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:
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
where
ev is the speed tracking deviation,
rv is the cylinder target speed,
z1 is the observed value of cylinder speed,
uv0 is the output value of the PD controller,
z2 is the observed value of cylinder acceleration,
kpv is proportional gain of the PD controller, and
kdv is the differential gain of the PD controller.
Based on the concept of frequency scale, the value of
kpv and
kdv can be determined by controller bandwidth
wcv, as shown:
The expression of the extended state observer is
where
z3 is the observed value of the additional state variable,
b0 is a rough approximation of the system control gain,
β1v,
β2v, and
β3v are the observer parameters, and
uv is the final output value of the control algorithm. According to the concept of frequency scale, the value of
β1v,
β2v, and
β3v can be determined by observer bandwidth
wov as shown:
Finally, the equation for the rest can be expressed as
As shown in the control algorithm, there are three controller parameters to be determined, which are controller bandwidth wcv, observer bandwidth wov, and system control gain b0. Controller bandwidth wcv determines controller response speed within a certain range; a larger wcv value can lead to better control performance. However, when the value of wcv becomes too large, the whole system may become unstable. As a consequence, the value of wcv 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 wov determines the tracking speed of ESO; the larger the value of wov, the faster the estimation of disturbance. However, a large wov 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 wcv or wov 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 wcv and wov 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 wcv and wov 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,
wcv in Equation (10) and
wov in Equation (12) are modified as follows:
where
is the amplification factor of the controller bandwidth and
is the amplification factor of the observer bandwidth. Both amplification factors will be adjusted according to the speed tracking deviation
ev and its derivative
, as shown in
Figure 8.
Firstly, input normalization is performed to unify the input of the fuzzy control algorithm as follows:
where
E and ∆
E are defined within the same range of [−1, 1]. After this step, different ranges of
ev and
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
ev and
, as shown in
Figure 9.
Afterward, Mamdani-type fuzzy reasoning is used; the speed controller fuzzy rules table is shown in
Table 1, 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
where
and
are the variable inputs and
is the output of the
ith 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
and
, as shown in
Figure 10.
The centroid defuzzification method is applied during fuzzy decoupling:
where
and
are the membership function values of
and
, 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
ev and
. Secondly, an appropriate set of LADRC parameters is selected according to test results. After that, the value of
is increased until the system starts to become unstable in each section, then the value of
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
where
ep is the pressure tracking deviation,
rp is the target chamber pressure,
z1 is the observed cylinder chamber pressure value,
up0 is the output value of the P controller, and
kpp is the proportional gain of the P controller.
The expression of the extended state observer is
where
z2 is the observed value of the additional state variable,
b0 is a rough approximation of the system control gain,
β1p and
β2p are the observer parameters, and
up is the final output value of the control algorithm. According to the concept of frequency scale, the value of
β1p and
β2p can be determined by observer bandwidth
wop, as shown:
Finally, the equation for the remaining part can be expressed as
Combined with the fuzzy algorithm,
wop in Equation (10) is modified as follows:
where
is the amplification factor of observer bandwidth, which will be adjusted according to speed tracking deviation
ep and its derivative
. 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
kpp and
are shown in
Table 2.