Adaptive Robust Failure Compensation Control for Servo System Driven by Twin Motors

In the context of failure control of servo system driven by twin motors, there are still no available results to compensate unknown actuator failures which seem inevitable in practice by adaptive backstepping technique. Therefore, to rise the reliability of the system, we aim at addressing such a problem by proposing an adaptive robust actuator failure compensation control scheme based on backstepping technique for servo system driven by twin motors. Unlike the traditional backstepping, the estimation of unknown coefficient of intermediate state variable is introduced in coordinate changes. In addition, matching and non-matching uncertainties have been fully considered in the controller design. Simulation results show that the designed controller can ensure the boundedness of all the signals no matter actuator failures occur or not.


I. INTRODUCTION
The servo system has been widely used in many fields including industry, military and vehicle [1] etc. To get higher movement performances, servo system driven by twin motors is usually used and the performance of such a servo system has been received extensive attention [2]- [4]. Unknown nonlinearities in input signal such as backlash and dead zone are deeply studied [4]. It is clear that the purpose is to eliminate their influences and to rise the controlled performance.
Besides improving the performance of the system under partial control information missing [5], [6], more and more attention has been paid to the problem of system reliability including the studies on fault detect, fault tolerant control etc [7]- [17]. Actuator failure is an common fault and seems inevitable in practice control systems. Such failures which may lead to instability or even catastrophic accidents are often uncertain in time, value and pattern. The reliable control becomes more difficult under such unknown actuator failures. To address this problem, several schemes have been proposed in recent twenty years. Compared to other methods, an adaptive approach [7]- [14] can handle system parametric uncertainties by online estimating unknown parameters with update laws. As a promising approach, backstep-The associate editor coordinating the review of this manuscript and approving it for publication was Yan-Jun Liu.
ping technique [8]- [10] and [18]- [20] has been widely used in the controller design of a class of nonlinear systems. In [8]- [10], backstepping technique is used in the design of adaptive failure compensation scheme for a class of nonlinear systems. Transient performance was guaranteed by using prescribed performance bound in [8] while backlash hysteresis existing in practice of actuator failure systems was studied deeply in [9]. In [10], an important result which removes the assumption on finite failure number was established. However, in the context of the failure control of servo system driven by twin motors [2], there is still no results available based on adaptive approaches. In this note, we aim to address such a problem and propose an adaptive failure compensation control scheme technique for servo system driven by twin motors. Clearly, unknown parameter a = K L J L of x 3 in (8) makes traditional backstepping technique become incapable in the design of adaptive controller. Our idea is to introduce the estimation of ϑ = 1 a in the virtual control design and the coordinate changes (14). And then, we use proj to guarantee this estimation ϑ being bounded. To compensate the effects caused by matching and non-matching uncertainties ω 1 and ω 2 , the property of inequality about tanh(·) is used. Finally, an adaptive robust compensation control scheme is proposed and all uncertainties caused by unknown parameters, matching and non-matching terms and unknown actuator failures can be compensated successfully. The main contributions of this paper are as follows: (1) The control problem is investigated for servo system driven by twin motros with unknown parameters, matching and non-matching uncertainties, unknown actuator failures.
(2) In addition, we use inequality of tanh(·) to handle the effects caused by matching and non-matching uncertainties.
(3) Moreover, unlike the traditional backstepping technique the estimation of ϑ = 1 a is introduced in coordinate changes to compensate unknown parameter a as the coefficient of x 3 . It is shown that the proposed adaptive robust controller can ensure the boundedness of all the signals of the closed loop whether or not actuator failures occur.
The rest of the paper is organized as follows. In section 2, we formulate the servo system driven by twin motors with unknown actuator failures. An robust adaptive control scheme and the main results about stability are proposed in section 3. The simulation studies is given in section 4. Finally, the paper is concluded in section 5.

