Comparison of Advanced Multivariable Control Techniques for Axial-Piston Pump

Abstract

This article is devoted to a comparison of two advanced control techniques applied to the same plant. The plant is a certain type of axial-piston pump. A linear-quadratic (LQR) controller and an H-infinity (H_∞) controller were synthesized to regulate the displacement volume of the pump. The classical solution to such a problem is to use a hydro-mechanical controller (by pressure, flow rate, or power) but, in the available sources, there are solutions that implement proportional-integral-derivative (PID), LQR, model predictive control (MPC), etc. Unlike a classical solution, in our case, the hydro-mechanical controller is replaced by an electro-hydraulic proportional valve, which receives a reference signal from an industrial microcontroller. It is used as the actuator of the pump swash plate. The pump swash plate swivel angle determines the displacement volume, respectively, and the flow rate of the pump. The microcontroller is capable of embedding various control algorithms with different structures and complexities. The developed LQR and H_∞ controllers are compared in the simulation and real experiment conditions. For this purpose, the authors have developed a laboratory experimental test bench, enabling a real-time function of the control system via USB/CAN communication. Both controllers are compared under different pump loading modes. Also, this paper contributes an uncertain model of an axial-piston pump with proportional valve control that is obtained from experimental data. Based on this model, the robust stability of the closed-loop system is investigated by comparing the structured singular value (μ). The investigations show that both control systems achieved robust stability. Moreover, they can tolerate up to four times larger uncertainties than modeled ones. The system with the H_∞ controller attenuates approximately at least 30 times the disturbances with frequency up to 1 rad/s while the system with the LQR controller attenuates at least 10 times the same disturbances.

1. Introduction

The digital transformation in the industry does not ignore the specific branch of hydraulic drives. The transformation of hydraulic drives is evident in the introduction of digital control, which is the reason for forming a new direction called “digital hydraulics” [1,2]. This has motivated a number of research studies and developments. Initially, these have been aimed at developing pilot digital control of various types of hydraulic control devices—pressure valves, flow control valves, directional control valves, etc. The introduction of pilot digital control in research aims to replace the proportional electrical control by means of small two-way two-position (2/2) valves connected in parallel, which are controlled by a high-frequency switching signal. The goal is to reduce the cost of the proportional hydraulic devices with the possibility of high response, compactness, and flexibility in the schematic solutions of the hydraulic drive [3,4]. This idea finds practical application in a number of produced devices such as an electro-hydraulic steering unit for mobile machines [5]. Even more, their capabilities for automated remote control are expanding.

There are not a few examples of industrial hydraulic drives with digital control. Later, this idea was transferred to more complex hydraulic devices, such as devices for regulating the displacement volume of pumps and motors. In this case, the goal of the digital transformation is to replace existing hydro-mechanical controllers [6,7,8], implementing different control laws (flow rate, pressure, and power) with digitally controlled regulators [9]. This allows a combination of all the listed control laws in one controller, which most often represents a high-response proportional hydraulic spool valve with digital control. Automation is developing simultaneously with hydraulics but at a faster pace. In this area, the existing controllers implemented as analog circuits of electronic amplifiers [10] have been replaced by embedded control systems or real-time control systems [11,12,13,14]. Their use for control of hydraulic drives significantly improves energy efficiency [15]. To date, this has motivated a number of studies to develop advanced control algorithms for digitally controlled hydraulic control devices, including controllers for axial-piston pumps and motors [3,16]. One part of the research is devoted to the development of digital P, PI, and PID regulators to replace the classic analog ones realized in the electronic amplifiers of the proportional valve [17,18,19] used to regulate the displacement volume. However, with the development of control algorithms, there are developments in embedded control systems for axial-piston machines based on so-called advanced control algorithms [20,21,22,23,24,25]. Unlike classical ones, these are based on optimality criteria on the plant transfer matrix [26,27,28,29,30]. This allows a better control performance at the expense of a larger computing resource. Such algorithms are the LQR, H_∞, and μ-regulator [30].

The authors have developed a control system for one of the most commonly used types of axial-piston pumps designed for open-circuit hydraulic systems. The purpose of the developed system is real-time control of pump displacement volume (respectively, the flow rate to the hydraulic system) by replacing a conventional hydro-mechanical controller with a proportional spool valve. The valve’s analog electronic amplifier has been replaced with an industrial controller. The control laws are realized in a mobile workstation in a suitable programming environment and the connection with the controller is made through CAN-USB communication. Different types of controllers were studied and were published by the authors.

