Robust Adaptive Backstepping Control Design for a Nonlinear ...
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeC10.5
Robust Adaptive Backstepping Control Design for a Nonlinear Hydraulic-Mechanical System M. Choux∗,∗∗ , H.R. Karimi∗ , G. Hovland∗ , M.R. Hansen∗ , M. Ottestad∗ and M. Blanke∗∗ Abstract— The complex dynamics that characterize hydraulic systems make it difficult for the control design to achieve prescribed goals in an efficient manner. In this paper, we present the design and analysis of a robust nonlinear controller for a Nonlinear Hydraulic-Mechanical (NHM) System. The system consists of an electrohydraulic servo valve and two hydraulic cylinders. Specifically, by considering a part of the dynamics of the NHM system as a norm-bounded uncertainty, two adaptive controllers are developed based on the backstepping technique that ensure the tracking error signals asymptotically converge to zero despite the uncertainties in the system according to the Barbalat lemma. The resulting controllers are able to take into account the interval uncertainties in Coulomb friction parameters and in the internal leakage parameters in the cylinders. Two adaptation laws are obtained by using the Lyapunov functional method and inequality techniques. Simulation results demonstrate the performance and feasibility of the proposed method. Fig. 1.
I. INTRODUCTION Hydraulics -the science of liquid flow- is a very old discipline, which has commanded new interest in last decades, especially in the area of hydraulic control, and fills a substantial portion of the field of control. The dynamics of hydraulic systems are highly nonlinear. Furthermore, the system may be subjected to non-smooth and discontinuous nonlinearities due to control input saturation, directional change of valve opening, friction, and valve overlap. Hydraulic control components and systems are found in many mobile, airborne, and offshore applications [19]. In some applications, the main emphasis is placed on valves for servo control. Hydraulic control valves are devices that use mechanical motion to control a source of fluid power. They vary in arrangement and complexity, depending upon their function. Because control valves are the interface from mechanical (electrical) to fluid in hydraulic systems, their performance is under scrutiny, especially when system difficulties occur. During the last decade, backstepping based designs have emerged as powerful tools for stabilizing nonlinear systems for tracking and regulation purposes [16]. The main advantage of these designs is the systematic construction of a Lyapunov function for the closed loop, allowing the analysis of its stability properties. The adaptive version of these designs, especially the tuning functions design, offers the possibility to synthesize in a systematic way controllers for a wide class of nonlinear systems (those under the strictfeedback form) whose structure is known but with unknown ∗ Faculty of Technology and Science, The University of Agder, Serviceboks 509, N-4898 Grimstad, Norway. ∗∗ Department of Electrical Engineering, Technical University of Denmark, Kgs. Lyngby, Copenhagen, Denmark.
[email protected] or [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
A nonlinear hydraulic-mechanical (NHM) system.
parameters ([9], [10], [16], [17], [18], [29], [30]). They also offer the possibility to analyze transient behavior of the closed loop in the absence of uncertainties. Despite the fact that the robustness of the tuning functions design has been studied extensively for linear systems ([7], [8]), much more is to be done in the nonlinear case ([3], [15], [24]). Recently, in [11] feedback linearization, supervisory control and H∞ control were used to design an adaptive control law for a class of nonlinear dynamical systems based on wavelet networks. In order to simulate the practical potential of the proposed control scheme in [11], an adaptive controller was designed for output tracking of a transverse flux permanent magnet machine by utilizing high gain observers and radial basis function networks in [12]. On the other hand, several reported works (see for instance [6], [20], [25]) provide linear control theory for the control of hydraulic systems and much of the work in the control of hydraulic systems uses feedback linearization or backstepping techniques (see for instance [22], [26], [28]). In [2] the non-linear adaptive control was applied to the force control of an active suspension driven by a double-rod cylinder. They demonstrated that non-linear control schemes can achieve a much better performance than conventional linear controllers. They considered the parametric uncertainties of the cylinder only. The results are also extended to the trajectory tracking in [1]. In [27], a discontinuous projection based adaptive robust controller is constructed for the high performance robust motion control of a typical electro-hydraulic servosystem driven by a double-rod
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WeC10.5 hydraulic cylinder. Recently, in [27], the high performance robust motion control of electro-hydraulic servo-systems driven by double-rod hydraulic actuators was investigated. A two-stage nonlinear robust controller using the Lyapunov redesign method was established for velocity tracking control of hydraulic elevators in [14], and theoretical analysis and experimental evaluation of a Lyapunov-based control scheme to regulate the impacts of a hydraulic actuator that comes in contact with a nonmoving environment was proposed in [23]. In this paper, the authors derive a mathematical model describing the dynamics of an electrohydraulic servo valve and two hydraulic cylinders for a nonlinear hydraulic-mechanical (NHM) system. Considering a part of the dynamics of the NHM system as a norm-bounded uncertainty, two robust nonlinear adaptive controllers are developed based on the backstepping technique that ensure the tracking error signals asymptotically converge to zero despite the uncertainties in the system. Interval uncertainties in Coulomb friction parameters and in the internal leakage parameters in the cylinders are considered in the design. Two adaptation laws are obtained by using the Lyapunov functional method and inequality techniques. Therefore, the paper not only constructs a practical robust nonlinear adaptive controller for the NHM systems but also extends the theoretical results of an adaptive backstepping control algorithm to a class of nonlinear continuous uncertain systems which are described in a semi-strict feedback form. Simulation results are obtained to illustrate the effectiveness of the proposed method. The paper is structured as follows. In Section II, the derived mathematical models of the mechanical and hydraulic parts of the system considered are presented. The controller design, using a robust adaptive backstepping method, is given in Section III. The simulation results of the proposed method are provided in Section IV. Conclusions and future works are given in Section V.
TABLE I Description of parameters in electro-mechanical system. Parameter m, x k1 , k2 d1 , d2 m1 , m2 A1 , A2 ϕ1 , ϕ2 y1 , y2 y01 , y02 ρ β Cd , A d w pcr q1 , q2 q3 , q4 q5 , q6 Q1 , Q2 Q3 , Q4 x1 , x2 p p1 , p2 p3 , p4 C1 , C2 D1 , D2 c1 , c2 F1 , F2 Fr1 , Fr2 V1 , · · · , V, 4
Restriction flow Q1 − Cd wx1 Q2 − Cd wx2 Q3 − Cd wx1 Q4 − Cd wx2
II. DYNAMIC FORMULATION AND PROBLEM STATEMENT In this paper the hydraulic-mechanical system shown in Fig. 1 is considered [5]. The system parameters used in the model description are tabulated in Table I. The goal of the controller is to make the mass position x track the reference. A robust controller is needed for the directional valve positions x1 , x2 using the rod measurements y1 , y2 , y˙1 , y˙2 and the time-varying pressure measurements p1 , p2 , p3 , p4 . Robustness is required against: 1) Change in Coulomb friction parameters C1 and C2 in the cylinders and 2) Change in internal leakage parameters c1 and c2 in the cylinders. The equations are divided into the following groups: Flow continuity, restriction flow, operation mode, actuator equilibrium, mechanical coupling and dynamic pressure. The tank pressure is assumed to be zero. Flow continuity at nodes (For x1 > 0 and x2 > 0) Q1 − q1 − q5 Q2 − q2 − q6 Q3 − q5 − q3 Q4 − q6 − q4
=0 =0 =0 =0
Description Weight and position of mass Spring constants Damper constants Weight of piston and rod, cylinder 1, 2 Cross-sectional inlet area of cylinder 1, 2 Area ratio, cylinder 1, 2 Piston positions, zero at middle of cylinder Half the length of the cylinders Density of hydraulic oil [kg/m3 ] Bulk modulus of oil [Pa] Discharge coefficient and area for valves Orifice dimension for control valves Crack pressure for valves Flow into cylinders Flow out of cylinders Flow through v1 and v2 Flow through directional valves, piston side Flow through directional valves, rod side Spool positions of directional valves Constant pump pressure Piston side pressures at cylinders Rod side pressures at cylinders Coulomb friction coefficients for cylinders Viscous friction coefficients for cylinders Leakage coefficients for cylinders External forces acting on the cylinder rods Friction forces inside cylinders Chambers volumes in the cylinders
q
2
qρ 2
qρ
(p − p1 )
=0
(p − p2 )
=0 (2)
2 ρ p3
=0
q
2 ρ p4
=0
q
2
Operation mode q5 − Cd1 Ad1 q6 − Cd2 Ad2
qρ
(p1 − p3 ) = 0
2 ρ (p2
(3)
− p4 ) = 0
Actuator equilibrium (A1 p1 − ϕ1 A1 p3 ) − Fr1 − F1 − m1 y¨1 −(A2 p2 − ϕ2 A2 p4 ) − Fr2 + F2 − m2 y¨2 Fr1 − C1 sgn(y˙ 1 ) − D1 y˙ 1 Fr2 − C2 sgn(y˙ 2 ) − D2 y˙ 2
=0 =0 =0 =0
(4)
Mechanical coupling
(1)
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F1 − F2 − m¨ x k1 (y1 − x) + d1 (y˙ 1 − x) ˙ − F1 k2 (x − y2 ) + d2 (x˙ − y˙ 2 ) − F2
=0 =0 =0
(5)
WeC10.5 Dynamic pressure V1 β p˙ 1
− q1 + c1 (p1 − p3 ) + A1 y˙ 1 V1 − A1 (y01 + y1 ) V2 β p˙ 2 − q2 + c2 (p2 − p4 ) − A2 y˙ 2 V2 − A2 (y02 − y2 ) V3 β p˙ 3 + q3 + c1 (p3 − p1 ) − ϕ1 A1 y˙ 1 V3 − ϕ1 A1 (y01 − y1 ) V4 β p˙ 4 + q4 + c2 (p4 − p2 ) + ϕ2 A2 y˙ 2 V4 − ϕ2 A2 (y02 + y2 )
=0 =0 =0 =0 =0 =0 =0 =0
(6)
[10]. This idea is adopted to design a nonlinear controller for position tracking in the NHM system. Defining the state variables as [x, x, ˙ y1 , y˙ 1 , y2 , y˙ 2 , p3 , p4 , p1 , p2 ] =[z1 , z2 , z3 , z4 , z5 , z6 , z7 , z8 , z9 , z10 ] and the inputs [u1 , u2 ] = [x1 , x2 ], equations (1)-(7) can be rewritten as z˙2 =
The following assumption completes the description of the NHM system (1)-(6). Assumption 1. The uncertain friction and leakage parameters Ci and ci , i = 1, 2 lie in the following known intervals:
z˙4 = z˙6 =
i=1 3 X i=1 2 X
a1i zi
(8)
a2i zi + f1 (z4 ) + a24 z7 + a25 z9 + ∆2 (z4 )
(9)
a3i zi + a33 z5 + f2 (z6 ) + a34 z8 + a35 z10
i=1
C i ≤ Ci ≤ C i , ci ≤ ci ≤ ci
+ ∆5 (z6 )
in other words, Ci and ci can be represented as follows: Ci = C0i + fCi , ci = c0i + fci
6 X
(10)
z˙7 = g71 (z3 )z4 + g72 (z3 , z7 , z9 ) + g73 (z3 , z7 )u1 + ∆7 (z3 , z7 , z9 )
(7)
(11)
z˙8 = g81 (z5 )z6 + g82 (z5 , z8 , z10 ) + g83 (z5 , z8 )u2
where C0i =
+ ∆8 (z5 , z8 , z10 )
1 1 (C i + C i ), c0i = (ci + ci ) 2 2
z˙9 = g91 (z3 )z4 + g92 (z3 , z7 , z9 ) + g93 (z3 , z9 )u1
with |fCi | ≤ 4Ci and |fci | ≤ 4ci where
+ ∆3 (z3 , z7 , z9 )
(13)
z˙10 = g101 (z5 )z6 + g102 (z5 , z8 , z10 ) + g103 (z5 , z10 )u2
1 1 (C i − C i ), 4ci = (ci − ci ) 2 2 III. ROBUST M ULTIVARIABLE C ONTROL P ROBLEM 4Ci =
+ ∆6 (z5 , z8 , z10 )
Fig. 2 shows the control structure for a multivariable control system, where z1 , ..., z10 are the state variables defined later. The goal is to control the position x of the
Fig. 2.
