Proceedings of 2010 IEEE Student Conference on Research and Development (SCOReD 2010), 13 - 14 Dec 2010, Putrajaya, Malaysia
PID Controller Design for an Industrial Hydraulic Actuator with Servo System S Md Rozali1
MF Rahmat2, N Abdul Wahab2, R Ghazali2, and Zulfatman2
1
Faculty of Electrical Engineering Universiti Teknikal Malaysia Melaka 76109 Durian Tunggal,Melaka,Malaysia
[email protected]
2
Faculty of Electrical Engineering Universiti Teknologi Malaysia UTM Skudai 81310,Johor,Malaysia
[email protected],
[email protected] process. Actuation time, hydraulic fluid supply pressure range, acting type, over torque protection, local position indication and integral pushbuttons and controls are among the important specifications for electro-hydraulic valve actuators. A suitable controller needs to be designed in order to acquire the highest performance of the electro-hydraulic actuator. The controller design requires the best mathematical model of the system under control. Thus, a method of identifying the system needs to be chosen so that the best accuracy of the model can be obtained. Model identification of electro-hydraulic position servo system based on hardware-inthe-loop simulation environment of Real-time Workshop(RTW) and system identification toolbox in Matlab has been proposed [2],[3]. Nonlinear hybrid controller composed of a proportional controller, a fuzzy controller and a classical PID controller for the model attained has been introduced. Similar work of modeling of electro-hydraulic actuator also had been done [1]. In contrast, the proposed PID controller designed for the model obtained is applied in simulation mode only without applying it in real time mode. Experimental work on recursive identification of an electrohydraulic system that represented by a discrete-time model in open-loop and closed-loop configurations is presented [4]. The unknown parameters of the system based on ARX model is estimated by using Recursive Least Square (RLS) method while model validation is verified by residual analysis. The electro-hydraulic system is modeled as linear periodic systems perturbed along periodic input and output trajectories [5]. By define the valid range of the linear approximation, periodic repetitive controller is designed and implemented within this range. In contrary with the previous work where step and sinusoidal signal is injected to the system, this research applied Pseudo binary random sequence (PRBS) signals as input to the electro-hydraulic actuator in order to model the system. The root mean square (RMS) errors between the outputs of actual plant and one-step predicted values belonging to models with different orders have been compared [6]. A development of model- based Fault Detection, Identification and Estimation (FDIE) scheme has been done for the condition monitoring of the EHA system [7]. It combines the use of a model of EHA system with the
Abstract-Electro-hydraulic system (EHS) consists of several dynamic parts which are widely used in motion control application. These dynamic parts need to be controlled to determine direction of the motion. Mathematical model of EHS is required in order to design a controller for the system. In this paper, system identification technique is used for system modeling. Model of the system is estimated by using System Identification Toolbox in Matlab. This process began with collection of input and output data from experimental works. The data collected is used for model estimation. Auto Regressive with eXogeneous input (ARX) model is chosen as model structure of the system. Based on the input and output data of the system, best fit criterion and correlation analysis of the residual is analyze to determine the adequate model for representing the EHS system. By using Ziegler-Nichols tuning method, PID controller is designed for the model chosen through simulation in Simulink. In order to verify this controller, it is applied to the real time system and the performance of the system is monitored. The result obtained shows that the output of the system with controller in simulation mode and experimental works is almost similar. The output of the system also tracked the input given successfully. Keywords: Electro-hydraulic system, System identification, ARX model, PID controller
I.