The main contribution of this article is the development of a two output, one input axial-piston model with unstructured output uncertainties. Also, another contribution is the results from the comparison of two advanced control techniques that showed high control performance of the same hydraulic plant. These are the LQR and H_∞ controllers. The main idea of the comparison is to analyze their robust stability and the robust performance of the displacement volume control of an axial-piston pump. The study of both controllers was carried out under real experiment and numerical simulation conditions.

The other contribution of this paper is the development of a structure of an electro-hydraulic system with displacement volume control of existing hydraulic cylinders or motors that have feedback with respect to velocity and load. In contrast to the existing systems, our development uses a standard open-loop circuit axial-piston pump, which has various advantages. Such type of axial-piston pump is not recommended for hydraulic systems with secondary control. In addition, this significantly reduces the cost of the drive system. Another novelty is the designed controllers that ensure robust stability of the closed-loop system on the basis of an estimated uncertain model.

This article is organized as follows: Section 2 presents a short plant description and the main results for the estimated plant model; in Section 3, a comparison of the frequency domain between advanced multivariable control systems is shown; Section 4 presents the comparison of the experimental results obtained with the design control algorithms; and Section 5 presents a short conclusion.

2. Plant Uncertainty Modeling

2.1. Plant Description

A system has been developed that enables the experimental study of different types of control laws for real-time displacement volume regulation of an axial-piston pump. The system was realized by the authors as a laboratory test bench. The test bench consists of two subsystems—hydraulic and control. Figure 1 shows a combined diagram of the test bench showing both the hydraulic and control subsystems. A detailed description of the laboratory system was given in a previous work by the authors [11] but only a brief one is given in this article.

The hydraulic subsystem consists of a reservoir and a motor pump group and a loading part. The tank volume is 130 L and serves as a supporting frame for the bench, as it allows both the motor pump group and the loading part to be placed on its cover for compactness. The pump is known in the practice of type A10VSO hydraulic drives with a displacement volume of 18 cm³. This type of variable displacement axial-piston pump is intended for hydraulic systems with open circulation of the working fluid and is not suitable for hydrostatic transmissions for mobile machines or industrial hydraulic systems with secondary control [17]. The electric motor driving the pump is three-phase asynchronous with a power of 7.5 kW and a nominal rotation frequency of 1450 min⁻¹. The displacement volume of the pump is controlled by a type VT-DFP electro-hydraulic proportional spool valve [31]. This type of valve is selective for this type of pump and it is installed modularly in the same place as the type DR (Druck Regler) conventional hydro-mechanical pressure controller [32]. In order to use the hydro-mechanical controller as a pressure limiter in case the electro-hydraulic control fails, it is connected in parallel to the proportional valve. This parallel connection between the DR and the VT-DFP is realized by a specially designed hydraulic block mounted on the pump. On the other hand, the block allows the inclusion of a pressure transducer in the pump output. The loading part consists of a precision hydraulic throttle check valve with manual control connected to the pump discharge pipeline to change the pressure load.

The control subsystem consists of an electronic amplifier VT-5041 (Bosch Rexroth, Lohr am Main, Germany), an industrial controller MC012-022 (Danfoss, Nordborg, Denmark), and a portable workstation. In the environment of the MATLAB/Simulink^® (ver. R2009b) software product, various control algorithms are implemented that form a signal according to the assignment to the industrial controller MC012-022 [33]. The connection between the workstation and the controller is made through a USB/CAN interface CG150. The controller calculates the parameters of the control pulse-width modulated signal supplied to the power stage of the electronic amplifier. The electronic amplifier drives the valve’s proportional solenoid and processes the spool position feedback signal in the valve. In this way, the control subsystem enables real-time control through an algorithm implemented by a Simulink^® model with a 10 ms sampling time, ensuring precise control of the pump parameters. The laboratory test bench realization is presented in Figure 2.

