Anh-Tu Nguyen
email: tnguyen@uphf.fr

Antoine Dequidt

Van-Anh Nguyen

Laurent Vermeiren

Michel Dambrine

Fuzzy Descriptor Tracking Control with Guaranteed L ∞ Error-Bound for Robot Manipulators

Keywords: Robot manipulators, fuzzy control, tracking control, Lyapunov stability, L∞ performance, linear matrix inequality

This paper is concerned with the nonlinear tracking control design for robot manipulators. In spite of the rich literature in the field, the problem has not yet been addressed adequately due to the lack of an effective control design. Using a descriptor fuzzy model-based framework, we propose a new approach to design a feedback-feedforward control scheme for robot manipulators in a general form. The goal is to guarantee a small level of an L∞-gain specification to improve the tracking performance while significantly reducing the numerical complexity for real-time implementation. Based on Lyapunov stability arguments, the control design is formulated as a convex optimization problem involving linear matrix inequalities. Numerical experiments performed with a high-fidelity manipulator benchmark model, embedded in the Simscape Multibody TM environment, demonstrate the effectiveness of the proposed control solution over existing standard approaches.

I. INTRODUCTION

Nowadays, robot manipulators are widely used in all areas of industry for process automation such as material handling, welding, painting, cutting, grinding, etc. These industrial robots have several advantages, including high speed, compactness, accuracy and reliability. For applications such as arc welding, laser cutting or machining, one of the most important tasks is to obtain a satisfactory tracking performance of predefined reference trajectories [START_REF] Dawson | Robot Manipulator Control: Theory and Practice[END_REF]. It is well-known that tracking control is more challenging than stabilization or regulation problems. First, the tracking controller must drive the system output toward a desired reference with specified closed-loop properties. Second, the stabilization/regulation issue can be viewed as a special case of tracking control, for which the desired trajectory is constant in time. Moreover, tracking control of robot manipulators is challenging due to complex nonlinearities, coupling dynamics effects, unknown disturbances, and modeling uncertainties [START_REF] Baek | A new adaptive sliding-mode control scheme for application to robot manipulators[END_REF][START_REF] Sun | Decentralized robust tracking control for 2-degree of freedom planar robot manipulator subject to disturbances and uncertainties[END_REF][START_REF] Li | Robust control of two-link manipulator with disturbance torque and time-varying mass loads[END_REF].

Up to now, numerous tracking control approaches have been proposed for robot manipulators, including proportionalintegral-derivative (PID) control [START_REF] Jafarov | A new variable structure PID-controller design for robot manipulators[END_REF][START_REF] Arteaga-Pérez | An alternative proof to the asymptotic stability of PID controllers for regulation of robot manipulators[END_REF][START_REF] Pan | Efficient PID tracking control of robotic manipulators driven by compliant actuators[END_REF], fuzzy logic control [START_REF] Marwan | Real-time on line tuning of fuzzy controller for two-link rigid-flexible robot manipulators[END_REF][START_REF] Gaidhane | Design of interval type-2 fuzzy precompensated PID controller applied to two-DoF robotic manipulator with variable payload[END_REF], observer-based backstepping control [START_REF] Sahu | Sampled-data extended state observer-based backstepping control of two-link flexible manipulator[END_REF], sliding mode control [START_REF] Baek | A new adaptive sliding-mode control scheme for application to robot manipulators[END_REF][START_REF] De Jesús Rubio | Sliding mode control of robotic arms with deadzone[END_REF][START_REF] Zhang | Continuous finite-time control for uncertain robot manipulators with integral sliding mode[END_REF], H ∞ control [START_REF] Makarov | Modeling and preview H ∞ design for motion control of elastic-joint robots with uncertainties[END_REF], fault-tolerant tracking control [START_REF] Van | Finite time fault tolerant control for robot manipulators using time delay estimation and continuous nonsingular fast terminal sliding mode control[END_REF][START_REF] Meng | Multi-model switching-based fault tolerant control for planar robot manipulators[END_REF], adaptive control [START_REF] Nojavanzadeh | Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators[END_REF][START_REF] Yang | Adaptive parameter estimation and control design for manipulators with finite-time convergence[END_REF][START_REF] Zhang | Adaptive projection neural network for kinematic control of redundant manipulators with unknown physical parameters[END_REF][START_REF] Sun | Adaptive fuzzy tracking control of flexible-joint robots with full-state constraints[END_REF], neural networks control [START_REF] Zhang | Adaptive neural network control of coordinated robotic manipulators with output constraint[END_REF][START_REF] Zhang | Three recurrent neural networks and three numerical methods for solving a repetitive motion planning scheme of redundant robot manipulators[END_REF][START_REF] Jin | Robot manipulator control using neural networks: A survey[END_REF][START_REF] Zhang | Adaptive neural control for robotic manipulators with output constraints and uncertainties[END_REF], boundary vibration control [START_REF] He | Boundary vibration control for a flexible Timoshenko robotic manipulator[END_REF], reinforcement learning control [START_REF] Ouyang | Reinforcement learning control of a single-link flexible robotic manipulator[END_REF], Riccati equation-based approach [START_REF] Nasiri | Observer-based robust control for flexible-joint robot manipulators: A state-dependent Riccati equation-based approach[END_REF], etc. Despite great theoretical advances in robotics control, single-axis PID control still remains the predominant method for industrial robot manipulators due to its simple structure, easy tuning and convenient implementation [START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF]). However, without taking into account the coupling between different robot joints, linear PID control may lead to a significant loss of performance even closed-loop instability under various practical configurations [START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF]. Computed torque control (CTC) technique has been widely applied in robotics [START_REF] Dawson | Robot Manipulator Control: Theory and Practice[END_REF][START_REF] Song | A computed torque controller for uncertain robotic manipulator systems: Fuzzy approach[END_REF][START_REF] Buondonno | Efficient computation of inverse dynamics and feedback linearization for VSA-based robots[END_REF][START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF]). An alternative solution to improve the tracking performance is to combine a feedforward action based on the manipulator dynamics together with a feedback control action. However, the resulting feedback-feedforward control structures suffer the robustness issue with respect to the modeling uncertainty and the external disturbance forces. To overcome this major drawback, fuzzy control has been proposed for the robot tracking problem [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF][START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF][START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF][START_REF] Nguyen | Nonlinear tracking control with reduced complexity of serial robots: A robust fuzzy descriptor approach[END_REF]. However, the control design complexity and the tracking precision may be further improved for generic robot manipulators.

Over the past decades, fuzzy tracking control has been actively investigated for nonlinear systems [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. The fruitful results can be classified into two major categories: fuzzy adaptive control scheme [START_REF] Chen | Adaptive fuzzy output tracking control of MIMO nonlinear uncertain systems[END_REF][START_REF] Zhou | Adaptive output-feedback fuzzy tracking control for a class of nonlinear systems[END_REF][START_REF] Wang | Performance-based adaptive fuzzy tracking control for networked industrial processes[END_REF], Li et al. 2017) and fuzzy model-based control scheme [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF][START_REF] Jia | Fuzzy H ∞ tracking control for nonlinear networked control systems in T-S fuzzy model[END_REF][START_REF] Zhang | H ∞ step tracking control for networked discrete-time nonlinear systems with integral and predictive actions[END_REF], Pan et al. 2020[START_REF] Nguyen | Robust set-invariance based fuzzy output tracking control for vehicle autonomous driving under uncertain lateral forces and steering constraints[END_REF]. Despite a great theoretical interest, fuzzy adaptive control usually requires online complex adaptation laws to deal with unknown parameters and nonlinearities, which may induce real-time difficulties. Since robot modeling has been well-established [START_REF] Dawson | Robot Manipulator Control: Theory and Practice[END_REF], fuzzy model-based control approaches would be more appropriate for high-speed motion tracking of industrial manipulators as recently shown in [START_REF] Nguyen | Nonlinear tracking control with reduced complexity of serial robots: A robust fuzzy descriptor approach[END_REF][START_REF] Jonnalagadda | Nonlinear control design using Takagi-Sugeno fuzzy applied to under-actuated visual servo system[END_REF]. Note that the existing fuzzy model-based approaches mostly rely on the well-known H ∞ specification to guarantee the tracking performance which can lead to the following drawbacks, especially for complex industrial manipulators. First, an explicit requirement and consideration of a reference model for tracking control design may induce unnecessary numerical complexities.

Second, the effects of both known tracking reference and unknown external forces on the tracking error have been treated in the same fashion, i.e., unknown disturbances. This may result in conservative control results. Third, the H ∞ specification represents the energy-to-energy gain of the close-loop system. However, in most of practical situations as robotics control, the disturbances, e.g., friction torques, are rather amplitude-bounded than energy-bounded. Hence, within an H ∞ control formulation, it is not obvious to characterize the amplitude-bound of the tracking error, which is a crucial tracking performance index. Motivated by the above practical and theoretical issues in manipulator control, this paper proposes a fuzzy model-based control approach for high-precision and high-speed motion tracking problem of generic robot manipulators. The main contributions of the proposed fuzzy tracking control framework can be summarized as follows.

• We exploit the specific descriptor form of robot manipulators to derive a feedback-feedforward tracking control scheme. The feedforward control action aims at accounting for the effects of known disturbances to improve the tracking performance.

Based on a fuzzy descriptor tracking error dynamics and Lyapunov arguments, the feedback control action is designed to guarantee some predefined closed-loop properties. The feedback control design is recast as an optimization problem with linear matrix inequality (LMI) constraints, effectively solved with available numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]).

• In contrast to the standard H ∞ tracking performance, a new L ∞ -gain specification is taken into the control design to guarantee a tracking error-bound as small as possible according to the solvability of the optimization-based control problem. Moreover, the new framework offers a simple and systematic approach not only for the control design but also for the real-time implementation. This feature is particularly interesting for industrial robot applications.

• The effectiveness of the proposed tracking control approach is clearly demonstrated with a high-fidelity manipulator benchmark developed by ABB Robotics [START_REF] Moberg | A benchmark problem for robust control of a multivariable nonlinear flexible manipulator[END_REF]. A comparative study between different notable tracking control approaches is performed to emphasize the interests of the new control framework.

The paper is organized as follows. The tracking control problem is formulated in Section II. The fuzzy model-based feedback control design is presented in Section III. Section IV provides numerical experiments obtained with a 2-DoF robot benchmark.

Concluding remarks are reported in Section V.

Notation. Ω r denotes the set of numbers {1, 2, ..., r}. For a matrix X, X denotes its transpose, X 0 means that X is positive definite, HeX = X + X , and λ min (X), λ max (X) denotes respectively the minimal and maximal eigenvalues of a symmetric matrix X. For a vector v ∈ R n , we denote its 2-norm as v = √ v v, and v i is its ith entry. For a function f : R → R n , its L ∞ -norm is defined as f ∞ = sup t∈R f (t) , and B ∞ is the set of bounded functions f . diag(X 1 , X 2 ) denotes a block-diagonal matrix composed of X 1 , X 2 . I is the identity matrix of appropriate dimension. stands for the terms deduced by symmetry. The argument of a function is omitted when its meaning is clear.

