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Abstract

The stroke control of a hydrostatic transmission (HST) has an important role for heavy-duty, on-highway, and off-road vehicles like agricultural and construction equipments along with mining and forestry vehicles to increase energy efficiency and power variability. The author proposes a multiple Model Predictive Controller (MPC) and multiple Model Reference Adaptive Control (MRAC) as a new control strategy for stroke control of HST, and compare their performances. The proposed controllers are based on a linear HST system mathematical model to track complex stroke reference trajectories of primary and secondary regulations under the realistic disturbances such as engine and load torques as a result of type of soil and field road conditions. The results of the Processor in the Loop simulations are encouraging in terms of the MPC having advantages over the MRAC and classical control strategies like being able to be designed for multi-input-multi-output systems, insensitive to external stochastic disturbances such as white noise as well as modeling errors and uncertainties to be implemented on an HST for real-time implementation.

Keywords

heavy-duty vehicles hydrostatic transmission stroke control multiple model predictive control primary and secondary regulation disturbance rejection

Introduction

A hydrostatic transmission (HST) can be defined as a ‘pump-controlled motor’.¹ It has a shaft as mechanical input, one or more shafts as mechanical output, and along with a number of internal volumetric machines like a hydraulic pump driven by an engine and one or more hydraulic motors driven by oil discharged pump.^2,3 The hydraulic pump converts the input mechanical energy into pressurized fluid and the motor(s) convert(s) it back to mechanical energy to drive the vehicle.⁴ Meanwhile, the energy is transmitted from the pump to the motor by oil through flexible lines or hydraulic hoses. HST can operate at the maximum pressure rating of high-performance pumps with the help of these lines or hoses.⁵ HSTs are classified into two categories: the ones with fixed displacement pumps and motors; and the other ones with variable displacement pumps and/or motors.⁶ HST transmits power from the prime mover to the load like a gearbox in case of having fixed displacement pumps and motors. However, it can regulate power in case of having variable displacement pumps and/or motors.⁷ Even though HST has lower efficiency comparing with stepped transmissions; insuring uninterrupted power to wheels during shifting, improved maneuverability, and dynamic braking; operating over a wide range of torque/speed ratios; delivering an infinitely variable or stepless output speed from a constant input speed and a constant speed from a variable input speed such that achieving the vehicle’s stepless speed regulation; producing a maximum torque output from a minimum power input; transmitting high power with low inertia; remaining stalled and undamaged or overheated under full load; and providing flexible equipment layout in design of a vehicle or a machine due to the connection of the pump and motor with flexible hoses are amongst the most important advantages of it.^3,5

HSTs have been widely used in heavy-duty, on-highway, and off-road vehicles like agricultural and construction equipments along with mining and forestry vehicles to increase energy efficiency and power variability for more than half a century.^1,8–10 They have a very large usage area like tractor,^11,12 forklift,¹³ excavator,¹⁴ gas turbine,¹⁵ ATV,¹⁶ hydraulic elevator system,¹⁷ robotic systems,^18–22 wind turbine.^9,23

Linear quadratic gaussian algorithms based on a linear state-space model of an HST for the velocity regulator design of hydrostatic transmissions were suggested.²⁴ A velocity tracking controller for HSTs was designed utilizing gain scheduled full state feedback was designed.²⁵ Linear Quadratic Regulator (LQR) is designed based on the linear state-space model of the secondary-regulation HST system.²⁶ A sliding-mode approach with disturbance compensation by a nonlinear reduced-order disturbance observer for the tracking control of a hydrostatic drive train was proposed.²⁷ Then they proposed a backstepping second-order sliding mode control approach for the tracking control of the motor angular velocity of the HST.²⁸ Comparison of the performances of a new type of self-tuning fuzzy PID (Proportional Integral Derivative) controller and classical PID controller for speed control of the HST system was presented.²⁹ A trajectory tracking control application of motor angular velocity and pressure difference using Lyapunov techniques in combination with a disturbance observer for a drive chain with HST was presented.³⁰ The generalized control law for the HST ratio for a hybrid driveline including kinetic energy storage with constant pressure system was investigated.³¹ The performance of a gain scheduled LQR with feedforward control and disturbance compensation applied to a wind turbine with HST were investigated.³² A hydrostatic variable-ratio system of a wind power turbine drive train to produce constant electric power at varying wind speeds by controlling the variable displacement hydraulic pump was presented.³³

The importance of usage of MPC has been grown up in on-line applications with the help of modern computers. The design formulation to have a completely multivariable framework with hard and soft constraints, and overcoming the problems of model uncertainties and external disturbances are amongst the most important advantages of MPC.³⁴ It has been preferred in the usage of various agricultural³⁵ and automotive^36–38 applications. Two MPCs based on linear state-space model of propulsion system were implemented on a New Holland combine harvester to develop a cruise control system by controlling engine speed and the pump setting. The first one minimizes the speed error and the control effort, in addition to them, another one minimizes the engine speed.³⁹ Gradient and Newton-Raphson-based fast nonlinear model predictive controllers are investigated and compared both for the tracking of the HST’s motor angular velocity.⁴⁰ Two distributed nonlinear model predictive controllers were designed in order to track the speed of the hydromotors of the HST and their performances were compared with PID.⁴¹ A nonlinear model predictive controller for the energy management of nonhybrid HSTs in terms of the velocity-tracking and fuel economy was designed.⁴² The design and evaluation of the linear MPC method were designed to track engine speed, engine torque, accumulator pressure, and vehicle speed; and compared with the PID-based system under two typical driving cycles in a hydraulic hybrid powertrain system.^43,44 An MPC to track a desired hydraulic motor speed while constraining efficient engine operation by regulating the throttle command, pump displacement, and valve opening was devised.⁴⁵ A predictive neural network controller was applied to control the rotational speed of the hydraulic engine of HST.⁴⁶

In this study, a multiple MPC consisting of two linear MPCs based on linear HST system mathematical model¹⁰ was designed as a new control strategy for stroke control of HST on the strength of the advantages of MPC. Also, a Multiple Model Reference Adaptive Controller (MRAC) was proposed in order to compare the performance of both controllers. The performance of the controllers applied to the nonlinear HST model was tested under the deterministic and stochastic disturbances like engine and load torques as a result of the type of soil and field road conditions and a complex reference trajectory being the combination of both ramp and step functions in order to obtain the system response in primary and secondary regulations of both normal and optimal drive modes.⁴⁷ Processor in the Loop (PIL) simulations were carried out in order to handle computational errors and communication delays during the simulations. The results of the simulations are encouraging in terms of the MPCs to be implemented on an HST for real-time implementation.

Mathematical modeling of hydrostatic transmission

There are lots of studies about the modeling of HST in the literature.^{4,8,10,30,48–51} Among them, the mathematical model describing the dynamics of an HST with variable displacement pump and motor¹⁰ is used on the advantages of MPC design.

According to the HST system in Figure 1, the relations about engine, hydraulic pump, hydraulic motor, and hose must be given. The rotation of the engine and the pump can be described as follows:

(J_{p} + J_{d}) {\overset{\cdot}{ω}}_{e} = T_{d} - T_{fp} - T_{p}

(1)

Figure 1.