II. MODELS AND PROBLEM STATEMENT
We consider the servo system driven by twin motors [2] shown in Figs.1. The mathematical model can be described as followṡ where θ L (t), ω L (t), J L , K L are angle position, angular velocity, moment of inertia from load transformed into motor side, stiffness coefficient of transmission mechanism, respectively. The effect caused by load is not directly shown in above system model (1). Instead, it has been considered in parameters θ L (t), ω L (t), J L . Parameters θ j (t), ω j (t), J mj , K Tj , u j (t) (j = 1, 2) are angle position, angular velocity, moment of inertia, electromagnetic torque coefficient and control signal of jth motor, respectively. w 1 (t), w 2j (t)(j = 1, 2) are unknown nonlinear modeling errors and denote matching, non-matching uncertainties. The effects generated by reducers and the backlash caused by gears are ignored in the establishment of the mathematical model of the servo system driven by twin motors shown in Figs.1. So the mathematical model (1) is an ideal model. In order to simplify system model, we let Then we havė To obtain the strict feedback structure, letting Then we havė All interior parameters of these twin motors are the same. Namely, K T 1 = K T 2 , J m1 = J m2 . Then we havė Letting System model can be rewritten aṡ where w 2 (t) = w 21 (t) + w 22 (t). We let 2a = b 1 and b = (b 1 , b 2 ) T , theṅ Namely,ẋ where known function Remark 1: The following difficults exist in the controller design.
• The unknown parameter ϑ = 1/a will be estimated to handle the unknown parameter a in term ax 3 . In addition b 1 = 2a is also estimated by designing an adaptive operator in the proposed adaptive control law to compensate the unknown effect caused by 2ax 1 .
• Compare to common triangular system, there is a nonzero coefficient composed by multiple parameter a in front of system state x 3 in the second state differential equation. Such an unknown parameter makes the controller design become more and more difficult and traditional backstepping becomes unavailable. We will eliminate the effect of this unknown parameters by designing an adaptive estimator of 1 a and introduce this estimator in the proposed controller. As we all know, actuator failure is inevitable in the servo system driven by motors. Such as aging of electrical components and traction loss from input to output always exist. Similar to [8]- [10], failure of the jth actuator at time instant t jf can be modeled as follows where σ j ,ū j , t jf are unknown constants and 0 ≤ σ j ≤ 1.
The signal ν j is the input of the j − th actuator. An actuator with its input equal to its output, i.e. u j = ν j is regarded as a failure-free actuator. Easily, we can get the following situations: The jth actuator works normally.
The jth actuator is called partial loss of effectiveness.
• σ j = 0, It indicates u j =ū j . The ith actuator is called total loss of effectiveness. σ j ,ū j are unknown and can be seen as the time-dependent jump parameters. A specific failure corresponds to a group of values of σ j ,ū j . With the failure model given in (11), system (10) can be rewritten aṡ To design the adaptive control law, the following assumptions are made.
Assumption 1: There is at least one actuator being not total loss of effectiveness. Any actuator can change only from normal to partial failure or total failure and only fails once.
From Assumption 1, we know that there is a finite time instant T f and no new failure will occur after T f . Assumption 2: Unknown parameters ρ, a, b lie in a known bounded set, respectively and these bounded sets don't include zero.
From (7) and the actual meanings of parameters K L , J L , J mj , K Tj , the sign of a, b, ρ are known.

