IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL 56, NO 9, SEPTEMBER 2009 3523 Robust Trajectory Tracking for an Electrohydraulic Actuator Alexander G Loukianov, Jorge Rivera, Yuri V Orlov, and Edgar Yoshio Morales Teraoka Abstract—Various robust control techniques, such as integralblock, sliding-mode, and H -infinity controls, are combined to design a controller, forcing an electrohydraulic actuator which is driven by a servovalve to track a chaotic reference trajectory This approach enables one to compensate the inherent nonlinearities of the actuator and reject matched external disturbances and attenuate mismatched external disturbances The capabilities of the approach are illustrated in a simulation study Index Terms—Electrohydraulics, H -infinity control, variable structure systems I I NTRODUCTION N UMEROUS synchronization problems that come from practice are typically considered as tracking problems In this paper, such an interpretation of a synchronization problem is exemplified with robust chaotic tracking for an electrohydraulic actuator Nowadays, electrohydraulic actuators are very important tools for industrial processes This is mainly due to their fast response and great power-supply capacity with respect to the mass or volume they occupy However, controlling electrohydraulic systems presents a formidable problem since their dynamics are highly nonlinear Therefore, the investigation of the position or force control for electrohydraulic actuators is of great interest from both academic and industrial perspectives This paper is motivated by coffee-harvest automation, where electrohydraulic actuators seem very useful for shaking the tree branches In particular, the shaking action, being viewed a chaotic system, is very attractive for producing a broadband spectrum Various control techniques, including traditional proportional-integral differential (PID) controllers [1], recursive Lyapunov designs [2], adaptive neural-network-based controllers [3], and partial feedbacks [4], have been used to control Manuscript received January 15, 2008; revised November 5, 2008 First published November 25, 2008; current version published August 12, 2009 This work was supported by CONACYT México under Project 46069 A G Loukianov is with the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, C.P 45091, México (e-mail: louk@gdl.cinvestav.mx) J Rivera is with the Departamento de Electrónica, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, C.P 44430, México (e-mail: jorge.rivera@cucei.udg.mx) Y V Orlov is with the Department of Electronics and Telecomunications, Mexican Scientific Research and Advanced Studies Center of Ensenada, Carretera Tijuana-Ensenada, B.C 22860, México (e-mail: yorlov@cicese.mx) E Y Morales Teraoka is with the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, C.P 45091, México, and also with Tohoku University, Sendai 980-8577, Japan (e-mail: mty7810@gmail.com) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org Digital Object Identifier 10.1109/TIE.2008.2010207 either the position or force of a hydraulic actuator driven by a servovalve On the other hand, a fruitful and relatively simple approach, particularly when dealing with nonlinear plants subjected to perturbations, is based on the use of a variable structure control technique with sliding mode (SM) [5]–[10] A sliding mode controller was proposed in [11] to control the hydraulic actuator force In this case, the relative degree turns out to be two It is, however, well known that the SM controller ensures the robustness of the closed-loop system only with respect to matched perturbations The SM concept, combined with the block control (BC) technique [12], was shown to provide a robust stabilization of a nonlinear system in the presence of mismatched perturbations, and it was used to design a discontinuous controller for the hydraulic actuator in [13] Since the latter controller typically provides a conservative result for improving