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State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach
ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach Jue Yu 1, Jian Zhuang 1,n, Dehong Yu 1 School of Mechanical Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 June 2014 Received in revised form 25 July 2014 Accepted 18 August 2014
This paper concerns a state feedback integral control using a Lyapunov function approach for a rotary direct drive servo valve (RDDV) while considering parameter uncertainties. Modeling of this RDDV servovalve reveals that its mechanical performance is deeply influenced by friction torques and flow torques; however, these torques are uncertain and mutable due to the nature of fluid flow. To eliminate load resistance and to achieve satisfactory position responses, this paper develops a state feedback control that integrates an integral action and a Lyapunov function. The integral action is introduced to address the nonzero steady-state error; in particular, the Lyapunov function is employed to improve control robustness by adjusting the varying parameters within their value ranges. This new controller also has the advantages of simple structure and ease of implementation. Simulation and experimental results demonstrate that the proposed controller can achieve higher control accuracy and stronger robustness. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: State feedback control Integral control Direct drive servo valve Lyapunov function Parameter uncertainty
1. Introduction A servovalve is a vital device in achieving a precise motion control in an electrohydraulic servo system; its performance is deeply affected by the adopted control strategy. In recent years, a variety of control techniques applied in the electrohydraulic field have emerged. In Ref. [1], a sliding-mode controller is applied to force the hydraulic actuator to slide along a predefined surface in the state space. This control method can manifest robustness because control signals are independent of the controlled system model. However, the chattering phenomenon introduced by fast switching inputs cannot be avoided, and this may degrade system performance. In Ref. [2], a feedback linearization control is applied to improve system performance based on the idea of transforming nonlinear dynamics into a linear form with state feedback. This method has shown improved performance over traditional linear control methods, but it also has an important limitation; i.e., robustness could be weakened in the presence of parameter uncertainty or unmodeled dynamics. In Ref. [3], an improved adaptive control method was considered by updating controller gains to compensate for parameter fluctuations, but this requires a great deal of calculation. Both repetitive control [4] and iterative
n
Corresponding author. E-mail address: [email protected] (J. Yu). 1 Tel.: þ86 15102930290.
learning control [5] can be applied in electrohydraulic systems with periodic action trajectories. These two methods also need a large storage space and a fixed movement starting position. The Lyapunov method has also been widely used to enhance system stability in the hydraulic field. In Ref. [6], a backstepping technique with the Lyapunov control function is applied to improve robustness of the electrohydraulic control system. In Ref. [7], a robust control meeting the Lyapunov stability condition is used to control the output pressure of a cylinder. Both have a powerful ability to ensure system stability and accuracy. In Ref. [8], a parametric optimization method is applied to enhance control valve performance. In Ref. [9], a robust H1 controller extended with an integral action is used to control an electro-hydraulic servo system. Intelligent control methods include controllers using a neural network [10], evolution strategies [11] and fuzzy control [12]. These methods integrate artificial intelligence into the electrohydraulic control system, making controllers adapt to the nonlinear terms. These intelligent controllers have shown improved performance over traditional control methods in experimental tests; however, they always need large memory space and heavy computation. A large gap exists between theoretical analysis and online application especially when facing a shortage of funds. Despite the development of a vast array of advanced control strategies, the PID controller [13] is by far the most widely used within industry. Its popularity stems from its simple structure, easy operation and applicability in a wide variety of operating
http://dx.doi.org/10.1016/j.isatra.2014.08.006 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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scenarios. However, traditional PID controllers cannot exhibit consistent control performance when encountering complex, time-varying and uncertain systems. To this, many improvements have been proposed, that have already demonstrated satisfactory control effects; however, selection is always based on the cost of increased algorithm complexity. In addition to the PID controller, the state feedback controller [14] is also widely used when the system model and state variables can be obtained accurately and timely. Through this controller, the poles of the closed-loop system can be placed arbitrarily; therefore, it can guarantee system stability and specify the transient response of state variables. However, given the proportional-plus-derivative nature of the static state feedback controller, its steady-state disturbance rejection is not good enough. Although the response is quick and smooth, there is always an offset in the plant output when there is nonzero disturbance. Additionally, this controller is deeply influenced by the accuracy of the parameters; system robustness may be reduced by parameter errors. Typically, an integral action [15] and a state observer [16] are used to compensate for the abovementioned shortcomings. This paper proposes a RDDV servovalve [17,18] with a rotary spool and a rotary driver. The spool, directly driven by a brushless DC motor (BLDC), rotates within a certain angle range in the valve chamber, and its rotation angle modulates fluid flow through the control ports of the valve. Taking the RDDV servovalve position control as the aim, this paper analyzes the load on the spool and proposes a corresponding state feedback controller. The mechanics modeling of the RDDV servovalve reveals that its mechanical performance is deeply influenced by flow torques and friction torques; however, these torques are uncertain, mutable and sometimes cannot even be obtained due to the nature of fluid flow. To address this problem while still preserving excellent dynamic performance [19], an integral action and a Lyapunov direct method are proposed to improve the prototype controller. This integral action can eliminate steady-state tracking error by introduce an integral nature to the controller. The Lyapunov direct method is utilized to account for parameter uncertainties. More specifically, this paper first describes the servovalve structure, including mechanical structure and driving mode, then analyzes the loading condition on the spool, determines the impact caused by pressure drops on the spool and builds its mathematical model. Second, this paper proposes a state feedback integral controller to assign system poles to the desired places and eliminate the steady-state error. Due to the coefficients of frictions and flow torques cannot be determined accurately; therefore, a switching control law is then proposed to determine those estimated parameters that represent their uncertain counterparts. This switching control law is derived using a Lyapunov approach and can enhance system stability. Finally, simulations and experiments using a prototype of the RDDV servovalve are carried out to verify the compensating effect of the proposed controller. It is worth noting that, despite the analysis object of this paper being the RDDV servovalve, the proposed control algorithm can also be applied to traditional hydraulic valves because their spool loading conditions are the same. This paper is organized as follows: Section 2 describes the mathematical model of the RDDV servovalve under study. Section 3 shows the Lyapunov function based state feedback integral control method. Section 4 presents simulation and experimental results. Finally, the conclusions are presented in Section 5.
2. Modeling of RDDV The controlled plant analyzed in this paper is composed of a RDDV servovalve and a BLDC motor as the driver. Their mechanical models are analyzed below in detail.
Fig. 1. Three-dimensional diagram of the RDDV.
Fig. 2. Working principle diagram of RDDV servovalve.
2.1. Modeling of valve structure The proposed RDDV servovalve is a three-position four-way control valve as shown in Figs. 1 and 2. It is composed of a rotary spool, a static sleeve, a valve body, valve covers and other accessories [18]. The flow state of this servovalve is controlled by the spool rotation angle; thus, the load on the spool is of vital importance. As introduced in Refs. [17,20], resistances on the spool mainly include steady-state flow torque, transient-state flow torque and friction torques. Due to the similar flow passage structure, these resistances are similar to those of traditional slide valves, but with force analysis turned into torque analysis. Thus, mechanical models of traditional slide valves can be cited for the RDDV servovalve. Steady-state flow torque is generated when fluid flows through an orifice because the change of momentum will reflect torque, preventing spool rotation. Referring to Figs. 1 and 2, at orifices A and B, the fluid flows perpendicularly to the spool axis; however, at orifices P and T, the vena contracta in which the fluid flows into (out of) the chamber is at a jet angle to the spool axis. The jet force acts on the spool in the same direction as the fluid can be divided
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into a circumferential component and a radial component. By symmetrically locating orifices on the circumference of the sleeve, the radial direction force can be compensated. The circumferential component of the jet force Fsc can be solved by Bernoulli’s equation [20].
