An Adaptive Prescribed Performance Position Tracking Controller for Hydraulic Systems

Abstract

Unknown time-varying parameters, along with mismatched and matched disturbances, exist in hydraulic systems, worsening position tracking performance and even destabilizing systems. To address this issue, this article proposes an adaptive full-state prescribed performance position tracking control for hydraulic systems subject both to unknown time-varying parameters and to mismatched and matched disturbances. First, a smooth nonlinear term is skillfully introduced into the controller design so that it can simultaneously cope with both unknown time-varying parameters and disturbances. Next, by integrating the adaptive technique and the prescribed performance function, an adaptive full-state prescribed performance position tracking controller is developed for hydraulic systems in which both the transient and steady performance of all the control errors can be prescribed. A stability analysis then confirms both the prescribed transient performance and the asymptotic steady performance of all the control errors. Finally, the superiority of the proposed controller is also validated by comparison with simulation results.

1. Introduction

Hydraulic systems have long played a role in industrial applications due to their large force/torque output capabilities and small size-to-power ratios [1,2,3,4,5,6,7,8,9,10]. However, the achievement of high-precision tracking control for hydraulic systems is hard because of the existence of both high nonlinearities and large uncertainties. To surmount this problem, many controls have been developed for hydraulic systems during recent years. An output-feedback nonlinear robust control was proposed for hydraulic systems in [11], in which an extended state observer (ESO) was adopted to estimate unknown system states and compensate for unknown disturbances. Subsequently, in [12], Na et al. employed a finite-time sliding mode state observer and presented a novel unknown system dynamics estimator-based control method for hydraulic systems, confirming the validity of their approach through extensive experimental results. In [13], Ba et al. developed a sliding mode observer (SMO) that was designed to compensate for unknown dynamics in hydraulic systems. Excellent zero-error control performance was also found to hold theoretically in [13]. However, the assumption that the boundedness of the second-order derivatives of disturbances must be met might have limited the actual application of SMO in [13]. Remarkably, the designs of all the above controllers neglected to account for the large parametric uncertainties which result from component wear and temperature change in hydraulic systems, the existence of which can worsen control performance and even cause systems to be destabilized.

To improve robustness against both unknown disturbances and parametric uncertainties and, furthermore, to achieve expected control performance, a typical control framework named adaptive robust control was developed in [14] and subsequently successfully utilized in hydraulic systems [15,16,17,18,19]. In [20], Wang et al. exploited an ESO-based nonlinear adaptive control method for single-rod hydraulic systems, and obtained experimental results which confirmed the merits of the proposed approach. A two-ESO-based adaptive robust control method was exploited for hydraulic systems in [21], in which ESOs were utilized to estimate both the unmeasured velocity value and unknown disturbances. An output-feedback-based adaptive-gain super-twisting sliding mode control framework was exploited for hydraulic systems in [22], in which the ESO was utilized to estimate the unknown system states and disturbances. However, the adaptive controllers reported in [11,12,13,14,15,16,17,18,19,20,21,22] only obtained bounded-error tracking performance in theory. Moreover, these adaptive controllers always assumed that unknown parameters existing in hydraulic systems were constant. However, temperature change and component wear, both unknown time-varying parameters and disturbances, exist widely in hydraulic systems. The problem of how to cope with both unknown time-varying parameters and time-varying disturbances is still, then, a challenge for hydraulic systems.

The other motivation for this paper is that all the aforementioned controllers were designed to achieve enhancement of steady performance in hydraulic systems while ignoring transient performance. To meet the requirements of prescribed transient performance for hydraulic systems, Xu et al. combined ESO and a prescribed performance function (PPF) in [23,24,25], then overcame the problem of motion control for state-constrained hydraulic systems in [26], in which the constraints of all system states could not be violated and tracking performance could always be kept in a prescribed region using their proposed controller. The authors of [27] used the Barrier Lyapunov Function and the PPF to develop an output feedback-based adaptive prescribed performance controller for hydraulic systems, achieving the prescribed transient performance of the tracking error without the violation of full-state constraints. Guo et al. exploited an adaptive prescribed performance backstepping controller for hydraulic manipulators in [28], in which large coupled uncertainties were estimated and compensated for using a neural network. In contrast, in previous controllers [23,24,25,26,27,28], only the preset performance of the tracking error could be assured, while unknown parameters needed to be assumed to remain time-invariant. The question of how to design an adaptive prescribed performance control method for hydraulic systems with unknown time-varying parameters while obtaining the prescribed performance of all the control errors is therefore still an open one.

Motivated by the above facts, in this study we propose an adaptive full-state prescribed performance position tracking controller for hydraulic systems which delivers both the prescribed transient performance and the asymptotic steady performance of all the control errors. Compared to previous studies which sought to obtain control results for hydraulic systems, the contributions of this paper can be stated as follows:

(1)

An adaptive full-state prescribed performance position tracking controller is developed for hydraulic systems, in which both the boundedness of all the closed-loop signals and the asymptotic steady performance of the position tracking error can be ensured.

