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

Authors: Faguang Wang; Hongmei Wang; Yong Zhang; Xijin Guo
Journal: Journal of Sensors
 (2018)

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## Abstract

A minimally invasive surgery robot is difficult to control when actuator saturation exists. In this paper, a Takagi-Sugeno fuzzy model-based controller is designed for a minimally invasive surgery robot with actuator saturation, which is difficult to control. The contractively invariant ellipsoid theorem is applied for the actuator saturation. The proposed scheme can be derived using theH-infinity control theorem and parallel distributed compensation. The 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. The proposed scheme is simulated in a Novint Falcon haptic device system.


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## Body

## 1. Introduction

In minimally invasive robotic surgery, the work space is limited precisely. The mechanical structure and the electrical characteristics can be additional constraint boundaries for system inputs and outputs. Therefore, 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. The output overshoot cannot be suppressed well, even becoming unstable [1, 2]. This 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. The 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. The 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 [10].The T-S fuzzy theorem, which is an important part of the fuzzy control theorem, was proposed by Takagi and Sugeno in 1985 [11]. It is utilized in both system stable control and model identification [11, 12]. Specifically, it has a good performance in nonlinear system control [12]. The T-S fuzzy model, which is the summation of the product of the T-S fuzzy local models and their corresponding membership functions, can approximate the nonlinear system under an arbitrary degree of accuracy [12]. The T-S fuzzy model is nonlinear, and its controller design is difficult. Consequently, a procedure called parallel distributed compensation (PDC) makes the T-S fuzzy controller design easier. PDC was proposed by Wang et al. in 1995 [13]. The fuzzy sets of the PDC controller are similar with the fuzzy model. Under each controller rule, a controller can be designed for the local T-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 Lyapunov function. Although PDC provides a design procedure for the T-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 (LMIs) solved this kind of problem. LMI was defined by Willems in 1971 [14]. Neestorv and Nemirovskii [15] proposed an interior point method, which can directly solve the LMI convex optimization problem. LMI was first applied to the T-S fuzzy system stabilization analysis in [16]. In the succeeding years, LMI became the focus of increasing number of researchers [17]. Except for a few special cases without analytical solution, LMIs are generally efficient [11]. Furthermore, the LMI toolbox software provides a direct shortcut to the computer solution of LMIs. Because PDC and LMIs provide a better T-S fuzzy system controller design, they are applied in the proposed controller design in this paper.For the advantage of the T-S fuzzy theorem, it was widely used to deal with the actuator saturation problem. In [18], based on the T-S fuzzy model, a robust dissipative controller was designed for the multiple-input multiple-output (MIMO) system with saturated time-delay input and parameter uncertainty. Their results show that the closed-loop system can be stable, but the stabilization time cannot be guaranteed. For this problem, [19, 20] provided a finite-time control by optimal control and estimated the attraction domain. The combination of the T-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. This study was continued in her work by BMIs [8]. Consequently, many researchers focused on actuator saturation problems, where the T-S fuzzy based controller design method was popular. The Lyapunov 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 LMIs. Meanwhile, a predetermined H∞ norm was satisfied.This paper is organized as follows. The basic T-S fuzzy theorem is presented in Section2. The proposed solution for the actuator saturation problem is presented in Section 3. The simulation is presented in Section 4. The paper concludes in Section 5.


