A Singular Perturbation Theory-Based Composite Control Design for a Pump-Controlled Hydraulic Actuator with Position Tracking Error Constraint

: Pump-controlled hydraulic actuators (PHAs) contain slow mechanical and fast hydraulic dynamics, and thus singular perturbation theory can be adopted in the control strategies of PHAs. In this article, we develop a singular perturbation theory-based composite control approach for a PHA with position tracking error constraint. Disturbance observers (DOBs) are used to estimate the matched and mismatched uncertainties for online compensation. A sliding surface-like error variable is proposed to transform the second-order mechanical subsystem into a ﬁrst-order error subsystem. Consequently, the position tracking error constraint of the PHA is decomposed into the output constraint of the ﬁrst-order error subsystem and the stabilizing of the ﬁrst-order hydraulic subsystem. Slow and fast control laws can be easily designed without using the backstepping technique, thus simplifying the control design and reducing the computational burden to a large extent. Theoretical analysis veriﬁes that desired stability properties can be achieved by an appropriate selection of the control parameters. Simulations and experiments are performed to conﬁrm the efﬁcacy and practicability of the proposed control strategy.


Introduction
Pump-controlled hydraulic actuators (PHAs) have been widely used in industrial applications [1][2][3] due to their high power-to-weight ratio, energetic efficiency, simple structure, compactness, and ease of maintenance [4][5][6].However, ensuring highperformance output tracking control is a primary demand for PHAs.Unfortunately, parametric uncertainties, sophisticated friction force, and other disturbances can degrade control performance.Load disturbances, for example, not only affect velocity dynamics but can also cause piston vibrations and pressure fluctuations.To mitigate these effects, many advanced control strategies have been successfully applied in hydraulic systems, such as sliding mode control [7], robust adaptive control [8], and robust integral of the sign of the error (RISE) [9].While these methods are effective, they rely on high gains to achieve uncertainty attenuation, which may result in large design conservativeness and amplify the noise effect.
As an alternative, observer-based methods, such as disturbance observer (DOB) [10] and extended state observer (ESO) [11], provide estimates of uncertainties for online compensation in controller synthesis, thus avoiding high robust feedback gains and reducing design conservativeness.Observer-based methods have drawn increasing attention in recent years.In [12], a disturbance observer-based state estimator (DOBSE) was designed to asymptotically estimate the disturbance without specifying the boundedness of the disturbance.In [13], a fixed-time disturbance observer was developed to asymptotically estimate the disturbance within a given period.In [14], two ESOs were coordinated to concurrently estimate the unavailable states and matched and mismatched uncertainties with only output information.In [15], a finite-time extended state observer (FTESO) was proposed to achieve the asymptotic estimation of disturbances and unavailable states in finite time.In [16], extended sliding mode observers (ESMOs) were proposed to achieve faster convergence, higher accuracy, and better robustness against uncertainties by comparison with traditional ESOs.
Aside from uncertainty attenuation, the outputs of PHAs are often required to track their desired trajectories within an acceptable range in the whole operation process.To achieve the output tracking error constraint, a barrier Lyapunov function (BLF) [17] was proposed in the backstepping framework for strict feedback nonlinear systems.The BLF contains a natural logarithm function of the output tracking error such that the BLF will be unbounded if the output tracking error violates the prescribed bound.Thus, the output tracking error constraint is achieved by designing a proper controller to guarantee the boundedness of the BLF.BLF-based control approaches have been widely applied in hydraulic systems, and some examples can be found in [18][19][20].
Based on the above discussion, we can design a BLF-based backstepping controller with observers to guarantee the output tracking error constraint of a PHA subject to matched and mismatched uncertainties.However, the drawback of this method is that the backstepping technique needs the calculation of the time derivatives of intermediate virtual control laws, which becomes extremely complicated in high-order systems.Moreover, the incorporated BLF exaggerates the design complexity and computational burden.Although dynamic surface control (DSC) [21] can avoid the problem of "explosion of complexity", the introduced filters also complicate the control structure and cause phase lag [22].Therefore, it is of practical importance to develop a BLF-based output tracking error constraint control method for PHAs without using the backstepping technique.
In this paper, we present a simple control design scheme for a PHA to achieve position tracking error constraint and uncertainty compensation without the backstepping technique.It starts with decomposing the PHA into slow mechanical and fast hydraulic subsystems from the viewpoint of time scale separation [23].Owing to this, singular perturbation theory [24] can be utilized in the control synthesis of the PHA.We decompose the actual control law into slow and fast components and then stipulate a restrictive condition to make them decoupled to control the corresponding subsystems.Consequently, the control task is reduced to investigating the two reduced-order subsystems.
The mechanical subsystem of the PHA is in a second-order integral chain form to describe the position and velocity dynamics.To achieve position tracking error constraint, the backstepping technique is still required in the synthesis of the slow control law.To circumvent this, we propose a sliding surface-like error variable to transform the secondorder mechanical subsystem into a first-order error subsystem.Via this transformation, the position tracking error constraint of the PHA is equivalent to the output constraint of the transformed first-order error subsystem.Therefore, a BLF-based slow control law can be easily designed without any intermediate virtual control law.Two DOBs are combined to estimate the uncertainties for online compensation.The fast control law is required to ensure the uniform stability of the first-order hydraulic subsystem.Compared with backstepping-based methods, the proposed composite control method is low-complexity and straightforward.Stability analysis via singular perturbation theory reveals that the position tracking error constraint and desired stability properties can be obtained by a proper selection of the control parameters.
The main contributions of this article are generalized below: (1) Position tracking error constraint is investigated for a PHA subject to uncertainties.Two DOBs are incorporated to provide estimates of the matched and mismatched uncertainties.(2) Unlike the existing works [14,25,26], the PHA is decomposed into slow mechanical and fast hydraulic subsystems from a two-time scale perspective.Therefore, the control task is reduced to investigating the two reduced-order subsystems.(3) Compared with the BLF-based control [27][28][29], the proposed sliding surface-like error variable transforms the position tracking error constraint of the third-order PHA into the output constraint of a first-order error subsystem and the stabilizing of the first-order hydraulic subsystem.Consequently, a composite controller can be easily designed without the use of the backstepping technique.(4) The proposed control approach has a simple control structure, low computational burden, and satisfactory control accuracy, which makes it promising in industrial applications.
The remainder of this article is organized as follows.The system is modeled in Section 2. In Section 3, the singular perturbation theory-based composite control design is proposed in detail.The closed-loop system stability analysis is conducted in Section 4. Simulation and experimental validation are displayed in Section 5. Section 6 concludes this paper.

