Practical adaptive robust tracking control of the dual-valve parallel electro-hydraulic servo system with reduced-order model

Abstract

To meet the increasing demand for trajectory tracking accuracy and high-efficiency requirements in modern mobile machinery, this paper proposes a practical adaptive robust control method based on the dual-valve parallel electro-hydraulic servo system. Existing tracking control strategies for such servo systems rely on a backstepping method and assume that full states are known, which is stringent in practice. To cope with the problem, we reduce the mathematical model order of the studied system using singular perturbation theory in this paper, and only the position feedback is required for the control implementation. Then, to achieve high accuracy tracking control, a direct adaptive robust control scheme combing with a dynamic flow allocation layer is adopted to synthesize the controller. With this method, parameter uncertainties and load disturbance are rejected, and valve characteristics are considered in the dynamic flow allocation layer to solve the flow redundancy problem, respectively. Convergence of the simplified system tracking results is proved theoretically. Extensive co-simulations and experimental tests are carried out to illustrate the effectiveness of the proposed strategy, both tracking precision and flow distribution can be achieved efficiently by simple parameter adjustments in the field.

Keywords

Direct adaptive robust control dual-valve hydraulic servo system singular perturbation theory flow allocation output feedback

Introduction

Hydraulic systems are pervasively used in large construction machinery (Huang et al., 2021; Tong et al., 2020), industrial equipment (Liu et al., 2020), and various kinds of mobile machinery (Padovani et al., 2020) due to their large power-to-size ratio, the ability to generate large force or torque, and simple structure. In existing hydraulic systems among mobile machinery, large-flow proportional valve with inferior dynamic performance is widely adopted, which is hard to meet the requirement of higher tracking accuracy and higher efficiency. A kind of mobile hydraulic servo system with larger flow rates, more precise tracking and positioning capability, and lower costs will be desired in the future.

In this paper, a dual-valve parallel electro-hydraulic servo (DVPEHS) scheme is discussed. Differ from conventional electro-hydraulic servo (EHS) systems, the DVPEHS system consists of a proportional valve and a servo valve, the proportional valve not only has a lower cost but also works in the load-sensitive system to reduce throttling loss, while the servo valve generally works with a constant pressure pump to enhance responsiveness. However, the researchers point out that the control law distribution about flow redundancy (Su et al., 2020) is a difficult problem and has raised extensive attention. One solution is synchronous control, in which the system input is assigned to the two valves simultaneously (Bai and Quan, 2016). However, the two valves should be matched for phase and gain to maintain consistency because of the inevitable inconsistencies in the machining and assembly, which is complicated in engineering practice. Another solution is asynchronous control, in which the valves are activated sequentially according to the required flow rate so that the positioning performance of the actuator can be guaranteed by a single servo valve. Typically, a harmonic control concept was proposed by Su et al. (2022), where a threshold value was predetermined. However, this method of flow distribution does not account for the differences in valve performance. Meanwhile, proper arrangement of the control law among multi-parts according to the time sequence is widely used in multiple-in-single-output systems, such as pump-valve-coordinated systems (Du et al., 2022; Lyu et al., 2020) and multi-cylinder parallel systems.

However, EHS systems usually suffer from load variations, parameter uncertainties, and modeling nonlinearities in practice. Numerous control strategies have been developed specifically, such as adaptive control (Guo et al.,2020, 2021) sliding mode control (Thomas et al., 2022; Zhao et al., 2020), and observer-based control (Du et al., 2022; Wang et al., 2022). Yao and Tomizuka (1997) proposed an adaptive robust control (ARC) scheme for EHS systems, using robust control as the basic framework to overcome uncertain disturbances and designing a discontinuous parameter adaptation law to compensate for parameter variations. Further research is conducted to enhance disturbance elimination performance around ARC (Na et al., 2020), where the disturbance is bounded and the bound should be known. However, most ARC algorithms are designed by means of the backstepping scheme, which brings about “differential explosion” (Nie et al., 2021). Although the dynamic surface control technique (Zhang and Shi, 2022) is introduced to avoid this issue, extra complexity of the filter cannot be ignored. Meanwhile, aforementioned nonlinear control methods require full system states to be known or measurable in the backstepping designs, which is hardly available in engineering practice. Some scholars have realized the singular dynamics of hydraulic systems (Barchi et al., 2021; Guo et al., 2022; Jing et al., 2020) and have shown improved hydraulic tracking control. Notably, to the authors’ best knowledge, the above control schemes for DVPEHS systems are rarely discussed and deserve ongoing exploration for practical implementation.

Motivated by the above discussions, this paper adopts a direct adaptive robust control scheme with reduced-order model and dynamic allocation to improve the tracking performance of the DVPEHS systems, where a small-flow servo valve is connected in parallel with a high-flow proportional valve. The main contributions are listed as follows:

First, by introducing the singular perturbation theory, we establish the mathematical model of the DVPEHS systems and reduce the model order. Only the system output (actuator displacement) is required for controller design. The backstepping process and state observers are effectively avoided, which largely reduces computational complexity and implementation costs.

Furthermore, we present the design of a direct adaptive robust control (DARC) law based on the reduced-order model, and the stability of the algorithm is proved. A virtual control flow rate is synthesized according to DARC, and the flow rates of the servo and proportional valves are allocated dynamically in the flow allocation layer according to the tracking trajectory and the hydraulic characteristics of the two valves.

Third, comprehensive simulation and experimental studies are conducted with a high-frequency response servo valve and a proportional valve. The feasibility and high tracking performance are verified by the results. Only a few dynamic flow allocation parameters need to be adjusted in the field to match the characteristics of the different valves.

The rest of this paper is organized as follows. In section “System description and full-order model,” the system architecture is introduced, and the derivation of the mathematical full-order model of the DVPEHS system is provided. The reduced-order model is established in section “Model reduction.” The adaptive robust control scheme with an integral sliding surface and the flow allocation layer is given in section “Controller design.” In section “Simulation and experiment tests,” the simulation and experiment tests are performed to verify the effectiveness of the proposed controller. Conclusions are drawn in section “Conclusion.”

