A secure iterative learning control method based on homomorphic encryption

Abstract

The present invention discloses a secure iterative learning control method based on homomorphic encryption. The method comprises: establishing a secure iterative learning control system; constructing a secure iterative learning controller; inputting an initial unencrypted input signal into a controlled object for control, and the controlled object outputting an initial unencrypted output signal; collecting data from a sensor and sequentially quantizing and encrypting the signal to obtain an initial encrypted output signal, and inputting the initial encrypted output signal into a cloud platform; determining whether the tracking error between the unencrypted output signal and a preset reference output signal reaches a preset indicator, repeating the iterative execution until the preset indicator is reached, completing the control of the controlled object, and terminating the control. The present invention performs signal transmission and control signal update in an encrypted manner, thereby ensuring the information security of the system while completing the established control task.

Description

Technical Field

[0001] The present invention relates to a secure iterative learning control method, and in particular to a secure iterative learning control method based on homomorphic encryption. Background Art

[0002] In industry, due to the scale of systems, many control solutions are required to rely on the public internet and cloud platforms. However, open signal transmission and proxy computing can raise potential security concerns. For example, malicious adversaries can exploit data obtained through eavesdropping to attack control systems, thereby disrupting their stable operation. Encrypting system signals can effectively protect data privacy and mitigate the threat of information leakage. Homomorphic encryption is an effective tool for this purpose. It allows operations such as homomorphic sums and multiplications to be performed on encrypted information, ensuring that the resulting data remains valid after decryption. However, a method that effectively achieves both point-by-point tracking of the desired output and privacy protection of system signals is currently lacking. Summary of the Invention

[0003] In order to solve the problems existing in the background technology, the present invention provides a secure iterative learning control method based on homomorphic encryption, which addresses the problem that existing iterative learning control schemes may expose sensitive information in network systems.

[0004] The technical solution adopted in the present invention is:

[0005] The safe iterative learning control method of the present invention comprises the following steps:

[0006] S1. Establish a safe iterative learning control system, including a cloud platform, actuators, controlled objects and sensors. The actuators and sensors are connected to the cloud platform network for communication, and the actuators and sensors are electrically connected to the controlled objects.

[0007] S2. Build a secure iterative learning controller in the cloud platform.

[0008] S3. After encrypting the preset initial control signal and the preset reference output signal, an initial encrypted control signal and an encrypted reference output signal are obtained respectively, and the initial encrypted control signal and the encrypted reference output signal are input into a storage unit in the cloud platform for storage; that is, the initialization process.

[0009] S4. In the initial stage, the preset initial encrypted input signal is input into the executor for decryption processing to obtain an initial unencrypted input signal.

[0010] S5. Input the initial unencrypted input signal into the controlled object to control the controlled object, and the controlled object outputs the initial unencrypted output signal; the sensor collects the initial unencrypted output signal output by the controlled object, and the sensor sequentially quantizes and encrypts the initial unencrypted output signal to obtain an initial encrypted output signal, and inputs the initial encrypted output signal into a storage unit in the cloud platform for storage.

[0011] S6. When the tracking error between the initial unencrypted output signal and the preset reference output signal reaches a preset index, the control of the controlled object is completed and the control is terminated; that is, the initial encrypted output signal can track the preset reference output signal, and the control of the controlled object is completed.

[0012] When the tracking error between the initial unencrypted output signal and the preset reference output signal does not reach the preset index, the initial encrypted control signal, the encrypted reference output signal and the initial encrypted output signal in the cloud platform are input into the secure iterative learning controller of the cloud platform, the secure iterative learning controller outputs the encrypted input signal of the next iteration, the encrypted input signal of the next iteration is input into the actuator for decryption and saturation processing in sequence, and the unencrypted input signal and encrypted control signal of the next iteration are obtained respectively, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly iterated to perform the same operations of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6, until the unencrypted input signal obtained in the next iteration is input into the controlled object to control the controlled object, and the tracking error between the unencrypted output signal output by the controlled object and the preset reference output signal reaches the preset index, then the control of the controlled object is completed and the control is terminated.

[0013] In step S1, the actuator and the sensor are specifically Raspberry Pi development boards; the controlled object is specifically a planar two-degree-of-freedom robotic arm; and the actuator and the sensor are electrically connected to the servos at the two joints of the planar two-degree-of-freedom robotic arm.