II. TRACKING CONTROL FORMULATION

We first introduce the general model of robot manipulators. Then, the related tracking control problem is formulated.

A. Descriptor Representation of Robot Manipulators

The rigid-body dynamics of an n degrees of freedom (DoF) robot manipulator can be expressed in the joint space as M(q)q(t) + N (q, q) q(t) + G(q) = Γ(t) + Γ d (t),

where q(t) ∈ R n is the vector of generalized coordinates in joint space, Γ(t) ∈ R n is the vector of generalized control forces, Γ d (t) ∈ R n is the vector of external generalized forces applied to the joints, M(q) ∈ R n×n is the inertia matrix, N (q, q) ∈ R n×n is the Coriolis/centripetal matrix plus the viscous friction coefficients of the joints, and G(q) ∈ R n represents the generalized gravity forces. Note that the dynamics (1) can account for many types of robot manipulators [START_REF] Dawson | Robot Manipulator Control: Theory and Practice[END_REF]. Since the vector field G(q) is smooth, we can then parameterize G(q) = P(q)q(t). The disturbance Γ d (t) is amplitudebounded, i.e., Γ d (t) ∈ B ∞ . Note that for the proposed fuzzy-model-based control approach, the matrices M(q), N (q, q) and P(q)q(t) characterizing the robot dynamics are assumed to be known. Let us denote x(t) = q (t) q (t) , the manipulator dynamics (1) can be rewritten in the following descriptor form:

E(x) ẋ(t) = A(x)x(t) + B(Γ(t) + Γ d (t)), (2)
where

E(x) = I 0 0 M(q) , A(x) = 0 I -P(q) -N (q, q)
, B = 0

I .
Note that due to the physical constraints, the state vector x(t) remains in a bounded set D x , defined as

D x = x ∈ R 2n : x i min ≤ x i (t) ≤ x i max . (3)
The given bounds x i min and x i max , i ∈ Ω 2n , characterize the workspace and the joint velocity limitations of the manipulator.

B. Feedback-Feedforward Structure for Tracking Control

The reference trajectory of robot manipulators can be completely defined by designers as

x r (t) = q r (t) q r (t) , where q r (t), qr (t) ∈ R n are respectively the reference position and velocity to be followed. For tracking control, we assume that x r is sufficiently smooth, i.e., q r (t) ∈ C 2 . We define the tracking error as e(t) = x(t)x r (t). Then, the error dynamics can be derived from model (2) as

E(x) ė(t) = A(x)e(t) + B(Γ(t) + d(t)) + R(x, x r , ẋr ) (4)
where

R(x, x r , ẋr ) = A(x)x r (t) -E(x) ẋr (t), and
d(t) = Γ d (t).
To improve the tracking control performance, the control input Γ(t) is decomposed into two parts as

Γ(t) = u fb (t) + u ff (t). (5)
The feedback control u fb (t), specified in Section III, is used to ensure predefined properties of the tracking error dynamics under disturbances and uncertainties. The feedforward control u ff (t) accounts for the effects of x r (t) on the tracking error e(t). From ( 4) and ( 5), the error dynamics is defined as

E(x) ė(t) = A(x)e(t) + B(u fb (t) + d(t)) + Q(u ff , x, x r ) (6)
where Q(u ff , x, x r ) = Bu ff (t) + R(x, x r , ẋr ). Since x r (t) and ẋr (t) are known, u ff (t) aims at mitigating the effects of these exogenous signals on the tracking error dynamics, i.e.,

Q(u ff , x, x r ) = Bu ff (t) -E(x) ẋr (t) + A(x)x r (t) = 0. (7)
Since matrix B is of full column rank, its pseudo-inverse B † exists and B † B = I. Premultiplying ( 7) with B † , the feedforward control law is derived as follows:

u ff (t) = B † (E(x) ẋr (t) -A(x)x r (t)). (8)
From ( 6) and ( 8), the tracking error dynamics is rewritten as

E(x) ė(t) = A(x)e(t) + B(u fb (t) + d(t)). (9)
We are now in the position to formulate the control problem related to the closed-loop tracking error system (9).

Problem 1. Determine a feedback control law u fb (t) such that the error dynamics (9) satisfies the following properties.

(P1) For zero-disturbance system, i.e., d(t) = 0, for ∀t ≥ 0, the zero solution of system ( 9) is exponentially stable with a predefined decay rate α > 0.

(P2) The closed-loop system ( 9) is input-to-state stable with respect to the disturbance d(t).

(P3) For a zero initial tracking error, i.e., e(0) = 0, we have

e(t) ≤ γ d(t) ∞ , t ≥ 0,
for some positive scalar γ.

The tracking error dynamics (9) verifying the properties (P1)-( P3) is said to be uniformly L ∞ -stable with a performance level γ. A similar concept of L ∞ -stablity can be found in [START_REF] Nguyen | Unknown input observers for simultaneous estimation of vehicle dynamics and driver torque: Theoretical design and hardware experiments[END_REF][START_REF] Nguyen | Avoiding unmeasured premise variables in designing unknown input observers for Takagi-Sugeno fuzzy systems[END_REF].

Hereafter, we provide a numerically tractable solution for the above L ∞ -gain tracking control problem.

III. L ∞ FUZZY DESCRIPTOR CONTROL DESIGN

This section first transforms the error tracking dynamics into an exact fuzzy descriptor form. Then, an LMI-based solution is proposed for Problem 1. Finally, based on a robust control scheme, complexity-reduced feedback control law is proposed.

A. Fuzzy Descriptor Representation of the Error Dynamics

For control design, system ( 9) is transformed into its equivalent fuzzy representation using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], Chapter 2). To this end, we first determine the vector of premise variables z(t) = z 1 (t) . . . z pe+p (t) ,

containing p e and p linearly independent nonlinearities in E(x) and A(x), respectively. Since x(t) ∈ D x , it follows that z i (t) ∈ z i min , z i max , where z i min and z i max , for ∀i ∈ Ω pe+p , are easily computed from the bounds given in (3). Moreover, each premise variable can be equivalently rewritten as

z i (t) = ψ i 0 (z i )z i min + ψ i 1 (z i )z i max ,
where

ψ i 0 (z i ) = z i max -z i (t) z i max -z i min , ψ i 1 (z i ) = 1 -ψ i 0 (z i ), i ∈ Ω pe+p .
As a consequence, the error dynamics ( 9) can be represented in the following fuzzy descriptor form [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF]:

re k=1 v k (z)E k ė(t) = r i=1 h i (z)A i e(t) + B(u fb (t) + d(t)). ( 10
)
The membership functions (MFs) are defined as follows [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]):

v k (z) = pe j=1 ψ j kj (z j ), h i (z) = pe+p j=pe+1 ψ j ij (z j ),
where k j , i j ∈ {0, 1}, (k, i) ∈ Ω re × Ω r , r e = 2 pe and r = 2 p . Note that

E k = E(x)| v k (z)=1 , A i = A(x)| hi(z)=1
and the MFs satisfy the following convex sum property:

re k=1 v k (z) = 1, v k (z) ≥ 0, k ∈ Ω re , r i=1 h i (z) = 1, h i (z) ≥ 0, i ∈ Ω r . (11)
Remark 1. Due to the nonsingularity of E(x), a standard state-space model can be directly recovered from the descriptor form (9) as ė(t) = E(x) -1 A(x)e(t) + E(x) -1 B(u fb (t) + d(t)). However, such a representation significantly complexifies the nonlinear control design since a large number of new premise variables is introduced, e.g., the input matrix E(x) -1 B is not anymore constant. Keeping the descriptor form (9) reduces the number of linear submodels of the fuzzy model ( 10), thus the number of design conditions. This allows reducing the computational burden, the control structure complexity and also the design conservatism [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF].

Let us define ē(t) = e (t) ė (t) . Then, the fuzzy descriptor system (10) can be rewritten as

Ē ė(t) = r i=1 re k=1 h i (z)v k (z) Āik ē(t) + B(u fb (t) + d(t)), (12)
where

Ē = I 0 0 0 , B = 0 B , Āik = 0 I A i -E k . ( 13
)
For the control design of system (12), we consider the nonlinear feedback control law of the form

u fb (t) = - r i=1 re k=1 h i (z)v k (z)K ik e(t), (14)
where the feedback gains K ik ∈ R n×2n are to be designed. From ( 10) and ( 14), the closed-loop error dynamics can be expressed on the following descriptor fuzzy form:

Ē ė(t) = r i=1 re k=1 h i (z)v k (z)( Āik -B Kik )ē(t) + Bd(t), (15)
where Kik = K ik 0 . This paper proposes two procedures to design the feedback control law ( 14) with different degrees of numerical complexity such that the closed-loop specifications as defined in Problem 1 are satisfied for system (15).

B. LMI-Based Feedback Control Design

The following result provides an LMI-based conditions to design a feedback control law ( 14) solving Problem 1.

Theorem 1. Consider the tracking error dynamics (9). Given a positive decay rate α, assume that there exist a positive definite matrix

P ∈ R 2n×2n , matrices Q i ∈ R 2n×2n , R i ∈ R 2n×2n , M ik ∈ R n×2n , for ∀(i, k) ∈ Ω r × Ω re ,
Ξ ik ≺ 0, ∀(i, k) ∈ Ω r × Ω re , (17)
where

Ξ ik = He    αP + Q i R i 0 A i P -BM ik -E k Q i -E k R i B 0 0 -αI    .
Then, the feedback law ( 14) with the gains K ik = M ik P -1 , for (i, k) ∈ Ω r × Ω re , is such that the properties given in Problem 1 are satisfied with the guaranteed L ∞ -gain γ = √ ν.

