Research on Dynamic Performance of Independent Metering Valves Controlling Concrete-Placing Booms Based on Fuzzy-LADRC Controller

Abstract

Motion control of truck-mounted concrete pump booms is an important part of concrete construction work. The quality of the boom movement will affect the efficiency and safety of construction operations. To make the boom move quickly and steadily, its hydraulic control system needs to be improved. In this article, we firstly combine independent metering control (IMC) and active disturbance rejection control (ADRC). The hydraulic system that controls the boom is meliorated by replacing the traditional proportional directional spool valve with four cartridge flow control valves to realize IMC. Meanwhile, a fuzzy linear ADRC algorithm is proposed for a new IMC hydraulic system where ADRC is linearized and combined with fuzzy logic to achieve better motion control performance of boom sections. According to the results of experiments conducted using all six boom sections, compared with the PID control algorithm, the fuzzy linear active disturbance rejection control (FLADRC) proposed in this article can improve the stability and rapidity of valves controlled by hydraulic cylinders by 10.0% to 66.7% in different working conditions.

1. Introduction

Truck-mounted concrete pumps play a crucial role in construction work, aiming to complete concrete placing tasks efficiently, quickly, and accurately [1]. To achieve this, the hydraulic-controlled boom system must effectively deploy and arrange the conveying pipe. The stability and rapidity of its movement determines the quality and efficiency of concrete placement. The motion of the boom sections is controlled by hydraulic cylinders between them, so the control performance of the hydraulic system directly affects the movement performance of the boom system.

In the hydraulic field, independent metering control (IMC) technology, which separates the mechanical connections between metering ports, is an effective approach to enhance control flexibility [2]. The IMC system can be designed with various valves and configurations [3]. Using four 2/2 valves is one of the most popular ways to realize an IMC system. Zhang, Q. used four spool-type two-way cartridge valves to realize the function of a four-way directional valve [4,5]. Bin, Y. used four proportional cartridge valves to control one cylinder and an extra valve to implement a regeneration mode [6,7,8]; based on this, Lu, L. used four proportional cartridge valves and one accumulator to minimize the energy consumption of the system while maintaining good motion tracking performance [9,10]. Shenouda, A. used four electro-hydraulic poppet valves to construct an IMC system on a mid-size tractor loader backhoe and first proposed the concept of continuously variable modes [11,12,13]. Andrea, V. used four valves to control a truck-mounted crane [14]. Kyujeong, C. conducted a simulation of a four-valve IMC on an excavator using AMESim [15], Jong-Chan, L. used a five-valve IMC on a 20-ton crawler excavator [16], and KaiLei, L. conducted simulation research of a five-valve IMC together with a pressure compensation valve [17]. In addition, based on a four-valve IMC system, some researchers used more valves to construct an IMC system; for example, STEAM [18] used eight valves and DFCU [19] used twenty valves.

Active disturbance rejection control (ADRC) is a novel model-free control strategy whose exciting performance has been demonstrated in the literature. Han, J. first proposed the concept of ADRC, which is an evolutionary style of PID [20]. ADRC has already been used in many kinds of hydraulic systems, such as electro-hydraulic position servo control systems [21], aircraft electro-mechanical actuators [22], hydraulic quadruped robots [23], hydraulic-driven winches [24], exoskeleton joints [25], hydraulic AGC systems [26], hydraulic active suspensions [27,28] and missile electro-hydraulic actuators [29].

Gao, Z. linearized an ADRC algorithm and proposed the concept of LADRC (linear ADRC) [30]. Unlike a traditional ADRC algorithm, which has 11 parameters to be adjusted according to the system configuration, LADRC only has 3 parameters that should be regulated. Application of LADRC in hydraulic controls has already been conducted by many researchers. Quan, L. studied a reduced-order model-based LADRC on a hydraulic servo system using singular value perturbation theory [31]. Huang, J. adopted a sixth-order LADRC in the closed-loop position control of a hydraulic cylinder in a multi-joint hydraulic manipulator [32]. Wang used an LADRC with a reconstructed model to accomplish high-precision motion control of a hydraulic position servo system [33]. Jin, K. applied an LADRC with acceleration feed-forward on displacement tracking of a valve-controlled asymmetric cylinder [34]. Liu, F. designed an LADRC controller with residual dead-zone compensation for a hydraulic position servo system of a weeding machine [35].

Combining LADRC with fuzzy control can further improve the dynamic performance of the system; for example, Zhou, X. used an FLADRC to control medium- and high-voltage distribution networks [36], Sun, C. used an FLADRC to control an industrial quadrotor UAV [37] and flying robot [38], Zhang, F. used an FLADRC to control a deep-sea AUV [39], Han, D. used an FLADRC to control autopilot attitude [40], Feng, X. used an FLADRC to control an IPMSM [41], and Qing, Y. used an FLADRC to control a permanent magnet–electromagnetic hybrid suspension platform [42].

The previous studies mostly focused on system configurations and control algorithms and utilized small-scale hardware-in-the-loop test setups. This article, however, presents an experiment conducted on a full-scale truck-mounted concrete pump with a six-section boom that can reach 63 m in length. We developed fuzzy-LADRC controllers to control the speed and pressure of the hydraulic cylinders on the boom.