(2)

By skillfully introducing a smooth nonlinear function, unknown time-varying parameters and disturbances are both handled simultaneously, thereby relaxing the assumption that unknown parameters should be constant or their derivatives should be bounded, which underlies existing adaptive control results [11,12,13,14,15,16,17,18,19,20,21,22].

(3)

Unlike with existing controllers that can only ensure the prescribed performance of the output error in [23,24,25,26,27,28], both the prescribed transient performance and the steady performance of all the control errors for hydraulic systems are guaranteed.

In the following paper, the system modeling of the hydraulic system and the controller design are presented in Section 2 and Section 3, respectively. The simulation results and the conclusion are presented in Section 4 and Section 5, respectively.

2. System Modeling

The considered hydraulic system is illustrated in Figure 1, in which it can be seen that the valve-controlled hydraulic axis causes the load to move.

The dynamics of the load can be denoted as

$$m_l{\ddot{y}}_l=A\left(P_1-P_2\right)-a_1(t){\dot{y}}_l-a_2(t)\tanh (c_1{\dot{y}}_l)-a_3(t)\left[\tanh (c_2{\dot{y}}_l)-\tanh (c_3{\dot{y}}_l)\right]+f(t)$$

(1)

where $m_l$ and $y_l$ denote the load mass and displacement; $P_1$ and $P_2$ denote the pressure values of the two chambers; $A$ denotes the effective piston area; $a_i(t)$ ($i=1,2,3$) are time-varying parameters that denote the amplitudes of viscous friction, Coulomb friction, and the Stribeck effect, respectively; $c_i$ denotes shape coefficients that approximate Coulomb friction and the Stribeck effect; and $f(t)$ denotes the unknown mismatched disturbance.

Following [29], the pressure dynamics of the hydraulic actuator can be denoted as

$$\begin{array}{l}{\displaystyle \frac{V_1{\dot{P}}_1}{{\beta }_e}}=-A{\dot{y}}_l-C_t(t)(P_1-P_2)+{\Delta }_1(t)+q_1\\ {\displaystyle \frac{V_2{\dot{P}}_2}{{\beta }_e}}=A{\dot{y}}_l-C_t(t)(P_2-P_1)+{\Delta }_2(t)-q_2\end{array}$$

(2)

where $V_1=V_{01}+A{y}_l$ and $V_2=V_{02}-A{y}_l$ denote the respective volumes of the two chambers, with V₀₁ and V₀₂ standing for the initial respective volumes of the two chambers; ${\beta }_e$ is the effective oil bulk modulus; $C_t(t)$ is the time-varying internal leakage coefficient; $q_1$ and $q_2$ denote, respectively, the supplied and return flow rate values; and ${\Delta }_1(t)$ and ${\Delta }_2(t)$ denote the unknown uncertainties.

Because a high-performance servo valve is utilized in the considered hydraulic system, the dynamics of the valve can be omitted. Therefore, and following [29], the flow rate values $q_1$ and $q_2$ can be obtained as follows:

$$\begin{array}{l}{q}_1=k_{q}u\left[sg(u)\sqrt{\left|P_s-P_1\right|}+sg(-u)\sqrt{\left|P_1-P_r\right|}\right]\\ q_2=k_{q}u\left[sg(u)\sqrt{\left|P_2-P_r\right|}+sg(-u)\sqrt{\left|P_s-P_2\right|}\right]\end{array}$$

(3)

where k_q is the flow gain, and $sg(\ast )$ is

$$sg(\ast )=\left\{\begin{array}{l}1,\ast \ge 0\\ 0,\ast <0\end{array}\right.$$

(4)

From (2), we get

$$\begin{array}{l}A({\dot{P}}_1-{\dot{P}}_2)={\displaystyle \frac{A{\beta }_e}{V_1}}\left[-A{\dot{y}}_l-C_t(t)(P_1-P_2)+{\Delta }_1(t)+q_1\right]\\ -{\displaystyle \frac{A{\beta }_e}{V_2}}\left[A{\dot{y}}_l-C_t(t)(P_2-P_1)+{\Delta }_2(t)-q_2\right]\\ =A{\beta }_e({\displaystyle \frac{1}{V_1}}{q}_1+{\displaystyle \frac{1}{V_2}}{q}_2)-{\beta }_e{A}^2({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}}){\dot{y}}_l\\ -{\beta }_{e}A({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}})C_t(t)(P_1-P_2)+A{\beta }_e\left[{\displaystyle \frac{{\Delta }_1(t)}{V_1}}+{\displaystyle \frac{{\Delta }_2(t)}{V_2}}\right]\end{array}$$

(5)

