Adaptive extended state observer-based flatness nonlinear output control for torque tracking of electrohydraulic loading system

Abstract

Electrohydraulic loading system is a torque servo system with high-accuracy and high-frequency response. In this paper, an adaptive extended state observer-based flatness nonlinear output control is proposed to improve the torque tracking performance of electrohydraulic loading system. This method combines a flatness concept, expected state, adaptive extended state observer and system output to develop a stable control system. Expected input feedforward based on the flatness property is designed to provide model compensation for bandwidth enhancement. An adaptive extended state observer is proposed to estimate the unmeasured states and the unmodeled dynamics. Based on estimated states and disturbances, state feedback control is developed to ensure the stability of closed-loop system, and to improve torque tracking accuracy and system robustness. The stability of the closed-loop system is proved by the Lyapunov stability theory. Extensive experiments were carried out to verify the performance of high-accuracy tracking of the proposed control strategy.

Keywords

Electrohydraulic system nonlinear control flatness adaptive extended state observer output control disturbance/uncertainty estimation and attenuation

Introduction

Owing to the advantages of high power-to-size ratio and large payload capability, electro-hydraulic system has a wide range of applications (Ba et al., 2016; Shiralkar and Kurode, 2016; Wang and Wang, 2013). It has been widely applied in robot manipulators (Dunnigan et al., 1997; Guo et al., 2008; Sirouspour and Salcudean, 2001), active suspensions (Sun et al., 2013), flight motion simulators (Han et al., 2015; Yao et al., 2013), load simulators (Han et al., 2014; Wang et al., 2013), and so forth. Conventional linear control has been commonly used in hydraulic systems owing to their simplicity, clear functionality and ease of implementation (Ishak et al., 2012; Liu and Daley, 2000). However, these methods can only guarantee local stability and it is difficult for them to meet high performance requirements. In some special scenarios, the electro-hydraulic servo system (EHSS) is required to achieve high tracking accuracy and high dynamic response. Some properties for EHSS, like inherent high-order nonlinear characteristics, parameter uncertainties and modeling uncertainties (Shiralkar and Kurode, 2016; Yao et al., 2013), make it relatively hard to develop high-performance closed-loop controllers. Therefore, more advanced control algorithms are really a necessity rather than a luxury (Alleyne and Liu, 1999).

Many advanced nonlinear controllers with strong robustness and self-adaptability have been developed to improve tracking performance for EHSS, such as feedback linearization (Seo et al., 2007; Won et al., 2015b), backstepping control (Kaddissi et al., 2007), passivity-based control (Kim et al., 2012), flatness-based control (Kim et al., 2015), adaptive control (Ahn et al., 2014), $H_{\infty}$ robust control (Guo Q et al., 2015), adaptive robust control (Mohanty and Yao, 2011; Yao et al., 2000), sliding mode variable structure control (Bessa et al., 2010; Guan and Pan, 2008), fuzzy control (Du and Zhang, 2009; Truong and Ahn, 2011) and iterative learning control (Wang et al., 2013). Among these methods, flatness-based control has not been applied to hydraulic control until recently. There were only a few researches on this type of control. This method is simpler and more efficient compared with other algorithms. The flatness property could help to explicitly analyze system model based on system state variables and system input. These variables can be expressed as a function of flat output and a finite number of its time derivatives (Fliess et al., 1995). The control input can be parameterized by flat output and its time derivatives for asymptotic tracking of the expected trajectories. The controller is designed with expected state variables that are derived from system dynamics model. Controllers of many nonlinear systems were designed based on the flatness property: helicopters (Koo and Sastry, 1999), hovercrafts (Sira-Ramrez and Ibanez, 2000) and piezoelectric actuators (Rodriguez-Fortun et al., 2013). Electrohydraulic servo systems have also been shown to be flat (Kim et al., 2013). Flatness-based nonlinear control that includes both feedforward control and feedback control was proposed for electrohydraulic position servo systems (Kim et al., 2015). This controller can reduce current input ripples in EHSSs and improve the expected trajectory-tracking performance. Uncertainties and disturbances may degrade the performance of flatness-based control. Disturbances and uncertainties widely exist in all control systems and would always bring harmful effects onto their performance without any doubt.

Currently, great attention has been brought into disturbance rejection in controller design. Many algorithms have been proposed (Cheng et al., 2016; Guo and Wen, 2010). In a broad sense, these algorithms can be categorized into two different type of controls, that is, robust control and adaptive control (Yang et al., 2016). Robust control could achieve the best possible robustness probably only by compromising the nominal performance. It is difficult for adaptive control to deal with complex disturbances and uncertainties. It would probably become unstable under some practical applications in the presence of unstructured uncertainty or disturbances. Disturbance/uncertainty estimation and attenuation (DUEA) techniques can alleviate some weakness for robust control and adaptive control described above (Yang et al., 2016). It is able to provide an alternative approach for robust and adaptive control to deal with uncertain systems. In recent years, DUEA techniques have been thoroughly studied and applied into electrohydraulic servo systems (Guo K et al., 2015; Kim et al., 2013; Won et al., 2015(a); Yao et al., 2014).

All aforementioned control strategies need full-state information. That is to say, in addition to the system output, other information including differential output and pressure is also needed in electrohydraulic servo control. For many practical hydraulic systems, output is the only available information owing to cost reduction purpose or structure restriction. Heavy measurement noise of the pressure and velocity measurements may also deteriorate the performance of full-state feedback controllers. Extended state observer (ESO) is one of the DUEA techniques. It can observe system states, and provide real-time estimation of “total disturbances” (Chen et al., 2016). It does not require too much model information and can estimate disturbances. This helped bring ESO into wide applications in a variety of different controllers (Cai et al., 2014; Cui et al., 2015; Guo et al., 2016; Yao et al., 2014). Generally, ESO can be only applied in integral-chain essential linear systems (Li et al., 2012) and cannot be applied into electrohydraulic systems directly. In addition, although ESO has been intensively studied in the literatures, only a few research works on ESO were performed involving uncertain or time-varying parameters.

