POSITION CONTROL OF AN ELECTRO-HYDRAULIC SERVOSYSTEM A non-linear backstepping approach
Claude Kaddissi, Jean-Pierre Kenné, Maarouf Saad École de Technologie Supérieure, 1100 Notre-Dame West street, Quebec H3C 1K3, Canada
Keywords: Electro-hydraulic systems modeling, Non-linear control, Backstepping.
Abstract: This paper studies the control of an electro-hydraulic servosystem using a non-linear backstepping based approach. These systems are known to be highly non-linear due to many factors as leakage, friction and especially the fluid flow expression through the servo-valve. Another fact, often neglected or avoided, is that such systems have a non-differentiable mathematical model for bi-directional applications. All these facets are pointed out in the proposed model of this paper. Therefore, the control of this class of systems should be based on non-linear strategies. Many experiments showed the failure of classic control with electro-hydraulic systems, unless operating in the neighborhood of a desired value or reference signal. The backstepping is used here to overcome all the non-linear effects. The model discontinuity is solved in this paper, by approximating the non-differentiable function by a sigmoid, so that the backstepping could be used without restrictions. In fact, simulation results show the effectiveness of the proposed approach in terms of guaranteed stability and zero tracking error.
1 INTRODUCTION
singularity, the traditional constant-gain controllers An electro-hydraulic system is composed of multiple have become inadequate. Lim (1997) applied simple components: A pump, which feeds the system with poles placement for a linearized model of an electro- fluid oil from an oil container; An accumulator, hydraulic system. Plahuta et al. (1997) tried a which acts as an auxiliary of energy integrated in the retroaction strategy for variable displacement hydraulic circuit; A relief valve on the other hand to hydraulic actuator and this by using two cascaded compensate the increase of pressure if any; A control loops for position and speed control. Zeng hydraulic actuator to drive a given load at a desired and Hu (1999) used a PDF algorithm (Pseudo- position, its displacement direction, speed and Derivative Feedback) where the integrator part of a acceleration are determined by a servo-valve. Note PID controller was placed in the direct path. that oil exiting the hydraulic actuator is driven Experiments and simulations showed that factors through the servo-valve back to the tank. resulting in dynamic variations are beyond the Electro-hydraulic systems became increasingly capacity of these controllers. And there are also current in many kinds of industrial equipments and many of these factors to take into account, such as processes. Such applications include rolling and paper mills, aircraft’s and all kinds of automation load variations, changes of transducers including cars industry where linear movements, fast characteristics and properties of the hydraulic fluid, response, and accurate positioning with heavy loads changes of servo components dynamics and other are needed. This is principally because of the great system components, etc. As a result, many robust power capacity with good dynamic response and and adaptive control methods have been used. Yu system solution that they can offer, as compared to and Kuo (1996) employed an indirect adaptive DC and AC motors. However, as a result of the ever- controller for speed feedback, based on pole demanding complexity with these applications, placement. Sliding mode controller has been used by considering non-linearity and mathematical model Fink and Singh (1998), in order to regulate the pressure drop due to the load across the actuator. On the other hand fuzzy logic control has been
270 Kaddissi C., Kenné J. and Saad M. (2004). POSITION CONTROL OF AN ELECTRO-HYDRAULIC SERVOSYSTEM - A non-linear backstepping approach. In Proceedings of the First International Conference on Informatics in Control, Automation and Robotics, pages 270-276 DOI: 10.5220/0001134402700276 Copyright c SciTePress POSITION CONTROL OF AN ELECTRO-HYDRAULIC SERVOSYSTEM - A non-linear backstepping approach