Block diagram of the HST system.

where $J_{d}$ and $J_{p}$ denote the moment of inertia of the diesel engine and pump, respectively; $ω_{e}$ is the engine angular velocity; $T_{d}$ is the engine torque; $T_{fp}$ is the friction aﬀecting the engine and $T_{p}$ is the torque from the pump to the engine. $ω_{e}$ is equal to the pump angular velocity $ω_{p}$ since the engine shaft and pump shaft are rigidly attached to each other. Instead of $J_{d}$ and $J_{p}$ , equivalent $J_{pd}$ can be utilized. Rewriting the equation (1) with respect to equations (2) and (3) describing $T_{fp}$ and $T_{p}$ , the differential equation of the pump-side of the HST can be obtained in equation (4).

T_{fp} = μ_{p} ω_{p} + K_{p}

(2)

T_{p} = \frac{D_{p} Δ p}{η_{tp}}

(3)

J_{pd} {\overset{\cdot}{ω}}_{p} = T_{d} - μ_{p} ω_{p} - K_{p} - \frac{D_{p} Δ p}{η_{tp}}

(4)

where $μ_{p}$ is coulomb friction coefficient, $K_{p}$ is viscous friction on the pump shaft, $D_{p}$ is the displacement of the pump, $Δ p$ is the pressure difference between charge and return pressure, $η_{tp}$ mechanical efficiency of the pump. The motor angular velocity $ω_{m}$ is directly related to the vehicle speed, since hydraulic motor directly actuates vehicle wheel as can seen in the equation below.

(J_{m} + J_{w}) {\overset{\cdot}{ω}}_{m} = T_{m} - T_{fm} - T_{L}

(5)

where $J_{w}$ and $J_{m}$ denote the moment of inertia of the wheel hydraulic motor, respectively; $T_{m}$ is the torque that the motor delivers to the traction model; $T_{fm}$ is the friction like Coulomb and viscous and $T_{L}$ is the load torque applied to the vehicle. To include vehicle dynamics, $J_{w}$ and $J_{m}$ are combined in equivalent $J_{mw}$ . Rewriting the equation (5) with respect to equations (6) and (7) describing $T_{m}$ and $T_{fm}$ , the differential equation of the motor-side of the HST can be obtained in equation (8).

T_{m} = D_{m} Δ p η_{tm}

(6)

T_{fm} = μ_{m} ω_{m} + K_{m} sign (ω_{m})

(7)

J_{mw} {\overset{\cdot}{ω}}_{m} = D_{m} Δ p η_{tm} - μ_{m} ω_{m} - K_{m} sign (ω_{m}) - T_{L}

(8)

where $D_{m}$ is the displacement of the motor and $η_{tm}$ is mechanical efficiency of the motor, $μ_{m}$ is coulomb friction coefficient on the motor shaft, $K_{m} sign (ω_{m})$ is viscous friction which is against the turning direction of $ω_{m}$ . The pressure in the hose can be determined by the ﬂows of oil into and out from the hose, which can be explained as follows:

p = \frac{β}{V} \int_{0}^{t} (Q_{p} - Q_{m} - Q_{v} - Q_{l}) d τ

(9)

where $V$ is the volume of the hose, $β$ is the bulk modulus, $Q_{p}$ is the ﬂow from the pump, $Q_{m}$ is the ﬂow into the motor, $Q_{v}$ is the ﬂow through the valve, and $Q_{l}$ is the leakage. $Q_{v}$ and $Q_{l}$ are negligible, as $Q_{l}$ is minimal and $Q_{v}$ remains inactive below 415 bar, with high pressure affecting swashplate angles beforehand. Equation (13) can be obtained by rewriting equation (9) in terms of equations (10)–(12) describing $Δ p$ , $Q_{p}$ , and $Q_{m}$ , respectively.

Δ p = p - p_{return}

(10)

Q_{p} = D_{p} ω_{p} η_{vp}

(11)

Q_{m} = \frac{ω_{m} D_{m}}{η_{vm}}

(12)

V \dot{Δ p} = β D_{p} ω_{p} η_{vp} - β \frac{ω_{m} D_{m}}{η_{vm}} - V {\overset{\cdot}{p}}_{return}

(13)

where $η_{vp}$ and $η_{vm}$ are the volumetric efficiency of the pump and motor respectively which are a function of displacement and angular velocity. In hydrostatic transmissions oil circulate in close loop, to prevent overheating and contamination problems, flushing pump is used in return line. The change in pressure of the return line is defined as ${\overset{\cdot}{p}}_{return}$ . Most of the working times of the vehicle are in forward and very low speeds, therefore one direction cause simplification on friction and remove $sign (ω_{m})$ and in low speeds torque efficiencies approximated as constant. The effect of the changes on the return line is negligible with respect to main line pressure because of that ${\overset{\cdot}{p}}_{return}$ is zero as a result of $p_{return}$ to be constant with a value of 20 bars.

There are three system variables to define the dynamic behavior of the system. Therefore the state equations of the system can be summarized by means of the equations (4), (8), (13) as follows:

{\overset{\cdot}{ω}}_{p} = \frac{T_{d}}{J_{pd}} - \frac{μ_{p} ω_{p}}{J_{pd}} - \frac{K_{p}}{J_{pd}} - \frac{D_{p} Δ p}{J_{pd} η_{tp}}

(14)

{\overset{\cdot}{ω}}_{m} = \frac{D_{m} Δ p η_{tm}}{J_{mw}} - \frac{μ_{m} ω_{m}}{J_{mw}} - \frac{K_{m} sign (ω_{m})}{J_{mw}} - \frac{T_{L}}{J_{mw}}

(15)

Δ \overset{\cdot}{p} = \frac{β D_{p} ω_{p} η_{vp}}{V} - \frac{β ω_{m} D_{m}}{V η_{vm}} - {\overset{\cdot}{p}}_{return}

(16)

The nonlinear state space model of the HST can be obtained based on equations (14)–(16). It consists of three first order nonlinear differential equations in state space form as shown below.