This paper proposed an adaptive extended state observer. The stability of the observer is demonstrated theoretically. A flatness nonlinear output control is proposed to improve torque tracking performance of electro-hydraulic loading system (EHLS) by synthesizing flatness-based control and adaptive state observer. This way, flatness control can be realized only by measuring the output. Robustness against disturbances is enhanced by disturbance/uncertainty estimation and compensation of adaptive extended state observer. The stability and tracking performance of the closed-loop system are proved based on Lyapunov theory.

The rest of this paper is organized as follows: the description and modelling of the electrohydraulic loading system are derived in the next section. Then, design of the proposed control law for electrohydraulic loading system is demonstrated, and the stability of the overall closed-loop system is derived. Next, experimental validation of the proposed scheme is given. The final section concludes this paper.

System description and dynamic model

The electrohydraulic loading system

Actuator has been widely used as servo driver in various servo control systems. Actuators are often subject to complicated loading conditions in their practical applications. They are usually required to be tested under practical dynamic loads to ensure that they meet the requirements in specific situations. This kind of test should be carried out in a specific load environment, which requires devices and techniques that can help accurately generate the same load condition as practice. The loading simulation technique for actuator test in lab has become more and more important, which can significantly help push actuator’s development, greatly saving experimental expense.

Torque can be generated between two relatively moving objects by friction. Friction may be used to simulate load torque this way, and the EHLS can be then developed. The structure is shown in Figure 1. The basic components in this system are hydraulic cylinders, hydraulic power, servo valve, friction plates, torque sensor, hydraulic swing motor and electric motor. In this system, the left part is the tested actuator system, which is a positioning servo control system; the right part called loading system is a torque servo control system that can generate torque to simulate aerodynamic load acting on the tested actuator. Friction plate A is connected with the shaft of tested actuator through a sliding key. It is able to move along the shaft and rotate together with it. Friction plate B is fixed onto the shaft of motor. It is driven by motor to keep rotating all the time at a constant speed, which would introduce a relative motion between two friction plates. Friction is then produced as both plates are connected with each other. Kinetic friction torque between friction plates can be transferred to the shaft of the tested hydraulic motor to serve as load. Hydraulic cylinders are fixed on the base and parallelly actuated by servo valve to load force on friction plates. The force acting onto friction plates can be adjusted by valve-controlled hydraulic cylinders, to change the loading torque needed. Actual loading torque from torque sensor will be fed back to controller for implementation of torque servo control.

Mathematical model

Friction is very complicated. A simple approach is studied by assuming friction force as a linear function of load. According to torus contact relation between friction plates A and B, friction torque is given by applying infinitesimal calculus.

T = \int_{R_{1}}^{R_{2}} r \cdot μ p 2 π rdr = \int_{R_{1}}^{R_{2}} r \cdot μ \frac{F}{π (R_{2}^{2} - R_{1}^{2})} 2 π rdr = μ RF

(1)

where $R = \frac{2}{3} \frac{{R_{2}}^{3} - {R_{1}}^{3}}{{R_{2}}^{2} - {R_{1}}^{2}}$ is equivalent friction radius [m], $R_{1}$ is internal radius of frictional contact area [m], $R_{2}$ is external radius of frictional contact area [m], $μ$ is friction coefficient, F is load force acting on a pair of friction pairs [N].

Load force acting on a pair of friction pairs can be expressed by

F = Ky

(2)

where K is the load stiffness [N/m], y is the position of hydraulic cylinder [m].

Friction torque is changed with the change of the load force. It can be accurately controlled by adjusting the valve-controlled hydraulic cylinder. The valve-controlled hydraulic cylinder shown in Figure 2 is a crucial element in this system. A basic set of equations describing the dynamic behavior of a valve-controlled hydraulic motor includes the following equations (Merritt, 1967).

The control flow rate equation of hydraulic valve can be written as

Q_{L} = C_{d} w x_{v} \sqrt{\frac{1}{ρ} (P_{s} - sign (x_{v}) P_{L})}

(3)

where $Q_{L}$ is load flow rate [m³/s], $C_{d}$ is discharge coefficient of servo valve, w represents valve spool area gradient [m], $x_{v}$ is spool position of the servo-valve [m], $ρ$ is the fluid density [kg/m³], $P_{s}$ is oil supply pressure [Pa], $P_{L}$ is load pressure [Pa].

By applying the continuity law to each chamber of hydraulic cylinder, the load flow rate continuity equation is given by

Q_{L} = A_{p} \overset{\cdot}{y} + C_{t} P_{L} + \frac{V_{t}}{4 β_{e}} {\overset{\cdot}{P}}_{L}

(4)

where $A_{p}$ is the effective pressure area of the piston [m²], $C_{t} = C_{ip} + C_{ep} / 2$ is the total leakage coefficient [m⁵/(N·s)] and $C_{ip}$ is the internal leakage coefficient [m⁵/(N·s)], $C_{ep}$ is the external leakage coefficient [m⁵/(N·s)], $V_{t}$ is the total equivalent control volume [m³], and $β_{e}$ is the effective volume elasticity modulus of hydraulic fluid [N/m²].

By applying Newton’s second law, dynamics of the piston can be described by

A_{p} P_{L} = m \overset{\cdot\cdot}{y} + B_{c} \overset{\cdot}{y} + K y + f

(5)

where m is the equivalent mass of load [kg], $B_{c}$ is the viscous damping coefficient of the pistons [N/(m/s)], $f$ is the friction or external load force [N].

In general, dynamic behavior of electric elements and servo valve are much higher than that of actuator in electro-hydraulic servo system. So, the relationship between input voltage and spool position of servo-valve is given by

x_{v} = k_{v} u

(6)

Defining state variable as $x_{1} = T$ , $x_{2} = \overset{\cdot}{T}$ and $x_{3} = P_{L}$ , combining equations (1) to (6), dynamics of EHLS can be then described in the following state-space form

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = \frac{1}{m} (- K x_{1} - B_{c} x_{2} + μ R K A x_{3} - μ R K f) \\ {\dot{x}}_{3} = \frac{4 β_{e}}{V_{t}} (- \frac{A}{μ R K} x_{2} - C_{t} x_{3} + C_{d} w k_{v} u \sqrt{1 / ρ} (P_{s} - s i g n (u) x_{3})) \end{matrix}

(7)

EHLS is a torque servo system. The output of this system is equal to the state $x_{1}$ . A flatness-based adaptive controller will be developed in the next section to meet the control requirements.