employed by Yongqian et al. (1998), the controller was based on a decision rules matrix of the error. Results were very closed to neural network based controller as in Kandil et al. 1999, where learning was accomplished through the output error retro- propagation and thus system knowledge was not necessary. Simulation results in these advanced strategies were very successful in most cases, aside minor transient state problems or small steady state error. However, stability is not guaranteed in these approaches. This is because in such cases stability is studied on discrete domains and one cannot expect system behaviour at limits. Another point to consider is that in spite of ensuring system robustness, most control strategies might generate a control law with high amplitude, which causes system saturation. Such is the case of feedback linéarizations. Therefore one should think to benefit of system non- linearities that offer more manoeuvrability to avoid saturating control signals. This paper proposes a backstepping approach, Figure 1: Schematic of hydraulic servo-system tacking into account all system non-linearities. A major advantage of the backstepping approach is its flexibility to build a control law by avoiding the amount of flow to the tank. A hydraulic rotary cancellation of useful non-linearities. In addition actuator drives the load. The actuator rotation is due modification is brought so the system model can be to the oil flow coming from the servo-valve, the used for bi-directional applications. Under such latter determines its direction, speed and acceleration modification the use of this approach ensures global through convenient position of its spool. The control stability of the system, and generates low amplitude signal being generated by the controller designed in control signal. this paper actuates the spool to the right position. We should note that the oil returns to the tank from the The paper is organised as follow. Section 2 presents servo-valve at atmospheric pressure and we assume the dynamic model of the electro-hydraulic system that the latter is a single stage servo-valve, critically with emphasis on non-linearity and non- centred and the orifices are matched and differentiability. In section 3, we shall present the symmetrical. design of a backstepping control strategy according to the system properties. In section 4 Simulation The dynamic equation for a servo-valve spool results and comparisons will be done. Some movement can be given by (LeQuoc et al., 1990): conclusions will be carried out in the last section. τ v A&v + Av = Ku (1)
Where u is the control input, K is the servo–valve 2 SYSTEM MODELING constant, τ v is its time constant and Av is the valve opening area. The flow rate from and to the servo- Consider the hydraulic system shown in figure 1. In valve, through the valve orifices, assuming small this hydraulic circuit the DC electric motor drives leakage, are given as: the pump at constant speed. The pump in turn delivers oil flow from the tank to the rest of the Ps − PL (2) Q1 = Q2 = Cd Av components. Normally, the pressure Ps at the pump ρ discharge depends on the load, however it is made constant due to the presence of an accumulator and a relief valve. In fact, the accumulator acts as an Where PL is the differential pressure due to load. additional pressure source in case needed. On the other side the relief valve compensates the pressure PL = P1 − P2 in the positive direction increase due to big loads, by returning the additional PL = P2 − P1 in the negative direction