The main objectives of this paper were to achieve low-vibration speed control for the hydraulic cylinders controlled by independent metering valves and to minimize the speed response time using the fuzzy-LADRC approach. The experimental results for speed response control indicate that fuzzy-LADRC controllers can effectively reduce fluctuations and improve cylinder speed response time during the motion control of the truck-mounted concrete pump boom.

2. System Configuration

2.1. The Test Rig

Due to limitations with the counterbalance valve, regeneration modes were not feasible. As a result, a five-valve or more IMC configuration could not be used in the boom hydraulic system. To overcome this, we have chosen a four-valve configuration for improved accuracy in motion control of the boom sections.

Cartridge flow control valves were used instead of spool valves, providing several advantages. For instance, a cartridge valve has a seat that reduces leakage, which is essential for the boom hydraulic system. Furthermore, a high-pressure and a low-pressure piping system are employed to supply oil to the IMC valves in the hydraulic cylinder system of each boom section, as the pressure of the truck-mounted concrete pump hydraulic system cannot be altered.

Figure 1 illustrates the schematic structure of the IMV-controlled hydraulic systems that regulate the motion of the boom sections (IMCHB).

As seen in Figure 1, the “P” line represents the high-pressure piping system, which carries compressed hydraulic fluid from the pump. The “T” line represents the low-pressure piping system that is connected to the tank. The four proportional flow valves have a one-way flow function in the neutral state.

The oil inlet proportional flow valves (6-2 and 6-3) allow fluid to flow from the working port to the high-pressure oil pipeline, while the oil return proportional flow valves (6-1 and 6-4) allow fluid to flow from the oil return pipeline to the working port. To prevent oil leakage, a check valve is installed between the oil return proportional flow valve and the oil return pipeline, and another check valve is installed between the oil inlet proportional flow valve and the working chamber of the hydraulic cylinder. These check valves ensure that the oil remains locked when the flow control valves are in a neutral position.

2.2. The Numerical Model

2.2.1. Modelling of Hydraulic System

According to Newton’s third theorem, the force on the piston rod of the hydraulic cylinder is:

$$F=P_1{A}_1-P_2{A}_2=m_p\ddot{y}+c_p\dot{y}+F_L$$

(1)

where F is the driving force of the valve-controlled cylinder; P₁ and P₂ denote pressures inside the two chambers of the cylinder; A₁ and A₂ are the ram areas of the two chambers; y is the piston rod displacement; m_p is the equivalent mass of the cylinder piston rod and its load; c_p is the equivalent damping coefficient of the cylinder piston rod and its load; F_L is the resultant external force that includes the cylinder piston rod external load, unmodeled friction, and the sum of other uncertain external disturbance forces.

Since the mathematical models of the two chambers of the hydraulic cylinder are the same, in order to avoid repeated modeling, assuming that Q₁ is the flow rate of the oil inlet chamber and Q₂ is the flow rate of the oil outlet chamber, the flow rate of the two chambers of the hydraulic cylinder can be expressed as

$$\begin{array}{l}{Q}_1=C_d{W}_1{x}_1\sqrt{2(P_s-P_1)/\rho }\begin{array}{cc}& x_1\ge 0\end{array}\\ Q_2=C_d{W}_2{x}_2\sqrt{2(P_2-P_r)/\rho }\begin{array}{cc}& x_2<0\end{array}\end{array}$$

(2)

where C_d is the flow coefficient; W₁ is the area gradient of the oil inlet valve; W₂ is the area gradient of the oil outlet valve; x₁ is the displacement of the oil inlet valve main spool; x₂ is the displacement of the oil outlet valve main spool; P_s is the system oil supply pressure; P_r is the system oil-return pressure; and ρ is the hydraulic oil density.

Due to established hydraulic system sealing technology, external leakage of the hydraulic cylinder can be neglected, so the flow continuity equations of the two chambers are

$$\begin{array}{l}{Q}_1=A_1 \dot{y} +C_{ip}(P_2-P_1)+\frac{V_1}{{\beta }_e}{\dot{P}}_1\\ Q_2=A_2\dot{y}+C_{ip}(P_1-P_2)-\frac{V_2}{{\beta }_e}{\dot{P}}_2\end{array}$$

(3)

where C_ip is internal leakage coefficient; V₁ and V₂ are the volume of the hydraulic cylinder inlet and outlet chambers, respectively; and β_e is the oil volume elastic modulus.

In order to establish the state space equation of the IMV-controlled linearized asymmetric hydraulic cylinder system (3), we have

$$\begin{array}{l}{Q}_1=K_{q1}{x}_1-K_{c1}{p}_1\begin{array}{cc}& x_1\ge 0\end{array}\\ Q_2=K_{q2}{x}_2+K_{c2}{p}_2\begin{array}{cc}& x_2<0\end{array}\end{array}$$

(4)

where

$$\begin{array}{l}{K}_{q1}=C_d{W}_1\sqrt{2(P_s-P_1)/\rho }\hspace{1em}{K}_{c1}=\frac{C_d{W}_1{x}_1}{\sqrt{2(P_s-P_1)/\rho }}\\ K_{q2}=C_d{W}_2\sqrt{2(P_2-P_{\mathrm{r}})/\rho }\hspace{1em}{K}_{c2}=\frac{C_d{W}_2{x}_2}{\sqrt{2(P_2-P_{\mathrm{r}})/\rho }}\end{array}$$