Flatness-based adaptive nonlinear output controller

Adaptive extended state observer design

For practical implementation purposes, an adaptive extended state observer is designed to estimate the full states.

Define parameters $a_{1} = \frac{K}{m}$ , $a_{2} = \frac{B_{c}}{m}$ , $a_{3} = \frac{μ RKA}{m}$ , $b_{1} = \frac{4 β_{e} A}{μ RK V_{t}}$ , $b_{2} = \frac{4 β_{e} C_{t}}{V_{t}}$ , $θ = \frac{4 β_{e} C_{d} w k_{v}}{V_{t} \sqrt{ρ}}$ , $g (x_{3}, u) = \sqrt{(P_{s} - sign (u) x_{3})}$ , $d = μ RKf$ . Hydraulic servo system is generally subjected to parametric uncertainties. For instance, parameters $β_{e}$ , $ρ$ are related to oil temperature and air content in the oil. Parameters $C_{d}$ , w, $k_{v}$ and $V_{t}$ are uncertain in hydraulic systems. So $θ$ is time-varying and uncertain. According to the definitions above and also by taking unmodeled dynamics into consideration, system (7) can be remodeled as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - a_{1} x_{1} - a_{2} x_{2} + a_{3} x_{3} + d \\ {\overset{\cdot}{x}}_{3} = - b_{1} x_{2} - b_{2} x_{3} + θ g (x_{3}, u) u + m (t) \end{matrix}

(8)

In order to accomplish the aforementioned design missions, major modeling uncertainty $m (t)$ is firstly extended as an additional state variable, that is, defining $x_{4} = m (t)$ , and system state is now extended to $x_{e} = {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}]}^{T}$ . The original plant (8) can be rewritten as the following, by using $h (t)$ as the time derivative of $x_{4}$

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - a_{1} x_{1} - a_{2} x_{2} + a_{3} x_{3} + d \\ {\overset{\cdot}{x}}_{3} = - b_{1} x_{2} - b_{2} x_{3} + θ (t) g (x_{3}, u) u + x_{4} \\ {\overset{\cdot}{x}}_{4} = h (t) \end{matrix}

(9)

The extended system model (9) can be then described as

{\begin{matrix} {\overset{\cdot}{x}}_{e} = A_{e} x_{e} + ϕ_{e} (x) + B_{e} θ g (u, x_{3}) u + υ_{e} (t) \\ y = C_{e} x_{e} \end{matrix}

(10)

where

\begin{matrix} A_{e} = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & a_{3} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}], B_{e} = {[\begin{matrix} 0 & 0 & 1 & 0 \end{matrix}]}^{T}, C_{e} = {[\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}]}^{T}, \\ ϕ_{e} (x) = [\begin{matrix} 0 \\ φ_{2} (x_{1}, x_{2}) \\ φ_{3} (x_{2}, x_{3}) \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ - a_{1} x_{1} - a_{2} x_{2} \\ - b_{1} x_{2} - b_{2} x_{3} \\ 0 \end{matrix}], υ_{e} (t) = {[\begin{matrix} 0 & d & 0 & h (t) \end{matrix}]}^{T} . \end{matrix}

Assumption 1: The function $g (u, x_{3})$ is Lipschitz with respect to $x_{3}$ in its practical range; $φ_{2} (x_{1}, x_{2})$ is Lipschitz with respect to $x_{1}$ and $x_{2}$ ; $φ_{3} (x_{2}, x_{3})$ is Lipschitz with respect to $x_{2}$ and $x_{3}$ .

To estimate the full states, a nonlinear observer is designed by

{\overset{\cdot}{\hat{x}}}_{e} = A_{e} {\hat{x}}_{e} + ϕ_{e} ({\hat{x}}_{e}) + B_{e} \hat{θ} g (u, {\hat{x}}_{3}) u + H (y - C_{e} {\hat{x}}_{e})

(11)

with the following parameter adaption law

\overset{\cdot}{\hat{θ}} = \frac{r}{ω_{0}^{2}} g (u, {\hat{x}}_{3}) u C_{e} ε - r σ (\hat{θ} - θ_{0})

(12)

where $H = {[\begin{matrix} h_{1} & h_{2} & h_{3} & h_{4} \end{matrix}]}^{T} = {[\begin{matrix} α_{1} ω_{0} & α_{2} ω_{0}^{2} & α_{3} ω_{0}^{3} & α_{4} ω_{0}^{4} \end{matrix}]}^{T}$ is adjustable parameter vector, $α_{1} > 0$ , $ω_{0} > 0$ , $r > 0$ and $σ > 0$ are adjustable parameter, $θ_{0}$ is the theoretical initial value of the parameter $θ$ .

Defining the estimation error as ${\tilde{x}}_{e} = x_{e} - {\hat{x}}_{e}$ , the dynamic of the state estimation error can be expressed as

\begin{matrix} {\overset{\cdot}{\tilde{x}}}_{e} = (A_{e} - H C_{e}) {\tilde{x}}_{e} + {\tilde{ϕ}}_{e} + υ_{e} (t) + B_{e} [θ g (u, x_{3}) u - \hat{θ} g (u, {\hat{x}}_{3}) u] \\ = (A_{e} - H C_{e}) {\tilde{x}}_{e} + {\tilde{ϕ}}_{e} + υ_{e} (t) + B_{e} (θ \tilde{g} u + \tilde{θ} \hat{g} u) \\ = (A_{e} - H C_{e}) {\tilde{x}}_{e} + δ_{e} + υ_{e} (t) + B_{e} \tilde{θ} g (u, {\hat{x}}_{3}) u \\ = (A_{e} - H C_{e}) {\tilde{x}}_{e} + δ_{e} + υ_{e} (t) + B_{e} \tilde{θ} g (u, {\hat{x}}_{3}) u \end{matrix}