the performance of a nonlinear perturbed system, one can additionally apply an H∞ controller However, a real direct implementation of such a controller is hardly possible for high-order dynamic systems because of the numerical complexity of solving Riccati equations In order to reduce this complexity, the integral BC/SM technique [14] is used in this paper to decompose the original system into several subsystems of lower dimensions, and the nonlinear H∞ -control method is then separately applied to the resulting lower order subsystems The position tracking problem, where a prescribed trajectory to follow is chaotic, is under study The plant model is a nonlinear system that presents the dynamics of an external cylinder load (a spring and a damper in parallel), a friction model, and an approximation of the servovalve dynamics Although such a plant model is greatly simplified compared with the actual system dynamics, it captures all essential features of the real dynamics The present model is of the relative degree of four The proposed control approach is as follows The plant model is first represented in the nonlinear block-controllable form [12] In order to design a nonlinear sliding surface, the BC technique, combined with an integral element [14], is then used to linearize the nominal unperturbed part of the plant dynamics and reject an unknown constant part of the mismatched perturbation in each block of the system After that, the H∞ control approach [15] is separately applied to each block to attenuate the mismatched disturbances in the SM dynamics A discontinuous control strategy, ensuring the stability of the SM, is finally proposed under an a priori known upper bound on the control signal Due to the nature of the proposed combined BC/SM/H∞ approach, the resulting controller is expected to yield the desired robustness properties against both matched and mismatched unknown disturbances with a priori known 0278-0046/$26.00 © 2009 IEEE Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply 3524 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL 56, NO 9, SEPTEMBER 2009 where PL is the load pressure, Vt is the total actuator volume, βe is the effective bulk modulus, Ctm is the coefficient of total leakage due to pressure, and QL is the turbulent hydraulic fluid flow through an orifice The relationship between spool-valve displacement xv and the load flow QL is given by QL = Cd wxv Fig Piston with load norm bounds Such a combination results in solving a finite number of reduced-order Riccati equations rather than solving a full-order Riccati equation The state measurements are assumed to be available, throughout The complete mathematical model used to describe the behavior of the electrohydraulic actuator consists of the dynamics of a hydraulic actuator disturbed by an external load and the dynamics of a servovalve This model is naturally separated into three parts, namely, the mechanical, hydraulic, and servovalve subsystems In what follows, these parts are briefly described The piston disturbed by an external load, being modeled as a spring and a damper in parallel attached to the piston, is shown in Fig By applying Newton’s law, the system dynamics are derived in the form (1) where xp is the piston position, vp = dxp /dt is the piston velocity, a is the acceleration of the piston, fi represents the acting forces, PL is the load pressure, Fr is the internal friction of the cylinder, m is the actuator mass, ks is the load spring stiffness, bd is the load viscous damping, and Λa is the piston area Introducing the state variables x1 = xp , x2 = vp , and x3 = PL and adding to (1) an unknown force M (t) as a perturbation, the first two state space equations of the plant are thus given by x˙ = x2 x˙ = (−ks x1 − bd x2 + Λa x3 − Fr (x2 ) − M (t)) m where Cd is the valve discharge coefficient, w is the spool-valve area gradient, P s is the supply pressure, and ρ is the hydraulic fluid density The spool area gradient for