[0014] In step S3, the preset initial control signal μ 0,i (t) and the preset reference output signal x d,j (t) After direct encryption processing, the initial encryption control signal is obtained. and encrypted reference output signal The details are as follows:

[0015]

[0016] Among them, μ 0,i (t) represents the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, that is, the preset initial control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the initial stage; represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, i = 1, 2, that is, the preset encrypted control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the initial stage; x d,j (t) represents the j-th preset reference output signal of the planar two-degree-of-freedom manipulator at time t, j = 1, 2, that is, the reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t; r 3,0,i represents the encrypted parameter of the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, r 2,j represents the encrypted parameter of the jth preset reference output signal of the planar two-degree-of-freedom manipulator at time t, r 3,0,i and represents the set of integers, r represents the set of integers Elements in It represents the jth encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t, that is, the encrypted reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t; the encrypted reference output signal It can be formulated and calculated on any secure platform and is not required to be completed on the sensor side.

[0017] In step S6, the initial encrypted control signal in the cloud platform is Encrypted reference output signal and the initial encrypted output signal Input into the secure iterative learning controller of the cloud platform, which then outputs the encrypted input signal for the next iteration. The security iterative learning controller is as follows:

[0018]

[0019] in, It represents the encrypted input signal i applied to the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, k = 0, 1, 2, ..., that is, the encrypted torque applied to each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the k+1th iteration. When k = 0, represents the i-th encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k-th iteration. When k = 0, It represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; represents the jth initial encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t+1; It represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t during the kth iteration. When k = 0, It represents the jth initial encrypted output signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; The encrypted element of the i-th row and j-th column of the learning gain matrix L is obtained by inputting the i-th row and j-th column element of the learning gain matrix L into the sensor and performing the same quantization and encryption process as the output signal.

[0020] The learning gain matrix L satisfies the following conditions:

[0021]

[0022] Where I represents the identity matrix; G(x k ) represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration The input dynamic matrix.

[0023] In step S6, the encrypted input signal of the next iteration is The input executor performs decryption and saturation processing in sequence to obtain the unencrypted input signal u for the next iteration. 1,i (t) and encrypted control signal The details are as follows:

[0024]

[0025] Among them, u k+1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration, i.e., the torque applied to each joint of the planar two-degree-of-freedom manipulator in plain text at time t during the k+1th iteration. When k = 0, u 1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage, u 0,i (t) and They represent the i-th initial unencrypted input signal and the preset initial encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the initial stage; s and l represent the first quantization parameter and the second quantization parameter, respectively; represents a first mapping function based on the first quantization parameter s and the second quantization parameter l; Represents a decryption operation; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. When k=0, represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; ε Λ () indicates encryption operation; represents a second mapping function based on the first quantization parameter s and the second quantization parameter l; sat() represents a saturation function; b u Represents the saturation parameter of the saturation function sat(), which is a positive constant; r 3,k+1,i It represents the i-th unencrypted input signal u applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. k+1,i (t) encryption parameters,

[0026] The obtained encrypted control signal is stored in the storage unit of the cloud platform for use in the next iteration; saturation processing can prevent overflow of the cryptographic algorithm; the secure iterative learning controller can enable the output of the controlled object to track the reference output, and prevent the input and output signals of the controlled object from being eavesdropped, thereby protecting the privacy of the controlled object.

[0027] The first quantization parameter s and the second quantization parameter l specifically satisfy the following quantization constraints:

[0028] -2 l-s-1 ≤u k+1,i (t)≤2 l-s-1 -2 -s

[0029] -2 l-s-1 ≤x k+1,j (t)≤2 l-s-1 -2 -s

[0030] (2 l-1 -1)(2n x (2 l-1 -1)+2 s )+1≤2 l-1

[0031] Among them, x k+1,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the k+1th iteration; the first quantization parameter s is the first quantization parameter that satisfies the quantization constraint. -s The smallest integer value, the second quantization parameter l is the second quantization parameter that satisfies the quantization constraint. l The largest integer value; n x Represents the dimension of the output signal of a planar 2-DOF manipulator.

[0032] The mapping function based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0033]

[0034] Represents the transition from the set of integers to the set of rational numbers The mapping function.