2.2. Uncertainty Model Identification

The numeric values of the axial-piston pump parameters very often are not exactly known and/or fully unknown. This makes it impossible to use only an approach of analytical modeling to obtain an appropriate pump model. The alternative way to build a mathematical model is to estimate the so-called “black box” model from experimental data [34]. Building such a model does not require information for pump parameters. Moreover, the model identified from experimental data allows not only a nominal plant model to be obtained but, in addition, it is possible to estimate the bounds of uncertainty in order to obtain an uncertainty model. It should be noted that the estimated model is valid only for the developed test bench. This is due to the fact that it is estimated from experimental data on the basis of the “black box” approach. The nominal axial-piston pump model is obtained in a previous author’s work [35]. The new contribution of this paper is the developed uncertainty axial-piston pump model. Further, this model is used to make analyses of the robust stability and performance of both control systems that are based on an LQR controller and an H_∞ controller. The nominal axial-piston pump model is introduced as a two-output single input discrete-time 6th order state space model

$$x(k+1)=A x(k)+B u(k)+K e(k)$$

(1)

$$y(k)=C x(k)+D u(k)+e(k)$$

(2)

where the state vector is denoted by $x(k)={\left(x_1,x_2,\cdots ,x_6\right)}^T$, the output vector is denoted by $y={\left(y_Q,y_p\right)}^T$ ($y_Q$ is the first output (pump flow rate in L/min), $y_p$ is the second output (pump output pressure in bar)), the control signal is denoted by $u$ (pulse width modulation signal in mV), and the residual error vector is denoted by $e={\left(e_Q,e_p\right)}^T$. The values of nominal model parameters are

$$\begin{array}{l}\mathrm{A}=\left(\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ -0.3& 0.43& 0.84& -0.01& 0.04& -0.03\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0.57& -11.17& 10.83& 0.72& -1.62& 1.9\end{array}\right),\mathrm{B}=\left(\begin{array}{c}0.005\\ 0.161\\ 0\\ 0.209\\ 0.046\\ 0.058\end{array}\right),\\ \mathrm{K}=\left(\begin{array}{cc}0& 0.075\\ 0.082& -0.015\\ 0.031& 0.031\\ 0.423& 0.099\\ -0.033& 0.259\\ 0.141& -0.029\end{array}\right),\mathrm{C}=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\end{array}\right)\mathrm{and}\mathrm{D}=\left(\begin{array}{c}0\\ 0\end{array}\right)\end{array}.$$

(3)

The matrix parameter values in Equation (3) describe the nominal plant dynamics in specific operation modes. The model sample time is $T_S=0.01\mathrm{s}$. It should be noted that the estimated model is valid only for the developed test bench. In Figure 3, a comparison between the measured outputs and model outputs is depicted. Also, the data sets presented in Figure 3 are different from the data sets used for the model estimation, and all data are centered. The obtained values of FIT between the measured and model outputs are sufficiently large. In Figure 4, the residual error test of the nominal model is presented. The results prove that the parameter’s estimates of the nominal model are unbiased. This means that the true parameter’s values fall with a probability tending to 1 in confidence intervals determined as three standard deviations from the estimated values. The confidence intervals are used to obtain the bounds of structured (parameter) uncertainty. However, the model (3) has many uncertain parameters that will complicate the robustness analysis of the control system. To overcome this issue, a model with so-called unstructured uncertainties can be derived. This is obtained by approximating the parameter (structured) uncertainty as uncertainty in frequency responses (unstructured uncertainty). This is carried out using the Gauss approximation formula.

The magnitude responses for both output channels (pump flow rate and pump output pressure) of the identified model are represented as a random variable with normal distribution. The parameters of it depend on the standard deviation of the model parameters ${\sigma }_{\theta }$. Thus, it forms a confidence region around each frequency calculated as

$$\begin{array}{l}\left|G_1(j\omega +3{\sigma }_{{\theta }_1})\right|=\left|G_{1,nom}(j\omega )(1+G_{\max ,1}(j\omega ){\Delta }_1)\right|,\\ \left|G_2(j\omega +3{\sigma }_{{\theta }_2})\right|=\left|G_{2,nom}(j\omega )(1+G_{\max ,2}(j\omega ){\Delta }_2)\right|\end{array},$$