(5)

According to (1)–(5), the state space equation of the studied system is

$$\left[\begin{array}{c}\dot{y}\\ \ddot{y}\\ {\dot{P}}_1\\ {\dot{P}}_2\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& A_{22}& A_{23}& A_{24}\\ 0& A_{32}& A_{33}& A_{34}\\ 0& A_{42}& A_{43}& A_{44}\end{array}\right]\left[\begin{array}{c}y\\ \dot{y}\\ P_1\\ P_2\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ B_{31}\\ 0\end{array}\begin{array}{c}0\\ 0\\ 0\\ B_{42}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ -\frac{1}{m}\\ 0\\ 0\end{array}\right]F_L$$

(6)

where

$$\begin{array}{l}{A}_{22}=-\frac{c_p}{m},A_{23}=\frac{A_1}{m},A_{24}=-\frac{A_2}{m},A_{32}=-\frac{A_1{\beta }_e}{V_1},A_{33}=-\frac{{\beta }_e(C_{ip}+K_{c1})}{V_1},A_{34}=\frac{{\beta }_e{C}_{ip}}{V_1}\\ A_{42}=\frac{A_2{\beta }_e}{V_2},A_{43}=\frac{{\beta }_e{C}_{ip}}{V_2},A_{44}=-\frac{{\beta }_e(C_{ip}+K_{c2})}{V_2},B_{31}=\frac{{\beta }_e{K}_{q1}}{V_1},B_{42}=-\frac{{\beta }_e{K}_{q2}}{V_2}\end{array}$$

(7)

Considering that the response dynamics of the valve used in the study are much higher than the actuator’s operation frequency band, valve spool displacement is considered as hydraulic system control input.

2.2.2. Modelling of External Forces

For the sake of illustration, in this paper the hydraulic cylinder that controls the movement of a boom section i (i = 1, 2, 3, 4, 5, 6) is called cylinder #i (i = 1, 2, 3, 4, 5, 6).

For cylinder #1, boom section 1 is articulated as shown in Figure 2, according to Newton–Euler equations and trigonometric function, the resultant external force applied on cylinder #1 F_AB can be expressed as

$${\mathit{F}}_{AB}=-\frac{{\mathit{M}}_f}{L_{OA}\cdot \sin \alpha }$$

(8)

where M_f is the resultant resistance moment of boom section 1; L_OA stands for the distance between O and A.

Similarly, for cylinder #i (i = 2, 3, 5) as shown in Figure 3, the resultant external force applied on cylinder #i F_DE can be expressed as

$$\begin{array}{l}{\mathit{F}}_{DE}=-\frac{{\mathit{F}}_C\cdot L_{BC}\cdot \sin (\gamma -\angle BCE)}{L_{BE}\cdot \sin \alpha }\\ \gamma =\mathrm{arccot}(\cot \angle BCE-\frac{\sin \alpha \cdot \sin (\angle EBC-\beta )}{\sin \beta \cdot \sin (\alpha +\angle BEC)\cdot \sin \angle BCE})\end{array}$$

(9)

where F_C is the resultant resistance force that boom section i (i = 2, 3, 5) applies on linkage EBC; L_BC stands for the distance between B and C; and L_BE stands for the distance between B and E.

For cylinder #4 as shown in Figure 4, the resultant external force applied on cylinder #4 F_DE can be expressed as

$$\begin{array}{l}{\mathit{F}}_{DE}=-\frac{{\mathit{F}}_C\cdot L_{BC}\cdot \sin (\gamma +\angle BCE)}{L_{BE}\cdot \sin (\alpha +\angle BEC)}\\ \gamma =\mathrm{arccot}(-\frac{\sin (\alpha +\angle BEC)\cdot \cos (\angle CBE-\beta )}{\sin \beta \cdot \sin \alpha \sin \angle BCE}-\cot \angle BCE)\end{array}$$

(10)

where F_C is the resultant resistance force that boom section 4 applies on linkage EBC; L_BC stands for the distance between B and C; and L_BE stands for the distance between B and E.

For cylinder #6 as shown in Figure 5, the resultant external force applied on cylinder #6 F_DB can be expressed as

$$\begin{array}{l}{\mathit{F}}_{DB}=\frac{{\mathit{F}}_{C\mathrm{B}}}{-\cos \alpha +\frac{\sin \alpha }{\sin \gamma }\cdot \cos (\varphi -\alpha -\gamma )}\\ \gamma =\arctan (\frac{\sin (\alpha -\varphi )}{\frac{\sin \varphi }{\sin \alpha }-\cos (\alpha -\varphi )})\end{array}$$

(11)

where F_C is the resultant resistance force that boom section 6 applies on linkage BC.

3. Control

3.1. LADRC Algorithm

ADRC, proposed by Han, J., uses several nonlinear functions such as fhan and fal in the design of its tracking differentiator (TD), extended state observer (ESO), and state error feedback. These nonlinear techniques effectively compensate for external disturbances and unknown model parts, but also increase the control algorithm’s complexity. Han, J. introduced the concept of “time scale” to categorize different systems and control processes. Based on the time scale concept, a single ADRC controller can be applied to systems with the same time scale, showcasing its potential for industrial applications. However, the need to tune up to 12 control parameters can hinder the use of ADRC.