[0035] The second mapping function based on the first quantization parameter s and the second quantization parameter l is specifically as follows

[0036] From the set of rational numbers A mapping function to the set of integers.

[0037] The decryption operation The details are as follows:

[0038]

[0039] λ=lcm(p1-1,p2-1)

[0040] Λ=p1p2

[0041] Wherein, λ and Λ represent the private key and public key respectively, η represents the modular inverse of the private key λ, ηλ = 1 mod Λ; p1 and p2 represent the first prime number and the second prime number respectively, the first prime number p1 and the second prime number p2 are the two largest prime numbers in the solution of gcd(p1p2, (p1-1)(p2-1)) = 1; the public key Λ and the private key λ are generated by the Paillier homomorphic encryption algorithm; the public key Λ can be publicly accessed, while the private key λ must be kept secret; gcd() and lcm() represent the greatest common divisor and the least common multiple respectively.

[0042] The public key Λ satisfies the following conditions:

[0043] 2 l-1 +2n x (2 l-1 -1) 2 ≤Λ

[0044] Among them, n x Represents the dimension of the output signal of a planar 2-DOF manipulator.

[0045] The encryption operation ε Λ () The details are as follows:

[0046]

[0047] Among them, r 1,k+1,jIndicates that at the k+1th iteration, the encryption operation ε Λ () encryption parameters,

[0048] Encryption operation ε Λ (·) and decryption operation is reversible, that is, for any plaintext m and random number r, there exists

[0049] In step S6, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly subjected to the same operations as those of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6. When repeating step S5, the unencrypted output signal obtained in the current iteration is sequentially quantized and encrypted by the sensor in each iteration to obtain the encrypted output signal of the current iteration, as follows:

[0050]

[0051] Among them, x k,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration; represents a quantization function based on a first quantization parameter s and a second quantization parameter l; It represents the jth unencrypted output signal x of the planar two-degree-of-freedom manipulator at time t during the kth iteration. k,j (t) quantized unencrypted output signal after quantization processing; It represents the jth quantized unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration. Encrypted output signal after encryption processing; r 1,k,j Indicates that at the kth iteration, the encryption operation ε Λ () encryption parameters,

[0052] The quantization function based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0053]

[0054] in, represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l; q represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l Elements in .

[0055] A set of rational numbers based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0056]

[0057] Among them, a l and a v are all 0-1 variables; v represents an element in the set {1,2,…,l-1}; quantizing the signal can support the safe iterative learning control law to run on a digital computer; the first quantization parameter s and the second quantization parameter l are used to adjust the resolution and length of the fixed-point number respectively. It can be found that the rational number set Includes resolution 2 -s Separated-2 l-s-1 and 2 l-s-1 -2 -s All rational numbers between .

[0058] In the step S6, the unencrypted input signal obtained in the current iteration is input into the controlled object to control the controlled object, specifically, the torque applied to each joint of the planar two-degree-of-freedom manipulator obtained in the current iteration is used as the torque at each joint of the planar two-degree-of-freedom manipulator to control the planar two-degree-of-freedom manipulator.

[0059] The tracking error between the unencrypted output signal of the controlled object and the preset reference output signal reaches the preset index, as follows:

[0060] |x d,j (t)-x k,j (t)|≤ε

[0061] Where ε represents the preset tracking error.

[0062] The beneficial effects of the present invention are:

[0063] The present invention designs a secure iterative learning control framework for network control systems, which is used to track the reference trajectory point by point during the operation cycle and protect the privacy of system information. The secure control method designed by the present invention can be applied to general nonlinear systems and does not require the use of relevant information such as the system's model structure and physical parameters. Signal transmission and control signal updates are performed in an encrypted manner, thereby ensuring the information security of the system while completing the established control tasks. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 This is a framework diagram of the safe iterative learning control method of the present invention;

[0065] Figure 2 This is a flowchart of the implementation of the safe iterative learning control method of the present invention. DETAILED DESCRIPTION

[0066] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0067] The safe iterative learning control method of the present invention comprises the following steps:

[0068] S1, such as Figure 1 As shown, a safe iterative learning control system is established, including a cloud platform, an actuator, a controlled object and a sensor. The actuator and the sensor are both connected to the cloud platform network for communication, and the actuator and the sensor are both electrically connected to the controlled object.