Proof. Multiplying inequality (18) by h i (z)v k (z) and summing up for i ∈ Ω r and k ∈ Ω re , it follows that

r i=1 re k=1 h i (z)v k (z)He(E k R i ) 0,
which can be rewritten as

He re k=1 v k (z)E k r i=1 h i (z)R i 0. (19)
Inequality ( 19) implies the regularity of r i=1 h i (z)R i . This allows us to define a Lyapunov candidate function of the form

V(ē) = ē Ē r i=1 h i (z) Pi -1 ē, Pi = P 0 Q i R i . ( 20
)
Remark that

Ē r i=1 h i (z) Pi -1 = r i=1 h i (z) Pi -1 Ē = P -1 0 0 0 .
Then, V(ē(t)), defined in (20), can be rewritten as V(ē(t)) = V(e(t)) = e (t)P -1 e(t), which is a positive definite function of e(t). We denote

ê(t) = r i=1 h i (z) Pi -1 ē(t).
The time-derivative of V(ē(t)) along the trajectory of system (15) can be expressed as

V(ē(t)) = He A z ē(t) + Bd(t) Ph ē(t) = He A z Ph ê(t) + Bd(t) ê(t) , (21)
where

A z = r i=1 re k=1 h i (z)v k (z)( Āik -B Kik ) and Ph = r i=1 h i (z) Pi .
From the matrix definition ( 13), the expressions of V(ē(t)) in ( 20) and of V(ē(t)) in ( 21), the following relation can be obtained after simple matrix multiplications:

V(ē(t)) + 2α V(ē(t)) -d (t)d(t) = ξ (t)Ξ hv ξ(t), (22)
with ξ = ê d and

Ξ hv = r i=1 re k=1 h i (z)v k (z)Ξ ik .
Using the convexity property (11) of the membership functions, it follows from (18) that Ξ hv ≺ 0. Combining this with ( 22), it follows that

V(ē(τ )) ≤ -2α(V(ē(τ )) -d(τ ) 2 ) ≤ -2α(V(ē(τ )) -d(τ ) 2 ∞ ), ∀τ > 0. ( 23
)
Multiplying both sides of inequality ( 23) by e 2αt , then integrating the result over [t 0 , t], we get

e 2αt V(ē(t)) ≤ e 2αt0 V(ē(t 0 )) + 2α d(t) 2 ∞ t t0 e 2ατ dτ = e 2αt0 V(ē(t 0 )) + d(t) 2 ∞ (e 2αt -e 2αt0 ). (24)
It follows from (24) that

V(ē(t)) ≤ e -2α(t-t0) V(ē(t 0 )) + d(t) 2 ∞ 1 -e -2α(t-t0) ≤ e -2α(t-t0) V(ē(t 0 )) + d(t) 2 ∞ . ( 25
)
By Schur complement lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], inequality ( 17) is equivalent to P -1 ν -1 I. Then, denote σ M = λ max (P -1 ),

the Lyapunov function V(ē) satisfies

ν -1 e(t) 2 ≤ V(ē(t)) ≤ σ M e(t) 2 . ( 26
)
It follows from ( 25) and ( 26) that

e(t) 2 ≤ νσ M e -2α(t-t0) e(t 0 ) 2 + ν d(t) 2 ∞ ,
which, in turn, implies that

e(t) ≤ √ νσ M e -α(t-t0) e(t 0 ) + √ ν d(t) ∞ . (27)
Inequality ( 27) guarantees the closed-loop properties (P1), (P2) and (P3) with γ = √ ν, defined in Problem 1.

Remark 2. For any initial condition e(t 0 ) and for ∀d(t) ∈ B ∞ , it follows from ( 27) that

lim sup t→∞ e(t) ≤ γ d(t) ∞ . (28)
It is clear from (28) that minimizing γ leads to a minimized value of e(t), thus a better tracking performance. Moreover, if the optimization problem ( 16) in Theorem 1 is feasible with an arbitrarily small γ, then the tracking error is arbitrarily small.

C. Reduced-Complexity Fuzzy Tracking Control

Theoretically, the design conditions in Theorem 1 can be applied to any robot manipulators whose dynamics can be described by (1). However, from a practical viewpoint, these conditions may lead to a major numerical limitation. Indeed, the complexity of the fuzzy model ( 10) is represented by the integers r and r e , which exponentially grow in function of the number of premise variables in A(x) and E(x) of system (2). Hence, for complex manipulators with high degrees of freedom n, the nonlinear feedback law ( 14) may be impractical due to the limited computational resources of embedded systems. In such situations, an effective reduced-complexity control approach is required to design the nonlinear controller ( 14). In particular, reducedcomplexity feedback controller is still able to guarantee the closed-loop properties defined in Problem 1. For this purpose, we rewrite some premise variables as follows (Nguyen, Coutinho, Guerra, Palhares & Pan 2020):

z j (t) = z jm + δ j (t)z jr , δ j (t) ∈ [-1, 1], (29)
with z jm = 1 2 (z j max + z j min ), z jr = 1 2 (z j maxz j min ), for some j ∈ Ω pe+p . Substituting ( 29) into ( 9), then applying the sector nonlinearity approach described in Section III-A, we can obtain the following fuzzy system:

r * e k=1 v k (z) Êk ė(t) = r * i=1 h i (z) Âi e(t) + B(u fb (t) + d(t)) (30)
where

Êk = E * k + ∆E k (t), k ∈ Ω r *
e , with r * e < r e , and Âi = A * i + ∆A i (t), i ∈ Ω r * , with r * < r. Note that the time-varying matrices ∆E k (t) and ∆A i (t) can be further parameterized as

∆E k (t) = H e ∆ e (t)W ek , ∆A i (t) = H a ∆ a (t)W ai , for ∀(k, i) ∈ Ω r * e × Ω r * .
The constant matrices of adequate dimensions H e , H a , W ek , W ai are constructed from z jr in (29) whereas matrices ∆ (t), ∈ {e, a}, are obtained from the uncertain terms δ j (t). Note also that ∆ (t)∆ (t) I, for ∈ {e, a}.

The obtention of (30) from the nonlinear descriptor system (2) is further illustrated in Section IV-B. Remark 3. The numerical complexity of system (30), represented by r * e and r * , exponentially decreases with respect to the number of the premise variables considered as uncertain terms shown in (29). However, a large number of uncertain terms may introduce more conservatism to the control design of system (30). Note that any time-varying premise variable z j (t), for j ∈ Ω pe+p , can be represented in the form (29). However, in this paper we propose, as far as possible, to exploit the relation (29) to deal with the premise variables depending on the angular velocities. Then, the membership functions of the descriptor fuzzy systems can be constructed with premise variables only depending on the joint positions. These latter can be easily measured with low-cow position sensors. This fact would be useful for the development of a robust fuzzy output tracking control of robot manipulators without requiring velocity sensors.

The following theorem provides an LMI-based solution to design a reduced-complexity feedback control law

u fb (t) = - r * i=1 r * e k=1 h i (z)v k (z)K * ik e(t), (31)
such that the uncertain fuzzy system (30) satisfies the closed-loop properties specified in Problem 1.

Theorem 2. Consider the tracking error dynamics (30). Given a positive decay rate α, assume that there exist a positive definite matrix P ∈ R 2n×2n , matrices

Q i ∈ R 2n×2n , R i ∈ R 2n×2n , M ik ∈ R n×2n ,
   Ξ * ik S ik H -S ik W ik 0 -S ik    ≺ 0, ∀(i, k) ∈ Ω r * × Ω r * e , (33)
where S ik = diag(φ a ik I, φ e ik I) and

Ξ * ik = He    αP + Q i R i 0 A * i P -BM ik -E * k Q i -E * k R i B 0 0 -αI    , W ik = W ai P 0 0 -W ek Q i -W ek R i 0 , H = 0 H a 0 0 H e 0 .
Then, the reduced-complexity controller (31) with the gains

K * ik = M ik P -1 , for (i, k) ∈ Ω r * × Ω r *
e , is such that the properties defined in Problem 1 are satisfied with the guaranteed L ∞ -gain γ = √ ν.

Proof. Multiplying (33) with h i (z)v k (z) ≥ 0 and summing up for all

(i, k) ∈ Ω r * × Ω r * e , it follows that    Ξ * hv S hv H -S hv W hv 0 -S hv    ≺ 0, (34)
where

Π hv = r * i=1 r * e k=1 h i (z)v k (z)Π ik , with Π ∈ {Ξ * , S, W}.
Applying the Schur complement lemma, we can prove that (34) is equivalent to

Ξ * hv + H S hv H + W hv S -1 hv W hv ≺ 0. ( 35
)
Denote ∆(t) = diag(∆ a (t), ∆ e (t)). Since ∆ (t)∆(t) I, using the following matrix fact:

He(X Y) X SX + Y S -1 Y,
with S = S hv , X = H and Y = ∆(t)W hv , it follows from (35) that

Ξ * hv + He(H ∆(t)W hv ) ≺ 0. ( 36
)
Consider a Lyapunov candidate function of the form (20) with r = r * . Then, following similar arguments as in the proof of Theorem 1 and from ( 36), it can be shown that

V(ē(t)) + 2αV(ē(t)) -2ανd (t)d(t) = ξ (t)Σ hv ξ(t) < 0, with Σ hv = Ξ * hv + He(H ∆(t)W hv ).
The guarantee of properties (P1), (P2) and (P3) in Problem 1 with respect to system (30) can be proved as in the proof of Theorem 1.

Remark 4. A decay rate performance is incorporated in the Lyapunov stability conditions of Theorems 1 and 2 to improve the tracking performance. A large value of the decay rate α leads to a quick error convergence [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]). However, this may induce aggressive closed-loop behaviors. The tuning of the positive decay rate α can be easily performed using error-and-trial method to obtain a satisfactory tracking control performance in practice.

Remark 5. The control design in Theorem 1 (respectively Theorem 2) is recast as an optimization problem (16) (respectively (32)) under LMI constraints. Such convex optimization problems can be effectively solved with standard numerical solvers [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In this paper, the feedback gains of the tracking fuzzy controllers ( 14) and ( 31) are computed using YALMIP toolbox and SDPT3 solver [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in Matlab[END_REF]). Remark 6. Solving the optimization problems in Theorems 1 and 2 may yield excessively high feedback gains, which are impractical for real-world applications. To prevent this practical issue, the following LMI constraints can be incorporated in these convex optimization problems:

P µI, τ ≤ εµ, ( 37
) τ I M ik M ik τ I 0, ∀(i, k) ∈ Ω r × Ω re , (38)
where µ, τ are additional scalar decision variables, and ε is a given Euclidean norm-bound of the matrix gains K ik . Indeed, by Schur complement, inequality ( 38) is equivalent to

M ik M ik τ 2 I, ∀(i, k) ∈ Ω r × Ω re .
Then, it follows from (37) that

M ik M ik ε 2 µ 2 I ε 2 P 2 . ( 39
)
Pre-and postmultiplying (39) with P -1 , it follows that

P -1 M ik M ik P -1 ε 2 I, which is equivalent to K ik ≤ ε.
Remark 7. An integral action can be easily introduced in the feedback control structure to improve the tracking performance.

To this end, it suffices to include the integral state x I (t), whose dynamics can be defined as

ẋI (t) = Ce(t),
with C = I 0 ∈ R n×2n . This amounts substituting in the convex optimization problems A i , B and E k by the following respective partitioned matrices:

A i 0 C 0 , B 0 , E k 0 0 I .
The following numerical experiments are obtained with the corresponding integral fuzzy feedback control scheme.

IV. ILLUSTRATIVE RESULTS AND DISCUSSIONS

To highlight the interests of the proposed approach, we provides illustrative results obtained with the 2-DoF manipulator benchmark developed by ABB Robotics [START_REF] Moberg | A benchmark problem for robust control of a multivariable nonlinear flexible manipulator[END_REF]. Then, objective comparisons with notable control approaches are also given.