INTRODUCTION
A. Electro-Hydraulic Actuator System Electro-hydraulic actuator converts electrical signal to hydraulic power [1]. It is used for delivering high actuation forces and high power. It is widely used since it has simple construction, low cost small size-to-power ratios and be able to apply very large torques and forces with fast response time. On the other hand, it is highly non-linear system with uncertain dynamics which complicate the development of high performance closed-loop controllers. Thus, the mathematical representation of the system cannot sufficiently represent the practical system [2]. Basically, single-rod and double-rod actuator with either proportional valve or servo valve are used. Since electro-hydraulic actuator can provide precise movement, high power capability, fast response characteristics and good positioning capability, its applications are important in the field of robotics, suspension systems and industrial
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multiple-model estimation algorithm to evaluate the reason and severity of a fault present. Based on radial basis function (RBF) neural network, a research has been done to develop a model reference adaptive PID control to improve the control performance of hydraulic parallel robot [8]. RBF neural network is also used to identify the hydraulic servo system online and then regulate the PID parameters on-line which makes the system more adaptive. In order to control the velocity of hydraulic actuator, robust design techniques are suggested [9]. The effectiveness of the controllers for the system is illustrated through simulations for several values of the model’s uncertain parameters. Large variation of the effective bulk modulus of the working fluid which is due to the absence of a heat exchanger is overcome by adaptive control scheme [10]. The error equations for velocity, acceleration and jerk which are generated in the design procedure of the standard backstepping control scheme are used in this new scheme. Dynamic model of the entire actuator incorporating highly non-linear hydraulic functions and the LuGre dynamic friction model is used to arrive at a suitable controller [11]. By applying an adaptive observer in the controller the use of acceleration measurement is avoided. On the other hand, state estimation of electro-hydraulic actuator by using proportional integral observer is presented [12]. The proposed approach utilizes the complete model of the system including the nonlinear dynamics and the performance is demonstrated with the simulated results. Repetitive control which is new control scheme is proposed where the controlled variables follow periodic reference commands [13]. Derivation of sufficient conditions for the stability of repetitive control systems and modified repetitive control systems is done by applying the small gain theorem and the stability theorem for time-lag systems. Combination of various robust control techniques such as integral-block, sliding mode and H-infinity control is implemented to design controller forcing electro-hydraulic system to track a chaotic reference trajectory [14]. This approach enables one to compensate the inherent nonlinearities of the actuator and reject matched external disturbances and attenuate mismatched external disturbances. A discontinuous controller which is able to track force reference trajectory system is designed by applying sliding mode control technique [15]. Instead of fixed boundary layers, sliding mode controller with varying boundary layers has been proposed to improve the tracking performance of a nonlinear EHA position servo system [16]. By using a method called nonlinear Quantitative feedback Theory (QFT), a robust time-invariant controller is designed for EHS [17]. In order to form a discontinuous controller for hydraulic valve, a new control strategy which combines the classical fuzzy control theory and tri-state valve technology is presented [18]. An integrated hybrid design, consisting of two independent sets of cylinder controllers (a feed-forward controller and a fuzzy tracking controller) and one coordinate fuzzy controller is also proposed for a dual-cylinder electro-hydraulic lifting system which is used to achieve a synchronized positioning objective
with unbalanced loadings, uncertainties and disturbances [19]. Block control, sliding mode control and integral control techniques has been combined to design a controller in order to force an EHA system driven by servo valve to track a given chaotic trajectory [20]. Thus, it is able to compensate the inherent nonlinearities of the actuator and to reject external constant disturbances. B. Modeling of Electro-Hydraulic Actuator System A mathematical model of electro-hydraulic actuator consists of the dynamics of the system disturbed by an external load and the dynamics of a servo valve. The linear differential equations that describe the actuator-valve dynamics are given by [9] • 1 (1) v p (t ) = [AP (t ) − bv (t ) − F (t )] L
m
p
L
4β P L (t ) = K f x v (t ) − K tp PL (t ) − Av p (t ) V
[
•
]
(2)
Where v p = piston velocity PL = hydraulic pressure FL = external load disturbance x v = spool valve displacement A = piston surface area m = mass of theload
β = effective bulk mod ulus b = viscous damping coefficient V = total volume of hydraulic oil in the piston chamber and the connecting lines
For zero initial conditions, the Laplace Transform of the equations (1) – (2) produced the following input-output relation: (3) U p ( s ) = H ( s ) X v ( s ) + H L ( s ) FL ( s ) Where H (s) =
4 β AK f
(4)
( ms + b)(Vs + 4 β K tp ) = 4bA 2
H L ( s) =
− 4βK tp − sV
(5)
(ms + b)(Vs + 4 βK tp ) + 4bA2
The transfer function of solenoid can be approximated by the servo valve spool position gain denoted by k v . Thus, inputoutput relation (3) can be rewritten as (6)
U p ( s ) = H ( s) k vVin ( s ) + H L ( s ) FL ( s )
Where
Vin ( s ) = laplace transform of the control voltage vin (t )
Using equations (1)-(2), linear system with uncertain structure is derived in state space form as d (7) x (t ) = A ( q ) x (t ) _ B ( q )v (t ) + D F (t ) dt
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0
0
in
0
L
response is also recorded and compared with the output response attained from the simulation mode.