[0069] In step S1, the actuator and the sensor are specifically Raspberry Pi development boards; the controlled object is specifically a planar two-degree-of-freedom robotic arm; and the actuator and the sensor are electrically connected to the servos at the two joints of the planar two-degree-of-freedom robotic arm.

[0070] S2. Build a secure iterative learning controller in the cloud platform.

[0071] S3, such as Figure 2 As shown, the preset initial control signal and the preset reference output signal are first encrypted to obtain the initial encrypted control signal and the encrypted reference output signal respectively, and the initial encrypted control signal and the encrypted reference output signal are input into the storage unit in the cloud platform for storage; that is, the initialization process.

[0072] In step S3, the preset initial control signal μ 0,i (t) and the preset reference output signal x d,j (t) After direct encryption processing, the initial encryption control signal is obtained. and encrypted reference output signal The details are as follows:

[0073]

[0074] Among them, μ 0,i (t) represents the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, that is, the preset initial control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the initial stage; represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, i = 1, 2, that is, the preset encrypted control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the initial stage; x d,j (t) represents the j-th preset reference output signal of the planar two-degree-of-freedom manipulator at time t, j = 1, 2, that is, the reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t; r 3,0,i represents the encrypted parameter of the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, r 2,jrepresents the encrypted parameter of the jth preset reference output signal of the planar two-degree-of-freedom manipulator at time t, r 3,0,i and represents the set of integers, r represents the set of integers Elements in It represents the jth encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t, that is, the encrypted reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t; the encrypted reference output signal It can be formulated and calculated on any secure platform and is not required to be completed on the sensor side.

[0075] S4. In the initial stage, the preset initial encrypted input signal is input into the executor for decryption processing to obtain an initial unencrypted input signal.

[0076] S5. Input the initial unencrypted input signal into the controlled object to control the controlled object, and the controlled object outputs the initial unencrypted output signal; the sensor collects the initial unencrypted output signal output by the controlled object, and the sensor sequentially quantizes and encrypts the initial unencrypted output signal to obtain an initial encrypted output signal, and inputs the initial encrypted output signal into a storage unit in the cloud platform for storage.

[0077] S6. When the tracking error between the initial unencrypted output signal and the preset reference output signal reaches a preset index, the control of the controlled object is completed and the control is terminated; that is, the initial encrypted output signal can track the preset reference output signal, and the control of the controlled object is completed.

[0078] When the tracking error between the initial unencrypted output signal and the preset reference output signal does not reach the preset index, the initial encrypted control signal, the encrypted reference output signal and the initial encrypted output signal in the cloud platform are input into the secure iterative learning controller of the cloud platform, the secure iterative learning controller outputs the encrypted input signal of the next iteration, the encrypted input signal of the next iteration is input into the actuator for decryption and saturation processing in sequence, and the unencrypted input signal and encrypted control signal of the next iteration are obtained respectively, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly iterated to perform the same operations of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6, until the unencrypted input signal obtained in the next iteration is input into the controlled object to control the controlled object, and the tracking error between the unencrypted output signal output by the controlled object and the preset reference output signal reaches the preset index, then the control of the controlled object is completed and the control is terminated.

[0079] In step S6, the initial encrypted control signal in the cloud platform is Encrypted reference output signal and the initial encrypted output signal Input into the secure iterative learning controller of the cloud platform, which then outputs the encrypted input signal for the next iteration. The security iterative learning controller is as follows:

[0080]

[0081] in, It represents the encrypted input signal i applied to the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, k = 0, 1, 2, ..., that is, the encrypted torque applied to each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the k+1th iteration. When k = 0, represents the i-th encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k-th iteration. When k = 0, It represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; represents the jth initial encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t+1; It represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t during the kth iteration. When k = 0, It represents the jth initial encrypted output signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; The encrypted element of the i-th row and j-th column of the learning gain matrix L is obtained by inputting the i-th row and j-th column element of the learning gain matrix L into the sensor and performing the same quantization and encryption process as the output signal.

[0082] The learning gain matrix L satisfies the following conditions:

[0083]

[0084] Where I represents the identity matrix; G(x k ) represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration The input dynamic matrix.