III. DESIGN OF EVENT-TRIGGER CONTROLLER
The control objective is to design a robust adaptive failure compensation control scheme to guarantee all signals bounded under any failure of actuators and to realize the output signal y = x 1 tracking to the reference signal y r effectively. To obtain a suitable control law and update laws for controller parameters based on the backstepping approach, we first make the following coordinate changes where z 1 is tracking error and α i (i = 1, 2, 3) is virtual control. ϑ is the estimation of parameter ϑ = 1 a . Virtual control α i is an actively constructed control variable. In step i, our objective is to design a virtual control law α i−1 which makes z i tends to zero. Then we will give the recursive design steps by backstepping approaches.
Remark 2: Different from the general lower triangular system whose controller can be designed by using backstepping, an unknown parameter a as the coefficient of x 3 exists. Thus the traditional backstepping can not be used for the controller design. To solve this problem, an estimation of ϑ = 1/a is introduced in coordinate changes. Then the uncertainty caused by unknown parameter coefficient a can be compensated and backstepping technique can be carried out.
Step 1: From system model (12) and coordinate transformation (14), we haveż Considering the following Lyapunov function The derivative of Lyapunov function iṡ Virtual control α 1 can be chosen as where c 1 is a positive constant. Then we havė Step 2: From system model, we know that the derivative of Clearly, we know α 1 is dependent on variables x 1 and y r . So we can geṫ With (14) we can geṫ Note thatθ = ϑ −θ, then aθα 2 = a(ϑ −θ)α 2 = α 2 − aθα 2 Considering the following Lyapunov function whereâ is the estimation of parameter a andb is the estimation of parameter b.ã = a−â andb = b−b denote estimation errors. ϑ , a are positive constants and b is a positive definite design matrix. With (19) and (23), the derivative of Choosing α 2 as where c 2 is a positive constant. So we havė Note that the following properties of function tanh(·) Well, as we know, all continuous approximate functions of sign(·) have similar property, for example sg(·) in [21]. With VOLUME 9, 2021 (28), we can get Then we havė Step 3: The derivative of z 3 iṡ Considering the following Lyapunov function The derivative of V 3 iṡ Following we analyze the term z 3ż3 Then α 3 can be chosen as Remark 3: We will use proj(·) to guaranteeθ being bounded. So above bound functionδ 1 (x 1 , x 2 , x 3 ) can be found easily.
With (33)-(35), we havė Step 4: The derivative of z 4 iṡ The derivative of α 3 iṡ Different from traditional backstepping technique, in the following we will give the control law and adaptive update laws of unknown parameters. Similar to [7]- [9], the structure of adaptive failure compensation controller can be written as where κ is a desired parametric vector and is a known vector. Both are 3 dimensional vectors. They can be described as Because κ is unknown, it can not be directly used in the ν j design. Instead, we use its estimation to generate the input signal ν j . Then we can get Control Laws: whereκ be its estimation. Let T i is the time instant of actuator failure occurrence and set Q iT denotes the actuators of total failure in interval (T i , T i+1 ](i = 0, 1, · · · , f ) and Q iT ∪Q iT = 1, 2. Let T f = +∞ and T 0 = 0. In time interval (T 0 , T 1 ], all actuators work normally. In interval (T i , T i+1 ], vectors κ and should be chosen to satisfy By fixing as κ can be chosen as α can be regarded as the virtual control signal in this step and chosen as and update laws arė where κ is a positive definite matrix. l κ is a positive constant and κ 0 is a constant vector. proj(·) denotes a projection operator. It can guarantee that the estimationsb,â,θ are all bounded. We choose Lyapunov function in time interval Note that Then with virtual control α given in (48), we havė With update laws (50), we can geṫ Note that where Theorem 1: Consider the servo system driven by twin motors (1), with unknown parameters and unknown actuator failures described by (11). Under the Assumption 1 to Assumption 4 and with the control laws (43)-(48) and the update laws (49), all signals of the closed-loop system are bounded.
Proof: From (56), in time interval [0, T 1 ] signalsκ, z i are bounded. Note thatθ,ã,b are bounded due to the proj(·) operator in update laws. Therefor we can get V is bounded in interval [0, T 1 ].
Note that the difference between V (T + 1 ) and V (T − 1 ) is only the coefficients in front of the termκ T κκ . Since all the possible jumping on κ are bounded, V (T + 1 ) is bounded, then V (T − 2 ) is bounded. Similar to the above analysis, we can get V (T − j+1 ) is bounded from the bound of V (T + j ). Also in time interval (T f , ∞), it can be shown that V (t) is bounded. Then we have z,θ,κ,b,ã bounded in [0, ∞]. Further more, all signals are bounded including virtual control α i (i = 1, 2, 3), α state x i and input signal ν i .
Case 1. All actuators work normally during the operation of servo system. Namely no failure occurs. Figs.2-3 show   the simulation results. The tracking performance including tracking error is shown in Fig.2 and the output signal is given in Fig.3.
Case 2. We suppose at t = 10, the first actuator loses its effectiveness by an unknown value 90%. The respective has been shown in Fig.4 including tracking error and output signal. Fig.5 shows the compare of the tracking error between these first two cases.   Case 3. We suppose at t = 10, the first actuator is stuck at an unknown value 30. The tracking error and output signal are shown in Fig.6.
Case 4. A comprehensive failures is considered in this case. We suppose the first actuator loses its effectiveness by an unknown value 50% at unknown time instant t = 10 and the second actuator is stuck at an unknown value 20 at t = 20. The tracking error and output signal are shown in Fig.7.
Based on the above simulation results, we can get that the tracking performance of the system can be achieved successfully whether or not failures occur.

V. CONCLUSION
An adaptive robust control scheme is proposed by using backstepping techniques to compensate unknown actuator failures for servo system driven by twin motors. The boundedness of all signals of closed-loop system can be guaranteed whether or not failures occur. The simulation studies also have verified the established theoretical results. JUN