(13)

where ${\tilde{ϕ}}_{e} = ϕ_{e} (x_{e}) - ϕ_{e} ({\hat{x}}_{e}) = {[\begin{matrix} 0 & {\tilde{φ}}_{2} & {\tilde{φ}}_{3} & 0 \end{matrix}]}^{T}$ , ${\tilde{φ}}_{2} = φ_{2} (x_{1}, x_{2}) - φ_{2} ({\hat{x}}_{1}, {\hat{x}}_{2})$ , ${\tilde{φ}}_{3} = φ_{3} (x_{2}, x_{3}) - φ_{3} ({\hat{x}}_{2}, {\hat{x}}_{3})$ , $\tilde{g} = g (x_{3}, u) - g ({\hat{x}}_{3}, u)$ , $δ_{e} = {[\begin{matrix} 0 & δ_{2} & δ_{3} & 0 \end{matrix}]}^{T} = {[\begin{matrix} 0 & {\tilde{φ}}_{2} & {\tilde{φ}}_{3} + θ \tilde{g} u & 0 \end{matrix}]}^{T}$ .

Defining the new variable $ε = {[\begin{matrix} ε_{1} & ε_{2} & ε_{3} & ε_{4} \end{matrix}]}^{T} = {[\begin{matrix} {\tilde{x}}_{1} & \frac{{\tilde{x}}_{2}}{ω_{0}} & \frac{{\tilde{x}}_{3}}{ω_{0}^{2}} & \frac{{\tilde{x}}_{4}}{ω_{0}^{3}} \end{matrix}]}^{T}$ , the estimation error dynamics (13) is then transformed into a singularly perturbed form as

\overset{\cdot}{ε} = ω_{0} A_{ε} ε + η (ω_{0}, x_{e}, {\hat{x}}_{e}, u) + Δ (ω_{0}, t) + \frac{\tilde{θ} g (u, {\hat{x}}_{3}) B_{e} u}{ω_{0}^{2}}

(14)

where

A_{ε} = [\begin{matrix} - α_{1} & 1 & 0 & 0 \\ - α_{2} & 0 & a_{3} & 0 \\ - α_{3} & 0 & 0 & 1 \\ - α_{4} & 0 & 0 & 0 \end{matrix}], η (ω_{0}, x_{e}, {\hat{x}}_{e}, u) = {[\begin{matrix} 0 & \frac{δ_{2}}{ω_{0}} & \frac{δ_{3}}{ω_{0}^{2}} & 0 \end{matrix}]}^{T}, Δ (ω_{0}, t) = {[\begin{matrix} 0 & \frac{d}{ω_{0}} & 0 & \frac{h (t)}{ω_{0}^{3}} \end{matrix}]}^{T} .

Based on assumption 1 and definitions of $φ_{2} (x_{1}, x_{2})$ , $φ_{3} (x_{2}, x_{3})$ and $g (x_{3}, u)$ , there exists a set of known constants $c_{1}$ , $c_{2}$ , $c_{3}$ , $c_{4}$ and $c_{5}$ such that

{\begin{matrix} | {\tilde{φ}}_{2} | = | φ_{2} (x_{1}, x_{2}) - φ_{2} ({\hat{x}}_{1}, {\hat{x}}_{2}) | \leq c_{1} | {\tilde{x}}_{1} | + c_{2} | {\tilde{x}}_{2} | \\ | {\tilde{φ}}_{3} | = | φ_{3} (x_{2}, x_{3}) - φ_{3} ({\hat{x}}_{2}, {\hat{x}}_{3}) | \leq c_{3} | {\tilde{x}}_{2} | + c_{4} | {\tilde{x}}_{3} | \\ | \tilde{g} | = | g (x_{3}, u) - g ({\hat{x}}_{3}, u) | \leq c_{5} | {\tilde{x}}_{3} | \end{matrix}

(15)

Choose $α_{1}$ , $α_{2}$ and $α_{3}$ such that $A_{ε}$ is Hurwitz. Then, there exists a positive definite matrix $P_{ε}$ satisfying the following Lyapunov equation

{A_{ε}}^{T} P_{ε} + P_{ε} A_{ε} = - I

(16)

$\tilde{θ} = θ - \hat{θ}$ is defined as the parameter estimation error. Following Lyapunov candidate function is taken into consideration

V_{ε} = ε^{T} P_{ε} ε + \frac{1}{r} {\tilde{θ}}^{2}

(17)

There must exist a positive constant $γ_{η}$ that satisfies the following situation

‖ η (ω_{0}, x_{e}, {\hat{x}}_{e}, u) ‖ = ‖ {[\begin{matrix} 0 & \frac{δ_{2}}{ω_{0}} & \frac{δ_{3}}{ω_{0}^{2}} & 0 \end{matrix}]}^{T} ‖ \leq γ_{η} ‖ ε ‖

(18)

Time derivative for equation (17) is then given as

\begin{matrix} {\overset{\cdot}{V}}_{ε} = {\overset{\cdot}{ε}}^{T} P_{ε} ε + ε^{T} P_{ε} \overset{\cdot}{ε} + \frac{2}{r} \tilde{θ} \overset{\cdot}{\tilde{θ}} \\ = ω_{0} ε^{T} ({A_{ε}}^{T} P_{ε} + P_{ε} A_{ε}) ε + 2 η (ω_{0}, x_{e}, {\hat{x}}_{e}, u) P_{ε} ε \\ + 2 Δ (ω_{0}, t) P ε + \frac{2}{ω_{0}^{2}} \tilde{θ} g (u, {\hat{x}}_{3}) u P_{ε} B_{e} ε + \frac{2}{r} \tilde{θ} \overset{\cdot}{\tilde{θ}} \end{matrix}

(19)

