency. Maccari discussed the vibration control of van der pol oscillator and a parametrically excited nonlinear system by integrating time-delayed state feedback controller [5, 6]. The author has constructed what he defined by 'stability chart' for the amplitude and phase of the periodic phase-locked solutions. Then, he showed that it is possible to calculate the feedback gains and time-delays required to generate specific vibration amplitude. Morrison et al. [7] investigated the nonlinear Mathieu equation at 2:1 parametric resonance and time-delayed positive-position feedback controller concluded that involving time-delay into an undamped parametrically excited system increases the linear damping of the system. Time-delayed position-velocity controller was utilized to suppress the free vibrations of van der Pol oscillator [8], where the authors applied the two variable expansion methods to obtain two Hopf curves that are plotted in control-gain time-delay plane. They showed that the obtained Hopf-curves include a region at which the system vibrations are eliminated and the oscillation amplitude could be suppressed by appropriately setting the loop delay within this region. Shao et al. [9] investigated the effects of time-delays on the dynamics of a nonlinear Microelectromechanical system. Time-delayed position-velocity feedback controller is applied to suppress the system vibrations. The authors concluded that the existence of time-delays in the case of negative feedback destabilize the system motions, while the time-delays enhance the controller efficiency at positive-position and negative-velocity feedback control. The nonlinear vibrations of a nonlinear system having quadratic, cubic, quartic, and quantic nonlinearity have been discussed in Ref. [10]. Time-delayed position-velocity feedback controller is employed to suppress the system vibrations. The authors reported that suitable choice of the time-delays and feedback gains reduces the system resonant peaks. Different time-delayed control algorithms to suppress the nonlinear vibrations of a cantilever beam model when excited externally and parametrically have been studied in Refs [11-13, 14]. Saeed and El-Ganaini [15] applied time-delayed position-velocity feedback controller to mitigate the nonlinear oscillations of a horizontally supported nonlinear Jeffcott-rotor systems. They illustrated simple and concrete method to calculate the optimal values of time-delays at both the negative and positive feedback control. Then, they showed that involving time-delays in the control loop can enhance the controller effectiveness from 80% to 95%.

In this work, the lateral vibration of the vertically supported nonlinear Jeffcott-rotor system [16] is studied. The position-velocity (PD) feedback controller is proposed to control the considered system vibrations. The time-delays (''_1and' '''_2) in the closed loop are included in the system model. Sensitivity of the system response curves to the controller gains and loop delays are explored. Then, stability chart of the controlled system is illustrated to show the stable and unstable solutions regions in ''_1 ''_2-plane. The effects of increasing both the position gain (''_1) and velocity gain (''_2) on the stability chart topology is illustrated. The analysis revealed the possibility of selecting time-delays values in some way to improve the controller efficiency in suppressing the system vibrations. The reported optimal values of time-delays are illustrated analytically and were confirmed numerically. Then, we derived too (two) simple algebraic equations that can be used to calculate the optimal control parameters (i.e. ''_1,''_2,''_1, and ''_2) for the best control performance.

Mathematical model and asymptotic analyses

The nonlinear differential equations governing the lateral vibrations in X_1and X_2 directions of a vertically supported Jeffcott-rotor system shown in Fig. 1 are given as follow:

(1.1)

(1.2)

Where, m is the disc mass, c_x,c_y are the linear damping coefficients at X_1and X_2 directions, R_(x_1 ),R_(x_2 ) are the nonlinear restoring force at X_1and X_2 directions, e_d denotes the disc eccentricity, '' is the disc spinning speed, and t is the time. The restoring force is considered as a nonlinear function of the distance r of the rotating disc from the geometric center G as shown in Fig. 1b as [16]:

(2)

Accordingly, the restoring forces in X_1 and X_2 directions are given as:

(3.1)

(3.1)

Substituting equations (3) into equations (1), we get

(4.1)

(4.2)

By introducing the dimensionless parameters t^*=''t,''='(k_1'm),x_1^*=x_1'e,x_2^*=x_2'e into equations (4), and omitting the asterisks for brevity, we get

(5.1)

(5.2)

where ''_1=c_x/'(mk_1 ),''_2=c_y/'(mk_1 ),''=(e^2 k_2)/k_1 ,f=e_d/e,''= ''/'' .

By integrating a time-delayed conventional PD-controller to the system equations (5) as shown in Fig. 1c, the governing equations of the whole system become as follows:

(6.1)

(6.2)

where' '''_1 x_1 (t-''_1)+''_2 x ''_1 (t-''_2) and ''_1 x_2 (t-''_1)+''_2 x ''_2 (t-''_2) represent the control forces at X_(1 )and X_(2 )directions, respectively.