The following can also be obtained:

$$\begin{array}{l}A({\dot{P}}_1-{\dot{P}}_2)=k_{q}A{\beta }_e({\displaystyle \frac{S_1}{V_1}}+{\displaystyle \frac{S_2}{V_2}})u-{\beta }_e{A}^2({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}}){\dot{y}}_l\\ -{\beta }_{e}A({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}})C_t(t)(P_1-P_2)+A{\beta }_e\left[{\displaystyle \frac{{\Delta }_1(t)}{V_1}}+{\displaystyle \frac{{\Delta }_2(t)}{V_2}}\right]\end{array}$$

(6)

where

$$\begin{array}{l}{S}_1=sg(u)\sqrt{\left|P_s-P_1\right|}+sg(-u)\sqrt{\left|P_1-P_r\right|}\\ S_2=sg(u)\sqrt{\left|P_2-P_r\right|}+sg(-u)\sqrt{\left|P_s-P_2\right|}\end{array}$$

(7)

Using (1) and (6), constructing the variables $\varsigma =\left[{\varsigma }_1;{\varsigma }_2;{\varsigma }_3\right]=\left[y_l;{\dot{y}}_l;A(P_1-P_2)/m_l\right]$ yields

$$\left\{\begin{array}{l}{\dot{\varsigma }}_1={\varsigma }_2\\ {\dot{\varsigma }}_2={\varsigma }_3-{\displaystyle \frac{a_1(t){\varsigma }_2+a_2(t)\tanh (c_1{\varsigma }_2)+a_3(t)\left[\tanh (c_2{\varsigma }_2)-\tanh (c_3{\varsigma }_2)\right]}{m_l}}+D_1(t)\\ {\dot{\varsigma }}_3=G_{1}u-G_2-G_3{C}_t(t)+D_2(t)\end{array}\right.$$

(8)

in which

$$\left\{\begin{array}{l}{D}_1(t)={\displaystyle \frac{f(t)}{m_l}},D_2(t)={\displaystyle \frac{A{\beta }_e}{m_l}}\left[{\displaystyle \frac{{\Delta }_1(t)}{V_1}}+{\displaystyle \frac{{\Delta }_2(t)}{V_2}}\right]\\ G_1={\displaystyle \frac{k_{q}A{\beta }_e}{m_l}}({\displaystyle \frac{S_1}{V_1}}+{\displaystyle \frac{S_2}{V_2}}),G_2={\beta }_e{A}^2({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}}){\displaystyle \frac{{\varsigma }_2}{m_l}},G_3={\beta }_e({\displaystyle \frac{1}{V_1}}+{\displaystyle \frac{1}{V_2}}){\varsigma }_3\end{array}\right.$$

(9)

Uniting (9), and defining the unknown time-varying parameter set $\theta (t)={\left[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6\right]}^T={\left[a_1(t),a_2(t),a_3(t),D_1(t),C_t(t),D_2(t)\right]}^T$, we obtain

$$\left\{\begin{array}{l}{\dot{\varsigma }}_1={\varsigma }_2\\ {\dot{\varsigma }}_2={\varsigma }_3-{\theta }^T(t)H_1\\ {\dot{\varsigma }}_3=G_{1}u-G_2-{\theta }^T(t)H_2\end{array}\right.$$

(10)

in which

$$\left\{\begin{array}{l}{H}_1={\left[{\displaystyle \frac{{\varsigma }_2}{m_l}},{\displaystyle \frac{\tanh (c_1{\varsigma }_2)}{m_l}},{\displaystyle \frac{\tanh (c_2{\varsigma }_2)-\tanh (c_3{\varsigma }_2)}{m_l}},1,0,0\right]}^T\\ H_2={\left[0,0,0,0,G_3,1\right]}^T\end{array}\right.$$

(11)

Our objective is to design a control input to obtain both the prescribed transient performance and the asymptotic steady performance for hydraulic systems subject to both unknown time-varying parameters and disturbances under Assumption 1, as follows:

Assumption 1.

The reference command ${\varsigma }_r$ meets $\left\{\left({\varsigma }_r,{\dot{\varsigma }}_r,{\ddot{\varsigma }}_r\right)\left|{\varsigma }_r^2+{\dot{\varsigma }}_r^2+{\ddot{\varsigma }}_r^2\le r_0\right.\right\}$ with $r_0>0$. The unknown time-varying parameter set $\theta (t)$ meets $‖\theta (t)‖\le r_1$ with $r_1>0$.

3. Adaptive Prescribed Performance Controller Design

3.1. Error Variable Transformation and Prescribed Performance Function Transformation

First, we construct the following error variables:

$$z_1={\varsigma }_1-{\varsigma }_r,z_2={\varsigma }_2-{\alpha }_{1f},z_3={\varsigma }_3-{\alpha }_{2f}$$

(12)

where ${\alpha }_i$ ($i=1,2$) denote the virtual controls; and ${\alpha }_{if}$ denote filtered signals of ${\alpha }_i$.