System Modeling
The schematic diagram of a PHA is depicted in Figure 1.As seen, the control voltage acts on the motor driver, and the servo motor drives the bidirectional pump to rotate.The pump outputs pressure oil to force the movement of the piston rod of the cylinder.Relief valves are installed to restrict the maximum working pressure.A charging circuit is constructed to compensate for the external leakage.A load is connected to the piston rod.Pressure sensors and a rotary encoder are installed to record the pressures inside the chambers of the cylinder and the position of the piston rod, respectively.
The dynamics of the piston rod can be determined by Newton's law: where m is the total mass of the piston rod and the connected load; y is the position of the piston rod; A is the ram area of the cylinder; p 1 and p 2 are the pressures inside the two chambers of the cylinder; F f represents the sophisticated nonlinear friction, which comprises linear viscous friction and other nonlinear elements; F l is the load force; and F d denotes the lumped uncertainties due to parametric uncertainties, unmodeled nonlinearities, and load disturbances.
Neglecting the external leakage, the pressure dynamics in both chambers of the cylinder are given by [30]: . .
where V 01 and V 02 represent the initial control volumes of each chamber and the volumes of the connecting pipelines; β e is the effective bulk modulus; C tlc denotes the total leakage coefficient of the cylinder; q 1 and q 2 denote the flow rates into the two chambers of the cylinder; and q n1 and q n2 represent parametric uncertainties and modeling errors.The dynamics of the piston rod can be determined by Newton s law: ( ) where m is the total mass of the piston rod and the connected load; y is the position of the piston rod; A is the ram area of the cylinder; p1 and p2 are the pressures inside the two chambers of the cylinder; Ff represents the sophisticated nonlinear friction, which com prises linear viscous friction and other nonlinear elements; Fl is the load force; and Fd de notes the lumped uncertainties due to parametric uncertainties, unmodeled nonlineari ties, and load disturbances.
Neglecting the external leakage, the pressure dynamics in both chambers of the cyl inder are given by [30]: ( ) where V01 and V02 represent the initial control volumes of each chamber and the volumes of the connecting pipelines; βe is the effective bulk modulus; Ctlc denotes the total leakage coefficient of the cylinder; q1 and q2 denote the flow rates into the two chambers of the  It is inferred from Figure 1 that the flow rates q 1 and q 2 are determined by: where q p1 and q p2 are pump flow rates; q c1 and q c2 are charging flow rates; and q v1 and q v2 are the flow rates through the relief valves.We should note that in normal operation, the system pressure does not violate the safety limit value of the relief valves.Therefore, there is no flow rate through the relief valves, which means that: Referring to [31], the pump flow rate can be expressed by: where D is the pump volumetric displacement; ω is the rotary speed; C tlp represents the total leakage coefficient of the pump; and q n3 denotes the parametric uncertainties and modeling error.
Since the motor is directly connected to the pump, the pump rotational dynamics can be determined by the following equilibrium equation of moments [32]: where T m is the torque generated by the servo motor; J represents the inertia of the motor/pump; B is the coefficient of friction and viscosity; and T d represents the static friction at the pump/motor interface.In this study, the servo motor has an inner velocity closed-loop and fast control response; therefore, the rotational dynamics can be simplified to a static equation [31,33,34]: where K m is the scale input coefficient of the servo motor and u is the voltage input.Defining the state variables as T and combining (1)-( 9), the mathematical model of the studied PHA is given by: . where ; and the matched uncertainties d 2 = (q n1 + q n3 + q c1 )/(V 01 + Ax 1 ) − (q c2 + q n2 − q n3 )/(V 02 − Ax 1 ).Note that since V 01 and V 02 contain the volumes of the pipelines, no matter where the piston rod is, b 1 , b 2, and b 3 are always positive.Generally speaking, β e reaches an order of magnitude of 10 9 or higher; thus, ε is a positive constant much smaller than 1.It implies that the system (10) is in a singularly perturbed form, and thus singular perturbation theory can be utilized in control design.
The objective of this article is to develop a continuous control law u such that: (1) The position of the piston rod tracks a given reference trajectory within an expected range, i.e.,: where y d is a given reference trajectory and k b is a prescribed bound.(2) A satisfactory final position tracking error can be obtained.