(4)

where subscript “1” is used for the pump flow rate output and subscript “2” is used for the pump output pressure, $G_{1,nom}$ and $G_{2,nom}$ are the magnitude response corresponding to the nominal model (3), ${\sigma }_{{\theta }_1}$, ${\sigma }_{{\theta }_2}$ are the standard deviations of the nominal model parameters, $G_{\max ,1}(j\omega )$ and $G_{\max ,2}(j\omega )$ are the magnitude response bounds with respect to the pump flow rate and pump output pressure, respectively, and $\left|{\Delta }_1\right|\le 1$, $\left|{\Delta }_2\right|\le 1$ are scalar uncertainties. In order to parametrize functions $G_{\max ,1}(j\omega )$ and $G_{\max ,2}(j\omega )$, they are approximated by stable minimum phase discrete time filters. We use approximation with filters of 5th orders for both output channels. The obtained discrete-time transfer functions for $G_{\max ,1}(j\omega )$ and $G_{\max ,2}(j\omega )$ are

$$\begin{array}{l}{G}_{\max ,1}(z)={\displaystyle \frac{0.2389z^5-0.7547z^4+1.081z^3-0.9286z^2+0.4445z+0.08076}{z^5-2.15z^4+1.739z^3-0.3649z^2-0.821z+0.5987}},\\ G_{\max ,2}(z)={\displaystyle \frac{0.1289z^5-0.3532z^4+0.3597z^3-0.1724z^2+0.0353z+0.0016}{z^5-2.222z^4+1.953z^3-0.7306z^2-0.4909z+0.4904}}\end{array},$$

(5)

In Figure 5 and Figure 6, the exact maximum perturbations and approximated ones for both output channels are depicted. It is seen that the 5th order filters are approximated very well for the exact maximum deviation from the nominal model.

The family of bode plots of the developed uncertain model are presented in Figure 7 and Figure 8. They are obtained for 30 randomly chosen values of prescribed model uncertainty. As can be seen, the most significant uncertainty occurred for plant static gain. It should be noted that there is a significant difference between the smaller time constants of the plant model.

The structure of the estimated uncertain model is depicted in Figure 9. It will be used in the robustness analysis in Section 3.3 according to the framework based on the structured singular value [30] (Section 3.8), [36] (Chapter 11).

3. Advanced Multivariable Controller Design

In this section, two alternative control techniques for closed-loop optimization are presented. The scheme of the closed-loop control system is depicted in Figure 10, where the block named “Controller” is the LQG controller (linear quadratic regulator with Kalman filter as state estimator) or H_∞ controller (H-infinity controller). Both controllers are designed on the basis of the nominal state space model (3). The LQR appears as a solution to the dynamic programming problem for a linear plant model and quadratic cost function over the infinite time horizon. On the other hand, the H_∞ approach aims to minimize the worst-case impact of the exogenous closed-loop signals on weighted system performance. First, the designed controllers using the identified model of the axial-piston pump with respect to the flow rate are briefly presented [37,38]. Then, the controller performance is examined in the time domain and the frequency domain.

3.1. Linear Quadratic Controller Design

In this setting, the cost function is minimized

$$J_{LQR}={\displaystyle \sum }_{k=0}^{\infty }{q}_1{I}_e\left(k{T}_s\right)+q_2{C}^{T}x{\left(k{T}_s\right)}^{T}x\left(k{T}_s\right)C+r_{1}u\left(k{T}_s\right),$$

(6)

where $x$ is the current state of the axial-piston pump model at discrete time $t=k{T}_s$ seconds, with sampling time $T_s=0.01$ s, $u$ is the control signal sent to the spool valve coil amplifier

$$I_e\left(k{T}_s\right)={\displaystyle \sum }_{l=0}^k\left(Q_{ref}\left(l{T}_s\right)-Cx\left(l{T}_s\right)\right)T_s,$$

(7)

is the discrete time integral of the flow rate deviation from the desired level of $Q_{ref}$. The weights $q_1=500$, $q_2=0.001$, and $r_1=0.001$ are selected to reflect the relative importance of the performance terms and by observing the amplitude limits for the control signal in a simulation. The conditions for positive definiteness for the weight matrices as well as the controllability of the $\left(A,B\right)$ pair are satisfied. The resultant controller is obtained with

$$K_{LQR}=B^{T}PA/r_1,$$

(8)

where $P$ is a positive definite solution of the algebraic Riccati equation

$$A^{T}PA-P-A^{T}PB{\left(B^{T}PB+R\right)}^{-1}{B}^{T}PA+Q=0,$$

(9)

and determines the optimal cost as $J_{LQR}=x_0^{T}P{x}_0.$ The matrix is chosen as $Q=diag\left(q_1,q_2{C}^{T}C\right)$. The control signal from the LQR is obtained as state feedback