(12)
(14)
Parameters in this model are listed in Table III. Remark 1. Obviously, the state-space system (8)-(14) is not in the strict-feedback form. However, it is shown that the system can be rendered into the desired strict-feedback form if a part of the dynamics of the NHM system is treated as a norm-bounded uncertainty. In the following steps, we present our recursive backstepping design procedure via Lyapunov functions for the NHM system (8)-(14). Step 1. Let e1 (t) = z1 (t) − xr (t). The design procedure is beginning with defining the following Lyapunov-like function 1 e1 (t)2 (15) V1 (t) = 2 If z2 (t) = z2d (z1 (t), xr (t)) were considered as a control input, it would be able to seek z2d (z1 , xr ) to render the derivative of V˙ 1 (t) negative. One may write
Robust control system.
mass in order to follow a desired trajectory of position and velocity. In this section a robust multivariable control system is presented which reduces the sensitivity to changing parameters such as Coulomb friction and internal leakage. The backstepping approach [16] which is commonly known and relatively easy to implement will be used for the NHM system. The main contribution of this paper is the application of backstepping techniques to a NHM system. A recursive framework is proposed to construct a Lyapunov function and corresponding control actions for the system stabilization
V˙ 1 (t)
= e1 (t)(z2d (t) − x˙r (t))
(16)
= x˙r (t) − L1 e1 (t)
(17)
Let z2d (t)
then V˙ 1 (t) = −L1 e1 (t)2 ≤ 0, where L1 stands for a positive weighting parameter.
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Step 2. Let e2 (t) = z2 (t) − z2d (t). This results in: e˙1 = −L1 e1 + e2
(18)
WeC10.5 Now, in order to go one step ahead, a new Lyapunov-like function V2 (t) is defined as 1 q1 ˜ 2 V2 (t) = V1 (t) + e2 (t)2 + ∆ (19) 1 (t) 2 2 where q1 , the adaptation gain, is an arbitrary positive constant ˜ 1 (t) = ∆ ˆ1 − ∆ ¯ 1 (t) is the adaptation error which and ∆ ˆ 1 equals to an estimate of the will be defined later. ∆ unknown parameter ∆1 (z4 , z5 , z6 ) = a14 z4 + a15 z5 + a16 z6 ¯1 = treated as a norm-bounded uncertainty, with bound ∆ k∆1 (z4 , z5 , z6 )k∞ . By taking the derivative of the equation above ˜ 1∆ ˜˙ 1 V˙ 2 = V˙ 1 + e2 (z˙2 − z˙2d ) + q1 ∆
Let V4 (t) be defined as follows 1 q2 ˜ 2 V4 (t) = V3 (t) + e4 (t)2 + ∆ (27) 2 (t) 2 2 ˜ 2 (t) = ∆ ˆ 2 (t) − where q2 is an arbitrary positive constant, ∆ ¯ ˆ ∆2 is an adaptation error, ∆2 equals to an estimate of the ¯ 2 = k∆2 (z4 )k∞ . Noting unknown parameter ∆2 (z4 ) and ∆ (9) and the results in Step 3 ˜ 2∆ ˜˙ 2 V˙ 4 = V˙ 3 + e4 (z˙4 − z˙4d ) + q2 ∆ ≤−
+∆1 −
+e4
˜ 1∆ ˜˙ 1 + q1 ∆
by introducing a new positive weighting parameter L2 , V˙ 2 (t) can be rewritten as V˙ 2 = − ≤−
i=1 2 X
ˆ 1 + q1 ∆ ˜ 1∆ ˆ˙ 1 Li e2i + e2 ∆1 − ke2 k∆
a2i zi + f1 + a24 z7 + a25 z9 + ∆2 − z˙4d
z9d =
3 X
−1 a25
a2i zi + f1 + a24 z7 − z˙4d + e3 + L4 e4
i=1
(28) ˆ2 +∆
e4 ke4 k
by introducing a new positive parameter L4 , we obtain V˙ 4 ≤ −
4 X
˜ 2 q2 ∆ ˆ˙ 2 − ke4 k Li e2i + ∆
(29)
i=1
˜ 1 q1 ∆ ˆ˙ 1 − ke2 k Li e2i + ∆
(22)
Substituting the following relation
(23)
ˆ˙ 2 (t) = q −1 ke4 (t)k, ∆ ˆ 2 (0) = s20 ∆ (30) 2 P 4 2 results in V˙ 4 (t) ≤ − i=1 Li ei (t) where s20 is specified by the designer.
i=1
ˆ˙ 1 (t) = q −1 ke2 (t)k, ∆ 1
ˆ 1 (0) = s10 ∆
where s10 is specified by the designer, renders the derivative of the Lyapunov function V2 non-positive. Step 3. Let e3 (t) = z3 (t) − z3d (t). Its derivative is: e˙2 = −e1 − L2 e2 + a13 e3 − ∆∗1
(24)
ˆ 1 e2 where ∆∗1 = ∆1 − ∆ ke2 k Let V3 (t) be defined as follows 1 V3 (t) = V2 (t) + e3 (t)2 2 By taking the derivative of V3 (t), one may write
(25)
Step 5. Let e9 (t) = z9 (t) − z9d (t). This results in the error system: e1 e˙1 −L1 1 0 0 0 0 e e˙2 −1 −L2 a13 2 ∆∗1 0 0 = e3 − 0 e˙3 0 −a13 −L3 1 0 e4 0 0 −1 −L4 a25 e˙4 ∆∗2 e9 (31) ˆ 2 e4 where ∆∗2 = ∆2 − ∆ ke4 k Consider V9 (t) given by 1 q3 ˜ 2 V9 (t) = V4 (t) + e9 (t)2 + ∆ (32) 3 (t) 2 2 ˜ 3 (t) = ∆ ˆ 3 (t) − where q3 is an arbitrary positive constant, ∆ ¯ 3 is an adaptation error, ∆ ˆ 3 is an estimate of ∆3 (z3 , z7 , z9 ) ∆ and ∆¯3 = kδ3 k∞ . From (13) and the results in Step 4
V˙ 3 (t) = V˙ 2 (t) + e3 (t)e˙ 3 (t) 2 X
!
If the virtual control input z9 (t) = z9d (t) is chosen as
(21)
Substituting the following relation
≤−
3 X i=1
If the virtual control input z3 (t) = z3d (z2 (t), z1 (t), z1d (t), z2d (t)) is chosen as −1 ˆ 1 e2 a11 z1 + a12 z2 − z˙2d + e1 + L2 e2 + ∆ z3d = a13 ke2 k (20)
2 X
˜ 2∆ ˜˙ 2 Li e2i + e3 (t)e4 (t) + q2 ∆
i=1
= −L1 e21 + e1 e2 + e2 (a11 z1 + a12 z2 + a13 z3 z˙2d )
3 X
Li ei (t)2 + a13 e2 (t)e3 (t) + e3 (t)(z4 (t) − z˙3d (t))
i=1
By considering z4 (t) = z4d (t) =P z˙3d (t) − a13 e2 (t) − L3 e3 (t) 3 ˙ it is concluded that V3 (t) = − i=1 Li ei (t)2 , where L3 is a new positive weighting parameter.