System description and full-order model

The basic schematic of the studied DVPEHS system is given in Figure 1, which consists of a servo valve, a proportional valve, and a hydraulic actuator. The two valve spool displacements generate the flow, which in turn determines the displacement or speed of the hydraulic actuator. The dynamics of the load can be described by

m {\overset{\cdot\cdot}{x}}_{p} = A_{1} p_{1} - A_{2} p_{2} - b {\overset{\cdot}{x}}_{p} - f_{d} (t)

(1)

where $m$ is the mass of the load; $x_{p}$ is the displacement of the load; ${\overset{\cdot}{x}}_{p}$ is the load velocity; $p_{1}$ and $p_{2}$ denote the pressures of the piston and rod chambers of the cylinder; $A_{1}$ and $A_{2}$ are the ram areas of the piston and rod chambers, respectively; $b$ denotes the viscous friction coefficient combined with the modeled damping on the load and the cylinder rod, and $f_{d} (t)$ refers to the unmodeled friction dynamics and the unknown external disturbances of the mechanical systems.

Considering the oil compressibility and neglecting the external leakage of the cylinder, the flow continuity equations of the actuator can be expressed as

{\begin{matrix} β_{e}^{- 1} {\overset{\cdot}{p}}_{1} = V_{1}^{- 1} (t) [Q_{1} - A_{1} {\overset{\cdot}{x}}_{p} - C_{i} (p_{1} - p_{2})] \\ β_{e}^{- 1} {\overset{\cdot}{p}}_{2} = V_{2}^{- 1} (t) [A_{2} {\overset{\cdot}{x}}_{p} - Q_{2} + C_{i} (p_{1} - p_{2})] \end{matrix}

(2)

where $β_{e}$ is the effective bulk modulus; $V_{1} (t)$ is the total control volume of the piston chamber; $V_{2} (t)$ is the total control volume of the rod chamber; $C_{i}$ refers to the internal leakage coefficient of the cylinder. $Q_{1}$ is the total supplied flow rate of the piston chamber, and $Q_{2}$ is the return flow of the rod chamber. Furthermore, the flow rates $Q_{1}$ and $Q_{2}$ can be obtained from the valve characteristics as follows

{\begin{matrix} Q_{1} = Q_{s 1} + Q_{p 1} \\ Q_{2} = Q_{s 2} + Q_{p 2} \end{matrix}

(3)

where $Q_{s 1}$ and $Q_{p 1}$ are the supplied flow rates of the servo valve and proportional valve; $Q_{s 2}$ and $Q_{p 2}$ are the return flow rates through the two valves, respectively. Define function

sg (\cdot) = {\begin{matrix} 1, if \cdot > 0 \\ 0, if \cdot \leq 0 \end{matrix}

and them can be given as

{\begin{matrix} Q_{s 1} = k_{qs} x_{sv} [sg (x_{sv}) \sqrt{P_{s} - P_{1}} + sg (- x_{sv}) \sqrt{P_{1}}] \\ Q_{s 2} = k_{qs} x_{sv} [sg (x_{sv}) \sqrt{P_{2}} + sg (- x_{sv}) \sqrt{P_{S} - P_{2}}] \\ Q_{p 1} = k_{qp} x_{pv} [sg (x_{pv}) \sqrt{P_{s} - P_{1}} + sg (- x_{pv}) \sqrt{P_{1}}] \\ Q_{p 2} = k_{qp} x_{pv} [sg (x_{pv}) \sqrt{P_{2}} + sg (- x_{pv}) \sqrt{P_{S} - P_{2}}] \end{matrix}

(4)

where $k_{qs} = C_{d} ω_{s} \sqrt{2 / ρ}$ , $C_{d}$ is the flow coefficient, $ω_{s}$ is the area gradient of the servo valve, and $ρ$ is the oil density. And $k_{qp} = C_{d} ω_{p} \sqrt{2 / ρ}$ , $ω_{p}$ is the area gradient of the proportional valve. Define $p_{L} = p_{1} - p_{2}$ and $Q_{L} = (Q_{1} + Q_{2}) / 2$ , and assume $V_{1} (0) = V_{2} (0)$ at the beginning, with proper simplification according to (Sun et al., 2013), the load pressure dynamics of the DVPEHS system can be expressed as

β_{e}^{- 1} {\overset{\cdot}{p}}_{L} = Q_{L} - A_{1}^{'} {\overset{\cdot}{x}}_{p} - C_{i} p_{L}

(5)

where $V_{t} = (V_{1} (0) + V_{2} (0))$ represents the initial total control volume, $A_{1}^{'} = (A_{1} + A_{2}) / 2$ . The $Q_{L}$ is specifically determined by $Q_{s 1}$ , $Q_{s 2}, Q_{p 1}, Q_{p 2}$ . In order to make the expression concise, the linearization method is applied around the zero point of the valves, and the result is described as follows

Q_{L} = Q_{Ls} + Q_{Lp} = k_{q 1} x_{sv} - k_{c 1} p_{L} + k_{q 2} x_{pv} - k_{c 2} p_{L}

(6)

where $Q_{Ls}$ and $Q_{Lp}$ represent the load flow rate associated with the servo valve and the proportional valve, respectively. $k_{q 1}$ and $k_{q 2}$ represent the linear flow gains of the proportional and servo valves. $x_{sv}$ is the spool displacement of servo valve, and $x_{pv}$ is the spool displacement of proportional valve. $k_{c 1}$ and $k_{c 2}$ are called the flow-pressure coefficient of the valves. Combining equations (1)–(6), the considered system can be described as

{\begin{matrix} m {\overset{\cdot\cdot}{x}}_{p} = A_{1} p_{1} - A_{2} p_{2} - b {\overset{\cdot}{x}}_{p} - f_{d} (t) \\ \frac{1}{4} V_{t} β_{e}^{- 1} {\overset{\cdot}{p}}_{L} = k_{q 1} x_{sv} + k_{q 2} x_{pv} - A_{1}^{'} {\overset{\cdot}{x}}_{p} - C_{t} p_{L} \end{matrix}

(7)

where $C_{t} = C_{i} + k_{c 1} + k_{c 2}$ .

To achieve fast and accurate tracking tasks for the system (7), we need to overcome several obstacles, which are analyzed as follows:

First, the system parameters in equation (1) may change slowly over time, such as the viscous damping coefficient $b$ and unknown external disturbance $f_{d} (t)$ , which have an impact on the system dynamic response.