To get the filtering signals ${\alpha }_{if}$, a set of nonlinear filters is presented as follows:

$${\dot{\alpha }}_{if}=-{\omega }_i{\epsilon }_i-{\displaystyle \frac{{\epsilon }_i{\gamma }_i^2}{{\epsilon }_i{\gamma }_i\tanh \left[{\epsilon }_i/h(t)\right]+h(t)}},{\alpha }_{if}(0)={\alpha }_i(0),i=1,2$$

(13)

in which ${\epsilon }_i={\alpha }_{if}-{\alpha }_i$ denote the filtering errors; ${\omega }_i>0$; $h(t)>0$ meets ${\displaystyle {\int }_0^{t}h(\nu )}\mathrm{d}\nu \le \overline{h}<+\infty$, $\forall t\ge 0$, with a constant $\overline{h}>0$; and ${\gamma }_i>0$.

To achieve the full-state prescribed transient performance of all the control errors, the following performance indexes hold:

$$-{\Phi }_i(t)<z_i<{\Phi }_i(t),i=1,2,3$$

(14)

with the prescribed performance functions ${\Phi }_i(t)$ being designed as

$${\Phi }_i(t)=({\Phi }_{i0}-{\Phi }_{i\infty })e^{-ct}+{\Phi }_{i\infty },i=1,2,3$$

(15)

where ${\Phi }_{i0}$, ${\Phi }_{i\infty }$, and c are positive constants; and ${\Phi }_{i0}>{\Phi }_{i\infty }$.

To promote the subsequent controller design, we define the transformation variables as follows:

$${\chi }_i=z_i/{\Phi }_i(t),i=1,2,3$$

(16)

3.2. Controller Design

The specific prescribed performance controller design procedure is provided below.

Step 1: Differentiating ${\chi }_1$ in (16) yields

$$\begin{array}{l}{\dot{\chi }}_1={\displaystyle \frac{{\dot{z}}_1{\Phi }_1(t)-z_1{\dot{\Phi }}_1(t)}{{\Phi }_1^2(t)}}\\ ={\displaystyle \frac{{\Phi }_2(t){\chi }_2+{\alpha }_1+{\epsilon }_1-{\dot{\varsigma }}_r-{\chi }_1{\dot{\Phi }}_1(t)}{{\Phi }_1(t)}}\end{array}$$

(17)

Choosing the Lyapunov Function $L_1={\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{1}{1-{\chi }_1^2}}$ yields

$${\dot{L}}_1={\displaystyle \frac{{\chi }_1{\dot{\chi }}_1}{1-{\chi }_1^2}}={\displaystyle \frac{{\chi }_1\left[{\Phi }_2(t){\chi }_2+{\alpha }_1+{\epsilon }_1-{\dot{\varsigma }}_r-{\chi }_1{\dot{\Phi }}_1(t)\right]}{{\Phi }_1(t)(1-{\chi }_1^2)}}$$

(18)

The virtual control ${\alpha }_1$ is designed as follows:

$${\alpha }_1={\dot{\varsigma }}_r+{\chi }_1{\dot{\Phi }}_1(t)-{\epsilon }_1-{\varpi }_1{k}_1(1-{\chi }_1^2){\chi }_1$$

(19)

where ${\varpi }_1>0$ and $k_1>0$.

Putting (19) into (18), we get

$${\dot{L}}_1={\displaystyle \frac{{\chi }_1{\dot{\chi }}_1}{1-{\chi }_1^2}}={\displaystyle \frac{{\Phi }_2(t){\chi }_2{\chi }_1}{{\Phi }_1(t)(1-{\chi }_1^2)}}-{\displaystyle \frac{{\varpi }_1{k}_1{\chi }_1^2}{{\Phi }_1(t)}}$$

(20)

Step 2: Differentiating ${\chi }_2$ in (16) yields

$$\begin{array}{l}{\dot{\chi }}_2={\displaystyle \frac{{\dot{z}}_2{\Phi }_2(t)-z_2{\dot{\Phi }}_2(t)}{{\Phi }_2^2(t)}}\\ ={\displaystyle \frac{{\Phi }_3(t){\chi }_3+{\alpha }_2+{\epsilon }_2-{\theta }^T(t)H_1-{\dot{\alpha }}_{1f}-{\chi }_2{\dot{\Phi }}_2(t)}{{\Phi }_2(t)}}\end{array}$$

(21)

Choosing the Lyapunov Function $L_2=L_1+{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{1}{1-{\chi }_2^2}}$ yields

$${\dot{L}}_2={\displaystyle \frac{{\chi }_2\left[{\Phi }_3(t){\chi }_3+{\alpha }_2+{\epsilon }_2-{\theta }^T(t)H_1-{\dot{\alpha }}_{1f}-{\chi }_2{\dot{\Phi }}_2(t)\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\Phi }_2(t){\chi }_2{\chi }_1}{{\Phi }_1(t)(1-{\chi }_1^2)}}-{\displaystyle \frac{{\varpi }_1{k}_1{\chi }_1^2}{{\Phi }_1(t)}}$$