\underline{\overset{\cdot}{x}} = f (\underline{x}) + g (\underline{x}) \underline{u} + φ \underline{d}

(17)

where

\begin{matrix} f (\underline{x}) = (\begin{matrix} 0 \\ - \frac{μ_{m} x_{2}}{J_{mw}} - \frac{K_{m} sign (x_{2})}{J_{mw}} \\ - \frac{μ_{p} x_{3}}{J_{pd}} - \frac{K_{p}}{J_{pd}} \end{matrix}), g (\underline{x}) = (\begin{matrix} \frac{β x_{3} η_{vp}}{V} & - \frac{β x_{2}}{V η_{vm}} \\ 0 & \frac{x_{1} η_{tm}}{J_{mw}} \\ - \frac{x_{1}}{J_{pd} η_{tp}} & 0 \end{matrix}), \\ φ = (\begin{matrix} 0 & 0 \\ 0 & - \frac{1}{J_{mw}} \\ \frac{1}{J_{pd}} & 0 \end{matrix}), \underline{x} = (\begin{matrix} p \\ ω_{m} \\ ω_{p} \end{matrix}), \underline{u} = (\begin{matrix} D_{p} \\ D_{m} \end{matrix}) and \underline{d} = (\begin{matrix} T_{d} \\ T_{L} \end{matrix}) . \end{matrix}

Model predictive control

MPC comprises an optimization problem that is solved by the quadratic programming method. The MPC control action with the estimates of $x (k)$ at the time $k$ is obtained by solving the optimization problem as follows⁵²:

\begin{matrix} Δ u (k ∣ k), . . ., \min_{Δ u (m - 1 + k ∣ k)} {\sum_{i = 0}^{p - 1} (\sum_{j = 1}^{n_{y}} ∣ w_{i + 1, j}^{y} \\ (y_{j} (k + i + 1 ∣ k) - r_{j} (k + i + 1)) ∣^{2} \\ + \sum_{j = 1}^{n_{u}} ∣ w_{i, j}^{Δ u} Δ u_{j} (k + i ∣ k) ∣^{2} + \sum_{j = 1}^{n_{u}} ∣ w_{i, j}^{u} \\ (u_{j} (k + i ∣ k) ∣^{2} - u_{jtarget} (k + i)) ∣^{2})} \end{matrix}

(18)

subject to:

\begin{matrix} x (k + 1) & = Ax (k) + B_{u} u (k) + B_{v} v (k) + B_{d} n_{d} (k) \\ y (k) & = Cx (k) + D_{u} u (k) + D_{v} v (k) + D_{d} n_{d} (k) \end{matrix}

(19)

\begin{matrix} u_{j, \min} (i) \leq u_{j} (k + i ∣ k) \leq u_{j, \max} (i) \\ Δ u_{j, \min} (i) \leq Δ u_{j} (k + i ∣ k) \leq Δ u_{j, \max} (i) \\ y_{j, \min} (i) \leq y_{j} (k + i + 1 ∣ k) \leq y_{j, \max} (i) \\ Δ u (k + h ∣ k) = 0 \\ h = m, . . ., p - 1 \\ i = 0, . . ., p - 1 \end{matrix}

(20)

In equation (18); .‘ $()_{j}$ ’ denotes the $j - th$ component of an input-output vector, ‘ $(k + i ∣ k)$ ’ denotes the value predicted for time $k + i$ based on the information available as time $k$ , $r (k)$ is the current sample of the output reference. The sequence of input increments being $Δ u (k ∣ k), . . ., Δ u (m - 1 + k ∣ k)$ , finally $u (k) = u (k - 1) + Δ u {(k ∣ k)}^{*}$ , where $Δ u {(k ∣ k)}^{*}$ is the element of the optimal sequence. $w_{i, j}^{Δ u}$ , $w_{i, j}^{u}$ , $w_{i, j}^{y}$ are nonnegative weights for the corresponding variable. The smaller $w$ , the less important is the behavior of the corresponding variable to the overall performance index. Vector $u_{jtarget} (k + i)$ is a setpoint for the input vector. One typically uses $u_{jtarget}$ if the number of inputs is greater than the number of outputs, as a lower-priority setpoint. In equation (19); $A$ is the system matrix, $B_{u}$ is the input control matrix, $B_{v}$ is the measured disturbance matrix, $B_{d}$ is the unmeasured disturbance matrix, $C$ is the output matrix, $D_{u}$ is the feedthrough input control matrix, $D_{v}$ is the feedthrough measured disturbance matrix, and $D_{d}$ is the feedthrough unmeasured disturbance matrix. $x (k)$ is the plant state vector, $u (k)$ is the manipulated variables vector, $v (k)$ is the measured disturbances vector, $n_{d} (k)$ is white Gaussian noise as unmeasured disturbance under the assumption of disturbance models is unit gain, and $y (k)$ is the output vector. In equation (20); $u_{j, \min}$ , $u_{j, \max}$ , $Δ u_{j, \min}$ , $Δ u_{j, \max}$ , $y_{j, \min}$ , $y_{j, \max}$ are the lower and upper bounds on the corresponding variables. As the prediction of the future trajectories of the model performed at the time $k = 0$ is considered and $n_{d} (i) = 0$ is set for all predictions instants $i$ , the prediction model is obtained in equation (21).

\begin{matrix} y (i ∣ 0) = C \\ [A^{i} x (0) + \sum_{h = 0}^{i - 1} (A^{i - 1} B_{u} (u (- 1) + \sum_{j = 0}^{h} Δ u (j)) + B_{v} v (h))] \\ + D_{v} v (i) \end{matrix}

(21)

MPC optimizes its control decisions based on future predictions of system behavior. These predictions rely on accurate state estimations, which are provided by the state observer.

State observer

A state observer is used for reducing the prediction error by updating the plant model states $x (k)$ , the disturbance model states $x_{d} (k)$ , and the measurement noise model states $x_{m} (k)$ at each sampling instant. State estimator can be used providing that the models used in the design of the MPC are observable.⁵² The output of the system is estimated before the states of the system in terms of the known parameters as follows:

\begin{matrix} \hat{y} (k) = C \hat{x} (k ∣ k - 1) + D_{v} v (k) + D_{d} \bar{C} {\hat{x}}_{d} (k ∣ k - 1) \\ + \tilde{C} {\hat{x}}_{m} (k ∣ k - 1) \end{matrix}

(22)

where ‘( $\tilde{C}$ )’ and ‘( $\bar{C}$ )’ denote the output matrices of the measurement noise and the disturbance models, respectively. As can be seen from equation (22) that the estimated output $\hat{y} (k)$ is calculated from the models and the states estimated at the previous sampling instant $\hat{x} (k ∣ k - 1)$ . Then, the states at the current sampling instant can be calculated from $\hat{y} (k)$ as follows:

\begin{matrix} [\begin{matrix} \hat{x} (k ∣ k) \\ {\hat{x}}_{d} (k ∣ k) \\ {\hat{x}}_{m} (k ∣ k) \end{matrix}] = [\begin{matrix} \hat{x} (k ∣ k - 1) \\ {\hat{x}}_{d} (k ∣ k - 1) \\ {\hat{x}}_{m} (k ∣ k - 1) \end{matrix}] + K_{ob} (y (k) - \hat{y} (k)) \end{matrix}

(23)

where $K_{ob}$ is the observer gain obtained by Kalman filtering techniques, which indicates that the current states are estimated from the states estimated at the previous sampling instant and a correction term. The estimated states based on the error between the measured output $y (k)$ and the estimated output $\hat{y} (k)$ are updated by the correction term. The estimated states in equation (23) are used in the calculation of the optimal control law. Also, the estimated states are used for estimating the state at the next sampling instant as follows:

\begin{matrix} [\begin{matrix} \hat{x} (k + 1 ∣ k) \\ {\hat{x}}_{d} (k + 1 ∣ k) \\ {\hat{x}}_{m} (k + 1 ∣ k) \end{matrix}] = \\ [\begin{matrix} A \hat{x} (k ∣ k) + B_{u} u (k) + B_{v} v (k) + B_{d} \bar{C} {\hat{x}}_{d} (k ∣ k) \\ \bar{A} {\hat{x}}_{d} (k ∣ k) \\ \tilde{A} {\hat{x}}_{m} (k ∣ k) \end{matrix}] \end{matrix}

(24)

where ‘ $\tilde{(A}$ )’ and ‘( $\bar{A}$ )’ denote the system matrices of the measurement noise and the disturbance models, respectively. The estimates obtained in equation (24) are used as the estimates of the previous states $\hat{x} (k ∣ k - 1)$ in equation (23) at the next sampling period.