a cylindrical spool can be approximated simply as the circumference of the valve at each port Combining (3) and (4) results in the load pressure state equation + P s−sgn(xv )PL ρ By setting x4 = xv , the latter equation takes the form x˙ = −αx2 − βx3 + γx4 P s − sgn(x4 )x3 (5) (2) α = (4Λa βe /Vt ) β = (4Ctm βe /Vt ) γ = (4Cd wβe /Vt ) 1/ρ C Servovalve Subsystem By applying frequency response analysis and an HP Digital Dynamic Analyzer [17] to servovalve dynamics, it is established that the following second-order linear model: 2.4315 × 105 s2 + 6.2529 × 102 s + 2.5676 × 105 1/τ x4 (s) = Ka u(s) s + (1/τ ) where u is the control input, τ = 1/573 s−1 is the time constant, and Ka > Then, the dynamics of the servovalve subsystem can be approximated as Ka x˙ = − x4 + u τ τ The dynamics of the cylinder are derived in [16] for a symmetric actuator Defining the load pressure PL to be the pressure across the actuator piston, the derivative of the load pressure is given by the total load flow through the actuator divided by the fluid capacitance (3) (6) matches the measured frequency response B Hydraulic Subsystem Vt ˙ PL = −Λa x˙ p − Ctm PL + QL 4βe 4βe Cd wxv Vt with the following constant parameters: A Mechanical Subsystem fi = −ks xp − bd vp + Λa PL − Fr (v) (4) 4βe P˙L = (−Λa v − Ctm PL ) Vt II M ATHEMATICAL M ODEL ma = P s − sgn(xv )PL ρ (7) Then, using (2), (5), and (7), we obtain the plant model as x˙ = x2 x˙ = (−ks x1 − bd x2 + Λa x3 − w(x2 , t)) m x˙ = − αx2 − βx3 + γ P s − sgn(x4 )x3 x4 x˙ = − Ka x4 + u τ τ Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply (8) LOUKIANOV et al.: ROBUST TRAJECTORY TRACKING FOR AN ELECTROHYDRAULIC ACTUATOR where x = (x1 , , x4 )T , w(x2 , t) = Fr (x2 ) + M (t), and the control input is bounded by |u| < u0 (9) 3525 where v1 is an auxiliary virtual control (or the second part of the virtual control x3 ) which is to be designed to reject the unknown perturbation w(z2 , t) Thus, substituting (13) in the last block (12) yields z˙2 = −a21 z1 − a22 z2 − m01 z01 + v1 + g2 w(z2 , t) with u0 as a positive constant III C ONTROLLER D ESIGN Assuming that all the state variables are available for measurement, the control problem to be addressed is as follows The position of the actuator x1 is to track a reference trajectory r(t) generated by the Chen chaotic attractor [18] and the subsystem (12), (14), with the state vector ξ1 = (z01 , z1 , z2 )T , is presented in the form ξ˙1 = A1 ξ1 + B11 w(z2 , t) + B12 v1 ⎡ z(t) ˙ = r(t)y(t) − bz(t) ⎣ −m01 ⎡ ⎤ ⎣0⎦ ⎡ ⎤ ⎣0⎦ g2 ⎡ ⎤ ⎣1⎦ A1 = (10) where a, b, and c are constant parameters In the sequel, we present a control algorithm, solving the problem B12 = A Nonlinear Sliding Surface B11 = The sliding surface design procedure consists of three steps Step 1: The position tracking error C1T = z1 = x1 − r(t) is first introduced as a new variable, with z2 as its derivative, i.e., ˙ z2 = z˙1 = x2 − r(t) −a21 ⎤ ⎦ −a22 To achieve the robustness property of the subsystem (15) against the disturbance w(z2 , t), the optimal robust control technique [15] is applied The desired value v1d of the virtual control v1 in (15) is thus chosen to be in the form T Pε,1 ξ1 = −k01 z01 − k1 z1 − k2 z2 v1d = −B12 Next, the integral block t z01 = z1 dt (11) −∞ (15) where r(t) ˙ = a[y(t) − r(t)] y(t) ˙ = (c − a)r(t) − r(t)z(t) + cy(t) (14) (16) where Pε,1 is a positive definite solution of the following Riccati equation: T = Pε,1 A1 + AT Pε,1 + C1 C1 + ε1 I is designed After that, the first two blocks of (8), with the integrator (11), can be represented in the new variables z01 , z1 , and z2 as + Pε,1 T T B11 B11 − B12 B12 Pε,1 γ12 (17) with some positive ε1 and γ1 The parameters k01 , k1 , and k2 are thus determined by the solution of (17) Note that virtual control law attenuates the influence of the external disturbances w on the output z1 of the closed loop (15), (16) in the sense that the inequality z˙01 = z1 z˙1 = z2 z˙2 = − a21 z1 − a22 z2 + ¯b2 x3 + d¯2 (t) + g2 w(z2 , t) (12) ∞ ∞ where a21 = ks /m; a22 = bd /m; ¯b2 = Λa /m; d2 (t) = ră(t); g2 = 1/m; and w(z2 , t) = (−ks /m)r(t)−(bd /m)r(t)− w(x2 , t)|x2 =z2 +r(t) ˙ Following the BC technique [12], the variable x3 , considered in (12) as a virtual control, is selected so as to compensate the known perturbation term d¯2 (t) and introduce the new term −m01 z01 with a design parameter m01 > holds for all square integrable disturbances w and some attenuation level γ1 > The design parameter m01 is additionally utilized to alleviate solving the Riccati equation (17) If the new variable −d¯2 (t) − m01 z01 + v1 x3 = ¯b−1 z3 = v1 − v1d (13) z12 dt ≤ γ12 w2 dt Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply (18) 3526 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL 56, NO 9, SEPTEMBER 2009 is introduced, then substituting v1 = ¯b2 x3 + d¯2 (t) + m01 z01 (13) and (16) in (18) results in the following nonlinear transformation: z3 = ¯b2 x3 + d¯2 (t) + (m01 + k01 )z01 + k1 z1 + k2 z2 (19) where ξ3 = (z03 , z3 )T ; A3 = −m03 ; B31 = −m3 k g2 ; Using the solution Pε,3 of the Riccati equation [this equation is similar to (17)] B32 = which is equivalent to ; and C3T = T = Pε,3 A3 + AT Pε,3 + C3 C3 + ε3 I ¯b2 x3 = −d¯2 (t) − (m01 + k01 )z01 − k1 z1 − k2 z2 + z3 (20) + Pε,3 By substituting (20) in (12), one derives T T B31 B31 − B32 B32 Pε,3 γ32 (26) the desired value v3d of the virtual control v3 is defined by z˙01 = z1 T v3d = −B32 Pε,3 ξ3 = −k03 z03 − k3 z3 z˙1 = z2 z˙2 = − (m01 + k01 )z01 − (a21 + k1 )z1 − (a22 + k2 )z2 + z3 + g2 w(z2 , t) Step 2: Using the nonlinear transformation (19), a straightforward algebra reveals z˙3 = f¯3 (ξ1 , z3 ) + ¯b3 (x)x4 + d¯3 (t) + k2 g2 w(z2 , t) (21) (27) where k03 and k3 are positive parameters Introducing the new variable z4 = v3 − v3d (28) and substituting v3 = ¯b3 (x)x4 + f¯3 (ξ1 , z3 )+ d¯3 (t)+m03 z03 + m3 z3 (22) and (27) in (28) yield z4 = ¯b3 (x)x4 + f¯3 (ξ1 , z3 ) + d¯3 (t) where + (k03 + m03 )z03 + (k3 + m3 )z3 f¯3 (ξ1 , z3 ) = − ((k2 − β)(m01 + k0 )) z01 − (−βk1 − k01 − m01 + k2 (a21 + k1 )) z1 − ¯b2 α − βk2 − k1 + k2 (a22 + k2 ) z2 z˙03 = z3 − (β − k2 )z3 ¯b3 (x) = ¯b2 γ Then, combining (23) and (24) with (27) and (28), one derives z˙3 = − (m03 +k03 )z03 − (k3 +m3 )z3 +k2 g2 w(z2 , t)+z4 P s − sgn(x4 )x3 Step 3: Since the function ¯b3 (x) = ¯b2 γ P s − sgn(x4 )x3 is discontinuous, we select the sliding variable s to depend on the signum of x4 d¯3 (t) = − a21 βr − (¯b2 α + a22 β + a21 )r˙ − (β + a22 )ă r ăr Let us now choose the virtual control x4 in (21) to cancel the dynamic term f¯3 (ξ1 , z3 ) + d¯3 (t) and introduce the new dynamics −m03 z03 − m3 z3 , i.e., t z3 dt −∞ + d¯3 (t) + ¯b+ (x)x4 + d¯3 (t) + ¯b− (30) (x)x4 √ √ ¯ ¯− ¯ where ¯b+ (x) = b2 γ P s − x3 and b3 (x) = b2 γ P s + x3 Thus, the closed-loop system z˙3 = − m03 z03 − m3 z3 + v3 + k2 g2 w(z2 , t) s˙ = (23) (24) Being represented in matrix form, the latter equations are as follows: ξ˙3 = A3 ξ3 + B31 w(z2 , t) + B32 v3 (25) (29) z4− = f¯3 (ξ1 , z3 ) + (k03 + m03 )z03 + (k3 + m3 )z3 and substituting (22) in (21) results in z˙03 = z3 for x4 > for x4 < z4+ = f¯3 (ξ1 , z3 ) + (k03 + m03 )z03 + (k3 + m3 )z3 (22) where v3 is an auxiliary virtual control (or the second part of the virtual control x4 ) and m03 and m3 are design parameters Introducing the integral z03 = z4+ , z4− , with ¯ ¯ x4 = − ¯b−1 (x) f3 (ξ1 , z3 ) + d3 (t) + ¯b−1 (x)(−m03 z03 − m3 z3 + v3 ) s= ¯ f¯4 (ξ, z4 ) + ¯b+ (x)u + d4 (t), − ¯ ¯ ¯ f4 (ξ, z4 ) + b4 (x)u + d4 (t), for x4 > for x4 < (31) with ¯ ¯b+ (x) = γ b2 Ka τ P s − x3 , ¯ ¯b− (x) = γ b2 Ka τ P s + x3 where ξ = (ξ1 , ξ3 )T , f¯4 (ξ, z4 ), and d¯4 (t) = d(d¯3 (t))/dt are continuous functions Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply LOUKIANOV et al.: ROBUST TRAJECTORY TRACKING FOR AN ELECTROHYDRAULIC ACTUATOR B Discontinuous Control where Taking into account (9), the discontinuous control law is defined as −u0 ¯b+ (x) −u0 ¯b− (x) u= −1 −1 sign z4+ , for x4 > sign z4− , for x4 < (32) In the rest of the