Robust security predictive control method against spoofing attacks in injection molding process

Abstract

This patent discloses a robust and safe predictive control method for injection molding processes to resist deception attacks, belonging to the field of industrial process control. The method includes: Step 1: Establishing a state-space model of the injection molding process subjected to a deception attack; Step 2: Transforming the constructed state-space model of the injection molding process subjected to a deception attack into an extended system state-space model; Step 3: Designing a robust and safe predictive control law based on the above-mentioned injection molding process subjected to a deception attack; Step 4: Constructing the Lyapunov function of the system; Step 5: Combining the robust positive definite invariant set and the terminal constraint set, providing sufficient conditions to ensure system stability; Step 6: Solving for the control law gain. This invention can resist deception attacks on network-controlled injection molding processes, ensuring system security. Simultaneously, this method can ensure that the system effectively tracks the set trajectory in the first production batch, improving the production efficiency of the injection molding process.

Description

Technical Field

[0001] This invention belongs to the field of industrial process control and relates to a robust safety predictive control method for injection molding processes to resist deception attacks. Background Technology

[0002] With the continuous growth in demand for personalized products, more and more intermittent production processes are being deployed in networked control systems to improve control efficiency. However, the ensuing cybersecurity challenges make the need for efficient and safe control of mass production processes increasingly urgent. To address this, this paper proposes a robust safety predictive control method for injection molding processes susceptible to deception attacks, ensuring system stability under such attacks. Considering the impact of deception attacks, an extended deception attack model incorporating the output error integral is established. This model effectively improves the system's control accuracy while ensuring the reliability of the controller design. Based on the design model, a robust safety predictive tracking controller is designed to ensure system safety and stability. Through robust safety invariant sets and terminal constraint sets, the linear matrix inequality conditions guaranteeing system stability are derived. The controller gain is solved in real-time using online computation, enabling dynamic adjustment of the control input, thus overcoming the limitations of traditional offline methods in effectively handling the real-time dynamic characteristics of the system. Previous research has focused on iterative learning methods for controlling injection molding processes susceptible to deception attacks. However, traditional iterative learning control often requires multiple batches to optimize the controller because the initial batch has not accumulated sufficient historical data for learning and optimization. This frequently leads to defective products in the initial batch, resulting in wasted production resources. Furthermore, this not only increases unnecessary energy consumption in the initial batch but also reduces product quality. Therefore, it is essential to research a control method for injection molding processes susceptible to deception attacks that combines stability, speed, and robustness.

[0003] Currently, the mainstream control method for injection molding processes subjected to deception attacks is the iterative learning method. Under ideal conditions, this control method can effectively control multi-stage batch processes. However, in actual production, due to the influence of various factors, the control effect of iterative learning will be greatly reduced. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a robust and safe predictive control method for injection molding processes to resist deception attacks. This method can still operate stably even when the system is subjected to uncertainties, unknown external interference, or the controller is affected by deception attacks, thereby ensuring the safe and stable operation of the equipment.

[0005] This invention aims to achieve robust performance in the control system while retaining the advantages of rolling optimization in predictive control. It applies robust predictive principles in the early design phase, fully considering the system's uncertainties, external disturbances, and the impact of deception attacks on the controller. The injection molding process subjected to deception attacks is represented in a state-space format. Then, the output tracking error and the output error integral are introduced into the state space to establish a new, extended state-space model.

[0006] Finally, the system stability conditions based on LMI constraints are given, and the control gain that can simultaneously stabilize the system is obtained.

[0007] This invention is achieved through the following technical solution:

[0008] Step 1: Establish a state-space model of the injection molding process subjected to a deception attack: The input-output model of the injection molding process with uncertainty is as follows: During the injection molding process, it will be related to the valve opening ( The corresponding jet speed () The model is: (1) Jet speed ( The corresponding nozzle pressure () The model is: (2) Define discrete during injection molding State variables at time 1 Control variables and output variables : (3) Based on (1), (2) and (3), the state-space equations are obtained as follows: (4) In the formula, , , , Representing the discrete nature of the system respectively The uncertain state matrix at time t, the uncertain control input matrix, and the uncertain output matrix, where , , and These represent the state constant matrix and the control input constant matrix of corresponding dimensions, respectively. and Represents the discrete nature of the system The system state and control input parameters at time t must satisfy the following conditions: Among them, uncertainty disturbances satisfy ,symbol , , They are all constant matrices; Considering the impact of deception attacks and external disturbances on the system, we can obtain the following from equation (4): (5) In the formula, Represents the discrete nature of the system External disturbances at any time Represents the discrete nature of the system A time controller being subjected to a spoofing attack can be represented as: (6) In the formula Represents the discrete nature of the system Deceptive attack signals at all times, satisfying , Represents the discrete nature of the system Deceptive attack energy at all times The deception attack indicator representing the system satisfies: (7) then This indicates that the system has been subjected to a deception attack. At that time, the system was running normally; According to equations (5), (6), and (7), we can obtain: (8) Step 2: Transform the constructed state-space model of the injection molding process subjected to deception attack into an extended system state-space model; Define the system settings as The system is discrete The tracking error at time t is: (9) In the formula The system represents discrete Output at any moment; Define the integral of the output error over time: (10) In the formula The representative value is the maximum duration that the system can achieve in a single run, satisfying the following conditions: According to equation (10), the integral of the output error at the current time can be expressed as: (11) The system is discretet The deviation in time can be defined as: (12) Therefore, we can conclude that: (13) (14) In the formula , , , These represent the system in discrete states. t State deviation in incremental states at time step, input and spoofing attacks; Based on the above formula, the extended incremental equation is as follows: (15) In the formula, , , , , , , Represents discrete t The constantly expanding system state Represents discrete t System control input at any given time, , These represent the system in discrete states. t Uncertain state matrix and uncertain control input matrix at time t. and Represents the perturbation and output constant matrix, where , ,symbol and These represent the state constant matrix and control input constant matrix of corresponding dimensions, respectively. The matrix represents the uncertainty constants. Represents discrete t The system state and control uncertainties at time t must satisfy... ; Step 3: Design a robust safety predictive control law for the injection molding process subjected to the above-mentioned deception attack: (16) In the formula Representing the control law gain, in order to construct the closed-loop system, we substitute (16) into (15) to obtain the state-space model of the closed-loop system as follows: (17) Based on the above extended model (17), the system optimization problem is transformed into the following min-max optimization problems: (18) (19) In the formula For the stage cost function, the symbol is... It is a positive scalar. and These are weighted matrices of the corresponding dimensions of the system state variables and control inputs, respectively. Represents the terminal cost function; Step 4: Construct the Lyapunov function for the injection molding process subjected to a deception attack: Construct the following Lyapunov function: (20) In the formula It is a positive definite matrix; Step 5: Combining the robust positive definite invariant set and the terminal constraint set, give sufficient conditions to ensure the stability of the system; Robust positive definite invariant set: if a scalar exists make: (twenty one) Then set It is a robust positive definite invariant set. Even if the system faces uncertain parameters and external disturbances, as long as the initial state or the controlled state is within this set, the system state will not diverge, thus ensuring the robustness of the system under disturbances. Theorem 1: For a system, if there exists a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (twenty two) in ,So It is a robust positive definite invariant set; Terminal constraint set: if set It is a robust positive definite invariant set, and it exists Class functions and Lyapunov functions ,satisfy: (twenty three) (twenty four) Then set It is a set of terminal constraints. To ensure the stability of the closed-loop system, it is usually required that the predicted state at the end of the time domain falls into this set of terminal constraints. If the system contains a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (25) in ,So It is simply a set of terminal constraints; By combining the robust safety predictive control law (16), the closed-loop system model (17), the robust safety performance index (18), the Lyapunov function (20), the robust positive definite invariant set (21), and the terminal constraint set (24), sufficient conditions that can guarantee system stability are obtained, and the system stability conditions are transformed into linear matrix inequality forms using the linear matrix inequality technique, namely the linear matrix inequality conditions (22) and (25) in Theorem 1 and Theorem 2. Step Six: Solve for the control law gain ; Based on the system's state invariant set, we can obtain: (26) Input constraints must satisfy the following conditions: (27) Then it is easy to obtain: (28) In the formula In optimization, this represents the bounds of the control input constraints; therefore, the following optimization problem is proposed: (29) Under the premise of ensuring system stability and strict state constraints, the optimal control law gain is solved online. The control input is updated in real time (6) to ensure stable, safe and reliable operation of the system in the event of a deception attack. At the same time, the state space model (15) constructed by the state deviation and output error enables the designed robust safety predictive controller to quickly track the set trajectory and achieve high-precision tracking of the first batch of injection molding processes.

