Research Article T-S Fuzzy-Based Optimal Control for Minimally Invasive Robotic Surgery with Input Saturation

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1 Journal of Sensors Volume 218, Article ID , 9 pages Research Article -S Fuzzy-Based Optimal Control for Minimally Invasive Robotic Surgery with Input Saturation Faguang Wang, 1,2 Hongmei Wang, 1 Yong Zhang, 1 and Xijin Guo 1 1 School of Information and Control Engineering, China University of Mining and echnology, No. 1 Daxue Road, Xuzhou , China 2 Collaborative Innovation Center of Intelligent Mining Equipment, China University of Mining and echnology, No. 1 Daxue Road, Xuzhou , China Correspondence should be addressed to Hongmei Wang; whm99@cumt.edu.cn Received 23 December 217; Accepted 25 March 218; Published 23 April 218 Academic Editor: Mucheol Kim Copyright 218 Faguang Wang et al. his is an open access article distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A minimally invasive surgery robot is difficult to control when actuator saturation exists. In this paper, a akagi-sugeno fuzzy model-based controller is designed for a minimally invasive surgery robot with actuator saturation, which is difficult to control. he contractively invariant ellipsoid theorem is applied for the actuator saturation. he proposed scheme can be derived using the H-infinity control theorem and parallel distributed compensation. he result is rebuilt in the form of linear matrix inequalities for easier calculation by computer. Meanwhile, the uniformly ultimately bounded stable and the prescribed H-infinity control performance can be guaranteed. he proposed scheme is simulated in a Novint Falcon haptic device system. 1. Introduction In minimally invasive robotic surgery, the work space is limited precisely. he mechanical structure and the electrical characteristics can be additional constraint boundaries for system inputs and outputs. herefore, the controller of the surgery robot has a rigid input saturation requirement. However, if input saturation occurs, the output performance of the controlled object cannot satisfy the designed requirement, which can result in the decline of the closed-loop system response. he output overshoot cannot be suppressed well, even becoming unstable [1, 2]. his situation is prohibited in a minimally invasive robotic surgery. Many researchers investigated the input saturation problem and provided some solutions. Buckley [3] proposed an anti-reset windup method for the integral saturation problem. he error between the controller output and object input was used as a complementary feedback for the controlled system. Hanus et al. [4] proposed the condition technique, where the controller output continuously tracks for a new reference input and is located out of the saturation area to avoid the saturation case. he pole-placement method was applied to the antiwindup control, which assigns the poles of nonlinear system in the desired disk for stable analysis in [5]. Although the variable structure antiwindup controller showed a good performance in the integrator windup case [6], some preset parameters were needed that were difficult to determine because expert knowledge was required. Some antiwindup controllers were also based on observer [7], internal model control [8], saturation feedback control [9], and dynamic complement [1]. he -S fuzzy theorem, which is an important part of the fuzzy control theorem, was proposed by akagi and Sugeno in 1985 [11]. It is utilized in both system stable control and