Second, system (7) includes modeling errors, such as linearization, ignorance of leakages, and hydraulic parametric uncertainties, which are time-variant and hard to identify (Jelali and Kroll, 2002).

Third, the DVPEHS system is a multiple-input single-output system with different input characteristics subjected to the two kinds of valves.

Last but not least, system (7) has a dimension of four including an internal dynamic (Yao et al., 2000), and the designing controller with the backstepping method suffers from the problem of differential explosion.

Model reduction

As shown in equation (6), the dynamic properties of the DVPEHS system are influenced by both the mechanical and hydraulic systems, which have different time scales resulting in differences in their responsiveness. This feature can be described by the standard singular perturbation system, which is described as follows

{\begin{matrix} \overset{\cdot}{x} = f (t, x, z, ε), x (t_{0}) = ξ (ε) \\ ε \overset{\cdot}{z} = g (t, x, z, ε), z (t_{0}) = η (ε) \end{matrix}

(8)

where $x \in R^{n_{1}}, z \in R^{n_{2}}, ε > 0, t \in R$ , and $ε$ is an extremely small parameter as the singular perturbation parameter.

Note 1 (Khalil, 2001). Considering the singular perturbation system given in equation (8), and assuming that the following conditions are satisfied for all $(t, x, z, ε) \in [0, + \infty) \times B_{η} \times B_{x} \times [0, ε_{0}]$ , where $B_{η} \in R^{n_{1}} and B_{x} \in R^{n_{2}}$ :

(1) $z = h (t, x)$ is the unique root of $g (t, x, z, 0) = 0$ on $[t_{0}, t_{1}]$ for $x$ in $B_{η}$ , and $z$ in $B_{x}$ ;

(2) The function $h (t, x)$ and the Jacobian $\partial g (t, x, z, 0) / \partial z$ have continuous first partial derivatives with respect to their arguments. The initial data $ξ (ε)$ and $η (ε)$ depend smoothly on $ε$ .

(3) The reduced problem $\overset{\cdot}{x} = f (t, x, z, 0)$ has a unique solution $\bar{x}$ , defined on $[t_{0}, t_{1}]$ , and $‖ \bar{x} (t) ‖ \leq r_{1} < r$ for all $t \in [t_{0}, t_{1}]$ .

(4) The origin of the boundary-layer model $dy / d δ = g (t, x, y + h (t, x), 0)$ is exponentially stable, uniformly in $(t, x)$ , where $y = z - h (t, x), δ = t / ε$ .

Then, according to Tikhonov’s theorem (Khalil, 2001), there exists $ε_{0} > 0$ for all $ε_{0} > ε$ , the system (8) is exponentially stable in origin. Moreover, given any $t_{b} > t_{0}$ , there is $ε_{1} \leq ε_{0}$ such that $z (t, ε) - h (t, x) = O (ε)$ holds.

Let us focus on DVPEHS system model equation (6), the oil effective bulk modulus β_e is large enough (0.7∼1.4 GPa) in general cases (Manring and Fales, 2020). Hence, it is reasonable to define $ε = 1 / β_{e}$ , and equation (7) can be expressed as

{\begin{matrix} m {\overset{\cdot\cdot}{x}}_{p} = A_{1} p_{1} - A_{2} p_{2} - b {\overset{\cdot}{x}}_{p} - f_{d} (t) \\ ε {\overset{\cdot}{p}}_{L} = η (k_{q 1} x_{sv} + k_{q 2} x_{pv} - A_{1}^{'} {\overset{\cdot}{x}}_{p} - C_{t} p_{L}) \end{matrix}

(9)

where $η = 4 / V_{t}$ . For simplicity, define $x = [x_{1}, x_{2}]^{T} = [x_{p}, {\overset{\cdot}{x}}_{p}]^{T}, z = p_{L}$ , and equations (7) and (9) may constitute the standard singular perturbation form, where

\begin{matrix} f (t, x, z, ε) = [\begin{matrix} x_{2} \\ \frac{(A_{1} p_{1} - A_{2} p_{2} - b x_{2} - f_{d} (t))}{m} \end{matrix}] \\ and \\ g (t, x, z, ε) = η (k_{q 1} x_{sv} + k_{q 2} x_{pv} - A_{1}^{'} {\overset{\cdot}{x}}_{p} - C_{t} p_{L}) \end{matrix}

According to Note 1, solving the algebraic equation $g (t, x, z, ε) = 0$ , and the equation root is called “quasi-steady-state” of load pressure. The result is exhibited as

{\bar{p}}_{L} = \frac{k_{q 1} x_{sv} + k_{q 2} x_{pv} - A_{1}^{'} {\overset{\cdot}{x}}_{p}}{C_{t}}

(10)

It is obvious that the algebraic equation has a unique real root in the simplified model equation (9). Define $e_{p} = p_{L} - {\bar{p}}_{L}$ , new time scale $δ = t / ε$ , and derive $e_{p}$ with respect to $δ$ yields. In the boundary-layer theory, we regard the “quasi-steady-state” ${\bar{p}}_{L}$ as a constant value. So, we can obtain the boundary-layer model of equation (9) as

\begin{matrix} \frac{d e_{p}}{d δ} = ε \frac{d p_{L}}{dt} - ε \frac{d {\bar{p}}_{L}}{dt} \\ = η [k_{q 1} x_{sv} + k_{q 2} x_{pv} - A_{1}^{'} {\overset{\cdot}{x}}_{p} - C_{t} (e_{p} + {\bar{p}}_{L})] \\ = - η C_{t} e_{p} \end{matrix}

(11)

Remark 1. The coefficient $η C_{t}$ is always positive and continuous because of the internal leakage and the pressure-flow coefficients of the two valves. Meanwhile, the initial data $ξ (ε)$ and $η (ε)$ are also impossible to mutate in a physical sense. Otherwise, only a uniformly exponentially stable equilibrium point as the origin point exists in the boundary-layer model equation (11). Both the first, the second, and the fourth conditions in Note 1 are satisfied.