The role of the state observer is critical for ensuring that MPC receives reliable state estimations. Without the observer, MPC would rely solely on noisy and possibly incomplete sensor measurements, leading to inaccurate predictions and suboptimal control actions. The observer compensates for these uncertainties by reducing the impact of sensor inaccuracies, estimating the effect of disturbances that are not directly observable, ensuring a more robust and stable control response providing refined state estimates for better future trajectory calculations.

In order to implement the MPC and state observer algorithms, the following steps are followed:

State estimation: The state observer, based on Kalman filtering techniques, estimates the system states at each sampling instant using the measurement updates and disturbance models, which were given in equations (23) and (24). In this manner, it ensures that MPC receives accurate state estimations, leading to more effective control actions.

Prediction model update: The estimated states are used to update the prediction model given in equation (19), ensuring accurate future trajectory estimation.

MPC optimization: Using the updated states, the MPC optimization problem whose cost function was defined in equation (18) is solved at each sampling instant to determine the optimal control inputs.

Control application: The first element of the computed optimal input sequence is applied to the system, and the process repeats at the next time step.

This structured approach ensures real-time adaptability and robustness of the control scheme.

Multiple model predictive control

Multiple MPC can be used together when there is a set of pre-designed MPC’s based on current operating conditions, measured outputs, references, and disturbances and a need to switch the appropriate MPC. The active controller solves a quadratic program to determine optimal manipulated variables, while inactive controllers perform state estimation to ensure seamless switching. This approach enhances control performance under varying conditions by addressing nonlinearities with linear control techniques. However, it requires one active controller at all times and does not allow runtime updates to custom constraints. Controller design and review must be performed externally before configuration.⁵²

Model reference adaptive control

The adaptive controller is designed using the Model Reference Adaptive Control (MRAC) strategy, which operates by dynamically adjusting the controller parameters to ensure that the actual plant output closely follows the output of a reference model subjected to the same input.⁵³ The fundamental block diagram of an MRAC system is illustrated in Figure 2, $y_{m}$ represents the output of the reference model, while $y$ denotes the output of the actual plant. The difference between these outputs is defined as the error, $e$ , in equation (25).

e (t) = y (t) - y_{m} (t)

(25)

Figure 2.

Structure of MRAC⁵⁴.

The MIT rule is widely applicable for designing controllers within an MRAC framework for various systems. According to the MIT rule, a cost function is defined as follows:

J (θ) = \frac{e^{2}}{2}

(26)

The parameter $θ$ is updated to minimize the cost function, and this adjustment is made in the direction of the negative gradient of $J (θ)$ .

\frac{d θ}{dt} = - α \frac{\partial J}{\partial θ} = - α e \frac{\partial e}{\partial θ}

(27)

The partial derivative term, $\frac{\partial e}{\partial θ}$ , known as the sensitivity derivative, indicates how the error changes with respect to $θ$ . Equation (27) describes the time-dependent adjustment of $θ$ to ensure that $J (θ)$ in equation (26) approaches zero. Here, $α$ represents the adaptation gain, a positive constant that determines the rate of parameter adjustment.

Processor-in-the-Loop simulation results and discussion

Linear models were derived from linearizing the nonlinear mathematical model in equation (4) about operating points. Operating points were evaluated by means of defining states with their initial values, inputs with their constraints, and output for both primary and secondary regulations. The states are $p$ , $ω_{m}$ , and $ω_{p}$ ; inputs are $D_{p}$ and $D_{m}$ ; output is stroke, $i_{s}$ , which is the ratio of $ω_{m}$ to $ω_{p}$ . The operating point values of the states, inputs, and output for each regulation are given in Table 1.

Table 1.

Operating points of primary and secondary regulations.

States, Inputs and Output	Primary regulation	Secondary regulation
$p (bar)$	83	110
$ω_{m} (rpm)$	350	647
$ω_{p} (rpm)$	1647	531
$D_{p} (c m^{3} / rev)$	40	147
$D_{m} (c m^{3} / rev))$	160	130
$i_{s}$	0.21	1.22

Pump and motor displacements as model inputs are in the region of 0–147 and 68–160 cm³, respectively. $T_{d}$ and $T_{L}$ are kept constant as 2000 and 800 Nm, respectively. The deterministic and stochastic torque values are average probable values assigned from Carlsson.¹⁰ The values of the other HST system parameters can be found in Table 2.

Table 2.

The values of the HST system parameters.

System parameters of HST	Unit	Value
$J_{mw}$	Nms²	5.02
$J_{pd}$	Nms²	0.1723
$K_{m}$	Nm	0
$K_{p}$	Nm	0
$V$	m³	5.98 × 10⁻⁴
$β$	Pa	3.7 × 10⁶
$η_{tm}$	–	0.97
$η_{tp}$	–	0.92
$μ_{m}$	Nms	5
$μ_{p}$	Nms	10

The linear state space models obtained around the operating points above can be seen as follows:

\underline{\overset{\cdot}{x}} (t) = A_{pri} \underline{x} (t) + B_{pri} \underline{u} (t) \underline{\overset{\cdot}{x}} (t) = A_{\sec} \underline{x} (t) + B_{\sec} \underline{u} (t)

\begin{array}{l} y (t) = \underline{{C_{p r i}}^{T}} \underline{x} (t) + D_{p r i} \underline{u} (t) \\ y (t) = \underline{{C_{s e c}}^{T}} \underline{x} (t) + D_{s e c} \underline{u} (t) \end{array}

where $\underline{x}$ and $\underline{u}$ are the state and input column vectors, respectively; $y$ is the output; $A$ and $B$ are the system and input matrices, respectively; $\underline{C^{T}}$ and $\underline{D^{T}}$ are the output and feedthrough row vectors, respectively; $()_{pri}$ and $()_{\sec}$ denote for the linear model derived for primary and secondary regulations, respectively. The symbolic expressions and numerical values of the elements of the specified matrices and vectors can be found as follows:

\underline{x} = [\begin{matrix} p \\ ω_{m} \\ ω_{p} \end{matrix}],

\underline{u} = [\begin{matrix} D_{p} \\ D_{m} \end{matrix}],

y = i_{s}

A_{pri} = [\begin{matrix} 0 & - 1.337 \times 10^{5} & 2.842 \times 10^{4} \\ 3.092 \times 10^{- 5} & - 0.996 & 0 \\ - 0.0002526 & 0 & - 58.04 \end{matrix}],