$$u\left(k{T}_s\right)=K_{LQR}\widehat{x}\left(k{T}_s\right),$$

(10)

with $\widehat{x}$ representing an asymptotic state estimate $\widehat{x}\to x$, as $k\to \infty$. For state estimation, the designed Kalman filter is used as an observer

$$\widehat{x}\left(\left(k+1\right)T_s\right)=A\widehat{x}\left(k{T}_s\right)+Bu\left(k{T}_s\right)+L\left(y\left(k{T}_s\right)-C\widehat{x}\left(k{T}_s\right)\right),$$

(11)

where the observer gain matrix is

$$L=\left(AD{C}^T+G{Q}_e{H}^T\right)\left(CD{C}^T+R_e+H{Q}_e{H}^T\right).$$

(12)

The covariance matrix of state disturbance $Q_e=I_2$ is taken as a 2x2 identity matrix, the covariance of measurement noise is set as $R_e=0.01I_2$, and $D$ is the steady state covariance that is obtained as a solution of the discrete algebraic Riccati equation.

3.2. H_∞ Controller Design

With the design of the H_∞ controller, the infinity norm of the closed-loop system is minimized

$$J_H={\Vert \begin{array}{c}{W}_p{T}_{yQref}\\ W_u{T}_{uQref}\end{array}\Vert }_{\infty },$$

(13)

where $Q_{ref}$ represents the exogenous flow rate command signal and $T_{y{Q}_{ref}}$ is defined as the transfer matrix in the $Z$ domain

$$\left(\begin{array}{c}{Q}_{ref}\left(z\right)-y_Q\left(z\right)\\ I_e\left(z\right)\\ -y_P\left(z\right)\end{array}\right)=T_{y{Q}_{ref}}\left(z\right)Q_{ref}\left(z\right),$$

(14)

where $y_Q$ is the flow rate output of the model and $y_P$ is the pressure output signal from the model, $I_e$ is the integral of the flow rate tracking error, and $z=e^{-s T_s}$ is the complex frequency variable. The transfer matrix $T_{y{Q}_{ref}}$ is obtained from the nominal state space model (3). The weighting matrix

$$W_p=diag\left(\mathcal{Z}\left({\displaystyle \frac{s+50}{5 s+50}}\right),\mathcal{Z}\left({\displaystyle \frac{s+0.025}{5 s+0.025}}\right),0.005\right),$$

(15)

specifies the performance requirements for the signals of the closed-loop system in the frequency domain. Similarly, the weighting matrix for the control signal is set at $W_u=0.011$. The controller

$$K_H=argmin J_H,$$

(16)

is found using the linear matrix inequality (LMI) optimization method.

3.3. Robust Stability and Performance Analysis

Figure 11 compares the singular values of both controllers. Note that the integrator block is not included as part of the controller in this examination and is left as an external block. Both controllers have similar amplification for frequencies between 0.01 and 100 rad/s, though the singular value for the LQR is higher than the H_∞ controller for all frequencies between 5 and 20 dB. Therefore, the feedback gain of the LQR will be stronger and we would expect better performance of the closed loop with this controller. The LQR has a resonance frequency between 60 and 70 rad/s with a peak amplification of 60 dB. And the H_∞ controller has a peak amplification of 48 dB at 10 rad/s. The LQR demonstrates considerable amplification in the high-frequency range, indicating that eventual sensor noises will increase the oscillation in the control signal. In the low-frequency range, the amplification of the LQR is at 53 dB and for H_∞ it is at 45 dB.

In Figure 12, Figure 13 and Figure 14, the results from the robustness analysis in the time domain and frequency domain of both control systems are presented. The closed-loop characteristics are evaluated for 30 randomly chosen values of a prescribed model uncertainty. In Figure 12, the step responses of the closed-loop system with both controllers are plotted. The H_∞ controller led the system to a shorter rise time compared to the LQR. The response of the LQR contains an evident oscillatory component with a frequency of about 10 Hz. The settling time of both controllers is similar, at about 0.6 s. Hence, both controllers are tuned to achieve similar performance in the simulation and further examination of this can be carried out in an experimental setting. Additionally, it is seen from Figure 12 that parametric disturbances do not lead to changes in the quality or range of the responses.