Remark 2. According to Note 1, designing proper control law can determine the real load pressure $p_{L},$ which resulting $p_{L}$ keeping very close to its “quasi-steady-state” ${\bar{p}}_{L}$ with quantitative error $p_{L} - {\bar{p}}_{L} = O (ε)$ . For $t_{b} > t_{0}$ , there is $ε_{1} \leq ε_{0}$ such that $p_{L} (t, ε) - {\bar{p}}_{L} = O (ε)$ holds. Thus, if the “quasi-steady-state” ${\bar{p}}_{L}$ in the reduced-order model is stable, the stability of $p_{L}$ will be ensured.

Define ${\bar{p}}_{1}$ and ${\bar{p}}_{2}$ as the “quasi-steady-states” of $p_{1}$ and $p_{2}$ , respectively. Combining equations (7) and (9), and the dynamic model of DVPEHS system can be expressed by the state-space form as following

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ m {\overset{\cdot}{x}}_{2} = {\bar{p}}_{1} A_{1} - {\bar{p}}_{2} A_{2} - b x_{2} - f_{d}^{'} (t) \end{matrix}

(12)

In equation (12), $f_{d}^{'} (t) = ({\bar{p}}_{1} - p_{1}) A_{1} - ({\bar{p}}_{2} - p_{2}) A_{2} + f_{d}$ . Let ${\bar{p}}_{1} = {\bar{p}}_{2} + {\bar{p}}_{L}$ , substitute equation (10) into equation (12), so that the system model can be written as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ m {\overset{\cdot}{x}}_{2} = - ξ x_{2} + α_{n} - d \end{matrix}

(13)

where $ξ = b + A_{1} A_{1}^{'} / C_{t}$ represents the lumped damping coefficient, $d = f_{d}^{'} (t) - (A_{1} - A_{2}) {\bar{p}}_{2}$ represent the combination of external disturbance and simplification errors of reduced-order model. Otherwise, $α_{n} = A_{1} / C_{t} (k_{q 1} x_{sv} + k_{q 2} x_{pv})$ is defined as a nominal control law.

In the practical system, it is difficult to obtain the actual pressure signals in the presence of noise. The reduced-order model equation (13) can be introduced to synthesize the pressure dynamics into the disturbance term, enabling the controller design method based on the displacement feedback merely. However, the disturbance term $d$ has a more complex form with modeling error, pressure dynamics, and external disturbance, which is required to be suppressed by a proper control algorithm.

Controller design

In light of the aforementioned challenges, this paper proposes a novel controller with a two-part structure, as illustrated in Figure 2. The two parts, named the tracking layer and allocation layer, work collaboratively to generate the valve commands. The specific process is as follows: (1) in the tracking layer, the desired actuator force is determined according to the DARC to ensure guaranteed tracking performance and (2) in the allocation layer, the control commands $u_{s}$ and $u_{p}$ are synthesized with the consideration of tracking accuracy and different valve characteristics. The details are elaborated in the subsequent sections.

Direct adaptive robust controller with reduced-order model

The desired force $α_{n}$ is generated according to the DARC algorithm proposed by Yao and Tomizuka (1997), so that the actuator position can track the desired trajectory with guaranteed transient performance and accuracy. Considering the definition of the system reduced-order model equation (13), the unknown slow-varying parameters are defined as follows: $θ = [θ_{1}, θ_{2}, θ_{3}]^{T} = [m, ξ, \bar{d}]^{T}, d = \bar{d} + Δ$ , in which the $\bar{d}$ represents the slowly varying component, while $Δ$ is the rest part.

Assumption 1. Both $θ$ and $Δ$ are bounded, which satisfy

{\begin{matrix} θ = {θ_{i} : θ_{imin} \leq θ_{i} \leq θ_{imax}, i = 1, 2, 3} \\ | Δ | \leq Δ_{\max} \end{matrix}

(14)

Define $θ_{\max} = [θ_{1 \max}, θ_{2 \max}, θ_{3 \max}]^{T}$ as the upper bound of $θ$ , and $θ_{\min} = [θ_{1 \min}, θ_{2 \min}, θ_{3 \min}]^{T}$ as the lower bound of $θ$ . Define the tracking error of the hydraulic actuator as $e_{1} = x_{1} - x_{d}$ , where $x_{d}$ is the expected position trajectory. And define a sliding mode–like variable $s$ in the form of

s = c_{1} e_{1} + {\overset{\cdot}{x}}_{1} + c_{2} \int_{0}^{t} e_{1} d τ

(15)

in which $c_{1}$ and $c_{2}$ are the positive coefficients. The sliding surface indicates that the system sliding mode dynamics are independent of the parameter uncertainties and disturbance, thereby, it is crucial to design a controller to achieve the sliding surface, and the stability analysis will be addressed later.

Based on equation (15), the dynamics of $s$ can be expressed as

\begin{matrix} θ_{1} \overset{\cdot}{s} = θ_{1} (c_{1} {\overset{\cdot}{e}}_{1} + {\overset{\cdot}{x}}_{2} + c_{2} e_{1}) \\ = θ_{1} (c_{1} {\overset{\cdot}{e}}_{1} + c_{2} e_{1}) + α_{n} - θ_{2} x_{2} - (θ_{3} + Δ) \end{matrix}

(16)

where ${\overset{\cdot}{e}}_{1}$ is the velocity error, and $α_{n}$ is the nominal control law. In the reduced-order model, the virtual control law $α_{n}$ represents the driving force generated by the hydraulic actuator, which can be given by

\begin{matrix} α_{n} = α_{a} + α_{s} \\ α_{a} = - {\hat{θ}}_{1} (c_{1} {\overset{\cdot}{e}}_{1} + c_{2} e_{1}) + {\hat{θ}}_{2} x_{2} + {\hat{θ}}_{3} \\ α_{s} = α_{s 1} + α_{s 2} = - k_{1} s + α_{s 2} \end{matrix}