B_{pri} = [\begin{matrix} 1.448 \times 10^{5} & - 3.482 \times 10^{4} \\ 0 & 1.224 \\ - 39.97 & 0 \end{matrix}],

\underline{{C_{pri}}^{T}} = [\begin{matrix} 0 & 0.005799 & - 0.001232 \end{matrix}],

\underline{{D_{pri}}^{T}} = [\begin{matrix} 0 & 0 \end{matrix}],

A_{\sec} = [\begin{matrix} 0 & - 1.199 \times 10^{5} & 1.301 \times 10^{5} \\ 2.511 \times 10^{- 5} & - 0.996 & 0 \\ - 0.0009274 & 0 & - 58.04 \end{matrix}],

B_{\sec} = [\begin{matrix} 5.701 \times 10^{4} & - 6.186 \times 10^{4} \\ 0 & 1.746 \\ - 57 & 0 \end{matrix}],

\underline{{C_{\sec}}^{T}} = [\begin{matrix} 0 & 0.01798 & - 0.02191 \end{matrix}],

\underline{{D_{\sec}}^{T}} = [\begin{matrix} 0 & 0 \end{matrix}] .

The variability of volumetric efficiencies poses challenges, as pump efficiency is defined at a fixed speed (2100 rpm) while typical operation occurs at 1100–1600 rpm, and motor efficiency varies with rotation speed except at high speeds where displacement decreases. To address this, rotational dependency adjustments are made between the pump and motor. Thus, two-dimensional lookup tables with consistent structures for both components were used as detailed in Tables 3 and 4.¹⁰

Table 3.

Volumetric efficiency for the pump.¹⁰

$ω_{p}$ (rpm)		0	1100	1350	1600	2500
$D_{p} (c m^{3} / rev)$	0	0.56	0.56	0.58	0.59	0.59
	85	0.84	0.84	0.86	0.87	0.87
	147	0.9	0.9	0.92	0.93	0.93

Table 4.

Volumetric efficiency for the motor.¹⁰

$ω_{m}$ (rpm)		0	600	1200	2000	4000
$D_{m} (c m^{3} / rev)$	68	0.77	0.77	0.88	0.97	0.97
	105	0.81	0.81	0.91	0.97	0.97
	160	0.85	0.85	0.94	0.97	0.97

As it is known that controllers designed at different operating points by using linear control techniques are switched in solving nonlinear control problems, Multiple MPC Controllers block is used to have a better control of $i_{s}$ when crossing from primary regulation to secondary regulation or vice versa.⁵² In Figure 3, the Multiple MPC Controllers block receives $i_{s}$ , reference of $i_{s}$ and the switching signal at each control instant. The switching signal provides selection of the active controller from two MPC controllers designed at the operating points of first and secondary regulations. Switching is performed when the difference between the normalized value of $D_{p}$ and $D_{m}$ is positive. Then switching signal becomes 2 indicating that the MPC designed for secondary regulation is active, otherwise 1 for primary regulation is active. Finally, the active controller solves a linear quadratic programming problem to determine the optimal $D_{p}$ and $D_{m}$ .

Figure 3.

Block diagram created in MATLAB Simulink of the closed loop system.

Processor-in-the-Loop (PIL) simulation involves using a laptop to simulate the nonlinear plant and a target hardware or processor, where the controllers are deployed, as illustrated in Figure 4. Communication between the two is established via a USB-serial link. The target hardware used in this study is the BeagleBone Black, a cost-effective, compact board equipped with an AM335× ARM Cortex-A8 processor running at 1 GHz, 512 MB DDR3 RAM, 4 GB of on-board eMMC flash storage, and multiple connection interfaces including USB, Ethernet, and HDMI.⁵⁵ PIL simulation allows the evaluation of a candidate control algorithm’s performance directly on the target hardware. Unlike standard simulations, it accounts for hardware constraints such as limited memory resources and target-specific code behavior. Through PIL simulation, object code is tested using test vectors developed within a Simulink model, with the code deployed on the actual hardware. This approach enables verification of real code behavior, assessment of code coverage, and measurement of execution time. A key aspect of PIL simulation is that it does not run in real time. Simulink and the deployed object code exchange all input/output data at each sampling period, making communication delays more critical than computation time. Additionally, PIL simulation can help identify deviations in performance caused by the compilation process by comparing the model’s behavior with the deployed object code.⁵⁶

Figure 4.

Schematic diagram of the hardware used in the PIL simulations.⁵⁶

The Multiple MPC was designed in terms of prediction horizon $p$ , control horizon $m$ , input weights $w^{u}$ , output weights $w^{y}$ , input constraints as specified below and taking the manipulated variables rates as 1 for each regulation.

p_{pri} = p_{\sec} = 10

m_{pri} = m_{\sec} = 2

{w_{pri}}^{u} = [\begin{matrix} 10^{- 3} & 1 \end{matrix}]

{w_{pri}}^{y} = 10^{5}

{w_{\sec}}^{u} = [\begin{matrix} 1 & 10^{- 4} \end{matrix}]

{w_{\sec}}^{y} = 10^{4}

0 \leq {D_{p}}_{pri} = {D_{p}}_{\sec} \leq 147

68 \leq {D_{m}}_{pri} = {D_{m}}_{\sec} \leq 160

A Multiple MRAC was designed based on the transfer functions obtained by linear state space models in terms of the output $i_{s}$ and inputs $D_{p}$ and $D_{m}$ for each regulation.

G_{pri, Dp} (s) = \frac{i_{s, pri} (s)}{D_{p, pri} (s)} = \frac{0.04925 s^{2} + 0.1201 s + 1.551}{s^{3} + 59.03 s^{2} + 69.12 s + 247.1}

(28)

G_{pri, Dm} (s) = \frac{i_{s, pri} (s)}{D_{m, pri} (s)} = \frac{0.007099 s^{2} + 0.3949 s - 0.3731}{s^{3} + 59.03 s^{2} + 69.12 s + 247.1}

(29)

G_{\sec, Dp} (s) = \frac{i_{s, \sec} (s)}{D_{p, \sec (s)}} = \frac{1.249 s^{2} + 2.427 s + 3.059}{s^{3} + 59.03 s^{2} + 181.5 s + 294.9}

(30)

G_{\sec, Dm} (s) = \frac{i_{s, \sec} (s)}{D_{m, \sec} (s)} = \frac{0.03139 s^{2} + 0.5372 s - 3.339}{s^{3} + 59.03 s^{2} + 181.5 s + 294.9}

(31)

Firstly, an MRAC was designed using equation (28) as follows:

{\overset{. . .}{i}}_{s, pri} + 59.03 {\overset{\cdot\cdot}{i}}_{s, pri} + 69.12 {\overset{\cdot}{i}}_{s, pri} + 247.1 i_{s, pri} = 0.0492 {\overset{\cdot\cdot}{D}}_{p, pri} + 0.1201 {\overset{\cdot}{D}}_{p, pri} + 1.551 D_{p, pri}

{\overset{. . .}{y}}_{1, pri} + 59.03 {\overset{\cdot\cdot}{y}}_{1, pri} + 69.12 {\overset{\cdot}{y}}_{1, pri} + 247.1 y_{1, pri} = 0.04925 {\overset{\cdot\cdot}{u}}_{1, pri} + 0.1201 {\overset{\cdot}{u}}_{1, pri} + 1.551 u_{1, pri}

{\overset{. . .}{y}}_{1, pri} = - a_{11, pri} {\overset{\cdot\cdot}{y}}_{1, pri} - a_{12, pri} {\overset{\cdot}{y}}_{1, pri} - a_{13, pri} y_{1, pri} + b_{11, pri} {\overset{\cdot\cdot}{u}}_{1, pri} + b_{12, pri} {\overset{\cdot}{u}}_{1, pri} + b_{13, pri} u_{1, pri}

(32)

The reference model of the Multiple MRAC was chosen for all inputs and regulations in equation (32).