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is the source pressure, C is the flow Ps = P1 + P2 d 3 BACKSTEPPING BASED discharge coefficient and ρ is the fluid oil mass NON-LINEAR CONTROL density. Since oil viscosity might vary with temperature, it In this section, the non-linear backstepping, as should be considered in the actuator dynamics along presented by Hassan (2002), will be used to control with oil leakage. Thus we give the compressibility the electro-hydraulic system presented in figure 1. equation as: The control law is derived based on a lyapunov function, to ensure an input-output stability of the V P − sign(A )P system. This method has been employed before by P& = C A s v L − D θ& − C P (3) 2β L d v ρ m L L Ursu and Popescu (2002) with a reduced order model of an electro-hydraulic system, for position control; and with a 4th order model for load pressure We define C as the load leakage coefficient, is L β control, all with positive desired trajectories or the fluid bulk modulus, θ is the output angular sign(x4 ) = 1. Later and for better tracking position, V is the oil volume under compression in th characteristics the backstepping was applied to a 5 one chamber of the actuator and D is the actuator m order electro-hydraulic system model (Ursu, F. et al. volumetric displacement. 2003). However the same assumption was held (i.e. a positive reference for position). Conversely, it is Now we consider the hydraulic actuator equation of very vital to consider the case of a trajectory that motion given by Newton’s first law. Neglecting the takes positive and negative values. Else, the system frictional torque we have: application will loose generality, and will bypass a large number of applications where electro-hydraulic Jθ&& = Dm (P1 − P2 ) − Bθ& −TL (4) systems are used. One of those is electro-hydraulic active suspension, where a hydraulic actuator has to TL is the load torque, B viscous damping coefficient ensure a minimum vertical displacement of the car and J the actuator inertia. body. Thus, it is evident that under road stochastic Note that in equations (3) the non-linear term due to fluctuations, the servo-valve will direct the flow to the flow expression and the non-differentiable sign the actuator in either ways, depending if the car is function that stands for changing in motion direction, crossing a bump or a pothole. Here, this method will are at the origin of such systems complexity. be used with the system model (5) for position control and appropriate modification will be brought Finally choosing, to account for a positive-negative varying reference. x1 = θ , x2 = ω = θ& , x3 = PL , x4 = Av as state We denote by ei = xi − xid for i = 1,...,4 the error variables; the system can be easily described with a between each state variable and its desired trajectory. 4th order non-linear state space model. Let us choose a candidate lyapunov function defined by, x1 = x2 2 & ρ1e1 V1 = (6) x&2 = wa x3 − wbx2 − wc 2 x&3 = pa x4 Ps − x3.sign(x4 ) − pbx3 − pc x2 (5) Then its derivative is given by, x&4 = −ra x4 + rbu y = x 1 V&1 = ρ1e1 (x&1 − x&1d ) = ρ1e1 (e2 + x2d − x&1d )
Where ra, rb, pa, pb, pc, wa, wb, wc, are appropriate Thus, taking constants given by, x2d = x&1d − k1e1 (7) Renders, 1 K 2βCd 2βC L 2 r = , r = , p = , p = , V&1 = −k1 ρ1e1 + ρ1e1e2 (8) a τ b τ a b V v v V ρ In a second step we shall take, 2βDm Dm B TL pc = , wa = , wb = , wc = 2 V J J J ρ 2e2 V = V + (9) 2 1 2
272 POSITION CONTROL OF AN ELECTRO-HYDRAULIC SERVOSYSTEM - A non-linear backstepping approach
And its derivative,
V&2 = V&1 + ρ 2e2e&2 2 = −k1ρ1e1 + e2 [ρ1e1 − ρ 2 wb x2 + ρ 2 wa e3
+ ρ 2 wa x3d − ρ 2 wc − ρ 2 x&2d ] Figure 2: Figure 3: sign function sigmoid function Here, taking
w ρ 1 k w b 1 2 c 1 ρ p k x3d = x 2 − e1 + x&2d − e2 + u = r x + x − 3 a e P − sign(x )x − 4 e wa ρ 2 wa wa ρ 2 wa wa a 4 &4d 3 s 4 3 4 rb ρ 4 ρ 4 (10) (16) Will give, We should get the following result, 2 2 V&2 = −k1ρ1e1 − k2e2 + ρ2wae2e3 (11) 2 2 2 2 V&4 = −k1ρ1e1 − k2e2 − k3 ρ3e3 − k4e4 < 0 (17) Next, consider ρ e2 Unfortunately, this is not the case because the V = V + 3 3 (12) 3 2 2 control signal u is a discontinuous signal, which involves the derivative of x4d that contains a sign(x4) Its derivative is given by, function. Thus it is impossible to generate such control signal. For that reason and to remedy this
problem, we introduce the sigmoid function defined V&3 = V&2 + ρ3e3e&3 by (see figure 2 and 3), 2 2 = −k1 ρ1e1 − k 2e2 −ax +e3[ρ2wae2 + ρ3(− pc x2 − pb x3 1− e sigm(x) = (18) −ax + pa Ps − sign(x4 )x3 (e4 + x4d ) − x&3d )] 1+ e By choosing, This is a continuously differentiable function with