(17)

where $α_{a}$ is called the model compensation term, and $α_{s}$ is the robust feedback term. $α_{s}$ contains linear feedback term $α_{s 1}$ and nonlinear feedback term $α_{s 2}$ dominating the model uncertainties. $\overset{\hat{=}}{θ} [{\hat{θ}}_{1}, {\hat{θ}}_{2}, {\hat{θ}}_{3}]^{T}$ is the vector of the estimated parameters. Besides, $k_{1}$ is the positive feedback gain. $α_{s 2}$ should satisfy the following conditions

\begin{matrix} (1) s α_{s 2} \leq 0 \\ (2) s (ϕ^{T} \tilde{θ} + α_{s 2} - Δ) \leq ε_{1} \end{matrix}

(18)

where $ϕ = [- (c_{1} {\overset{\cdot}{e}}_{1} + c_{2} e_{1}), x_{2}, 1]^{T}$ , and $ε_{1}$ is a design parameter. Define $\tilde{θ} = [{\tilde{θ}}_{1}, {\tilde{θ}}_{2}, {\tilde{θ}}_{3}]^{T}$ , in which ${\tilde{θ}}_{i} = {\hat{θ}}_{i} - θ_{i}$ represents the parameter estimation error. The unknown parameter vector $θ$ is estimated online using the discontinuous projection adaptive law defined as

\hat{\overset{\cdot}{θ}} = {Proj}_{\hat{θ}} (- Γ ϕ s)

(19)

where $Γ$ is a positive-definite diagonal adaption rate matrix, and the discontinuous projection is defined as

Pro j_{{\hat{θ}}_{i}} (•_{i}) = {\begin{matrix} 0 & if {\hat{θ}}_{i} = θ_{imax} and • > 0 \\ 0 & if {\hat{θ}}_{i} = θ_{imin} and • < 0 \\ •_{i} & otherwise \end{matrix}

(20)

In summary, the following theoretical results can be obtained about the DARC controller.

Theorem 1. For system (13), considering Assumption 1, the definition in equation (15) and the parameter estimates $θ$ updating by the adaption law defined in equation (19), with the control law defined as equation (17), guarantees the following:

(1) All signals in the system can be guaranteed bounded. Furthermore, the suitable positive-definite function $V_{s}$ is defined as

V_{s} = \frac{1}{2} θ_{1} s^{2}

(21)

which is bounded by

V_{s} (t) \leq e^{- \frac{2 k_{1}}{θ_{1}} t} V_{s} (0) + \frac{θ_{1}}{2 k_{1}} ε_{1} [1 - e^{- \frac{2 k_{1}}{θ_{1}} t}]

(22)

(2) In the presence of parameter uncertainties, asymptotic output tracking is achieved based on the result in (1).

Proof. Differentiating $V_{s} (t)$ and noting equations (16), (17), and (19), the result can be shown as

\begin{matrix} {\overset{\cdot}{V}}_{s} (t) = - k_{1} s^{2} + s (ϕ^{T} \tilde{θ} + α_{s 2} - Δ) \\ \leq - \frac{2 k_{1}}{θ_{1}} V_{s} (t) + ε_{1} \end{matrix}

(23)

which indicates that s will be bounded in a finite time by comparison lemma. In order to prove (2), based on $V_{s}$ , choose another Lyapunov function $V_{a}$ , which is defined as

V_{a} = V_{s} (s) + \frac{1}{2} {\tilde{θ}}^{T} Γ^{- 1} \tilde{θ}

(24)

Deducing the derivative of $V_{a}$ as follows

\begin{array}{l} {\dot{V}}_{a} (t) = s (α_{s 2} + {\tilde{θ}}^{T} ϕ - k_{1} s) + {\tilde{θ}}^{T} Γ^{- 1} \hat{\dot{θ}} \leq - k_{1} s^{2} + {\tilde{θ}}^{T} \\ [ϕ s + Γ^{- 1} P r o j_{\hat{θ}} (- Γ ϕ s)] \\ \leq 0 \end{array}

(25)

Considering the ${\overset{\cdot}{V}}_{a},$ the asymptotical tracking results can be proved using the same method as in Yao et al. (1994). According to the LaSalle invariance principle, if $V_{s} \equiv 0$ , then $s \equiv 0, \overset{\cdot}{s} = 0$ , corresponding $x_{1} \to x_{d}$ . The derivative of the sliding surface defined by equation (15) can be given as

\overset{\cdot}{s} = c_{1} {\overset{\cdot}{e}}_{1} + {\overset{\cdot\cdot}{x}}_{1} + c_{2} e_{1} = 0

(26)

Furthermore, if an appropriate selection of constants $c_{1}$ and $c_{2}$ is meeting such that the characteristic polynomial in equation (26) is strictly Hurwitz, the tracking error $e_{1}$ will converge to zero asymptotically (Sinekli and Coban, 2017).

Dual-valve flow allocation

The adaptive robust motion controller also has generated the desired force $α_{a}$ and $α_{s}$ under the certain operating condition. The objective of the flow allocation layer is to attain the control commands for the two valves to provide the desired driving force.

As the dynamic responses of the valves in the study exceed the actuator’s operation frequency band, it is reasonable to use a linear model to describe the relationships between the control commands ( $u_{s}$ , $u_{p}$ ) and the corresponding valve displacement. The result of the servo valve can be represented as $x_{sv} = K_{v 1} u_{s}$ . Because of the proportional directional valve dead zone, the relationship between the control signal and valve displacement can be expressed by the following linear piecewise function

x_{pv} = {\begin{matrix} K_{v 2} (u_{p} - u_{upbound}), u_{p} > u_{upbound} \\ 0, u_{lowbound} \leq u_{p} \leq u_{upbound} \\ K_{v 2} (u_{p} - u_{lowbound}), u_{p} < u_{lowbound} \end{matrix}

(27)

where the $u_{upbound}$ and $u_{lowbound}$ represent the dead bands. And define $k_{vs} = K_{v 1} k_{q 1}$ and $k_{vp} = K_{v 2} k_{q 2}$ as the flow-signal coefficients of the servo valve and proportional valve.

