LPV Model-based Robust Controller Design of Electro-hydraulic Servo Systems

LPV Model-based Robust Controller Design of Electro-hydraulic Servo Systems

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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 15 (2011) 421 – 425 www.elsevier.com/locate/procedia

Advanced in Control Engineering and Information Science

LPV Model-based Robust Controller Design of Electrohydraulic Servo Systems Falu Wenga, Yuanchun Dingb, Minkang Tangb a* b

a Faculty of Electrical Engineering and Automation, Jiangxi University of Science and Technology,Jiangxi, 341000, China. School of Resources and Environmental Engineering, Jiangxi University of Science and Technology, Jiangxi, 341000, China.

Abstract

In this paper, a Lyapunov-based control algorithm is developed for position tracking control of a class of electro-hydraulic servo systems driven by double-rod hydraulic actuators. By utilizing the nonlinear servo valve flow-pressure characteristics of the hydraulic system, the electro-hydraulic servo system is firstly described as a linear parameter varying (LPV) model. Then the sufficient strict linear matrix inequality (LMI) conditions for the robust stabilizability of the uncertain systems are proposed in terms of LMIs. By solving those LMIs, a parameter-dependent controller is established for the closed-loop system to be stable with a prescribed level of disturbance attenuation. Finally, the numerical simulations demonstrate the usefulness and advantages of the proposed controllers. © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011] Keywords: Robust stabilizability; Uncertainty; Parameter-dependent controller; Electro-hydraulic servo system;

1. Introduction Electro-hydraulic servo systems have been widely used in industry by virtue of their small size to power ratios and the ability to apply very large torques and forces; for example, they have been used in robot manipulators [1], hydraulic elevators [2], and active suspension systems [3], etc. However, the

* Corresponding author. Tel.: +086-13397970846; fax: +086-797-8312259. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.08.080

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dynamic behavior of hydraulic systems is highly nonlinear due to phenomena such as nonlinear servo valve flow-pressure characteristics and variations in control fluid volumes and associated stiffness, which, in turn, cause difficulties in the control of such systems. Therefore, the study of hydraulic systems has received much attention and various analysis and synthesis methods have been developed over the past years; such as sliding mode adaptive control [4], robust control [5], adaptive robust control [6], etc.. In this paper, by utilizing the nonlinear servo valve flow-pressure characteristics, the electro-hydraulic servo system is described as a linear parameter varying (LPV) model. Based on this LPV model, the parameter-dependent controller is established for the stabilization of the closed-loop system. The obtained controllers are very simple and more easily implemented than those adaptive controllers. The numerical example is given to show the effectiveness of the proposed controllers. 2. PROBLEM FORMULATION AND DYNAMIC MODELS The system under consideration is depicted in Fig.1. The goal is to have the inertia load to track any specified motion trajectory as closely as possible. x

P1

P2

k

m Q1

Q2

PL = P1 − P2

i

QL = Ps

1 ( Q1 + Q2 ) 2

Pr

The dynamics of the inertia load can be described by mx&& = PL Ω − bx& − kx + ω (t ) ,

(1)

where x , m , Ω , b , and k represent the displacement, the mass of the load, the ram area of the cylinder, the damping coefficient, and the effective stiffness of spring, respectively, PL = P1 − P2 is the load pressure of the cylinder. The actuator (or the cylinder) dynamics can be written as [7] Vt & (2) PL = −Ωx& − Ctm PL + QL , 4β e where Vt , β e , and Ctm are the total volume of the cylinder, the effective bulk modulus, and the coefficient of the total internal leakage, respectively. QL = ( Q1 + Q2 ) 2 is the load flow, by Merritt [7] QL = Cd wxv

Ps − sgn( xv ) PL

ρ

,

(3)

where Cd , w , Ps , ρ , xv are discharge coefficient, spool valve area gradient, supply pressure of the fluid, fluid Density, and spool valve displacement, respectively. The spool valve displacement xv is related to the current input i by a first-order system given by [8]

τ v x&v = − xv + K v i

(4)

where τ v and K v are the time constant and gain of the servo-valve respectively. In practice, the mass, damping and stiffness are usually subjected to possible time-varying perturbations. From (1)–(4), the entire uncertain system can be expressed in the state space form as

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( E + ΔE ) q& (t ) = ( A(θ ) + ΔA) q(t ) + Bu (t ) + Bω ω (t ), Z (t ) = Cz q(t ),