[0009] The advantages and effects of this invention are as follows: This invention addresses injection molding processes susceptible to deception attacks by proposing a robust and safe predictive control method. Specifically targeting highly concealed deception attacks in industrial networks, this invention embeds a random attack index model directly into the system's state space, combining it with robust safety invariant sets and terminal constraint sets to provide stability conditions for the system under normal operating conditions. This design cuts off the path for attacks to propagate to future batches via historical data, ensuring input-state stability (ISS) and mean-square safety of the closed-loop system even under active attack environments. By introducing an output error integral term and constructing an extended state-space model on the time axis, this invention overcomes the limitations of traditional iterative learning control that relies on historical batch data. This technology ensures that the system achieves trajectory tracking in the first production batch, effectively reducing quality fluctuations and resource waste in the initial production phase. Attached Figure Description

[0010] Figure 1 To compare the output of the proposed method with that of the traditional two-dimensional method; Figure 2 To determine the absolute error between the proposed method and the traditional two-dimensional method contrast; Figure 3 To compare the control input of the proposed method with that of the traditional two-dimensional method; Figure 4 To compare the root mean square error (RMS) of the proposed method with that of the traditional two-dimensional method; Figure 5 The root mean square error of the proposed method is denoted as . Figure 6 For the proposed method The phase trajectory; Figure 7 A simplified diagram of the injection molding process: (a) Injection stage, (b) Holding pressure stage, (c) Cooling stage, (d) Mold release stage; Figure 8 This is a flowchart of the steps of the present invention. Detailed Implementation

[0011] The present invention will be further explained below with reference to the accompanying drawings and embodiments.

