Output Feedback Control for Pneumatic Muscle Joint System With Saturation Input

Pneumatic muscle as a relatively new pneumatic component is often applied to some precise and flexible control systems. The precise control accuracy puts forward higher requirements for the controller design of the pneumatic muscle system. To guarantee good performance, unknown input saturation which seems inevitable must be fully considered in the controller design. In this paper, such a control problem is investigated for a pneumatic muscle joint system with input saturation and external disturbance. The auxiliary signals are introduced to compensate for the effect caused by unknown saturation. Furthermore, a filter is constructed to estimate unmeasured system states. Then an output feedback control scheme has been proposed by these auxiliary signals and states filter. Finally, simulation studies are used to verify the effectiveness of the proposed control scheme.


I. INTRODUCTION
Pneumatic muscle [1], [2] is a new pneumatic component and has been used in precise and flexible control systems due to its good characteristics, for example, light-weight, low cost, high power-weight ratio, and high power volume ratio. Especially in the fields of rehabilitation medicine, virtual reality, and bionic robot and so on, production efficiency has been improved by using pneumatic muscle which seems similar to human skeletal muscle. However, the pneumatic muscle has uncertainties caused by strong nonlinear and gas compressibility [1], which leads to its controlling becoming more complicated. So it is difficult to achieve higher control accuracy. To improve the performance of the pneumatic muscle system, modern control theory has been used in the controller design and system analysis in recent twenty years. Several results which mainly aim to the estimation of unknown system parameters by using state feedback technique has been obtained. But the control schemes based on output feedback approach are always every limited due to the complexity of the practical system.
Uncertainties [3]- [20] which may affect the system performance heavily seem inevitable in practical systems.
The associate editor coordinating the review of this manuscript and approving it for publication was Yan-Jun Liu.
Such uncertainties may be caused by unknown parameters [3]- [7], [11]- [20], external disturbance [3], [4], [7], [8], modelling errors [5], [6], unknown actuator failures [9]- [20] and unknown strong nonlinear input including hysteresis, deadzone and saturation. Different methods and techniques are used to deal with different uncertainties. Adaptive estimation by designing update law is utilized to estimate linearized parameters while unknown modelling errors are handled by some inequalities when its upper bound being a known function. To compensate for the uncertainties caused by external disturbance, we usually assume that such disturbance is bounded by an unknown positive constant. By estimating this upper bound, the disturbance can be restrained effectively. Dead-zone and backlash hysteresis are usually approximated by a linear function. The approximation error is assumed bounded by an unknown constant. Then it will be treated as an external disturbance.
Compared with these above uncertainties, saturation is a potential problem for actuators of control systems. It often severely limits system performance, giving rise to undesirable inaccuracy or leading instability. Input saturation [3] is inevitable in a pneumatic muscle joint system due to the physical limitations of components. Because parameter in the saturation model is usually unknown and such uncertainties caused by unknown saturation can not be estimated VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ by constructing online estimator, the controller design of systems with input saturation become more and more difficult. To solve such a problem, we consider the control for pneumatic muscle joint system with input saturation and external disturbances in this paper. At the same time, considering system states are difficult to measure in practice, the proposed control scheme is an output feedback control scheme. The auxiliary signals are introduced to reduce the influence of unknown saturation input nonlinearity and filters are designed to estimate the unknown system states. The main contributions of this paper can be summarized as follows: (I) The control problem is investigated for pneumatic muscle joint systems with input saturation and external disturbance; (II) Filters are constructed to estimate the unknown system states and an output feedback control scheme is proposed to guarantee the stability of systems; (III) In contrast to existing results, the auxiliary signals r 1 , r 2 are introduced to deal with the unknown saturation. The uncertainties caused by unknown saturation can be compensated successfully. The paper is organized as follows: Section II describes a controlled system model with unknown saturation and unknown external disturbance. Section III presents the designed output feedback control law and analysis of the closed-loop system. Simulation results are given in Section IV to verify the effectiveness of the proposed control scheme. Finally, the paper is concluded in Section V.

II. PROBLEM STATEMENT
Based on the dynamic model of the Lagrangian form of the pneumatic muscle joint [2], we have where J is the moment of inertia of the pneumatic muscle joint. θ is the rotation angle of the pneumatic muscle joint. b v is the damping coefficient of the pneumatic muscle joint system. ϑ(t) represents external disturbances. b 1 , b 2 represents the radius of the pneumatic muscle joint. F 1 , F 2 are the pulling force on two pneumatic muscles and can be described by where C 1 , C 2 , C 3 , C 4 represent parameters in the mathematical model of aerodynamic muscles. ε 1 , ε 2 are the contraction rate of the pneumatic muscle and given as where ε 0 and l 0 represent the initial contraction rate and initial length of the pneumatic muscle, respectively. In equation (2), P 1 (t) and P 2 (t) are the pressure value of the pneumatic muscle. They are described by where k 0 is the proportionality factor. k u is the voltage coefficient. u 0 is the initial voltage. P 0 is the initial pressure of the pneumatic muscle. P(t) is the pressure change of the pneumatic muscle. We suppose that the joint radius is the gear radius of the joint. Then we have b 1 (1) (2) (3) and (4), the mathematical model of the pneumatic muscle joint system can be rewritten as is bounded by an unknown constant. Therefore we havë Then the system model can be rewritten aṡ where y is the output signal and where x 1 , x 2 , y and u are system states, output and input. d 1 , b 0 are known constants. Remark 1: As we all know, external disturbance as a common uncertainty is inevitable in practical systems. In the pneumatic muscle joint system, we use d(t) representing external disturbance and such disturbance satisfies where D max > 0 is an unknown constant. According to the practical actuator of the pneumatic muscle joint system. The following saturation of actuator is considered.
where u M > 0 is an unknown constant. u (v) and v are the output and input of actuator, respectively. With the saturation model (10), the controlled system is reorganized aṡ Remark 2: Saturation input is the most common strong nonlinearity of actuators in practical systems. In the pneumatic muscle joint system, the unknown saturation is inevitable due to the limitation of the inflation catheter. So it must be fully considered in the controller design and stability analysis.
To propose the controller design the following assumptions are made.
Assumption 1: The reference signal y r and its i-th(i = 1, 2) order derivatives are continuous and bounded.