As $θ$ is a piecewise constant, $\overset{\cdot}{θ}$ equals zero almost all the time, i.e., $\overset{\cdot}{\tilde{θ}} \approx - \overset{\cdot}{\hat{θ}}$ . Using the theorem assumption $P_{ε} B_{e} = C_{e}$ (Besançon, 2000; Jung et al., 2006) and substituting equation (12) into equation (19), one obtains

\begin{matrix} {\overset{\cdot}{V}}_{ε} \leq - ω_{0} ‖ ε ‖^{2} + 2 λ_{max} (P_{ε}) ‖ η (ω_{0}, x_{e}, {\hat{x}}_{e}, u) ‖ ‖ ε ‖ \\ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ + 2 σ \tilde{θ} (\hat{θ} - θ_{0}) \leq - ω_{0} ‖ ε ‖^{2} \\ + 2 γ_{η} λ_{max} (P_{ε}) ‖ ε ‖^{2} + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ \\ + 2 σ \tilde{θ} (\hat{θ} - θ_{0}) \end{matrix}

(20)

where

σ \tilde{θ} (\hat{θ} - θ_{0}) = - \frac{1}{2} σ {\tilde{θ}}^{2} - \frac{1}{2} σ {(\hat{θ} - θ_{0})}^{2} + \frac{1}{2} σ {(θ - θ_{0})}^{2}

(21)

By substituting equation (21) into equation (20), the following inequality can be generated

\begin{matrix} {\overset{\cdot}{V}}_{ε} \leq - ω_{0} ‖ ε ‖^{2} + 2 γ_{η} λ_{max} (P_{ε}) ‖ ε ‖^{2} + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ \\ + 2 σ \tilde{θ} (\hat{θ} - θ_{0}) \leq - ω_{0} ‖ ε ‖^{2} + 2 γ_{η} λ_{max} (P_{ε}) ‖ ε ‖^{2} \\ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ - σ {\tilde{θ}}^{2} + σ {(θ - θ_{0})}^{2} \\ \leq - [\frac{(ω_{0} - 2 γ_{η} λ_{max} (P_{ε}))}{λ_{max} (P_{ε})} ε^{T} P_{ε} ε + σ {\tilde{θ}}^{2}] \\ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ + σ {(θ - θ_{0})}^{2} \end{matrix}

(22)

If $ω_{0}$ is selected appropriately so that $ω_{0} > 2 γ_{η} λ_{max} (P_{ε})$ , equation (22) can be written as

{\overset{\cdot}{V}}_{ε} \leq - τ V_{ε} + ζ

(23)

where $τ = min {\frac{(ω_{0} - 2 γ_{η} λ_{max} (P_{ε}))}{λ_{max} (P_{ε})}, σ r}$ , in which $λ_{max} (P_{ε})$ denote the maximum eigenvalues of a matrix, $ζ = 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ + σ {(θ - θ_{0})}^{2}$ .

According to equation (23), equation (24) can be obtained with a comparison lemma.

V_{ε} \leq V_{ε} (0) \exp (- τ t) + \frac{ζ}{τ} [1 - \exp (- τ t)]

(24)

Remark 1: When $Δ (ω_{0}, t) = {[\begin{matrix} 0 & \frac{d}{ω_{0}} & 0 & \frac{h (t)}{ω_{0}^{3}} \end{matrix}]}^{T} \neq 0$ , the tracking error $\tilde{x}$ and the scaled state estimation error $ε$ are bounded. The ESO has an exponentially convergence transient performance with the exponentially converging rate $τ$ and the final estimation error being able to be adjusted $τ$ and $ζ$ freely in a known form. So, the designed adaptive ESO is stable and the estimation error can be arbitrarily reduced by increasing the bandwidth. While large bandwidth will lead to control saturation owing to the largely disturbance compensation, the designed observer bandwidth should be compromised to ensure both the suppression of external load and the stability of augmented system. When $Δ (ω_{0}, t) = {[\begin{matrix} 0 & \frac{d}{ω_{0}} & 0 & \frac{h (t)}{ω_{0}^{3}} \end{matrix}]}^{T} = 0$ , ${\overset{\cdot}{V}}_{ε} \leq - τ V_{ε}$ . In the absence of unknown external disturbance and time-varying modeling uncertainties, then the final estimation error is able to achieve asymptotic convergence to zero, that is, $\tilde{x} \to 0$ as $t \to 0$ .

Flatness-based nonlinear controller design

Considering a nonlinear system

\overset{\cdot}{x} = f (x, u)

(25)

where x is the state and u is the input with the same dimensions as the output y.

If there exists an output $y = P (x, u, \overset{\cdot}{u}, \dots, u^{(p)})$ such that $x = Q (y, \overset{\cdot}{y}, \dots, y^{(q)})$ and $u = R (y, \overset{\cdot}{y}, \dots, y^{(q + 1)})$ can be expressed as a function of y and a finite number of its time derivatives, system $\overset{\cdot}{x} = f (x, u)$ is then called flat. Output y is called the flat output.

According to the flatness property, the system model could be analyzed by formally calculating the system variables as functions of the flat output and its time derivatives.

System state variables as the flatness functions can be obtained from y, $\overset{\cdot}{y}$ and $\overset{\cdot\cdot}{y}$ .

{\begin{matrix} x_{1} = y \\ x_{2} = \overset{\cdot}{y} \\ x_{3} = \frac{1}{a_{3}} (a_{1} y + a_{2} \overset{\cdot}{y} + \overset{\cdot\cdot}{y} - d) \end{matrix}

(26)

The control input can be designed based on the flatness property for the asymptotic tracking. According to system (8), the control input can be written by

u = \frac{1}{θ g (x_{3}, u)} (b_{1} x_{2} + b_{2} x_{3} + {\overset{\cdot}{x}}_{3} - m (t))

(27)

The goal of this control system is to accurately track desired output torque $y_{d}$ . So, the desired states can be expressed by

{\begin{matrix} x_{1 d} = y_{d} \\ x_{2 d} = {\overset{\cdot}{y}}_{d} \\ x_{3 d} = \frac{1}{a_{3}} (a_{1} y_{d} + a_{2} {\overset{\cdot}{y}}_{d} + {\overset{\cdot\cdot}{y}}_{d} - d) \end{matrix}

(28)

System with desired states can be written as

{\begin{matrix} {\overset{\cdot}{x}}_{1 d} = x_{2 d} \\ {\overset{\cdot}{x}}_{2 d} = - a_{1} x_{1 d} - a_{2} x_{2 d} + a_{3} x_{3 d} + d \\ {\overset{\cdot}{x}}_{3 d} = - b_{1} x_{2 d} - b_{2} x_{3 d} + θ g (x_{3}, u) u_{d} + m (t) \end{matrix}