A. Two Degree-of-Freedom Robot Manipulator

The studied manipulator is depicted in Fig. 1. The first arm rotates about the z-axis. The second arm is attached to the first arm by a revolute joint. The robot parameters are given in Table I. Let us denote q 1 (t) and q 2 (t) the relative joint coordinates of the first and the second arms, respectively, and q 0 (t) = q 1 (t) + q 2 (t) the absolute joint coordinate of the second arm. The torques at joints 1 and 2 are respectively denoted as Γ 1 (t) and Γ 2 (t). The manipulator dynamics can be expressed in the form

(1) with G(q) = P(q)q(t) and , where

M(q) = c 1 + 2c 2 cos q 2 + I a1 c 3 + c 2 cos q 2 c 3 + c 2 cos q 2 c 3 + I a2 , N (q, q) = -2c 2 q2 sin q 2 + f v1 -c 2 q2 sin q 2 c 2 q1 sin q 2 f v2 , P(q) =
c 1 = m 1 r 2 1 + I 1 + m 2 L 2 1 + m 2 r 2 2 + I 2 , c 2 = m 2 L 1 r 2 , c 3 = m 2 r 2 2 + I 2 , c 4 = m 1 gr 1 + m 2 gL 1 , c 5 = m 2 gr 2 . Let us define the state x(t) = q 1 (t) q 2 (t) q1 (t) q2 (t) , the control input Γ(t) = Γ 1 (t) Γ 2 (t) . The disturbance input Γ d (t) = Γ d1 (t) Γ d2 (t)
is composed of the dry friction torques and the external forces as indicated in Appendix. The manipulator dynamics can be represented in the form (2) with

E(x) =       1 0 0 0 0 1 0 0 0 0 c 1 + 2c 2 z 1 c 3 + c 2 z 1 0 0 c 3 + c 2 z 1 c 3       , B =       0 0 0 0 1 0 0 1       , A(x) =       0 0 1 0 0 0 0 1 z 2 z 3 2z 4 -f v1 z 4 z 3 z 3 z 5 -f v2      
, where the premise variables are defined as follows:

z 1 = cos q 2 , z 4 = c 2 q2 sin q 2 , z 5 = -c 2 q1 sin q 2 , z 2 = -c 4 sin q 1 q 1 c 5 sin q 0 q 0 , z 3 = -c 5 sin q 0 q 0 . (40)

B. Fuzzy Descriptor Modeling for Robot Tracking Control

The set D x in (3) of the benchmark robot is defined with

|q 1 (t)| ≤ π rad, |q 2 (t)| ≤ π rad, | q1 (t)| ≤ π/2 rad/s, | q2 (t)| ≤ π/2 rad/s.
Fig. 1. Two degree-of-freedom benchmark manipulator.

These state limitations are determined to guarantee a large workspace together with the related physical robot constraints.

1) Exact Fuzzy Descriptor Model: From the physical parameters in Table I and the above workspace data, the bounds of the premise variables in (40) can be easily computed

z 1 max = -z 1 min = 1, z 2 max = z 3 max = 255.7,
z 2 min = -3384.5, z 3 min = -1177, z 4 max = -z 4 min = 188.5, z 5 max = -z 5 min = 188.5.

Using the sector nonlinearity method shown in Section III-A, an equivalent fuzzy descriptor representation (10) with 2 5 = 32 linear submodels (r e = 2 and r = 16) of the studied manipulator, called E-Model, can be derived. The details of these 32 submodels and the corresponding MFs are standard and omitted here for brevity.

2) Reduced-Complexity Fuzzy Descriptor Model: For illustration, the premise variables z i , i ∈ {2, 4, 5}, can be rewritten in the form (29). This leads to a fuzzy model (30) with 2 2 = 4 linear submodels (r * = 2 and r * e = 2), called R-Model. The state-space matrices of this fuzzy model are given by

E * 1 = E(z 1 min ), E * 2 = E(z 1 max ), ∆E 1 (t) = ∆E 2 (t) = 0, A * 1 =       0 0 1 0 0 0 0 1 z 2m z 3 min 2z 4m -f v1 z 4m z 3 min z 3 min z 5m -f v2       , A * 2 =       0 0 1 0 0 0 0 1 z 2m z 3 min 2z 4m -f v1 z 4m z 3 max z 3 min z 5m -f v2       , ∆ a (t) = diag(δ 2 (t), δ 4 (t), δ 5 (t)), H a =    0 0 1 0 0 0 1 0 0 0 0 1    , W a =    z 2r 0 0 0 0 0 2z 4r z 4r 0 0 z 5r 0    .
The corresponding MFs of R-Model are given by

v 1 (z) = z 1 -z 1 min z 1 max -z 1 min , v 2 (z) = 1 -v 1 (z), h 1 (z) = z 3 -z 3 min z 3 max -z 3 min , h 2 (z) = 1 -h 1 (z).
The control results obtained with the two above manipulator fuzzy models are discussed below.

C. Numerical Results and Discussions

Solving directly the optimization problems in Theorems 1 and 2 with a decay rate α = 15 yields impractical control laws ( 14) or (31) with a gain-amplitude order of 10 9 . We thus redesign the control gains by including additional LMI constraints as explained in Remark 6 to guarantee that K ik ≤ 10 6 , for ∀(i, k) ∈ Ω r × Ω re . Details on the obtained control solutions are not given here for brevity. A summary on the control design of both manipulator fuzzy models is depicted in Table II. Remark that using relation ( 29), the number of linear subsystems of the fuzzy descriptor representation is significantly reduced from 32

to 4. This leads to a major advantage for real-time control implementation without fear of causing closed-loop instability since Theorem 2 allows designing a robust controller (31) with respect to uncertainties ∆ a (t) and ∆ e (t), induced by the complexity reduction. Moreover, the number of scalar decision variables N var involved in LMI-based optimizations, representing their numerical complexity, is also drastically reduced for the control of R-Model. The L ∞ -gain γ obtained for both fuzzy models are very small, thus, a good level of tracking performance is expected for both control solutions. Note that no feasible solution can be found for any manipulator fuzzy representation with less than four linear submodels since the uncertainties caused by (29) become too large. Hence, in practice there is a tradeoff between numerical complexity reduction and modeling uncertainty as mentioned in Remark 3. The numerical experiments are performed with a refined robot model that takes into account the joint elasticity and dry friction effect, see Appendix. This high-fidelity simulation model is embedded in Simscape Multibody TM environment. The trajectory reference in the Cartesian space is a circle with radius R traj = 0.4 (m) and center coordinates O(0, 1.8) (m), which has to be performed in 3.2 (s) with a quintic polynomial of time. The joint trajectory is computed using manipulator inverse kinematics from the Cartesian trajectory. We now compare the tracking performance obtained with the following four controllers:

• LI32 controller is derived from Theorem 1 and E-Model,

• LI4 controller is derived from Theorem 2 and R-Model,

• PID controller and CTC controller, whose structures and design tunings have been chosen according to [START_REF] Paccot | A review on the dynamic control of parallel kinematic machines: Theory and experiments[END_REF].

The relative circularity errors c = OPe(t) -Rtraj Rtraj

, where P e (t) is the tool center point (TCP), obtained with the four controllers are depicted in Fig. 2. Observe that both proposed LI32 and LI4 controllers provide a high-precision tracking performance with respect to the considered trajectory reference. In particular, despite its reduced-complexity feature, controller LI4 offers similar, or even better, control performance compared to the other ones. These facts are also confirmed by the small tracking errors obtained with both proposed controllers as indicated in Fig. 3. To further illustrate the tracking performance of different controllers, let us consider the following performance indices:

• ∆R M = max t∈[0,T ]
OP e (t) -R traj (mm): maximal circular error, where T = 4 (s) is the simulation duration,

• ∆e x,M and ∆e y,M (mm): tracking errors with respect to the tool center point P e (t) in x and y directions,

• E iMaAV (mrad): maximum absolute value (MaAV) of the ith joint position tracking error, defined by

E iMaAV = max t∈[0,T ] |q i (t) -q ir (t)|, i = 1, 2.
Fig. 4 represents the values of these performance indices obtained with the considered controllers. This chart corroborates the good performance obtained by the proposed control laws compared with existing controllers. Especially, the tradeoff between the controller complexity and the related performance is also clarified. The simplest PID controller provides the worst values for all considered performance indices whereas more complex control schemes such as CTC or LI32 yield better tracking performance. The proposed controller LI4 appears as a good compromise leading to a simple enough control structure for real-time validation without any significant degradation of tracking performance. Concerning the actuator solicitations, Fig. 5 depicts the corresponding response of both control torques. Moreover, Table III shows the maximal and the root-mean-square (RMS) values of the two actuator torques of the benchmark manipulator. We can observe that all the torque values are of similar magnitudes for the four considered controllers.

V. CONCLUDING REMARKS

A systematic approach for tracking control design of robot manipulators has been proposed. Based on a fuzzy descriptor representation of generic manipulator systems, this approach makes use of Lyapunov stability arguments to guarantee an L ∞ -gain performance of the closed-loop tracking error dynamics. Exploiting a robust control scheme, we propose an effective way to significantly reduce the numerical complexity of the control structure, leading to a major advantage for real-time control implementation. The control design is recast as convex optimization problems, which can be easily solved with standard solvers.

The practical tracking performance of the proposed approach is clearly demonstrated with a high-fidelity 2-DoF benchmark manipulator. A comparative study with several prominent tracking control schemes is also performed to show the interests of the new approach. In particular, it has been shown that despite a significant complexity reduction, the robust fuzzy controller provides a similar tracking performance compared to the exact fuzzy controller. Hence, the proposed robust fuzzy-model-based control method offers a more effective solution for the tracking control problem of robot manipulators, especially within industrial context. Future works focus on the extension of the proposed approach to the design of a robust fuzzy output tracking control for robot manipulators and real-time validations. Moreover, fuzzy robot tracking control in interaction with human operators is also a promising research topic.

APPENDIX. MANIPULATOR BENCHMARK MODEL

The dynamics of the rigid-link flexible-joint manipulator model used for validation purposes is composed of actuator dynamics (subscript a) and linkage dynamics (subscript b) as

M a qa + N a qa + Q t = Γ + Γ f M b (q b )q b + N b (q b , qb ) qb + G(q b ) -Q t = J (q b )F e , (41)
where Q t = B t ( qaqb ) + K t (q aq b ), q a = q a1 q a2 is the vector of actuator angular positions, M a = diag(I a1 , I a2 )

is the matrix of motor inertia, N a = diag(f v1 , f v2 ) is the matrix of viscous friction, M b (q b ) and N b (q b , qb ) are respectively the linkage inertia and Coriolis/centripetal matrices, K t = diag(k t1 , k t2 ) is the matrix of joint stiffness, B t = diag(b t1 , b t2 ) is the joint damping matrix, and J (q b ) is the robot Jacobian such that the TCP velocity is expressed as ẊPe (t) = J (q b ) qb (t).

The disturbances are composed of the dry friction torque Γ f (t) acting on the actuators and the external force F e (t) acting on the TCP. The dry friction torque Γ f (t) = Γ f 1 (t) Γ f 2 (t) is modeled as follows [START_REF] Moberg | A benchmark problem for robust control of a multivariable nonlinear flexible manipulator[END_REF]:

Γ f i (t) = Γ ci (µ i + (1 -µ i )cosh -1 (β i qai (t)))tanh(α i qai (t)),
where Γ ci is the Coulomb friction, µ i is the Stribeck friction constant, and α i , β i are the constants of the smooth friction model for the ith joint. The external force used for validation is assumed to be tangential to the TCP trajectory as F e (t) = f e (t) ẊPe (t) ẊPe (t) -1 ,

  and a positive scalar ν, solutions of the following optimization problem:

Fig. 2 .

 2 Fig. 2. Time history of the relative circularity errors c for the four compared controllers PID, CTC, LI32 and LI4.

Fig. 4 .