Assumption 1:
The desired trajectory y d (t) and its derivatives up to the third order are bounded and continuous.

Assumption 2:
The uncertainties d 1 and d 2 are bounded and there are positive constants ∆ 1 and ∆ 2 such that: .
Remark 1: Note that the lumped uncertainties d 1 involve nonlinear friction F f , which is normally expressed by a discontinuous model to describe the friction effect that the friction cutovers around zero velocity.For instance, the most widely used discontinuous friction model [35] is defined by: where v s is the Stribeck velocity; f s is the stiction force; m usually takes the value of 1 or 2; and f c and f v are the Coulomb friction coefficient and viscous coefficient, respectively.
However, the discontinuity leads to the friction force being non-differentiable on zero velocity, which impedes the development of friction compensation approaches in hydraulic systems.To handle this problem, a smooth nonlinear friction model is proposed in [36] and is defined by: to represent various friction effects, in which γ 1 -γ 6 are positive parameters, the term tanh(γ 2 v)tanh(γ 3 v) stands for the Stribeck effect, the term γ 4 tanh(γ 5 v) represents the Coulomb friction, the term γ 6 v denotes the viscous friction, and γ 1 + γ 4 is the stiction force.Theoretical analysis has been performed in [37] to prove that a proper selection of the parameters γ 1 -γ 6 such that: then, the smooth friction model ( 14) behaves similarly to the discontinuous friction model ( 13).
In addition, verification experiments have been conducted in [38] to reveal that the smooth friction model ( 14) can give an excellent description of the actual friction.Therefore, similar to [37,39,40], it is acceptable and practicable to employ this smooth nonlinear friction model ( 14) to describe the actual friction in the PHA system.Consequently, the boundedness of the first-order derivative of the nonlinear friction is guaranteed.
Remark 2: Assumption 1 is frequently made in the tracking control of nonlinear systems [13,14,41].This assumption ensures the smoothness of the reference trajectories, which prevents the abrupt vibrations of system states [13].Assumption 2 is generally required in observer-based control approaches [14,29,42,43].From a practical point of view, it takes into account that the energy and the change rate of practical nonlinear dynamics are limited.Therefore, Assumptions 1-2 are reasonable.

Design of the DOBs
In this paper, two DOBs are designed, as follows, to estimate the matched and mismatched uncertainties: where l 1 and l 2 > 0 are the observer gains.Define the estimation errors as: It is obtained from ( 16) that: .
Since the observer gains l 1 and l 2 in ( 16), and the perturbation parameters ε are positive constants, there are always positive constants h 1 and h 2 such that: Then, the estimation error dynamics is rewritten as:

Singular Perturbation Theory-Based Composite Controller Design
The mathematical model of the PHA (10) and the estimation error dynamics (20) can be represented in the following singularly perturbed form: where: Introduce a new time scale τ = t/ε and the system dynamics ( 21)-( 22) can be represented in τ time scale as: As seen, the dynamics of ξ in the τ time scale are scaled down by the small parameter ε.Therefore, ξ dynamics can be regarded as a slow subsystem and λ dynamics as a fast subsystem.
Owing to this, singular perturbation theory can be adopted to obtain a reducedorder model, which is in an integral chain form to facilitate control design and uncertainty compensation.It starts with the assumption that ε = 0, and then ( 22) becomes an algorithm equation: The solution λ is called the "quasi-steady-state" of the fast variable λ, which can be calculated as: Introduce an error: to represent the discrepancy between the actual states and their quasi-steady-states.It is known from ( 26)-( 27) that: Then, the PHA system ( 10) is transformed into the following singular perturbed system in the (ξ, η 1 ) coordinates: Therefore, the singular perturbation theory reduces the control problem of the thirdorder PHA to that of a second-order integral chain system.However, since u is excluded from the η 1 dynamics, the convergence performance of η 1 cannot be arbitrarily adjusted.
Motivated by [24], we develop a singular perturbation theory-based composite control design method with: where u s and u f represent its slow and fast control laws, respectively.We intend to decouple the control laws and let them act on the corresponding subsystems.It is feasible by stipulating a restrictive condition that: By repeating the procedure in (25), we can obtain that the quasi-steady-state λ that becomes the solution of ψ(t, ξ, λ, u s , 0) = 0.It is easy to obtain that: Recall that x 3 = λ 1 + η 1 .This modification (32) makes the PHA system (10) transform into the following singularly perturbed system: As seen, the slow control law u s is moved to the slow ξ subsystem by λ 1 , while the fast control law u f is retained in the fast η 1 subsystem.Therefore, they can be separately designed for different control objectives.
The slow control law u s is designed to achieve the position tracking error constraint of the reduced-order integral chain system (33).We first define a tracking error vector e = [e 1 , e 2 ] T , in which: Its dynamics are given by: .e = e 2 To constrain the position tracking error e 1 within the prescribed bound k b , where k b > |e 1 (0)|, we first propose a sliding surface-like error variable: where r > 0 is a design parameter.
Remark 3: It can be seen that the bounds k b and ρ depend on the initial condition e 1 (0), which means that they may be chosen with different values when tracking different reference trajectories.It is the limitation of the proposed sliding surface-like error variable.
It can be inferred from Lemma 1 that the position tracking error constraint of ( 33)-( 34) is equivalent to the construction of u s to guarantee that |s(t)| < ρ.
Differentiating s with respect to t, we obtain the following first-order error subsystem: .
To achieve the output constraint of (44), we design a BLF-based slow control law: where k 1 > 0 is a feedback gain and − s 4(ρ 2 −s 2 ) a 2 1 + 1 is a robust term to suppress the effects of a 1 η 1 and η 2 .
u f is designed to make the η 1 subsystem converge fast and finally converge to a small region around zero.As a result, the effect of the term a 1 η 1 on the ξ subsystem can be made small.In the meantime, u f must satisfy the restrictive condition (31).To this end, u f can be designed as: where k 2 > 0 is the feedback gain.The control framework of the closed-loop system is presented in Figure 2.

Remark 4:
The singular perturbation theory introduces a two-time scale perspective to decompose the original third-order PHA into a second-order integral chain slow subsystem and a first-order fast subsystem.A singular perturbation theory-based composite control method is developed to separate the control law into fast and slow components, which function as the control inputs of the corresponding subsystems by stipulating a restrictive condition.A sliding surface-like error variable is proposed to further simplify the control task to the output constraint control of a transformed firstorder error subsystem and stabilize the first-order fast subsystem.Consequently, the control laws can be easily designed without any intermediate virtual control law.On the contrary, BLF-based backstepping controllers require a calculation of the time derivatives of virtual control laws, which becomes extremely complicated in high-order systems.Therefore, the proposed control approach greatly simplifies the controller design and reduces the computational burden.where k2 > 0 is the feedback gain.
The control framework of the closed-loop system is presented in Figure 2.

Remark 4:
The singular perturbation theory introduces a two-time scale perspective to decompo the original third-order PHA into a second-order integral chain slow subsystem and a first-ord fast subsystem.A singular perturbation theory-based composite control method is developed to se arate the control law into fast and slow components, which function as the control inputs of t corresponding subsystems by stipulating a restrictive condition.A sliding surface-like error vari ble is proposed to further simplify the control task to the output constraint control of a transforme first-order error subsystem and stabilize the first-order fast subsystem.Consequently, the contr laws can be easily designed without any intermediate virtual control law.On the contrary, BL based backstepping controllers require a calculation of the time derivatives of virtual control law which becomes extremely complicated in high-order systems.Therefore, the proposed control a proach greatly simplifies the controller design and reduces the computational burden.