˜ 3∆ ˜˙ 3 V˙ 5 = V˙ 4 + e9 (z˙9 − z˙9d ) + q3 ∆ ≤−
Step 4. Let e4 (t) = z4 (t) −
z4d (t).
This results in:
e˙3 = −a13 e2 − L3 e3 + e4
4 X
Li e2i + a25 e4 e9 + e9 (g91 z4 + g92
i=1
(26)
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˜ 3∆ ˜˙ 3 +g93 u1 + ∆3 − z˙9d ) + q3 ∆
WeC10.5 If the control law u1 (t) is chosen as −1 ˜ 3 e9 u1 = g91 z4 + g92 − z˙9d + a25 e4 + L5 e9 + ∆ g93 ke9 k (33) ˙ ˙ ˜ ˆ by considering the expression ∆3 (t) = ∆3 (t), then we have V˙ 9 ≤ −
4 X
˜ 3 q3 ∆ ˆ˙ 3 − ke9 k Li e2i − L5 e29 + ∆
V˙ 6 = V˙ 2∗ + e5 (z˙5 − z˙5d ) ≤ −L1 e21 − L6 e22 + a15 e2 e5 + e5 (z6 − z˙5d ) If z6 (t) = z6d (t) is chosen as z6d = z˙5d − a15 e2 − L7 e5
(34)
i=1
(41)
by introducing a new positive parameter L7 , V˙ 6 (t) is simplified to
Substituting the following relation ˆ˙ 3 (t) = q −1 ke9 (t)k, ∆ 3
From Step 6, its time derivative is
ˆ 3 (0) = s30 ∆
V˙ 6 (t) ≤ −L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 ≤ 0
(35)
results in V˙ 9 (t) ≤ −
4 X
Li ei (t)2 − L5 e9 (t)2
Step 8. Let e6 (t) = z6 (t) − z6d (t). This results in:
i=1
e˙5 = −a15 e − 2 − L7 e5 + e6
where s30 is specified by the designer. Step 6. In the next steps, the aim is to synthesize an actual control law for u2 (t). This can be done by the same backstepping design via Lyapunov functions as in results above. Again, a modified Lyapunov function V2 (t) in (19) is defined as q4 ˜ 1 2 (36) Vˆ2∗ (t) = V1 (t) + e2 (t)2 + ∆ 4 (t) 2 2 ˜ 4 (t) = ∆ ˆ 4 (t) − where q4 is an arbitrary positive constant, ∆ ¯ 4, ∆ ˆ 4 equals to an estimate of the unknown parame∆ ¯4 = ter ∆4 (z3 , z4 , z6 ) = a13 z3 + a14 z4 + a16 z6 and ∆ k∆4 (z3 , z4 , z6 )k∞ . By taking the derivative of the equation above ˙ ˜˙ 4 ˜ 4∆ Vˆ2∗ = V˙ 1 + e2 (z˙2 − z˙2d ) + q4 ∆ ≤ −L1 e21 + e1 e2 + e2 (a11 z1 + a12 z2 + a15 z5 ˜˙ 4 ˜ 4∆ +∆4 (z3 , z4 , z6 ) − z˙2d ) + q4 ∆ Similarly, if z5 (t) = z5d (t) is chosen as −1 ˆ 4 e2 z5d = a11 z1 + a12 z2 − z˙2d + e1 + L6 e2 + ∆ a15 ke2 k (37) then V˙ 2∗ (t) can be rewritten as V˙ 2∗
˜ 1 (q4 ∆ ˆ˙ 4 − ke2 k) ≤ −L1 e21 − L6 e22 + ∆
≤ −L1 e21 − L6 e22 − L7 e5 (t)2 + e5 e6 + e6 (a31 z1 + a32 z2 ˜ 5∆ ˜˙ 5 +a33 z5 + f2 + a34 z8 + a35 z10 + ∆5 − z˙6d ) + q5 ∆ ˜ 5 (t) = ∆ ˆ 5 (t) − where q5 is an arbitrary positive constant, ∆ ¯ 5 and ∆ ¯ 5 = k∆5 (z6 )k∞ . If z10 = z d (t) is chosen as ∆ 10 −1 d a31 z1 + a32 z2 + a33 z5 + f2 z10 = a35 (44) ˆ 5 e6 +a34 z8 − z˙6d + e5 + L8 e6 + ∆ ke6 k by introducing a new positive weighting parameter L8 , then ˜ 5 (q5 ∆ ˆ˙ 5 − ke6 k) V˙ 7 ≤ −L1 e21 − L6 e22 − L7 e25 − L8 e26 + ∆ Substituting the following relation ˆ˙ 5 (t) = q −1 ke6 (t)k, ∆ 5
ˆ 5 (0) = s50 ∆
(45)
V˙ 7 (t) ≤ −L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 − L8 e6 (t)2 ≤ 0 ˆ 4 (0) = s40 ∆
(38)
where s40 is specified by the designer, renders the derivative of the Lyapunov function V2∗ non-positive. Step 7. Let e5 (t) = z5 (t) − z5d (t). Its derivative is: e˙2 = −e1 − L6 e2 + a15 e5 − ∆∗4 ∆∗4
Again, consider the following Lyapunov function with its time derivative q5 ˜ 1 2 (43) V6 (t) = V5 (t) + e6 (t)2 + ∆ 5 (t) 2 2 and ˜ 5∆ ˜˙ 5 V˙ 6 = V˙ 5 + e6 (z˙6 − z˙6d ) + q5 ∆
results in
Substituting the following relation ˆ˙ 4 (t) = q −1 ke2 (t)k, ∆ 4
(42)
(39)
ˆ 4 e2 ∆ ke2 k
where = ∆4 − Now, let V5 (t) be defined as follows 1 V5 (t) = V2∗ (t) + e5 (t)2 2
(40)
where s50 is specified by the designer. d Step 9. Let e10 (t) = z10 (t) − z10 (t). This results in the error system: e e˙1 −L1 1 0 0 0 1 0 e2 ∗ e˙2 −1 −L a 0 0 ∆ 2 15 = e5 − 4 0 e˙5 0 −a15 e2 −L7 1 0 e6 e˙6 0 0 −1 −L8 a35 ∆∗5 e10 (46) ˆ 5 e6 where ∆∗ = ∆5 − ∆
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5
ke6 k
WeC10.5 Consider V10 (t) given by 1 q6 ˜ 2 V10 (t) = V6 (t) + e10 (t)2 + ∆ (47) 6 (t) 2 2 ˜ 6 (t) = ∆ ˆ 6 (t) − where q6 is an arbitrary positive constant, ∆ ¯ ˆ ∆6 is an adaptation error, ∆6 is an estimate of the unknown parameter ∆6 (z5 , z8 , z10 ) and ∆¯6 = k∆6 (z5 , z8 , z10 )k∞ . From (14) and the results in Step 8 d ˜ 6∆ ˜˙ 6 V˙ 10 = V˙ 6 + e10 (z˙10 − z˙10 ) + q6 ∆
≤ −L1 e21 −L6 e22 −L7 e25 −L8 e26 +a35 e6 e10 +e10 (g101 z6 +g102 d ˜ 6∆ ˜˙ 6 +g103 u2 + ∆6 − z˙10 ) + q6 ∆
If u2 (t) is chosen −1 d g101 z6 + g102 − z˙10 + a35 e6 + L9 e10 u2 = g103 ˜ 6 e10 (48) +∆ ke10 k
Fig. 3.
˜˙ 6 (t) = ∆ ˆ˙ 6 (t), then we have by considering the expression ∆ V˙ 10 ≤ −L1 e21 − L6 e22 − L7 e25 − L8 e26 − L9 e210 ˜ 6 q6 ∆ ˆ˙ 6 − ke10 k +∆
(49)
Simplified NHM system.
¯ i for the control necessary to know the precise values of ∆ ¯ design. Thus we can suppose that ∆i are constants large ¯ i −→ ∞ , such that the inequalities V˙ i (t) ≤ 0 enough, i.e., ∆ will always hold.