III. DESIGN OF ADAPTIVE CONTROLLERS
In order to obtain the output feedback control law, we rewrite the system asẋ where With (9), we have ||D(t)|| ≤ D max . Note that x is unavailable and only y is measured. So we design the filters to estimate x and generate available signals for controller design. The filters is constructed aṡ where k = [k 1 , k 2 ] T is a vector such that the matrix A 0 = A − kE T 1 is Hurwitz. Namely, there exits a P such that PA 0 + A T 0 P = −2I , P = P T > 0. By the filters, the x is estimated asx The derivative ofx(t) iṡ Now we consider the state estimation error Define the Lyapunov function V ThenV

Remark 3: Because P is a constant matrix and D(t) is bounded by D max , PD(t) 2 is bounded by a constant. Then from (20), we have theV being bounded by − T + PD(t) 2 . Although V is not monotonically decrease, V is bounded and its upper bound depends on PD(t) . Then we can get the estimation error = x(t) −x(t) is bounded.
Note that y is the only available in the controller design. With (11), the derivative of y iṡ y =ẋ 1 Note that 1 = x 1 −x 1 and from (14)-(16), we havė In the above formula, ν 0,2 , ξ 2 , 2 denote the second entries of ν 0 , ξ, . ν 0,1 is the first entries of ν 0 , ν 0,2 is the second entries of Aν 0 . v is the control input signal which will be designed. Considering the saturation shown in (10), we know u(v) can not be approximated by a linear function of v. Such an approximation is usually used in the dead-zone nonlinear input. To compensate for the uncertainties caused by unknown saturation input, the following auxiliary variables are introduced.ṙ where C k1 , C k2 are positive constants. u = u(t) − v represents the input of the auxiliary system (23). By constructing this auxiliary system, the signals r 1 and r 2 are introduced to smooth the saturation function. Before designing the controller, we first perform the following coordinate transformations.
Step 1: Starting with the error z 1 , we obtaiṅ Then we chosen the α 1 is Define the Lyapunov function V 1 as where C 1 , e 1 are positive parameters. Then the derivative of Step 2: We derive the error z 2 Choosing the control law v as

IV. STABILITY ANALYSIS
We now establish the boundedness of all signals in the closed loop system under the proposed output feedback control scheme. The following theorem about output feedback control of pneumatic muscle joint system with saturation input can be achieved. Theorem 1: Consider the pneumatic muscle joint system shown in (1), with saturation input (10), an output feedback controller (29). Under Assumption 1, all signals of the closedloop system are bounded under the control of the proposed control scheme.
Proof: Firstly, defining the Lyapunov function V 2 as The derivative of V 2 iṡ By using Young's inequality where d 1 > 0 is a design parameter. Then the derivative of V 2 can be rewritten aṡ where D max is bound of D(t). Let Note that Then we can getV where So we can get all signals in closed-loop systems are all bounded.

V. SIMULATION STUDIES
We now apply the proposed control scheme to the following 2nd-order system described aṡ   where x 1 , x 2 are system states, y is the output signal. u is the output of the saturation actuator while v is the input signal. are taken as: x 1 (0) = 1.5, x 2 (0) = 0, r 1 (0) = r 2 (0) = 0, ξ (0) = 0, (0) = 0, λ(0) = 0. Fig.1 represents tracking error and the state x 2 is shown in Fig.2. Fig.3 and Fig.4 show the auxiliary signals r 1 and r 2 . Fig.5 shows the signal v which is designed by the proposed control law (29) and the signal u(t) given by changing of saturation (10). Clearly, we can get that all signals of the systems are bounded under the control of the proposed output feedback control scheme.

VI. CONCLUSION
The control problem is investigated for a pneumatic muscle joint system with unknown input saturation and external disturbance. With the auxiliary signals and states filter, an output feedback control scheme has been proposed by using backstepping. The uncertainties caused by unknown saturation and external disturbance can be compensated and the stability of closed-loop systems can be guaranteed by the proposed control scheme. Finally, simulation studies are used to verify the effectiveness of the proposed control scheme. In our future work, we will consider the estimation of unknown parameter and to obtain the adaptive control scheme.