(8)
y(t ) = C0 x(t )
Where
A. Model estimation Electro-hydraulic actuator system that is used in this work is an open-loop system with draw wire sensor acts as position sensor to track the location of the load. Multi sine signal with three different frequencies that is represented by equation (9) and illustrated in Figure 1 is chosen as the input to the system [21]. This signal is preferable compared to original sinusoidal signal in order to capture the dynamic characteristic of the system. (9) multi sin e = cos( 0.5t s ) + cos( 2t s ) + cos(5t s )
⎡v p (t ) ⎤ x(t ) = ⎢ ⎥ ⎣ PL (t )⎦ A/ m ⎡− b / m ⎤ A0 (q) = ⎢ ⎥ ⎣− 4q1 A / V − 4q1q2 / V ⎦ ⎡0 ⎤ B0 (q) = ⎢ ⎥ ⎣4q1q3 k v / V ⎦ ⎡− 1 / m⎤ D0 = ⎢ ⎥ ⎣0 ⎦ C0 = [1 0]
3
The objectives of this paper are to represent a mathematical model of electro-hydraulic actuator (EHA) system using system identification technique. It is followed by designing suitable PID controller for the system in simulation and realtime mode. II.
2
1
0
-1
-2
EXPERIMENTAL AND SIMULATION SETUP
-3
The electro-hydraulic system that is used in this study is composed of a single-rod hydraulic cylinder driven by a direct servo valve Bosch Rexroth 4WREE6, 40 lpm flow rate at 70 bars. The dimension of hydraulic cylinder is 63mm/30mm/300mm. Piston position is measured by using 300mm draw wire sensor while 100bar pressure transducers are attached to measure the pressure into and from the cylinder. NI PCI 6221 card is used as interface between Matlab programs in PC with electro-hydraulic test bed. Mathematical modeling is a description of a system in terms of equations. It can be divided into two parts; physical modeling and system identification. In this research, system identification technique is applied to attain the model of the system. Multi sine input is injected to the EHS through Simulink to capture the position of the load which is recorded by draw wire sensor. The collected input and output data is stored in workspace Matlab. This data is used for model estimation and validation part. Validation process is done in order to compare the estimated model output with the real output from the experiments [22]. By looking at the best fit criterion parameter, the validated model will be accepted [1]. Open-loop testing is done in simulation mode by inject step and sine input to the model obtained so that a suitable controller can be designed to improve the performance of the system. Based on Ziegler-Nichols tuning method PID controller is designed by inserted the calculated parameters in PID block in Simulink and examined the output result. In simulation mode, the PID controller is connected to the discrete transfer function model in Simulink block only. The output response is observed and recorded. It is followed by inserted the similar PID in a real-time system where it is located in the forward path of real-system. The different between this mode and the previous mode is the latter used the electro-hydraulic actuator system itself instead of discrete transfer function model in the previous one. The output
0
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800
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1400
1600
Figure 1. Multi sine signal with three different frequencies
This signal is injected to the electro-hydraulic system and the output of the system is recorded. Several sets of 1500 numbers of input and ouput data with sampling time 50ms are collected for model estimation and validation. B. Design of Controller PID (proportional integral derivative) control is one of the earlier control strategies. Since it has simple control structure which was understood by plant operators and which they found relatively easy to tune, it still have wide range of applications in industrial control. By tuning the value of Kp, Ki and Kd of the PID controller, the performance of the system such as rise time, overshoot, settling time and steady state error can be improved. Though the output of the system with this controller will never reach zero steady state error, Kp or proportional controller is used to assure the output reach the reference input. Ki or integral controller is given to the system in order to obtain zero or very small steady state error. Derivative controller or Kd will improve the speed performance of the system. Sometimes derivative action may not be required since the proportional and integral action already produce good output response [23]. The tuning value of Kp, Ki and Kd are determined by using Ziegler-Nichols tuning method [23],[24]. The critical gain, Kcr and critical period of oscillation, Tcr need to be determine before the tuning process. The value of Kp, Ki and Kd is adjusted from this two parameters based on Ziegler-Nichols tuning rules.
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III.
RESULTS
A. Model Estimation
TABLE I.
THE CALCULATED VALUE OF KP, KI AND KD FOR PID CONTROLLER
Figure 2 shows the relationship between the input, u1 and the output, y1 of the electro-hydraulic system when multi sine input is injected to it.
Input Step Sine
Kp 0.276 1.8
Ki 0.028 0.03
Kd 0.09 0.01
Input and output signals
y1
50
Figure 3 shows the block diagram of ARX model without controller with step input. The same model also injected with sine input.