[0085] In step S6, the encrypted input signal of the next iteration is The input executor performs decryption and saturation processing in sequence to obtain the unencrypted input signal u for the next iteration. 1,i (t) and encrypted control signal The details are as follows:

[0086]

[0087] Among them, u k+1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration, i.e., the torque applied to each joint of the planar two-degree-of-freedom manipulator in plain text at time t during the k+1th iteration. When k = 0, u 1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage, u 0,i (t) and They represent the i-th initial unencrypted input signal and the preset initial encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the initial stage; s and l represent the first quantization parameter and the second quantization parameter, respectively; represents a first mapping function based on the first quantization parameter s and the second quantization parameter l; Represents a decryption operation; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. When k=0, represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; ε Λ () indicates encryption operation; represents a second mapping function based on the first quantization parameter s and the second quantization parameter l; sat() represents a saturation function; b u Represents the saturation parameter of the saturation function sat(), which is a positive constant; r 3,k+1,i It represents the i-th unencrypted input signal u applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. k+1,i (t) encryption parameters,

[0088] The obtained encrypted control signal is stored in the storage unit of the cloud platform for use in the next iteration; saturation processing can prevent overflow of the cryptographic algorithm; the secure iterative learning controller can enable the output of the controlled object to track the reference output, and prevent the input and output signals of the controlled object from being eavesdropped, thereby protecting the privacy of the controlled object.

[0089] The first quantization parameter s and the second quantization parameter l specifically satisfy the following quantization constraints:

[0090] -2 l-s-1 ≤u k+1,i (t)≤2 l-s-1 -2 -s

[0091] -2 l-s-1 ≤xk+1,j (t)≤2 l-s-1 -2 -s

[0092] (2 l-1 -1)(2n x (2 l-1 -1)+2 s )+1≤2 l-1

[0093] Among them, x k+1,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the k+1th iteration; the first quantization parameter s is the first quantization parameter that satisfies the quantization constraint. -s The smallest integer value, the second quantization parameter l is the second quantization parameter that satisfies the quantization constraint. l The largest integer value; n x Represents the dimension of the output signal of a planar two-degree-of-freedom manipulator;

[0094] Mapping function based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0095]

[0096] Represents the transition from the set of integers to the set of rational numbers The mapping function.

[0097] The second mapping function based on the first quantization parameter s and the second quantization parameter l is specifically as follows

[0098]

[0099] From the set of rational numbers A mapping function to the set of integers.

[0100] Decryption operation The details are as follows:

[0101]

[0102] λ=lcm(p1-1,p2-1)

[0103] Λ=p1p2

[0104] Where λ and Λ represent the private key and public key respectively, η represents the modular inverse of the private key λ, ηλ = 1 mod Λ; p1 and p2 represent the first prime number and the second prime number respectively, the first prime number p1 and the second prime number p2 are the two largest prime numbers in the solution of gcd(p1p2, (p1-1)(p2-1)) = 1; the public key Λ and the private key λ are generated by the Paillier homomorphic encryption algorithm; the public key Λ can be publicly accessed, while the private key λ must be kept secret; gcd() and lcm() represent the greatest common divisor and the least common multiple respectively;

[0105] The public key Λ satisfies the following conditions:

[0106] 2 l-1 +2n x (2 l-1 -1) 2 ≤Λ

[0107] Among them, n x Represents the dimension of the output signal of a planar 2-DOF manipulator.

[0108] Encryption operation ε Λ () The details are as follows:

[0109]

[0110] Among them, r 1,k+1,j Indicates that at the k+1th iteration, the encryption operation ε Λ () encryption parameters,

[0111] Encryption operation ε Λ (·) and decryption operation is reversible, that is, for any plaintext m and random number r, there exists

[0112] In step S6, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly subjected to the same operations as those of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6. When repeating step S5, the unencrypted output signal obtained in the current iteration is quantized and encrypted by the sensor in each iteration to obtain the encrypted output signal of the current iteration, as follows:

[0113]

[0114] Among them, x k,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration; represents a quantization function based on a first quantization parameter s and a second quantization parameter l; It represents the jth unencrypted output signal x of the planar two-degree-of-freedom manipulator at time t during the kth iteration. k,j (t) quantized unencrypted output signal after quantization processing; It represents the jth quantized unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration. Encrypted output signal after encryption processing; r 1,k,j Indicates that at the kth iteration, the encryption operation ε Λ () encryption parameters,

[0115] Quantization function based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0116]

[0117] in, represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l; q represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l Elements in .