G_{m} (s) = \frac{Y_{m} (s)}{U_{c} (s)} = \frac{5 s^{2} + 100 s + 500}{s^{3} + 25 s^{2} + 200 s + 500}

(33)

{\overset{. . .}{y}}_{m} = - 25 {\overset{\cdot\cdot}{y}}_{m} - 200 {\overset{\cdot}{y}}_{m} - 500 y_{m} + 5 {\overset{\cdot\cdot}{u}}_{c} + 100 {\overset{\cdot}{u}}_{c} + 500 u_{c}

{\overset{. . .}{y}}_{m} = - a_{m 1} {\overset{\cdot\cdot}{y}}_{m} - a_{m 2} {\overset{\cdot}{y}}_{m} - a_{m 3} y_{m} + b_{m 1} {\overset{\cdot\cdot}{u}}_{c} + b_{m 2} {\overset{\cdot}{u}}_{c} + b_{m 3} u_{c}

(34)

The control law of the MRAC was chosen as in equation (35).

u_{1, pri} = θ_{11, pri} u_{c} - θ_{12, pri} y_{1, pri}

(35)

Substituting equation (35) into equation (32):

\begin{matrix} {\overset{. . .}{y}}_{1, pri} = - a_{11, pri} {\overset{\cdot\cdot}{y}}_{1, pri} - a_{12, pri} {\overset{\cdot}{y}}_{1, pri} - a_{13, pri} y_{1, pri} + b_{11, pri} (θ_{11, pri} {\overset{\cdot\cdot}{u}}_{c} + {\overset{\cdot\cdot}{θ}}_{11, pri} u_{c} - θ_{12, pri} {\overset{\cdot\cdot}{y}}_{1, pri} - {\overset{\cdot\cdot}{θ}}_{12, pri} y_{1, pri}) \\ + b_{12, pri} (θ_{11, pri} {\overset{\cdot}{u}}_{c} + {\overset{\cdot}{θ}}_{11, pri} u_{c} - θ_{12, pri} {\overset{\cdot}{y}}_{1, pri} - {\overset{\cdot}{θ}}_{12, pri} y_{1, pri}) + b_{13, pri} (θ_{11, pri} u_{c} - θ_{12, pri} y \end{matrix}

\begin{matrix} {\overset{. . .}{y}}_{1, pri} = - (a_{11, pri} + b_{11, pri} θ_{12, pri}) {\overset{\cdot\cdot}{y}}_{1, pri} - (a_{12, pri} + b_{12, pri} θ_{12, pri}) {\overset{\cdot}{y}}_{1, pri} \\ - (a_{13, pri} + b_{13, pri} θ_{2, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{2, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}) y_{1, pri} + b_{11, pri} θ_{11, pri} {\overset{\cdot\cdot}{u}}_{c} + b_{12, pri} θ_{11, pri} {\overset{\cdot}{u}}_{c} \\ + (b_{11, pri} {\overset{\cdot\cdot}{θ}}_{1, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri}) u_{c} \end{matrix}

\begin{matrix} {\overset{. . .}{y}}_{1, pri} = - (a_{11, pri} + b_{11, pri} θ_{12, pri}) {\overset{\cdot\cdot}{y}}_{1, pri} - (a_{12, pri} + b_{12, pri} θ_{12, pri}) {\overset{\cdot}{y}}_{1, pri} \\ - (a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}) y_{1, pri} + b_{11, pri} θ_{11, pri} {\overset{\cdot\cdot}{u}}_{c} + b_{12, pri} θ_{11, pri} {\overset{\cdot}{u}}_{c} \\ + (b_{11, pri} {\overset{\cdot\cdot}{θ}}_{11, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri}) u_{c} \end{matrix}

(36)

Defining $p = \frac{d}{dt}$ and substituting in equation (36), plant output can be obtained in equation (37).

\begin{matrix} p^{3} y_{1, pri} = - (a_{11, pri} + b_{11, pri} θ_{12, pri}) p^{2} y_{1, pri} - (a_{12, pri} + b_{12, pri} θ_{12, pri}) p y_{1, pri} \\ - (a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}) y_{1, pri} + b_{11, pri} θ_{11, pri} p^{2} u_{c} + b_{12, pri} θ_{11, pri} p u_{c} \\ + (b_{11, pri} {\overset{\cdot\cdot}{θ}}_{11, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri}) u_{c} \end{matrix}

y_{1, pri} = \frac{b_{11, pri} θ_{11, pri} p^{2} + b_{12, pri} θ_{11, pri} p + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{11, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri}}{p^{3} + (a_{11, pri} + b_{11, pri} θ_{12, pri}) p^{2} + (a_{12, pri} + b_{12, pri} θ_{12, pri}) p + a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}} u_{c}

(37)

The error can be obtained substituting equation (37) in equation (25).

\begin{matrix} e_{1, pri} = \\ \frac{b_{11, pri} θ_{11, pri} p^{2} + b_{12, pri} θ_{11, pri} p + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{11, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri}}{p^{3} + (a_{11, pri} + b_{11, pri} θ_{12, pri}) p^{2} + (a_{12, pri} + b_{12, pri} θ_{12, pri}) p + a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}} u_{c} - y_{m} \end{matrix}

(38)

The sensitivity derivatives can be obtained using equations (27) and (38).

