Model prediction controller for penetration mechanism of seabed cone penetration test system

The system of seabed Cone Penetration Test (CPT) is a kind of equipment for offshore investigations. Seabed CPT is used to assess multiple physical properties of marine sediments. Constant-rate penetration mechanism is the core part of seabed CPT, which adopts dual-cylinder drive to penetrate the probe rod into the seafloor sediments at a constant speed (2cm/s). At present, the domestic CPT and its penetration equipment almost all rely on imports. In the process of system design, we found that the traditional PID control of the hydraulic cylinder position control accuracy is low, so we need to study higher precision position tracking control method to achieve uniform penetration of the probe rod. In this paper, a Linear Time-Varying Model Predictive Controller (LTV-MPC) is introduced, which can control two hydraulic cylinders to penetrate probe rod into the seabed sediment with smaller penetration rate error under the condition of changing penetration resistance.

electro-hydraulic servo system is given.Then the controller is designed and simulated based on the software of MATLAB, and the simulation results are discussed and summarized.

Model of electro-hydraulic servo system
Here, Figure 1 reveals the penetration mechanism model of seabed CPT, Figure 2 reveals the principle of electro-hydraulic servo system and the valve control cylinder model is shown in Figure 3.According to Figure 3, the state variable of the hydraulic servo system is defined as .
The state-space equation of the valve-controlled cylinder system is: ( where is output diaplacement, is current input of servo valve, and are the areas of roded cavity and rodless cavity respectively, and are the pressure of cylinder left and right cavity respectively, and are the hydraulic oil flow of the left and right chambers of the servo valve , , ,

Design of controller
The design of LTV-MPC can be divided into four steps: linearization of nonlinear system model, the realization of the prediction function, optimize the solution of the objective function, feedback mechanism design.The specific design process is as follows: The nonlinear state space expression of the electro-hydraulic servo system is: (2) By expanding the nonlinear function near the real-time state trajectory into a Taylor series, keeping only the first-order term and ignoring the higher-order term, we can obtain: (3) where, and are respectively Jacobian matrices with respect to and .Combining (2) and ( 3), (4) Let , , and use the first-order difference separation to get the discrete state space expression as follows: ( where, is the error quantity, , and is predicted by the model, .Define the new state quantity: (6) Substitute (6) into Equation (5) to get the expression for the new state space: (7) where (8) is shortened to , is shortened to .
The output of the system at future time is expressed in matrix form as follows: (9) where It can be seen that the state quantity and output quantity obtained in the prediction time domain can be calculated by the current state quantity of the system and the control increment in the control time domain.
Design an objective function that can reflect the tracking performance of the controlled system as follows: (11) where is the relaxation variable, so that the objective function does not enter the dead zone in the process of optimization solution process.
The objective function optimization problem is: where, is the positive definite matrix, .Satisfy dynamic constraints: (13) Satisfy time domain constraints: where, is the expected output, , , , is the weight matrix.

MEIE-2023 Journal of Physics: Conference Series 2591 (2023) 012027
The optimal control increment sequence in the control time domain is obtained by optimizing the solution of the system state at the time and the control quantity at the previous time in the control cycle.The first quantity of the sequence is applied to the system as the actual control increment, i.e. (15 The system performs this control quantity until the next moment, at which time the system repredicts the output of the next segment in the time domain based on the state information, and a new control increment sequence is obtained through the optimization process.And so on until the system completes the control process.

Simulation results and analysis
To test the control effect of LTV-MPC, the simulation is conducted with Matlab / Simulink software.The model parameters of electro-hydraulic servo system are shown in Table 1, The expected displacement is a step signal with a slope of 2 cm/s.The simulation principle is shown in Figure 4.The observation effect of the extended state observer is shown in Figure 5 and Figure 6.

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Figures 5-6 show the displacement observation error and the total disturbance observation error (the disturbance value is set to ).It can be seen that the maximum error value of displacement state observation is 5.23e-5 mm (error rate is 1.31e-5%), and the maximum error value of total disturbance observation is 2.024 N (error rate is 0.169%).Both of them are small, which is beneficial to improve the control accuracy of the controller.Figures 7-8 show the displacement tracking error of PID and LTV-MPC.Figure 7 shows the maximum error value of displacement tracking gradually increases and the tracking ability is lost.Figure 8 shows the maximum error value of displacement tracking is 0.046 mm.It can be seen that the model predictive controller has strong trajectory tracking ability.
In addition, we simulated the controller's tracking of sinusoidal signals with different amplitudes and frequencies, and the results were shown in Figure 9 and Figure 10.  Figure 9 and Figure 10 reveal that the controller is greatly affected by amplitude and frequency changes.The main reason is that the total disturbance value added in the system is six times that of the original system.The controller is unstable when tracking sinusoidal signals with amplitude above 0.45 m and frequency above 9.5 Hz, which indicates that we should optimize sampling time, control step size, prediction step size, constraint range, loss function weight and other parameters according to expectations in the actual controller design process, in order to ensure that the MPC has a high trajectory tracking ability.We guess that the change rate of the target trajectory increases with the increase of frequency and amplitude, which will affect the displacement tracking error.

Conclusions
In this paper, an LVT-MPC applied to seabed CPT is proposed and the design process of the controller is presented.The controller is simulated by Matlab/Simulink software.The results show that the controller has good control precision.But the controller's ability to track signals with high frequency and high replication is weak.As for how to observe the displacement tracking effect of the controller in the CPT penetration mechanism, it needs to be observed through subsequent supplementary experiments.

Figure 10 .
Figure 10.The relation between error and frequency (0.2 m).

Table 1 .
Physical model parameter table of position servo system.