2 2 Journal of Sensors model identification [11, 12]. Specifically, it has a good performance in nonlinear system control [12]. he -S fuzzy model, which is the summation of the product of the -S fuzzy local models and their corresponding membership functions, can approximate the nonlinear system under an arbitrary degree of accuracy [12]. he -S fuzzy model is nonlinear, and its controller design is difficult. Consequently, a procedure called parallel distributed compensation (PDC) makes the -S fuzzy controller design easier. PDC was proposed by Wang et al. in 1995 [13]. he fuzzy sets of the PDC controller are similar with the fuzzy model. Under each controller rule, a controller can be designed for the local -S model. By summation of the product of these controllers and their corresponding membership functions, the total controller of a nonlinear system can be represented. System stability is usually considered by the yapunov function. Although PDC provides a design procedure for the -S fuzzy controller, the calculation of controller gains for global stability is still difficult, particularly in the presence of many controller rules. A numerical optimization method called linear matrix inequalities (MIs) solved this kind of problem. MI was defined by Willems in 1971 [14]. Neestorv and Nemirovskii [15] proposed an interior point method, which can directly solve the MI convex optimization problem. MI was first applied to the -S fuzzy system stabilization analysis in [16]. In the succeeding years, MI became the focus of increasing number of researchers [17]. Excepor a few special cases without analytical solution, MIs are generally efficient [11]. Furthermore, the MI toolbox software provides a direct shortcut to the computer solution of MIs. Because PDC and MIs provide a better -S fuzzy system controller design, they are applied in the proposed controller design in this paper. For the advantage of the -S fuzzy theorem, it was widely used to deal with the actuator saturation problem. In [18], based on the -S fuzzy model, a robust dissipative controller was designed for the multiple-input multipleoutput (MIMO) system with saturated time-delay input and parameter uncertainty. heir results show that the closedloop system can be stable, but the stabilization time cannot be guaranteed. For this problem, [19, 2] provided a finitetime control by optimal control and estimated the attraction domain. he combination of the -S fuzzy model and optimal control show an animated controller design of a nonlinear system with actuator saturation [21]. For optimal control, Hu et al. [22] provided a useful control method for the saturation system with actuator saturation based on a contractively invariant ellipsoid. his study was continued in her work by BMIs [8]. Consequently, many researchers focused on actuator saturation problems, where the -S fuzzy based controller design method was popular. he yapunov stability criterion-based PDC fuzzy controller was designed for the actuator saturation system [23, 24]. Many researchers contributed in completing this theorem [25, 26]. In this paper, the result in Hu et al. [22] was adopted and rebuilt into a set of MIs. Meanwhile, a predetermined H norm was satisfied. his paper is organized as follows. he basic -S fuzzy theorem is presented in Section 2. he proposed solution for the actuator saturation problem is presented in Section 3. he simulation is presented in Section 4. he paper concludes in Section General -S Fuzzy Model and Control In this paper, the considered nonlinear system with disturbance W t is Xt = f X t + gxt Ut + Wt, 1 where X t R n 1, f X t R n 1, g X t R n m, U t R m 1, and W t W b and W b is the boundary of disturbance. In the -S fuzzy model, the premise variables of the -S fuzzy rules must be measurable, and they can represent some properties of the nonlinear system. herefore, a suitable selection of these variables is very importanor the accuracy and reliability of the -S fuzzy model. When the -S fuzzy premise variables have been defined, a corresponding local model can be provided by these variable values. he ith rule of the -S fuzzy system for a