two-conducting mode and pulse-width modulation (PWM) method are applied here. If the inductance of the stator coils is neglected [21] (the ratio between the inductance and the resistance is approximately 0.001 in this study), the mathematical model can be derived as follows:
F sc ¼ 2C v C d A0 Δp cos γ
2Rs I s þ 2kE ω þ 2V D ¼ K PWM U dc
ð1Þ
where Cv is the velocity coefficient, Cd is the flow coefficient, and Δp is the pressure drop through the orifice. γ is the jet angle at the orifice; its value range can be obtained by experiments. A0 is the orifice flow area; it can be calculated from Fig. 2. A0 ¼ 2La R sin θ=2 La R θ ð2Þ where La is the axial length of the orifice, R is the spool radius, and θ is the effective orifice opening angle. Due to the spool rotation range is small ( 151r θ r151); the term sin(θ/2) in Eq. (2) can be replaced by θ/2. Steady-state flow torque Ts can be calculated by using Eqs. (1) and (2). T s ¼ 2ðF s1c þ F s2c Þ r ¼ 4RrC v C d La cos
γ s Δps θ þ 4RrC v C d La cos γ r Δpr θ ð3Þ
where s-suffix indicates variables at orifice P, r-suffix indicates variables at orifice T, and r is the effective arm length of the flow torques. Transient-state flow torque is due to the spool motion because the acceleration of the fluid in the chamber generates an additional force on the spool. Its magnitude Ft at one single orifice can be calculated by applying Newton’s second law [20]. pffiffiffiffiffiffiffi pffiffiffiffiffiffi ð4Þ F t ¼ 2ρLa Lc C d R Δp ω where Lc is the chamber damping length. The direction of transient-state flow force is the same as the spool angular velocity in chamber P, but it is opposite to the spool angular velocity in chamber T [17,20]. Thus, the resultant transient-state flow torque of all four orifices of the RDDV servovalve Tt can be calculated as qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi T t ¼ 2 2ρrLa Lc C d R ω Δp r Δp s ð5Þ The friction torques when the spool rotates in the valve chamber include viscous friction torque, seal friction torque and friction torque caused by the lateral unbalance force. The resultant friction can be simplified as a combination of Coulomb friction and viscous friction; thus, the friction torques can be described as follows. T f ¼ T c signðωÞ þ T v ω
ð6Þ
where Tf is the resultant friction torque, Tc is the Coulomb torque, sign() is the sign function, Tv is the viscous torque coefficient. Tc and Tv are uncertain and time-varying because these friction torques are all deeply influenced by the fluid pressure in the valve chamber, which fluctuates with changing load. From the above-introduced torques, the total torque that the spool has to overcome can be summarized [17]: T L ¼ signðωÞ T c þ 4RrC v C d La θ cos γ s Δps þ cos γ r Δpr qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi þT v ω þ 2 2ρrLa Lc C d Rω Δp r Δp s ð7Þ
2.2. Model of BLDC motor The RDDV servovalve is closed-loop controlled. One of the spool ends is coupled with an angular displacement sensor and the other is coupled with a motor to be driven. Thus, the model of the BLDC motor is indispensable for controller design and performance prediction of the servovalve control system. The
ð8Þ
where Rs is the phase resistance, IS is the circuit equivalent current, kE is the back EMF coefficient, VD is the voltage drop of the MOSFET, KPWM is the input PWM duty cycle, and Udc is the input DC voltage. The motor torque is an impetus torque, promoting the motor shaft and the spool rotation. The mechanical model of the coupling system can be described as below: kT I s T L ¼ J θ€
ð9Þ
where kT is the motor torque coefficient and J is the rotational inertia of the BLDC motor and the spool. 2.3. System modeling Because the spool position response is of vital important to the control effect and the servovalve performance criterion, it is chosen as the control objective. The control system can be represented in a state space form as shown in Eqs. (7)–(9): ( x_ 1 ðtÞ ¼ x2 ðtÞ ð10Þ T U dc u þ C 0 signðx2 ðtÞÞ þC 1 x1 ðtÞ þ C 2 x2 ðtÞ x_ 2 ðtÞ ¼ k2JR s where x2 ðtÞ ¼ θ_ ðt Þ ¼ ωðt Þ; u ¼ K PWM UVdcD ; C 0 ¼ TJc ; 4RrC v C d La cos γ s Δps þ cos γ r Δpr ; C1 ¼ J h pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi kT kE þ T v Rs þ 2 2ρrLa Lc C d R Δpr Δps Rs C2 ¼ : JRs x1 ðtÞ ¼ θðt Þ;
Due to the nature of fluid flow, parameters Tc, Tv, γs and γr in Eq. (10) are uncertain and mutable; their exact values are too difficult to determine. However, in the published literature, CFD simulation and experiments [17], C0, C1 and C2 calculated by these uncertain parameters can be determined to float within their value ranges. The system robustness, although reduced by parameter uncertainties, should be improved by the proposed controller.
3. Controller design It can be clearly concluded from the modeling analysis that spool control response is deeply influenced by resisting torques; however, these parameters, i.e., Tc, Tv, γs, γr, are uncertain and mutable. To achieve fast and satisfactory spool position responses, this section will develop a robust state feedback controller based on a Lyapunov function approach. Taking the spool position as the output, i.e., y ¼ x1 , its dynamic characteristics can be obtained by Eq. (10): y€ ¼
kT U dc u þ C 0 signðy_ Þ þ C 1 yþ C 2 y_ 2JRs
ð11Þ
Then, for a static state feedback controller applying the system model analyzed above, the output error can be adopted: u¼
2JRs C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d þ s1 e_ þ s2 e kT U dc
ð12Þ
where yd is the desired output, and e ¼ yd y is the output error. The control problem is to design u such that y-yd as t-1 for zero steady errors. However, due to the approximation and
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Fig. 3. Functional diagram of the proposed state feedback integral control.
measurement errors of model parameters, steady-state errors may be produced by the above controller. To remedy this limitation, typically an integral action is added into the static state feedback control. Its schematic diagram is shown in Fig. 3, where a new state variable q defined as the integral of e is adopted. Z t Z t q¼ e dt ¼ yd y dt ð13Þ 0
0
The control input u can be designed by Eqs. (11) and (13): u¼
2JRs C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d þ k1 e_ þ k2 eþ k3 q kT U dc
ð14Þ
The system output dynamics can be obtained by combining Eqs. (11) and (14) e€ þk1 e_ þ k2 e þ k3 q ¼ 0
ð15Þ
K ¼[k1, k2, k3] is chosen to ensure that the characteristic equation p3 þ k1 p2 þ k2 p þ k3 ¼ 0 has all its roots strictly in the left-half complex plant, guaranteeing asymptotical stability of the system. The output u in Eq. (14) may be considered to be composed of two parts: one is a PID controller, i.e., k1 e_ þ k2 e þ k3 q; it brings the proportional-integral-derivative nature to the controller. The other is a state feedback controller, i.e., C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d . This eliminates the right-hand term of Eq. (11). As observed from the controller, system status and control robustness are both deeply influenced by parameters C0, C1, C2. However, their values are inaccurate and mutable because of the influence of the fluid flow characteristic and the friction during parameter measurement. Only the maximum and minimum values of C0, C1 and C2 can be applied, i.e., C 0 A ½C 0 min ; C 0 max , C 1 A ½C 1 min ; C 1 max , C 2 A ½C 2 min ; C 2 max . These extreme values are obtained with the measurement parameters of Δps, Δpr and the parameter ranges of Tc, Tv, γs and γr. To improve the consequent system performance degradation brought by uncertain parameters, a Lyapunov function-based controller is proposed in this section. Due to the uncertainty of these parameters, the actual control u can only be calculated by their estimates; then, Eq. (14) should be rewritten as i 2JRs h ^ u¼ C 0 signðy_ Þ C^ 1 y C^ 2 y_ þ y€ d þ k1 e_ þ k2 eþ k3 q ð16Þ kT U dc where Ĉ0, Ĉ1, and Ĉ2 are the estimated values of C0, C1, and C2, respectively. The actual system dynamic response can be obtained by substituting Eq. (16) into Eq. (11): e€ þk1 e_ þ k2 e þ k3 q ¼ C^ 0 C 0 signðy_ Þ þ C^ 1 C 1 y þ C^ 2 C 2 y_ ð17Þ A combined tracking error measure es(t) is defined: es ðt Þ ¼ e_ ðt Þ þ λ1 eðt Þ þ λ0 qðt Þ
ð18Þ
Table 1 List of system parameters. Quantity
Symbol
Value
Spool radius Effective arm length of flow torque Fluid mass density Phase resistance Velocity coefficient Flow coefficient Axial length of orifice Damping length of chamber Back EMF coefficient Motor torque coefficient Motor input DC voltage Rotational inertia of motor and spool Supply pressure Tank pressure Viscous torque coefficient Coulomb torque Jet angle at orifice P Jet angle at orifice T Estimated viscous torque coefficient
R r ρ Rs Cv Cd La Lc kE kT Udc J ps pr Tv Tc γs γr T^ v