(22)

The virtual control ${\alpha }_2$ is designed as follows:

$${\alpha }_2={\widehat{\theta }}^T{H}_1+{\dot{\alpha }}_{1f}+{\chi }_2{\dot{\Phi }}_2(t)-{\epsilon }_2-{\varpi }_2{k}_2(1-{\chi }_2^2){\chi }_2-{\displaystyle \frac{{\Phi }_2^2(t){\chi }_1(1-{\chi }_2^2)}{{\Phi }_1(t)(1-{\chi }_1^2)}}+{\alpha }_{2s}$$

(23)

where ${\varpi }_2>0$ and $k_2>0$; $\widehat{\theta }$ denotes the estimate of $\theta$; and the term ${\alpha }_{2s}$ is denoted as

$${\alpha }_{2s}=-{\displaystyle \frac{{\chi }_2{\delta }_2^2}{{\chi }_2{\delta }_2\tanh \left[{\chi }_2/h(t)\right]+{\Phi }_2(t)(1-{\chi }_2^2)h(t)}}$$

(24)

where ${\delta }_2>0$.

Putting (23) into (22), we get

$${\dot{L}}_2=-{\displaystyle \sum _{i=1}^2\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\Phi }_3(t){\chi }_3+{\alpha }_{2s}+{\widehat{\theta }}^T(t)H_1-{\theta }^T(t)H_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}$$

(25)

Step 3: Differentiating ${\chi }_3$ in (16) yields

$$\begin{array}{l}{\dot{\chi }}_3={\displaystyle \frac{{\dot{z}}_3{\Phi }_3(t)-z_3{\dot{\Phi }}_3(t)}{{\Phi }_3^2(t)}}\\ ={\displaystyle \frac{G_{1}u-G_2-{\theta }^T(t)H_2-{\dot{\alpha }}_{2f}-{\chi }_3{\dot{\Phi }}_3(t)}{{\Phi }_3(t)}}\end{array}$$

(26)

We choose the Lyapunov Function, as follows:

$$L_3=L_2+{\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{1}{1-{\chi }_3^2}}+{\displaystyle \frac{1}{2}}{({\theta }_{\max }-\widehat{\theta })}^T{\Gamma }^{-1}({\theta }_{\max }-\widehat{\theta })$$

(27)

where ${\theta }_{\max }$ is composed of the maximum value of each element in the set $\theta$; and $\Gamma \in {ℝ}^{6\times 6}$ denotes the adjustable positive parameter adaption matrix.

We therefore obtain the following:

$$\begin{array}{l}{\dot{L}}_3=-{\displaystyle \sum _{i=1}^2\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\Phi }_3(t){\chi }_3+{\alpha }_{2s}+{\widehat{\theta }}^T(t)H_1-{\theta }^T(t)H_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}\\ +{\displaystyle \frac{{\chi }_3\left[G_{1}u-G_2-{\theta }^T(t)H_2-{\dot{\alpha }}_{2f}-{\chi }_3{\dot{\Phi }}_3(t)\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}-{\displaystyle \frac{1}{2}}{({\theta }_{\max }-\widehat{\theta })}^T{\Gamma }^{-1}\dot{\widehat{\theta }}\end{array}$$

(28)

The control input $u$ is designed as follows:

$$u={\displaystyle \frac{G_2+{\widehat{\theta }}^T{H}_2+{\dot{\alpha }}_{2f}+{\chi }_3{\dot{\Phi }}_3(t)-{\varpi }_3{k}_3(1-{\chi }_3^2){\chi }_3+u_s}{G_1}}-{\displaystyle \frac{{\Phi }_3^2(t){\chi }_2(1-{\chi }_3^2)}{G_1{\Phi }_2(t)(1-{\chi }_2^2)}}$$

(29)

where ${\varpi }_3>0$ and $k_3>0$; and the term $u_s$ is denoted as

$$u_s=-{\displaystyle \frac{{\chi }_3{\delta }_3^2}{{\chi }_3{\delta }_3\tanh \left[{\chi }_3/h(t)\right]+{\Phi }_3(t)(1-{\chi }_3^2)h(t)}}$$

(30)

where ${\delta }_3>0$.