Figure 13 presents the family of output sensitivities of the closed-loop system with both controllers and the inverted transfer function of the corresponding weighting filter $W_e$. The output sensitivities indicate the system response with respect to output loading disturbances. It is seen that both controllers achieve good attenuation of low-frequency disturbances up to 1 rad/s (the LQR controller reduces the disturbance influence by approximately 10 times and the H_∞ controller by approximately 30 times).

Figure 14 examines the family of complementary sensitivity functions of the closed-loop system with both controllers, where the bandwidth of the system can be observed, which is around 3–4 rad/s. The bandwidth of the H_∞ controller appears a little larger than the LQR, extending up to 10 rad/s. The complementary sensitivity functions are also robust against parametric disturbances due to uncertainty.

The investigation of the system’s robust stability is performed with the help of the structure singular value (μ), which looks for higher parametric disturbance to make the closed-loop system unstable [30] (Section 3.8), [36] (Chapter 11). For the aim of the robust stability analysis, the closed-loop system with the uncertain model is transformed into a so-called $M-\Delta$ structure, shown in Figure 15.

In Figure 15, block $M=F_l(G_{nom},K_{•})$ denotes the transfer function matrix of the nominal part of the control system that is separated from the uncertain part of the control system denoted by $\Delta$. $F_l(G_{nom},K_{•})$ denotes a transfer matrix of a lower linear fractional transformation of the nominal plant and controller. It can be shown that, in case of uncertain model (4),

$$F_l(G_{nom},K_{•})=-G_{\max }{T}_{nom}$$

(17)

where $T_{nom}={\left|I+G_{nom}{K}_{•}\right|}^{-1}{G}_{nom}{K}_{•}$ is the nominal complementary sensitivity, $G_{nom}={\left[\begin{array}{cc}{G}_{1,nom}& G_{2,nom}\end{array}\right]}^T$ is the transfer matrix of the nominal plant, and $K_{•}$ is the transfer matrix of the LQR or H_∞ controller. The exact value of μ for the structure depicted in Figure 16 cannot be calculated analytically; hence, only the upper and the lower bounds are approximated. However, in our case, both bounds are pretty close and overlapping. Therefore, only the upper bound of μ is plotted in Figure 14.

As can be seen, both systems preserve robust stability for the frequency range of 0.01 to 200 rad/s. The peak μ value is below −10 dB or around 0.3, which means the system would preserve its stability even for a 300% higher norm of uncertainty. The peak μ is achieved at 70 rad/s for the LQR and at 100 rad/s for the H_∞ controller.

Finally, it should be noted that the developed uncertain model is estimated in a mode of maximal load condition. The simulation results show that the designed control systems are sufficiently insensitive with respect to model uncertainties. This provides motivation to implement designed control algorithms in actual control devices and to test them in real experiments under various modes.

4. Experimental Study

To compare the experimental results for both controllers, additional low pass filtering is applied to the measured flow rate because the differences between the performances of both controllers are small and can remain hidden in the measurement noise. The low pass filtering is tuned to not interfere with the bandwidth of the closed-loop system to keep the important features of the transient response. The low pass filter is specified as a discrete-time transfer function with a unit static gain and pole at 0.7, leading to an aperiodic filter response

$$W\left(z\right)={\displaystyle \frac{0.3}{z-0.7}}.$$

(18)

At the sampling frequency of 10 ms, pole 0.7 corresponds to 35 rad/s, which is around 10 times (1 decade) higher than the closed-loop bandwidth. The system is examined experimentally for varying loading disturbance applied as a restriction of the pump outlet with the help of a throttle check valve with manual adjustment. The throttle valve scale is precisely calibrated in a millimeter range, so four loading steps corresponding to 2, 3, 4, and 5 mm of flow restriction are set. During the pump loading, the swash-plate reaction is transferred over to the spool valve, which requires compensation from the control system, since many nonlinearities in the axial-piston pump model control system will face different realization of model uncertainty.

For the highest restriction corresponding to 2 mm, the highest pressure of up to 100 bar is actually seen in the pump outlet peaks and the gain of the pump with respect to the flow rate will be highest. In comparison, in Figure 17, a very similar performance for both controllers is seen with a settling time around 0.1–0.2 s during increasing flow rate demand and 0.2–0.3 s during decreasing flow rate. In this mode of loading, the flow rate range of control is between 14 to 18 L/min, restricted by the capabilities of the electronic amplifier for driving the spool valve coil. The observed random irregularities in the flow rate can be attributed to sensor noise, flow meter implementation as a displacement volume machine, and possible flow transients.