Substituting the following relation ˆ˙ 6 (t) = q −1 ke10 (t)k, ∆ 6
ˆ 6 (0) = s60 ∆
IV. SIMULATION RESULTS (50) TABLE II VALUES OF THE SYSTEM PARAMETERS
results in V˙ 8 (t) ≤
−L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 −L8 e6 (t)2 − L9 e10 (t)2
where s60 is specified by the designer. Integrating both inequalities of V˙ 5 (t) and V˙ 8 (t) in Steps 5 and 9 from zero to t, it yields, respectively, V5 (0) ≥ V5 (t) + F1 (t) ≥ F1 (t) and
Parameter k1
Value = 1000 N/m
Parameter d1
Value = 1000 Ns/m
m1 ρ Cd p c1 Cd1 y01 L2 = L3
= 6 kg = 900 kg/m3 = 0.65 = 80 bar =1 = 0.65 = 0.1 m = 20
A1 β Ad1 C1 ϕ1 w L1 q
= 1450 mm2 = 12665 bar = 0.5 mm2 = 50 N = 0.5 = 7 mm = 10 = 0.5
V8 (0) ≥ V8 (t) + F2 (t) ≥ F2 (t) where Z F1 (t) := 0
t
5 X [ Li ei (s)2 + L6 e9 (s)2 ] ds i=1
and Z F2 (t) :=
t
[L1 e1 (s)2 + L6 e2 (s)2 + L7 e5 (s)2
0
+L8 e6 (s)2 + L9 e10 (s)2 ] ds Since V5 (0) and V8 (0) are positive and finite, limt−→∞ Fi (t) exist and are finite. Thus according to Barbalat lemma (see [13] and [21]), it is easily concluded that limt−→∞ ei (t) = 0. Remark 3. It is worth noting that the unknown but existing ¯ i are only introduced in the paper and it is not constants ∆
In this section, to verify the performance of the suggested controller, computer simulation of the NHM system with the parameters given in Table II is realized. The simulation algorithm was implemented in Matlab/Simulink. The adaptive backstepping algorithm has a fairly high complexity. The number of states in the controller is similar to the number of states in the system. NHM systems are difficult to simulate even without controllers, because of a stiff system and nonlinear, discontinuous friction models. Hence, the simulation results in this paper show a reduced order version of the adaptive backstepping controller applied to the model of Fig. 3. The equations (9), (11) and (13) were used for the reduced order controller using the first half of the backstepping procedure in Section III. The second half, in which the controller output u2 is calculated can be omitted if the position y is fixed and y1 is tracked. In addition, the discontinuous sign function in the friction
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Fig. 4. Simulation of piston position y1 with 10 % uncertainty in leakage parameter (thick line), with known parameter (thin line) and with 1% uncertainty in leakage parameter using a non-adaptive controller (dashed line).
Fig. 6.
Simulation of spool valve position.
Fig. 7. Simulation of errors. e1 = y1 − yref (thick line), e2 = y˙1 − y˙1 desired (thin line) and e4 = p1 − p1 desired . Fig. 5.
Simulation of pressures p1 (thick line) and p3 (thin line).
model was replaced by a dynamic model. The LuGre [4] dynamic friction model was used. The results in Fig. 4-7 show for a step response of 0.01m the piston position y1 , the two chamber pressures of the cylinder, the actuated spool position and the errors ei (t) = zi (t) − zid (t) using an adaptive controller with uncertain leakage parameter (10% uncertainty). In addition, in Fig. 4, the position is given with a non-adaptive controller and in the cases of known leakage parameter. The results show that the adaptive backstepping controller is able to track the desired position and velocity, while bringing the error functions ei (t) to zero. V. CONCLUSIONS AND FUTURE WORKS A. Conclusions This paper dealt with the nonlinear control of an NHM system consisting of an electro-hydraulic servo valve and two hydraulic cylinders. Considering a part of the dynamics
as a norm-bounded uncertainty, two adaptive controllers were proposed for this nonlinear system via backstepping. Asymptotical converge to zero tracking error was achieved despite uncertainties. Design using Lyapunov based arguments and inequality techniques prevented the drawbacks of linearization methods. Simulation for a reduced order model validated the design. B. Future works Work continues on simulation of the complete model as well as experimental results. The authors believe that the controller for the complete system will be possible, by careful tuning of the controller and selection of initial state values. The developed controller strategy takes into account non-linearities associated with hydraulic dynamics. Measurement time delay with parametric uncertainties could be considered as well and so could uncertainty in nonlinearities from external disturbances and uncompensated friction. Practical implementation of the proposed controller
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WeC10.5 TABLE III PARAMETERS a11 = −m−1 (k1 + k2 ) a13 = m−1 k1 a15 = m−1 k2 a21 = k1 m1 −1 a23 = −k1 m1 −1 a25 = A1 m1 −1 a32 = d2 m2 −1 a34 = ϕ2 A2 m2 −1 a35 = −A2 m2 −1 ∆5 (x) = −fC2 m2 −1 sgn(x) g71 (x) = β(y01 − x)−1 g91 (x) =
−β y01 +x
a12 a14 a16 a22 a24 a31 a33
(d1 + d2 ) = −1 m = m−1 d1 = m−1 d2 = m1 −1 d1 = −ϕ1 A1 m1 −1 = k2 m2 −1 = −k2 m2 −1
∆2 (x) = −fC1 m1 −1 sgn(x) g81 (x) = y −β +x 02
g101 (x) =
β y02 −x
−1 fi (x) = ((−Di − di )x − C0i sgn(x))m qi , i = 1, 2
β (Cd1 Ad1 ρ2 (z − y) + c01 (z − y)) ϕ1 A1 (y01 −x) q −βCd w 2 g73 (x, y) = ϕ A y ρ 1 1 (y01 −x) q β g82 (x, y, z) = ϕ A (y +x) (Cd2 Ad2 ρ2 (z − y) + c02 (z − y)) 2 2 02 q −βCd w 2 g83 (x, y) = ϕ A y ρ 2 2 (y02 +x) q −β g92 (x, y, z) = A (y +x) (Cd1 Ad1 ρ2 (z − y) + c01 (z − y)) 1 01 q 2 dw g93 (x, y) = A βC (p − y) ρ 1 (y01 +x) q −β g102 (x, y, z) = A (y −x) (Cd2 Ad2 ρ2 (z − y) + c02 (z − y))
g72 (x, y, z) =
2
∆7 (x, y, z) ∆8 (x, y, z) ∆3 (x, y, z) ∆6 (x, y, z)
02
q
βCd w 2 (p − y) A2 (y02 −x) ρ βfc1 = ϕ A (y −x) (z − y) 1 1 01 c2 = ϕ A βf (z − y) 2 2 (y02 +x) −βfc1 = A (y +x) (z − y) 1 01 c2 = A −βf (z − y) 2 (y02 −x)
g103 (x, y) =
is currently being performed. Initial results show that valve dynamics can be significant and the proposed backstepping controller could be extended with two more states (valve spool position and velocity). R EFERENCES [1] A. Alleyne, A systematic approach to the control of electrohydraulic servosystems. Proc. the American Control Conference, Philadelphia, USA, pp. 833-837, 1998. [2] A. Alleyne and J.K. Hedrick, Nonlinear adaptive control of active suspension. IEEE Trans. on Control Systems Technology, vol. 3, pp. 94-101, 1995. [3] B. Aloliwi and H. Khalil, Adaptive output feedback regulation of a class of nonlinear systems: Convergence and robustness. IEEE Trans. Aut. Cont., vol. 42, pp. 1714-1716, 1997. ˚ om and C. Canudas de Wit, Revisiting the LuGre Friction [4] K.J. Astr¨ Model: Stick-Slip Motion and Rate Dependence, IEEE Control Systems Magazine, Vol. 28, No. 6, pp. 101-114, 2008. [5] O. Egeland and J.T. Gravdahl, Modeling and Simulation for Automatic Control, Marine Cybernetics. 2002. [6] P.M. FitzSimons and J.J. Palazzolo, Part i: Modeling of a one-degreeof-freedom active hydraulic mount; part ii: Control. ASME Journal of Dynamic Systems, Measurement, and Control, vol. 118, pp. 439-448, 1996. [7] F. Ikhouane and M. Krstic, Robustness of the tuning functions adaptive backstepping design for linear systems. IEEE-Trans. Aut. Cont., vol. 43, pp. 431-437, 1998. [8] F. Ikhouane and M. Krstic, M. Adaptive backstepping with parameter projection: Robustness and asymptotic performance. Automatica, vol. 34, pp. 429-435, 1998. [9] A. Isidori, Nonlinear Control Systems, Third Edition. Springer, London, 1995.
[10] Z.P. Jiang and D.J. Hill, A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics, IEEE Trans. on Automatic Control, vol. 44, no. 9, pp. 1705-1711, 1999. [11] H.R. Karimi, B. Lohmann, B. Moshiri and P.J. Maralani, Waveletbased identification and control design for a class of non-linear systems, Int. J. Wavelets, Multiresoloution and Image Processing, vol. 4, no. 1, pp. 213-226, 2006. [12] H.R. Karimi and A. Babazadeh, Modeling and output tracking of transverse flux permanent magnet machines using high gain observer and RBF Neural network, ISA Transactions, vol. 44, pp. 445-456, 2005. [13] H.K. Khalil, Nonlinear Systems, New York: Macmillan 1992. [14] C.-S. Kim, K.-S. Hong and M.-K. Kim, Nonlinear robust control of a hydraulic elevator: experiment-based modeling and two-stage Lyapunov redesign, Control Engineering Practice, vol. 13, pp. 789803, 2005. [15] H.S. Kim, K.I. Lee and Y.M. Cho, Robust two-stage nonlinear control for a hydraulic servo-system, Int. J. Control, vol. 75, no. 7, pp. 502516, 2002. [16] M. Krstic, I. Kanellakopaulos and P.V. Kokotovic, Nonlinear and Adaptive Control Design, Willey, New York, 1995. [17] P.V. Kokotovic and A. Murat, Constructive nonlinear control: a historical perspective, Automatica, vol. 37, no. 5, pp. 637-662, 2001. [18] W. Lin and T. Shen, Robust passivity and feedback design for minimum-phase nonlinear systems with structural uncertainty, Automatic, vol. 35, no. 1, pp. 35-48, 1999. [19] H.E. Merritt, Hydraulic control systems, John and Sons, Inc., New York, 1967. [20] A.R. Plummer and N.D. Vaughan, Robust adaptive control for hydraulic servosystems, ASME J. Dynamic System, Measurement and Control, vol. 118, pp. 237-244, 1996. [21] V.M. Popov, Hyperstability of control system, Berlin: Springer-Verlag, 1973. [22] L.D. Re and A. Isidori, Performance enhancement of nonlinear drives by feedback linearization of linear-bilinear cascade models. IEEE Trans on Control Systems Technology, vol. 3, pp. 299-308, 1995. [23] P. Sekhavat, N. Sepehri and Q. Wu, Impact stabilizing controller for hydraulic actuators with friction: Theory and experiments, Control Engineering Practice, vol. 14, pp. 1423-1433, 2006. [24] J. Stoev, J.Y. Choi and J. Farrell, Adaptive control for output feedback nonlinear systems in the presence of modeling errors. Automatica, vol. 38, pp. 1761-1767, 2002. [25] T.C. Tsao and M. Tomizuka, Robust adaptive and repetitive digital control and application to hydraulic servo for noncircular machining. ASME J. Dynamic Systems, Measurement, and Control, vol. 116, pp. 24-32, 1994. [26] R. Vossoughi and M. Donath, Dynamic feedback linerization for electro-hydraulically actuated control systems. ASME J. of Dynamic Systems, Measurement, and Control, vol. 117, pp. 468-477, 1995. [27] B. Yao, F. Bu and G.T.C. Chiu, Non-linear adaptive robust control of electro-hydraulic systems driven by double-rod actuators, Int. J. Control, vol. 74, no. 8, pp. 761-775, 2001. [28] H. Yu, Z.-J. Feng and X.-Y. Wang, Nonlinear control for a class of hydraulic servo system, J. Zhejiang Univ. Science, vol. 5, no. 11, pp. 1413-1417, 2004. [29] M. Zapateiro, N. Luo, H.R. Karimi and J. Vehi, Vibration control of a class of semiactive suspension system using neural network and backstepping techniques, Mechanical Systems and Signal Processing, Special Issue on Inverse Problems, doi:10.1016/j.ymssp. 2008.10.003, 2008 (in press). [30] M. Zapateiro, N. Luo and H.R. Karimi, Neural network-backstepping control for vibration reduction in a magnetorheological suspension system, Solid State Phenomena, vol. 147-149, pp. 839-844, 2009.