ρ2wa the following properties, ( pc x2 + pb x3 + x&3d − e2 − k3e3) ρ3 x4d = (13) a > 0 pa Ps − sign(x4 )x3 1 if ax → ∞ We have, sigm(x) = 0 if x = 0 (19) 2 2 2 V&3 = −k1 ρ1e1 − k 2 e2 − k3 ρ 3e3 -1 if ax → −∞ + ρ p e e P − sign(x )x (14) 3 a 3 4 a 4 3 and,
Finally we consider, dsigm(x) 2ae −ax ρ e 2 = (20) V = V + 4 4 (15) dx (1+ e −ax ) 2 4 3 2
Next we derive equation (15) and obtain, Note, that the rate at which sigm(x) converges to 1 or –1 depends on the slope ‘a/2’. V&4 = V&3 + ρ4e4e&4 Now we can rewrite the equation (13) as, 2 2 2 = −k1 ρ1e1 − k 2 e2 − k3 ρ 3e3 ρ2wa +e [ρ (−r x + r u) − ρ x ( pcx2 + pbx3 + x&3d − e2 − k3e3) 4 4 a 4 b 4 &4d ρ x = 3 (21) p e P sign(x )x ] 4d + ρ3 a 3 s − 4 3 pa Ps − sigm(x4)x3
By setting the control u as stated by equation (16) Therefore we can solve for control signal u in (16) by
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Figure 5: Control signal
equation (22), which drives the servo-valve spool x4 Figure 4: Position vs. desired position and to the convenient positions, far from saturating tracking error amplitudes. Once desired position reached, x4 holds at zero hence no oil will flow to the actuator. 1 ρ p k 3 a 4 u = ra x4 + x&4d − e3 Ps − sigm(x4 )x3 − e4 Next we will consider the case of a ‘sine function’ rb ρ4 ρ4 for the desired position with 1 amplitude and 2.5 (22) rad/s frequency. Hence at zero crossing the valve That gives, spool will have to move in the negative direction as well as the actuator. Figure 6 and 7 shows the V& = −k ρ e2 − k e2 − k ρ e2 − k e2 < 0 comparison of the control signal when the model in 4 1 1 1 2 2 3 3 3 4 4 (5) is simulated with the sign function and the sigmoid function respectively. It is obvious in the We can state now, that V&4 < 0 for every ei ≠ 0 , first case, that when the actuator changes its thus we conclude that the control law found in (22) direction the discontinuous sign term gives infinite renders the system globally asymptotically stable. control amplitude. While, a smooth control is obtained with the proposed sigmoid function. In the next section we present the simulation of the Furthermore, as seen from figure 8 the sigmoid controlled system. converges exponentially towards a sign with constant a = 2500. Finally as seen from figure 9; the backstepping used with the new system model, 4 SIMULATIONS AND RESULTS achieves a perfect tracking of the desired position. This approach is compared with the results of a We shall present in this section, simulation results to classical PID controller based on pole placement. It reveal the backstepping efficiency and robustness in is obvious; the PID fails to achieve good tracking, all cases (see appendix for control parameters which results in large steady state error. values). We conclude that, the backstepping effectiveness is First, let us choose a desired trajectory for position not affected by the approximation we made. The that tends to a constant, latter allows a smooth continuous control signal as in figure 7, which justify at the end of our work the t choice for this approach and the proposition we − brought to the system model. tr x1d = x1 f 1− e > 0 (23) 5 CONCLUSION Thus, x1d → x1 f when t → ∞ , with a time constant tr. Where we set x1f = 0.25rad and tr = 0.1 sec. In this paper we studied the position control of an electro-hydraulic servo-system. The control law we Graphics are given in figures 4 and 5. Figure 4, established is based on the non-linear backstepping. shows how the system output x1 follows the desired Our goal was to account for the system non-linearity trajectory with an excellent transient state and a zero and non-differentiability, and to show how system tracking error in steady state. On the other hand stability is globally guaranteed. We saw how figure 5 shows a smooth control signal generated by mathematical model non-differentiability prohibits
274 POSITION CONTROL OF AN ELECTRO-HYDRAULIC SERVOSYSTEM - A non-linear backstepping approach
Figure 7: Control signal when sigmoid is used
Figure 6: Control signal and actuator speed when sign function is used
Figure 8: sigm(x4) compared to sign(x4)