section, the stability analysis is presented To prove the stability of the closed-loop subsystem (31) with (32), we first consider x4 > and a Lyapunov function candidate such as V+ = + z Then V˙ + = z4+ f¯4 (ξ, z4 ) − u0 sign z4+ + d¯4 (t) ≤− z4+ 3527 T Pε,1 A1 − B12 B12 ⎡ = ⎣ −(m01 + k01 ) ⎡ ⎤ B11 = ⎣ ⎦ g2 ⎡ ⎤ B12 = ⎣ ⎦ ⎡ ⎤ 0 A13 = ⎣ 0 ⎦ T A3 − B32 B32 Pε,3 u0 − f¯4 (ξ, z4 ) + d¯4 (t) −(m03 + k03 ) = For x4 < 0, we have V − V˙ − B31 = − z = = z4− f¯4 (ξ, z4 ) − u0 sign z4− + d¯4 (t) ≤ − z4− ⎤ ⎦ −(a21 + k1 ) −(a22 + k2 ) −(m3 + k3 ) k2 g2 Using the Lyapunov function u0 − f¯4 (ξ, z4 ) + d¯4 (t) ξiT Pε,i ξi Vs = i=1,3 Hence, under the following assumption: and calculating its derivative along solutions to (34) and (35) yield u0 > f¯4 (ξ, z4 ) + d¯4 (t) + δ where δ √ > 0, both functions V + and V − meet the condition ˙ V ≤ −δ 2V , which is well known [5], to ensure that the state vector of the closed-loop system reaches the sliding surface s = in a finite time V˙ s = T Pε,i ξiT Ai − Bi2 Bi2 T Pε,i ξi + ξiT Pε,i i=1,3 T × Ai − Bi2 Bi2 Pε,i ξi + 2ξiT Pε,i Bi1 w(z2 , t) + 2ξ1T Pε,1 A13 ξ3 (36) C SM Dynamics Once achieved, the sliding motion on s = is governed by the SM equation [5] From the structure of (34) and (35), it follows that Bi2 = qi Bi1 , i = 1, z˙01 = z1 with q1 = z˙1 = z2 z˙2 = − (m01 + k01 )z01 − (a21 + k1 )z1 − (a22 + k2 )z2 z˙03 = z3 z˙3 = − (m03 + k03 )z03 − (k3 + m3 )z3 + k2 g2 w(z2 , t) (33) where the parameters k0 , k1 , k2 , k03 , and k3 are defined by the solutions Pε,1 and Pε,3 to the appropriate Riccati equations To analyze stability, (33) is represented as ξ˙1 = A1 − T B12 B12 Pε,1 ξ1 + B11 w(z2 , t) + A13 ξ3 T ξ˙3 = A3 − B32 B32 Pε,3 ξ3 + B31 w(z2 , t) (34) (35) (37) Using (37), we rewrite (17) and (26) as T Pε,i Ai − Bi2 Bi2 + g2 w(z2 , t) + z3 1 and q3 = g2 k2 g2 T T Pε,i + Pε,i Ai − Bi2 Bi2 Pε,i T T = −CiT Ci − εi I − Pε,i Bi1 Bi1 Pε,i − Pε,i Bi2 Bi2 Pε,i γi = −CiT Ci − εi I − T + qi2 Pε,i Bi1 Bi1 Pε,i , γi2 i = 1, (38) To estimate the crossing term 2ξ1T Pε,1 A13 ξ3 in (36), we use the following relation: X T Y + Y T X ≤ X T RX + Y T R−1 Y, Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply R > 3528 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL 56, NO 9, SEPTEMBER 2009 Set X = Pε,1 ξ1 ; Y = A13 ξ3 ; and R = I2 , and then 2ξ1T Pε,1 A13 ξ3 ≤ ξ1T Pε,1 Pε,1 ξ1 + ξ3T AT 13 A13 ξ3 (39) Define, now, the region ξ1 < r1 , r1 > such that the perturbation term w(z2 , t) is bounded by w(z2 , t) ≤ b1 ξ1 + b2 , with b1 > and b2 > 0∀ ξ1 < r1 (40) T Setting ωi = Bi1 Pε,i ξi , i = 1, and then substituting (38), (39), and (40) in (36) result in V˙ s ≤ −ξiT CiT Ci ξi − εi ξiT ξi − i=1,3 2(R13) + qi γi2 T × ξiT Pε,i Bi1 Bi1 Pε,i ξi + ξiT Pε,i Bi1 Fig w(z2 , t) + 2ξ1T Pε,1 A13 ξ3 −ci ξi ≤ − εi ξi + 2b1 ξ1 2 − 2(R13) + qi γi2 ωi Hence, a solution ξi (t) is ultimately bounded by [22] ξi (t) ≤ ¯bi , ¯bi = μi + d3 ξ3 (41) i=1,3 for d3 ≤ c3 (42) , b1 − qi2 i = 1, (43) where ci = λmin (CiT Ci ); i = 1, 3; d1 = λmax (Pε,1 Pε,1 ); and d3 = λmax (AT 13 A13 ) The condition (42) is satisfied [see the parameters of (34) and (25)], while the condition (43) for given d1 , c1 , and q1 can be achieved by appropriately choosing (or adjusting) the region ξ1 < r1 in (40) and the parameters γ1 and γ3 in the Riccati equations (17) and (26) Further, from the definition of ωi , it follows that ωi ≤ αi ξi , with αi = TP λi max Pε,i Bi1 Bi1 ε,i (44) Taking into account (44), (41) yields V˙ s ≤ −εi ξi + 2b2 αi ξi − εi θi ξi + εi θi ξi i=1,3 = − εi (1 − θi ) ξi − εi (1 − θi ) ξi − (εi θi ξi − 2b2 αi ) ξi i=1,3 ≤ (45) i=1,3 provided that εi θi ξi − 2b2 αi > 0, λmax (Pε,i ) , λmin (Pε,i ) i = 1, + 2b2 ωi 2b1 + d1 ≤ c1 and γi ≤ 2b2 αi = μi εi θ i + 2b2 ωi + d1 ξ1 −εi ξi ≤ Therefore, the derivative (45) is negative definite for all ξi > i=1,3 + b1 ω i Block diagram of the closed-loop system < θi < 1, i = 1, On the other hand, from the H∞ -optimal-control methodology [15], it follows that the feedback, which is designed based on proper