(5)

u (t ) = K (θ ) q(t ), ⎡0 1 0 0 ⎤ ⎡0 ⎤ ⎡0 ⎤ ⎢ ˆ ˆ ⎥ ⎢0 ⎥ ⎢1 ⎥ ⎡ ⎤ Vt C w −k − b Ω 0 ⎥ where E = diag ⎢1, mˆ , , τ v ⎥ , A (θ ) = ⎢ , B = ⎢ ⎥ , Bω = ⎢ ⎥ , A34 = d θ , ⎢ 0 −Ω −C A ⎥ ⎢0 ⎥ ⎢0 ⎥ 4β e ρ ⎣ ⎦ tm 34 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 0 0 − 1 ⎥⎦ ⎣0 ⎦ ⎣ Kv ⎦ 1 1 θ = ps − sgn( xv ) PL , mˆ = ( m + m ) , Δm = α mˆ , α ≤ α < 1 , α = ( m − m ) ( m + m ) , kˆ = k + k , 2 2 1 Δk = δ kˆ , δ ≤ δ < 1 , δ = k − k k + k , bˆ = b + b , Δb = λ bˆ , λ ≤ λ < 1 , λ = b − b b + b , 2 ˆ f T . f ∈ R 4 denotes the column vector with ith element to be 1 and ˆ f T + λ bf ΔE = α mˆ f 2 f 2T , ΔA = δ kf i 2 1 2 2

(

)(

)

(

(

)

(

(

)

)(

)

)

others to be 0. q = [ x, x& , PL , xv ] are state variables. m, k , c m, k , c are the lower(upper) bounds of the

mass, stiffness and damping respectively. Obviously, θ can be obtained online for Ps , sgn( xv ) , and PL can

be measured online. Assuming θ1 ≤ θ ≤ θ2 , we can describe A (θ ) as convex sets 2

A (θ ) = ∑ϖ i Ai , i = 1, 2 ,

(6)

i =1

ϖ 1 = (θ 2 − θ ) (θ 2 − θ1 ) , ϖ 2 = 1 − ϖ 1 , and Ai = A (θi )( i = 1, 2 ) are the vertices of A (θ ) . K (θ ) is the parameter-dependent control law need to be design, which can be describe in the following convex sets 2

K (θ ) = ∑ϖ i K i ,

(7)

i =1

where K i ( i = 1, 2 ) are the vertices of K (θ ) . Lemma 1: let μ , ν and F be real matrices of suitable dimensions with F T F ≤ I . Then, for any scalar σ >0 μ Fν + ν T F T μ T ≤ σμμ T + σ Tν Tν . (8) 3. PARAMETER-DEPENDENT CONTROLLER DESIGN Theorem 1: The system (5) without uncertainties is asymptotical stabilizable with the performance z 2 < γ ω 2 for all non-zero ω ∈ L2 [ 0, ∞ ) , if there exist P > 0 , S , Z i , scalar β satisfying the following

LMIs

⎡ψ 11 ⎢ ∗ Ψi = ⎢ ⎢∗ ⎢ ⎢⎣ ∗

ψ 12 Bω SCz T ⎤ ⎥ ψ 12 β Bω 0 ⎥ ∗ −γ 2I 0 ∗ ∗ −I

⎥ ⎥ ⎥⎦

< 0 , i = 1, 2,

(9)

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Where ψ 11 = Ai S T + SAiT + BZ i + Z iT BT , ψ 12 = P − ES T + β SAiT + β Z iT BT , ψ 22 = − β SE T − β ES T . Furthermore, the state-feedback controller gain matrices are described as (7) with K i = Z i S −T , i = 1, 2 .

(10)

Theorem 2: The system (5) is robustly stabilizable with the performance z

2

<γ ω

2

for all non-

zero ω ∈ L2 [ 0, ∞ ) , if there exist P > 0 , S , Z i , scalar β and positive scalars υ1 ,υ2 ,υ3 satisfying the following LMIs ⎡ ϒi M12 ⎤ ⎢ ⎥ < 0 , ( i = 1, 2 ) , ⎣* M 22 ⎦

(11)

Where T

T

ϒi = Ψ i + υ1Γ1Γ1T + υ2 Γ 2 Γ 2T + υ3Γ 3Γ 3T , Γ1 = α mˆ ⎡⎣ − f 2T − β f 2T 0 0 ⎤⎦ , Γ 2 = δ kˆ ⎡⎣ f 2T β f 2T 0 0 ⎤⎦ , T T T Γ3 = λ bˆ ⎡⎣ f 2T β f 2T 0 0 ⎤⎦ , Μ12 = [ H1 H 2 H 3 ] , H1 = ⎡⎣ 0 f 2T S T 0 0 ⎤⎦ , H 2 = ⎡⎣ f1T S T 0 0 0 ⎤⎦ , T

H 3 = ⎡⎣ f 2T S T 0 0 0 ⎤⎦ , M 22 = diag {−υ1 , −υ2 , −υ3 } , Ψ i (i = 1, 2) are the same defines as theorem 1. Furthermore, the state-feedback controller gain matrix description is the same as(10).