(b) (c)

Fig. 1 (a, b) Jeffcott-rotor system, and (c) schematic diagram of the controlled Jeffcott-rotor system

Asymptotic analysis and slow-flow governing equations

In this section, the multiple scales perturbation technique [17] is applied seeking an approximate solution to the equations (6) as follows:

(7.1)

(7.2)

Where '' is a small dimensionless perturbation parameter used as a book-keeping parameter, T_j=''^j t,j=0,1 are two time scales that describe the fast and slow scales of the system motions. The time derivatives in equations (6) can be expressed in terms of T_0 and T_1 as:

(8)

The system parameters are scaled according to their values as follows:

(9)

Substituting equations (7-9) into equations (6), and equating coefficients of like powers of '', we have

O ( ):

(10.1)

(10.2)

O ( ):

(11.1)

(11.2)

The solution of equations (10), can be expressed as

(12.1)

(12.2)

where, the coefficients Q_1 (T_1) and Q_2 (T_1) are unknown functions of T_0 and T_1. Substituting equations (12) into equations (11), yields

(13.1)

(13.2)

Before we proceed further, the possible resonance cases must determine at this approximation order, which is primary resonance (i.e. '''1). So, the closeness of the considered resonance case can be described by introducing the detuning parameters '' according to

(14)

Inserting an equation (14) into the small-divisor and secular terms of equations (13), yields the following solvability conditions

(15.1)

(15.2)

To analyze equations (15), we can express the unknown functions Q_1 (T_1) and Q_2 (T_1 ) in polar form as:

(16)

Substituting equations (16) into equations (15) with returning each scaled parameter to the original form (i.e. '' ''_1=''_1/'' ,'' ''_2=''_2/'' ,'' ''=''/'' ,'' ''_1=''_1/'' ,'' ''_2=''_2/'' ,f ''=f/'' ,'' ''=''/'' ,T_1=''t), we get the following slow-flow modulating equations governing the modulation-amplitudes and the modified phases of the system.

(17.1)

(17.2)

(17.3)

(17.4)

Where ''_1=''_1-''t,''_2=''_2-''t. By examining the first terms of equations (17), it is noted that the integration of the time-delayed position-velocity feedback controller to the system has modified its linear damping coefficients ''_1,''_2 to the equivalent coefficients ''_e1= ''_1-''_1 sin'(''_1 )+''_2 cos'(''_2 ) and ''_e2= ''_2-''_1 sin'(''_1 )+''_2 cos'(''_2 ), while the detuning parameter '' is modified to ''_e= ''-1/2[''_1 cos'(''_1 )+''_2 sin'(''_2 )]. Accordingly, it is possible to select the loop-delays (''_1and ''_2) in some way to maximize ''_e1and ''_(e2,) then enhancing the controller efficiency.

Steady-state lateral oscillations and stability investigations

Setting a ''_1=a ''_2='' ''_1='' ''_2=0 into equations (17), we get four nonlinear algebraic equations. These equations describe the steady state whirling motions of the considered system in terms of the controller parameters (''_1,''_2,''_1,''_2 ). Let (a_10,a_20,''_10,''_20) is the steady state solution of these algebraic equations. To study the stability of that solution (a_10,a_20,''_10,''_20), we examine the solution behaviors under small perturbations a_11,a_21,''_11, and ''_21. Therefore, we assume

(18)

Substituting equation (18) into equations (17) and expanding for small a_11,a_21,''_11, and ''_21, keeping the linear terms only (linearization of nonlinear system (17) around the equilibrium point (a_10,a_20,''_10,''_20), we have

(19)

The above square matrix is the system Jacobian matrix. Thus, the stability of the steady state oscillations depends on the eigenvalues of the above matrix. Accordingly, one can obtain the following eigenvalue equations.

(20)

Expanding the determinant (20), yields

(21)

Where '' denotes the Eigenvalues of the Jacobian matrix, and ''_1,''_2,''_3 and ''_4 are the coefficient of equation (21). Applying Routh-Hurwitz stability criterion, the necessary and sufficient conditions for the solution (a_10,a_20,''_10,''_20) to be stable are

(22)