[0012] This invention relates to a robust security predictive control method for resisting deception attacks in the injection molding process, the specific steps of which are as follows: Step 1: Establish a state-space model of the injection molding process subjected to a deception attack: The input-output model of the injection molding process with uncertainty is as follows: During the injection molding process, it will be related to the valve opening ( The corresponding jet speed () The model is: (1) Jet speed ( The corresponding nozzle pressure () The model is: (2) Define discrete during injection molding State variables at time 1 Control variables and output variables : (3) Based on (1), (2) and (3), the state-space equations are obtained as follows: (4) In the formula, , , , Representing the discrete nature of the system respectively The uncertain state matrix at time t, the uncertain control input matrix, and the uncertain output matrix, where , , and These represent the state constant matrix and the control input constant matrix of corresponding dimensions, respectively. and Represents the discrete nature of the system The system state and control input parameters at time t must satisfy the following conditions: Among them, uncertainty disturbances satisfy ,symbol , , They are all constant matrices; Considering the impact of deception attacks and external disturbances on the system, we can obtain the following from equation (4): (5) In the formula, Represents the discrete nature of the system External disturbances at any time Represents the discrete nature of the system A time controller being subjected to a spoofing attack can be represented as: (6) In the formula Represents the discrete nature of the system Deceptive attack signals at all times, satisfying , Represents the discrete nature of the system Deceptive attack energy at all times The deception attack indicator representing the system satisfies: (7) then This indicates that the system has been subjected to a deception attack. At that time, the system was running normally; According to equations (5), (6), and (7), we can obtain: (8) Step 2: Transform the constructed state-space model of the injection molding process subjected to deception attack into an extended system state-space model; Define the system settings as The system is discrete The tracking error at time t is: (9) In the formula The system represents discrete Output at any moment; Define the integral of the output error over time: (10) In the formula The representative value is the maximum duration that the system can achieve in a single run, satisfying the following conditions: According to equation (10), the integral of the output error at the current time can be expressed as: (11) The system is discrete t The deviation in time can be defined as: (12) Therefore, we can conclude that: (13) (14) In the formula , , , These represent the system in discrete states. t State deviation in incremental states at time step, input and spoofing attacks; Based on the above formula, the extended incremental equation is as follows: (15) In the formula, , , , , , , Represents discretet The constantly expanding system state Represents discrete t System control input at any given time, , These represent the system in discrete states. t Uncertain state matrix and uncertain control input matrix at time t. and Represents the perturbation and output constant matrix, where , ,symbol and These represent the state constant matrix and control input constant matrix of corresponding dimensions, respectively. The matrix represents the uncertainty constants. Represents discrete t The system state and control uncertainties at time t must satisfy... ; Step 3: Design a robust safety predictive control law for the injection molding process subjected to the above-mentioned deception attack: (16) In the formula Representing the control law gain, in order to construct the closed-loop system, we substitute (16) into (15) to obtain the state-space model of the closed-loop system as follows: (17) Based on the above extended model (17), the system optimization problem is transformed into the following min-max optimization problems: (18) (19) In the formula For the stage cost function, the symbol is... It is a positive scalar. and These are weighted matrices of the corresponding dimensions of the system state variables and control inputs, respectively. Represents the terminal cost function; Step 4: Construct the Lyapunov function for the injection molding process subjected to a deception attack: Construct the following Lyapunov function: (20) In the formula It is a positive definite matrix; Step 5: Combining the robust positive definite invariant set and the terminal constraint set, give sufficient conditions to ensure the stability of the system; Robust positive definite invariant set: if a scalar exists make: (twenty one) Then set It is a robust positive definite invariant set. Even if the system faces uncertain parameters and external disturbances, as long as the initial state or the controlled state is within this set, the system state will not diverge, thus ensuring the robustness of the system under disturbances. Theorem 1: For a system, if there exists a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (twenty two) in ,So It is a robust positive definite invariant set; Terminal constraint set: if set It is a robust positive definite invariant set, and it exists Class functions and Lyapunov functions ,satisfy: (twenty three) (twenty four) Then set It is a set of terminal constraints. To ensure the stability of the closed-loop system, it is usually required that the predicted state at the end of the time domain falls into this set of terminal constraints. If the system contains a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (25) in ,So It is simply a set of terminal constraints; By combining the robust safety predictive control law (16), the closed-loop system model (17), the robust safety performance index (18), the Lyapunov function (20), the robust positive definite invariant set (21), and the terminal constraint set (24), sufficient conditions that can guarantee system stability are obtained, and the system stability conditions are transformed into linear matrix inequality forms using the linear matrix inequality technique, namely the linear matrix inequality conditions (22) and (25) in Theorem 1 and Theorem 2. Step Six: Solve for the control law gain ; Based on the system's state invariant set, we can obtain: (26) Input constraints must satisfy the following conditions: (27) Then it is easy to obtain: (28) In the formula In optimization, this represents the bounds of the control input constraints; therefore, the following optimization problem is proposed: (29) Under the premise of ensuring system stability and strict state constraints, the optimal control law gain is solved online. The control input is updated in real time (6) to ensure stable, safe and reliable operation of the system in the event of a deception attack. At the same time, the state space model (15) constructed by the state deviation and output error enables the designed robust safety predictive controller to quickly track the set trajectory and achieve high-precision tracking of the first batch of injection molding processes.

[0013] Example 1: This invention proposes a robust safety predictive control method for injection molding processes subjected to deception attacks. This method can effectively solve the control performance problems in the injection molding process. The input-output model for an injection molding process with uncertainties is as follows: In the injection molding stage, the injection velocity (IV) corresponding to the valve opening (VO) is modeled as follows:

[0014] The model for nozzle pressure (NP) corresponding to injection velocity (IV) is as follows:

[0015] During the pressure holding phase, the models for valve opening (VO) and nozzle pressure (NP) are as follows:

[0016] Define discrete during injection molding State variables at time 1 Control variables and output variables :

[0017] Then, the following uncertain state-space model can be obtained, which takes into account deception attacks and interference;

[0018] In the formula , , Representing the discrete nature of the system respectively The uncertain state matrix at time t, the uncertain control input matrix, and the uncertain output matrix, where , , and These represent the state constant matrix and the control input constant matrix of corresponding dimensions, respectively. and Represents the discrete nature of the system The system state and control input parameters at time t must satisfy the following conditions: Among them, uncertainty disturbances satisfy ,symbol , , They are all constant matrices, where