nonlinear system is as follows: he ith rule: If z 1 t is M i1,, and z p t is M ip, then Xt = A i Xt + B i Ut, 2 where i =1,2,, is the rule number, A i R n n and B i R n m are the local model parameter matrices, z 1, z 2,, z p are the premise variables, and M i1,, M ip are the fuzzy sets. he nonlinear system can be approximated by the overall -S fuzzy system Xt = h i Zt A i Xt + B i Ut, 3 where Z t = z 1 t z 2 t z p t and h i Z t = μ i Z t / μ i Z t, μ i Z t = p θ ij z j t in which θ ij Z t is the grade of the membership of z j t in M ij. Note that the membership function should satisfy the following equation: h i Zt, h i Zt =1 Currently, the nonlinear system of (1) is changed to the -S fuzzy model (3). Because the local -S model of (2) is linear, its feedback controller can be designed easily as U j = K j Xt, 5 where K j R m n and j =1,2,,. By PDC and similar fuzzy rules of (2), the overall -S fuzzy controller is designed as Ut = h j Zt K j Xt 6 4

3 Journal of Sensors 3. H Robust Controller Design for Input Saturation he traditional -S fuzzy theorem cannot provide results in the control problem for a nonlinear system with input saturation. In this study, the proposed method can solve the input saturation problem. he controlled closed-loop system can be uniformly ultimately bounded (UUB) stable, and the prescribed H-infinity norm can be guaranteed. Furthermore, the result is shown in MIs, which can be directly solved by programming. Considering the saturation and uncertainty, the nonlinear system of (1) can be where Xt = f X t + gxt Sat Ut + Wt, 7 Sat Ut = sgn Ut b, Ut > b, Ut, Ut b, b >, is the boundary value of the saturation function, and the other symbols are similar with (1). Following (3), the -S fuzzy model can be where Xt = h i A i Xt + B i Sat Ut, 9 Ut = h j Zt U j t 1 Based on the work of Hu et al. [1], for the feedback controller, U j t = K j X t, if the ellipsoid ε P,1 is contractively invariant, and matrixes P >, K i, and H (m n matrix) make the initial value of X ε P,1 and ε P,1 l H, 11 where the polyhedron l H X t R n H j X t 1, j 1, m and H j is the jth row of H. he saturation feedback controller Sat U t satisfies Sat U j t = Sat K j Xt co D r K j Xt + D r H jxt, r 1, 2 m, where D r is an m-by-m diagonal matrix (diagonal elements are or 1) and D r = I D r with r 1, 2 m. If it is only applied to the -S fuzzy local model, the global stability of the nonlinear system cannot be satisfied. For global stability, some changes are needed. By (7) and (9), the nonlinear system with disturbance is as follows: 8 12 Xt = f X t + gxt Sat Ut + Wt = h i Zt A i Xt + B i Sat Ut + f X t h i Zt A i Xt + gxt h i Zt B i Sat Ut + Wt 13 Equation (12) can be guaranteed if each D r satisfies the following inequality: X h i Zt = where A i Xt + B i h j Zt + f X t h i Zt + gxt h i Zt A i Xt B i D r K j Xt + D r H jxt h j Zt D r K j Xt + D r H jxt + Wt h i Zt h j Zt A i + B i D r K j + D r H j Xt + Δf + Δg + Wt, Δf = f X t h i Zt A i Xt, Δg = h i Zt h j Zt gxt B i D r K j + D r H j Xt Some δ Ai 1, δ Bi 1, A t, and B t can be found and satisfy the following inequality: Δf = A t Xt, h i Zt ΔA i Xt h i Zt Δg h j Zt B t D r K j + D r H j Xt, where ΔA i = δ Ai A t and ΔB i = δ Bi B t. H control performance is defined as δ Ai A t Xt X t QX t dt t < ρ, 17 f W t Wtdt 3