## 2. General T-S Fuzzy Model and Control

In this paper, the considered nonlinear system with disturbanceWt is
(1)Ẋt=fXt+gXtUt+Wt,where Xt∈Rn×1, fXt∈Rn×1, gXt∈Rn×m, Ut∈Rm×1, and Wt≤Wb and Wb is the boundary of disturbance. In the T-S fuzzy model, the premise variables of the T-S fuzzy rules must be measurable, and they can represent some properties of the nonlinear system. Therefore, a suitable selection of these variables is very important for the accuracy and reliability of the T-S fuzzy model. When the T-S fuzzy premise variables have been defined, a corresponding local model can be provided by these variable values. The ith rule of the T-S fuzzy system for a nonlinear system is as follows:Theith rule: If z1t is Mi1, …, and zpt is Mip, then
(2)Ẋt=AiXt+BiUt,where i=1,2,…,L is the rule number, Ai∈Rn×n and Bi∈Rn×m are the local model parameter matrices, z1,z2,…,zp are the premise variables, and Mi1,…,Mip are the fuzzy sets. The nonlinear system can be approximated by the overall T-S fuzzy system
(3)Ẋt=∑i=1LhiZtAiXt+BiUt,where Zt=z1tz2t⋯zpt and hiZt=μiZt/∑i=1LμiZt, μiZt=∏j=1pθijzjt in which θijZt is the grade of the membership of zjt in Mij. Note that the membership function should satisfy the following equation:
(4)hiZt≥0,∑i=1LhiZt=1.Currently, the nonlinear system of (1) is changed to the T-S fuzzy model (3). Because the local T-S model of (2) is linear, its feedback controller can be designed easily as
(5)Uj=KjXt,where Kj∈Rm×n and j=1,2,…,L. By PDC and similar fuzzy rules of (2), the overall T-S fuzzy controller is designed as
(6)Ut=∑j=1LhjZtKjXt.


