Designing Low-Chattering Sliding Mode Controller for

2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI)

Designing Low-Chattering Sliding Mode Controller for an Electrohydraulic System S. Das Mahapatra, R.Saha, D. Sanyal Mechanical Engineering Department Jadavpur University Kolkata, India

A. Sengupta, U.Bhattacharyya

S. Sanyal

Construction Engineering Department Jadavpur University Kolkata, India

Electrical Engineering Department Jadavpur University Kolkata, India

Abstract— Sliding mode controllers of first and second orders have been designed for linear motion tracking for an eletrohydraulic actuation system having deadband and other nonlinear features like cylinder stiction, hysterisis and Stribeck characteristics. An input linearized structure has been developed to formulate sliding mode controllers, or SMC, of orders 1 and 2 for the nonlinear system. The applied SMC is coupled with a simple integral compensator for attenuating both the chatter of the input signal and any steady state error of the output. Realtime experiments have been carried out that show the first-order controller as better above 1.4Hz tracking and the second-order controller as better below 1.4Hz. Keywords — Sliding mode control; Phase-plane; Chattering; Robustness; Electrohydraulic system; Real-time experiments

I. INTRODUCTION

S

LIDING-MODE controller [1], or SMC, is regarded as a robust solutions for tackling uncertainties and system nonlinearities [2-3]. The SMC is designed in terms of a sliding variable that is a linear combination of the tracking error and its rates involving Routh coefficients and this variable defines a sliding input. For a nonlinear system, an input linearization model needs to be evolved for extracting the command voltage from the sliding input. The SMC drives the system dynamics through a reaching phase to keep it subsequently in a close proximity of a designed curve defining the sliding phase. This phase is realized by a high-frequency change of the input structure, called input chattering. For the independence of this phase from the system features and being designed as a curve in the phase-plane with opposite signs of sliding variable and its derivative, the dynamics continue up to achieving the zero of the sliding variable. The Routh coefficients ensure subsequent progress to zero tracking error. Different strategies have been implemented in the past to attenuate the chattering. Some such strategies are adaptive fuzzy PI SMC [2] and integral compensation [4]. Successful execution of SMC for electrohydraulic actuation system, or EHAS [5,6], is an active research in view of their highly nonlinear features. These systems have the highest power-toweight ratio [7-8] and higher fault tolerance than electrical actuation system. These make them suitable for use in harsh and remote working environments. The nonlinearities arise from the static friction, or stiction, as well as the dynamic friction with Stribeck and hysterisis effects in the pistoncylinder contact surface, square-root pressure-flow relation in the metering port of the valves irrespective of servo or proportional types and large deadband for proportional valves [9-10]. The ideal low-cost solution for rugged operations is the 978-1-4799-1769-3/16/$31.00 ©2016 IEEE

combination of industry-grade actuator and proportional valve, since these demand lower level for oil-filtration. But controller design for such nonlinear systems is more challenging due to the deadband and higher stiction. In this present study, SMC of first and second orders have been implemented coupled with an integral control of the sliding variable with a view to eliminate the input chatter without incurring the penalty of steady state error in the position tracking. This structure has been termed as SMIC. The developed controller has been used for a real-time study in an existing laboratory-scale electrohydraulic setup [3]. In the next section, this setup together with the input linearized modeling of the system is described followed by detailing of the controller architecture in Section III. Real-time results in Section IV corroborate the proposed structures as appropriate for the system. The achievements are summarized in the concluding section. II. SYSTEM DESCRIPTION AND MODELING An existing laboratory-scale setup [3] for real-time motion tracking experiments of a single-rod double-acting cylinder is depicted in Fig. 1 together with Table 1. The hydraulic power pack HPP supplies high pressure oil through the proportional valve PV and the cylinder C back to a tank. Difference of pressures P1 and P2 in the cylinder chambers Ch1 and Ch2 respectively with corresponding areas Aa1 and Aa 2 mentioned in inset of Fig. 1 produces the effective force to generate motion x of the piston-rod. A compression spring S is attached with the piston to provide external force by virtue of its stiffness k and pre-compression length  with initial position x0 . The acceleration for the mass m moving together with the piston can be expressed as

x  { P1 Aa1  P2 Aa 2  k ( x  x0   )  F f } / m ,

(1)

where F f is the friction acting against the piston and pump and tank pressures being PP and PT . In terms of the pump and tank pressures PP and PT , the control voltage V to the PV with deadband of Vd , a simple incompressible model for the chamber pressures [3] can be written for V | Vd | as

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P1  PP  [( Aa1 x ) /{CvA (V  | Vd |)}] 2 ,

(2a)

2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI) 2 and P2  PT  [( Aa 2 x ) /{CvB (V  | Vd |)}] .