4 4 Journal of Sensors where is the terminal time, Q is a positive-definite matrix, and ρ is the prescribed H norm, which is greater than and less than 1. If ρ is minimized, the effect of W t on X t is minimized. Considering the initial condition, (17) can be changed as X t QX t dt < X PX + ρ W t Wtdt, 18 where P is some symmetric positive-definite weighting matrix. Set the yapunov function as Vt = X t PX t, 19 where P >. For simplicity, (t) is omitted in the following sections. hen, the following theorem can be obtained. heorem 1. Under condition (11), if controller (12) is applied to a nonlinear system (7) and there exists a positive-definite matrix P >, such that the following matrix inequalities PA i + A i P + PB i D r K j + D r H j B i P + A t A t + D r K j + D r H j +2PP + ρ 2 PP + Q i < + D r K j + D r H j B t B t D r K j + D r H j are satisfied for each i, j, and r, then the closed-loop system is UUB, and the H control performance (18) is guaranteed as prescribed ρ 2. Proof 1. Using (14), the derivative of V t is Vt = X PX + X PX = X P l + h i A i X + B i Sat U + W + PX h i h j A i + B i D r K j + D r H j l h i A i X + B i Sat U + W I=1 h i h j A i + B i D r K j + D r H j PX X P X + Δf + Δg + W h i h j X PA i + B i D r K j + D r H j + A i + B i D r K j + D r H j + Δf Δf Using (16), (21) can be X + Δf + Δg + W P X + X PPX + Δg Δg + X PPX + X PW + W PX 2 21 V = h i h j X PA i + B i D r K j + D r H j + A i + B i D r K j + D r H j + X PPX + B t D r K j + D r H j X P X + A t X A t X B t D r K j + D r H j X + X PPX + X PW + W PX h i h j X + D r K j + D r H j PA i + A i P + PB i D r K j + D r H j B i P + A t A t + D r K j + D r H j B t B t D r K j + D r H j +2PP X ρ 1 PX ρw ρ 1 PX ρw + ρ 2 W W + ρ 2 X PPX h i h j X + D r K j + D r H j + D r K j + D r H j ++2PP Under the ith rule, if there is X t h 2 i Q i Xtdt W t Wtdt where Q i >and h iq i PA i + A i P + PB i D r K j + D r H j B i P + A t A t B t B t D r K j + D r H j X + ρ 2 W W + ρ 2 X PPX 22 X t Q i Xtdt < ρ, 23 W t Wtdt >, there is X t h iq i Xtdt t < ρ 24 f W t Wtdt for the nonlinear system. If h iq i = Q, (24) can be the H control performance defined in (17). herefore, for each simulation time, the H control performance defined as (17) can be guaranteed. hen, in the total simulation, a prescribed H norm can be guaranteed. Meanwhile, (18) can be as follows: t= t= If X h i Q i X dt < X PX + ρ t= t= W W dt PA i + A i P + PB i D r K j + D r H + D rk j + D r H B i P there is + A t A t + D r K j + D r H B t B t D r K j + D r H +2PP < ρ 2 PP Q i, 25 26

5 Journal of Sensors 5 V < h i h j X ρ 2 PP Q i X + ρ 2 W W + ρ 2 X PPX = X ρ 2 PP h i Q i + ρ 2 X PPX = X h i Q i X + ρ 2 W W X + ρ 2 W W 27 By integrating (27), the following result can be obtained: V V < t= t= X h i Q i X + ρ 2 W W dt 28 Figure 1: Novint Falcon. he yapunov function (19) can be when the system is stable. herefore, t= t= X h i Q i X dt < X PX + ρ 2 t= t= W W dt he above equation is the H performance defined in (25), which satisfies (18). Note that the H norm is changed to ρ 2. End of proof. 29 Motion platform c 휃 3i 휃 2i 휃 1i a b y z x ouch point In heorem 1, it is difficult to find the solution for (2). herefore, the transformation of inequality (2) into MIs is necessary. By setting R = P 1 and left and right multiplication of R, (2) can be A i R + RA i + B i D r K j + D r H j R + RD r K j + D r H j + RA t A t R + RD r K j + D r H j +2+ρ 2 + RQ i R < B t B t B i D r K j + D r H j R et K i = K i R, Ĥ i = H i R, and Q i = A t A t + Q i 1.By Schur complements, (3) can be changed as A B t D r K j + D r Ĥ j B t D r K j + D r Ĥ j I R Q i R <, where A = A i R + RA i + B i D r K j + D r Ĥ j + D r K j + D r Ĥ j B i + 2+ρ 2 I, R, K j, and Ĥ j are unknown. Condition (11) is equal to 1 Ĥ ik Ĥ ik R 3 31, 32 where Ĥ ik is the kth row of Ĥ i. herefore, the following theorem is given. heorem 2. For nonlinear system (7) with controller (12), if there exists min ρ 2, subject to 1 Ĥ ik Ĥ ik R >, Figure 2: Mathematical model of the single chain. R A B t D r K j + D r Ĥ j B t D r K j + D r Ĥ j I, R Q i R <, then the closed-loop system is UUB and the H control performance (18) is guaranteed as prescribed ρ 2. By this theorem, the H optimization problem can be changed to a constrained optimization problem by MIs. 33