8 10 3 m 6.1 10 3 m 880 kg m 3 0.6 Ω 0.98 0.69 1.2 10 3 m 2 10 2 m 0.04 0.08 24 V 5.4 10 5 kg m2 3 MPa 1 MPa [0, 1.2 10 2] [0.02, 0.24 N m] [651, 721] [501, 581] 3 10 3
Estimated Coulomb torque
T^ c r^ s r^ r VD
0.11 N m
Estimated Jet angle at orifice P Estimated Jet angle at orifice T Voltage drop of the MOSFET
71.61 56.41 0.03 V
To ensure system stability, the combined error es(t) should be calculated as a Hurwitz polynomial by choosing proper positive gains λ0 and λ1, with dynamics given by e_ s ðt Þ þKes ðt Þ ¼ eð2Þ ðt Þ þ λ1 þ K e_ ðt Þ þ λ0 þ K λ1 eðt Þ þ λ0 Kqðt Þ ¼ 0 ð19Þ If the gain K is positive and real, Eq. (19) implies that tracking error es(t) goes to zero as time goes to infinity, for all initial conditions. This combined tracking error es(t) can be applied to the RDDV servovalve by combining it and the system tracking error dynamics as described in Eq. (17) e_ s ðt Þ þK es ðt Þ ¼ e€ þ k1 e_ þ k2 e þ k3 q ¼ C^ 0 C 0 signðy_ Þ ð20Þ þ C^ 1 C 1 y þ C^ 2 C 2 y_ Thus, the following equations should be satisfied 8 > < λ 1 þ K ¼ k1 λ0 þ K λ1 ¼ k2 > :λ K ¼k 0
ð21Þ
3
A candidate quadratic function V(t) with the use of es(t) is defined V ðt Þ ¼ 12 e2s ðt Þ
ð22Þ
Obviously, the Lyapunov function V(t) is nonnegative definite. The time derivative of V can be obtained by utilizing the above
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(
analysis V_ ðt Þ ¼ es ðtÞ e_ s ðt Þ ¼ Ke2s ðt Þ þ Γ ðt Þ
ð23Þ
C^ 1 ¼ (
where
Γ ðt Þ ¼ es ðt Þ
h
C^ 0 C 0 signðy_ Þ þ C^ 1 C 1
i yþ C^ 2 C 2 y_
C^ 2 ¼
C 1 min ;
es ðtÞ y 4 0
C 1 max ;
otherwise
C 2 min ;
es ðtÞ y_ 4 0
C 2 max ;
otherwise
5
ð25Þ
ð26Þ
Due to Ke2s ðt Þ r 0, system stability can be guaranteed by
Γ ðt Þ r 0. Then, the objective is to define update laws for Ĉ0, Ĉ1
and Ĉ2. These can easily be decided depending on the sign of es(t), y and y_ . ( C 0 min ; es ðtÞ y_ 4 0 C^ 0 ¼ ð24Þ C 0 max ; otherwise
Table 2 List of control parameters. Symbol
Value
Symbol
Value
s1 k1 k3 λ1 kp kd
1.2e3 2.2e3 4e5 2e3 6.8 0.018
s2 k2 λ0 K ki
3.6e5 4.02e5 2e3 2e2 0.13
Hence, system accuracy and stability are guaranteed by an integral action and a switching control law of uncertain parameters. Obviously, if confidence intervals are expanded with other things equal, system robustness improves but compensation accuracy declines, and vice versa. It is necessary to enhance parameter measuring accuracy to improve accuracy and stability synchronously. The confidence intervals of the varying parameters should be adjusted according to the actual condition to realize a good tradeoff between stability and tracking accuracy. If all parameter values and confidence intervals are obtained, the RDDV drive signal can be generated via the following steps: 1. Measure the spool angular displacement x1, the spool angular velocity x2, and the pressure drops Δps and Δpr. 2. Calculate es(t), q as defined in Eqs. (13) and (18). 3. Calculate Ĉ0, Ĉ1 and Ĉ2 according to Eqs. (24)–(26). 4. Calculate the control output u as defined in Eq. (16), and implement it.
Fig. 4. Step response using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼ C0 max, C1 ¼C1 max, C2 ¼ C2 max, (b) C0 ¼C0 min, C1 ¼ C1 min, C2 ¼ C2 min, (c) C0 ¼ (C0 min þ C0 max)/2, C1 ¼(C1 min þC1 max)/2, C2 ¼(C2 min þC2 max)/2, (d) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
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4. Simulation and experiment results 4.1. Simulation results In this section, the performance of the proposed controller is illustrated by simulation results with the use of MATLAB. System parameters, except for ps and pr, are obtained through actual measurements and are listed in Table 1. In the first simulation test, a step signal with amplitude of 0.2 rad is adopted as reference. The proposed controller is compared to the PID control, the static state feedback control and the typical state feedback integral control. The PID controller has the same structure as the latter part of the proposed controller described in Eq. (14). The static state feedback control is calculated using Eq. (12). The state feedback integral control output is obtained using Eq. (16) and fixed estimated parameter values listed in Table 1. The proposed controller is calculated using the working steps described in the above section. The control parameters of these controllers are listed in Table 2. To verify the effect
Fig. 7. Step response using state feedback integral control and (I) C¼ Cmax, (II) C¼ (2Cmax þ Cmin)/3, (III) C ¼(Cmax þ2Cmin)/3, (IV) C¼ Cmin.
Fig. 5. Control parameters used in the proposed controller.
Fig. 6. Combined tracking error es(t) and control signal u used in the proposed controller.
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of these four controllers, the actual system parameters C0, C1 and C2 are set as different static values, i.e., maximum, minimum, median and estimated parameters in their feasible intervals. Control signals are calculated with estimated parameters or intervals. These control signals cannot exceed their physical constraints, i.e., u A ð 1; 1Þ. The tracking trajectories under this condition are shown in Fig. 4. It can be observed from Fig. 4 that the PID controller has an unsatisfied effect due to the lack of state compensation. It is too difficult for the PID controller to make a good tradeoff between the overshoot and the response speed. Because of their poor robustness, the static state feedback control and the state feedback integral control cannot be viewed as acceptable. As shown in Fig. 4(c) and (d), their response curves are satisfied when errors between the estimated parameters C^ 0 , C^ 1 and C^ 2 used in the control algorithms and the actual system parameters C0, C1 and C2 are small; however, their responses deteriorate remarkably when estimated parameters differ from the actual. As shown in Fig. 4(a) and (b), steady accuracy and robustness using these two controllers are obviously weakened. Due to the integral action,
Fig. 8. Step response using the proposed control and (I) C¼ Cmax, (II) C¼ (2Cmax þCmin)/3, (III) C¼ (Cmax þ 2Cmin)/3, (IV) C¼Cmin.
7
steady accuracy using the state feedback integral control is better than when using the static state feedback control. Compared with the other three controllers, the proposed controller using a Lyapunov function approach exhibits excellent robustness and fine dynamic responsibility. As shown in Fig. 4, its step response curves change slightly under different parameter errors. During its control process, the estimated parameters C^ 0 , C^ 1 and C^ 2 are dynamically changed to make the derivative of the Lyapunov function negative definite. Fig. 5 shows the estimated parameter values used in Fig. 4(a); they vary in the confidence interval. In Fig. 6, behaviors of the combined tracking error es(t) and the control signal u used in Fig. 4(a) are described; it is clear that es(t) has a fast convergence performance. It can also be observed from the above simulation that the response speeds of the proposed controller and the state feedback integral control are slower than that of the state feedback control and that a steady-state error also exists when using the state feedback integral control shown in Fig. 4(b). These are caused by small control parameters that are set to avoid large overshoots.
Fig. 10. Position response under square wave parameter fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
Fig. 9. Square wave parameters C0, C1 and C2.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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Fig. 11. Sine wave parameters C0, C1 and C2.
Fig. 12. Position response under sine wave parameter fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
To highlight the integral action, a new set of parameters with a large integral parameter, i.e., k1 ¼8e2, k2 ¼ 2e5, k3 ¼1.6e7, λ0 ¼4e4, λ1 ¼4e2, K ¼4e2, are applied. The other settings in this simulation are the same as shown above. Tracking trajectories using the typical state feedback integral control and the proposed controller under different parameter conditions are shown in Figs. 7 and 8, respectively. As is obviously from Figs. 7 and 8, the same conclusion as the first simulation test can be obtained. Additionally, with large parameter values, the response processes are speeded up remarkably. In the second simulation test, the actual system parameters C0, C1 and C2 are set as variable values, i.e., square-wave and sinewave fluctuations, to emulate the situations of the practical field. All other things being equal, the same simulation as the first simulation test is implemented. The system parameters and tracking trajectories under square wave parameter fluctuation are shown in Figs. 9 and 10, respectively. System parameters and tracking trajectories under sine wave parameter fluctuation are shown in Figs. 11 and 12, respectively.