 4 Fig. 3. Time history of the joint position errors.

Fig. 5 .

 5 Fig. 5. Time history of the control torques.

  and positive scalars ν, φ a ik , φ e ik , for∀(i, k) ∈ Ω r * × Ω r *e , solutions of the following optimization problem:

	minimize ν	(32)
	subject to (17) and

TABLE II CONTROL

 II DESIGN OF TWO FUZZY DESCRIPTOR MODELS.

	Fuzzy model Nb. submodels	Design	Nvar	γ
	E-Model	32	Theorem 1	1562	8.06 × 10 -5
	R-Model	4	Theorem 2	222	7.75 × 10 -6

TABLE III MAXIMAL

 III AND RMS ACTUATOR TORQUES AFTER GEARBOX (IN NM) FOR DIFFERENT CONTROLLERS.

	5,000
	4,000
	3,000
	2,000
	1,000
	0
	-1,000
	-2,000	0	0.5	1	1.5	2	2.5	3	3.5	4
						Time (s)
						Controller Γ 1max	Γ 1RMS	Γ 2max	Γ 2RMS
							PID		4804	1282	2165	476.3
						CTC		4484	1273	2103	463.2
						LI32		4478	1275	2131	471.1
							LI4		4709	1279	2138	473.6

where f e (t) ∈ 310, 490 (N) is the force magnitude. All the robot parameters are taken from the experimental benchmark [START_REF] Moberg | A benchmark problem for robust control of a multivariable nonlinear flexible manipulator[END_REF]) of an industrial manipulator developed by ABB Robotics.

Note that the manipulator model ( 41) has four degrees-of-freedom, including the joint deformation between each actuator and its corresponding arm [START_REF] Makarov | Modeling and preview H ∞ design for motion control of elastic-joint robots with uncertainties[END_REF]. This additional and weakly damped dynamics introduces vibrations due to the excitation of the mechanical resonances. For control design, the joints are assumed to be rigid, i.e., q a = q b = q. This leads to the control-based manipulator model with matrices M(q) = M a + M b (q), N (q, q) = N a + N b (q, q), G(q) given in Section IV-A. Moreover, the disturbance input is defined as Γ d (t) = J (q)F e (t) + Γ f (t).

Conflict of Interest. The authors declare that they have no conflict of interest.


Van-Anh Nguyen
email: nguyen.trananhtu@gmail.com.

Anh-Tu Nguyen

Antoine Dequidt

Laurent Vermeiren

Michel Dambrine

• Lau- Rent Vermeiren

Nonlinear Tracking Control with Reduced Complexity of Serial Robots: A Robust Fuzzy Descriptor Approach

Keywords: Fuzzy control, Descriptor fuzzy systems, Serial robots, Robot manipulators, Tracking control, Linear matrix inequality (LMI)

This paper presents a nonlinear tracking control approach for a two degrees of freedom (2-DoF) serial manipulator. The design goal is to achieve a guaranteed H ∞ tracking performance while keeping the designed controller as simple as possible for real-time implementation. To this end, the descriptor Takagi-Sugeno fuzzy modeling is used to describe the nonlinear dynamics of the robot. Then, based on Lyapunov stability theory, we propose conditions to design fuzzy controllers for trajectory tracking purposes. The control design procedure is reformulated as an optimization problem under linear matrix inequality constraints which can be effectively solved with semidefinite programming technique. Numerical experiments carried out with the Simscape Multibody TM library of Matlab R clearly demonstrate the effectiveness of the proposed approach in terms of tracking control and numerical simplicity.

Introduction

Nowadays, robot manipulators are widely used in all areas of industry for process automation like materials handling, welding, painting, cutting, grinding, etc. These industrial robots have several advantages such as high speed, compactness, accuracy and reliability. For path following applications such as arc welding, laser cutting or machining, one of the most important tasks is to obtain a high-performance tracking of some predefined reference trajectories [START_REF] Lu | Design and optimization of interval type-2 fuzzy logic controller for delta parallel robot trajectory control[END_REF][START_REF] Pan | Adaptive commandfiltered backstepping control of robot arms with compliant actuators[END_REF][START_REF] Nojavanzadeh | Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators[END_REF]. Tracking control problems are generally considered as more difficult than stabilization or regulation control problems since, in addition to guaranteeing stability of the closed-loop system, specifications on the tracking performance have to be respected for different types of reference trajectories which may be not standard as step or ramps.

Up to now, designing robust tracking controllers for robotics systems is known as a challenging control task [START_REF] Pan | Adaptive commandfiltered backstepping control of robot arms with compliant actuators[END_REF][START_REF] Nojavanzadeh | Adaptive fractional-order non-singular fast terminal sliding mode control for robot manipulators[END_REF][START_REF] Tsai | Robust tracking control for a wheeled mobile manipulator with dual arms using hybrid sliding-mode neural network[END_REF][START_REF] Huang | Backstepping holonomic tracking control of wheeled robots using an evolutionary fuzzy system with qualified ant colony optimization[END_REF][START_REF] Ouyang | PD with sliding mode control for trajectory tracking of robotic system[END_REF]. This may be explained by several factors. First, robot manipulators have a real nonlinear behaviour. Second, these robots are always subject to uncertain dynamics including unknown friction torques, uncertain inertia, or external disturbances, etc. The latter may limit the practical use of feedback linearization control approach, see for instance [START_REF] Hedberg | Comparing feedback linearization and Jacobian linearization for LQ control of an industrial manipulator[END_REF]. Therefore, a robust control methodology is essential for the tracking control of manipulators. Among the proposed control approaches available in the literature, it can be notably mentioned the sliding mode control theory in [START_REF] Tsai | Robust tracking control for a wheeled mobile manipulator with dual arms using hybrid sliding-mode neural network[END_REF][START_REF] Pi | Trajectory tracking control of a 6-DOF hydraulic parallel robot manipulator with uncertain load disturbances[END_REF][START_REF] Jafarov | A new variable structure PID-controller design for robot manipulators[END_REF], the backstepping control method together with metaheuristic tools in [START_REF] Huang | Backstepping holonomic tracking control of wheeled robots using an evolutionary fuzzy system with qualified ant colony optimization[END_REF], or adaptive control schemes [START_REF] Purwar | Adaptive output feedback tracking control of robot manipulators using position measurements only[END_REF][START_REF] Hu | Adaptive backstepping trajectory tracking control of robot manipulator[END_REF][START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF].

Another approach, more systematic, is the modelbased fuzzy control methodology [START_REF] Tseng | Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model[END_REF][START_REF] Nguyen | Lyapunovbased robust control design for a class of switching nonlinear systems subject to input saturation: application to engine control[END_REF][START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF]. It relies on the use of Takagi-Sugeno (T-S) fuzzy models introduced in [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF], which has since then attracted a great attention from academic researchers and engineers, see for instance the following monographs [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Zhang | Fuzzy Modeling and Fuzzy Control[END_REF][START_REF] Lilly | Fuzzy Control and Identification[END_REF][START_REF] Lendek | Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models[END_REF]. T-S models can be represented by a set of IF-THEN fuzzy rules with consequent parts being local linear representations. They can also be defined in state-space form as convex sums of several local linear submodels with weights called membership functions. They have been proved to constitute a class of universal approximators [START_REF] Nguyen | Fuzzy control systems: Past, present and future[END_REF]. In par-ticular, for sufficiently smooth nonlinear systems, an equivalent representation in T-S fuzzy form can be obtained semiglobally [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF].

With the choice of a particular control structure called parallel distributed compensation (PDC) [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF] allowing to preserve the convex structure of T-S fuzzy systems, Lyapunov's direct method can be readily applied to derive stability conditions expressed in the form of linear matrix inequalities (LMIs) which can be efficiently solved with available numerical solvers [START_REF] Nguyen | Driverautomation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. This approach has been successfully applied to control problems in different fields, including biology [START_REF] Chen | Robust optimal referencetracking design method for stochastic synthetic biology systems: T-S fuzzy approach[END_REF], transport [START_REF] Nguyen | Driverautomation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF][START_REF] Hu | Fuzzy guaranteed cost tracking control for a flexible air-breathing hypersonic vehicle[END_REF], mechatronics [START_REF] Hung | Fuzzy virtual reference model sensorless tracking control for linear induction motors[END_REF] and so on. However, one point usually underestimated in most of existing works consists in the complexity inherent to this control approach. Indeed, using the sector nonlinearity approach, the obtained T-S fuzzy model has a number of fuzzy rules which increases exponentially with the number of nonlinearities existing in the original nonlinear system [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. This has several implications that restrict the use of T-S fuzzy control approach to simple manipulators with few degrees of freedom. The first ones is concerned with the control design. Indeed, according to the limitations of actual LMI solvers, no feasible solution can be found to design the feedback control gains, or the computational burden may be too costly that it is impossible to tune correctly the additional control parameters. Another point is that this complexity is echoed in the structure of the control law which reproduces the system one. Hence, more real-time computations are needed for the control, requiring more powerful (and also more costly) processors.

To solve the above control issues, we propose a systematic control framework based on simplified models described as uncertain descriptor T-S fuzzy models. Descriptor T-S fuzzy models were introduced in [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF]. For a large class of nonlinear systems including robot manipulators, using the sector nonlinearity approach, the obtained descriptor T-S fuzzy model is much simpler in terms of structure and/or of number of rules than the standard T-S fuzzy model, see for instance [START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF] for an robotics application. In order to simplify the model complexity, some premise variables can be interpreted as "modeling uncertainties". For this class of uncertain descriptor T-S fuzzy models, we propose a new tracking control structure based on the PDC concept in which integral and feedforward actions are integrated to improve the tracking performance. Both feedback and feedforward controller gains can be obtained solving an LMI-based optimization problem to guarantee a predefined level of performance in disturbance attenuation (with a maximal L 2 -gain) and in transient response (with a minimal exponential rate).

To illustrate this methodology, we apply it to the tracking control of a 2-DoF serial manipulator. The rest of the paper is organized as follows. The 2-DoF serial manipulator is presented in Section 2 together with a simulation model taking into account the elasticity and the friction of the robot joints. In Section 3, the control model is given and transformed in a descriptor T-S representation. An LMI-based robust control design for fuzzy T-S descriptor system is presented in Section 4. Section 5 presents the control design for the simplified T-S model. Section 6 deals with the application to the tracking control of the 2-DoF serial manipulator. Some numerical experiments are given showing the interests of the proposed approach. Concluding remarks are reported in Section 7.

Notation.Ω r denotes the number set {1, 2, . . . , r}. For a square matrix X, X denotes its transpose, X > 0 means that X is positive definite, X (i) denotes its ith row and HeX = X +X . diag(X 1 , X 2 ) denotes a blockdiagonal matrix composed of X 1 , X 2 . I is the identity matrix of appropriate dimension. The symbol " " stands for the terms deduced by symmetry. The scalar functions η 1 , . . . , η N satisfy the convex sum property if η i ≥ 0 for all i ∈ Ω r and

N i=1 η i = 1. For brevity, we denote X h = r i=1 h i X i , Y hv = N i=1 Ne k=1 h i v k Y ik and Z hhv = N i=1 N j=1
Ne k=1 h i h j v k Z ijk where X i , Y ik and Z ijk are matrices of appropriate dimension and the scalar functions h i and v k , i ∈ Ω r , k ∈ Ω Ne , of any argument satisfy the convex sum property.