For the throttling position of 3 mm from the valve scale (Figure 18), where the range of pump operation is between 18 to 24 L/min, a similar settling time is seen; however, the response of the LQR is more oscillatory at a high flow rate command compared to the H_∞ closed loop. The observed oscillations have a frequency of around 3 Hz and become mostly dampened with time. Similar oscillatory behavior was observed in the simulated step response. However, the amplitudes are not relevant for the pump application field because they are comparable to the unfiltered noise level. On the other hand, the H_∞ controller response is less oscillatory but experiences a longer settling time at low flow rates, at about 0.4 s.

At the loading corresponding to the throttling position of 4 mm (Figure 19), differences between the controllers are more pronounced. The H_∞ controller is slower than the LQR and more conservative as a response. The settling time for this controller reaches close to 1 s. On the other hand, the LQR is more oscillatory at a high flow rate with an amplitude of variation of about 0.5 L/min. Interestingly, the oscillations at a low flow rate range are highly dampened. This can be attributed to resonances related to the axial-piston pump constructive elements.

Finally, the performance at low pump loading is presented in Figure 20, allowing the highest flow rate up to a nominal of 27 L/min. Here, the static gain of the pump model with respect to the flow rate is the smallest because of the low pressure in the pump outlet. Therefore, it is seen that the H_∞ controller is unable to reach the flow rate command and works with a small steady-state error of about 0.5 L/min. On the other hand, the higher gain of the LQR allows the system to reach the commanded flow in the examined time window.

The following figures investigate further details of the control system dynamics with both designed controllers. A common trade-off in control theory is between the frequency spectra of the control signal and the achieved closed-loop performance. Hence, the examination of the control values calculated by the LQR for the different loading conditions is depicted in Figure 21. For the highest loading conditions corresponding to valve openings of 2 and 3 mm, well-damped control action with an amplitude range of about 100 mV in a steady state is seen. For loading conditions corresponding to a 4 mm valve opening, a large overshoot in the control transient response during an increased flow rate demand is observed. Similar behavior is also observed for a valve opening of 5 mm, with a large bias in the steady-state level too. This can be explained by varying open-loop static gain with loading, which requires additional control energy for a lower loading range.

Figure 22 shows the control signal for the system implementation with the H_∞ controller. Here, the control signal levels in the steady state for changing loading conditions are bundled together. A higher noise content can be seen at a high flow rate. The frequency of the steady state noise is also higher compared to the one observed in the LQR system. This can be explained by the higher magnitude response (Figure 11) of the H_∞ controller beyond 100 rad/s. Also, no overshoots in the H_∞ control action can be seen but rather an initial jump in response due to proportional gain followed by mild integral action until reaching the steady-state level. In conclusion, it can be said that the H_∞ controller exploits some higher frequency sensitivity of the open-loop system with respect to the flow rate to guarantee a fast response with less deviation in control amplitudes compared to the LQR.

In addition to the control signal, two additional signals (the LVDT feedback and pump output pressure) are examined. They are measured in real time. These signals reveal the propagation of the calculated control action throughout the electrohydraulic system components. First, Figure 23 examines the proportional spool valve position measured by the LVDT for the case of the LQR and, second, Figure 24 shows the same signal for the system implemented with the H_∞ controller. Generally, the spool valve follows the inverted control command. It is interesting to see that for the LQR with the lowest loading setting at a 5 mm opening, the valve position signal has a saturation at 0%, which explains the elevated steady-state level of the control signal for this loading with the integrator windup.

The available spool valve operating range is from 0 to about 50% due to limitations of the utilized electronic amplifier. Hence, it is also seen close to saturation conditions for some of the loading cases where a minimal flow rate is desired. A good feature of the H_∞ controller is that it operates the spool valve farther from its saturation zone (Figure 24), keeping the spool valve opening between 5 and 45% at the cost of increased frequency content of the control signal. Therefore, it can be concluded that the H_∞ controller better exploits the available dynamic capacity of the spool valve and, in a sense, compensates for the limitations imposed by the electronic amplifier saturation. The higher frequency content in the spool valve action is not problematic from a mechanical perspective.

