Optimal analysis of VSL following-up performance considering hydraulic system SAD control strategy

3.1. Quadratic optimal control analysis with linearity

The principle of optimal control is to find a control variable $u\left(t\right)$ that has a small value and can satisfy the minimum system error $x\left(t\right)$, so that the output of the system quickly follows the input and the energy consumption is low. The minimum value of the index can be obtained from the Pontryagin principle, and the essence of the quadratic optimal control is the approximation of the feedback $K\left(t\right)$ of the original system. In this study, using the feedback of the optimal regulator to approximate the optimization, the simplified optimal control rate can be expressed as:

The algebraic equation for the Riccati matrix can be given as:

where, $A$ denots the EHS system matrix, $B$ is the control matrix of the EHS system, $Q$ and $R$ are the weighting matrices of the EHS system, herein $Q=$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{\mathrm{275000,1},1\right\}$, $R=$0.0001, $P$ is the numerical solution of the equation.

Therefore, the optimized system matrix can be described as:

Using Matlab to simulate the transfer function under quadratic optimal control, the step response curve of the EHS system after quadratic optimal control with linearity is revealed, as shown in Fig. 5. It can be seen from Fig. 5 that the EHS control system can reach stability in 4.8 s. After the control system undergoes secondary optimization control, the stabilization time is shortened by 15.3 s, and the control performance is greatly improved, but unfortunately the time is relatively long.

Fig. 5Step response curve of the EHS system after quadratic optimal control with linearity

3.2. Pole optimized configuration

Introducing the state feedback gain matrix $K=\left[\begin{array}{lll}{K}_1& K_2& K_3\end{array}\right]$, then the characteristic polynomial can be expressed as:

The core content of analyzing the performance of the control system is to determine the dominant pole, where the effect of the far pole is ignored. Therefore, the EHS control system is considered to be equivalent to a second-order control system containing a main pair of poles. Determining the location of the expected dominant pole from the representation of the dynamic indicators ${\sigma }_p$ % and $t_s$, it will be as follows:

where, ${\sigma }_p$ % represents the maximum overshoot, $\xi$ is the damping (0 $<\xi <$ 1). Assuming that the allowable error of the EHS control system is set to 5 %, there is:

where, $t_s$ is the adjustment time (s), ${\omega }_n$ denots the undamped frequency (rad/s). The expected dominant pole of an EHS system can be expressed as:

where, ${\lambda }_{\mathrm{1,2}}$ are the EHS system expects to dominate the poles. Considering that the maximum overshoot of the EHS control system is limited to ${\sigma }_p$% ≤ 5 %, and the adjustment time is set to $t_s\le$ 0.5 s, the expected dominant poles (${\lambda }_1$ and ${\lambda }_2$) of the EHS system can be given by the following Eq. (18). Analyzing this equation, the following settings are specified:

Herein, the clear value is $\xi =$ 0.707, $\xi {\omega }_n=$ 6.0 = 6, then there is:

Let the third pole ${\lambda }_3=10R_e\left[{\lambda }_1\right]=-60$ be set, so the following expression is:

Established by $\alpha \left(s\right)={\alpha }^{\mathrm{*}}\left(s\right)$, and then obtain the following expression:

Fig. 6 shows the step response curve of the pole optimal configured EHS control system obtained by the Simulink module simulation. As can be seen from Fig. 6, the EHS system can be stabilized within 1.3 seconds. After the EHS control system is optimized by the pole configuration, the stabilization time is shortened by 20.3 seconds, and the control performance is significantly improved, but the overshoot phenomenon occurs prematurely.

Fig. 6Step response curve of the EHS system after pole optimal configured control

3.3. Simulation analysis of classical PID control

The simple structure and convenient parameter calibration of classical PID control make it widely used in many fields. In this chapter, the proportional coefficient $K_p$ is firstly determined, and the integral constant and differential constant are set to zero. The proportional gain gradually increases from zero until the response curve oscillates, and then the value decreases slowly until the oscillation stops, 30 %-70 % of the proportional gain is used as the system proportional coefficient, and 60 % is taken in this study. The proportional coefficient is determined, and the larger value is taken as the initial value of the system integral constant. After several adjustments, the system stops oscillating, and 150 %-180 % of the integral constant is taken as the final value of the integral constant of the system, which is set as 165 %. In the case of EHS system output force, the differential constant $K_d$ is usually chosen to be 0. The parameters of the classical PID controller are selected as $K_p=$11.17, $K_i=$63.11, $K_d=$0. The step response curve of classic PID control of EHS system simulated by Simulink is shown in Fig. 7.