Inspired by the time scale concept, Gao, Z. introduced the “frequency scale” concept. This associates ADRC parameters with frequency, making the physical sense of the parameters more intuitive. Furthermore, state error feedback and the extended state observer are implemented in a linear form. With the bandwidths of the feedback controller and extended state observer as their unique parameters, only three control parameters need to be tuned.

3.2. Control System Design

Our experiments have shown that a fixed set of controller parameters may not be optimal for systems under varying conditions. To achieve more precise and rapid control of boom cylinder speeds, we have integrated a fuzzy algorithm with the LADRC (linear active disturbance rejection control) method. This allows us to adjust the bandwidth of the feedback controller and extended state observer based on the system state.

The proposed control system as shown in Figure 6 consists of a fuzzy-LADRC speed controller, which regulates the inlet proportional flow control valve according to remote control commands and calculated cylinder speeds. The LADRC controller parameters are dynamically modified to produce an appropriate output to drive the inlet proportional flow control valve.

Additionally, a fuzzy-LADRC pressure controller is used to control the outlet proportional flow control valve, with the goal of reducing pressure fluctuations in the hydraulic cylinder chambers.

3.3. Speed Controller Design

During the boom extension process, the external load of the hydraulic cylinder changes over time under the influence of friction forces, including friction torque between boom sections and connecting pieces and friction between the piston rod and the piston cylinder. Meanwhile, different arm postures will result in different center of gravity positions, so the gravity moments will also change over time, as shown in Figure 2, Figure 3, Figure 4 and Figure 5. In addition, because of the large scale of the test rig, many unpredicted influencing factors such as hydraulic system vibration [43] can lead to unstable cylinder dynamic performance, so the system model developed in Section 2 has model uncertainty as well as parameter uncertainty.

With all these uncertain influencing factors in mind, the LADRC algorithm was chosen to control the hydraulic cylinder speed because of its significant disturbance rejection capability. Furthermore, to increase speed control precision, we combined a fuzzy control algorithm together with LADRC, making the bandwidths of the feedback controller and extended state observer adjustable under different working conditions and stages.

For the system described in Equation (7), our goal was to generate suitable control input such that the piston rod speed can track the specified speed command as well as pressure target, so the rebuild state space equation of the system is as follows:

$$\begin{array}{l}\left[\begin{array}{l}\ddot{y}\\ {\dot{P}}_1\\ {\dot{P}}_2\end{array}\right]=\left[\begin{array}{ccc}-\frac{c_p}{m}& \frac{A_1}{m}& -\frac{A_2}{m}\\ -\frac{A_1{\beta }_e}{V_1}& -\frac{{\beta }_e(C_{ip}+K_{c1})}{V_1}& \frac{{\beta }_e{C}_{ip}}{V_1}\\ \frac{A_2{\beta }_e}{V_2}& \frac{{\beta }_e{C}_{ip}}{V_2}& -\frac{{\beta }_e(C_{ip}+K_{c2})}{V_2}\end{array}\right]\left[\begin{array}{l}\dot{y}\\ P_1\\ P_2\end{array}\right]+\left[\begin{array}{cc}0& 0\\ \frac{{\beta }_e{K}_{q1}}{V_1}& 0\\ 0& -\frac{{\beta }_e{K}_{q2}}{V_2}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}-\frac{1}{m}\\ 0\\ 0\end{array}\right]F_L\\ \\ Y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{l}\ddot{y}\\ P\\ P\end{array}\right]\end{array}$$

(12)

According to Equation (8), a second-order LADRC should be used to control the speed of the boom cylinders. To overcome the existing contradiction between stability and rapidity, a fuzzy-LADRC algorithm is applied in the speed control of cylinders, as shown in Figure 7.

The control algorithm shown in Figure 7 is composed of three parts; the formula of the PD controller is

$$\begin{array}{l}{e}_v=r_v-z_1\\ u_{v0}=k_p{e}_v-k_d{z}_2\end{array}$$

(13)

where e_v is the speed tracking deviation, r_v is the cylinder target speed, z₁ is the observed value of cylinder speed, u_v0 is the output value of the PD controller, z₂ is the observed value of cylinder acceleration, k_pv is proportional gain of the PD controller, and k_dv is the differential gain of the PD controller.

Based on the concept of frequency scale, the value of k_pv and k_dv can be determined by controller bandwidth w_cv, as shown:

$$\begin{array}{l}{k}_{pv}=w_{c\mathrm{v}}{}^2\\ k_{dv}=2w_{cv}\end{array}$$

(14)

The expression of the extended state observer is

$$\begin{array}{l}\left[\begin{array}{l}{\dot{z}}_1\\ {\dot{z}}_2\\ {\dot{z}}_3\end{array}\right]=\left[\begin{array}{ccc}-{\beta }_{1v}& 1& 0\\ -{\beta }_{2v}& 0& 1\\ -{\beta }_{3v}& 0& 0\end{array}\right]\left[\begin{array}{l}{z}_1\\ z_2\\ z_3\end{array}\right]+\left[\begin{array}{c}0\\ b_0\\ 0\end{array}\right]u_v+\left[\begin{array}{c}{\beta }_{1v}\\ {\beta }_{2v}\\ {\beta }_{3v}\end{array}\right]{\dot{y}}_{}\\ Y=\left[\begin{array}{l}{z}_1\\ z_2\\ z_3\end{array}\right]\end{array}$$