solutions of the Riccati equations (17) and (26), attenuates the influence of the external disturbances w on the output error z1 , z1 = x1 − r(t) Finally, a closed-loop block diagram is shown in Fig IV S IMULATIONS Although for the control law design we used a fourth-order model (8), in order to simulate the closed-loop system, we consider a servovalve model as a second-order transfer function (6) Hence, the plant state space is given by x˙ = x2 x˙ = (−ks x1 − bd x2 + Λa x3 − w(x2 , t)) m x˙ = − αx2 − βx3 + γ P s − sgn(x4 )x3 x4 x˙ = x5 x˙ = − an x4 − bn x5 + cn u where an , bn , and cn are suitable constants computable with (6) The unmodeled servovalve dynamics can cause switching oscillations Such oscillations are named “chattering” [5] The chattering comes out as low control accuracy, vibrations in mechanical parts, and undesirable heat losses in electric power circuits To avoid the chattering, the following observer is designed: Ka ˆ4 + u + l1 (x4 − x ˆ4 ) x ˆ˙ = − x τ τ Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply LOUKIANOV et al.: ROBUST TRAJECTORY TRACKING FOR AN ELECTROHYDRAULIC ACTUATOR 3529 where the positive parameter l1 is chosen such that the estimate error system + l1 x ¯4 τ x ¯˙ = − is stable Thus, the resulting controller is −u0 −u0 u= ¯b+ (x) ¯b− (x) −1 −1 sign zˆ4+ , sign zˆ4− , for x4 > for x4 < ˆ4 ) (29) and zˆ4− = η − (ξ1 , z03 , z3 , where zˆ4+ = η + (ξ1 , z03 , z3 , x x ˆ4 ) (30) The performance of the controller constructed in the previous section is now observed in simulations Robustness features of the closed-loop system are studied through disturbing the electrohydraulic actuator by the signal w(t) = 1500 + 240 sin(12t) + Fr (x2 ) where the used static friction model Fr [2] includes Karnopp’s stick–slip friction [19], [20] and the Stribeck effect [21] Fig Friction model used in the system model, including Karnopp and Stribeck models Fr (x2 ) = Fr,1 (x2 ) + Fr,2 (x2 ) + Fr,3 (x2 ) + Fr,4 (x2 ) (46) with Fr,1 (x2 ) = if x2 < −0.01, then −125e( 0.013 (x2 +0.01)) −100 else, it is zero Fr,2 (x2 ) = if x2 > −0.01 and x2 < 0.01 and Δx2 (δ) < x2 < 0.01 and Δx2 (δ) > then 225; else, it is zero Fr,3 (x2 ) = if x2 > −0.01 and then −225; else, it is zero Fr,4 (x2 ) = if x2 > −0.01 then + 125e(− 0.013 (x2 −0.01)) + 100 else, it is zero, where Δx2 (δ) = x2 (δk) − x2 (δ(k − 1)), i.e., the increment of x2 in discrete time A typical velocity–friction plot of such model is shown in Fig Regarding Karnopp’s friction, there are two key points, namely, a stick phase occurs when the velocity is within a small critical velocity range, and there is a maximum value for friction when the mass under consideration sticks The parameter values were taken from [13]: Parameter value m 24 kg; 16 010 N/m; ks 310 N/(m/s); bd 1.03 × 107 Pa; Ps 3.26 × 10−4 m2 ; Λa α 1.51 × 1010 N/m3 ; β s−1 ; √ √ γ 7.28 × 108 [ kg/(ms2 m)]; 0.947; ka τ 0.0017 s; 2.5676 × 105 ; an 6.2529 × 102 ; bn 2.4315 × 105 cn Fig Output tracking with control law designed in [13] ¯−1 Using these parameters, the value √ of b4 (x) is esti¯b4 (x) = (γ¯b2 Ka/τ ) P s ± x3 = (γΛa Ka/ mated as √ P s ± x = (7.28 · 108 ·√3.2 · 10−4 · 0.947)/(24 · 0.0017) mτ ) √ 1.03 · 107 ± x3 = 5.4 · 106 1.03 · 107 ± x3 From this, we can conclude that (¯b4 (x(t)))−1 < 1, which guarantees |u(t)| < u0 in (32) The parameters for the proposed controller are as follows: (m01 + k01 ) 133.33; 490.83; k1 38; k2 (m03 + k03 ) 0.423; 420.91; (k3 + m3 ) 6.5e7; u0 1; γ1 0.1; 119.22; γ3 7.63; 10; l1 δ 0.0001 Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply 3530 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL 56, NO 9, SEPTEMBER 2009 V C ONCLUSION A robust controller design is proposed for forcing the electrohydraulic actuator to approximately track desired chaotic position reference trajectories in the presence of both matched and mismatched disturbances The controller is developed on the basis of the BC SM approach and H∞ -control methodology Good performance of the