## 3.H∞ Robust Controller Design for Input Saturation

The traditional T-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. The controlled closed-loop system can be uniformly ultimately bounded (UUB) stable, and the prescribedH-infinity norm can be guaranteed. Furthermore, the result is shown in LMIs, which can be directly solved by programming.Considering the saturation and uncertainty, the nonlinear system of (1) can be
(7)Ẋt=fXt+gXt⋅SatUt+Wt,where
(8)SatUt=sgnUt⋅b,Ut>b,Ut,Ut≤b,b>0, is the boundary value of the saturation function, and the other symbols are similar with (1). Following (3), the T-S fuzzy model can be
(9)Ẋt=∑i=1LhiAiXt+BiSatUt,where
(10)Ut=∑j=1LhjZtUjt.Based on the work of Hu et al. [1], for the feedback controller, Ujt=KjXt, if the ellipsoid εP,1 is contractively invariant, and matrixes P>0, Ki, and H (m×n matrix) make the initial value of X0∈εP,1 and
(11)εP,1⊂ℓH,where the polyhedron ℓH≔Xt∈Rn:HjXt≤1,j∈1,m and Hj is the jth row of H. The saturation feedback controller SatUt satisfies
(12)SatUjt=SatKjXt∈coDrKjXt+Dr−HjXt,r∈1,2m,where Dr is an m-by-m diagonal matrix (diagonal elements are 0 or 1) and Dr−=I−Dr with r∈1,2m.If it is only applied to the T-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:
(13)Ẋt=fXt+gXtSatUt+Wt=∑i=1LhiZt⋅AiXt+BiSatUt+fXt−∑i=1LhiZtAi⋅Xt+gXt−∑i=1LhiZtBiSatUt+Wt.Equation (12) can be guaranteed if each Dr satisfies the following inequality:
(14)Ẋ≤∑i=1LhiZtAiXt+Bi∑j=1LhjZtDrKjXt+Dr−HjXt+fXt−∑i=1LhiZtAiXt+gXt−∑i=1LhiZtBi∑j=1LhjZtDrKjXt+Dr−HjXt+Wt=∑i=1L∑j=1LhiZthjZtAi+BiDrKj+Dr−HjXt+Δf+Δg+Wt,where
(15)Δf=fXt−∑i=1LhiZtAiXt,Δg=∑i=1LhiZt∑j=1LhjZtgXt−Bi⋅DrKj+Dr−HjXt.SomeδAi≤1, δBi≤1, At, and Bt can be found and satisfy the following inequality:
(16)Δf=∑i=1LhiZtΔAiXt≤∑i=1LhiZtδAiAtXt≤AtXt,Δg≤∑j=1LhjZtBtDrKj+Dr−HjXt,where ΔAi=δAiAt and ΔBi=δBiBt.H∞ control performance is defined as
(17)∫0tfXTtQXtdt∫0tfWTtWtdt<ρ,where tf is the terminal time, Q is a positive-definite matrix, and ρ is the prescribed H∞ norm, which is greater than 0 and less than 1. If ρ is minimized, the effect of Wt on Xt is minimized. Considering the initial condition, (17) can be changed as
(18)∫0tfXTtQXtdt<XT0PX0+ρ∫0tfWTtWtdt,where P is some symmetric positive-definite weighting matrix. Set the Lyapunov function as
(19)Vt=XTtPXt,where P>0. For simplicity, (t) is omitted in the following sections. Then, the following theorem can be obtained.Theorem 1.
Under condition (11), if controller (12) is applied to a nonlinear system (7) and there exists a positive-definite matrix P>0, such that the following matrix inequalities
(20)PAi+AiTP+PBiDrKj+Dr−Hj+DrKj+Dr−HjT⋅BiTP+AtTAt+DrKj+Dr−HjTBtTBtDrKj+Dr−Hj+2PP+ρ−2PP+Qi<0are satisfied for each i, j, and r, then the closed-loop system is UUB, and the H∞ control performance (18) is guaranteed as prescribed ρ2.Proof 1.