In Fig. 8, the EHS system has some improved performance through PID control. When a unit step signal is input, the EHS system tends to a stable state within 1.55 s, and its input and output signals have good matching performance, but the EHS system is also accompanied by a certain amount of overshoot.

Fig. 7Step response curve of the classic PID control of the EHS system simulated by Simulink

3.4. Simulation analysis of SAD control

To further improve the control effect of the control system, on the basis of the SAD control algorithm, the classical PID is comprehensively optimized, and the optimal scheme of the overall performance of the system is proposed, and the SAD controller is analyzed and designed in combination with the specific structure. The designed block diagram of the proposed SAD third-order controller in this paper is shown in Fig. 8.

Fig. 8Designed block diagram of the proposed SAD third-order controller

The SAD controller consists of three modules, the control parameter Following Differentiator (TD), the error feedback Extended State Controller (ESC) and the EHS Control Law (CL). Based on the system state space expression, the controlled object is set to be third-order, where the TD is third-order, and the ESC is fourth-order. Seeking to improve the dynamics of this control system, use state feedback to configure the desired poles. If the zero-point configuration of the control system can be realized, the dynamic performance of the control system can be improved. However, the zero point configuration needs to use the differential signal. The differential cannot filter out noise, nor is it used for zero point configuration, while the following differential can filter out the noise well, and the differential signal is zero point configuration. The TD follows a given signal and obtains its differential equation. The transition process is arranged by TD to prevent the overshoot caused by the sudden change of the given signal, so that the controlled objects approach the target smoothly, thereby improving the stability of the control system. By eliminating the poles of the system with the configured zeros, the system can be viewed as a control system close to unity. This sets the string level object to the following expression:

Assuming that $p_2\left(s\right)$ is known, the virtual control volume is:

The above Eq. (21) can be written as:

Once the virtual control quantity $U\left(t\right)$ is clarified, the actual control quantity acts as:

where, $p_1\left(s\right)$ and $p_2\left(s\right)$ are polynomials.

Fig. 9Zero-pole diagram of the controlled object exported by Matlab

The virtual control volume $U\left(t\right)$ is determined by applying a improved controller design of Self-Turbulent Flow (STF) algorithm approach to the control subsystem $y=U\left(t\right)/P_1\left(s\right)$, which thus allows for a certain range of uncertainties to exist. Assuming that the mathematical model of the controlled object contains zeros and poles, its general form can be described as:

Set the virtual control quantity $U\left(t\right)=q\left(s\right)u$, then the system becomes $y=U\left(t\right)/P\left(s\right)$, then the actual control quantity becomes $u=U\left(t\right)/q\left(s\right)$. If the controlled object conforms to the minimum phase system, the $q\left(s\right)$ belongs to a stable polynomial and is known. The SAD controller can be used to determine the virtual control quantity $U\left(t\right)$, and the actual control quantity $u\left(t\right)$ can be obtained by solving the system $u=U\left(t\right)/q\left(s\right)$ with $U\left(t\right)$ as the input. Through the above analysis, the zero-pole diagram of the controlled object is derived using Matlab, as shown in Fig. 9.

As can be seen from Fig. 9, the system model has no zeros and poles in the positive part of the complex plane and is considered as a minimum phase system. In this case, the virtual control variable $U\left(t\right)$ is determined by the active disturbance rejection controller, and $U\left(t\right)$ is used as the input to solve the system $u=U\left(t\right)/q\left(s\right)$ to obtain the actual control variable. The transfer function of the control system is known to satisfy the cubic polynomial $p_3\left(s\right)=s_3+a_3{s}_2+a_{2}s+a_1$. The parameters $a_1$, $a_2$ and $a_3$ are known, and the STF algorithm of the third-order TD differentiator is written as:

where, $r$ is a fast factor, $h$ denotes the sampling step size.

The error feedback Extended State Controller (ESC) follows the output of the control system, monitors the state of each stage of the research object and the total perturbation of the EHS system in real time, and then compensates the perturbation accordingly. If the system suffers from unknown disturbances, the control object is uncertain. This indeterminate object is expressed as:

where, $f(x,\dot{x},\cdot \cdot \cdot ,x^{(n-1)},t)$ is system unknown function, $w\left(t\right)$ is the unknown perturbation, $b$ is a control gain, $u$is the system input, $y$ is the system output.