System Modeling

This section provides a brief description on the studied serial manipulator, see Fig. 1. For brevity, the system nomenclature used in this work is given in Table 1. The first arm is of length L 1 and mass m 1 and rotates about the z-axis. The second arm, of length L 2 and mass m 2 , is attached to the first arm by a revolute joint at point O 2 . The rotational angle of first arm, measured clockwise, is denoted q b1 , that of second arm q b2 , and q b12 = q b1 +q b2 . The notations q a1 and q a2 represent the rotational angles of the first and second actuators. The motor torques are respectively denoted by Γ 1 and Γ 2 . To provide more realistic simulations in Section 6, the dynamics induced by the elastic joints [START_REF] Spong | Modeling and control of elastic joint robots[END_REF] between the actuators and the linkages is taken into account in the simulation model, as well as dry friction appearing on revolute joints using Stribeck friction model [START_REF] Andersson | Friction models for sliding dry, boundary and mixed lubricated contacts[END_REF].

The dynamics of the actuator is represented as

Γ = M a qa + N a qa + Γ t -Γ f , (1)
with q a = q a1 q a2 is the state, Γ f = Γ f 1 Γ f 2 is the vector of dry friction torques, M a = diag(I a1 , I a2 ) is the inertia matrix, and N a = diag(f v1 , f v2 ) the matrix of viscous friction. The elastic joints are modeled as

Γ t = K t (q a -q b ) + B t ( qa -qb ), (2)
where

K t = diag(k t1 , k t2 ), B t = diag(b t1 , b t2
) with k ti and b ti are respectively the stiffness and the damping of the ith joint. The friction torques are described by the Stribeck friction model as follows:

Γ f i = Λ ai if | qai | > 0 (slip), Λ bi if qai = 0 (stick),
where

Λ ai = -Γ f ci + (Γ f si -Γ f ci )e -( qai vs ) 2 sign( qai ), Λ bi = -min(|Γ i -Γ ti | , Γ f si )sign(Γ i -Γ ti ).
Finally, the linkage dynamics are given by

Γ t = M b (q b )q b + N b (q b , qb ) qb + P b (q b )q b , (3)
where

M b = c 1 + 2c 2 cos q b2 c 3 + c 2 cos q b2 c 3 + c 2 cos q b2 c 3 , N b = 2c 2 qb2 sin q b2 c 2 qb2 sin q b2 -c 2 qb1 sin q b2 0 , P b = -c 4 sin q b1 q b1 -c 5 sin q b12 q b12 -c 5 sin q b12 q b12 -c 5 sin q b12 q b12 -c 5 sin q b12 q b12 , with c 1 = m 1 r 2 1 + I 1 + m 2 L 2 1 + m 2 r 2 2 + I 2 , c 2 = m 2 L 1 r 2 , c 3 = m 2 r 2 2 + I 2 , c 4 = m 1 gr 1 + m 2 gL 1 , c 5 = m 2 gr 2 .
It should be noted that the above serial manipulator model has 4-DoF, i.e., two from the actuators and two others for the robot arms. The system modeling includes the transmission deformation between an actuator and its corresponding arm [START_REF] Makarov | Modeling and preview H ∞ control design for motion control of elastic-joint robots with uncertainties[END_REF]. This additional and weakly damped dynamics introduces vibrations due to the excitation of the mechanical resonances resulting from the deformations. However, for control purposes, we assume that the actuator positions q a1 and q a2 coincide with the arm positions q b1 and q b2 . As a consequence, the nonlinear dynamics can be computed from (1), ( 2) and ( 3) with q = q a = q b in the following form:

Γ = M (q)q + N (q, q) q + P (q)q -Γ f , (4)
with

M = M a + M b , N = N a + N b , P = P b .
In the following section, an equivalent representation in a fuzzy descriptor form of model ( 4) is presented. Actuator inertia (kgm 2 )

System Description and Reference Model

This section provides a new representation of the considered serial manipulator and define the reference model corresponding to the desired dynamical behaviour of the system. To this end, let us define the state vector as

x p = [q 1 q 2 q1 q2 ] ∈ R nx , the input vector u = [Γ 1 Γ 2 ] ∈ R nu , the controlled output vector y = [q 1 q 2 ]
, and the disturbance vector

d = Γ f ∈ R n d .
The dynamics of the manipulator (4) can be represented in the following regular descriptor form:

E p (z) ẋp (t) = A p (z) x p (t) + B p u(t) + B d d(t), y(t) = C p x p (t), (5)
where

E p (z) =     1 0 0 0 0 1 0 0 0 0 Îa1 + 2c 2 z 1 c 3 + c 2 z 1 0 0 c 3 + c 2 z 1 Îa2     , C p =     1 0 0 1 0 0 0 0     , A p (z) =     0 0 1 0 0 0 0 1 z 2 z 3 2z 4 -f v1 z 4 z 3 z 3 z 5 -f v2     , B p = B d =     0 0 0 0 1 0 0 1     , with Îa1 = I a1 + c 1 , Îa2 = I a2 + c 3 and z 1 = cos q 2 , z 2 = -c 4 sincq 1 -c 5 sincq 12 , (6)
z 3 = -c 5 sincq 12 , z 4 = c 2 q2 sin q 2 , z 5 = -c 2 q1 sin q 2 .
By applying the sector nonlinear approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF][START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF], the following T-S fuzzy representation can be derived:

E pv ẋp (t) = A ph x p (t) + B ph u(t) + B d d(t), (7)
y(t) = C p x(t),
where

E pv = Ne k=1 v k (z)(E pk ), [A ph B ph ] = N i=1 h i (z)[A pi B pi ].
Note that the T-S fuzzy descriptor system ( 7) is an equivalent form of (5) in a given bounded subset of the state space. The construction of this model is given in more details in Section 6.1. It can be shown that the matrix E pv is non-singular for all z. To guarantee a desired tracking performance, we consider the following reference model:

ẋr (t) = A r x r (t) + B r r(t), (8)
where x r (t) ∈ R nx is the reference state to be followed by the robot state x p (t) and r(t) ∈ R nr is the bounded reference input. The matrices A r and B r are designed according to the desired closed-loop dynamics of the serial manipulator. The tracking error is defined by

e(t) = x p (t) -x r (t).
From ( 7) and ( 8), the following tracking error dynamics can be obtained:

E pv ė(t) = A ph e(t) + (A ph -E pv A r )x r (t) (9) + B ph u(t) -E pv B r r(t) + B d d(t).
Defining the augmented state vector x(t) = [e(t) x r (t) ] and the exogenous input vector w(t) = [d(t) r(t) ] , the two dynamics ( 8) and ( 9) can be now gathered in the following regular descriptor representation:

E v ẋ(t) = A hv x(t) + B h u(t) + D v w(t), (10)
where

E v = E pv 0 0 I , D v = B d -E pv B r 0 B r , (11)
B h = B ph 0 , A hv = A ph A ph -E pv A r 0 A r .
Next, the descriptor system (10) will be used to design a tracking controller for the studied manipulator robot.

4 Tracking Control for Descriptor Fuzzy Systems

Design Specifications

We consider the class of controllers given by

u(t) = N i=1 Ne k=1 h i (z)v k (z) (K ik e(t) + L ik r(t)) = [K hv 0] x(t) + [0 L hv ] w(t). (12)
Our control goal is to determine the gain matrices K ik and L ik of controller [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] such that system (10) verifies the following closed-loop performance specifications.

-(Asymptotic stability). For w(t) = 0, the zero solution of the closed-loop system [START_REF] Hedberg | Comparing feedback linearization and Jacobian linearization for LQ control of an industrial manipulator[END_REF] under the control effect of ( 12) is asymptotically stable. -(Decay rate performance). Given a positive scalar α and for w(t) = 0, the tracking error e(t) converges exponentially towards the origin with a decay rate less than α. -(H ∞ tracking performance). Given a positive definite matrix Q and a positive scalar γ, the inequality

t f 0 e(t) Qe(t)dt ≤ γ 2 t f 0 w(t) w(t)dt (13)
holds for any t f > 0, and zero initial conditions.

Remark 1 Inequality (13) implies that if w(t) is a energybounded signal (w(t) ∈ L 2 ) then, the same holds for the tracking error, i.e., e(t) ∈ L 2 . The coefficient γ is called a bound for the L 2 -gain of the system. The smaller γ is, the more accurate is the following of the reference trajectory. Note that matrix Q, usually taken as an identity matrix [START_REF] Van Der Schaft | L 2 -gain analysis of nonlinear systems and nonlinear state-feedback H ∞ control[END_REF], allows additional degree-of-freedom in the control design.

Remark 2 An integral action can be easily introduced in the control law to achieve a zero error in steady state for a constant exogenous input w(t) by considering an additional state vector x I (t) equals to the integral of the output tracking error, i.e., ẋI (t) = C p e(t). Denoting x(t) = [e x I x r ] , the control law becomes

u(t) = K hv e(t) + H hv x I (t) + L hv r(t) = [ Khv 0] x(t) + [0 L hv ] w(t),
where Khv = [K hv H hv ], while, in the system description [START_REF] Purwar | Adaptive output feedback tracking control of robot manipulators using position measurements only[END_REF] and [START_REF] Hu | Adaptive backstepping trajectory tracking control of robot manipulator[END_REF], matrix E pv should be replaced by diag(E pv , I), matrix A ph by

  A ph 0 A ph -E pv B r C p 0 0 0 0 A r   ,
and so on.

LMI-Based Control Design Conditions

LMI-based constraints allowing the computation of the control gain matrices K ik and L ik are given in the following theorem.

Theorem 1 If there exist n x × n x -matrices P 11 > 0,

P 22 > 0, Q 11ij , Q 12ij , Q 21ij , Q 22ij , X 11ij , X 12ij , X 21ij , X 22ij , matrices M jk ∈ R nu×nx , L jk ∈ R nu×nr
, and a positive scalar γ such that

Ξ iik < 0, Ξ ijk + Ξ jik < 0, ( 14
)
where ∀i, j ∈ Ω r , i < j and k ∈ Ω Ne . The block matrix Ξ ijk is defined as

Ξ ijk =           S 11 S 21 S 22 S 31 S 32 S 33 S 41 S 42 S 43 S 44 0 0 B T d 0 -γ 2 I 0 0 S 63 B T r 0 -γ 2 I P 11 0 0 0 0 0 -Q -1           , (15)
and

S 11 = He(Q 11ij + αP 11 ), S 21 = Q 21ij + Q 12ij , S 31 = A pi P 11 + B pi M jk -E pk Q 11ij + X 11ij , S 41 = -Q 21ij + X 12ij , S 22 = He(Q 22ij + αP 22 ), S 32 = (A pi -E pk A r )P 22 -E pk Q 12ij + X 21ij , S 42 = A r P 22 -Q 22ij + X 22ij , S 43 = -X 21ij -X 12ij E pk , S 44 = -He(X 22ij ), S 33 = -He(E pk X 11ij ), S 63 = (B pi L jk ) T -(E pk B r ) T .
Then, the control law [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] with K ik = M ik P -1 11 and L ik is such that the specifications given in Section 4.1 are satisfied.