\frac{d e_{1, pri}}{d θ_{11, pri}} = \frac{(b_{11, pri} p^{2} + b_{12, pri} p + b_{13, pri}) u_{c}}{p^{3} + (a_{11, pri} + b_{11, pri} θ_{12, pri}) p^{2} + (a_{12, pri} + b_{12, pri} θ_{12, pri}) p + a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}}

(39)

\frac{d e_{1, pri}}{d θ_{12, pri}} = \frac{- (b_{11, pri} p^{2} + b_{12, pri} p + b_{13, pri}) y_{1, pri}}{p^{3} + (a_{11, pri} + b_{11, pri} θ_{12, pri}) p^{2} + (a_{12, pri} + b_{12, pri} θ_{12, pri}) p + a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}}

(40)

The relations between the parameters of the plant and the reference model are obtained equating equations (36) and (34).

a_{m 1} = a_{11, pri} + b_{11, pri} θ_{12, pri}

(41)

a_{m 2} = a_{12, pri} + b_{12, pri} θ_{12, pri}

(42)

a_{m 3} = a_{13, pri} + b_{13, pri} θ_{12, pri} + b_{11, pri} {\overset{\cdot\cdot}{θ}}_{12, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{12, pri}

(43)

b_{m 1} = b_{11, pri} θ_{11, pri} = > θ_{11, pri} = \frac{b_{m 1}}{b_{11, pri}}

(44)

b_{m 2} = b_{12, pri} θ_{11, pri}

(45)

b_{m 3} = b_{11, pri} {\overset{\cdot\cdot}{θ}}_{11, pri} + b_{12, pri} {\overset{\cdot}{θ}}_{11, pri} + b_{13, pri} θ_{11, pri} = (b_{11, pri} p^{2} + b_{12, pri} p + b_{13, pri}) θ_{11, pri}

(46)

Substituting equation (44) into equation (46):

b_{11, pri} p^{2} + b_{12, pri} p + b_{13, pri} = \frac{b_{11, pri} b_{m 3}}{b_{m 1}}

(47)

Substituting equations (41)–(43), (47) into equations (39) and (40):

\frac{d e_{1, pri}}{d θ_{11, pri}} = \frac{b_{11, pri} b_{m 3} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}}

\frac{d e_{1, pri}}{d θ_{12, pri}} = \frac{- b_{11, pri} b_{m 3} y_{1, pri}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}}

\begin{matrix} \frac{d θ_{11, pri}}{dt} & = - α_{1, pri} \frac{b_{11, pri} b_{m 3} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} e_{1, pri} = - α_{1, pri} \frac{b_{11, pri} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, pri} & = - γ_{1, pri} \frac{a_{m 3} u_{c}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{1, pri} \end{matrix}

\begin{matrix} \frac{d θ_{12, pri}}{dt} & = - α_{1, pri} \frac{- b_{11, pri} b_{m 3} y_{1, pri}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} e_{1, pri} = α_{1, pri} \frac{b_{11, pri} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} y_{1, pri}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, pri} & = - γ_{1, pri} \frac{a_{m 3} y_{1, pri}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{1, pri} \end{matrix}

The Multiple MRAC was designed as shown below by applying the same procedure of the first MRAC design.

u_{1, \sec} = θ_{11, \sec} u_{c} - θ_{12, \sec} y_{1, \sec}

\begin{matrix} \frac{d θ_{11, \sec}}{dt} & = - α_{1, \sec} \frac{b_{12, \sec} b_{m 3} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, \sec} & = - α_{1, \sec} \frac{b_{12, \sec} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, \sec} & = - γ_{1, \sec} \frac{a_{m 3} u_{c}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{1, \sec} \end{matrix}

\begin{matrix} \frac{d θ_{12, \sec}}{dt} & = - α_{1, \sec} \frac{- b_{12, \sec} b_{m 3} y_{1, \sec}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, \sec} & = α_{1, \sec} \frac{b_{12, \sec} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} y_{1, \sec}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{1, \sec} & = - γ_{1, \sec} \frac{a_{m 3} y_{1, \sec}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{1, \sec} \end{matrix}

u_{2, pri} = θ_{21, pri} u_{c} - θ_{22, pri} y_{2, pri}

\begin{matrix} \frac{d θ_{21, pri}}{dt} & = - α_{2, pri} \frac{b_{21, pri} b_{m 3} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, pri} & = - α_{2, pri} \frac{b_{2, pri} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, pri} & = - γ_{2, pri} \frac{a_{m 3} u_{c}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{2, pri} \end{matrix}

\begin{matrix} \frac{d θ_{22, pri}}{dt} & = - α_{2, pri} \frac{- b_{21, pri} b_{m 3} y_{2, pri}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, pri} & = α_{2, pri} \frac{b_{21, pri} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} y_{2, pri}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, pri} & = - γ_{2, pri} \frac{a_{m 3} y_{2, pri}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{2, pri} \end{matrix}

u_{2, \sec} = θ_{21, \sec} u_{c} - θ_{22, \sec} y_{2, \sec}

\begin{matrix} \frac{d θ_{21, \sec}}{dt} & = - α_{2, \sec} \frac{b_{21, \sec} b_{m 3} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, \sec} & = - α_{2, \sec} \frac{b_{2, \sec} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} u_{c}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, \sec} & = - γ_{2, \sec} \frac{a_{m 3} u_{c}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{2, \sec} \end{matrix}

\begin{matrix} \frac{d θ_{22, \sec}}{dt} & = - α_{2, \sec} \frac{- b_{21, \sec} b_{m 3} y_{2, \sec}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, \sec} & = α_{2, \sec} \frac{b_{21, pri} b_{m 3}}{a_{m 3} b_{m 1}} \frac{a_{m 3} b_{m 1} y_{2, \sec}}{(p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}) b_{m 1}} \\ e_{2, \sec} & = - γ_{2, \sec} \frac{a_{m 3} y_{2, \sec}}{p^{3} + a_{m 1} p^{2} + a_{m 2} p + a_{m 3}} e_{2, \sec} \end{matrix}

The controller parameters of the Multiple MRAC was chosen as $γ_{1, pri} = 90$ , $γ_{2, pri} = 90$ , $γ_{1, \sec} = 100$ , and $γ_{2, \sec} = 71$ during the PIL simulations. The switch mechanism approach of the proposed Multiple MRAC in terms of primary and secondary regulations was applied same as the Multiple MPC.

It is aimed that HST undergoes primary and secondary regulations as mentioned in Introduction section in a desired and controlled manner through constrained multi-input single-output multiple MPC and MRAC. Firstly, the performance of the proposed Multiple MPC and MRAC controllers were tested in case of the complex trajectory and deterministic disturbances. $T_{d}$ was changed from 2000 to 2300 Nm, $T_{L}$ was changed from 800 to 1000 Nm as shown in Figure 5(b) and (c), and reference of $i_{s}$ was changed from 0.2 to 1.9 in the combination of step and ramp functions as shown in Figure 5(b) in order to track the reference trajectories of primary and secondary regulations⁴⁷ under deterministic disturbances. The closed loop nonlinear models with integrated proposed MPC and MRAC were simulated 180 s with stiff differential equation solver with trapezoidal rule + backward differentiation formula in Matlab. The graphs for the PIL simulations under the specified conditions can be found in Figures 5 to 8.

Figure 5.

Stroke responses of the Multiple MPC and MRAC (a) and deterministic torque disturbances (b, c).

Figure 6.

Stroke responses of the Multiple MPC and MRAC (a) and volumetric efficiencies of pump and motor (b, c) in case of deterministic torque disturbances.

Figure 7.

Stroke response of the Multiple MPC and MRAC (a) and angular speeds of pump and motor (b, c) in case of deterministic torque disturbances.

Figure 8.