Using (14), the derivative of Vt is
(21)V̇t=XTPẊ+ẊTPX=XTP∑I=1lhiAiX+BiSatU+W+∑i=1lhiAiX+BiSatU+WTPX≤XTP∑i=1L∑j=1LhihjAi+BiDrKj+Dr−HjX+Δf+Δg+W+∑i=1L∑j=1LhihjAi+BiDrKj+Dr−HjX+Δf+Δg+WTPX≤∑i=1L∑j=1LhihjXTPAi+BiDrKj+Dr−Hj+Ai+BiDrKj+Dr−HjTPX+ΔfTΔf+XTPPX+ΔgTΔg+XTPPX+XTPW+WTPX.
Using (16), (21) can be
(22)V̇≤∑i=1L∑j=1LhihjXTPAi+BiDrKj+Dr−Hj+Ai+BiDrKj+Dr−HjTPX+AtXTAtX+XTPPX+BtDrKj+Dr−HjXTBtDrKj+Dr−HjX+XTPPX+XTPW+WTPX=∑i=1L∑j=1LhihjXTPAi+AiTP+PBiDrKj+Dr−Hj+DrKj+Dr−HjTBiTP+AtTAt+DrKj+Dr−HjT⋅BtTBtDrKj+Dr−Hj+2PPX−ρ−1PX−ρWTρ−1PX−ρW+ρ2WTW+ρ−2XTPPX≤∑i=1L∑j=1LhihjXTPAi+AiTP+PBiDrKj+Dr−Hj+DrKj+Dr−HjTBiTP+AtTAt+DrKj+Dr−HjTBtTBt⋅DrKj+Dr−Hj++2PPX+ρ2WTW+ρ−2XTPPX.
Under theith rule, if there is
(23)∫0tfXTthi2QiXtdt∫0tfWTtWtdt≤∫0tfXTtQiXtdt∫0tfWTtWtdt<ρ,where Qi>0 and ∑i=1LhiQi>0, there is
(24)∫0tfXTt∑i=1LhiQiXtdt∫0tfWTtWtdt<ρfor the nonlinear system. If ∑i=1LhiQi=Q, (24) can be the H∞ control performance defined in (17). Therefore, for each simulation time, the H∞ control performance defined as (17) can be guaranteed. Then, in the total simulation, a prescribed H∞ norm can be guaranteed. Meanwhile, (18) can be as follows:
(25)∫t=0t=tfXT∑i=1LhiQiXdt<XT0PX0+ρ∫t=0t=tfWTWdt.
If(26)PAi+AiTP+PBiDrKj+Dr−H+DrKj+Dr−HTBiTP+AtTAt+DrKj+Dr−HTBtTBtDrKj+Dr−H+2PP<−ρ−2PP−Qi,there is
(27)V̇<∑i=1L∑j=1LhihjXT−ρ−2PP−QiX+ρ2WTW+ρ−2XTPPX=XT−ρ−2PP−∑i=1LhiQiX+ρ2WTW+ρ−2XTPPX=−XT∑i=1LhiQiX+ρ2WTW.
By integrating (27), the following result can be obtained:
(28)Vtf−V0<∫t=0t=tf−XT∑i=1LhiQiX+ρ2WTWdt.
The Lyapunov function (19) can be 0 when the system is stable. Therefore,
(29)∫t=0t=tfXT∑i=1LhiQiXdt<XT0PX0+ρ2∫t=0t=tfWTWdt.
The above equation is theH∞ performance defined in (25), which satisfies (18). Note that the H∞ norm is changed to ρ2.
End of proof.In Theorem1, it is difficult to find the solution for (20). Therefore, the transformation of inequality (20) into LMIs is necessary. By setting R=P−1 and left and right multiplication of R, (20) can be
(30)AiR+RAiT+BiDrKj+Dr−HjR+RDrKj+Dr−HjTBiT+RAtTAtR+RDrKj+Dr−HjTBtTBtDrKj+Dr−HjR+2+ρ−2+RQiR<0.LetK^i=Ki⋅R, H^i=Hi⋅R, and Q^i=AtTAt+Qi−1. By Schur complements, (30) can be changed as
(31)A∗BtDrK^j+Dr−H^jTRBtDrK^j+Dr−H^j−I0R0−Q^i<0,where A∗=AiR+RAiT+BiDrK^j+Dr−H^j+DrK^j+Dr−⋅H^jTBiT+2+ρ−2I,R,K^j, and H^j are unknown. Condition (11) is equal to
(32)1H^ikH^ikTR≥0,where H^ik is the kth row of H^i. Therefore, the following theorem is given.Theorem 2.
For nonlinear system (7) with controller (12), if there exists
(33)minρ2,subjecttoA∗BtDrK^j+Dr−H^jTRBtDrK^j+Dr−H^j−I0R0−Q^i<0,1H^ikH^ikTR≥0,R>0,then the closed-loop system is UUB and the H∞ control performance (18) is guaranteed as prescribed ρ2.By this theorem, theH∞ optimization problem can be changed to a constrained optimization problem by LMIs.