Proof Using the extended state vector x * = [x ẋ ] , the closed-loop system can be represented in a singular descriptor form:

E * ẋ * = A * hhv x * + B * hv w, (16)
where

E * = I 0 0 0 , A * hhv = 0 I A hhv -E v , B * hv = 0 B hv , A hhv = A hv + B h [K hv 0], B hv = D v + B h [0 L hv ].
Let us define

P * hh = P 0 Q hh X hh , P = P 11 0 0 P 22 , (17)
where

Q hh = Q 11hh Q 12hh Q 21hh Q 22hh , X hh = X 11hh X 12hh X 21hh X 22hh .
From the definition of Ξ ijk in [START_REF] Vermeiren | Motion control of planar parallel robot using the fuzzy descriptor system approach[END_REF], some simple computations lead to

Ξ hhv =   Ξ 11 B * hv -γ 2 I 0 C * z P * hh 0 -Q -1   , (18)
where C * z = [I 0 0 0] and Ξ 11 = He(P * hh A * hv +αP * hh E * ). Since the membership functions satisfy the convex sum property, it follows from conditions ( 14) that

Ξ hhv < 0. ( 19
)
Note that Ξ 11 can be represented in the form

Ξ 11 = He(Q hh + αP) A hv P -E v Q hh + X hh -He(E v X hh )
.

Inequality [START_REF] Lilly | Fuzzy Control and Identification[END_REF] leads to He(E v X hh ) > 0. As a consequence, X hh and P * hh are non-singular matrices since E v is non-singular. Consider the following Lyapunov function candidate:

V (x * ) = x * E * P * -1 hh x * .
The derivative of the function V along the trajectories of ( 16) is given by

V (x * ) = ẋ * E * P * -1 hh x * + x * P * - hh E * ẋ * . ( 20
)
Note that due to the special structure of P * hh in ( 17), one has E * T Ṗ * -1 hh = 0. This means that there are no time-derivatives of the membership functions in the expression of V . It should be noted that dealing with unknown time-derivatives of the membership functions in the T-S fuzzy control framework still remains an open research issue, see [START_REF] Guerra | Non-quadratic local stabilization for continuous-time Takagi-Sugeno models[END_REF][START_REF] Nguyen | Improved lmi conditions for local quadratic stabilization of constrained Takagi-Sugeno fuzzy systems[END_REF][START_REF] Nguyen | An augmented system approach for LMI-based control design of constrained Takagi-Sugeno fuzzy systems[END_REF].

From ( 16) and ( 20), we can obtain the equality V (x * ) + e Qe -γ 2 w w + 2αV (x * ) = ξ Ω hhv ξ, where

ξ = P * -1 hh x * w , Ω hhv = Ξ 11 + P * hh C * z QC * z P * hh B * hv -γ 2 I .
By Schur complement lemma, we can show that ( 19) is equivalent to Ω hhv < 0. This means that V (x * ) + e Qe -γ 2 w w + 2αV (x * ) < 0.

By standard arguments, the properties of the closedloop system can then be easily established.

5 Complexity Reduction for Real-Time Control

Reduction of Fuzzy Rules

This section presents a way to reduce the number of rules for the generalized form T-S fuzzy systems. The rule reduction is related to the computational effort for controller design utilizing LMIs. To deal with this problem, some nonlinearities appearing in the initial model ( 5) are considered as parameter uncertainties.

Applying again the sector nonlinear approach, an uncertain descriptor T-S representation can be derived:

Êpv ẋp (t) = Âph x p (t) + Bph u(t) + B d d(t), (22)
y(t) = C h x(t),
where

Êpv = Ne k=1 v k (z)(E pk + ∆E pk ), [ Âph Bph ] = N i=1 h i (z)[A pi + ∆A pi B pi + ∆B pi ].
The system uncertainties can be represented in the form

∆E pk = H e ∆ e W epk , ∆A pi = H a ∆ a W api , ∆B pi = H b ∆ b W bpi , k ∈ Ω Ne , i ∈ Ω r ,
where ∆ ∆ (t) ≤ I with ∈ {e, a, b}.

LMI-Based Robust Control Design Conditions

The following theorem provides LMI conditions to design a robust PDC controller [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] for the uncertain descriptor system [START_REF] Nguyen | Driverautomation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF].

Theorem 2 If there exist n x × n x -matrices P 11 > 0,

P 22 > 0, Q 11ij , Q 12ij , Q 21ij , Q 22ij , X 11ij , X 12ij , X 21ij , X 22ij , matrices M jk ∈ R nu×nx , L jk ∈ R nu×nr
Ψ iik < 0, Ψ ijk + Ψ jik < 0, ( 23
)
where ∀i, j ∈ Ω r , i < j and k ∈ Ω Ne . The block matrix Ψ ijk is defined as

Ψ ijk = Ψ 1ijk Ψ 2ijk Ψ 3ijk , with Ψ 1ijk =          
0 0 B d 0 -γ 2 I 0 0 S 63 B r 0 -γ 2 I P 11 0 0 0 0 0 -Q -1           , Ψ 2ijk =   T 11 T 12 0 0 0 0 0 T 21 0 0 0 0 T 26 0 T 31 T 32 T 33 T 34 0 T 36 0   , Ψ 3ijk = diag(-τ a ijk I, -τ b ijk I, -τ e ijk I)
, where

S * 33 =S 33 + τ a ijk H a H a + τ b ijk H b H b + τ e ijk H e H e , T 11 =W api P 11 , T 21 = W bpi M jk , T 31 = -W epk Q 11ij , T 12 = W api P 22 , T 32 = -W epk (A r P 22 -Q 12ij ), T 33 = -W epk X 11ij , T 34 = -W epk X 12ij , T 26 = W bpi L jk , T 36 = -W epk B r .
Then, the control law [START_REF] Pan | Composite learning robot control with guaranteed parameter convergence[END_REF] with K ik = M ik P -1 11 and L ik is such that the specifications given in Section 4.1 are satisfied for the uncertain case.

Proof Following the same line as in the proof of Theorem 1, we can prove that ( 21) is satisfied if

Ξhhv = Ξ hhv + ∆Ξ hhv < 0,
where Ξ hhv is defined in [START_REF] Zhang | Fuzzy Modeling and Fuzzy Control[END_REF]. The uncertain term ∆Ξ hhv is defined as follows:

∆Ξ hhv = H ∆(t)Ψ 2hhv + Ψ 2hhv ∆(t) H,
where

∆(t) = diag(∆ a (t), ∆ b (t), ∆ e (t)), H =   0 0 H a 0 0 0 0 0 0 H b 0 0 0 0 0 0 H e 0 0 0 0   .
Let S hhv = -Ψ 3hhv > 0. By completion of squares, it follows easily that

Ξ hhv + ∆Ξ hhv ≤ Ξ hhv + H S hhv H + Ψ 2hhv ∆(t) S -1 hhv ∆(t)Ψ 2hhv ≤ Ξ hhv + H S hhv H + Ψ 2hhv S -1
hhv Ψ 2hhv . By Schur complement lemma, it can be easily shown that Ξ hhv + ∆Ξ hhv < 0 if Ψ hhv < 0. This latter is ensured by inequalities [START_REF] Chen | Robust optimal referencetracking design method for stochastic synthetic biology systems: T-S fuzzy approach[END_REF], which completes the proof.

Remark 3 A large value of the guaranteed decay rate α or a small value of the H ∞ gain γ may lead to unacceptable control gain. This problem can be easily dealt with in the feedback design by introducing additional LMI constraints in the optimization problem. The decision variable M ik is linked with the control gains by the expression M ik = K ik P 11 . To prevent excessively high feedback gains, it can be imposed that

M ik M ik ≤ 2 I nu , i ∈ Ω N , k ∈ Ω Ne , (24)
for some positive scalar . Condition ( 24) is equivalent to the following LMIs:

2 I nu M ik M ik I nx ≥ 0, i ∈ Ω N , k ∈ Ω Ne .
Remark 4 The computational complexity of the LMIbased optimizations can be evaluated by the number of scalar decision variables N var and the number of rows of all considered LMI conditions N row . These numbers corresponding to Theorem 1 are given by

N var = 6 + n x (1 + 3N 2 + N e N 2 + n d ), N row = 6n x (1 + N N e ) + n x N (8n x N + N e n u ).
Therefore, we can reduce the computational complexity by reducing the number of rules N and N e .

Application to Robot Fuzzy Tracking Control

The T-S fuzzy descriptor control approach is applied in the following for the trajectory tracking of the 2-DoF robot. Numerical results obtained with the simulation robot model presented in Section 2 are then provided to demonstrate the practical performance of the proposed tracking controllers.

Fuzzy Descriptor Modeling for Tracking Control

The following joint limitations are considered for the studied 2-DoF manipulator:

|q 1 | ≤ π rad, |q 2 | ≤ π rad, | q1 | ≤ 15 rad/s, | q2 | ≤ 15 rad/s. ( 25
)
The above limits are defined such that the serial robot can operate in a large workspace while taking into account its physical characteristics.

Exact Fuzzy Descriptor Representation

The premise variables z i defined in ( 6) are bounded on the subset defined by [START_REF] Hung | Fuzzy virtual reference model sensorless tracking control for linear induction motors[END_REF]. Let z i min and z i max be respectively the lower and upper bounds of the premise z i on that subset. Using the parameter values given in Table 1, it can be obtain the following bounds: Using the sector nonlinearity approach [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF], an exact T-S fuzzy descriptor representation for the control-based robot model ( 5) can be easily obtained. The derivation of this 2 5 = 32 rules T-S fuzzy representation is standard and omitted here due to the space limitation.

z 1 max = -z 1 min = 1, z 2 max = z 3 max = 4.

Fuzzy Modeling with Reduced Complexity

Theoretically, the control approach proposed in Section 4 can be used to deal with T-S descriptor systems with any number of rules. However, a large number of rules may cause costly computational burden for realtime control implementation. Therefore, it is worth simplifying the T-S fuzzy model to cope with the hardware limitations. To achieve this goal, some system nonlinearities will be considered as modeling uncertainties in order to reduce the number of premise variables, and thus the number of submodels. Then, these uncertainties can be dealt with using the robust control scheme proposed in Theorem 2. We present here an example leading to a T-S fuzzy system with 8 rules from system [START_REF] Nguyen | Driverautomation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF]. To this end, we consider

z 4 (t) = z 4 max f 1 (t), z 5 (t) = z 5 max f 2 (t), with |f i (t)| ≤ 1, i = 1, 2. Let us define ω i1 = z i -z i min z i max -z i min , ω i2 = 1 -ω i1 , i ∈ Ω 3 .
The membership functions of the above T-S fuzzy descriptor system are expressed as follows:

v 1 (z) = ω 11 , h 1 (z) = ω 21 ω 31 , h 2 (z) = ω 22 ω 31 , v 2 (z) = ω 12 , h 3 (z) = ω 21 ω 32 , h 4 (z) = ω 22 ω 32 .
Then, the model ( 5) can be represented in the form [START_REF] Nguyen | Driverautomation cooperative approach for shared steering control under multiple system constraints: Design and experiments[END_REF] with the nominal matrices given by

E p1 = E p (z 1 min ), E p2 = E p (z 1 max ), A p1 = A p (z 2 min , z 3 min ), A p2 = A p (z 2 max , z 3 min ), A p3 = A p (z 2 min , z 3 max ), A p4 = A p (z 2 max , z 3 max ), B pi = B p , i ∈ Ω 4 ,
and

∆E pv = 0, ∆B ph = 0, ∆ pa = diag(f 1 (t), f 2 (t)),
H a = 0 0 1 0 0 0 0 1 , W aph = 0 0 2z 4 max z 4 max 0 0 z 5 max 0 .