[0019] Furthermore, to facilitate the evaluation of the system's tracking performance, the absolute error and root mean square tracking error are defined as follows:

[0020]

[0021] When the probability of a deception attack is set to 0.5, the boundaries between the deception attack energy and external interference are both set to... Weight matrix The initial set point was ,when Gradually increase to It is sloping;

[0022] This invention simulates an injection molding process susceptible to deception attacks. The simulations compare the performance of the traditional two-dimensional iterative control method and the robust safety predictive control method proposed in this invention. The performance of the two methods is compared as follows: Figure 1-6 As shown.

[0023] Figure 1The performance of the proposed method and the comparative method in tracking the reference trajectory over multiple cycles is compared. The results show that the proposed method can quickly and stably track the setpoint from the first cycle and rapidly achieves high-precision tracking in subsequent cycles. In contrast, the comparative method failed to track the setpoint in the initial cycle, and its performance converged slowly in the batch direction, requiring more iterations to gradually approach and eventually achieve stable tracking. Figure 2 The performance advantages of the proposed method are demonstrated: compared with the comparative method that exhibits large and divergent errors in the early loops, the proposed method maintains a significantly lower and faster convergence error level even in the initial loops. Figure 3 This demonstrates that the proposed method maintains a higher and more stable level of control input across all cycles, ensuring rapid tracking of the system output.

[0024] Figure 4 and Figure 5 This demonstrates that the proposed method maintains an extremely low and stable root mean square error level from the early iterations, exhibiting both extremely fast convergence speed and excellent performance stability. In contrast, the comparative methods show errors in the early iterations, have slow convergence speed, and require more than ten iterations to achieve a similar low error level. Figure 6 Phase trajectory diagrams for the first to fifth cycles are shown. The results demonstrate that the proposed control strategy exhibits a stable and rapidly converging trajectory even in the initial cycle, with the state increment remaining within a small range, thus avoiding the initial divergence or large oscillations common in traditional methods. Furthermore, the compact distribution of the multi-cycle trajectories indicates that the proposed method effectively suppresses the effects of network spoofing attacks and random disturbances, consistently maintaining the convergence direction and system stability.

[0025] In summary, the method designed in this invention can effectively resist the adverse effects of deception attacks on the control system, and provides a brand-new technical approach for the design of multi-stage intermittent control systems.

[0026] Injection molding of plastic products is a common multi-stage batch process. Figure 7 The four important stages of injection molding are injection, holding pressure, cooling, and demolding.

[0027] In summary, this invention uses the injection molding process as an example to verify the stability and effectiveness of the designed controller. Simulation results show that the designed controller effectively guarantees the system's tracking performance under spoofing attacks and can still operate stably, thereby ensuring the safe and stable operation of the equipment.