(29)

Instead of states in equation (27), the desired control input can be rewritten by using the desired states

u_{d} = \frac{1}{θ g (x_{3 d}, u)} (b_{1} x_{2 d} + b_{2} x_{3 d} + {\overset{\cdot}{x}}_{3 d} - m (t))

(30)

Now let the tracking error $e = {[\begin{matrix} e_{1} & e_{2} & e_{3} \end{matrix}]}^{T}$ be defined as

{\begin{matrix} e_{1} = x_{1 d} - x_{1} \\ e_{2} = x_{2 d} - x_{2} \\ e_{3} = x_{3 d} - x_{3} \end{matrix}

(31)

The control law is designed by

u = \frac{θ g (x_{3 d}, u) u_{d} + Le}{θ g (x_{3}, u)}

(32)

Theorem 1: Considering system (8) and the desired system (29), as long as L is properly chosen, the control input (32) can stabilize the system (8) at the origin. The stability of the control system will be proved using Lyapunov asymptotic stability theorem.

Defining $x = {[\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}]}^{T}$ , the system (8) can be rewritten by

\overset{\cdot}{x} = Ax + B θ g (x_{3}, u) u + υ (t)

(33)

where $A = [\begin{matrix} 0 & 1 & 0 \\ - a_{1} & - a_{2} & - a_{3} \\ 0 & - b_{1} & - b_{2} \end{matrix}]$ , $B = {[\begin{matrix} 0 & 0 & 1 \end{matrix}]}^{T}$ , $υ (t) = {[\begin{matrix} 0 & d & m (t) \end{matrix}]}^{T}$ .

Defining $x_{d} = {[\begin{matrix} x_{1 d} & x_{2 d} & x_{3 d} \end{matrix}]}^{T}$ , the system (29) can be rewritten by

{\overset{\cdot}{x}}_{d} = A x_{d} + B θ g (x_{3 d}, u_{d}) u_{d} + υ (t)

(34)

Combining equation (33) and equation (34), the time derivative of e can be expressed by

\begin{matrix} \overset{\cdot}{e} = {\overset{\cdot}{x}}_{d} - \overset{\cdot}{x} \\ = A x_{d} + B θ g (x_{3 d}, u_{d}) u_{d} - (Ax + B θ g (x_{3}, u) u) \\ = Ae + B θ g (x_{3 d}, u_{d}) u_{d} - B θ g (x_{3}, u) \frac{θ g (x_{3 d}, u_{d}) u_{d} + Le}{θ g (x_{3}, u)} \\ = (A - BL) e \end{matrix}

(35)

Properly choose gain L so that the matrix $A - BL$ is Hurwitz. Thus, there exists a positive definite matrix P satisfying the following Lyapunov equation definite.

{(A - BL)}^{T} P + P (A - BL) = - Q

(36)

where $Q = Q^{T}$ is a positive definite matrix.

\overset{\cdot}{V} = e^{T} [{(A - BL)}^{T} P + P (A - BL)] e \leq - e^{T} Qe

(37)

Thus, $\overset{\cdot}{V}$ is negative definite. By the Lyapunov stability theory, the tracking error e exponentially converges to zero. The tracking error can be governed by the controller parameters.

The actual implementing control law (32) need full states information. However, in this paper only the torque signal is available in the loading system. Using estimated parameter $\hat{θ}$ and ${\hat{x}}_{4}$ , the desired system (29) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{1 d} = x_{2 d} \\ {\overset{\cdot}{x}}_{2 d} = - a_{1} x_{1 d} - a_{2} x_{2 d} + a_{3} x_{3 d} + d \\ {\overset{\cdot}{x}}_{3 d} = - b_{1} x_{2 d} - b_{2} x_{3 d} + \hat{θ} g (x_{3}, u) u_{d} + {\hat{x}}_{4} \end{matrix}

(38)

According to equation (38), the desired control input can be rewritten as

u_{d} = \frac{1}{\hat{θ} g (x_{3 d}, u)} (b_{1} x_{2 d} + b_{2} x_{3 d} + {\overset{\cdot}{x}}_{3 d} - {\hat{x}}_{4})

(39)

Since the estimated states $\hat{x}$ are used in the control law (32) instead of the states x, except $x_{1}$ , the control input can be obtained by

u = \frac{\hat{θ} g (x_{3 d}, u) u_{d} + L \hat{e}}{\hat{θ} g ({\hat{x}}_{3}, u)}

(40)

Remark 2: In the controller, the load torque d can be considered as a system parameter (Kim et al., 2015). It may be known or obtained by error-and-try method. The major uncertainty/disturbance $m (t) = x_{4}$ is estimated by the adaptive extended state observer.

Remark 3: The defined parameter $θ$ is bounded and positive, that is, $θ \in Ω_{θ} = [θ_{min}, θ_{max}]$ and $θ_{min} > 0$ .The estimated parameter $\hat{θ}$ might go to zero, causing the singular value of the controller. So, the parameters $σ$ should be properly chosen as far as possible so that $\hat{θ}$ is always positive.

Stability analysis of closed-loop stability

Using estimated parameter $\hat{θ}$ and ${\hat{x}}_{4}$ , the desired system (38) can be rewritten in a compact form as

{\overset{\cdot}{x}}_{d} = A x_{d} + B \hat{θ} g (x_{3 d}, u_{d}) u_{d} + \hat{υ} (t)

(41)

where $\tilde{υ} (t) = υ (t) - \hat{υ} (t) = {[\begin{matrix} 0 & 0 & {\tilde{x}}_{4} \end{matrix}]}^{T}$ .