Fig. 13. Step response under an output fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
As can easily be observed from Figs. 10 and 12, the proposed controller exhibits more robust and higher accuracy than other controllers. Due to the lack of state compensation, the PID controller has a large overshoot and fluctuation. Similarly due to the poor state compensation caused by parameter errors, the state feedback control and the state feedback integral control both exhibit poor robustness. In the third simulation test, a step signal with amplitude of 0.2 rad is adopted as input and the system parameters C0, C1 and C2 are set as sine-wave fluctuations as used in the above simulation. Additionally, an output fluctuation is added to these control systems; i.e., the outputs are decreased by 0.05 rad at 0.1 s. Tracking trajectories using these four controllers are shown in Fig. 13. From the response curves shown in Fig. 13, the proposed controller can return to the stable position rapidly and smoothly. However, the state feedback control and the state feedback integral control both need longer settling time; their responses are remarkably influenced by output fluctuation. In the fourth simulation test, a sinusoidal signal with amplitude of 0.2 rad and a frequency of 20 Hz is adopted as reference. Control
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Fig. 14. Sinusoidal response using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼(C0 min þC0 max)/2, C1 ¼ (C1 min þ C1 max)/2, C2 ¼ (C2 min þ C2 max)/2, (b) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
parameters are set to the same parameters as those used in the first simulation. The actual system parameters C0, C1 and C2 are set as different values, i.e., median and estimated parameters in their feasible intervals. Tracking trajectories are shown in Fig. 14. It can be observed from Fig. 14 that the proposed controller is superior to the state feedback integral control in amplitude and phase characteristic. However, both lag behind the state feedback control. This is due to the response characteristic being largely determined by the selected control parameters. In the fifth simulation test, the frequency response characteristics of these four controllers are examined to validate the response characteristics of the proposed controller. A sinusoidal signal with an amplitude of 0.2 rad is adopted as the reference, the control parameters are set to the same parameters as those used in the fourth simulation. The actual system parameters C0, C1 and C2 are set as different values, i.e., maximum, minimum and estimated parameters in their feasible intervals. Their Bode diagrams under different parameter errors are shown in Fig. 15. It can be observed from Fig. 15 that the frequency response performance of the PID controller is worse than that of other controllers. This is due to the lack of state compensation. The state feedback control can achieve the best amplitude characteristics and the best phase characteristics in the low frequency part through large control parameter values. Similarly to PD control, its response speed and amplitude characteristics can be improved through large parameter values; however, its amplitude frequency curve changes significantly under different parameter errors. The same conclusion about this controller, i.e., poor robustness,
Fig. 15. Frequency responses using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼C0 max, C1 ¼ C1 max, C2 ¼ C2 max, (b) C0 ¼C0 min, C1 ¼ C1 min, C2 ¼ C2 min, (c) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
as described above can also be obtained in this simulation. Both frequency response curves of the state feedback integral control and the proposed controller changes slightly under different parameter errors. Their frequency response performances can be further improved through large control parameter values. The proposed controller exhibits better response performance than the
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Position command
PWM DSP-based Servo controller
Motor driver
BLDC
AD
y
AD
p
RDDV
Cylinder
Angle U
FPGA-based Data Acquisition
Pressure U
Fig. 16. Control diagram of RDDV servovalve.
Fig. 18. Position response under ps ¼ 3 MPa using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. Fig. 17. Prototype of RDDV on a test stand.
state feedback integral control across the entire frequency range under the same parameters. The application of the Lyapunov function not only can improve system robustness but also can improve frequency response. In all the above simulation tests, the proposed controller shows an advantage over other controllers in accuracy, stabilization time and robustness. Under the condition that some parameters are inaccurate, mutable or cannot even be obtained, the proposed controller may be the most viable option. 4.2. Real-time control experiment In this section, the performance of the proposed controller is illustrated by experimental testing on a prototype of the RDDV servovalve. Referring to the functional diagram in Fig. 16 and the test rig in Fig. 17, the experimental setup is mainly composed of a prototype of the RDDV servovalve, an angular displacement sensor, a BLDC motor, fluid pressure transducers, a hydraulic circuit and a control system. The custom-made servovalve has been assembled to allow external control of the spool position through a BLDC motor and of angle measurement through an angular sensor. A prime mover and a relief are used to deliver filtered oil at a constant supply pressure to the whole system. A cylinder is chosen as the actuator; its varying load brings uncertainty and change to the system. The control system hardware mainly includes Data Acquisition (DAQ) cards, a DSP controller and a motor driver. The DAQ cards based on Xilinx FPGA XC3S200 and analog–digital converter AD7656 collect sensor data including the spool angle, orifice pressures, and flow rate, etc. at a
Fig. 19. Position response under ps ¼ 6 MPa using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
20 kHz/16 bit sample rate, and then transfer these data to the controller for feedback and to the master PC for analyzing controller efficiency. The DSP controller based on TI Digital Signal Controller TMS320F28335 is used to operate the control algorithm and provide the controlling quantity at 5 kHz. The motor driver is used to amplify the controlling quantity to drive the BLDC motor. The system and control parameters used in experimental tests are also listed in Tables 1 and 2. Before the tests, the spool position is set at an intermediate position; the cylinder position is set at a predefined position to enhance experimental repeatability. Then, a step input with
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amplitude of 0.2 rad under variable cylinder load is implemented to the RDDV servovalve. The trajectories using these controllers under the system supply pressure at ps ¼ 3 MPa and ps ¼ 6 MPa as shown in Figs. 18 and 19, respectively. It can be observed from Figs. 18 and 19 that the PID controller has a longer adjust time. The state feedback control without integral action also cannot be satisfied for a large steady-state error. The state feedback integral control can eliminate the steadystate tracking error; however, its accuracy and settling time are still worse than those of the proposed controller. The proposed controller shows advantages over other controllers in accuracy and stabilization time. Additionally, the robustness of the proposed controller is superior to that of controllers with fixed parameters. 5. Conclusion In this paper, a novel state feedback integral control using a Lyapunov function approach, which accounts for parameter uncertainties, has been presented for a rotary direct drive servovalve. This control not only can eliminate steady-state error of the system inherently by introducing an integral action, but also can improve robustness by adjusting parameters via a Lyapunov method. The structure and algorithm of the control strategy are very simple, enabling it to be easily implemented into embedded systems. Simulation and experimental results show that this novel control has satisfactory tracking properties and robustness, especially when the measured or calculated parameters deviate seriously from actual values. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant no. 51375363) and the Science and Technology Planning Project of Xi’an (Grant no. CX12504). References
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[2] Mintsa HA, Venugopal R, Kenné JP, Belleau C. Feedback linearization-based position control of an electrohydraulic servo system with supply pressure uncertainty. IEEE Trans Control Syst Technol 2012;20(4):1092–9. [3] Sadeghieh A, Sazgar H, Goodarzi K, Lucas C. Identification and real-time position control of a servo-hydraulic rotary actuator by means of a neurobiologically motivated algorithm. ISA Trans 2012;51(1):208–19. [4] Liao HH, Roelle MJ, Chen JS, Park S, Gerdes JC. Implementation and analysis of a repetitive controller for an electro-hydraulic engine valve system. IEEE Trans Control Syst Technol 2011;19(5):1102–13. [5] Zhao SD, Wang J, Wang LH, Hua CJ, He YP. Iterative learning control of electrohydraulic proportional feeding system in slotting machine for metal bar cropping. Int J Mach Tools Manuf 2005;45(7):923–31. [6] Kim HM, Park SH, Song JH, Kim JS. Robust position control of electro-hydraulic actuator systems using the adaptive back-stepping control scheme. Proc Inst Mech Eng Part I J Syst Control Eng 2010;224(6):737–46. [7] Park SH, Lee JM, Kim JS. Robust control of the pressure in a control-cylinder with direct drive valve for the variable displacement axial piston pump. Proc Inst Mech Eng Part I J Syst Control Eng 2009;223(4):455–65. [8] Champagne RP, Boyle SJ. Optimizing valve actuator parameters to enhance control valve performance. ISA Trans 1996;35(3):217–23. [9] Vladimir M, Željko Š, Mario E. Robust position control synthesis of an electrohydraulic servo system. ISA Trans 2010;49(4):535–42. [10] Boubaker D, Abdelaziz B. A stable neural adaptive force controller for a hydraulic actuator. Proc Inst Mech Eng I: J Syst Control Eng 2003;217 (4):303–10. [11] Yu J, Zhuang J, Yu DH. Parameter optimization of PID controller based on complex system genetic algorithm in electro-hydraulic servo control system. World congress on intelligent control and automation 2012:30–5. [12] Truong DQ, Ahn KK. Force control for hydraulic load simulator using selftuning grey predictor—fuzzy PID. Mechatronics 2009;19(2):233–46. [13] Ahn KK, Truong DQ, Thanh TQ, Lee BR. Online self-tuning fuzzy proportionalintegral-derivative control for hydraulic load simulator. Proc Inst Mech Eng Part I J Syst Control Eng 2008;222(2):81–95. [14] Schulte H, Hahn H. Fuzzy state feedback gain scheduling control of servopneumatic actuators. Control Eng Pract 2004;12(5):639–50. [15] Krikelis NJ. State feedback integral control with ‘intelligent’ integrators. Int J Control 1980;32(3):465–73. [16] Li HB, Sun ZQ, Chow MY, Sun FC. Gain-scheduling-based state feedback integral control for networked control systems. IEEE Trans Ind Electron 2011;58(6):2465–72. [17] Yu J, Zhuang J, Yu DH. Modeling and analysis of a rotary direct drive servo valve. Chin J Mech Eng-En 2014;27(5):1064–74. [18] Yu J, Zhuang J. Hydraulic rotary valve with square wave shaped spool. China patent 201110366373.7; 2011. [19] Ayalew B, Kulakowski BT. Modeling supply and return line dynamics for an electrohydraulic actuation system. ISA Trans 2005;44(3):329–43. [20] Merritt HE. Hydraulic control systems. New York: John Wiley & Sons; 1967. [21] Moseler O, Isermann R. Application of model-based fault detection to a brushless DC motor. IEEE Trans Ind Electron 2000;47(5):1015–20.