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Robust Adaptive Backstepping Control Design for a Nonlinear Hydraulic-Mechanical System M. Choux∗,∗∗ , H.R. Karimi∗ , G. Hovland∗ , M.R. Hansen∗ , M. Ottestad∗ and M. Blanke∗∗ Abstract— The complex dynamics that characterize hydraulic systems make it difficult for the control design to achieve prescribed goals in an efficient manner. In this paper, we present the design and analysis of a robust nonlinear controller for a Nonlinear Hydraulic-Mechanical (NHM) System. The system consists of an electrohydraulic servo valve and two hydraulic cylinders. Specifically, by considering a part of the dynamics of the NHM system as a norm-bounded uncertainty, two adaptive controllers are developed based on the backstepping technique that ensure the tracking error signals asymptotically converge to zero despite the uncertainties in the system according to the Barbalat lemma. The resulting controllers are able to take into account the interval uncertainties in Coulomb friction parameters and in the internal leakage parameters in the cylinders. Two adaptation laws are obtained by using the Lyapunov functional method and inequality techniques. Simulation results demonstrate the performance and feasibility of the proposed method. Fig. 1.
I. INTRODUCTION Hydraulics -the science of liquid flow- is a very old discipline, which has commanded new interest in last decades, especially in the area of hydraulic control, and fills a substantial portion of the field of control. The dynamics of hydraulic systems are highly nonlinear. Furthermore, the system may be subjected to non-smooth and discontinuous nonlinearities due to control input saturation, directional change of valve opening, friction, and valve overlap. Hydraulic control components and systems are found in many mobile, airborne, and offshore applications [19]. In some applications, the main emphasis is placed on valves for servo control. Hydraulic control valves are devices that use mechanical motion to control a source of fluid power. They vary in arrangement and complexity, depending upon their function. Because control valves are the interface from mechanical (electrical) to fluid in hydraulic systems, their performance is under scrutiny, especially when system difficulties occur. During the last decade, backstepping based designs have emerged as powerful tools for stabilizing nonlinear systems for tracking and regulation purposes [16]. The main advantage of these designs is the systematic construction of a Lyapunov function for the closed loop, allowing the analysis of its stability properties. The adaptive version of these designs, especially the tuning functions design, offers the possibility to synthesize in a systematic way controllers for a wide class of nonlinear systems (those under the strictfeedback form) whose structure is known but with unknown ∗ Faculty of Technology and Science, The University of Agder, Serviceboks 509, N-4898 Grimstad, Norway. ∗∗ Department of Electrical Engineering, Technical University of Denmark, Kgs. Lyngby, Copenhagen, Denmark.
[email protected] or [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
A nonlinear hydraulic-mechanical (NHM) system.
parameters ([9], [10], [16], [17], [18], [29], [30]). They also offer the possibility to analyze transient behavior of the closed loop in the absence of uncertainties. Despite the fact that the robustness of the tuning functions design has been studied extensively for linear systems ([7], [8]), much more is to be done in the nonlinear case ([3], [15], [24]). Recently, in [11] feedback linearization, supervisory control and H∞ control were used to design an adaptive control law for a class of nonlinear dynamical systems based on wavelet networks. In order to simulate the practical potential of the proposed control scheme in [11], an adaptive controller was designed for output tracking of a transverse flux permanent magnet machine by utilizing high gain observers and radial basis function networks in [12]. On the other hand, several reported works (see for instance [6], [20], [25]) provide linear control theory for the control of hydraulic systems and much of the work in the control of hydraulic systems uses feedback linearization or backstepping techniques (see for instance [22], [26], [28]). In [2] the non-linear adaptive control was applied to the force control of an active suspension driven by a double-rod cylinder. They demonstrated that non-linear control schemes can achieve a much better performance than conventional linear controllers. They considered the parametric uncertainties of the cylinder only. The results are also extended to the trajectory tracking in [1]. In [27], a discontinuous projection based adaptive robust controller is constructed for the high performance robust motion control of a typical electro-hydraulic servosystem driven by a double-rod
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WeC10.5 hydraulic cylinder. Recently, in [27], the high performance robust motion control of electro-hydraulic servo-systems driven by double-rod hydraulic actuators was investigated. A two-stage nonlinear robust controller using the Lyapunov redesign method was established for velocity tracking control of hydraulic elevators in [14], and theoretical analysis and experimental evaluation of a Lyapunov-based control scheme to regulate the impacts of a hydraulic actuator that comes in contact with a nonmoving environment was proposed in [23]. In this paper, the authors derive a mathematical model describing the dynamics of an electrohydraulic servo valve and two hydraulic cylinders for a nonlinear hydraulic-mechanical (NHM) system. Considering a part of the dynamics of the NHM system as a norm-bounded uncertainty, two robust nonlinear adaptive controllers are developed based on the backstepping technique that ensure the tracking error signals asymptotically converge to zero despite the uncertainties in the system. Interval uncertainties in Coulomb friction parameters and in the internal leakage parameters in the cylinders are considered in the design. Two adaptation laws are obtained by using the Lyapunov functional method and inequality techniques. Therefore, the paper not only constructs a practical robust nonlinear adaptive controller for the NHM systems but also extends the theoretical results of an adaptive backstepping control algorithm to a class of nonlinear continuous uncertain systems which are described in a semi-strict feedback form. Simulation results are obtained to illustrate the effectiveness of the proposed method. The paper is structured as follows. In Section II, the derived mathematical models of the mechanical and hydraulic parts of the system considered are presented. The controller design, using a robust adaptive backstepping method, is given in Section III. The simulation results of the proposed method are provided in Section IV. Conclusions and future works are given in Section V.