4. NUMERICAL EXAMPLE

Simulation results are obtained for the system showing in figure 1 having the following actual parameters: [ m, m ] = [ 200,300] kg , Ps = 10MPa , Ω = 5550mm 2 , ⎡⎣ b , b ⎤⎦ = [16, 22] N / (mm / s ) , K v = 0.45 , τ v = 0.00636 , Ctm = 15mm5 Ns , Vt = 1.75 × 106 mm3 , β e = 700 N mm 2 , ⎡⎣ k , k ⎤⎦ = [ 65, 75] N mm , Cd w ρ = 3.42 ×104 mm3 N s . By defining Max( PL ) = 90% × Ps = 9 MPa , we obtain θ1 = 1, θ 2 = 10 . Assuming ⎡ S1 0 ⎤ S= ⎢ ⎥ , Z1 = [ Z11 0] , Z 2 = [ Z 21 0] , ⎣ S 2 S3 ⎦

where S1 ∈ R 3×3 , S3 ∈ R1×1 , S2 , Z11 , Z 21 ∈ R1×3 , solving the LMIs(11), we obtain the solution as follows:

β = 0.41 , K1 = [ -1.6421 0.0413 - 0.0071 0] , K 2 = [ - 0.9212 - 0.0154 - 0.0031 0] . For description in brevity, we denote this designed controller as controller I thereafter. By the Corollary 1 in [9], we obtain the fixed gain controller II: K = [ -1.5541 0.0023 - 0.0072 0] . In order to verify the dynamics of the closed-loop system, the step response simulation results are demonstrated in Fig.2 and Fig.3. It is observed from Fig. 2 that the controller I obtained in this paper has much less overshoot than the controller II. Fig.3 shows the smooth input signal obtained by controller I. At the same time, there is a parameter β for us to adjust to obtain the satisfying performance. 5. CONCLUSION

The problem of robust controller synthesis for a class of electro-hydraulic servo systems driven by double-rod hydraulic actuators is investigated in this research. Firstly, by introducing a linear varying

Falu Weng et al.Y./ Procedia Engineering 15 (2011) 421 – 425 F. Weng, L. Liang, Ding/ Procedia Engineering 00 (2011) 000–000

parameter, the system is described as a linear parameter varying (LPV) model. Secondly, based on the LPV model, the sufficient strict LMI conditions for the robust stabilizability of the uncertain systems is proposed by using parameter-dependent Lyapunov method. Then by solving those LMIs, the desired parameter-dependent controller is obtained. Finally, the numerical examples are given to show the effectiveness of the proposed theorems.

Fig. 2. Trajectory tracking (actual position: dot line, desired: dark line)

Fig. 3. servo valve control input

Acknowledge

This work was supported by the National Natural Science Foundation (No.51064008) and Jiangxi provincial Science Foundation (No.GJJ08273) of China. References [1] Raade JW, Kazerooni H. Analysis and design of a novel hydraulic power source for mobile robots. IEEE Trans. Autom. Sci. Eng. 2005;2:226-232. [2] Daohang S, Vladimir BB, Huayong Y. New model and sliding mode control of hydraulic elevator velocity tracking system. Simul. Practice Theory. 2002;9:365-385. [3] Qiao F, Sun S, Sun J, and Zhu Q. Sliding Mode Control Design of Active Vehicle Suspension Systems with Two-Time Scale Submodels. Adv. Sci. Lett. 2011;4:953-957. [4] Guan C, Pan S. Adaptive sliding mode control of electro-hydraulic system with nonlinear unknown parameters. Control Engineering Practice. 2008;6:1275-1284. [5] Mao WJ, Zhang YY. Robust Control of Linear Continuous-Time Polytopic Systems and Application to an Electro-Hydraulic Servo System. Intelligent Robotics and Applications, Lecture Notes in Computer Science. 2008;5314:401-409. [6] Guan C, Pan S. Nonlinear Adaptive Robust Control of Single-Rod Electro-Hydraulic Actuator With Unknown Nonlinear Parameters. IEEE Transactions on Control systems Technology. 2008;16:434-445. [7] Merritt HE. Hydraulic Control Systems. Wiley, New York; 1967. [8] Alleyne A. Nonlinear force control of an electro-hydraulic actuator. Proceedings of the Japan/USA Symposium on Flexible Automation. Boston, USA;1996, p. 193-200. [9] Yu J, Ding Y, and Weng F. Robust Control for Electro-hydraulic Servo Systems Driven by Double-rod Actuators. The 3rd IEEE International Conference on Advanced Computer Control. Harbin, China; 2011. p. 160-164.

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