(15)

where z₃ is the observed value of the additional state variable, b₀ is a rough approximation of the system control gain, β_1v, β_2v, and β_3v are the observer parameters, and u_v is the final output value of the control algorithm. According to the concept of frequency scale, the value of β_1v, β_2v, and β_3v can be determined by observer bandwidth w_ov as shown:

$$\begin{array}{l}{\beta }_{1v}=3w_{ov}\\ {\beta }_{2v}=3w_{ov}{}^2\\ {\beta }_{3v}=w_{ov}{}^3\end{array}$$

(16)

Finally, the equation for the rest can be expressed as

$$u_v=\frac{u_{v0}-z_3}{b_0}$$

(17)

As shown in the control algorithm, there are three controller parameters to be determined, which are controller bandwidth w_cv, observer bandwidth w_ov, and system control gain b₀. Controller bandwidth w_cv determines controller response speed within a certain range; a larger w_cv value can lead to better control performance. However, when the value of w_cv becomes too large, the whole system may become unstable. As a consequence, the value of w_cv should be adjusted according to the system’s transient response requirement and should be limited within a certain range so that precise measurement of the process variable can be obtained. Observer bandwidth w_ov determines the tracking speed of ESO; the larger the value of w_ov, the faster the estimation of disturbance. However, a large w_ov will lead to higher sensitivity to system noise and cause the observer to oscillate. Therefore, its value also depends on the acceptable noise threshold or the sampling delay that makes the state observer oscillate. System control gain b₀ represents the properties of the object; it can be derived from the initial acceleration of system step response.

Increasing the value of w_cv or w_ov will cause the high-band gain of the system to become larger, thereby making the system’s anti-interference ability worse, which makes the value of w_cv and w_ov a trade-off between rapidity and stability. What’s more, during different processes of cylinder motion, the value will change rapidly over a wide area, so the proper ranges of w_cv and w_ov can be different at different times. As a result, good control performance cannot be obtained with a set of fixed parameters.

With this in mind, combining a fuzzy control algorithm together with an LADRC would be a good choice to reach a compromise between rapidity and stability. Combined with the fuzzy algorithm, w_cv in Equation (10) and w_ov in Equation (12) are modified as follows:

$$\begin{array}{l}{w}_{cv}={\alpha }_{cv}{w}_{cv}{}_0\\ w_{ov}={\alpha }_{ov}{w}_{ov}{}_0\end{array}$$

(18)

where ${\alpha }_{cv}$ is the amplification factor of the controller bandwidth and ${\alpha }_{ov}$ is the amplification factor of the observer bandwidth. Both amplification factors will be adjusted according to the speed tracking deviation e_v and its derivative $\dot{e_v}$, as shown in Figure 8.

Firstly, input normalization is performed to unify the input of the fuzzy control algorithm as follows:

$$\begin{array}{l}E=\frac{E_{\max }-E_{\min }}{e_{\max }-e_{\min }}(e-\frac{e_{\max }+e_{\min }}{2})+\frac{E_{\max }+E_{\min }}{2}\\ \Delta E=\frac{\Delta E_{\max }-\Delta E_{\min }}{\Delta e_{\max }-\Delta e_{\min }}(\Delta e-\frac{\Delta e_{\max }+\Delta e_{\min }}{2})+\frac{E{\Delta }_{\max }+\Delta E_{\min }}{2}\end{array}$$

(19)

where E and ∆E are defined within the same range of [−1, 1]. After this step, different ranges of e_v and $\dot{e_v}$ under different hydraulic cylinder working conditions can be transformed into unified ranges.

Next, during the fuzzy process, the inputs of the fuzzy control algorithm are expressed as seven linguistic variables: largely negative (NB), moderately negative (NM), slightly negative (NS), neutral (Z), slightly positive (PS), moderately positive (PM), and largely positive (PB). In this study, the triangle membership function with high sensitivity is applied to both e_v and $\dot{e_v}$, as shown in Figure 9.

Afterward, Mamdani-type fuzzy reasoning is used; the speed controller fuzzy rules table is shown in

Table 1. Fuzzy rules base of speed controller.

${\mathit{\alpha }}_{\mathit{c}\mathit{v}}$$,{\mathit{\alpha }}_{\mathit{o}\mathit{v}}$		${\mathit{e}}_{\mathit{v}}$
		NB	NM	NS	Z	PS	PM	PB
$\dot{{\mathit{e}}_{\mathit{v}}}$	NB	M, Z	VB, S	VB, VB	B, VB	B, M	M, Z	S, Z
	NM	M, Z	B, VS	B, VB	M, VB	M, M	M, Z	S, Z
	NS	S, Z	M, VS	M, B	S, B	M, B	M, Z	S, Z
	Z	S, Z	M, Z	M, B	Z, M	M, B	M, Z	S, Z
	PS	S, Z	M, Z	M, B	S, B	M, B	M, VS	S, Z
	PM	S, Z	M, Z	M, M	M, VB	B, VB	B, VS	M, Z
	PB	S, Z	M, Z	B, M	B, VB	VB, VB	VB, S	M, Z

, where M, B, S, VB, and VS indicate moderate, large, small, extremely large, and extremely small, respectively.