[1] Yang L, Yang S, Richard B. Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system. IEEE/ ASME Trans Mechatron 2013;18(1):1–10.
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State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach Jue Yu 1, Jian Zhuang 1,n, Dehong Yu 1 School of Mechanical Engineering, Xi’an Jiaotong University, No. 28, Xianning West Road, Xi’an, Shaanxi, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 6 June 2014 Received in revised form 25 July 2014 Accepted 18 August 2014
This paper concerns a state feedback integral control using a Lyapunov function approach for a rotary direct drive servo valve (RDDV) while considering parameter uncertainties. Modeling of this RDDV servovalve reveals that its mechanical performance is deeply influenced by friction torques and flow torques; however, these torques are uncertain and mutable due to the nature of fluid flow. To eliminate load resistance and to achieve satisfactory position responses, this paper develops a state feedback control that integrates an integral action and a Lyapunov function. The integral action is introduced to address the nonzero steady-state error; in particular, the Lyapunov function is employed to improve control robustness by adjusting the varying parameters within their value ranges. This new controller also has the advantages of simple structure and ease of implementation. Simulation and experimental results demonstrate that the proposed controller can achieve higher control accuracy and stronger robustness. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: State feedback control Integral control Direct drive servo valve Lyapunov function Parameter uncertainty
1. Introduction A servovalve is a vital device in achieving a precise motion control in an electrohydraulic servo system; its performance is deeply affected by the adopted control strategy. In recent years, a variety of control techniques applied in the electrohydraulic field have emerged. In Ref. [1], a sliding-mode controller is applied to force the hydraulic actuator to slide along a predefined surface in the state space. This control method can manifest robustness because control signals are independent of the controlled system model. However, the chattering phenomenon introduced by fast switching inputs cannot be avoided, and this may degrade system performance. In Ref. [2], a feedback linearization control is applied to improve system performance based on the idea of transforming nonlinear dynamics into a linear form with state feedback. This method has shown improved performance over traditional linear control methods, but it also has an important limitation; i.e., robustness could be weakened in the presence of parameter uncertainty or unmodeled dynamics. In Ref. [3], an improved adaptive control method was considered by updating controller gains to compensate for parameter fluctuations, but this requires a great deal of calculation. Both repetitive control [4] and iterative
n
Corresponding author. E-mail address: [email protected] (J. Yu). 1 Tel.: þ86 15102930290.
learning control [5] can be applied in electrohydraulic systems with periodic action trajectories. These two methods also need a large storage space and a fixed movement starting position. The Lyapunov method has also been widely used to enhance system stability in the hydraulic field. In Ref. [6], a backstepping technique with the Lyapunov control function is applied to improve robustness of the electrohydraulic control system. In Ref. [7], a robust control meeting the Lyapunov stability condition is used to control the output pressure of a cylinder. Both have a powerful ability to ensure system stability and accuracy. In Ref. [8], a parametric optimization method is applied to enhance control valve performance. In Ref. [9], a robust H1 controller extended with an integral action is used to control an electro-hydraulic servo system. Intelligent control methods include controllers using a neural network [10], evolution strategies [11] and fuzzy control [12]. These methods integrate artificial intelligence into the electrohydraulic control system, making controllers adapt to the nonlinear terms. These intelligent controllers have shown improved performance over traditional control methods in experimental tests; however, they always need large memory space and heavy computation. A large gap exists between theoretical analysis and online application especially when facing a shortage of funds. Despite the development of a vast array of advanced control strategies, the PID controller [13] is by far the most widely used within industry. Its popularity stems from its simple structure, easy operation and applicability in a wide variety of operating
http://dx.doi.org/10.1016/j.isatra.2014.08.006 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.
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scenarios. However, traditional PID controllers cannot exhibit consistent control performance when encountering complex, time-varying and uncertain systems. To this, many improvements have been proposed, that have already demonstrated satisfactory control effects; however, selection is always based on the cost of increased algorithm complexity. In addition to the PID controller, the state feedback controller [14] is also widely used when the system model and state variables can be obtained accurately and timely. Through this controller, the poles of the closed-loop system can be placed arbitrarily; therefore, it can guarantee system stability and specify the transient response of state variables. However, given the proportional-plus-derivative nature of the static state feedback controller, its steady-state disturbance rejection is not good enough. Although the response is quick and smooth, there is always an offset in the plant output when there is nonzero disturbance. Additionally, this controller is deeply influenced by the accuracy of the parameters; system robustness may be reduced by parameter errors. Typically, an integral action [15] and a state observer [16] are used to compensate for the abovementioned shortcomings. This paper proposes a RDDV servovalve [17,18] with a rotary spool and a rotary driver. The spool, directly driven by a brushless DC motor (BLDC), rotates within a certain angle range in the valve chamber, and its rotation angle modulates fluid flow through the control ports of the valve. Taking the RDDV servovalve position control as the aim, this paper analyzes the load on the spool and proposes a corresponding state feedback controller. The mechanics modeling of the RDDV servovalve reveals that its mechanical performance is deeply influenced by flow torques and friction torques; however, these torques are uncertain, mutable and sometimes cannot even be obtained due to the nature of fluid flow. To address this problem while still preserving excellent dynamic performance [19], an integral action and a Lyapunov direct method are proposed to improve the prototype controller. This integral action can eliminate steady-state tracking error by introduce an integral nature to the controller. The Lyapunov direct method is utilized to account for parameter uncertainties. More specifically, this paper first describes the servovalve structure, including mechanical structure and driving mode, then analyzes the loading condition on the spool, determines the impact caused by pressure drops on the spool and builds its mathematical model. Second, this paper proposes a state feedback integral controller to assign system poles to the desired places and eliminate the steady-state error. Due to the coefficients of frictions and flow torques cannot be determined accurately; therefore, a switching control law is then proposed to determine those estimated parameters that represent their uncertain counterparts. This switching control law is derived using a Lyapunov approach and can enhance system stability. Finally, simulations and experiments using a prototype of the RDDV servovalve are carried out to verify the compensating effect of the proposed controller. It is worth noting that, despite the analysis object of this paper being the RDDV servovalve, the proposed control algorithm can also be applied to traditional hydraulic valves because their spool loading conditions are the same. This paper is organized as follows: Section 2 describes the mathematical model of the RDDV servovalve under study. Section 3 shows the Lyapunov function based state feedback integral control method. Section 4 presents simulation and experimental results. Finally, the conclusions are presented in Section 5.
2. Modeling of RDDV The controlled plant analyzed in this paper is composed of a RDDV servovalve and a BLDC motor as the driver. Their mechanical models are analyzed below in detail.
Fig. 1. Three-dimensional diagram of the RDDV.
Fig. 2. Working principle diagram of RDDV servovalve.