Putting (29) into (28), we get

$$\begin{array}{l}{\dot{L}}_3=-{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}+{\widehat{\theta }}^T(t)H_1-{\theta }^T(t)H_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}\\ +{\displaystyle \frac{{\chi }_3\left[u_s+{\widehat{\theta }}^T(t)H_2-{\theta }^T(t)H_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}-{({\theta }_{\max }-\widehat{\theta })}^T{\Gamma }^{-1}\dot{\widehat{\theta }}\end{array}$$

(31)

Defining ${\epsilon }_{\theta }=\theta (t)-{\theta }_{\max }$, accordingly, we get

$$\begin{array}{l}{\dot{L}}_3=-{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}\\ -{({\theta }_{\max }-\widehat{\theta })}^T{\Gamma }^{-1}\left[\dot{\widehat{\theta }}+{\displaystyle \frac{\Gamma H_1{\chi }_2}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{\Gamma H_2{\chi }_3}{{\Phi }_3(t)(1-{\chi }_3^2)}}\right]\end{array}$$

(32)

We construct the following parameter updated law:

$$\dot{\widehat{\theta }}=-{\displaystyle \frac{\Gamma H_1{\chi }_2}{{\Phi }_2(t)(1-{\chi }_2^2)}}-{\displaystyle \frac{\Gamma H_2{\chi }_3}{{\Phi }_3(t)(1-{\chi }_3^2)}}-\kappa \Gamma \widehat{\theta }h(t)$$

(33)

where $\kappa >0$.

The following is now obtained:

$${\dot{L}}_3=-{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}+\kappa {({\theta }_{\max }-\widehat{\theta })}^T\widehat{\theta }h(t)$$

(34)

The developed controller diagram of the hydraulic system is presented below (Figure 2).

3.3. Main Result and Stability Analysis

The main result of this paper is provided below.

Theorem 1.

Under Assumption 1, the control law (29) and the parameter updated law (33), by picking up the proper control parameters $k_i$, ${\omega }_j$, $\Gamma$, $\kappa$, ${\Phi }_{i0}$, ${\Phi }_{i\infty }$, and c for $i=1,2,3$ and $j=1,2$, we determine the following: (1) the boundedness of all the closed-loop signals holds; (2) both the prescribed transient performance and the asymptotic steady performance of all the control errors hold.

Proof of Theorem 1.

We construct a new Lyapunov Function, as follows:

$$L=L_3+{\displaystyle \frac{1}{2}}{\epsilon }_1^2+{\displaystyle \frac{1}{2}}{\epsilon }_2^2$$

(35)

According to (13), (34), and (35), we obtain

$$\begin{array}{l}\dot{L}=-{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}+\kappa {({\theta }_{\max }-\widehat{\theta })}^T\widehat{\theta }h(t)\\ -{\displaystyle \sum _{i=1}^2{\omega }_i{\epsilon }_i^2}-{\displaystyle \sum _{i=1}^2\frac{{\epsilon }_i^2{\gamma }_i^2}{{\epsilon }_i{\gamma }_i\tanh \left[{\epsilon }_i/h(t)\right]+h(t)}}-{\displaystyle \sum _{i=1}^2{\epsilon }_i{\dot{\alpha }}_i}\end{array}$$

(36)

We note that

$$\begin{array}{l}{({\theta }_{\max }-\widehat{\theta })}^T\widehat{\theta }=-{({\theta }_{\max }-\widehat{\theta })}^T({\theta }_{\max }-\widehat{\theta })+{({\theta }_{\max }-\widehat{\theta })}^T{\theta }_{\max }\\ \le -{({\theta }_{\max }-\widehat{\theta })}^T({\theta }_{\max }-\widehat{\theta })+{({\theta }_{\max }-\widehat{\theta })}^T({\theta }_{\max }-\widehat{\theta })+{{\theta }_{\max }}^T{\theta }_{\max }/4\\ \le {{\theta }_{\max }}^T{\theta }_{\max }/4\end{array}$$

(37)

We now obtain

$$\begin{array}{l}\dot{L}=-{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}+M_{1}h(t)\\ -{\displaystyle \sum _{i=1}^2{\omega }_i{\epsilon }_i^2}-{\displaystyle \sum _{i=1}^2\frac{{\epsilon }_i^2{\gamma }_i^2}{{\epsilon }_i{\gamma }_i\tanh \left[{\epsilon }_i/h(t)\right]+h(t)}}-{\displaystyle \sum _{i=1}^2{\epsilon }_i{\dot{\alpha }}_i}\end{array}$$

(38)

where $M_1=\kappa {{\theta }_{\max }}^T{\theta }_{\max }/4$.

From Assumption 1, the compact set ${\Xi }_1=\left\{\left({\varsigma }_r,{\dot{\varsigma }}_r,{\ddot{\varsigma }}_r\right)\left|{\varsigma }_r^2+{\dot{\varsigma }}_r^2+{\ddot{\varsigma }}_r^2\le r_0\right.\right\}$ holds. Also, the compact set ${\Xi }_2=\left\{V(t)\left|V(t)\le r_2\right.\right\}$ holds with $r_2>0$. Hence, the set ${\Xi }_1\times {\Xi }_2$ is compact. Accordingly, there are positive constants ${\gamma }_1$ and ${\gamma }_2$ such that $\left|{\dot{\alpha }}_1\right|\le {\gamma }_1$ and $\left|{\dot{\alpha }}_2\right|\le {\gamma }_2$ on ${\Xi }_1\times {\Xi }_2$.