The error feedback ESC in this paper is fourth-order. Under linear conditions, the fourth-order ESC algorithm of the control system is written as:

where, ${\beta }_{01}$, ${\beta }_{02}$, ${\beta }_{03}$, ${\beta }_{04}$ are the ESC parameters, $e$ is the feedback error term.

The transfer relationship from input to output of the error feedback ESC is expressed as:

In the control system, the ESC contains four output variables, $z_1$ follows the system output $y$, $z_2$ follows $y^{\text{'}}$, $z_3$ follows $y^{\text{'}\text{'}}$, and $z_4$ follows the system integral disturbance, and the disturbance is compensated by the feedforward method. In order to achieve a predetermined estimation accuracy, a larger gain coefficient is selected, that is, a larger value of the gain coefficients ${\beta }_{01}$, ${\beta }_{02}$, ${\beta }_{03}$, ${\beta }_{04}$ is required to satisfy the high-gain ESC mode. Based on the four optimization methods and control strategy system models of classical PID control, quadratic optimization control, pole configuration and SAD, the EHS system is simulated by Simulink, and the step response curves of the unit step signal under the four control strategies are compared and analyzed, as shown in Fig. 10.

Fig. 10Step response curve of unit step signal under four control strategies

In Fig. 10, the SAD performance is the most stable after the system is optimized. The response speed of PID is 0.6 seconds slower than that of SAD control, and the response is accompanied by a large amount of overshoot. The response time of the pole configuration is 0.3 seconds longer than that of the SAD control. Compared with the PID control, although the overshoot is significantly reduced, the overshoot still exists. Both quadratic optimal control and SAD control do not have overshoot, but the response time of SAD control is 4.8 seconds shorter than that.

4.1. Experiment setup

To verify the effectiveness of the proposed four control strategies, a hydraulic platform has been established, which is referred Fig. 11 for details. For testing the proposed controllers, the experimental following-up error performance is shown by quantitative acquisition and monitoring in this section. The verification platform consists of a hydraulic system including a double-rod hydraulic cylinder, a linear encoder (accuracy class ±5.0 μm) for position and velocity information, two oil pressure sensors (accuracy class 0.1 MPa) for measuring differential pressure, a servo valve (rated flow rate 18 L/min at 0.6 MPa) with a bandwidth higher than 120 Hz, drive shaft, mass load $m=$50 kg, hydraulic supply with $P_s=$ 1.0 MPa, and measurement and control system. The measurement and control system consists of monitoring software, real-time control software, A/D card (Advantech PCI-1716), D/A card (Advantech PCI-1723), counter card (Heidenhain IK-220), all these cards are 16-bit. The monitoring software is programmed with NI LabWindows/CVI, and the real-time control software is compiled with Microsoft Visual Studio 2005 plus Ardence RTX7.0. Ardence RTX 7.0 is used to provide a real-time working environment for real-time control software under Windows XP operating system. To implement the proposed controller and other comparison controllers defined below, the discretized C++ code was programmed. The control sampling time for the control is 0.4ms. The following-up errors of the four control strategies within 1.0s are revealed in Fig. 12. The following-up errors of the four control strategies within 4.0s are revealed in Fig. 13. Where, Case1 represents the pole optimized configuration, Case2 represents the quadratic optimal control, Case3 represents the classical PID control, Case4 represents the SAD control.

In Figs. 12 and 13, the following-up performance of the Case 4 controller is better than that of the others, which is clearly evident. It is also noted that the following-up performance of the Case 2 controller is relatively weak, which indicates that the parameter uncertainty has a great influence on the high frequency band of the controller. Similarly, the following-up performance of the conventional PID controller (Case 1) is the least ideal compared with that of the others.

Fig. 11The EHS system experiment platform

Fig. 12Following-up errors within 1.0 s

Fig. 13Following-up errors within 4.0 s

4.2. Discussion of the results

From these results above, especially looking to Figs. 12 and 13, is it certain that it is possible to conclude that, the following-up of the controller of Case 3 is greatly in-creased compared with that of Case 1 and Case 2. An improved STF algorithm is introduced into the EHS system with LOSP optimized for the SAD control strategy, which is a novel move. Another interesting consideration is the comparison of the gains of the Cases, it is possible to state that the following up performance of the Case 4 controller is better than that of Case 3. On the other side, it is verified that the variable hydraulic LOSP acquisition and monitoring proposed in this paper can be effectively and stability adjusted by a pre-designed EHS control system.