2.1. Modeling of valve structure The proposed RDDV servovalve is a three-position four-way control valve as shown in Figs. 1 and 2. It is composed of a rotary spool, a static sleeve, a valve body, valve covers and other accessories [18]. The flow state of this servovalve is controlled by the spool rotation angle; thus, the load on the spool is of vital importance. As introduced in Refs. [17,20], resistances on the spool mainly include steady-state flow torque, transient-state flow torque and friction torques. Due to the similar flow passage structure, these resistances are similar to those of traditional slide valves, but with force analysis turned into torque analysis. Thus, mechanical models of traditional slide valves can be cited for the RDDV servovalve. Steady-state flow torque is generated when fluid flows through an orifice because the change of momentum will reflect torque, preventing spool rotation. Referring to Figs. 1 and 2, at orifices A and B, the fluid flows perpendicularly to the spool axis; however, at orifices P and T, the vena contracta in which the fluid flows into (out of) the chamber is at a jet angle to the spool axis. The jet force acts on the spool in the same direction as the fluid can be divided
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into a circumferential component and a radial component. By symmetrically locating orifices on the circumference of the sleeve, the radial direction force can be compensated. The circumferential component of the jet force Fsc can be solved by Bernoulli’s equation [20].
two-conducting mode and pulse-width modulation (PWM) method are applied here. If the inductance of the stator coils is neglected [21] (the ratio between the inductance and the resistance is approximately 0.001 in this study), the mathematical model can be derived as follows:
F sc ¼ 2C v C d A0 Δp cos γ
2Rs I s þ 2kE ω þ 2V D ¼ K PWM U dc
ð1Þ
where Cv is the velocity coefficient, Cd is the flow coefficient, and Δp is the pressure drop through the orifice. γ is the jet angle at the orifice; its value range can be obtained by experiments. A0 is the orifice flow area; it can be calculated from Fig. 2. A0 ¼ 2La R sin θ=2 La R θ ð2Þ where La is the axial length of the orifice, R is the spool radius, and θ is the effective orifice opening angle. Due to the spool rotation range is small ( 151r θ r151); the term sin(θ/2) in Eq. (2) can be replaced by θ/2. Steady-state flow torque Ts can be calculated by using Eqs. (1) and (2). T s ¼ 2ðF s1c þ F s2c Þ r ¼ 4RrC v C d La cos
γ s Δps θ þ 4RrC v C d La cos γ r Δpr θ ð3Þ
where s-suffix indicates variables at orifice P, r-suffix indicates variables at orifice T, and r is the effective arm length of the flow torques. Transient-state flow torque is due to the spool motion because the acceleration of the fluid in the chamber generates an additional force on the spool. Its magnitude Ft at one single orifice can be calculated by applying Newton’s second law [20]. pffiffiffiffiffiffiffi pffiffiffiffiffiffi ð4Þ F t ¼ 2ρLa Lc C d R Δp ω where Lc is the chamber damping length. The direction of transient-state flow force is the same as the spool angular velocity in chamber P, but it is opposite to the spool angular velocity in chamber T [17,20]. Thus, the resultant transient-state flow torque of all four orifices of the RDDV servovalve Tt can be calculated as qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi T t ¼ 2 2ρrLa Lc C d R ω Δp r Δp s ð5Þ The friction torques when the spool rotates in the valve chamber include viscous friction torque, seal friction torque and friction torque caused by the lateral unbalance force. The resultant friction can be simplified as a combination of Coulomb friction and viscous friction; thus, the friction torques can be described as follows. T f ¼ T c signðωÞ þ T v ω
ð6Þ
where Tf is the resultant friction torque, Tc is the Coulomb torque, sign() is the sign function, Tv is the viscous torque coefficient. Tc and Tv are uncertain and time-varying because these friction torques are all deeply influenced by the fluid pressure in the valve chamber, which fluctuates with changing load. From the above-introduced torques, the total torque that the spool has to overcome can be summarized [17]: T L ¼ signðωÞ T c þ 4RrC v C d La θ cos γ s Δps þ cos γ r Δpr qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi þT v ω þ 2 2ρrLa Lc C d Rω Δp r Δp s ð7Þ
2.2. Model of BLDC motor The RDDV servovalve is closed-loop controlled. One of the spool ends is coupled with an angular displacement sensor and the other is coupled with a motor to be driven. Thus, the model of the BLDC motor is indispensable for controller design and performance prediction of the servovalve control system. The
ð8Þ
where Rs is the phase resistance, IS is the circuit equivalent current, kE is the back EMF coefficient, VD is the voltage drop of the MOSFET, KPWM is the input PWM duty cycle, and Udc is the input DC voltage. The motor torque is an impetus torque, promoting the motor shaft and the spool rotation. The mechanical model of the coupling system can be described as below: kT I s T L ¼ J θ€
ð9Þ
where kT is the motor torque coefficient and J is the rotational inertia of the BLDC motor and the spool. 2.3. System modeling Because the spool position response is of vital important to the control effect and the servovalve performance criterion, it is chosen as the control objective. The control system can be represented in a state space form as shown in Eqs. (7)–(9): ( x_ 1 ðtÞ ¼ x2 ðtÞ ð10Þ T U dc u þ C 0 signðx2 ðtÞÞ þC 1 x1 ðtÞ þ C 2 x2 ðtÞ x_ 2 ðtÞ ¼ k2JR s where x2 ðtÞ ¼ θ_ ðt Þ ¼ ωðt Þ; u ¼ K PWM UVdcD ; C 0 ¼ TJc ; 4RrC v C d La cos γ s Δps þ cos γ r Δpr ; C1 ¼ J h pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi kT kE þ T v Rs þ 2 2ρrLa Lc C d R Δpr Δps Rs C2 ¼ : JRs x1 ðtÞ ¼ θðt Þ;
Due to the nature of fluid flow, parameters Tc, Tv, γs and γr in Eq. (10) are uncertain and mutable; their exact values are too difficult to determine. However, in the published literature, CFD simulation and experiments [17], C0, C1 and C2 calculated by these uncertain parameters can be determined to float within their value ranges. The system robustness, although reduced by parameter uncertainties, should be improved by the proposed controller.
3. Controller design It can be clearly concluded from the modeling analysis that spool control response is deeply influenced by resisting torques; however, these parameters, i.e., Tc, Tv, γs, γr, are uncertain and mutable. To achieve fast and satisfactory spool position responses, this section will develop a robust state feedback controller based on a Lyapunov function approach. Taking the spool position as the output, i.e., y ¼ x1 , its dynamic characteristics can be obtained by Eq. (10): y€ ¼
kT U dc u þ C 0 signðy_ Þ þ C 1 yþ C 2 y_ 2JRs
ð11Þ
Then, for a static state feedback controller applying the system model analyzed above, the output error can be adopted: u¼
2JRs C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d þ s1 e_ þ s2 e kT U dc
ð12Þ
where yd is the desired output, and e ¼ yd y is the output error. The control problem is to design u such that y-yd as t-1 for zero steady errors. However, due to the approximation and
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Fig. 3. Functional diagram of the proposed state feedback integral control.