Results and discussions

In this section, the steady state lateral vibration of a nonlinear vertically suspended Jeffcott-rotor system before and after control are investigated via solving the nonlinear algebraic equations (17) when a ''_1=a ''_2='' ''_1='' ''_2=0. The Newton-Raphson algorithm is employed to solve these equations at the system parameters: f=0.025,''=0.05,''_1=''_2=0.015 ,''_1=0.0 ,''=1+'',''_1=0.1,''_2=0.02,''_1=''_2=0.0, unless otherwise is mentioned. The response curves are obtained by varying the detuning parameter ('') or the disc eccentricity (f) as a bifurcation control parameter and plotting the corresponding vibration amplitudes (a_1,a_2). In addition, the obtained solution stability is reported by checking the conditions (22). Then, the obtained solution is plotted as solid lines if it is stable and as dashed lines if it is unstable as shown in Figs 2-3, 9-11, 14. Moreover, to confirm the accuracy of the obtained bifurcation diagrams, the non-autonomous system equations (6) have been solved numerically using the Matlab solvers DDE23 (when ''_1,''_2>0.0) and DDE45 (when ''_1=''_2=0.0), then the steady state vibration amplitudes that obtained numerically are illustrated in the bifurcation diagrams as big-dots as shown in Figs 3, 4, 10, and 13. Frequency Response Curve (FRC) of the Jeffcott rotor system before the control (i.e. ''_1=''_2=0.0) is displayed in Fig. 2 at different levels of disc eccentricity (f). The figure shows that the system responds as a semi-linear system as long as f'0.01, while at f>0.01, the nonlinearity dominates the system response, and the bistable vibrating amplitudes appear at a specific range of the disc spinning speed. Accordingly, the nonlinear Jeffcott rotor system can execute one of two whirling orbits depending on the initial position of the rotating disc. Fig. 3 compares the system FRC before controling and after applying negative position-velocity feedback controller at zero time-delays (i.e. ''_1=0.1,''_2=0.02, and ''_1=''_2=0.0). The figure illustrates that the connection of the proposed controller to the system has eliminated the bistable solution interval (i.e. Eliminates jump phenomenon) and reduced the vibration peaks to approximately one half, while Fig. 4 illustrates the system FRC before and after applying positive-position and negative-velocity feedback controller (i.e. ''_1=-0.1,''_2=0.02, and ''_1=''_2=0.0). Figs. 3 and 4 show that the positive-position negative-velocity feedback controller has high efficiency in mitigating the system vibrations than the negative-position negative-velocity controller. Stability of the controlled system as a function of time-delays are illustrated as stable and unstable regions in ''_1 ''_2-plane in Fig. 5. Fig. 5a shows the system stability chart at negative-position negative-velocity feedback controller (i.e. ''_1=0.1,''_2=0.02), while Fig. 5b displays the stability chart at positive-position negative-velocity feedback controller (i.e. ''_1=-0.1,''_2=0.02). Generally, Fig. 5 illustrates that the stable solutions region is repeated periodically in ''_1 ''_2-plane and its topology depends on the values of both positions and velocity gains. The effects of increasing the velocity gain (''_2) on the stability chart at ''_1=0.1,-0.1 are illustrated in Figs. 6 and 7, respectively. The figures show how the controller gains alter the stability chart topology, where the stable and unstable regions are longitudinal areas parallel to ''_2-axis at ''_1>''_2, while at ''_1<''_2 they become transversal regions parallel to ''_1-axis. At ''_1=''_2, the unstable solutions regions become periodic-isolated areas as shown in Figs. 6b and 7b. Fig. 8 illustrates the controlled system frequency-response curves at different values of the time-delay (''_1) that selected according to Fig. 5a when ''_2=''/2. The figure shows the great influences of the time-delays on the controlled system FRC. It is noticed from Fig. 8 that the best conditions for vibration suppression occur when ''_1=3''/2 and ''_2=''/2 (i.e. when the time- delays are corresponding to the center of stable solutions region of Fig. 5a). The influences of the time-delays that selected according to the stability chart shown in Fig. 5b (i.e.' '''_1=-0.1,''_2=0.02) on the controlled system FRC are illustrated in Fig. 9. The figure confirms that the best frequency response curve occurs at '''_1='''_2=''/2 (i.e. when time-delays are selected within the center of stable solutions region shown in Fig. 5b). However, the existence of time delays in the control loop can destabilize the system under control; it is possible to enhance the controller efficiency in suppressing the system vibrations if time-delays are selected to be within the center of the stable solutions region. The system frequency-response curves before controling and after applying the time-delayed position-velocity controller is compared analytically and numerically in Fig. 10. The optimal value of the time-delays in the case of negative feedback control was employed. Comparing Figs. 3 with 10, the figures illustrate that the time-delay has improved the controller efficiency, effectiveness. Figs 11 and 12 illustrate the Jeffcott-rotor temporal vibrations before and after control according to the points that marked as a bistable- and unique-solution on Fig. 10. The two figures were obtained at the system parameters ''=0.035,''_1=''_2=''_1=''_2=0 (without control), and ''=0.035,''_1=0.1,''_2=0.02,''_1=3''/2,''_2=''/2 (with control). Figs 11 and 12 were obtained via numerically integrating the system original equations (6) utilizing the DDE23 Matlab solver at the initial conditions x_1=x_2=x ''_1=x ''_2=0 and x_1=1,x_2=-1,x ''_1=x ''_2=0, respectively. The figures confirm the sensitivity of the system to the initial conditions before control, where the system has two different whirling-orbits before control that depend on the initial conditions, while these bistable motion has been diminished to small unique whirling-orbit after control. To investigate the Jeffcott-rotor lateral vibrations at a wide range of disc eccentricity (f), the system eccentricity-response curve before control, after control, and after optimal time-delayed control is illustrated in Fig. 13 at ''=0. The figure demonstrated that involving time-delays in the control loop enhanced the controller performance in suppressing the system vibrations.