closed-loop system is obtained in spite of strong disturbances affecting the system R EFERENCES Fig Output tracking with proposed control Fig Control action The proposed control law is compared with that obtained in [13], which was designed to force the electrohydraulic actuator to follow a chaotic reference signal r(t) in the presence of a constant disturbance, by means of high gains The reference trajectoryx(t) is generated by the Chen chaotic attractor (10) with a = 35, b = 3, and c = 28 The states r, y, and z of (10) are multiplied for a suitable constant (0.002) in order to meet the actuator amplitude restrictions Fig shows the trajectory of the electrohydraulic actuator driven by the controller from [13] Fig shows the actuator position under the control scheme proposed in this paper One can observe that the proposed controller is more robust to external disturbances and parameter variations compared with that of [13] because the tracking is more accurate under the presence of the same disturbance Finally, Fig shows the control action corresponding to the proposed discontinuous control law [1] A Alleyne and R Liu, “On the limitations of force tracking control for hydraulic active suspensions,” ASME J Dyn Syst., Meas Control, vol 121, no 2, pp 184–190, 1999 [2] A Alleyne and R Liu, “A simplified approach to force control for electrohydraulic systems,” Control Eng Pract., vol 8, no 12, pp 1347–1356, Dec 2000 [3] B Daachi, A Benallegue, and N K M’Sirdi, “A stable neural adaptive force controller for a hydraulic actuator,” in Proc Int Conf Robot Autom., 2001, vol 4, pp 3465–3470 [4] B Ayalew and K W Jablokow, “Partial feedback linearising forcetracking control: Implementation and testing in electrohydraulic actuation,” IET Control Theory Appl., vol 1, no 3, pp 689–698, May 2007 [5] V I Utkin, J Guldner, and J Shi, Sliding Modes in Electromechanical Systems London, U.K.: Taylor & Francis, 1999 ˘ [6] K Abidi and A Sabanovic, “Sliding-mode control for high-precision motion of a piezostage,” IEEE Trans Ind Electron., vol 54, no 1, pp 629–637, Feb 2007 [7] W F Xie, “Sliding-mode-observer-based adaptive control for servo actuator with friction,” IEEE Trans Ind Electron., vol 54, no 3, pp 1517– 1527, Jun 2007 [8] A V Topalov, G L Cascella, V Giordano, F Cupertino, and O Kaynak, “Sliding mode neuro-adaptive control of electric drives,” IEEE Trans Ind Electron., vol 54, no 1, pp 671–679, Feb 2007 ˘ [9] Y Yildiz, A Sabanovic, and K Abidi, “Sliding-mode neuro-controller for uncertain systems,” IEEE Trans Ind Electron., vol 54, no 3, pp 1676– 1685, Jun 2007 [10] A G Loukianov, J M Cañedo, V I Utkin, and J Cabrera-Vazquez, “Discontinuous controller for power systems: Sliding-mode block control approach,” IEEE Trans Ind Electron., vol 51, no 2, pp 340–353, Apr 2004 [11] M Jerouane and F Lamnabhi-Lagarrigue, “A new robust sliding mode controller for a hydraulic actuator,” in Proc 40th IEEE CDC, Orlando, FL, 2001, pp 908–913 [12] A G Loukianov, “Robust block decomposition sliding mode control design,” Math Probl Eng., vol 8, no 4/5, pp 349–365, 2002 [13] M A Avila, A G Loukianov, and E N Sanchez, “Electrohydraulic actuator trajectory tracking,” in Proc ACC, Boston, MA, 2004, pp 2603–2608 [14] O G Rios-Gastelum, B Castillo-Toledo, and A G Loukianov, “Nonlinear block integral sliding mode control: Application to induction motor control,” in Proc Conf Decisions Control, Maui, HI, 2003, vol 3, pp 3124–3129 [15] L T Aguilar, Y Orlov, and L Acho, “Nonlinear H∞ -control of nonsmooth time-varying systems with application to friction mechanical manipulators,” Automatica, vol 39, no 9, pp 1531–1542, Sep 2003 [16] H E Merritt, Hydraulic Control Systems New York: Wiley, 1967 [17] R Liu, Nonlinear Control of Electrohydraulic Servosystems, Theory and Experiment Beijing, China: Tsinghua Univ., 1994 [18] G Chen and T Ueta, “Yet another chaotic attractor,” Int J Bifurcation Chaos, vol 9, no 7, pp 1465–1466, Jul 1999 [19] C Garcia, “Comparison of friction