Closed-Loop System Stability Analysis via Singular Perturbation Theory
Combing ( 20), ( 28), (34), and ( 44)-( 46), the closed-loop system can be represented as the following singularly perturbed form: . s = f (t, s, η, ε) ε .η = g(t, s, η, ε) (47) where: The stability analysis for the singularly perturbed system (47) is conducted in two steps: (1) Find two Lyapunov functions to investigate the stability properties of the boundary layer system and the reduced system; (2) Use the sum of the aforementioned Lyapunov functions as a composite Lyapunov function candidate to analyze the closed-loop system stability.
Step 1: The boundary layer system is obtained by setting ε = 0 in the τ time scale so that: where: Theorem 1: The boundary layer system ( 49) is asymptotically stable if the control parameters k 2 , h 1 , and h 2 are chosen such that 1 − P η 2 > 0, where P η is a symmetric positive definite matrix satisfying P η A η + A T η P η = −2I.
Proof.Select a Lyapunov function candidate: Differentiating W(η) along the solution of (49), we have: .
W is negative definite.Therefore, the boundary layer system ( 49) is asymptotically stable.
Proof.Select a barrier Lyapunov function candidate Differentiating V(s) along the solution of (52), we have: .
Step 2: Select a composite Lyapunov function candidate for the closed-loop system (47): Theorem 3: Consider the closed-loop system (47).If Assumptions 1-2 hold, for any given positive constant c such that the initial condition is (s(0), η(0)) ∈ Ω s × Ω η , where Ω s := {s ∈ R|V(s) ≤ c} and Ω η := η ∈ R 2 W(η) ≤ c , there are positive control parameters k 1 , k 2 , h 1 , and h 2 such that: (1) All closed-loop system signals are bounded, and the prescribed boundary of the position tracking error is never violated; (2) The position tracking error finally converges to a region around zero, which can be arbitrarily small by an appropriate selection of the control parameters.
Proof.There is a compact set The closed-loop system (47) can be regarded as a perturbed system of its reduced system and boundary layer system.The reduced and boundary layer systems have been proven to be asymptotically stable.However, the perturbation terms f (t, s, η, ε) − f (t, s, 0, 0) and g(t, s, η, ε) − g(t, s, η, 0) may cause the closed-loop system to become unstable.
s − re 2 , it can be inferred that there is a continuous function M such that: .
It is known from Lemma 1 that for all s ∈ Ω s ⊆ (−ρ, ρ), e 2 is bounded.It follows from Assumption 1 that ... y d is bounded.Therefore, the continuous function M has a maximum value on Ω v .Then, it can be easily deduced that there is a positive constant ϕ such that the perturbation term satisfies: Now, differentiating (54) along the solution of (47), we have: Applying Young's inequality to sη 2 ρ 2 −s 2 and ϕ P η 2 η 2 , we have: Therefore, (62) becomes: .
v < 0 on v(t) = 2c,∀t ≥ 0, which means that Ω v is a positive invariant set.Therefore, any trajectories starting from Ω s × Ω η will remain within Ω v .Since v(s, η) is bounded, its components V(s) and W(η) are also bounded, and it can be concluded that η is bounded and |s(t)| < ρ, ∀t ≥ 0. It follows from Lemma 1 that e 1 and e 2 are bounded by k b and r(2k b − |e 1 (0)|), which means that the position tracking error constraint is satisfied.According to Assumptions 1-2, it can be easily inferred from ( 17) and ( 35) that x 1 , x 2 , d1 , and d2 are bounded.Thus, it can be concluded from (45) that u s is also bounded.Consequently, the boundedness of λ 1 is guaranteed from (32), and x 3 = λ 1 + η 1 is also bounded.Additionally, it is known from ( 46) that u f is bounded.Finally, the boundedness of u = u s + u f is verified.Therefore, all closed-loop system signals are bounded, and the prescribed boundary of the position tracking error is never violated.
(2) It follows from (66) that: Therefore: It means that e 1 (t) finally converges to a compact set: which can be arbitrarily small by an appropriate selection of the control parameters k 1 , k 2 , h 1 , and h 2 such that σ is sufficiently large.
Remark 5: It is inferred from the stability analysis that the slow control law u s and the fast control law u f are designed to ensure the asymptotic stability of the reduced system and the boundary layer system, respectively.Considering that the perturbation terms may cause the closed-loop system to be unstable, some restrictive conditions are imposed on the control parameters to ensure the uniformly ultimate boundedness of the closed-loop system.Meanwhile, the position tracking error constraint of the original PHA is reserved, and a satisfactory final tracking accuracy can be achieved.