## 4. 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. The Novint Falcon structure is shown in Figure1.Figure 1
Novint Falcon.The Novint Falcon consists of the base, motion platform, and 3 chains. The single chain can be abstract presented in Figure2.Figure 2
Mathematical model of the single chain.Based on [27] and ignoring some nonessentials, the ith chain dynamic equation is
(34)τ=Dθθ¨+Hθ,θ̇θ̇+Gθ,where τ=τ1,τ2,τ3T is the moment of driving force for each link; θ=θ1i,θ2i,θ3iT is shown in Figure 2; Dθ=IT+JT−13mb+mcJ−1, where IT is the moment of inertia and mb and mc are masses of links b and c; Hθ,θ̇=Df+JT−13mb+mcJT−1′, where Df denotes the damped coefficient; Gθ=−Mg is the gravity moment;
(35)J=J11J12J13J21J22J23J31J32J33−1α1000α2000α3is the Jacobian matrix; Ji1=cosθ2isinθ3icosϕi−cosθ3isinϕi, Ji2=cosθ3icosϕi+cosθ2isinθ3isinϕi, Ji3=sinθ2isinθ3i, αi=sinθ2i−θ1isinθ3i, and ϕ are the angle between the chain and the coordinate frame (x,y,z). Because the exact parameters are difficult to determine, set IT and Mg as identity matrices and 3mb+mc=1 in this example. This 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. Therefore, the matrix Dθ is invertible. Set θ1i=x1t, θ2i=x2t, θ3i=x3t, τ=ut, and
(36)ẋ1t=x4t,ẋ2t=x5t,ẋ3t=x6t.Theθ of the other chains are set to be closed to require values, and the errors are distributed in the disturbance. Therefore, the system (34) can be as follows:
(37)ẋ4tẋ5tẋ6t=Ax4tx5tx6t+But+E,where A=−Dθ−1Hθ,θ̇, B=Dθ−1, and E=−Dθ−1Gθ. Considering actuator saturation, the corresponding ith fuzzy rule is as follows:Rulei.
IfA is Ai and B is Bi, then (37) is
(38)ẋ4tẋ5tẋ6t=Aix4tx5tx6t+BiSatut,where A1=Amaxθ1i,maxθ2i,maxθ3i, A2=Aminθ1i,maxθ2i,maxθ3i, A3=Amaxθ1i,minθ2i,maxθ3i,A4=Amaxθ1i,maxθ2i,minθ3i, A5=Aminθ1i,minθ2i,maxθ3i, A6=Aminθ1i,maxθ2i,minθ3i, A7=Amaxθ1i,minθ2i,minθ3i, and A8=Aminθ1i,minθ2i,minθ3i; Bi is calculated similarly, and E is treated as a disturbance. Although Ai depends on θ, the values of maxθ and minθ do not depend on the system input. Therefore, they can be used for the fuzzy rule definition. Set θ1i∈0,0.7854, θ2i∈0,0.7854, and θ3i∈1.9199,2.2689. Satut=sgnut⋅minut,10. The H∞ norm is 0.3. The reference value is
(39)x1rx2rx3rx4rx5rx6r=0.70.72.2000.Using Theorem2, the error system feedback controller gains can be calculated as follows:
(40)K1=−2.9×104,−1.8×105,2.6×104;3.0×104,6.5×104,−3.4×104;−7.9×102,1.2×105,7.7×103,K2=1.8×104,1.6×105,−1.4×104;−1.8×104,−3.2×104,2.1×104;−1.2×102,−1.3×105,−6.9×103,K3=−3.8×104,−4.7×105,2.2×104;1.4×104,1.1×105,−1.2×104;2.4×104,3.6×105,−1.1×104,K4=1.9×104,2.9×105,−7.6×103;−1.7×104,2.2×103,2.2×104;−1.4×103,−2.9×105,−1.4×104,K5=−2.0×104,−2.4×105,1.2×104;9.7×103,5.1×104,−9.5×103;9.8×103,1.8×105,−2.1×103,K6=8.4×103,1.3×105,−3.6×103;−7.5×103,−3.2×103,9.3×103;−9.1×102,−1.2×105,−5.7×103,K7=−1.9×103,−3.0×105,−1.4×104;4.4×103,−1.1×105,−1.1×104;−2.4×103,4.0×105,2.5×104,K8=9.2×102,6.5×104,2.4×103;−2.1×103,1.9×104,3.6×103;1.1×103,−8.4×104,−6.0×103.To test the robustness of the controller, a disturbance(41)Wt=sin12t300Tis added in (38). For comparison, a controller designed by Theorem 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. The simulation results are shown in the following figures.From Figures3–5, less overshooting can be observed compared with the reference paper. The proposed controller performed well with input saturation, and the H∞ performance can be achieved.Figure 3
The error ofx1.Figure 4
The error ofx2.Figure 5
The error ofx3.


## 5. Conclusion

In this paper, the T-S fuzzy control theorem and the ability of contractively invariant ellipsoid were applied for nonlinear systems with input saturation. The proposed method was described by LMIs, which can be calculated by a computer. The 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 prescribedH-infinity norm was satisfied. In the example, only eight rules of the T-S fuzzy system were applied; there were still more than one hundred LMIs that need to be calculated. Although a feasible solution of the LMIs can be guaranteed using the proposed method, a method with less LMIs is still necessary for further study.



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