models applied to a control valve,” Control Eng Pract., vol 16, no 10, pp 1231–1243, Oct 2008 [20] L Marton and B Lantos, “Modeling, identification, and compensation of stick–slip friction,” IEEE Trans Ind Electron., vol 54, no 1, pp 511–521, Feb 2007 [21] B Armstrong-Hélouvry, P Dupont, and C Canudas de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol 30, no 7, pp 1083–1138, Jul 1994 [22] H K Khalil, Nonlinear Systems Englewood Cliffs, NJ: Prentice-Hall, 1996 Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply LOUKIANOV et al.: ROBUST TRAJECTORY TRACKING FOR AN ELECTROHYDRAULIC ACTUATOR Alexander G Loukianov was born in Moscow, Russia, in 1946 He received the Dipl Eng degree from Polytechnic Institute, Moscow, Russia, in 1975 and the Ph.D degree in automatic control from the Institute of Control Sciences, Russian Academy of Sciences, Moscow, in 1985 He was with the Institute of Control Sciences in 1978, and was the Head of the Discontinuous Control Systems Laboratory from 1994 to 1995 In 1995–1997, he held a visiting position with the University of East London, London, U.K Since April 1997, he has been with the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, México, as a Professor of electrical engineering graduate programs In 1992–1995, he was in charge of an industrial project between his institute and the largest Russian car plant and also of several international projects supported by INTAS and INCO-COPERNICUS, Brussels He has published more than 90 technical papers in international journals and conferences and has served as a Reviewer for different international journals and conferences His research interests include nonlinear systems control and variable structure systems with sliding mode as applied to electric drives and power systems control, robotics, space, and automotive control Jorge Rivera was born in El Rosario, Sinaloa, México, in 1975 He received the B.Sc degree from the Technological Institute of the Sea, Mazatlán, México, in 1999 and the M.Sc and Ph.D degrees in electrical engineering from the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, México, in 2001 and 2005, respectively Since 2006, he has been with the Universidad de Guadalajara, Guadalajara, México, as a full-time Professor with the Departamento de Electrónica, Centro Universitario de Ciencias Exactas e Ingenierías, Guadalajara, Mexico His research interests include regulator theory, sliding-mode control, discrete-time nonlinear control systems, and their applications to electrical machines 3531 Yuri V Orlov received the M.S degree from the Mechanical–Mathematical Faculty, Moscow State University, Moscow, Russia, in 1979, the Ph.D degree in physics and mathematics from the Institute of Control Science, Moscow, in 1984, and the Dr.Sc degree in physics and mathematics from Moscow Aviation Institute, Moscow, in 1990 From 1979 to 1992, he was with the Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia He was also a part-time Professor with Moscow Aviation Institute from 1989 to 1992 He has been a Full Professor with the Department of Electronics and Telecommunications, Mexican Scientific Research and Advanced Studies Center of Ensenada, Ensenada, México, since 1993 His research interests include mathematical methods in control, analysis, and synthesis of nonlinear nonsmooth discontinuous time-delay distributed parameter systems and their applications to electromechanical systems Edgar Yoshio Morales Teraoka was born in Guadalajara, México, in 1978 He received the B.S degree in electromechanical engineering from Universidad Panamericana, Guadalajara, in 2001 and the M.S degree in electrical engineering from the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, Mexico He is currently working toward the Ph.D degree in electric and communications engineering at Tohoku University, Sendai, Japan Authorized licensed use limited to: CINVESTAV IPN Downloaded on January 29, 2010 at 15:48 from IEEE Xplore Restrictions apply