TABLE I Description of parameters in electro-mechanical system. Parameter m, x k1 , k2 d1 , d2 m1 , m2 A1 , A2 ϕ1 , ϕ2 y1 , y2 y01 , y02 ρ β Cd , A d w pcr q1 , q2 q3 , q4 q5 , q6 Q1 , Q2 Q3 , Q4 x1 , x2 p p1 , p2 p3 , p4 C1 , C2 D1 , D2 c1 , c2 F1 , F2 Fr1 , Fr2 V1 , · · · , V, 4
Restriction flow Q1 − Cd wx1 Q2 − Cd wx2 Q3 − Cd wx1 Q4 − Cd wx2
II. DYNAMIC FORMULATION AND PROBLEM STATEMENT In this paper the hydraulic-mechanical system shown in Fig. 1 is considered [5]. The system parameters used in the model description are tabulated in Table I. The goal of the controller is to make the mass position x track the reference. A robust controller is needed for the directional valve positions x1 , x2 using the rod measurements y1 , y2 , y˙1 , y˙2 and the time-varying pressure measurements p1 , p2 , p3 , p4 . Robustness is required against: 1) Change in Coulomb friction parameters C1 and C2 in the cylinders and 2) Change in internal leakage parameters c1 and c2 in the cylinders. The equations are divided into the following groups: Flow continuity, restriction flow, operation mode, actuator equilibrium, mechanical coupling and dynamic pressure. The tank pressure is assumed to be zero. Flow continuity at nodes (For x1 > 0 and x2 > 0) Q1 − q1 − q5 Q2 − q2 − q6 Q3 − q5 − q3 Q4 − q6 − q4
=0 =0 =0 =0
Description Weight and position of mass Spring constants Damper constants Weight of piston and rod, cylinder 1, 2 Cross-sectional inlet area of cylinder 1, 2 Area ratio, cylinder 1, 2 Piston positions, zero at middle of cylinder Half the length of the cylinders Density of hydraulic oil [kg/m3 ] Bulk modulus of oil [Pa] Discharge coefficient and area for valves Orifice dimension for control valves Crack pressure for valves Flow into cylinders Flow out of cylinders Flow through v1 and v2 Flow through directional valves, piston side Flow through directional valves, rod side Spool positions of directional valves Constant pump pressure Piston side pressures at cylinders Rod side pressures at cylinders Coulomb friction coefficients for cylinders Viscous friction coefficients for cylinders Leakage coefficients for cylinders External forces acting on the cylinder rods Friction forces inside cylinders Chambers volumes in the cylinders
q
2
qρ 2
qρ
(p − p1 )
=0
(p − p2 )
=0 (2)
2 ρ p3
=0
q
2 ρ p4
=0
q
2
Operation mode q5 − Cd1 Ad1 q6 − Cd2 Ad2
qρ
(p1 − p3 ) = 0
2 ρ (p2
(3)
− p4 ) = 0
Actuator equilibrium (A1 p1 − ϕ1 A1 p3 ) − Fr1 − F1 − m1 y¨1 −(A2 p2 − ϕ2 A2 p4 ) − Fr2 + F2 − m2 y¨2 Fr1 − C1 sgn(y˙ 1 ) − D1 y˙ 1 Fr2 − C2 sgn(y˙ 2 ) − D2 y˙ 2
=0 =0 =0 =0
(4)
Mechanical coupling
(1)
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F1 − F2 − m¨ x k1 (y1 − x) + d1 (y˙ 1 − x) ˙ − F1 k2 (x − y2 ) + d2 (x˙ − y˙ 2 ) − F2
=0 =0 =0
(5)
WeC10.5 Dynamic pressure V1 β p˙ 1
− q1 + c1 (p1 − p3 ) + A1 y˙ 1 V1 − A1 (y01 + y1 ) V2 β p˙ 2 − q2 + c2 (p2 − p4 ) − A2 y˙ 2 V2 − A2 (y02 − y2 ) V3 β p˙ 3 + q3 + c1 (p3 − p1 ) − ϕ1 A1 y˙ 1 V3 − ϕ1 A1 (y01 − y1 ) V4 β p˙ 4 + q4 + c2 (p4 − p2 ) + ϕ2 A2 y˙ 2 V4 − ϕ2 A2 (y02 + y2 )
=0 =0 =0 =0 =0 =0 =0 =0
(6)
[10]. This idea is adopted to design a nonlinear controller for position tracking in the NHM system. Defining the state variables as [x, x, ˙ y1 , y˙ 1 , y2 , y˙ 2 , p3 , p4 , p1 , p2 ] =[z1 , z2 , z3 , z4 , z5 , z6 , z7 , z8 , z9 , z10 ] and the inputs [u1 , u2 ] = [x1 , x2 ], equations (1)-(7) can be rewritten as z˙2 =
The following assumption completes the description of the NHM system (1)-(6). Assumption 1. The uncertain friction and leakage parameters Ci and ci , i = 1, 2 lie in the following known intervals:
z˙4 = z˙6 =
i=1 3 X i=1 2 X
a1i zi
(8)
a2i zi + f1 (z4 ) + a24 z7 + a25 z9 + ∆2 (z4 )
(9)
a3i zi + a33 z5 + f2 (z6 ) + a34 z8 + a35 z10
i=1
C i ≤ Ci ≤ C i , ci ≤ ci ≤ ci
+ ∆5 (z6 )
in other words, Ci and ci can be represented as follows: Ci = C0i + fCi , ci = c0i + fci
6 X
(10)
z˙7 = g71 (z3 )z4 + g72 (z3 , z7 , z9 ) + g73 (z3 , z7 )u1 + ∆7 (z3 , z7 , z9 )
(7)
(11)
z˙8 = g81 (z5 )z6 + g82 (z5 , z8 , z10 ) + g83 (z5 , z8 )u2
where C0i =
+ ∆8 (z5 , z8 , z10 )
1 1 (C i + C i ), c0i = (ci + ci ) 2 2
z˙9 = g91 (z3 )z4 + g92 (z3 , z7 , z9 ) + g93 (z3 , z9 )u1
with |fCi | ≤ 4Ci and |fci | ≤ 4ci where
+ ∆3 (z3 , z7 , z9 )
(13)
z˙10 = g101 (z5 )z6 + g102 (z5 , z8 , z10 ) + g103 (z5 , z10 )u2
1 1 (C i − C i ), 4ci = (ci − ci ) 2 2 III. ROBUST M ULTIVARIABLE C ONTROL P ROBLEM 4Ci =
+ ∆6 (z5 , z8 , z10 )
Fig. 2 shows the control structure for a multivariable control system, where z1 , ..., z10 are the state variables defined later. The goal is to control the position x of the
Fig. 2.
(12)
(14)
Parameters in this model are listed in Table III. Remark 1. Obviously, the state-space system (8)-(14) is not in the strict-feedback form. However, it is shown that the system can be rendered into the desired strict-feedback form if a part of the dynamics of the NHM system is treated as a norm-bounded uncertainty. In the following steps, we present our recursive backstepping design procedure via Lyapunov functions for the NHM system (8)-(14). Step 1. Let e1 (t) = z1 (t) − xr (t). The design procedure is beginning with defining the following Lyapunov-like function 1 e1 (t)2 (15) V1 (t) = 2 If z2 (t) = z2d (z1 (t), xr (t)) were considered as a control input, it would be able to seek z2d (z1 , xr ) to render the derivative of V˙ 1 (t) negative. One may write
Robust control system.
mass in order to follow a desired trajectory of position and velocity. In this section a robust multivariable control system is presented which reduces the sensitivity to changing parameters such as Coulomb friction and internal leakage. The backstepping approach [16] which is commonly known and relatively easy to implement will be used for the NHM system. The main contribution of this paper is the application of backstepping techniques to a NHM system. A recursive framework is proposed to construct a Lyapunov function and corresponding control actions for the system stabilization
V˙ 1 (t)
= e1 (t)(z2d (t) − x˙r (t))
(16)
= x˙r (t) − L1 e1 (t)
(17)
Let z2d (t)
then V˙ 1 (t) = −L1 e1 (t)2 ≤ 0, where L1 stands for a positive weighting parameter.
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Step 2. Let e2 (t) = z2 (t) − z2d (t). This results in: e˙1 = −L1 e1 + e2
(18)
WeC10.5 Now, in order to go one step ahead, a new Lyapunov-like function V2 (t) is defined as 1 q1 ˜ 2 V2 (t) = V1 (t) + e2 (t)2 + ∆ (19) 1 (t) 2 2 where q1 , the adaptation gain, is an arbitrary positive constant ˜ 1 (t) = ∆ ˆ1 − ∆ ¯ 1 (t) is the adaptation error which and ∆ ˆ 1 equals to an estimate of the will be defined later. ∆ unknown parameter ∆1 (z4 , z5 , z6 ) = a14 z4 + a15 z5 + a16 z6 ¯1 = treated as a norm-bounded uncertainty, with bound ∆ k∆1 (z4 , z5 , z6 )k∞ . By taking the derivative of the equation above ˜ 1∆ ˜˙ 1 V˙ 2 = V˙ 1 + e2 (z˙2 − z˙2d ) + q1 ∆
Let V4 (t) be defined as follows 1 q2 ˜ 2 V4 (t) = V3 (t) + e4 (t)2 + ∆ (27) 2 (t) 2 2 ˜ 2 (t) = ∆ ˆ 2 (t) − where q2 is an arbitrary positive constant, ∆ ¯ ˆ ∆2 is an adaptation error, ∆2 equals to an estimate of the ¯ 2 = k∆2 (z4 )k∞ . Noting unknown parameter ∆2 (z4 ) and ∆ (9) and the results in Step 3 ˜ 2∆ ˜˙ 2 V˙ 4 = V˙ 3 + e4 (z˙4 − z˙4d ) + q2 ∆ ≤−
+∆1 −
+e4
˜ 1∆ ˜˙ 1 + q1 ∆
by introducing a new positive weighting parameter L2 , V˙ 2 (t) can be rewritten as V˙ 2 = − ≤−
i=1 2 X
ˆ 1 + q1 ∆ ˜ 1∆ ˆ˙ 1 Li e2i + e2 ∆1 − ke2 k∆
a2i zi + f1 + a24 z7 + a25 z9 + ∆2 − z˙4d
z9d =
3 X
−1 a25
a2i zi + f1 + a24 z7 − z˙4d + e3 + L4 e4
i=1
(28) ˆ2 +∆
e4 ke4 k
by introducing a new positive parameter L4 , we obtain V˙ 4 ≤ −
4 X
˜ 2 q2 ∆ ˆ˙ 2 − ke4 k Li e2i + ∆
(29)
i=1
˜ 1 q1 ∆ ˆ˙ 1 − ke2 k Li e2i + ∆
(22)
Substituting the following relation
(23)
ˆ˙ 2 (t) = q −1 ke4 (t)k, ∆ ˆ 2 (0) = s20 ∆ (30) 2 P 4 2 results in V˙ 4 (t) ≤ − i=1 Li ei (t) where s20 is specified by the designer.
i=1
ˆ˙ 1 (t) = q −1 ke2 (t)k, ∆ 1
ˆ 1 (0) = s10 ∆
where s10 is specified by the designer, renders the derivative of the Lyapunov function V2 non-positive. Step 3. Let e3 (t) = z3 (t) − z3d (t). Its derivative is: e˙2 = −e1 − L2 e2 + a13 e3 − ∆∗1
(24)
ˆ 1 e2 where ∆∗1 = ∆1 − ∆ ke2 k Let V3 (t) be defined as follows 1 V3 (t) = V2 (t) + e3 (t)2 2 By taking the derivative of V3 (t), one may write
(25)
Step 5. Let e9 (t) = z9 (t) − z9d (t). This results in the error system: e1 e˙1 −L1 1 0 0 0 0 e e˙2 −1 −L2 a13 2 ∆∗1 0 0 = e3 − 0 e˙3 0 −a13 −L3 1 0 e4 0 0 −1 −L4 a25 e˙4 ∆∗2 e9 (31) ˆ 2 e4 where ∆∗2 = ∆2 − ∆ ke4 k Consider V9 (t) given by 1 q3 ˜ 2 V9 (t) = V4 (t) + e9 (t)2 + ∆ (32) 3 (t) 2 2 ˜ 3 (t) = ∆ ˆ 3 (t) − where q3 is an arbitrary positive constant, ∆ ¯ 3 is an adaptation error, ∆ ˆ 3 is an estimate of ∆3 (z3 , z7 , z9 ) ∆ and ∆¯3 = kδ3 k∞ . From (13) and the results in Step 4
V˙ 3 (t) = V˙ 2 (t) + e3 (t)e˙ 3 (t) 2 X
!