(2b)

In Fig. 1, single and double arrow heads represent respectively the piston extension for V | Vd | and retraction for V   | Vd | . Corresponding to a demand x d and the LVDT-acquired response x , the objective of the controller is to achieve low tracking error

xe  xd  x .

1-SMC and 2-SMC s in (4) is defined respectively as , (5a)

and s2  xe ,

(5b)

with a positive error coefficient s x in (5a).The generalized form of (4) can be rewritten with help of the above equations as

(3) For designing the sliding mode controller, the above equation can be expressed with help of (1), (2a) and (2b) as,

sn(n)  un  dn ,

s1  x e  s x x e ,

s1  u1  d1 ,

(5c)

s2  u2  d 2 ,

(5d)

with the linearized input and the disturbance models respectively as

(4)

where n denotes the order of the sliding variable s and (n) is the nth order derivative of the corresponding s with the control input u n and the disturbance d n for an nth order SMC.

2 2 un   sgn(V )( x 2 / m){( Aa31 / CvA )  ( Aa32 / CvB )} /

{V  sgn(V ) | Vd |}2 ,

(6a)

d1  [{k ( x0   )  PP Aa1  PT Aa2  F f }  kx] / m  xd  s x x e for V | Vd | , (6b) d1  [{k ( x0   )  PP Aa2  PT Aa1  F f }  kx] / m  xd  s x x e

for V   | Vd | ,

(6c)

and d 2  d1  s x x e ,

(6d)

where CvA and CvB are the valve coefficients respectively for Ports A and B. Assuming these ideal relationships between the position error and the control voltage as acceptable, the voltage extraction model from (6a) can be written as 2 2 V  sgn(xe )[ x 2 {( Aa31 / CvA )  ( Aa32 / CvB )} /(m | u |)]1/ 2  Vd

.(7) The controller structure for calculating this voltage is presented next. III. CONTROLLER ARCHITECTURE Figure 1. Laboratory set up and the flow details of an electrohydraulic motion control system with components listed in Table 1.

Proposing a controller structure comprised of an SMC u s and an integral part u I , the control input in (7) is written as

Table 1: Component specifications of the experipental set-up of Figure 1. Component

Make

Proportional valve PV Rexroth with integral control electronics ICE Single-rod cylinder C

Rexroth

Linear variable differential transformer LVDT, L Spring, S

HBM

Local Made

u  us  u I .

Specifications 4WREE 10 E50-23/G24K31/A1V; 50 lpm at 1.0 MPa total pressure drop at metered ports; 2:1 port area, 10% deadband CD250D40/20-200A1X/ 01CGDHT, 2.5 kg piston mass 0-200mm range with 0-10V analog output

Real-time system RTS NI systems

20kN/m stiffness, 0.4m Total Lngth 1GHz, 32-bit NI-cRIO 9081

Input module IM

NI systems

10VA/D NI-cRIO 9215

Output module OM

NI systems

10V D/A NI-cRIO 9263

(8)

Within the maximum and minimum bounds of  s and  s , the SMC in (8) is designed with help of (5a) and (5b) as

us   s sgn { ( sn )} ,

(9a)

where  ( s1 )  s1 ,

(9b)

and  ( s2 )  s2   s s2 with  s  0 and p / q  1 .

317

p/q

,

(9c)

2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI)

The need of chattering attenuation for the SMC has been described by a Matlab simulation presented in Figs. 2 and 3 for a system of the form (4) with a random disturbance input

d (t )  sin{2 .rand (1,1)} ,

(10a)

given as rand(1,1) available as a built-in command in Matlab that provides random numbers between 0 and 1 at different instant of time t at which (4) would be integrated. Of course, the sin function converts the random distribution between ±1. It is well-known that for

 s | d | ,

(10b)

Figure 2. 1-SMC control input and phase plane plot.

the sliding variable would stabilize to near-zero value [11]. The Matlab simulation has been obtained with  s equal to 2000 and d equal to 1500. A time step of integration 10-3s has been chosen that is consistent with the sampling speed employed in the real-time system. Figs. 2 (a), (3b) and (3d) of the corresponding control inputs exhibit high-frequency chatter corresponding SMC (9a) with  s equal to ±2000. This is seen to appear following the initial period of decrease of the sliding variable to near-zero value apparent in Figs. 2(b), (3a) and (3c). During the chattering phase, small and large oscillations of s and s respectively are apparent from the phase-plane plot in the s  s plane of Figs. 2(c), (3a) and (3c). Prior to the near-origin chatter in this plane, the superimposed effect of disturbance over a nearly constant s is quite evident.

Figure 3. 2-SMC control input and phase plane.