Figure 9: Position vs. desired position and tracking error successful control. Introducing the sigmoid function for backstepping and PID brought solution to the non-differentiable aspect and gave the model a smoother expression allowing Hassan, K. (2002). Non-Linear Systems (Third ed.): successful control via the same strategy. Comparison Prentice Hall, Upper Saddle River NJ07458. with classical PID showed the effectiveness of this Kandil, N., LeQuoc, S., & Saad, M. (1999). On-Line control strategy, as it ensures perfect tracking with Trained Neural Controllers for Nonlinear Hydraulic small transient and steady state error. Our future System. 14th World Congress of IFAC, 323-328. work consists of real time implementation of this LeQuoc, S., Cheng, R. M. H., & Leung, K. H. (1990). control law to reveal its effectiveness and bring Tuning an electrohydraulic servovalve to obtain a high improvements if necessary. We are looking as well amplitude ratio and a low resonance peak. Journal of to industrial applications of electro-hydraulic Fluid Control, 20(3Mar), 30-49. systems, especially in cars industry. Our fields of Lim, T. J. (1997). Pole placement control of an interests are basically electro-hydraulic active electrohydraulic servo motor. Paper presented at the suspension control and power transmission control Power Electronics and Drive Systems, 1997. systems, for better ride quality. Those are the most Proceedings., 1997 International Conference on. competent nowadays applications for hydraulic Plahuta, M. J., Franchek, M. A., & Stern, H. (1997). servo-systems; especially that road means of Robust controller design for a variable displacement transports are facing a great competition on behalf of hydraulic motor. American Society of Mechanical the air and maritime transportations means. Engineers, The Fluid Power and Systems Technology Division (Publication) FPST. Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition, Nov 16-21 1997, 4, 169-176. REFERENCES Ursu, I., F. Popescu, & Ursu, F. (2003, May 29-31,). Control synthesis for electrohydraulic servo Fink, A., & Singh, T. (1998). Discrete sliding mode mathematical model. Paper presented at the controller for pressure control with an Proceedings of the CAIM 2003, the 11th International electrohydraulic servovalve. Paper presented at the Conference on Applied and Industrial Mathematics, control Applications, 1998. Proceedings of the 1998 Oradea, Romania. IEEE International Conference on.
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Ursu, I., & Popescu, F. (2002, October 11-13,). APPENDIX Backstepping control synthesis for position and force nonlinear hydraulic servoactuators. Paper presented at Table 1: Hydraulic servo-system parameters the Book of Abstracts of the CAIM 2002, 10th International Conference on Applied and Industrial Servo-valve time constant τ 3.18x10-3 sec Mathematics,. Pitesti, Romania, October 11-13, p. 16. v
Yongqian, Z., LeQuoc, S., & Saad, M. (1998). Nonlinear 2 fuzzy control on a hydraulic servo system. Paper Servo-valve amplifier gain K 0.0397cm /V presented at the American Control Conference, 1998. Proceedings of the 1998. Flow discharge coefficient Cd 0.63 Yu, W.-S., & Kuo, T.-S. (1996). Robust indirect adaptive control of the electrohydraulic velocity control Fluid bulk modulus β 7.995x103 daN/cm2 systems. IEE Proceedings: Control Theory and Applications, 143(5), 448-454. 3 Actuator chamber volume V 135.4 cm Zeng, W., & Hu, J. (1999). Application of intelligent PDF
control algorithm to an electrohydraulic position servo 2 system. IEEE/ASME International Conference on Supply pressure PS 68.94 daN/cm Advanced Intelligent Mechatronics, AIM Proceedings of the 1999 IEEE/ASME International Conference on Fluid mass density ρ 85 g/cm3 5 Advanced Intelligent Mechatronics (AIM '99), Sep 19- 981x10 5 Sep 23 1999, 233-238. Leakage coefficient CL 0.09047 cm /daN.s
Actuator displacement D 2.802 cm3/rad m
Viscous damping coeff. B 0.766 daN.s.cm
Actuator Inertial load J 0.0481 daN.cm.s2
Actuator load torque TL 11.2 daN.cm
Table 2: Control parameters
k1 10
k2 2.5
k3 900
k4 800
ρ 400 1
ρ 2 0.05
ρ3 0.003
ρ 1700 4
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