The flow allocation can be divided into two parts: the first one is called the static allocation layer, and the other is called the dynamic allocation layer. As expressed in equation (28), in the static allocation layer, when the required flow rate of the system is small, only the servo valve is enabled, and when the flow rate exceeds the set value, the proportional valve is activated. Define $Q_{s} = k_{q 1} x_{sv}$ and $Q_{p} = k_{q 2} x_{pv}$ , the total flow rate can be expressed as

Q = \frac{C_{t}}{A_{1}} α_{n} = {\begin{matrix} Q_{s} + Q_{p}, {\overset{\cdot}{x}}_{d} > {\overset{\cdot}{x}}_{dset} \\ Q_{s}, - {\overset{\cdot}{x}}_{dset} \leq {\overset{\cdot}{x}}_{d} \leq {\overset{\cdot}{x}}_{dset} \\ Q_{s} + Q_{p}, {\overset{\cdot}{x}}_{d} < - {\overset{\cdot}{x}}_{dset} \end{matrix}

(28)

However, in actual systems, the flow rates supplied by the proportional valve and servo valve are apart from each other and exhibit distinct dynamic characteristics. The proportional valve with poor dynamic is more suited for speed control scenarios, and the servo valve has high zero-point accuracy, large pressure gain, and flow gain, which ensures a fast system response. It is rewarding to assign reasonable flow rates to the proportional and servo valves when they work together. In the dynamic allocation layer, to achieve this goal, according to the total virtual control law $α_{n}$ , define a virtual control law $α_{a 1}$ described as

α_{a 1} = - {\hat{θ}}_{1} (c_{1} {\overset{\cdot}{e}}_{1} + c_{2} e_{1}) + {\hat{θ}}_{2} {\overset{\cdot}{x}}_{p}

(29)

which represents the nominal flow rate according to the reference. Define $α_{a 2} = {\hat{θ}}_{3}$ , $α_{d} = α_{a 2} + α_{s}$ , so that the total virtual control law is rearranged as follows

α_{n} = α_{a} + α_{s} = α_{a 1} + α_{d}

(30)

The term $α_{d}$ contains not only the fast-transient components, that is, $α_{s}$ , but also some slow parts. Therefore, a filter is adopted to figure out the low-frequency component, which is regarded as a part of the control law belonging to the proportional valve. The low-pass filter is implemented as shown in equation (31).

α_{df} = λ_{1} \frac{1}{τ_{f} s + 1} α_{d}

(31)

where $λ_{1} = w_{1} k_{vp}^{2} / w_{2} k_{vs}^{2} + w_{1} k_{vp}^{2}$ . The ratio $w_{1} / w_{2}$ represents the proportion of control law for the proportional valve and servo valve, which can be adjusted in fields to resist flow saturation. As a result, the control law can be expressed as

{\begin{matrix} u_{s} = \frac{C_{t} (α_{d} - α_{d f})}{(k_{v s} A_{1})} \\ u_{p} = \frac{C_{t} (α_{a 1} + α_{d f})}{(k_{v p} A_{1})} + s g (α_{a 1} + α_{d f}) u_{u p b o u n d} + s g (- α_{a 1} - α_{d f}) u_{l o w b o u n d} \end{matrix}

(32)

Remark 3. The virtual control law in equation (29) is directly related to the tracking errors and model compensation, which comprise both low- and high-frequency components. In the flow allocation layer, the valve characteristics are considered. As described in equation (32), the low-frequency component of the total virtual control law is assigned to the proportional valve, whose bandwidth is lower compared with the servo valve, while the rest is assigned to the servo valve. And from the result in Theorem 1, it is easy to know that the DVPEHS system, together with the proposed control method, can achieve asymptotic tracking and all states will be stable if the reduced-order system (13) augmented with the developed DARC algorithm is stable.

Simulation and experiment tests

Both simulation and experimental tests are conducted in this section. Two kinds of error-evaluated indices are established, which are the absolute maximum error (AME) index and the mean square error (MSE) index. The definitions are in form of following

I_{AME} = max_{i = 1, \dots, N} {| e_{1} (i) |}

(33)

I_{MSE} = \frac{1}{N} \sum_{i = 1}^{N} {[e_{1} (i)]}^{2}

(34)

where $N$ is the recorded data number representing periods in simulation or experiments. To verify the efficiency of the proposed approach, three methods were compared in both simulation and experiment tests as follows:

(1) C1: The DVPEHS system with the proposed system controller in this study, the parameters of the controller are listed as following: $k_{1} = 330, 000, c_{1} = 220,$ $c_{2} = 1, ε = 0.5, γ_{1} = 1000, γ_{2} = 10, γ_{3} = 1, w_{1} = 4, w_{2} = 1$ . The initial values of $θ$ is $[30, 50, 000, 0]^{T}$ , and the values of $θ_{\max}$ and $θ_{\min}$ are $θ_{\max} = [100,$ $80, 000, 1000]^{T}, θ_{\min} = {[20, 20, 000, - 1000]}^{T}$ .

(2) C2: The DVPEHS system with a similar controller. In C2, the dynamic distribution of the flow is not considered, but only the static distribution of the flow according to the speed is conducted. The $α_{a}$ of the virtual control law is assigned to the proportional valve and the rest is assigned to the servo valve, which can be shown as

{\begin{matrix} u_{s} = \frac{C_{t} (α_{n} - α_{a})}{(k_{vs} A_{1})} \\ u_{p} = \frac{C_{t} α_{a}}{(k_{vp} A_{1})} + sg (α_{a}) u_{upbound} + sg (- α_{a}) u_{lowbound} \end{matrix}

(35)

The parameters in C2 are similar to C1, but $γ_{2} = 1 200$ , $c_{2} = 1.5$ .

(3) C3: The DVPEHS system with the typical proportional–integral–derivative (PID) controller, which is commonly used in industrial applications (Ang et al., 2005). The controller is given as

{\begin{matrix} u_{s} = k_{P 1} e_{1} + k_{I 1} \int_{0}^{t} e_{1} d τ + \frac{k_{D 1} d e_{1}}{dt} \\ u_{p} = k_{P 2} e_{1} + k_{I 2} \int_{0}^{t} e_{1} d τ + \frac{k_{D 2} d e_{1}}{dt} \end{matrix}

(36)

in which parameters are tuned for position tracking performance of three kinds of commands carefully, and the values of parameters are listed as follows: $k_{P 1} = 50, 000, 000,$ $k_{I 1} = 300, k_{D 1} = 2 400, k_{P 2} = 60, 000,$ $000, k_{I 2} = 10, k_{D 2} = 200$ .