Fig. 4 reveals that the 2-SMC reaches higher s at the intersection with the  s surface at b, c and d respectively with  s and p / q pairs of (10, 1/2), (10, 1/3) and (20, 1/2). Of course, a higher s helps in reducing s at a faster rate towards its zero target. However, a lower p / q may also imply continuing with a relatively higher s up to quite near the origin of the phase plane beyond e. This explains the choice of the p / q value of 1/3 as the optimum, as apparent from Fig. 4. The overall SMIC structure (8) is depicted in the Fig. 5 that contains a basic integral input over and above an SMC that is switched off for |s| . In terms of a time constant T , the integral input is defined as u I  (  s n dt ) / T n 1 .

Figure 4. 2-SMC phase plane plot with different values of

(11)

In terms of (11), the dead band voltage in (7) has been formulated in a simple manner as

Vd  u I / k ,

(12)

where k is taken as 1 ms-2V-1. The roll of (11) is envisaged as for removing the chattering problem for |s| without incurring steady state error. The real-time performance of this controller is presented next.

Figure 5. Control structure diagram.

318

s

and

p/q .

2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI)

IV. RESULTS AND DISCUSSIONS All the results obtained in the set up shown in Fig. 1 are given in Figs. 6 to 10 with the parameters required for voltage extraction in (7) have been taken as CvA 2CvB 2.1910 7 m3/(s.VPa) for the corresponding port area 0.00125m2 and 0.00093m2 with 2.5kg piston mass along with  s =2000 and  =0.12m. The SMC specific parameters have been taken as s x  2 (s-1) and  = 0.1m/s for the 1-SMIC and  s = 10 and 20 (s-1) and  = 0.1m for the 2-SMIC. Figs. 6 and 7 show the responses for sinusoidal demands of 0.02m amplitude and frequencies of 0.5, 1, 2 and 4Hz for both 1-SMIC and 2-SMIC as satisfactory. Corresponding voltage plot for 2-SMIC is depicted in Fig. 8, revealing higher voltage range for higher frequency leading to controller saturation at 10V for 4Hz demand. A comparison of the performance of these controllers has been depicted in Fig. 9 in terms of integral absolute error and control effort defined respectively as IAE  0T| x e | dt ,

(13a)

CE  0T V 2 dt ,

(13b)

Figure 8. Control voltage for 2-SMIC sinusoidal tracking demands.

Figure 6. 1-SMIC performance for sinusoidal tracking demands

Figure 9. IAE and CE performance for SMIC.

Figure 7. 2-SMIC performances for sinusoidal tracking demands. Figure 10. Phase plane plot for two step demand with 2SMIC.

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2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI)

The IAE values in Fig. 9 reveal the 1-SMIC performance to be better than 2-SMIC above 1.4 Hz frequency, whereas the below this frequency 2-SMIC has performed better. Again, the control effort is less for 2-SMIC in the range between 1.5 Hz and 3.0 Hz, beyond which 1-SMIC appears as more energy efficient. For the lower frequency tracking, the higher IAE and CE values can be attributed to higher friction forces. Due to the voltage saturation for higher frequency in Fig. 8, IAE is also increased in Fig. 9a. Fig. 10 depicts the system dynamics in the phase plane of the tracking error and its derivative for the 2-SMIC controller for the cases of piston extension and retraction by 0.02m. In comparison to Fig. 3, the finite-time delay in real hardware resulting in control switching at lower frequency along with error changes of larger magnitude is quite evident. The nature of the error oscillation in the vicinity of the sliding surface in Fig. 9 can be attributed to the nonlinear Stribeck and hysterisis effects of the dynamic friction [12]. It is well known that these nonlinear effects to be different during extension and retraction, especially for single-rod cylinder. This difference might have yielded different sliding behaviors during extension and retraction in Fig. 10. V. CONCLUSIONS A new controller structure has been proposed by combining SMC with integral control. Simple input-linearized SMC structures have been realized by employing simple system models for achieving ease of voltage extraction from the sliding input. These resulted in input-linearized SMIC structures. The integral controller has been proposed to mitigate the SMC related chattering by withdrawing it in the vicinity of zero sliding variable. Another reason of using the integral controller is to overcome the steady state error problem. A comparison of system performance with core 1SMC and 2-SMC forms has been carried out for real-time linear tracking motion in a laboratory set up of electrohydraulic actuation system. The controllers exhibited no chattering. The first-order and second-order schemes have yielded lower error and required lower control energy in different frequency ranges.

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VI. ACKNOWLEDGEMENTS We sincerely acknowledge the supports of ARDB New Delhi, ERIPR New Delhi and UGC New Delhi for the set-up realization and ERIPR New Delhi for the scholar support. VII. REFERENCES [1]

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