0
-50
0
500
1000
1500
2000
2500
input 10
To Workspace
u1
5 0 -5 -10
0.01944 z3 +0.6005 z 2 -1.014 z+0.5188 0
500
1000
1500
2000
z 4 -2.056 z 3 +1.186 z2 +0.1772 z-0.3061
2500
Time
Step
Scope
Discrete Transfer Fcn
Figure 2. The input-output data of EHA system
output To Workspace 1
These input and output data is divided into two parts. The first part is used to estimate the model of the system while the other part for model validation. Auto-regressive Exogenous (ARX) model is selected as the model structure of the system since it is the simplest model incorporating the stimulus signal. The estimation of ARX model is the most efficient of the polynomial estimation methods because it is the result of solving linear regression equations in analytic form. The solution also satisfies the global minimum of the loss function. The model estimated can be accepted when the best fit percentage is more than 90% [22]. Besides, from the residuals analysis, the auto correlation and cross correlation of input and output data should be in the range of confidence interval. Validation of the data shows that 92.8% best fit meaning that the estimated model is almost tracking the real output data from the experiments. All these process is done through System Identification Toolbox in Matlab. The polynomial model obtained is in the form of discrete time equation which is represented as follow
Figure 3. The model with step input
The output response with each step and sine input is illustrated in Figures 4 and 5 correspondingly. 120
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Figure 4. Response of the model with step input 15
10
A(q) = 1 − 2.056q −1 + 1.186q −2 + 0.177q −3 − 0.3061q −4 5
(10)
B(q) = 0.01944q −1 + 0.6005q −2 − 1.014q −3 + 0.5188q −4
0
The transfer function for ARX model is given by [21] G (q) =
-5
B(q) A(q)
-10
Thus, the transfer function for the system can be represented by G (q) =
−1
−2
−3
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100
Figure 5. Response of the model with sine input
−4
0.01944 q + 0.6005 q − 1.014 q + 0.5188 q 1 − 2.056 q −1 + 1.186 q − 2 + 0.177 q −3 − 0.3061q − 4
0
Referring to Figure 4, the system produced large steady-state error when step input is injected to it. Figure 5 also illustrates that the sine input caused the output of the system being unstable and oscillate. Although the model is accepted as the best model for the system, the stability of the model need to be improved.
(11)
B. PID Controller The critical gain for the model attained is Kc = 0.46 with critical period, Tc = 7.1481. From calculation based on Ziegler-Nichols tuning method and manual adjustment, the parameters of PID controller with different input are shown in Table I.
The block diagram of the system with PID controller injected with step input in simulation mode is shown in Figures 6. The same procedure is repeated with sine input. The
221
output responses with both inputs are shown in Figures 7 and 8.
Based on this figure the response of the system with step and sine input are revealed by Figures 10 and 11 respectively. 0.06
input
0.05
To Workspace 1 0.04
0.01944 z 3 +0.6005 z2 -1.014 z+0.5188
PID Step
z4 -2.056 z 3 +1.186 z 2 +0.1772 z-0.3061
Discrete PID Controller
Discrete Transfer Fcn
Scope
0.03
output
0.02
To Workspace 2 0.01
Figure 6. System with PID controller with step input
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0.06
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Figure 10. Response of real-time PID controller with step input (experiment)
0.05
0.04 0.03
0.03 0.02
0.02 0.01
0.01
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-0.01
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-0.02
Figure 7. Response of the system with PID controller with step input (simulation)
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Figure 8. Response of the system with PID controller with sine input (simulation)
Based on Figures 7 and 8, the output response of the system with step and sine input is improved by adding PID controller in the forward path of system. The system produced very small steady-state error with both inputs. The output of the system also tracked the input given to it. On the other hand, Figure 9 shows the similar PID is inserted in the forward path of the system in real-time mode. Displacement Tracking Reference
Voltage Input to the Valve
IV.
300
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PID Discrete PID Controller
Input
Position
Disp
Saturation Hydraulic System LVDT
CONCLUSION
System identification technique has been successfully applied to electro-hydraulic system in order to produce the best linear discrete model of the system. PID controller is designed effectively for the system and applied in both simulation and real-time mode. The value of Kp, Ki and Kd is determined by Ziegler-Nichols tuning method since this tuning method is used widely. Step and sine input are injected to the system and the simulation result shows that the output tracked the input. The real-time experiments also proved the result where the output obtained is almost similar with the output response from simulation mode.
To Workspace
Step
200
Figures 10 and 11 show that the outputs from real-time experiments are almost similar with the output attained from simulation which produce very small steady-state error and fast response time. It also can be seen that the output tracked the input with very small error. The setting value of Kp, Ki and Kd of the PID controller is acceptable and improve the performance of EHA system. This is proved by the increment of the best fir of the model from 92% to 98%. A slight different between input and output happened because the electro-hydraulic system which is nonlinear model is modeled in linear model and some nonlinearity and uncertainties characteristic are ignored.
0.02
Sine Wave
100
Figure 11. Response of real-time PID controller with sine input (experiment)
0.04
-0.03
0
Sensor Output Displacement (m)
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Figure 9. Real-time PID controller
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