The output rule of the fuzzy inference engine is

$$R^i:\mathrm{IF} e_d\left(k\right) is A_1^i,\mathrm{AND} \Delta e_d\left(k\right) is A_2^i, \mathrm{THEN} Y=E_i$$

(20)

where $A_1^i$ and $A_2^i$ are the variable inputs and $E_i={[E_1,E_2,E_3]}^T$ is the output of the i_th fuzzy rule.

Then, during de-fuzzy processing, outputs of the fuzzy control algorithm are expressed as six linguistic variables: neutral (Z), extremely small (VS), small (S), moderate (M), large (B), and extremely large (VB). In this study, the triangle membership function with high sensitivity is applied to both ${\alpha }_{cv}$ and ${\alpha }_{ov}$, as shown in Figure 10.

The centroid defuzzification method is applied during fuzzy decoupling:

$${[{\alpha }_{DOUT},{\alpha }_{DI},{\alpha }_{DD}]}^T=\frac{{\displaystyle \sum _{i=1}^p\left({\mu }_{A_1^i}(e_d){\mu }_{A_2^i}(\Delta e_d)\right)E_i}}{{\displaystyle \sum _{i=1}^p\left({\mu }_{A_1^i}(e_d){\mu }_{A_2^i}(\Delta e_d)\right)}}$$

(21)

where ${\mu }_{A_1^i}(e_d)$ and ${\mu }_{A_2^i}(\Delta e_d)$ are the membership function values of $e_d(k)$ and $\Delta e_d(k)$, respectively, and P is the number of fuzzy rules.

During the adjustment of the FLADRC-based speed controller fuzzy rules, many tests are conducted, which assure the best dynamic performance under different working conditions. Firstly, as shown in Table 1, the states of the system are divided into 49 sections according to the value of e_v and $\dot{e_v}$. Secondly, an appropriate set of LADRC parameters is selected according to test results. After that, the value of ${\alpha }_{cv}$ is increased until the system starts to become unstable in each section, then the value of ${\alpha }_{ov}$ is increased until the system starts to become unstable as well. Finally, compromise between the two fuzzy parameters is made to achieve better dynamic performance.

3.4. Pressure Controller Design

As shown in Equation (1), the pressure of the inlet and outlet chambers can affect each other through external load, so pressure vibrations on one side will cause vibrations on the other side at the same time. These vibrations can significantly affect cylinder speed stability, so the pressure curves on both sides should be as smooth as possible. Under the influence of external load, frictional resistance, and inaccurate modeling parameters, a pressure controller based on fuzzy-LADRC is designed to reduce the pressure ripple of both inlet and outlet chambers. Since the inlet proportional flow control valve is used to control the cylinder speed, we chose an outlet flow control valve to control the back pressure of the cylinder.

According to Equation (8), first-order LADRC should be used to control boom cylinder chamber pressure. To overcome the existing contradiction between stability and rapidity, a fuzzy-LADRC algorithm was applied, as shown in Figure 11.

The control algorithm shown in Figure 11 is composed of three parts. The formula of the P controller is

$$\begin{array}{l}{e}_p=r_p-z_1\\ u_{p0}=k_{pp}{e}_p\end{array}$$

(22)

where e_p is the pressure tracking deviation, r_p is the target chamber pressure, z₁ is the observed cylinder chamber pressure value, u_p0 is the output value of the P controller, and k_pp is the proportional gain of the P controller.

The expression of the extended state observer is

$$\left[\begin{array}{l}{\dot{z}}_1\\ {\dot{z}}_2\end{array}\right]=\left[\begin{array}{cc}-{\beta }_{1p}& 1\\ -{\beta }_{2p}& 0\end{array}\right]\left[\begin{array}{l}{z}_1\\ z_2\end{array}\right]+\left[\begin{array}{l}{b}_0\\ 0\end{array}\right]u_p+\left[\begin{array}{l}{\beta }_{1p}\\ {\beta }_{2p}\end{array}\right]p$$

(23)

where z₂ is the observed value of the additional state variable, b₀ is a rough approximation of the system control gain, β_1p and β_2p are the observer parameters, and u_p is the final output value of the control algorithm. According to the concept of frequency scale, the value of β_1p and β_2p can be determined by observer bandwidth w_op, as shown:

$$\begin{array}{l}{\beta }_{1p}=2w_{op}\\ {\beta }_{2p}=w_{op}{}^2\end{array}$$

(24)

Finally, the equation for the remaining part can be expressed as

$$u_v=\frac{u_{p0}-z_2}{b_0}$$

(25)

Combined with the fuzzy algorithm, w_op in Equation (10) is modified as follows:

$$w_{op}={\alpha }_{op}{w}_{op}{}_0$$

(26)

where ${\alpha }_{op}$ is the amplification factor of observer bandwidth, which will be adjusted according to speed tracking deviation e_p and its derivative $\dot{e_p}$. As with the speed controller expressed in Section 3.4, the inputs and output are characterized by the triangular membership function, as shown in Figure 9 and Figure 10. The fuzzy logic controller uses a Mamdani-type method while the centroid method is used for defuzzification. The fuzzy rules of k_pp and ${\alpha }_{op}$ are shown in

Table 2. Fuzzy rules base of chamber pressure controller.