If the virtual control input z9 (t) = z9d (t) is chosen as
(21)
Substituting the following relation
≤−
3 X i=1
If the virtual control input z3 (t) = z3d (z2 (t), z1 (t), z1d (t), z2d (t)) is chosen as −1 ˆ 1 e2 a11 z1 + a12 z2 − z˙2d + e1 + L2 e2 + ∆ z3d = a13 ke2 k (20)
2 X
˜ 2∆ ˜˙ 2 Li e2i + e3 (t)e4 (t) + q2 ∆
i=1
= −L1 e21 + e1 e2 + e2 (a11 z1 + a12 z2 + a13 z3 z˙2d )
3 X
Li ei (t)2 + a13 e2 (t)e3 (t) + e3 (t)(z4 (t) − z˙3d (t))
i=1
By considering z4 (t) = z4d (t) =P z˙3d (t) − a13 e2 (t) − L3 e3 (t) 3 ˙ it is concluded that V3 (t) = − i=1 Li ei (t)2 , where L3 is a new positive weighting parameter.
˜ 3∆ ˜˙ 3 V˙ 5 = V˙ 4 + e9 (z˙9 − z˙9d ) + q3 ∆ ≤−
Step 4. Let e4 (t) = z4 (t) −
z4d (t).
This results in:
e˙3 = −a13 e2 − L3 e3 + e4
4 X
Li e2i + a25 e4 e9 + e9 (g91 z4 + g92
i=1
(26)
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˜ 3∆ ˜˙ 3 +g93 u1 + ∆3 − z˙9d ) + q3 ∆
WeC10.5 If the control law u1 (t) is chosen as −1 ˜ 3 e9 u1 = g91 z4 + g92 − z˙9d + a25 e4 + L5 e9 + ∆ g93 ke9 k (33) ˙ ˙ ˜ ˆ by considering the expression ∆3 (t) = ∆3 (t), then we have V˙ 9 ≤ −
4 X
˜ 3 q3 ∆ ˆ˙ 3 − ke9 k Li e2i − L5 e29 + ∆
V˙ 6 = V˙ 2∗ + e5 (z˙5 − z˙5d ) ≤ −L1 e21 − L6 e22 + a15 e2 e5 + e5 (z6 − z˙5d ) If z6 (t) = z6d (t) is chosen as z6d = z˙5d − a15 e2 − L7 e5
(34)
i=1
(41)
by introducing a new positive parameter L7 , V˙ 6 (t) is simplified to
Substituting the following relation ˆ˙ 3 (t) = q −1 ke9 (t)k, ∆ 3
From Step 6, its time derivative is
ˆ 3 (0) = s30 ∆
V˙ 6 (t) ≤ −L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 ≤ 0
(35)
results in V˙ 9 (t) ≤ −
4 X
Li ei (t)2 − L5 e9 (t)2
Step 8. Let e6 (t) = z6 (t) − z6d (t). This results in:
i=1
e˙5 = −a15 e − 2 − L7 e5 + e6
where s30 is specified by the designer. Step 6. In the next steps, the aim is to synthesize an actual control law for u2 (t). This can be done by the same backstepping design via Lyapunov functions as in results above. Again, a modified Lyapunov function V2 (t) in (19) is defined as q4 ˜ 1 2 (36) Vˆ2∗ (t) = V1 (t) + e2 (t)2 + ∆ 4 (t) 2 2 ˜ 4 (t) = ∆ ˆ 4 (t) − where q4 is an arbitrary positive constant, ∆ ¯ 4, ∆ ˆ 4 equals to an estimate of the unknown parame∆ ¯4 = ter ∆4 (z3 , z4 , z6 ) = a13 z3 + a14 z4 + a16 z6 and ∆ k∆4 (z3 , z4 , z6 )k∞ . By taking the derivative of the equation above ˙ ˜˙ 4 ˜ 4∆ Vˆ2∗ = V˙ 1 + e2 (z˙2 − z˙2d ) + q4 ∆ ≤ −L1 e21 + e1 e2 + e2 (a11 z1 + a12 z2 + a15 z5 ˜˙ 4 ˜ 4∆ +∆4 (z3 , z4 , z6 ) − z˙2d ) + q4 ∆ Similarly, if z5 (t) = z5d (t) is chosen as −1 ˆ 4 e2 z5d = a11 z1 + a12 z2 − z˙2d + e1 + L6 e2 + ∆ a15 ke2 k (37) then V˙ 2∗ (t) can be rewritten as V˙ 2∗
˜ 1 (q4 ∆ ˆ˙ 4 − ke2 k) ≤ −L1 e21 − L6 e22 + ∆
≤ −L1 e21 − L6 e22 − L7 e5 (t)2 + e5 e6 + e6 (a31 z1 + a32 z2 ˜ 5∆ ˜˙ 5 +a33 z5 + f2 + a34 z8 + a35 z10 + ∆5 − z˙6d ) + q5 ∆ ˜ 5 (t) = ∆ ˆ 5 (t) − where q5 is an arbitrary positive constant, ∆ ¯ 5 and ∆ ¯ 5 = k∆5 (z6 )k∞ . If z10 = z d (t) is chosen as ∆ 10 −1 d a31 z1 + a32 z2 + a33 z5 + f2 z10 = a35 (44) ˆ 5 e6 +a34 z8 − z˙6d + e5 + L8 e6 + ∆ ke6 k by introducing a new positive weighting parameter L8 , then ˜ 5 (q5 ∆ ˆ˙ 5 − ke6 k) V˙ 7 ≤ −L1 e21 − L6 e22 − L7 e25 − L8 e26 + ∆ Substituting the following relation ˆ˙ 5 (t) = q −1 ke6 (t)k, ∆ 5
ˆ 5 (0) = s50 ∆
(45)
V˙ 7 (t) ≤ −L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 − L8 e6 (t)2 ≤ 0 ˆ 4 (0) = s40 ∆
(38)
where s40 is specified by the designer, renders the derivative of the Lyapunov function V2∗ non-positive. Step 7. Let e5 (t) = z5 (t) − z5d (t). Its derivative is: e˙2 = −e1 − L6 e2 + a15 e5 − ∆∗4 ∆∗4
Again, consider the following Lyapunov function with its time derivative q5 ˜ 1 2 (43) V6 (t) = V5 (t) + e6 (t)2 + ∆ 5 (t) 2 2 and ˜ 5∆ ˜˙ 5 V˙ 6 = V˙ 5 + e6 (z˙6 − z˙6d ) + q5 ∆
results in
Substituting the following relation ˆ˙ 4 (t) = q −1 ke2 (t)k, ∆ 4
(42)
(39)
ˆ 4 e2 ∆ ke2 k
where = ∆4 − Now, let V5 (t) be defined as follows 1 V5 (t) = V2∗ (t) + e5 (t)2 2
(40)
where s50 is specified by the designer. d Step 9. Let e10 (t) = z10 (t) − z10 (t). This results in the error system: e e˙1 −L1 1 0 0 0 1 0 e2 ∗ e˙2 −1 −L a 0 0 ∆ 2 15 = e5 − 4 0 e˙5 0 −a15 e2 −L7 1 0 e6 e˙6 0 0 −1 −L8 a35 ∆∗5 e10 (46) ˆ 5 e6 where ∆∗ = ∆5 − ∆
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5
ke6 k
WeC10.5 Consider V10 (t) given by 1 q6 ˜ 2 V10 (t) = V6 (t) + e10 (t)2 + ∆ (47) 6 (t) 2 2 ˜ 6 (t) = ∆ ˆ 6 (t) − where q6 is an arbitrary positive constant, ∆ ¯ ˆ ∆6 is an adaptation error, ∆6 is an estimate of the unknown parameter ∆6 (z5 , z8 , z10 ) and ∆¯6 = k∆6 (z5 , z8 , z10 )k∞ . From (14) and the results in Step 8 d ˜ 6∆ ˜˙ 6 V˙ 10 = V˙ 6 + e10 (z˙10 − z˙10 ) + q6 ∆
≤ −L1 e21 −L6 e22 −L7 e25 −L8 e26 +a35 e6 e10 +e10 (g101 z6 +g102 d ˜ 6∆ ˜˙ 6 +g103 u2 + ∆6 − z˙10 ) + q6 ∆
If u2 (t) is chosen −1 d g101 z6 + g102 − z˙10 + a35 e6 + L9 e10 u2 = g103 ˜ 6 e10 (48) +∆ ke10 k
Fig. 3.