'''' '''' '' '' '''' '''''' '''''' '''' '''' '''' '''' '''' '''' ''''

before and after control

(a) (b)

Figure 2. Uncontrolled system frequency-response curve at different values of the disc eccentricity f: (a) x_1 mode, (b) x_2 mode.

(a) (b)

Figure 3. The system frequency-response curve before and after control at: ''_1=0.1,''_2=0.02,''_1=''_2=0, (a) x_1mode, (b) x_2 mode

(a) (b)

Figure 4. The system frequency-response curve before and after control at: ''_1=-0.1,''_2=0.02,''_1=''_2=0, (a) x_1mode, (b) x_2 mode

(b)

Figure 5. Modified linear damping coefficients' '''_hori and ''_vert as functions of the time delays at control gains: (a) ''_1=0.1,' '''_2=0.02, (b)''_1=-0.1,' '''_2=0.02

(a) (b) (c)

Figure 6. Modified linear damping coefficients' '''_hori and ''_vert as functions of the time delays at control gains: (a) ''_1=0.1,' '''_2=0.05, (b)''_1=0.1,' '''_2=0.1, and (c)''_1=0.1,' '''_2=0.15.

(b) (c)

Figure 7. Modified linear damping coefficients' '''_hori and ''_vert as functions of the time delays at control gains: (a)''_1=-0.1,' '''_2=0.05, (b) ''_1=-0.1,' '''_2=0.1, and (c) ''_1=-0.1,' '''_2=0.15.

(b)

Figure 8. The controlled system frequency-response curve when ''_1=0.1,''_2=0.02 at different values of the time delays ''_1,''_2: (a) x_1mode, (b) x_2 mode

(b)

Figure 9. The controlled system frequency-response curve when ''_1=-0.1,''_2=0.02 at different values of the time delays ''_1,''_2: (a) x_1mode, (b) x_2 mode

(a) (b)

Figure 10. System frequency-response curve before control (''_1=''_2=0), and after control (''_1=0.1,''_2=0.02) with the optimal delayed times(' '''_1=3''/2,''_2=''/2 ): (a) x_1mode, (b) x_2 mode

(b) (c)

Figure 11. (a, b) time histories, and (c) long-time orbit plot at: ''=0.035,' '''_1=0.1,''_2=0.02,' '''_1=3''/2,''_2=''/2 and initial conditions x_1=x_2=x ''_1=x ''_2=0

(b) (c)

Figure 12. (a, b) time histories, and (c) long-time orbit plot at: ''=0.035,' '''_1=0.1,''_2=0.02,' '''_1=3''/2,''_2=''/2 and initial conditions x_1=1,x_2=-1,x ''_1=x ''_2=0

(b)

Figure 13. The controlled system eccentricity-response curve without control (''_1=''_2=0), with control (''_1=0.1,''_2=0.02), and with delayed-time control (''_1=0.1,''_2=0.02,' '''_1=3''/2,''_2=''/2 ) at ''=0.035: (a) x_1 mode, (b) x_2 mode

Concluded remarks

Within this work, time-delayed position-velocity controller is proposed to control the nonlinear whirling motions of a vertically suspended Jeffcott-rotor system. Multiple time scales perturbation method is utilized to obtain the slow-flow modulating equations that govern the whole system dynamics. The obtained equations are employed to investigate the effects of both the loop-delays and control gain on the system response curves. Based on the above discussions we may conclude the following:

The stable solutions area in ''_1 ''_2- plane is repeated periodically, where its topology depends on the controller gains.

However, the time-delay in the closed loop control system is one of the main reasons of the loop instability; it is possible to improve the position-velocity feedback controller efficiency via selecting the time-delays that maximize ''_e1 and ''_e2.

At specific values of the loop delays, the position-controller acts as a velocity-controller and vice versa.

The center of the stable-solutions area shown in ''_1 ''_2- plane represents the optimal values of loop delays for vibration suppression.

At small unintentional loop-delays, the integration of positive-position and negative-velocity feedback controller to the targeted system is the optimal gain selection for enhancing the controller efficiency.

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