Observe that the obtained nonlinear uncertain descriptor system has now only 3 premise variables z 1 (t), z 2 (t) and z 3 (t), and so 2 3 = 8 rules. Similarly, we can obtain a 4-rules (respectively 2-rules) T-S fuzzy model by considering z 2 (t), z 4 (t), z 5 (t) (respectively z 1 (t), z 2 (t), z 4 (t), z 5 (t)) as system uncertainties. It is clear that a linear uncertain descriptor system can be easily obtained following the same line while considering all premise variables as model uncertainties. The details of these uncertain T-S fuzzy descriptor systems are not given here for brevity.

Tracking Reference Model

The coordinates of the effector position P in the xy-plane are expressed by the forward kinematic model as follows, see Fig. 1:

x P = L 1 cos(q 1 ) + L 2 cos(q 1 + q 2 ), (26)
y P = L 1 sin(q 1 ) + L 2 sin(q 1 + q 2 ).

It is desired that the point P follows a circular trajectory reference with a radius of R = 0.2 m and a center defined as x P 0 = 0.35 m, y P 0 = 0.65 m. The corresponding position of P at time t is given as

x P (t) = x P 0 + R cos ψ(t), y P (t) = y P 0 + R sin ψ(t),

where ψ(t) is defined as a quintic polynomial so that the circular trajectory is executed in a duration of 1.5 s with a maximum curvilinear velocity of 1.57 m/s. The desired angular positions q 1d (t) and q 2d (t) are computed using the inverse kinematic model derived from the forward one [START_REF] Taniguchi | Fuzzy descriptor systems and nonlinear model following control[END_REF]. The state-space matrices of the reference model ( 8) are given as follows:

A r =     0 0 1 0 0 0 0 1 -3844 0 -124 0 0 -3844 0 -124     , B r =     0 0 0 0 0 0 0 0 0 0 0 0 3844 0 124 0 1 0 0 3844 0 124 0 1     .
This corresponds to two decoupled, second-order dynamics with a real double pole located at -62 rad/s to avoid the excitation of unmodeled high-frequency dynamics which appears in the simulation model ( 1)-( 3). The reference model input is defined as r(t) = q 1d (t) q 2d (t) q1d (t) q2d (t) q1d (t) q2d (t) .

This choice allows guaranteeing a sufficiently smoothness level for the reference trajectory. The detail is omitted here since the reference design is out of the scope of this paper. Fig. 2 End-effector trajectories for Case 1. Reference (solid black), trajectory with 32-rules controller (dotted blue), trajectory with 4-rules controller (dashed red).

Illustrative Results and Discussions

This section provides numerical results carried out with the robot simulation model presented in Section 2 embedded in Simscape Multibody TM environment of Matlab/Simulink software. For tracking purposes, an integral action is incorporated in the PDC control scheme as explained in Remark 2. All LMI problems are solved with Matlab using the Yalmip Toolbox [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF] and the solver SDPT3 [START_REF] Toh | SDPT3 -a Matlab software package for semidefinite programming, version 1.3[END_REF]. A comparative study between two tracking controllers derived from a full T-S fuzzy representation and a simplified one is given to emphasize the interest of the proposed robust control scheme for real-time applications. It should be stressed also that when solving LMI conditions in Theorem 2, feasible solutions can only be obtained with T-S fuzzy descriptor models having at least 4 rules. This means that a tradeoff between the complexity of the model and the design conservatism has to be found.

To demonstrate the robustness of the proposed approach, a comparison between results obtained for three cases is provided. The first case (Case 1) is the nominal one in which the robot is unloaded and free of contact. The second one (Case 2) occurs during a machining process. In this case, the external disturbance corresponds The reference trajectory in the x -y plane as well as those obtained for the robot effector are depicted in Fig. 2 for the nominal case. As shown on this figure, the obtained performance are similar for the two controllers. This can be confirmed by the circularity errors shown in Fig. 4) for which the maximal deviation with respect to the radius of the reference trajectory is about 1.3 mm for the two controllers. The circularity errors are also plotted in the two operating conditions (Cases 2 and 3) in Figs. 5 & 6. Here again the behaviour obtained for the two controllers are similar, the worst one being with respect to the external force applied (Case 2) for which the trajectory radius evolutes between -2.3 and 4.4 mm. Both cases show the robustness of the tracking performance to external disturbances (machining force) and uncertainties (load variation). The actuator torques are given in Fig. 7 for the 4-rules controller in the nominal case. The maximal required torques are 228 Nm for joint 1 and 47 Nm for joint 2. Joint velocities are shown in Fig. 8, and the friction forces applied on the two joints in Fig. 10. The tracking errors of angular positions are shown in Fig. 9 confirming the good tracking performance obtained using the proposed approach.

Concluding Remarks

A robust T-S fuzzy descriptor approach was applied to control a nonlinear 2-DoF serial manipulator. This approach can not only guarantee a high H ∞ tracking performance but also allow the complexity reduction of both control design and implementation of the controllers. Keeping the descriptor form stemmed from the mechanical modeling of robot manipulators while considering only essential nonlinearities for control design, the proposed nonlinear control approach allows dealing with more complex manipulators compared to the standard T-S control methodology. Application to a two DoF serial manipulator have clearly proved the effectiveness of the new approach despite the existence of unmodeled phenomena such as dry frictions, external disturbances, or parametric variations. Future works focus on the design of a robust output feedback scheme requiring only position measurements for real-time control implementation [START_REF] Nguyen | Sensor reduction for driver-automation shared steering control via an adaptive authority allocation strategy[END_REF].

Fig. 1

 1 Fig. 1 Description of a 2-DoF robot.

  , and positive scalars γ, τ a ijk , τ b ijk , τ e ijk such that

  79, z 2 min = -103.01, z 3 min = -22.07, z 4 max = z 5 max = -z 4 min = -z 5 min = 16.87.

Fig. 3

 3 Fig. 3 External force. Vertical (dotted blue) and horizontal (solid red) components during circular trajectory.

Fig. 4

 4 Fig. 4 Circularity errors in Case 1 obtained with 32-rules controller (dotted blue), with 4-rules controller (solid red).

Fig. 5 Fig. 6

 56 Fig. 5 Circularity errors in Case 2 obtained with 32-rules controller (dotted blue), with 4-rules controller (solid red).

2 Fig. 7 Fig. 8 2 Fig. 9

 27829 Fig.7Actuator torques in Case 1: actuator 1 (dotted blue), actuator 2 (solid red).

Fig. 10

 10 Fig. 10 Friction torques: joint 1 (dotted blue), joint 2 (solid red).

Table 1

 1 System Nomenclature.

	Symbol	Description	Value
	L 1 , L 2	Lengths of robot arms (m)	0.5, 0.5
	m 1 , m 2 I 1 , I 2	Masses of robot arms (kg) Inertia of robot arms (kgm 2 )	15, 9 0.31, 0.19
	r 1 , r 2	Distances between joint axis	0.25, 0.25
		and mass center (m)
	f v1 , f v2	Coefficients of viscous	0.14, 0.14
		friction (Nms/rd)
	g	Gravity acceleration (m/s 2 )	9.81
	I a1 , I a2

  = 20 and Q = diag(10 -5 , 10 -5 , 1, 1). For conciseness, the details on the 32-rules PDC controller obtained from Theorem 1 with the same values of α and Q are omitted. From Remark 4, the computation of the controller gains via a robust compensation scheme can be significantly simplified, see Table2.

				control results are obtained
				K 11 = 10 4 ×	-6.31 1.16 -0.0978 -0.0018 0.91 -6.513 0.0044 -0.0591	,
				H 11 = 10 4 ×	0.720 0.939 0.939 4.62	,
				K 12 = 10 3 ×	-9.73 -1.74 -0.469 -0.0896 -1.89 -3.76 -0.0968 -0.162	,
				H 12 = 10 3 ×	0.312 0.159 0.152 0.0302	,
				L 11 =	14.7 -1.13 0.47 -0.04 0.004 -3 × 10 -4 -6.37 15.2 -0.21 0.49 -0.002 3.9 × 10 -4 ,
				L 12 =	12.3 0.634 0.395 0.0205 0.0032 0.0002 3.82 11.8 0.123 0.380 0.0010 0.0031	,
	to an external force F e (t) tangential to the effector tra-	with α
	jectory applied on it
	F e (t) = -f e (t)	˙ (t) || (t)||	,
	with (t) = [ ẋP ẏP ] and f e (t) ∈ [138, 218] N, see
	Fig. 3. For the last case (Case 3), the robot manipu-
	lator must carry a mass of 5 kg. This corresponds to a
	variation of the effector mass m 2 of 55.6%. Note that,
	as explained in Section 2, in all these cases, the fric-
	tion and joint elasticity effects are taken into account
	in the simulation, although not considered in the con-
	trol model.
	For illustration, we take the case of a 4-rules uncer-
	tain T-S fuzzy descriptor model obtained by considering
	z 2 (t), z 4 (t) and z 5 (t) as system uncertainties. Solving
	the LMI conditions in Theorem 2 for this case gives
	control gains with very high values, see for instance
	K 11 = 10 9 ×	3.88 0.15 0.03 0.01 0.20 7.89 0.01 0.08	,
	H 11 = 10 9 ×	-3.88 -0.15 -0.21 -7.89	,
	K 12 = 10 8 ×	8.12 -0.03 0.08 -0.01 -0.04 0.01 -0.01 0.01	,
	H 12 = 10 8 ×	-8.13 0.03 0.04 -0.01	.
	It is clear that such control gains are impractical for
	real-time implementation. Hence, we redesign the feed-
	back gains by including the additional constraint (24)
	with = 10 4 as explained in Remark 3. The following

Table 2

 2 Computation Costs.

	Number of rules 32-rules 16-rules 8-rules 4-rules
	N var N row	5138 33816	1298 8728	338 2328	98 664
	Circularity error (mm)	-2 0 2 4			32-rules 4-rules
		0	0.5	1		1.5
				Time (s)

2.71, 0.50 k t1 , k t2 Stiffness constant ×10 4 (N/m) 6.4, 1.2 b t1 , b t2 Damping constant (Ns/m) 20.4, 3.8 Γ f s1 , Γ f s2 Static friction torque (Nm) 6, 3 Γ f c1 , Γ f c2 Coulomb friction torque (Nm) 4, 2 v s Sliding speed coefficient (m/s) 0.1