Hence, we get

$$\begin{array}{l}\dot{L}\le -{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}+{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}+M_{1}h(t)\\ -{\displaystyle \sum _{i=1}^2{\omega }_i{\epsilon }_i^2}-{\displaystyle \sum _{i=1}^2\frac{{\epsilon }_i^2{\gamma }_i^2}{{\epsilon }_i{\gamma }_i\tanh \left[{\epsilon }_i/h(t)\right]+h(t)}}+{\displaystyle \sum _{i=1}^2{\gamma }_i\left|{\epsilon }_i\right|}\end{array}$$

(39)

The following inequalities hold:

$$\left\{\begin{array}{l}{\displaystyle \frac{{\chi }_2\left[{\alpha }_{2s}-{{\epsilon }_{\theta }}^T{H}_1\right]}{{\Phi }_2(t)(1-{\chi }_2^2)}}\le h(t),{\displaystyle \frac{{\chi }_3\left[u_s-{{\epsilon }_{\theta }}^T{H}_2\right]}{{\Phi }_3(t)(1-{\chi }_3^2)}}\le h(t)\\ -{\displaystyle \sum _{i=1}^2\frac{{\epsilon }_i^2{\gamma }_i^2}{{\epsilon }_i{\gamma }_i\tanh \left[{\epsilon }_i/h(t)\right]+h(t)}}+{\displaystyle \sum _{i=1}^2{\gamma }_i\left|{\epsilon }_i\right|}\le 2h(t)\end{array}\right.$$

(40)

We now obtain

$$\begin{array}{l}\dot{L}\le -{\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+M_{2}h(t)-{\displaystyle \sum _{i=1}^2{\omega }_i{\epsilon }_i^2}\\ =-\Omega +M_{2}h(t)\end{array}$$

(41)

where $\Omega ={\displaystyle \sum _{i=1}^3\frac{{\varpi }_i{k}_i{\chi }_i^2}{{\Phi }_i(t)}}+{\displaystyle \sum _{i=1}^2{\omega }_i{\epsilon }_i^2}$ and $M_2=M_1+4$.

Integrating the two sides of (41), we get

$$L\left(t\right)+{\displaystyle {\int }_0^t\Omega \left(\nu \right)}d\nu \le L\left(0\right)+M_2{\displaystyle {\int }_0^{t}h(\nu )}d\nu \le L\left(0\right)+M_2\overline{h}$$

(42)

Hence, the boundedness of all the system signals holds, and $\Omega$ is uniformly continuous. Using Barbalat’s lemma [30], $\Omega \to 0$ as $t\to \infty$. Thus, all the control errors asymptotically can converge to zero. From (35), one obtains $\left|{\chi }_i\right|<1$ on ${\Xi }_1\times {\Xi }_2$ for $i=1,2,3$. Accordingly, one obtains $-{\Phi }_i(t)<z_i<{\Phi }_i(t)$.

Consequently, Theorem 1 holds. □

4. Simulation Results

In this section, the effectiveness of the designed controller is verified through simulation comparison using the software Matlab/Simulink 2016a. The time-varying parameters of the system are set as $a_1(t)=4000+20\sin (t)\mathrm{N}·\mathrm{s}/\mathrm{m}$, $a_2(t)=100+5\sin (t)\mathrm{N}·\mathrm{s}/\mathrm{m}$, $a_2(t)=100+2\sin (t)\mathrm{N}·\mathrm{s}/\mathrm{m}$, and $C_t(t)=7\times {10}^{-12}+{10}^{-13}\sin (t){\mathrm{m}}^5/\mathrm{N}/\mathrm{s}$; the disturbances are set as $d_1(t)=300+10\sin (0.5\pi t)\mathrm{N}$, ${\Delta }_1(t)=-{10}^{-7}\sin (\pi t){\mathrm{m}}^3/\mathrm{s}$, and ${\Delta }_2(t)={10}^{-7}\sin (\pi t){\mathrm{m}}^3/\mathrm{s}$. The other physical parameters of the hydraulic system are presented in

Table 1. Physical parameters of the hydraulic system.

Parameter	Value	Parameter	Value
ml (kg)	30	Ps (MPa)	10
Pr (MPa)	0.1	A (mm2)	904.778
V01 (m3)	3.98 × 10−5	V02 (m3)	3.98 × 10−5
${\beta }_e$ (MPa)	700	kq (${\mathrm{m}}^4/(\mathrm{s}\cdot \mathrm{V}\cdot \sqrt{\mathrm{N}})$)	7.937 × 10−8

. The sample time is set to 0.5 ms. The simulation time is set to 40 s.

Two controllers are compared to validate the effectiveness of the developed controller in the article.