Finally, the pump output pressure achieved with both controllers is shown in Figure 25 and Figure 26. The pump pressure in our setup is indicative of applied loading at the pump outlet with the throttle valve. Therefore, the mean value of the pressure signal increases with a reduction in the loading valve opening from 20 to 120 bar, where 120 bar is the set point for the system pressure relief valve. The differences in the outlet pressures for the LQR and H_∞ systems are mostly at a low loading range, i.e., a loading valve opening of 4 and 5 mm. The LQR system for loading 5 reacts with a pump output pressure between 15 and 20 bars, while the H_∞ system for the same loading reaches output pressure between 25 and 27 bars at the same flow rate setpoint. Hence, two close but different pressure outputs for the same flow rate level are seen, which cannot be observed in a steady-state pump characteristic. The explanation is in the way both controllers achieve the flow rate setpoint for this loading. The LQR chooses a “classical” low-frequency approach to slowly increase the control the signal until it reaches the desired setpoint, while the H_∞ controller is characterized by higher oscillations around the steady-state control level. Therefore, the conclusion here, which deserves more attention in future research, is that the steady-state pump parameters may depend on the frequency content or SNR of the applied control signal to the spool valve. The following metrics of the control system performance

$$Q_{Err}=\sqrt{{\displaystyle \sum _{k=0}^n{\left(Q_{ref}(k)-y_Q(k)\right)}^2}}$$

(19)

$$u_c=\sqrt{{\displaystyle \sum _{k=0}^{n}u{(k)}^2}}$$

(20)

are evaluated and presented in

Table 1. Performance indices obtained by experimental results.

	${\mathit{Q}}_{\mathit{E}\mathit{r}\mathit{r}}$		${\mathit{u}}_{\mathit{c}}$
	L/min		V
Loading	LQR	H∞	LQR	H∞
2	22.8572	19.6470	3.8577	3.4014
3	27.2118	25.5429	3.4245	2.8730
4	30.7123	27.5488	4.1221	3.6266
5	30.6101	34.8291	6.4099	3.4725

.

As can be seen, the flow rate error (19) of the control system with the H_∞ controller is a little bit smaller than the ones of the control system with the LQR controller. The only exception is the flow rate error for the mode with the lowest loading. This is in accordance with the obtained steady-state error (Figure 20). The index (20) of the control signal for the system with the H_∞ controller is smaller than the one for the control system with the LQR controller. This indicates that the H_∞ controller is more efficient.

5. Conclusions

The current paper was based on previous authors’ research on a test bench for axial-piston pump flow rate control using either linear quadratic or H_∞ criteria for optimization. The main contribution of the paper is the developed multivariable uncertainty model of the axial-piston pump. Another contribution is the evaluation of the robust stability and performance of the control system with these regulators in simulation and experiments to compare both approaches and to establish a recommendation for the future design of axial-piston pump control systems.

The axial-piston pump is a complex hydraulic unit and its dynamic model includes nonlinear terms reflecting swash plate swivel angle dynamics, working fluid action in the piston chambers, and trigonometric terms due to rotation. Additionally, the hydraulic actuator for changing, the electro-hydraulic proportional spool valve, and the electronic amplifier for it further complicate the dynamic behavior of the system. However, using a system identification approach, the system is presented as a “black box” linear time-invariant state space model with uncertainty.

The uncertain model represents the complexity of the actual system as an infinite set of linear time-invariant models related to various operating conditions for the pump. From the experimental data for various loading, it is seen that pump behavior is experiencing a level of variation that cannot be explained with a single linear model. Therefore, the comparison of the LQR and H_∞ controller response is performed using the uncertain model set. As a result, both controllers demonstrate good performance in simulation, satisfying the performance requirement, presenting a similar transient response and accuracy. Running the controllers in the test bench revealed interesting insights into their performance in relation to the loading pressure at the pump outlet. The LQR controller was characterized by a faster response and no steady-state error, at the cost of exciting some oscillation waveform for the middle level of loading. On the other hand, the H_∞ controller established a higher margin of stability of the system at the cost of lower steady-state accuracy and slower response to disturbances. Nevertheless, further optimization and tuning of both control structures is possible and it should not be definitively stated that one is better than the other. From a complexity standpoint, the order of the H_∞ controller is higher than the LQR and the design is also generally faster.