Co-simulation result

In this paper, the MATLAB/AMESim co-simulation platform is adopted to conduct a co-simulation. The DVPEHS system is constructed in AMESim, and then the control law is implemented by the S-function method in MATLAB/Simulink. The simulation time step is 0.001 seconds, and system parameters are listed in Table 1.

Remark 4. In general, the controller parameters were tuned as followings: first, the parameters of the sliding surface $c_{1}$ and $c_{2}$ were chosen according to the characteristics of the system. Then, the feedback gain of the sliding error was tuned just as a simple proportional controller. By increasing the gain $k_{1}$ , we could increase the rate of convergence, but too much gain would result in oscillation. Both $c_{1}$ and $k_{1}$ are important to improve the responsiveness. Regarding the flow allocation layer parameters, we focused on the parameter $w_{1}$ , which determines the flow rate ratio of the two valves. For simplicity, setting $w_{2} = 1$ , and adjust $w_{1}$ to achieve flow rate anti-saturation. The parameter of the dynamic filter was determined considering proportional valve dynamics in advance. The adaption law coefficients are determined according to the tracking errors, with larger errors requiring larger coefficients. A too small disturbance adaption value will lead to system instability. It is essential to note that selecting optimal control parameters can be a complex process that requires both theoretical analysis and experimentation.

Table 1.

Parameters of the test system.

Parameter	Description	Value	Parameter	Description	Value
$m$	Load mass	60 kg	$C_{t}$	Total leakage coefficient	$2.9 \times 10^{- 11} m^{3} / s / Pa$
$b$	Viscous friction	50 000 N s/m	$k_{vs}$	Servo valve gain	$6.67 \times 10^{- 5} m^{3} / s V$
$l$	Actuator stroke	350 mm	$k_{vp}$	Proportional gain	$8 \times 10^{- 5} m^{3} / s V$
$d_{1}$	Piston diameter	80 mm	$β_{e}$	Oil bulk modulus	700 MPa
$d_{2}$	Rod diameter	45 mm

Three different position trajectory commands are utilized in both simulations and experimental tests. The details of reference trajectories are listed as follows:

(1) Set 1: Fast point-to-point trapezoidal curve from 0.1 to 0.3 m, with a maximum velocity of 0.1 m/s, and velocity bursts are buffered by a first-order low-pass filter with a transfer function of $G (s) = 1 / (0.1 s + 1)$ ;

(2) Set 2: Slow point-to-point trapezoidal curve from 0.1 to 0.2 m, with a maximum velocity of 0.0125 m/s, and velocity bursts are buffered by a first-order low-pass filter in common with Set 1. A disturbance is added at 3 seconds to evaluate the disturbance rejection ability, which can be described as $d_{in} = 1000 sg (t - 3) + 300 \sin (2 π t) sg (t - 3) (N)$ .

(3) Set 3: A sinusoidal signal with an amplitude of 8 mm and a frequency of 1 Hz, which is important for testing the ability to track periodic trajectory.

The tracking error comparisons among Sets 1, 2, and 3 are shown in Figure 3, and the tracking performance indices are summarized in Table 2. The transient process in the first second is omitted. It can be concluded that both C1, C2, and C3 can achieve high performance of tracking accuracy at low speed. However, in terms of tracking the fast-trapezoidal curve, C1 outperforms C2 and C3; thus, the effectiveness of the proposed system controller in dealing with the fast response of the DVPEHS system could be verified.

Table 2.

Simulation results comparison.

Methods	Set 1			Set 2			Set 3
	C1	C2	C3	C1	C2	C3	C1	C2	C3
$I_{AME} (mm)$	0.35	1.52	0.58	0.24	0.26	0.44	0.23	0.38	0.57
$I_{MSE}$	$4.17 \times 10^{- 9}$	$4.53 \times 10^{- 8}$	$5.82 \times 10^{- 8}$	$1.97 \times 10^{- 11}$	$6.87 \times 10^{- 11}$	$3.03 \times 10^{- 10}$	$1.15 \times 10^{- 8}$	$5.66 \times 10^{- 8}$	$4.13 \times 10^{- 8}$

Figure 3(a) shows the reference and tracking errors in Set 1; smooth tracking could be achieved among three methods. Large tracking errors occurred when sudden changes in the speed signal or force disturbances in the system were detected. This is primarily because tracking performance relies heavily on disturbance estimation in the absence of system pressure information, which dynamics depend on the adaption law. In addition, C3 results converge slowly when the system tracks a fast trajectory, indicating the effect of a lack of estimation. Compared to C1, all disturbance estimate results are assigned to the proportional valve in C2, which dynamic is limited. Consequently, the feedback term $α_{s}$ becomes too large, such that the servo valve is saturated, leading to limited system dynamic performance. As a result, the $I_{AME}$ of C2 is more significant than C1.

Figure 3(b) illustrates the reference and tracking errors in Set 2, both three methods could achieve high performance position tracking and strong disturbance rejection ability. The maximum tracking error of C1 is nearly 0.24 mm, which is similar to C2 (0.26 mm) and smaller than C3 (0.44 mm). The better performance of C1 and C2 is owing to the highly precise disturbance adaption and high feedback gain $k_{1}$ . However, exaggeratedly high gains may result in system oscillations even unstable results. On the other hand, at low speed, the fast response of the servo valve is enough to ensure tracking accuracy, so that C1 and C2 are equally effective.

Comparative simulation results of the position tracking of Set 3 are illustrated in Figure 3(c), where the tracking reference and errors are provided. Although the proposed method exhibits the best tracking accuracy, the performance advantage is not particularly obvious compared with C2 and C3. The reason is that reducing the order of the system places restrictions on tracking high-frequency periodic signals. However, in actual mobile machinery, point-to-point trajectory signals are widely used, while sinusoidal signals with sharp speed changes are rarely used. Therefore, for the proposed method, the performance flaw of tracking high-frequency periodic signals will not limit its application in actual mobile machinery.

Table 2 displays the evaluation indices used to assess tracking performance. The $I_{AME}$ represents the algorithms’ transient performance, while the $I_{MSE}$ is used to evaluate the steady-state tracking errors. When tracking trapezoidal curves, C1outperforms the other algorithms, while the C3 lacks steady-state tracking accuracy and C2 shows poor responsiveness in transient. For sinusoidal curve, C1 has the smallest $I_{AME}$ , but the $I_{MSE}$ index is not much better than the other two algorithms.