Remark 6:
It can be inferred from ( 42) and ( 72) that the transient performance and the final tracking accuracy of the position tracking error can be improved by either increasing r or decreasing ρ.Increasing σ is an alternative approach to improve the final tracking accuracy.According to (67), σ can be tuned to be large by increasing k i (i = 1,2) or decreasing h i (i = 1,2).
To demonstrate the efficacy of the proposed approach, three comparable controllers are constructed as follows: (1) SPC: This is the proposed singular perturbation theory-based composite controller with the sliding surface-like error variable.The controller parameters of SPC are chosen as k 1 = 10 5 , k 2 = 5 × 10 −10 , h 1 = 5 × 10 5 , h 2 = 10 6 , ρ = 0.06, and r = 50.(2) RNC: This is a reduce-order model-based nonlinear controller proposed in [23].It is constructed as: The feedback gains are selected as L 1 = 2 × 10 6 and L 2 = 1.5 × 10 5 .The uncertainty estimates d1 and d2 are obtained from (16).All other parameters are chosen the same as SPC to form a fair comparison.
(3) BDC: This is a backstepping controller with dynamic surface control.The structure and control parameters of BDC were shown as follows: . . where and d1 and d2 were obtained from (16).The control parameters were selected as γ 1 = 50, γ 2 = 1000, γ 3 = 30, and τ 1 = τ 2 = 0.015.All other parameters are chosen the same as the proposed approach to form a fair comparison.
The simulation results displayed in Figures 3 and 4 illustrate that all controllers retained their position tracking errors within the prescribed bound, and SPC performed better than RNC and BDC, which illustrates the effectiveness of the proposed sliding surface-like error variable s in output tracking error constraint.The boundedness of s, e 1 , and e 2 is exhibited in Figure 5.Consistent with Lemma 1, it was constrained by ρ, k b , and r(2k b − |e 1 (0)|), respectively.Therefore, the position tracking error constraint was achieved.Moreover, it is inferred from (35) that x 1 and x 2 are bounded.The boundedness of x 3 is also shown in Figure 5. Additionally, the profiles of the uncertainty estimation are displayed in Figure 6.As seen, the estimates obtained by the DOBs in (16) tracked their actual values well.The control signal is shown in Figure 7, which was continuous and bounded.the uncertainties, which also helps to improve the control accuracy.However, we should note t overly large control parameters can cause oscillations, and thus the tuning process should be ca ful.   the uncertainties, which also helps to improve the control accuracy.However, we should note th overly large control parameters can cause oscillations, and thus the tuning process should be car ful.Remark 7: A stringent guideline of control parameter selection has been given in Theorem 3. It implies that for any given initial condition, there always are sufficiently large control parameters k 1 , k 2 , l 1 , and l 2 such that the desired control performance can be achieved.However, the guideline is too complex to follow in practical applications.More seriously, as commented in [47], the parameter values calculated from stability analysis may be unrealistic in practice since they normally take into account the bounding information of uncertainties and system parameters.In this paper, similar to the approach in [48], we start with small feedback gains and gradually increase them until satisfactory control performance is achieved.The observer gains are then tuned up to estimate the uncertainties, which also helps to improve the control accuracy.However, we should note that overly large control parameters can cause oscillations, and thus the tuning process should be careful.

Experimental Setup
An actual PHA, shown in Figure 8, is set up to test the practicality of the proposed control scheme.The PHA includes a Parker double-rod cylinder, a fixed displacement pump (Rexroth A10FZG006/10W), and a servo motor (Yaskawa SGM7G-30AFC61).A rotary encoder (Kübler 8.5000.8132.5000)and two pressure sensors (HYDAC EDS 3448-5-0100-Y00) are installed to measure the required position and pressure signals.The velocity signal is obtained by applying a backward difference algorithm to the position signal.Two second-order Butterworth filters with a cutoff frequency of 20 Hz are adopted to avoid measurement noise in the velocity and pressure signals [49].The measured signals are then sampled with a sampling period T s = 1 ms and transmitted to a host PC via a data acquisition module, Quanser Q8-usb.Control algorithms are constructed and executed in MATLAB/Simulink, which communicates with Quanser Q8-usb via the embedded QUARC to transmit the control data to the DC motor driver.

Experimental Setup
An actual PHA, shown in Figure 8, is set up to test the practicality of the proposed control scheme.The PHA includes a Parker double-rod cylinder, a fixed displacement pump (Rexroth A10FZG006/10W), and a servo motor (Yaskawa SGM7G-30AFC61).A rotary encoder (Kübler 8.5000.8132.5000)and two pressure sensors (HYDAC EDS 3448-5-0100-Y00) are installed to measure the required position and pressure signals.The velocity signal is obtained by applying a backward difference algorithm to the position signal.Two second-order Butterworth filters with a cutoff frequency of 20 Hz are adopted to avoid measurement noise in the velocity and pressure signals [49].The measured signals are then sampled with a sampling period Ts = 1 ms and transmitted to a host PC via a data acquisition module, Quanser Q8-usb.Control algorithms are constructed and executed in MATLAB/Simulink, which communicates with Quanser Q8-usb via the embedded QUARC to transmit the control data to the DC motor driver.

Experimental Results
To investigate the superiority of the proposed scheme, RNC and BDC were employed for comparison.To quantitatively evaluate their performances, we introduce three performance indices as follows: (1) The maximum absolute error: (2) The integral of the squared errors:

Experimental Results
To investigate the superiority of the proposed scheme, RNC and BDC were employed for comparison.To quantitatively evaluate their performances, we introduce three performance indices as follows: (1) The maximum absolute error: (2) The integral of the squared errors: (3) The integral of time multiplied by the absolute errors: where N represents the number of the sampled data.1.It is obvious that all the controllers constrained their position tracking errors within the prescribed bound.Nevertheless, SPC performed better than RNC and BDC with its position tracking error shown in Figure 10, which was farther away from the prescribed bound.The reason is because the term s 1 + a 2 1 /4 ρ 2 − s 2 in (45) will grow rapidly to provide a strong enough control action to prevent the position tracking error from approaching its bound.The profiles of s, e 1 , and e 2 of SPC are plotted in Figure 11.They were, respectively, bounded by ρ, k b , and r(2k b -|e 1 (0)|), which verifies the efficacy of the designed sliding surfacelike error variable s in ensuring that e 1 and e 2 are constrained by the known bounds.In addition, the boundedness of the uncertainty estimation and the control signal are verified in Figures 12 and 13.Case 2: To test the robustness of the compared controllers, a spring, shown in Figure 14, was installed to strengthen the mismatched uncertainties.The reference was chosen as y d (t) = 15 arctan(sin(πt/2))(1 − e −t )/0.7854 mm, and the spring was pre-compressed by 18 mm.This setting makes the spring compression vary from 3 mm to 33 mm when the actuator moves forward and backward.It means that the additive spring force was nonvanishing during the operation process.The tracking performances are displayed in Figure 15 and Table 2.As seen, all controllers constrained their position tracking errors within the prescribed bound under the strengthened mismatched uncertainties.Nevertheless, SPC performed better than RNC and BDC in both the transient and steadystate periods, which demonstrated the superiority of the proposed control strategy.s, e 1 , and e 2 of SPC were plotted in Figure 16 to confirm the robustness of the proposed control approach in state constraint under external disturbance.In addition, it can be inferred from force analysis that the friction and spring force have the same sign when the actuator moves forward, while the spring force counteracts partial friction when the actuator moves backward.It renders the lumped mismatched uncertainty asymmetrical.The estimate of d 1 , shown in Figure 17, illustrates that the incorporated DOB reproduced the asymmetry of the uncertainties.
ysis that the friction and spring force have the same sign when the actuator move ward, while the spring force counteracts partial friction when the actuator moves ward.It renders the lumped mismatched uncertainty asymmetrical.The estimate shown in Figure 17, illustrates that the incorporated DOB reproduced the asymme the uncertainties.ward, while the spring force counteracts partial friction when the actuator m ward.It renders the lumped mismatched uncertainty asymmetrical.The estim shown in Figure 17, illustrates that the incorporated DOB reproduced the asy the uncertainties.u y =  V and exerted on the control input during 4s and 8s.In this period, u* = u-ud was actually applied to control the PHA.The reference was chosen as yd(t) = 15 arctan(sin(πt))(1 − e −t )/0.7854 mm.The position tracking errors of the compared controllers are plotted in Figure 18, and their performance indices during 4s and 8s are collected in Table 3.In this case, SPC successfully retained its position tracking error within the prescribed bound while RNC and BDC failed, which illustrates the efficacy of the proposed sliding surface-like error variable in state constraint.s, e1, and e2 of SPC were drawn in Figure 19 to confirm that they were constrained by the bounds given in Lemma 1.In addition, it can be seen in Figure 20 that the DOB captured the dramatic changes in the estimate of d2 when the input disturbance was exerted.Case 3: To test the robustness of the controllers against the matched uncertainties, similar to [48], a disturbance was selected as 10 d d u y =  V and exerted on the control input during 4s and 8s.In this period, u* = u-ud was actually applied to control the PHA.The reference was chosen as yd(t) = 15 arctan(sin(πt))(1 − e −t )/0.7854 mm.The position tracking errors of the compared controllers are plotted in Figure 18, and their performance indices during 4s and 8s are collected in Table 3.In this case, SPC successfully retained its position tracking error within the prescribed bound while RNC and BDC failed, which illustrates the efficacy of the proposed sliding surface-like error variable in state constraint.s, e1, and e2 of SPC were drawn in Figure 19 to confirm that they were constrained by the bounds given in Lemma 1.In addition, it can be seen in Figure 20 that the DOB captured the dramatic changes in the estimate of d2 when the input disturbance was exerted.3.In this case, SPC successfully retained its position tracking error within the prescribed bound while RNC and BDC failed, which illustrates the efficacy of the proposed sliding surface-like error variable in state constraint.s, e 1 , and e 2 of SPC were drawn in Figure 19 to confirm that they were constrained by the bounds given in Lemma 1.In addition, it can be seen in Figure 20 that the DOB captured the dramatic changes in the estimate of d 2 when the input disturbance was exerted.

Conclusions
In this article, we present a singular perturbation theory-based composite control approach for a PHA with position tracking error constraint.The PHA can be separated into the second-order slow mechanical and first-order fast hydraulic subsystems from the two-time scale perspective.Slow and fast control laws are decoupled to work as the inputs of the corresponding subsystems.To facilitate position tracking error constraint control without the backstepping technique, a sliding surface-like error variable is proposed to transform the second-order mechanical subsystem into a first-order error subsystem, where a BLF-based slow control law can be easily designed without any intermediate virtual control law.Two DOBs are incorporated in the controller design to provide estimates of the uncertainties.The remaining fast control law is designed to make the first-order hydraulic subsystem asymptotically converge to its equilibrium point related to the slow control law.The theoretical analysis gives rigorous proof of the closed-loop system's stability.Simulations and experiments are conducted to test the efficacy and practicability of the developed control approach.

Figure 2 .
Figure 2. Control framework of the closed-loop system.

Figure 2 .
Figure 2. Control framework of the closed-loop system.