(1) AAPPC: This is the controller proposed in this paper, whose controller parameters are as follows: $k_1=500$, $k_2=300$, $k_3=200$, ${\omega }_1={\omega }_2=2000$, ${\gamma }_1={\gamma }_2=0.0001$, $\kappa =1\times {10}^{-7}$, $\Gamma =diag\{4\times {10}^7,2\times {10}^4,7\times {10}^5,70,5\times {10}^{-10},1\times {10}^6\}$, ${\varpi }_1={\varpi }_2={\varpi }_3=2$, ${\Phi }_{10}=0.0001$, ${\Phi }_{1\infty }=0.00005$, ${\Phi }_{20}=0.4$, ${\Phi }_{2\infty }=0.2$, ${\Phi }_{30}=150$, ${\Phi }_{3\infty }=50$, $c=1.5$, and $h(t)=1/(t^2+500)$. Also, the initial value of the unknown time-varying parameter set is set as $\widehat{\theta }(0)=[2000;1;1;0;1\times {10}^{-13};0]$.

(2) AAC: This controller is an adaptive controller without prescribed performance functions. The corresponding control parameters of AAC are the same as those of AAPPC. The goal of the comparison with AAC is to validate the effectiveness of full-state prescribed performance functions in AAPPC.

Two simulation cases were run to validate the effectiveness of the controller proposed in this paper.

Case 1: First a step-like reference command was employed to test the control performance of the developed control method. The corresponding simulation results are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. It can be seen from Figure 3 that the developed AAPPC can achieve excellent asymptotic tracking performance. The curves of tracking errors between AAPPC and AAC are shown in Figure 4, in which it can be seen that the steady-state tracking error of AAPPC controller is approximately 1 × 10⁻³ mm while that of AAC is approximately 2 × 10⁻³ mm. It is also obvious that AAC has good asymptotic tracking performance and steady-state tracking accuracy, owing to the employment of the time-varying parameter adaptive technique in AAPPC. In contrast, when the prescribed performance function is not used to constrain the control errors, the initial transient of its tracking error z₁ exceeds the set safety performance bounds in AAC. Furthermore, the tracking error z₁ of AAPPC is always within the set safety performance boundary range, which verifies the effectiveness of the prescribed performance mechanism in AAPPC. In addition, it can be seen in Figure 5 and Figure 6 that the estimates of the unknown time-varying parameters and disturbances of the system both have good convergence performance, thereby verifying the effectiveness of the designed parameter adaptive law in AAPPC. The control errors z₂ and z₃ of AAPPC are shown in Figure 7 and Figure 8, respectively, in which it can be seen that z₂ and z₃ are always within the set safety performance boundary range. The control input curve in AAPPC is shown in Figure 9. Its amplitude is within ±0.2 V, and it is smooth and continuous, which is conducive to actual application of the controller.

Case 2: To further verify the dynamic tracking performance of the designed AAPPC, a sine-like reference command was run. The corresponding simulation results are shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. It can be seen from Figure 10 that AAPPC still delivers excellent asymptotic tracking performance, with a steady-state tracking error of approximately 2 × 10⁻³ mm. The tracking error curves of AAPPC and AAC are shown in Figure 11, in which it can be seen that the tracking error z₁ of AAPPC is still always within the set safety performance boundary and surpasses that of AAC, verifying the performance consistency of the controller proposed in this paper. In addition, it can be seen in Figure 12 and Figure 13 that the estimates of the unknown time-varying parameters and disturbances of the system both still have good convergence performance, thereby verifying the effectiveness of the designed parameter adaptive law in AAPPC. The control errors z₂ and z₃ of AAPPC are shown in Figure 14 and Figure 15, respectively, in which z₂ and z₃ can be seen to be still within the set safety performance boundary range. In Figure 16, it can be seen that the control input curve in AAPPC is smooth and continuous. Consequently, Case 2 also verifies the merits of the presented control method.

5. Conclusions

In this paper, we present an adaptive full-state prescribed performance position tracking controller for hydraulic systems, in which both the transient and steady performance of all the control errors can be prescribed. First, a smooth nonlinear term was skillfully introduced into the controller design so that it could simultaneously cope with both unknown time-varying parameters and disturbances. Next, by co-designing the Lyapunov Function, we enabled both the prescribed transient performance and the asymptotic steady performance of all the control errors to be achieved with the exploited controller. Finally, the merits of the proposed controller were also validated by comparative simulation results. Compared to an existing adaptive controller without full-state prescribed performance functions, the steady tracking accuracy of the hydraulic system was found to increase by 50%.

However, the adaptive full-state prescribed performance position tracking controller for hydraulic systems developed in the present study relies on the initial values of all the control errors. This may lead to restrictions in application owing to the unknown initial state values. In the future, the possibility of an initial-restriction-free full-state prescribed performance position tracking controller for hydraulic systems may be worth studying.