Figure 3(d)–(f) show the control commands of the dual valves in Sets 1, 2, and 3. The control command of the proportional valve in C1 takes a larger share of the overall control law and is smoother than C2 and C3, which indicates that the proposed control algorithm can effectively work in the DVPEHS system. Figure 4 illustrates the curve of trapezoidal tracking under disturbance among C1, C2, and C3. Compared with C3, the disturbance rejection effect of C1 and C2, which contains the parameter adaption term, show a more minor error in the step response instantaneously. However, due to the high gain coefficient in the feedback term, the system exhibits greater vibrations than those controlled in C3.

Comparative experiments

The simulation results have effectively validated the performance of the controller. To verify the performance further, the developed algorithm is tested on the experimental rig, as outlined in this experimental subsection. The experimental platform consists of a hydraulic supplier, a DVPEHS system, a load simulation system, and a MATLAB xPC real-time control system. Figure 5 provides a comprehensive view of the experimental setup and its schematic diagram. The MATLAB xPC real-time control system is conducted to execute the tests, comprising a host industrial computer (IPC-7120; ADVANTECH), the PCI-bus measuring cards, and junction boxes. Commands are transferred to the valve amplifiers by the 16-bit D/A converters on the cards, while sensor signals are collected by 16-bit A/D converters. Specifically, a magnetic railing ruler (MSR 500) is mounted on the load to measure the displacement of the mass.

For the DVPEHS system implementation, a single-rod cylinder is used to drive the load mass. The cylinder is equipped with a Rexroth 4WRREH6 servo valve and a Rexroth 4WRA6 proportional valve, both operating in parallel. The cut-off frequency response of the 4WRREH6 valve is up to 80 Hz at full-valve stroke, and the step time (0%–100% stroke) of the 4WRA6 proportional valve is 80 ms. The maximum flow rate of the servo valves is $6.67 \times 10^{- 4} m^{3} / s$ , while the proportional valve is $8 \times 10^{- 4} m^{3} / s$ . To match the flow rate of the cylinder, the control signals of the two valves are limited in actual experiments. The maximum opening of the servo valve is restricted to ±30%, while the proportional valve is limited to ±80%.

Following the simulations, three sets of comparative experiments are carried out. The primary objective of the first two sets is to validate the point-to-point trajectory tracking performance of the algorithm, with the fastest velocity in the tests nearing the physical limit. Three control strategies are compared, the same as mentioned in the simulation part.

Remark 5. Due to the differences in physical parameters, the simulation parameters would cause the system vibration in experiments, so the parameters need to be tuned in the field. The critical adjustment was that $w_{1}$ was changed to 3 by trial and error, and other parameters are shown as follows: $k_{1} = 300, 000$ , $c_{1} = 200$ , $c_{2} = 2.5$ , and $γ_{1} = 1 400$ .

The details are depicted in Figure 7, and the error comparison results are summarized in Table 3, which are consistent with the trend of simulation results. Both control strategies track closely when the reference velocity increases, while the proposed algorithm is superior to the classical PID method and DARC without flow allocation. As shown in Figure 6(b), in Set 1 and Set 3, the proposed controller has the minimum tracking error, while in Set 2, C1 performed similarly to C2 and better than C3. In Set 2, disturbances like pressure dynamic change slowly, which can be well suppressed by the parameter adaption method. As a result, the proportional valve acts smoothly and the servo valve can ensure tracking performance. Figure 7 gives the adaption results of the uncertain parameters. During the system operation, the estimation of the load mass increased gradually, whereas the viscous friction coefficient $b$ exhibited fluctuations around its upper bound. The estimated value of the lumped disturbance term $d$ changed rapidly when the actuator changed the direction, attributable to the pressure dynamics that are linked to the actuator’s working direction. Despite these changes, both adaptive parameters remained bounded, which is consistent with our assumptions. Overall, simulation and experimental tests have verified the proposed algorithm performance, which can achieve high-precision tracking, low-cost system configuration, and simple computation.

Table 3.

Experimental results comparison.

Methods	Set 1			Set 2			Set 3
	C1	C2	C3	C1	C2	C3	C1	C2	C3
$I_{AME} (mm)$	0.94	3.65	1.44	0.16	0.16	0.24	1.04	4.13	1.07
$I_{MSE}$	$1.71 \times 10^{- 8}$	$2.81 \times 10^{- 7}$	$6.08 \times 10^{- 7}$	$3.02 \times 10^{- 10}$	$2.90 \times 10^{- 10}$	$1.54 \times 10^{- 8}$	$2.86 \times 10^{- 7}$	$2.92 \times 10^{- 6}$	$4.13 \times 10^{- 7}$

Conclusion

In this study, a practical adaptive robust tracking control algorithm is proposed for the application of positioning and trajectory tracking of the DVPEHS systems. The proposed control algorithm is based on the direct adaptive robust control scheme according to Yao and Tomizuka (1997), where a projection-type adaptive law in combination with a robust feedback term is conducted to attenuate parametric uncertainties and uncertain nonlinearities. Most importantly, the proposed method avoids the full-state feedback and differential explosion problem of the DARC algorithm by reducing the system model order with singular perturbation theory. The dynamic flow distribution method is introduced to solve the flow redundancy problem in the DVPEHS systems. The proposed controller is a particular asymptotic output feedback control strategy, omitting the pressure sensors of the system and reducing the computation complexity, which is convenient for practical application. Finally, comparative simulation and experimental tests are performed to demonstrate the effectiveness and advancement of the proposed control scheme.

However, the DARC can be applied to the DVPEHS system without considering hydraulic dynamics, which is reasonable in situations where the system working frequency is lower than the hydraulic system dynamics. The effectiveness of the parameter uncertainties and disturbance rejection relies on the estimation of the lumped disturbance, so improving disturbance estimation dynamics without overshoots is a topic that warrants further investigation. In the future, this new integrated controller will be put into an actual wheeled hydraulic manipulator for further validation in actual mobile machinery.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Qiang Zhang

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