Combining equation (33) and equation (41), the time derivative of e can be rewritten by

\begin{matrix} \overset{\cdot}{e} = {\overset{\cdot}{x}}_{d} - \overset{\cdot}{x} \\ = A x_{d} + B \hat{θ} g (x_{3 d}, u_{d}) u_{d} + \hat{υ} (t) - (Ax + B θ g (x_{3}, u) u + υ (t)) \\ = Ae + B \hat{θ} g (x_{3 d}, u_{d}) u_{d} - B θ g (x_{3}, u) u - \tilde{υ} (t) \\ = Ae + B \hat{θ} g (x_{3 d}, u_{d}) u_{d} - B \hat{θ} g ({\hat{x}}_{3}, u) u \\ - [B \tilde{θ} g ({\hat{x}}_{3}, u) + B θ \tilde{g}] u - \tilde{υ} (t) \\ = Ae + B \hat{θ} g (x_{3 d}, u_{d}) u_{d} - B \hat{θ} g ({\hat{x}}_{3}, u) \frac{\hat{θ} g (x_{3 d}, u_{d}) u_{d} + L \hat{e}}{\hat{θ} g ({\hat{x}}_{3}, u)} \\ - [B \tilde{θ} g ({\hat{x}}_{3}, u) + B θ \tilde{g}] u - \tilde{υ} (t) \\ = Ae - BL \hat{e} - [B \tilde{θ} g ({\hat{x}}_{3}, u) + B θ \tilde{g}] u - \tilde{υ} (t) \\ = Ae - BL \hat{e} - [B \tilde{θ} g ({\hat{x}}_{3}, u) + B θ \tilde{g}] u - \tilde{υ} (t) \end{matrix}

(42)

Considering $\hat{e} = x_{d} - \hat{x} = e + \tilde{x}$ , the time derivative of e can be rewritten as

\overset{\cdot}{e} = (A - BL) e - BL \tilde{x} - [B \tilde{θ} g ({\hat{x}}_{3}, u) + B θ \tilde{g}] u - \tilde{υ} (t)

(43)

Choose a Lyapunov function to prove the stability of the closed-loop system

V_{c} = e^{T} Pe + ε^{T} P_{ε} ε + \frac{1}{α} {\tilde{θ}}^{2}

(44)

${\hat{x}}_{3}$ is the estimated load pressure, which must satisfy the situation $- P_{s} \leq {\hat{x}}_{3} \leq P_{s}$ . So, the inequality $\sqrt{(P_{s} - sign (u) {\hat{x}}_{3})} \leq \sqrt{2 P_{s}} = κ$ is proved. Considering the hardware constraint, there exists the maximum control input ${| u |}_{max}$ . According to the definitions of the parameters $\tilde{x}$ and $ε$ , there exists a positive constant $γ_{ε}$ that satisfies the equation $‖ \tilde{x} ‖ \leq γ_{ε} ‖ ε ‖$ .

For convenient writing, a set of known constants is defined as

γ_{1} = γ_{ε} ‖ PB ‖ ‖ L ‖, γ_{2} = ω_{0}^{2} c_{5} χ θ_{max} ‖ PB ‖, γ_{3} = ω_{0}^{3} ‖ PB ‖, γ_{4} = κ χ {\tilde{θ}}_{max} ‖ PB ‖

(45)

where ${\tilde{θ}}_{max} = θ_{max} - θ_{min}$ , $χ = {| u |}_{max}$ .

Its time derivative along equation (44) is given as

\begin{matrix} {\overset{\cdot}{V}}_{c} = {\overset{\cdot}{e}}^{T} Pe + e^{T} P \overset{\cdot}{e} + {\overset{\cdot}{ε}}^{T} P_{ε} ε + ε^{T} P_{ε} \overset{\cdot}{ε} + \frac{2}{α} \tilde{θ} \overset{\cdot}{\tilde{θ}} \\ = e^{T} [{(A - BL)}^{T} P + P (A - BL)] e - 2 e^{T} PBL \tilde{x} - 2 e^{T} PB θ \tilde{g} u \\ - 2 e^{T} PB \tilde{θ} g ({\hat{x}}_{3}, u) u \\ - 2 e^{T} PB \tilde{υ} (t) + ω_{0} ε^{T} ({A_{ε}}^{T} P_{ε} + P_{ε} A_{ε}) ε + 2 Δ (ω, x, \hat{x}, u) P_{ε} ε \\ + \frac{2 P_{ε} ε B \tilde{θ} g ({\hat{x}}_{3}, u) u}{ω_{0}^{2}} + \frac{2}{α} \tilde{θ} \overset{\cdot}{\tilde{θ}} \\ \leq - ‖ e^{T} ‖ ‖ Q ‖ ‖ e ‖ + 2 ‖ e^{T} ‖ ‖ PB ‖ ‖ L ‖ ‖ \tilde{x} ‖ + 2 χ θ_{max} ‖ e^{T} ‖ \\ ‖ PB ‖ | \tilde{g} | + 2 χ {\tilde{θ}}_{max} g ({\hat{x}}_{3}, u) ‖ e^{T} ‖ ‖ PB ‖ \\ + 2 ‖ e^{T} ‖ ‖ PB ‖ ‖ \tilde{υ} (t) ‖ - (ω_{0} - 2 γ_{η} λ_{max} (P_{ε})) ‖ ε ‖^{2} \\ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ - σ {\tilde{θ}}^{2} + σ {(θ - θ_{0})}^{2} \\ \leq - ‖ e^{T} ‖ ‖ Q ‖ ‖ e ‖ + 2 γ_{1} ‖ e^{T} ‖ ‖ ε ‖ + 2 γ_{2} ‖ e^{T} ‖ ‖ ε ‖ \\ + 2 γ_{3} ‖ e^{T} ‖ ‖ ε ‖ + 2 γ_{4} ‖ e^{T} ‖ - σ {\tilde{θ}}^{2} \\ - (ω_{0} - 2 γ_{η} λ_{max} (P_{ε})) ‖ ε ‖^{2} + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ \\ + σ {(θ - θ_{0})}^{2} \\ \leq - (‖ Q ‖ - γ_{1} - γ_{2} - γ_{3}) ‖ e ‖^{2} \\ - [ω_{0} - 2 γ_{η} λ_{max} (P_{ε}) - γ_{1} - γ_{2} - γ_{3}] ‖ ε ‖^{2} - σ {\tilde{θ}}^{2} \\ + 2 γ_{4} ‖ e ‖ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ + σ {(θ - θ_{0})}^{2} \end{matrix}