Actuators 2023 ,
12, x FOR PEER REVIEW 16 of

Figure 7 .
Figure 7.Control input of SPC.Figure 7. Control input of SPC.

Figure 7 .
Figure 7.Control input of SPC.Figure 7. Control input of SPC.

Figure 8 .
Figure 8. Configuration of the PHA test rig.

Figure 8 .
Figure 8. Configuration of the PHA test rig.

Case 1 :
A low-frequency smooth reference trajectory y d (t) = 30 arctan(sin(πt/2)) (1 − e −t )/0.7854 mm was used for the controllers, and the prescribed bound k b = 1.2 mm.The experimental results are depicted in Figures 9 and 10 and Table ) will grow rapidly to provide a strong enough con action to prevent the position tracking error from approaching its bound.The profiles s, e1, and e2 of SPC are plotted in Figure11.They were, respectively, bounded by ρ, kb, a r(2kb-|e1(0)|), which verifies the efficacy of the designed sliding surface-like error varia s in ensuring that e1 and e2 are constrained by the known bounds.In addition, the bou edness of the uncertainty estimation and the control signal are verified in Figures 12 a 13 .

Figure 9 .
Figure 9. Position tracking profiles of each controller in Case 1.Figure 9. Position tracking profiles of each controller in Case 1.

Figure 9 .
Figure 9. Position tracking profiles of each controller in Case 1.Figure 9. Position tracking profiles of each controller in Case 1.

Figure 9 .
Figure 9. Position tracking profiles of each controller in Case 1.

Figure 10 .
Figure 10.Position tracking errors of each controller in Case 1.

Figure 10 .
Figure 10.Position tracking errors of each controller in Case 1.

Figure 11 .
Figure 11.Profiles of s, e 1 , and e 2 of SPC in Case 1.

Figure 14 .
Figure 14.The spring installed in Case 2.

Figure 15 .
Figure 15.Position tracking errors of each controller in Case 2.

Figure 14 .
Figure 14.The spring installed in Case 2.

Figure 14 .
Figure 14.The spring installed in Case 2.

Figure 15 .
Figure 15.Position tracking errors of each controller in Case 2.

Figure 15 .
Figure 15.Position tracking errors of each controller in Case 2.

Figure 17 .
Figure 17.Mismatched uncertainty estimation of SPC in Case 2.

Figure 17 .
Figure 17.Mismatched uncertainty estimation of SPC in Case 2.

Figure 17 .Case 3 :
Figure 17.Mismatched uncertainty estimation of SPC in Case 2.Case 3: To test the robustness of the controllers against the matched uncertainties, similar to[48], a disturbance was selected as u d = 10 .yd V and exerted on the control input during 4 s and 8 s.In this period, u* = u − u d was actually applied to control the PHA.The reference was chosen as y d (t) = 15 arctan(sin(πt))(1 − e −t )/0.7854 mm.The position tracking errors of the compared controllers are plotted in Figure 18, and their performance indices during 4 s and 8 s are collected in Table3.In this case, SPC successfully retained its position tracking error within the prescribed bound while RNC and BDC failed, which illustrates the efficacy of the proposed sliding surface-like error variable in state constraint.s, e 1 , and e 2 of SPC were drawn in Figure19to confirm that they were constrained by the bounds given in Lemma 1.In addition, it can be seen in Figure20that the DOB captured the dramatic changes in the estimate of d 2 when the input disturbance was exerted.

Figure 18 .
Figure 18.Position tracking errors of each controller in Case 3.

Figure 18 .
Figure 18.Position tracking errors of each controller in Case 3.

Figure 18 .
Figure 18.Position tracking errors of each controller in Case 3.

Figure 19 .
Figure 19.Profiles of s, e1, and e2 of SPC in Case 3.Figure 19.Profiles of s, e 1 , and e 2 of SPC in Case 3.

Figure 19 .
Figure 19.Profiles of s, e1, and e2 of SPC in Case 3.Figure 19.Profiles of s, e 1 , and e 2 of SPC in Case 3.

Figure 20 .
Figure 20.Matched uncertainty estimation of SPC in Case 3.

Figure 20 .
Figure 20.Matched uncertainty estimation of SPC in Case 3.
Actuators 2023, 12, x FOR PEER REVIEW 19 o experimental results are depicted in Figures 9 and 10 and Table1.It is obvious that all controllers constrained their position tracking errors within the prescribed bound.Nev theless, SPC performed better than RNC and BDC with its position tracking error sho in Figure10, which was farther away from the prescribed bound.The reason is beca

Table 1 .
Performance indices of the compared controllers during 16-20 s.

Table 2 .
Performance indices of the compared controllers during 16-20 s.

Table 2 .
Performance indices of the compared controllers during 16-20 s.

Table 2 .
Performance indices of the compared controllers during 16-20 s.

Table 3 .
Performance indices of the compared controllers during 4 s and 8 s.