˜˙ 6 (t) = ∆ ˆ˙ 6 (t), then we have by considering the expression ∆ V˙ 10 ≤ −L1 e21 − L6 e22 − L7 e25 − L8 e26 − L9 e210 ˜ 6 q6 ∆ ˆ˙ 6 − ke10 k +∆
(49)
Simplified NHM system.
¯ i for the control necessary to know the precise values of ∆ ¯ design. Thus we can suppose that ∆i are constants large ¯ i −→ ∞ , such that the inequalities V˙ i (t) ≤ 0 enough, i.e., ∆ will always hold.
Substituting the following relation ˆ˙ 6 (t) = q −1 ke10 (t)k, ∆ 6
ˆ 6 (0) = s60 ∆
IV. SIMULATION RESULTS (50) TABLE II VALUES OF THE SYSTEM PARAMETERS
results in V˙ 8 (t) ≤
−L1 e1 (t)2 − L6 e2 (t)2 − L7 e5 (t)2 −L8 e6 (t)2 − L9 e10 (t)2
where s60 is specified by the designer. Integrating both inequalities of V˙ 5 (t) and V˙ 8 (t) in Steps 5 and 9 from zero to t, it yields, respectively, V5 (0) ≥ V5 (t) + F1 (t) ≥ F1 (t) and
Parameter k1
Value = 1000 N/m
Parameter d1
Value = 1000 Ns/m
m1 ρ Cd p c1 Cd1 y01 L2 = L3
= 6 kg = 900 kg/m3 = 0.65 = 80 bar =1 = 0.65 = 0.1 m = 20
A1 β Ad1 C1 ϕ1 w L1 q
= 1450 mm2 = 12665 bar = 0.5 mm2 = 50 N = 0.5 = 7 mm = 10 = 0.5
V8 (0) ≥ V8 (t) + F2 (t) ≥ F2 (t) where Z F1 (t) := 0
t
5 X [ Li ei (s)2 + L6 e9 (s)2 ] ds i=1
and Z F2 (t) :=
t
[L1 e1 (s)2 + L6 e2 (s)2 + L7 e5 (s)2
0
+L8 e6 (s)2 + L9 e10 (s)2 ] ds Since V5 (0) and V8 (0) are positive and finite, limt−→∞ Fi (t) exist and are finite. Thus according to Barbalat lemma (see [13] and [21]), it is easily concluded that limt−→∞ ei (t) = 0. Remark 3. It is worth noting that the unknown but existing ¯ i are only introduced in the paper and it is not constants ∆
In this section, to verify the performance of the suggested controller, computer simulation of the NHM system with the parameters given in Table II is realized. The simulation algorithm was implemented in Matlab/Simulink. The adaptive backstepping algorithm has a fairly high complexity. The number of states in the controller is similar to the number of states in the system. NHM systems are difficult to simulate even without controllers, because of a stiff system and nonlinear, discontinuous friction models. Hence, the simulation results in this paper show a reduced order version of the adaptive backstepping controller applied to the model of Fig. 3. The equations (9), (11) and (13) were used for the reduced order controller using the first half of the backstepping procedure in Section III. The second half, in which the controller output u2 is calculated can be omitted if the position y is fixed and y1 is tracked. In addition, the discontinuous sign function in the friction
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WeC10.5
Fig. 4. Simulation of piston position y1 with 10 % uncertainty in leakage parameter (thick line), with known parameter (thin line) and with 1% uncertainty in leakage parameter using a non-adaptive controller (dashed line).
Fig. 6.
Simulation of spool valve position.
Fig. 7. Simulation of errors. e1 = y1 − yref (thick line), e2 = y˙1 − y˙1 desired (thin line) and e4 = p1 − p1 desired . Fig. 5.
Simulation of pressures p1 (thick line) and p3 (thin line).
model was replaced by a dynamic model. The LuGre [4] dynamic friction model was used. The results in Fig. 4-7 show for a step response of 0.01m the piston position y1 , the two chamber pressures of the cylinder, the actuated spool position and the errors ei (t) = zi (t) − zid (t) using an adaptive controller with uncertain leakage parameter (10% uncertainty). In addition, in Fig. 4, the position is given with a non-adaptive controller and in the cases of known leakage parameter. The results show that the adaptive backstepping controller is able to track the desired position and velocity, while bringing the error functions ei (t) to zero. V. CONCLUSIONS AND FUTURE WORKS A. Conclusions This paper dealt with the nonlinear control of an NHM system consisting of an electro-hydraulic servo valve and two hydraulic cylinders. Considering a part of the dynamics
as a norm-bounded uncertainty, two adaptive controllers were proposed for this nonlinear system via backstepping. Asymptotical converge to zero tracking error was achieved despite uncertainties. Design using Lyapunov based arguments and inequality techniques prevented the drawbacks of linearization methods. Simulation for a reduced order model validated the design. B. Future works Work continues on simulation of the complete model as well as experimental results. The authors believe that the controller for the complete system will be possible, by careful tuning of the controller and selection of initial state values. The developed controller strategy takes into account non-linearities associated with hydraulic dynamics. Measurement time delay with parametric uncertainties could be considered as well and so could uncertainty in nonlinearities from external disturbances and uncompensated friction. Practical implementation of the proposed controller
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WeC10.5 TABLE III PARAMETERS a11 = −m−1 (k1 + k2 ) a13 = m−1 k1 a15 = m−1 k2 a21 = k1 m1 −1 a23 = −k1 m1 −1 a25 = A1 m1 −1 a32 = d2 m2 −1 a34 = ϕ2 A2 m2 −1 a35 = −A2 m2 −1 ∆5 (x) = −fC2 m2 −1 sgn(x) g71 (x) = β(y01 − x)−1 g91 (x) =
−β y01 +x
a12 a14 a16 a22 a24 a31 a33
(d1 + d2 ) = −1 m = m−1 d1 = m−1 d2 = m1 −1 d1 = −ϕ1 A1 m1 −1 = k2 m2 −1 = −k2 m2 −1
∆2 (x) = −fC1 m1 −1 sgn(x) g81 (x) = y −β +x 02
g101 (x) =
β y02 −x
−1 fi (x) = ((−Di − di )x − C0i sgn(x))m qi , i = 1, 2
β (Cd1 Ad1 ρ2 (z − y) + c01 (z − y)) ϕ1 A1 (y01 −x) q −βCd w 2 g73 (x, y) = ϕ A y ρ 1 1 (y01 −x) q β g82 (x, y, z) = ϕ A (y +x) (Cd2 Ad2 ρ2 (z − y) + c02 (z − y)) 2 2 02 q −βCd w 2 g83 (x, y) = ϕ A y ρ 2 2 (y02 +x) q −β g92 (x, y, z) = A (y +x) (Cd1 Ad1 ρ2 (z − y) + c01 (z − y)) 1 01 q 2 dw g93 (x, y) = A βC (p − y) ρ 1 (y01 +x) q −β g102 (x, y, z) = A (y −x) (Cd2 Ad2 ρ2 (z − y) + c02 (z − y))
g72 (x, y, z) =
2
∆7 (x, y, z) ∆8 (x, y, z) ∆3 (x, y, z) ∆6 (x, y, z)
02
q
βCd w 2 (p − y) A2 (y02 −x) ρ βfc1 = ϕ A (y −x) (z − y) 1 1 01 c2 = ϕ A βf (z − y) 2 2 (y02 +x) −βfc1 = A (y +x) (z − y) 1 01 c2 = A −βf (z − y) 2 (y02 −x)
g103 (x, y) =
is currently being performed. Initial results show that valve dynamics can be significant and the proposed backstepping controller could be extended with two more states (valve spool position and velocity). R EFERENCES [1] A. Alleyne, A systematic approach to the control of electrohydraulic servosystems. Proc. the American Control Conference, Philadelphia, USA, pp. 833-837, 1998. [2] A. Alleyne and J.K. Hedrick, Nonlinear adaptive control of active suspension. IEEE Trans. on Control Systems Technology, vol. 3, pp. 94-101, 1995. [3] B. Aloliwi and H. Khalil, Adaptive output feedback regulation of a class of nonlinear systems: Convergence and robustness. IEEE Trans. Aut. Cont., vol. 42, pp. 1714-1716, 1997. ˚ om and C. Canudas de Wit, Revisiting the LuGre Friction [4] K.J. Astr¨ Model: Stick-Slip Motion and Rate Dependence, IEEE Control Systems Magazine, Vol. 28, No. 6, pp. 101-114, 2008. [5] O. Egeland and J.T. Gravdahl, Modeling and Simulation for Automatic Control, Marine Cybernetics. 2002. [6] P.M. FitzSimons and J.J. Palazzolo, Part i: Modeling of a one-degreeof-freedom active hydraulic mount; part ii: Control. ASME Journal of Dynamic Systems, Measurement, and Control, vol. 118, pp. 439-448, 1996. [7] F. Ikhouane and M. Krstic, Robustness of the tuning functions adaptive backstepping design for linear systems. IEEE-Trans. Aut. Cont., vol. 43, pp. 431-437, 1998. [8] F. Ikhouane and M. Krstic, M. Adaptive backstepping with parameter projection: Robustness and asymptotic performance. Automatica, vol. 34, pp. 429-435, 1998. [9] A. Isidori, Nonlinear Control Systems, Third Edition. Springer, London, 1995.
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