Normalized percentage of pump and motor displacements of the Multiple MPC (a) and MRAC (b) and switch signal of Multiple MPC and MRAC (c) in case of deterministic torque disturbances.

Finally, the performance of the proposed Multiple MPC and MRAC controllers were tested in case of the same complex trajectory as in the first scenario, but, stochastic disturbances. A uniformly distributed random numbers with a frequency of 1 kHz was used for a interval of $T_{d}$ as from 2000 to 2300 Nm, and $T_{L}$ from 800 to 1000 Nm. The same structure and parameters of the Multiple MPC and MRAC were used as in the first scenario during the PIL simulations; however, two Kalman estimators were designed for the each regulation and integrated as providing the same switch mechanism by choosing the process noise covariance and measurement noise covariance as 1 and noise cross covariance as 0 for the Multiple MRAC in order to overcome the stochastic disturbances. The closed loop nonlinear models with integrated proposed MPC and MRAC were simulated 180 s with stiff differential equation solver with trapezoidal rule + backward differentiation formula in Matlab. The graphs for the PIL simulations under the specified conditions can be found in Figures 9 to 12.

Figure 9.

Stroke responses of the Multiple MPC and MRAC (a) and stochastic torque disturbances (b, c).

Figure 10.

Stroke responses of the Multiple MPC and MRAC (a) and volumetric efficiencies of pump and motor (b, c) in case of stochastic torque disturbances.

Figure 11.

Stroke response of the Multiple MPC and MRAC (a) and angular speeds of pump and motor (b, c) in case of stochastic torque disturbances.

Figure 12.

Normalized percentage of pump and motor displacements of the Multiple MPC (a) and MRAC (b) and switch signal of Multiple MPC and MRAC (c) in case of stochastic torque disturbances.

Percentage maximum overshoot (MO), peak time ( $t_{p}$ ), and Mean Squared Error (MSE) can be obtained as shown in Tables 5 and 6 during the PIL simulations in case of deterministic and stochastic disturbances from Figures 5 to 12.

Table 5.

Maximum performance measures of the PIL simulations in case of deterministic disturbances.

Controller	MO	$t_{p}$	MSE
Multiple MPC	25	1	0.0024
Multiple MRAC	31	2.1	0.0068

Table 6.

Maximum performance measures of the PIL simulations in case of stochastic disturbances.

Controller	Maximum overshoot (%)	Peak time (s)	MSE
Multiple MPC	24	1	0.0048
Multiple MRAC	30	2.2	0.0116

The closed loop system with Multiple MRAC has higher maximum overshoot, peak time, and MSE values compared with the Multiple MPC during the PIL simulations for both deterministic and stochastic disturbances as can be seen in Tables 5 and 6. It can be clearly seen in Figure 5(a) that the proposed Multiple MPC has a superior reference tracking ability and it is insensitive to parameter changes. Due to the change of the volumetric efficiencies and so the angular speeds of the pump and motor in Figures 6(b–c), 7(b–c), 10(b–c), and 11(b–c), maximum overshoots, particularly for the Multiple MRAC, occurred. The proposed controllers to reject the deterministic disturbances in Figure 5(b) and (c) and stochastic disturbances in Figure 9(b) and (c) can be concluded.

Average Effort (AE), Maximum Effort (ME), and Energy Consumption (EC) metrics in terms of controller effort can be obtained from the PIL simulations for both deterministic and stochastic disturbances in Tables 7 and 8.

Table 7.

Controller performances during the PIL simulations in case of deterministic disturbances.

Controller	AE	ME	EC
Multiple MPC	0.79	1	1201.21
Multiple MRAC	0.48	0.97	493.89

Table 8.

Controller performances during the PIL simulations in case of stochastic disturbances.

Controller	AE	ME	EC
Multiple MPC	0.79	1	1201.72
Multiple MRAC	0.47	0.75	441.72

It can be seen from Tables 7 and 8 that Multiple MPC exhibits a higher average control effort under both deterministic and stochastic disturbances, indicating a more aggressive control strategy, while Multiple MRAC operates with lower effort, making it potentially advantageous for energy efficiency. The Multiple MPC has higher maximum control effort values, reflecting a more aggressive control approach, while the Multiple MRAC operates with smaller maximum values, which can lead to smoother control. The Multiple MPC consumes significantly more energy in both disturbance conditions, likely due to more frequent and larger control signals, while the Multiple MRAC shows better energy efficiency, consuming less energy overall. As a result, the Multiple MPC provides a faster and more dynamic control strategy, making it suitable for systems requiring rapid response to changes and maintains similar average and maximum control efforts, with a slight increase in energy consumption. So it adapts to new conditions while preserving its aggressive nature.

The whole PIL simulations conducted in this study demonstrate that computational errors and communication delays were not observed, as the control input differences between the model and the object code were consistently zero across all simulations. This confirms that the selection of BeagleBone Black Rev C as the target hardware was appropriate.

It is concluded in the result of PIL simulations that $i_{s}$ tracks the complicated reference trajectory under the disturbances with small overshoots peak time and MSE values by means of the proposed multiple MPC as comparing with multiple MRAC determining optimal $D_{p}$ and $D_{m}$ in primary and secondary regulations.

Conclusions

The stroke control of an HST has an important role in speed control for the whole system of an offroad vehicle such as a forest vehicle or a tractor since it provides to track reference trajectories of primary and secondary regulation for both normal and optimal drive modes. The difficulties of stroke control are based on different disturbances, that is, on a tractor such as type of soil and field road conditions. Classical controllers available in the literature can not provide satisfactory control performance since they are sensitive to these disturbances. In this study, a multiple MPC and MRAC were proposed as a new control strategy for stroke control of the HST, and their performances were compared. Therefore, it made possible to deal with unmeasured disturbances like engine and load torques with the help of the simulation studies. PIL simulations play a crucial role not only in testing and verifying the controller object code but also in evaluating the practical implications of controller design. In this study, the BeagleBone Black has proven to be a practical and effective choice for the target hardware. Moreover, the controllers implemented on this hardware are capable of managing multiple reference changes for system states under varying volumetric efficiencies of pump and motor, and both deterministic and stochastic disturbances, offering enhanced flexibility for tracking complex trajectories. In the meantime, the results of the PIL simulations are encouraging in terms of the proposed multiple MPC to be implemented on an HST for real-time application.

As future work, both stroke and engine speed control will be realized by means of MPCs designs in terms of minimizing fuel consumption.

Footnotes

Acknowledgements

The author would like to extend thanks to Prof. Dr. Can Özsoy and Prof. Dr. İlker Murat Koç for their valuable support of this research.

Handling Editor: Sharmili Pandian

ORCID iD

Hakan Ülker

Statements and Declarations

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Design and application of multiple model predictive control in stroke control of hydrostatic transmission

Abstract

Keywords

Introduction

Mathematical modeling of hydrostatic transmission

Model predictive control

State observer

Multiple model predictive control

Model reference adaptive control

Processor-in-the-Loop simulation results and discussion

Conclusions

Footnotes

Acknowledgements

ORCID iD

Statements and Declarations

References