Claims

1. A robust security predictive control method for injection molding processes to resist spoofing attacks, characterized in that: The specific steps are as follows: Step 1: Establish a state-space model of the injection molding process subjected to a deception attack: The input-output model of the injection molding process with uncertainty is as follows: During the injection molding process, it will be related to the valve opening ( The corresponding jet speed () The model is: (1) Jet speed ( The corresponding nozzle pressure () The model is: (2) Define discrete during injection molding State variables at time 1 Control variables and output variables : (3) Based on (1), (2) and (3), the state-space equations are obtained as follows: (4) In the formula, , , , Representing the discrete nature of the system respectively The uncertain state matrix at time t, the uncertain control input matrix, and the uncertain output matrix, where , , and These represent the state constant matrix and the control input constant matrix of corresponding dimensions, respectively. and Represents the discrete nature of the system The system state and control input parameters at time t must satisfy the following conditions: Among them, uncertainty disturbances satisfy ,symbol , , They are all constant matrices; Considering the impact of deception attacks and external disturbances on the system, we can obtain the following from equation (4): (5) In the formula, Represents the discrete nature of the system External disturbances at any time Represents the discrete nature of the system A time controller being subjected to a spoofing attack can be represented as: (6) In the formula Represents the discrete nature of the system Deceptive attack signals at all times, satisfying , Represents the discrete nature of the system Deceptive attack energy at all times The deception attack indicator representing the system satisfies: (7) then This indicates that the system has been subjected to a deception attack. At that time, the system was running normally; According to equations (5), (6), and (7), we can obtain: (8) Step 2: Transform the constructed state-space model of the injection molding process subjected to deception attack into an extended system state-space model; Define the system settings as The system is discrete The tracking error at time t is: (9) In the formula The system represents discrete Output at any moment; Define the integral of the output error over time: (10) In the formula The representative value is the maximum duration that the system can achieve in a single run, satisfying the following conditions: According to equation (10), the integral of the output error at the current time can be expressed as: (11) The system is discrete t The deviation in time can be defined as: (12) Therefore, we can conclude that: (13) (14) In the formula , , , These represent the system in discrete states. t State deviation in incremental states at time step, input and spoofing attacks; Based on the above formula, the extended incremental equation is as follows: (15) In the formula, , , , , , , Represents discrete t The constantly expanding system state Represents discrete t System control input at any given time, , These represent the system in discrete states. t Uncertain state matrix and uncertain control input matrix at time t. and Represents the perturbation and output constant matrix, where , ,symbol and These represent the state constant matrix and control input constant matrix of corresponding dimensions, respectively. The matrix represents the uncertainty constants. Represents discrete t The system state and control uncertainties at time t must satisfy... ; Step 3: Design a robust safety predictive control law for the injection molding process subjected to the above-mentioned deception attack: (16) In the formula Representing the control law gain, in order to construct the closed-loop system, we substitute (16) into (15) to obtain the state-space model of the closed-loop system as follows: (17) Based on the above extended model (17), the system optimization problem is transformed into the following min-max optimization problems: (18) (19) In the formula For the stage cost function, the symbol is... It is a positive scalar. and These are weighted matrices of the corresponding dimensions of the system state variables and control inputs, respectively. Represents the terminal cost function; Step 4: Construct the Lyapunov function for the injection molding process subjected to a deception attack: Construct the following Lyapunov function: (20) In the formula It is a positive definite matrix; Step 5: Combining the robust positive definite invariant set and the terminal constraint set, give sufficient conditions to ensure the stability of the system; Robust positive definite invariant set: if a scalar exists make: (21) Then set It is a robust positive definite invariant set. Even if the system faces uncertain parameters and external disturbances, as long as the initial state or the controlled state is within this set, the system state will not diverge, thus ensuring the robustness of the system under disturbances. Theorem 1: For a system, if there exists a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (22) in ,So It is a robust positive definite invariant set; Terminal constraint set: if set It is a robust positive definite invariant set, and it exists Class functions and Lyapunov functions ,satisfy: (23) (24) Then set It is a set of terminal constraints. To ensure the stability of the closed-loop system, it is usually required that the predicted state at the end of the time domain falls into this set of terminal constraints. If the system contains a matrix There exists a positive scalar scalar , , Two positive definite matrices and The following linear matrix inequality conditions are satisfied; (25) in ,So It is simply a set of terminal constraints; By combining the robust safety predictive control law (16), the closed-loop system model (17), the robust safety performance index (18), the Lyapunov function (20), the robust positive definite invariant set (21), and the terminal constraint set (24), sufficient conditions that can guarantee system stability are obtained, and the system stability conditions are transformed into linear matrix inequality forms using the linear matrix inequality technique, namely the linear matrix inequality conditions (22) and (25) in Theorem 1 and Theorem 2. Step Six: Solve for the control law gain K ; Based on the system's state invariant set, we can obtain: (26) Input constraints must satisfy the following conditions: (27) Then it is easy to obtain: (28) In the formula In optimization, this represents the bounds of the control input constraints; therefore, the following optimization problem is proposed: (29) Under the premise of ensuring system stability and strict state constraints, the optimal control law gain is solved online. The control input is updated in real time (6) to ensure that the system can operate stably, safely and reliably under the condition of being subjected to deception attack. The state space model (15) constructed by the state deviation and output error enables the designed robust safety predictive controller to quickly track the set trajectory and realize the first batch of high-precision tracking of the injection molding process.

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