${\mathit{k}}_{\mathit{p}\mathit{p}}$$,{\mathit{\alpha }}_{\mathit{o}\mathit{p}}$		${\mathit{e}}_{\mathit{d}}\left(\mathit{k}\right)$
		NB	NM	NS	Z	PS	PM	PB
$\mathbf{\Delta }{\mathit{e}}_{\mathit{d}}\left(\mathit{k}\right)$	NB	M, Z	VB, Z	VB, VB	B, VB	B, B	M, Z	S, Z
	NM	M, Z	B, Z	B, M	M, B	M, M	M, Z	S, Z
	NS	S, Z	M, Z	M, M	S, M	M, S	M, Z	S, Z
	Z	S, Z	M, Z	M, Z	Z, Z	M, Z	M, Z	S, Z
	PS	S, Z	M, Z	M, S	S, M	M, M	M, Z	S, Z
	PM	S, Z	M, Z	M, M	M, B	B, B	B, Z	M, Z
	PB	S, Z	M, Z	B, B	B, VB	VB, VB	VB, Z	M, Z

.

4. Experiment Setup

4.1. Hydraulic System Retrofit

A truck-mounted concrete pump with a six-section boom was used for experimentation in this article, as shown in Figure 12a. Traditionally, the boom cylinders are controlled by a proportional directional valve located at the pump base. This results in a time delay, as compressed fluid is pumped through long pipes to the cylinders. To reduce this delay, IMVs as described in Figure 1 were mounted near the cylinders (e.g., near the boom section 1 cylinder), as shown in Figure 12b. By placing the IMVs close to the cylinders, the response time of the cylinders is significantly reduced, providing a solid foundation for the control algorithm design.

4.2. Electronic Control System

A CAN bus was used to connect the electronic devices on the test rig, as shown in Figure 13. Operators used the joystick on the remote control to send speed control commands. The remote control communicated with the pump truck controller via Wi-Fi, allowing the controller to send boom movement speed commands via the CAN bus. An STM32 controller was used to receive speed commands and messages from the pressure and position sensors through analog input ports. The controller also sent calculated control current values to the corresponding coil drivers via the CAN bus.

5. Experiment Results

Placing concrete accurately requires the movement of each boom section [44], so motion control of each boom section was tested. According to the knowledge of skilled operators, unfolding and folding in 180 degrees (90 degrees for boom section 1) is the most common working condition of boom sections on truck-mounted pumps. With this in mind, every boom section’s most common working condition was chosen as their basic experiment.

As discussed in section 2, cylinder #2, cylinder #3, and cylinder #5 have the same linkage mechanism structure, so to eliminate repeated experimentation only cylinder #2 was tested during the experiment. Meanwhile, to show the advantage of FLADRC, a traditional PID, which is the most-used control algorithm, was applied under the same working conditions. The trial and error method was used in adjusting the PID; with enough experiments, we thought the best performance was achieved.

Additionally, the PID parameters for each link are different, which were adjusted to achieve the best dynamic performance. Parameters in this paper were all selected according to experiment results.

5.1. Experiment with Boom Section 1

The movement of boom section 1 is depicted in Figure 14. The unfolding process involves a movement from 0 degrees to 90 degrees, with the vectors of external force and cylinder #1 speed in opposite directions. To maintain the stability of the chamber pressure in cylinder #1, it is recommended to set the target back pressure as low as possible. However, during the folding process, where the movement is from 90 degrees to 0 degrees, the vectors of external force and speed are in the same direction. In this case, the target back pressure should be set higher to maintain the stability of cylinder #1′s chamber pressure.

According to relevant regulations, the target speed of cylinder #1 was set to 10 mm/s during the unfolding process and 14.5 mm/s during the folding process. The speed step response experiment results using the PID are shown in Figure 15 while the results using the LADRC are shown in Figure 16.

To make the comparison between PID and FLADRC more distinct, the speed error of cylinder #1 during the unfolding and folding process is shown in Figure 17 and Figure 18, respectively.

Startup time, which represents system rapidity, and fluctuation range, which relates to system stability, are the two parameters that operators are most concerned about. Thus, these two parameters were selected as the assessment indices of the control algorithm, as shown in

Table 3. Assessment indices of cylinder#1 motion control using PID and FLADRC.

	Unfolding		Folding
	PID	FLADRC	PID	FLADRC
Startup time(s)	9.6	6.3	3.6	3.6
Fluctuation range	7.3%	7.4%	9.0%	6.2%

.

For cylinder #1, the application of FLADRC resulted in a 34.4% reduction in the startup time of the unfolding process compared to the PID method. Additionally, the fluctuation range of cylinder #1’s folding motion using the FLADRC was 30.8% lower than when using the PID.