6 6 Journal of Sensors.1 Controlled by the proposed method of this paper Error of x1 (rad) ime (s) Controlled by the proposed method of the comparison paper Error of x1 (rad) ime (s) Figure 3: he error of x Numerical Example In minimally invasive surgery, haptic tracking is very important. Novint Falcon is a simple arm robot with force feedback ability, which benefited from parallel architecture, and can be applied in minimally invasive surgery. he Novint Falcon structure is shown in Figure 1. he Novint Falcon consists of the base, motion platform, and 3 chains. he single chain can be abstract presented in Figure 2. Based on [27] and ignoring some nonessentials, the ith chain dynamic equation is τ = D θ θ + H θ, θ θ + G θ, 34 where τ = τ 1, τ 2, τ 3 is the moment of driving force for each link; θ = θ 1i, θ 2i, θ 3i is shown in Figure 2; D θ = I + J 1 3m b + m c J 1, where I is the moment of inertia and m b and m c are masses of links b and c; H θ, θ = D f + J 1 3m b + m c J 1, where D f denotes the damped coefficient; G θ = M g is the gravity moment; J = J 11 J 12 J 13 J 21 J 22 J 23 J 31 J 32 J 33 1 α 1 α 2 35 α 3 is the Jacobian matrix; J i1 = cos θ 2i sin θ 3i cos ϕ i cos θ 3i sin ϕ i, J i2 = cos θ 3i cos ϕ i + cos θ 2i sin θ 3i sin ϕ i, J i3 = sin θ 2i sin θ 3i, α i = sin θ 2i θ 1i sin θ 3i, and ϕ are the angle between the chain and the coordinate frame (x, y, z). Because the exact parameters are difficult to determine, set I and M g as identity matrices and 3m b + m c =1in this example. his is reasonable in this paper because the error between the actual value and the required value can be distributed in the disturbance or parameter uncertainty for this system. herefore, the matrix D θ is invertible. Set θ 1i = x 1 t, θ 2i = x 2 t, θ 3i = x 3 t, τ = u t, and x 1 t = x 4 t, x 2 t = x 5 t, x 3 t = x 6 t he θ of the other chains are set to be closed to require values, and the errors are distributed in the disturbance. herefore, the system (34) can be as follows: x 4 t x 5 t x 6 t = A x 4 t x 5 t x 6 t 36 + Bu t + E, 37 where A = D θ 1 H θ, θ, B = D θ 1, and E = D θ 1 G θ. Considering actuator saturation, the corresponding ith fuzzy rule is as follows: Rule i. IfA is A i and B is B i, then (37) is x 4 t x 5 t x 6 t = A i x 4 t x 5 t x 6 t + B i Sat ut, 38 where A 1 = A max θ 1i,max θ 2i,max θ 3i, A 2 =A min θ 1i, max θ 2i,max θ 3i, A 3 = A max θ 1i,min θ 2i,max θ 3i, A 4 = A max θ 1i,max θ 2i,min θ 3i, A 5 = A min θ 1i,min θ 2i,max θ 3i, A 6 = A min θ 1i,max θ 2i,min θ 3i, A 7 =

7 Journal of Sensors 7 Error of x2 (rad) Error of x2 (rad) Controlled by the proposed method of this paper ime (s) Controlled by the proposed method of the comparison paper ime (s) Figure 4: he error of x 2. A max θ 1i,min θ 2i,min θ 3i, and A 8 = A min θ 1i,min θ 2i,min θ 3i ; B i is calculated similarly, and E is treated as a disturbance. Although A i depends on θ, the values of max θ and min θ do not depend on the system input. herefore, they can be used for the fuzzy rule definition. Set θ 1i, 7854, θ 2i, 7854, and θ 3i , Sat u t = sgn u t min u t,1. he H norm is.3. he reference value is x 1r x 2r x 3r x 4r x 5r x 6r = Using heorem 2, the error system feedback controller gains can be calculated as follows: K1 = , , ;3 1 4,65 1 4, ; ,12 1 5,77 1 3, K2 = ,16 1 5, ; , , ; , , , K3 = , , ;14 1 4,11 1 5, ;24 1 4,36 1 5, , K4 = ,29 1 5, ; ,22 1 3, ; , , , K5 = 2 1 4, , ;97 1 3,51 1 4, ;98 1 3,18 1 5, , 4 K6 = ,13 1 5, ; , , ; , , , K7 = , 3 1 5, ;44 1 3, , ; ,4 1 5,25 1 4, K8 = ,65 1 4, ; ,19 1 4, ;11 1 3, , o test the robustness of the controller, a disturbance Wt = sin 12t 3 is added in (38). For comparison, a controller designed by heorem 2 of [18] is applied. In [18], the controller was designed for the system with input saturation and disturbance. If the time delay is ignored in [18], its results can be applied in this example. he simulation results are shown in the following figures. 41 From Figures 3 5, less overshooting can be observed compared with the reference paper. he proposed controller performed well with input saturation, and the H performance can be achieved. 5. Conclusion In this paper, the -S fuzzy control theorem and the ability of contractively invariant ellipsoid were applied for nonlinear systems with input saturation. he proposed method was