(46)

Properly choose gains $ω_{0}$ and L satisfying the following conditions

‖ Q ‖ - γ_{1} - γ_{2} - γ_{3} > 0 ω_{0} - 2 γ_{η} λ_{max} (P_{ε}) - γ_{1} - γ_{2} - γ_{3} > 0

(47)

Equation (46) can be rearranged as

{\overset{\cdot}{V}}_{c} \leq - τ_{c} V_{c} + Ω

(48)

where $τ_{c} = min {‖ Q ‖ - γ_{1} - γ_{2} - γ_{3}, ω_{0} - 2 γ_{η} λ_{max} (P_{ε}) - γ_{1} - γ_{2} - γ_{3}, σ}$ , $Ω = 2 γ_{4} ‖ e ‖ + 2 λ_{max} (P_{ε}) ‖ Δ (ω_{0}, t) ‖ ‖ ε ‖ + σ {(θ - θ_{0})}^{2}$ .

According to equation (48) and stability theory, it is easy to ensure that the closed loop system is stable.

Experimental validation

A test instrument was set up, in order to validate torque tracking performance of the proposed novel loading method. As shown in Figure 3, this test instrument consisted of hydraulic cylinders, servo valves (FF106), a torque sensor (CYB802S,range: 0∼±150N·m, precision: ±0.1%FS ), friction plates, an electric motor, a hydraulic motor, an angle encoder, a PCI1710 card and a PCI1716 card.

Host-client structure was adopted in this friction loading servo control system. The host computer was an ordinary PC. It was used to build the man-machine interface and to process data with LabVIEW. The client computer was an industrial control computer from Advantech Co. Ltd, which was used to run the real-time servo control program. The xPC Target rapid prototype control technology was used in the friction loading servo control system. Simulink control model of system was set up in the host computer. This model was compiled into code for real-time operation through the compile. It was then loaded into the client computer to the control system. Sampling interval could reach 1ms.

The following three controllers were compared in order to validate the developed controller:

AESOBFNOC: Adaptive extended state observer-based flatness nonlinear output controller is proposed in this paper and described above. Control gains are given as follows $α_{1} = 4$ , $α_{2} = 6$ , $α_{3} = 4$ , $α_{4} = 1$ , $ω_{0} = 350$ , $r = 10000$ , $σ = 10$ , $l_{1} = 1.26 \times 10^{9}$ , $l_{2} = 3.35 \times 10^{6}$ , $l_{3} = 1.20 \times 10^{4}$ , $θ_{0} = 1.4143 \times 10^{6}$ .

ESOBBC: Extended state observer-based backstepping controller was proposed in Guo et al. (2016). The controller was designed based on backstepping approach. In addition, extended state observer was used to estimate not only the unmeasured states but also the modeling uncertainties. Control gains are given as follows: $k_{1} = 1200$ , $k_{2} = 3500$ , $k_{3} = 50000$ . Other parameters are the same as the parameters of the proposed controller.

PID: This is the traditional proportional–integral–derivative controller. Also, its gains tuned carefully via an error-and-try method are $k_{p} = 1.8$ , $k_{i} = 56$ and $k_{d} = 0.024$ , which denote the proportional gain, integral gain and derivative gain, respectively.

Those three controllers were firstly tested for a sinusoidal torque trajectory $T_{d} = 10 \sin (2 π \times t) + 15$ , to verify the performance of proposed controller. Expected torque trajectory and corresponding tracking performance of the proposed controller are shown in Figure 4(a). Tracking errors under these three controllers are shown in Figure 4(b), respectively. Figure 4(a) shows that electrohydraulic loading system with the proposed controller is able to track 1Hz torque command. It can be seen in Figure 4(b) that these tracking errors are about 0.6N·m, 0.8N·m and 1N·m, respectively. These three controllers show good performance when tracking low frequency torque signal. Tracking performance with AESOBFNOC is the best.

Compared with the other two controllers, performance of proposed controller does not stand out when tracking low frequency torque signal. A faster torque trajectory $T_{d} = 10 \sin (2 π \times 10 t) + 15$ was then given to test the performance of proposed controller. Tracking performance contrast for 10Hz torque command is shown in Figure 5. Figure 6 shows the states estimated by adaptive ESO. These three controllers are based on these estimated states. Figure 5(a) shows that tracking phase difference of system with AESOBFNOC and ESOBBC is smaller than that of system with PID controller. Figure 5(b) shows the corresponding tracking errors. Maximum tracking error of system with PID is about 6N·m. It is known that nonlinearity of novel electrohydraulic loading system will degrade the performance of PID. Maximum torque tracking errors of AESOBFNOC and ESOBBC are around 2.5N·m. But tracking error of AESOBFNOC is still slightly less than that of ESOBBC.

Tracking results show that under low frequency of 1Hz, proposed controller, ESOBBC and PID controller demonstrated similar control performances. As the frequency of the desired torque increases higher from 1Hz to 10Hz, the control performances of proposed controller are better than that of ESOBBC and PID. Obviously, the proposed AESOBFNOC and ESOBBC are better than PID controller in terms of both transient and final tracking errors since AESOBFNOC and ESOBBC can employ the system model to achieve accurate model-based compensation. Compared with ESOBBC, AESOBFNOC employed an input feedforward to enhance bandwidth. Adaptive ESO in AESOBFNOC is able to estimate uncertain parameters and compensate them in controllers.

Conclusions

In this paper, an adaptive extended state observer-based flatness nonlinear output control is proposed to improve the torque tracking performance of electrohydraulic loading system. Based on mathematical model of electrohydraulic loading system, flatness-based nonlinear control is designed. Adaptive extended state observer is designed to provide full-state information. Stability of closed-loop system is proved by Lyapunov stability theory. Experimental results demonstrate a good performance for the proposed controller. System dynamic loading performance is greatly improved by the proposed controller. In the future, control gains need to be further optimized. The adaptive scheme may be applied to improve the robustness of system.

Footnotes

Declaration of conflicting interest

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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