5.2. Experiment with Boom Section 2

During the unfolding process, boom section 2 moves from 0 degrees to 180 degrees, as shown in Figure 19. The vectors of external force applied on cylinder #2 and its speed are in opposite directions. Conversely, during the folding process, boom section 2 moves from 180 degrees to 0 degrees, and the vectors of external force applied on cylinder #2 and its speed are in the same direction. The back pressure setting standard of cylinder #2 was the same as that of cylinder #1.

Based on relevant regulations, cylinder #2’s speed target was set as 10 mm/s during the unfolding process and at 14 mm/s during the folding process. Step response experiment results are shown in Figure 20 and Figure 21.

The speed error of cylinder #2 during the unfolding and folding processes using PID and FLADRC is shown in Figure 22 and Figure 23.

The startup time and fluctuation range of the different control algorithms are shown in

Table 4. Assessment indices of cylinder #2 motion control using PID and FLADRC.

	Unfolding		Folding
	PID	FLADRC	PID	FLADRC
Startup time(s)	3.3	3.4	4.2	2.2
Fluctuation range	10.0%	9.0%	10.7%	7.8%

.

For cylinder #2, when the FLADRC was applied, the fluctuation range of the unfolding process was shortened by 10% compared with that using the PID, while both the startup time and fluctuation range of cylinder #2’s folding motion using the FLADRC was 42.6% and 27.1% less than that using the PID, respectively.

5.3. Experiment with Boom Section 4

The range of motion as well as vectors of external force and speed during the unfolding and folding processes of cylinder #4 were same as cylinder #2, so the back pressure setting standard of cylinder #4 was same as cylinder #2, as shown in Figure 24.

Based on relevant regulations, cylinder #4’s speed target was set as 10 mm/s during the unfolding process and 10 mm/s during the folding process. The step response experiment results are shown in Figure 25 and Figure 26.

The speed error of cylinder #4 during the unfolding and folding processes using PID and FLADRC is shown in Figure 27 and Figure 28.

The startup time and fluctuation range of the different control algorithms are shown in

Table 5. Assessment indices of cylinder #4 motion control using PID and FLADRC.

	Unfolding		Folding
	PID	FLADRC	PID	FLADRC
Startup time(s)	6.0	2.0	0.5	0.5
Fluctuation range	12%	7.0%	13%	11%

.

For cylinder #4, when the FLADRC was applied, both the startup time and fluctuation range of the unfolding process was shortened by 66.7% and 41.7% compared with using the PID, respectively, while the fluctuation range of cylinder #2’s folding motion using the FLADRC was 15.4% less than that using the PID.

5.4. Experiment with Boom Section 6

The range of motion, vectors of external force, and speed during the unfolding and folding processes of cylinder #6 were the same as those of cylinder #2. As a result, the back pressure setting standard of cylinder #6 was the same as that of cylinder #2, as depicted in Figure 29.

As per relevant regulations, the speed target for cylinder #6 was set at 13 mm/s during the unfolding process and 13 mm/s during the folding process. The step response experiment results are shown in Figure 30 and Figure 31.

The speed error of cylinder #6 during the unfolding and folding processes using PID and FLADRC are shown in Figure 32 and Figure 33.

The startup time and fluctuation range of the different control algorithms are shown in

Table 6. Assessment indices of cylinder #6 motion control using PID and FLADRC.

	Unfolding		Folding
	PID	FLADRC	PID	FLADRC
Startup time(s)	6.2	3.7	3.3	3.4
Fluctuation range	15.4%	12.3%	13.8%	12.3%

.

For cylinder#6, the application of the FLADRC resulted in a 40.3% reduction in the startup time and a 20.1% reduction in the fluctuation range of the unfolding process compared to the PID method. Meanwhile, the fluctuation range of cylinder #2’s folding motion using the FLADRC was 10.9% lower than when using the PID.

6. Discussion

According to the system state space Equation (8), the speed of the cylinder can be affected by various parameters that change under different working conditions, such as the equivalent mass m_p, equivalent damping coefficient c_p, and rod external load F_L. The FLADRC algorithm proposed in this paper can effectively eliminate the influence of both the external disturbance of the cylinder system and internal parameter uncertainty, resulting in better cylinder speed control performance and easing the contradiction between stability and rapidity. When the FLADRC algorithm is applied, at least one parameter in the startup time and fluctuation range can be improved compared to the PID algorithm, with an improvement range of 10% to 66.7% under different working conditions.

7. Conclusions

This paper presents the use of four cartridge valves to replace the traditional proportional directional spool valve in controlling the motion of truck-mounted concrete pump boom cylinders. Additionally, the FLADRC strategy was proposed to improve stability and rapidity in the presence of uncertain model parameters and external loads.

Mathematical modeling and experiments were conducted to verify the effectiveness of the FLADRC. When the FLADRC was applied, the motion fluctuation range of cylinders #2, #4, and #6 during the unfolding and folding processes and the motion fluctuation range of cylinder #1 during the folding process were smaller compared with using a PID. Because of more steady external forces, the fluctuation range of cylinder #1 during the unfolding process was not significantly improved by FLADRC. The startup time of cylinders #1, #4, and #6 during unfolding and cylinder #2 during folding were smaller compared with that when using PID due to the over-running load; startup time under other working conditions was not shortened by using the FLADRC.

The results show the superiority of FLADRC during the unfolding and folding of different boom sections. Multi-linkage joint control will be studied in our future works.