measurement errors of model parameters, steady-state errors may be produced by the above controller. To remedy this limitation, typically an integral action is added into the static state feedback control. Its schematic diagram is shown in Fig. 3, where a new state variable q defined as the integral of e is adopted. Z t Z t q¼ e dt ¼ yd y dt ð13Þ 0
0
The control input u can be designed by Eqs. (11) and (13): u¼
2JRs C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d þ k1 e_ þ k2 eþ k3 q kT U dc
ð14Þ
The system output dynamics can be obtained by combining Eqs. (11) and (14) e€ þk1 e_ þ k2 e þ k3 q ¼ 0
ð15Þ
K ¼[k1, k2, k3] is chosen to ensure that the characteristic equation p3 þ k1 p2 þ k2 p þ k3 ¼ 0 has all its roots strictly in the left-half complex plant, guaranteeing asymptotical stability of the system. The output u in Eq. (14) may be considered to be composed of two parts: one is a PID controller, i.e., k1 e_ þ k2 e þ k3 q; it brings the proportional-integral-derivative nature to the controller. The other is a state feedback controller, i.e., C 0 signðy_ Þ C 1 y C 2 y_ þ y€ d . This eliminates the right-hand term of Eq. (11). As observed from the controller, system status and control robustness are both deeply influenced by parameters C0, C1, C2. However, their values are inaccurate and mutable because of the influence of the fluid flow characteristic and the friction during parameter measurement. Only the maximum and minimum values of C0, C1 and C2 can be applied, i.e., C 0 A ½C 0 min ; C 0 max , C 1 A ½C 1 min ; C 1 max , C 2 A ½C 2 min ; C 2 max . These extreme values are obtained with the measurement parameters of Δps, Δpr and the parameter ranges of Tc, Tv, γs and γr. To improve the consequent system performance degradation brought by uncertain parameters, a Lyapunov function-based controller is proposed in this section. Due to the uncertainty of these parameters, the actual control u can only be calculated by their estimates; then, Eq. (14) should be rewritten as i 2JRs h ^ u¼ C 0 signðy_ Þ C^ 1 y C^ 2 y_ þ y€ d þ k1 e_ þ k2 eþ k3 q ð16Þ kT U dc where Ĉ0, Ĉ1, and Ĉ2 are the estimated values of C0, C1, and C2, respectively. The actual system dynamic response can be obtained by substituting Eq. (16) into Eq. (11): e€ þk1 e_ þ k2 e þ k3 q ¼ C^ 0 C 0 signðy_ Þ þ C^ 1 C 1 y þ C^ 2 C 2 y_ ð17Þ A combined tracking error measure es(t) is defined: es ðt Þ ¼ e_ ðt Þ þ λ1 eðt Þ þ λ0 qðt Þ
ð18Þ
Table 1 List of system parameters. Quantity
Symbol
Value
Spool radius Effective arm length of flow torque Fluid mass density Phase resistance Velocity coefficient Flow coefficient Axial length of orifice Damping length of chamber Back EMF coefficient Motor torque coefficient Motor input DC voltage Rotational inertia of motor and spool Supply pressure Tank pressure Viscous torque coefficient Coulomb torque Jet angle at orifice P Jet angle at orifice T Estimated viscous torque coefficient
R r ρ Rs Cv Cd La Lc kE kT Udc J ps pr Tv Tc γs γr T^ v
8 10 3 m 6.1 10 3 m 880 kg m 3 0.6 Ω 0.98 0.69 1.2 10 3 m 2 10 2 m 0.04 0.08 24 V 5.4 10 5 kg m2 3 MPa 1 MPa [0, 1.2 10 2] [0.02, 0.24 N m] [651, 721] [501, 581] 3 10 3
Estimated Coulomb torque
T^ c r^ s r^ r VD
0.11 N m
Estimated Jet angle at orifice P Estimated Jet angle at orifice T Voltage drop of the MOSFET
71.61 56.41 0.03 V
To ensure system stability, the combined error es(t) should be calculated as a Hurwitz polynomial by choosing proper positive gains λ0 and λ1, with dynamics given by e_ s ðt Þ þKes ðt Þ ¼ eð2Þ ðt Þ þ λ1 þ K e_ ðt Þ þ λ0 þ K λ1 eðt Þ þ λ0 Kqðt Þ ¼ 0 ð19Þ If the gain K is positive and real, Eq. (19) implies that tracking error es(t) goes to zero as time goes to infinity, for all initial conditions. This combined tracking error es(t) can be applied to the RDDV servovalve by combining it and the system tracking error dynamics as described in Eq. (17) e_ s ðt Þ þK es ðt Þ ¼ e€ þ k1 e_ þ k2 e þ k3 q ¼ C^ 0 C 0 signðy_ Þ ð20Þ þ C^ 1 C 1 y þ C^ 2 C 2 y_ Thus, the following equations should be satisfied 8 > < λ 1 þ K ¼ k1 λ0 þ K λ1 ¼ k2 > :λ K ¼k 0
ð21Þ
3
A candidate quadratic function V(t) with the use of es(t) is defined V ðt Þ ¼ 12 e2s ðt Þ
ð22Þ
Obviously, the Lyapunov function V(t) is nonnegative definite. The time derivative of V can be obtained by utilizing the above
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(
analysis V_ ðt Þ ¼ es ðtÞ e_ s ðt Þ ¼ Ke2s ðt Þ þ Γ ðt Þ
ð23Þ
C^ 1 ¼ (
where
Γ ðt Þ ¼ es ðt Þ
h
C^ 0 C 0 signðy_ Þ þ C^ 1 C 1
i yþ C^ 2 C 2 y_
C^ 2 ¼
C 1 min ;
es ðtÞ y 4 0
C 1 max ;
otherwise
C 2 min ;
es ðtÞ y_ 4 0
C 2 max ;
otherwise
5
ð25Þ
ð26Þ
Due to Ke2s ðt Þ r 0, system stability can be guaranteed by
Γ ðt Þ r 0. Then, the objective is to define update laws for Ĉ0, Ĉ1
and Ĉ2. These can easily be decided depending on the sign of es(t), y and y_ . ( C 0 min ; es ðtÞ y_ 4 0 C^ 0 ¼ ð24Þ C 0 max ; otherwise
Table 2 List of control parameters. Symbol
Value
Symbol
Value
s1 k1 k3 λ1 kp kd
1.2e3 2.2e3 4e5 2e3 6.8 0.018
s2 k2 λ0 K ki
3.6e5 4.02e5 2e3 2e2 0.13
Hence, system accuracy and stability are guaranteed by an integral action and a switching control law of uncertain parameters. Obviously, if confidence intervals are expanded with other things equal, system robustness improves but compensation accuracy declines, and vice versa. It is necessary to enhance parameter measuring accuracy to improve accuracy and stability synchronously. The confidence intervals of the varying parameters should be adjusted according to the actual condition to realize a good tradeoff between stability and tracking accuracy. If all parameter values and confidence intervals are obtained, the RDDV drive signal can be generated via the following steps: 1. Measure the spool angular displacement x1, the spool angular velocity x2, and the pressure drops Δps and Δpr. 2. Calculate es(t), q as defined in Eqs. (13) and (18). 3. Calculate Ĉ0, Ĉ1 and Ĉ2 according to Eqs. (24)–(26). 4. Calculate the control output u as defined in Eq. (16), and implement it.
Fig. 4. Step response using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼ C0 max, C1 ¼C1 max, C2 ¼ C2 max, (b) C0 ¼C0 min, C1 ¼ C1 min, C2 ¼ C2 min, (c) C0 ¼ (C0 min þ C0 max)/2, C1 ¼(C1 min þC1 max)/2, C2 ¼(C2 min þC2 max)/2, (d) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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4. Simulation and experiment results 4.1. Simulation results In this section, the performance of the proposed controller is illustrated by simulation results with the use of MATLAB. System parameters, except for ps and pr, are obtained through actual measurements and are listed in Table 1. In the first simulation test, a step signal with amplitude of 0.2 rad is adopted as reference. The proposed controller is compared to the PID control, the static state feedback control and the typical state feedback integral control. The PID controller has the same structure as the latter part of the proposed controller described in Eq. (14). The static state feedback control is calculated using Eq. (12). The state feedback integral control output is obtained using Eq. (16) and fixed estimated parameter values listed in Table 1. The proposed controller is calculated using the working steps described in the above section. The control parameters of these controllers are listed in Table 2. To verify the effect
Fig. 7. Step response using state feedback integral control and (I) C¼ Cmax, (II) C¼ (2Cmax þ Cmin)/3, (III) C ¼(Cmax þ2Cmin)/3, (IV) C¼ Cmin.
Fig. 5. Control parameters used in the proposed controller.
Fig. 6. Combined tracking error es(t) and control signal u used in the proposed controller.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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of these four controllers, the actual system parameters C0, C1 and C2 are set as different static values, i.e., maximum, minimum, median and estimated parameters in their feasible intervals. Control signals are calculated with estimated parameters or intervals. These control signals cannot exceed their physical constraints, i.e., u A ð 1; 1Þ. The tracking trajectories under this condition are shown in Fig. 4. It can be observed from Fig. 4 that the PID controller has an unsatisfied effect due to the lack of state compensation. It is too difficult for the PID controller to make a good tradeoff between the overshoot and the response speed. Because of their poor robustness, the static state feedback control and the state feedback integral control cannot be viewed as acceptable. As shown in Fig. 4(c) and (d), their response curves are satisfied when errors between the estimated parameters C^ 0 , C^ 1 and C^ 2 used in the control algorithms and the actual system parameters C0, C1 and C2 are small; however, their responses deteriorate remarkably when estimated parameters differ from the actual. As shown in Fig. 4(a) and (b), steady accuracy and robustness using these two controllers are obviously weakened. Due to the integral action,
Fig. 8. Step response using the proposed control and (I) C¼ Cmax, (II) C¼ (2Cmax þCmin)/3, (III) C¼ (Cmax þ 2Cmin)/3, (IV) C¼Cmin.
7
steady accuracy using the state feedback integral control is better than when using the static state feedback control. Compared with the other three controllers, the proposed controller using a Lyapunov function approach exhibits excellent robustness and fine dynamic responsibility. As shown in Fig. 4, its step response curves change slightly under different parameter errors. During its control process, the estimated parameters C^ 0 , C^ 1 and C^ 2 are dynamically changed to make the derivative of the Lyapunov function negative definite. Fig. 5 shows the estimated parameter values used in Fig. 4(a); they vary in the confidence interval. In Fig. 6, behaviors of the combined tracking error es(t) and the control signal u used in Fig. 4(a) are described; it is clear that es(t) has a fast convergence performance. It can also be observed from the above simulation that the response speeds of the proposed controller and the state feedback integral control are slower than that of the state feedback control and that a steady-state error also exists when using the state feedback integral control shown in Fig. 4(b). These are caused by small control parameters that are set to avoid large overshoots.