8 8 Journal of Sensors.15 Controlled by the proposed method of this paper Error of x3 (rad) ime (s).15 Controlled by the proposed method of the comparison paper Error of x3 (rad) ime (s) Figure 5: he error of x 3. described by MIs, which can be calculated by a computer. he proposed theorem was tested using the Novint Falcon, and a good result was achieved. Comparing the results, the proposed method of this paper has given a better result. Meanwhile, the prescribed H-infinity norm was satisfied. In the example, only eight rules of the -S fuzzy system were applied; there were still more than one hundred MIs that need to be calculated. Although a feasible solution of the MIs can be guaranteed using the proposed method, a method with less MIs is still necessary for further study. Conflicts of Interest he authors declare that they have no conflicts of interest. Acknowledgments his work was supported in part by the National Natural Science Foundation of China ( ) and the Fundamental Research Funds for the Central Universities (213QNB26). References [1]. Hu, A. R. eel, and. Zaccarian, Stability and performance for saturated systems via quadratic and nonquadratic yapunov functions, IEEE ransactions on Automatic Control, vol. 51, no. 11, pp , 26. [2] S. Sajjadi-Kia and F. Jabbari, Controllers for linear systems with bounded actuators: slab scheduling and anti-windup, Automatica, vol. 49, no. 3, pp , 213. [3] P. Buckley, Designing override and feedforward controls, Control Engineering, vol. 18, no. 8, pp , [4] R. Hanus, M. Kinnaert, and J.-. Henrotte, Conditioning technique, a general anti-windup and bumpless transfer method, Automatica, vol. 23, no. 6, pp , [5] F.-G. Wang, H.-M. Wang, S.-K. Park, and X.-S. Wang, inear pole-placement anti-windup control for input saturation nonlinear system based on akagi Sugeno fuzzy model, International Journal of Control, Automation and Systems, vol. 14, no. 6, pp , 216. [6] V. D. Hajare, A. A. Khandekar, and B. M. Patre, Discrete sliding mode controller with reaching phase elimination for IO systems, ISA ransactions, vol. 66, pp , 217. [7] W. iu, J. u, Z. Zhang, and S. Xu, Observer-based neural control for MIMO pure-feedback non-linear systems with input saturation and disturbances, IE Control heory & Applications, vol. 1, no. 17, pp , 216. [8] A. ekka, M. C. urner, and P. P. Menon, Anti-windup for a class of partially linearisable non-linear systems with application to wave energy converter control, IE Control heory & Applications, vol. 1, no. 18, pp , 216. [9] C. Hua, Y. Yang, and P. X. iu, Output-feedback adaptive control of networked teleoperation system with time-varying delay and bounded inputs, IEEE/ASME ransactions on Mechatronics, vol. 2, no. 5, pp , 215. [1]. Zaccarian, Y. i, E. Weyer, M. Cantoni, and A. R. eel, Anti-windup for marginally stable plants and its application to open water channel control systems, Control Engineering Practice, vol. 15, no. 2, pp , 27. [11]. akagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE ransactions on Systems, Man, and Cybernetics, vol. SMC-15, no. 1, pp , [12] K. anaka and H. O. Wang, Fuzzy Control Systems Design and Analysis, John Wiley & Sons Inc., New York, NY, USA, 21. [13] H. O. Wang, K. anaka, and M. F. Griffin, Parallel distributed compensation of nonlinear systems by akagi-sugeno fuzzy model, in Proceedings of 1995 IEEE International Conference on Fuzzy Systems, pp , Yokohama, Japan, 1995.

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