Fig. 10. Position response under square wave parameter fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
Fig. 9. Square wave parameters C0, C1 and C2.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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Fig. 11. Sine wave parameters C0, C1 and C2.
Fig. 12. Position response under sine wave parameter fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
To highlight the integral action, a new set of parameters with a large integral parameter, i.e., k1 ¼8e2, k2 ¼ 2e5, k3 ¼1.6e7, λ0 ¼4e4, λ1 ¼4e2, K ¼4e2, are applied. The other settings in this simulation are the same as shown above. Tracking trajectories using the typical state feedback integral control and the proposed controller under different parameter conditions are shown in Figs. 7 and 8, respectively. As is obviously from Figs. 7 and 8, the same conclusion as the first simulation test can be obtained. Additionally, with large parameter values, the response processes are speeded up remarkably. In the second simulation test, the actual system parameters C0, C1 and C2 are set as variable values, i.e., square-wave and sinewave fluctuations, to emulate the situations of the practical field. All other things being equal, the same simulation as the first simulation test is implemented. The system parameters and tracking trajectories under square wave parameter fluctuation are shown in Figs. 9 and 10, respectively. System parameters and tracking trajectories under sine wave parameter fluctuation are shown in Figs. 11 and 12, respectively.
Fig. 13. Step response under an output fluctuation using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
As can easily be observed from Figs. 10 and 12, the proposed controller exhibits more robust and higher accuracy than other controllers. Due to the lack of state compensation, the PID controller has a large overshoot and fluctuation. Similarly due to the poor state compensation caused by parameter errors, the state feedback control and the state feedback integral control both exhibit poor robustness. In the third simulation test, a step signal with amplitude of 0.2 rad is adopted as input and the system parameters C0, C1 and C2 are set as sine-wave fluctuations as used in the above simulation. Additionally, an output fluctuation is added to these control systems; i.e., the outputs are decreased by 0.05 rad at 0.1 s. Tracking trajectories using these four controllers are shown in Fig. 13. From the response curves shown in Fig. 13, the proposed controller can return to the stable position rapidly and smoothly. However, the state feedback control and the state feedback integral control both need longer settling time; their responses are remarkably influenced by output fluctuation. In the fourth simulation test, a sinusoidal signal with amplitude of 0.2 rad and a frequency of 20 Hz is adopted as reference. Control
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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Fig. 14. Sinusoidal response using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼(C0 min þC0 max)/2, C1 ¼ (C1 min þ C1 max)/2, C2 ¼ (C2 min þ C2 max)/2, (b) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
parameters are set to the same parameters as those used in the first simulation. The actual system parameters C0, C1 and C2 are set as different values, i.e., median and estimated parameters in their feasible intervals. Tracking trajectories are shown in Fig. 14. It can be observed from Fig. 14 that the proposed controller is superior to the state feedback integral control in amplitude and phase characteristic. However, both lag behind the state feedback control. This is due to the response characteristic being largely determined by the selected control parameters. In the fifth simulation test, the frequency response characteristics of these four controllers are examined to validate the response characteristics of the proposed controller. A sinusoidal signal with an amplitude of 0.2 rad is adopted as the reference, the control parameters are set to the same parameters as those used in the fourth simulation. The actual system parameters C0, C1 and C2 are set as different values, i.e., maximum, minimum and estimated parameters in their feasible intervals. Their Bode diagrams under different parameter errors are shown in Fig. 15. It can be observed from Fig. 15 that the frequency response performance of the PID controller is worse than that of other controllers. This is due to the lack of state compensation. The state feedback control can achieve the best amplitude characteristics and the best phase characteristics in the low frequency part through large control parameter values. Similarly to PD control, its response speed and amplitude characteristics can be improved through large parameter values; however, its amplitude frequency curve changes significantly under different parameter errors. The same conclusion about this controller, i.e., poor robustness,
Fig. 15. Frequency responses using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. (a) C0 ¼C0 max, C1 ¼ C1 max, C2 ¼ C2 max, (b) C0 ¼C0 min, C1 ¼ C1 min, C2 ¼ C2 min, (c) C0, C1 and C2 calculated by T^ c , T^ v , γ^ s and γ^ r in Table 1.
as described above can also be obtained in this simulation. Both frequency response curves of the state feedback integral control and the proposed controller changes slightly under different parameter errors. Their frequency response performances can be further improved through large control parameter values. The proposed controller exhibits better response performance than the
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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Position command
PWM DSP-based Servo controller
Motor driver
BLDC
AD
y
AD
p
RDDV
Cylinder
Angle U
FPGA-based Data Acquisition
Pressure U
Fig. 16. Control diagram of RDDV servovalve.
Fig. 18. Position response under ps ¼ 3 MPa using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller. Fig. 17. Prototype of RDDV on a test stand.
state feedback integral control across the entire frequency range under the same parameters. The application of the Lyapunov function not only can improve system robustness but also can improve frequency response. In all the above simulation tests, the proposed controller shows an advantage over other controllers in accuracy, stabilization time and robustness. Under the condition that some parameters are inaccurate, mutable or cannot even be obtained, the proposed controller may be the most viable option. 4.2. Real-time control experiment In this section, the performance of the proposed controller is illustrated by experimental testing on a prototype of the RDDV servovalve. Referring to the functional diagram in Fig. 16 and the test rig in Fig. 17, the experimental setup is mainly composed of a prototype of the RDDV servovalve, an angular displacement sensor, a BLDC motor, fluid pressure transducers, a hydraulic circuit and a control system. The custom-made servovalve has been assembled to allow external control of the spool position through a BLDC motor and of angle measurement through an angular sensor. A prime mover and a relief are used to deliver filtered oil at a constant supply pressure to the whole system. A cylinder is chosen as the actuator; its varying load brings uncertainty and change to the system. The control system hardware mainly includes Data Acquisition (DAQ) cards, a DSP controller and a motor driver. The DAQ cards based on Xilinx FPGA XC3S200 and analog–digital converter AD7656 collect sensor data including the spool angle, orifice pressures, and flow rate, etc. at a
Fig. 19. Position response under ps ¼ 6 MPa using (I) PID control, (II) state feedback control, (III) state feedback integral control, and (IV) the proposed controller.
20 kHz/16 bit sample rate, and then transfer these data to the controller for feedback and to the master PC for analyzing controller efficiency. The DSP controller based on TI Digital Signal Controller TMS320F28335 is used to operate the control algorithm and provide the controlling quantity at 5 kHz. The motor driver is used to amplify the controlling quantity to drive the BLDC motor. The system and control parameters used in experimental tests are also listed in Tables 1 and 2. Before the tests, the spool position is set at an intermediate position; the cylinder position is set at a predefined position to enhance experimental repeatability. Then, a step input with
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i
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amplitude of 0.2 rad under variable cylinder load is implemented to the RDDV servovalve. The trajectories using these controllers under the system supply pressure at ps ¼ 3 MPa and ps ¼ 6 MPa as shown in Figs. 18 and 19, respectively. It can be observed from Figs. 18 and 19 that the PID controller has a longer adjust time. The state feedback control without integral action also cannot be satisfied for a large steady-state error. The state feedback integral control can eliminate the steadystate tracking error; however, its accuracy and settling time are still worse than those of the proposed controller. The proposed controller shows advantages over other controllers in accuracy and stabilization time. Additionally, the robustness of the proposed controller is superior to that of controllers with fixed parameters. 5. Conclusion In this paper, a novel state feedback integral control using a Lyapunov function approach, which accounts for parameter uncertainties, has been presented for a rotary direct drive servovalve. This control not only can eliminate steady-state error of the system inherently by introducing an integral action, but also can improve robustness by adjusting parameters via a Lyapunov method. The structure and algorithm of the control strategy are very simple, enabling it to be easily implemented into embedded systems. Simulation and experimental results show that this novel control has satisfactory tracking properties and robustness, especially when the measured or calculated parameters deviate seriously from actual values. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant no. 51375363) and the Science and Technology Planning Project of Xi’an (Grant no. CX12504). References
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[1] Yang L, Yang S, Richard B. Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system. IEEE/ ASME Trans Mechatron 2013;18(1):1–10.
Please cite this article as: Yu J, et al. State feedback integral control for a rotary direct drive servo valve using a Lyapunov function approach. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.08.006i