[0118] A set of rational numbers based on the first quantization parameter s and the second quantization parameter l The details are as follows:

[0119]

[0120] Among them, a l and a v are all 0-1 variables; v represents an element in the set {1,2,…,l-1}; quantizing the signal can support the safe iterative learning control law to run on a digital computer; the first quantization parameter s and the second quantization parameter l are used to adjust the resolution and length of the fixed-point number respectively. It can be found that the rational number set Includes resolution 2 -s Separated-2 l-s-1 and 2 l-s-1 -2 -s All rational numbers between .

[0121] In step S6, the unencrypted input signal obtained in the current iteration is input into the controlled object to control the controlled object. Specifically, the torque applied to each joint of the planar two-degree-of-freedom manipulator obtained in the current iteration is used as the torque at each joint of the planar two-degree-of-freedom manipulator to control the planar two-degree-of-freedom manipulator.

[0122] The tracking error between the unencrypted output signal of the controlled object and the preset reference output signal reaches the preset index, as follows:

[0123] |xd,j (t)-x k,j (t)|≤ε

[0124] Where ε represents the preset tracking error.

[0125] The method of the present invention constructs a signal quantization and restoration method to support data transmission on the network and homomorphic operations on the cloud platform; develops an iterative learning control method based on quantization feedback to provide support for the integration of cryptographic algorithms and control schemes; designs a secure iterative learning control method based on homomorphic operations and builds a secure control framework for network systems, wherein the sensor is responsible for quantizing and encrypting the measured system output and uploading it to the cloud platform, the controller is responsible for updating the input instructions of the control system using the secure iterative learning control method, and the actuator is responsible for decrypting the downloaded control instructions and applying them to the controlled object.

[0126] The control algorithm of the present invention has very low requirements on the physical model information of the system. At the same time, the update of the safety control instructions only relies on the encrypted system signal, so it can ensure the privacy of the system signal while achieving the tracking control task.

Claims

1. A secure iterative learning control method based on homomorphic encryption technology, characterized by: The steps include: S1. Establish a secure iterative learning control system, including a cloud platform, actuators, a controlled object, and sensors. The actuators and sensors are connected to the cloud platform network for communication, and are electrically connected to the controlled object. S2. Build a secure iterative learning controller in the cloud platform; S3. Encrypting the preset initial control signal and the preset reference output signal to obtain an initial encrypted control signal and an encrypted reference output signal, respectively, and inputting the initial encrypted control signal and the encrypted reference output signal into the cloud platform for storage; S4, inputting the preset initial encrypted input signal into the executor for decryption processing to obtain an initial unencrypted input signal; S5. Inputting the initial unencrypted input signal into the controlled object to control the controlled object, and the controlled object outputs the initial unencrypted output signal; The sensor collects the initial unencrypted output signal of the controlled object, quantizes and encrypts the initial unencrypted output signal in sequence to obtain the initial encrypted output signal, and then inputs the initial encrypted output signal into the cloud platform for storage; S6. When the tracking error between the initial unencrypted output signal and the preset reference output signal reaches a preset index, the control of the controlled object is completed and the control is terminated; When the tracking error between the initial unencrypted output signal and the preset reference output signal does not reach the preset index, the initial encrypted control signal, the encrypted reference output signal and the initial encrypted output signal in the cloud platform are input into the secure iterative learning controller of the cloud platform, the secure iterative learning controller outputs the encrypted input signal of the next iteration, the encrypted input signal of the next iteration is input into the actuator for decryption and saturation processing in sequence, and the unencrypted input signal and encrypted control signal of the next iteration are obtained respectively, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly iterated to perform the same operations of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6, until the unencrypted input signal obtained in the next iteration is input into the controlled object to control the controlled object, and the tracking error between the unencrypted output signal output by the controlled object and the preset reference output signal reaches the preset index, then the control of the controlled object is completed and the control is terminated.

2. The secure iterative learning control method based on homomorphic encryption technology according to claim 1, characterized in that: In step S1, the actuator and the sensor are specifically Raspberry Pi development boards; the controlled object is specifically a planar two-degree-of-freedom robotic arm; and the actuator and the sensor are electrically connected to the servos at the two joints of the planar two-degree-of-freedom robotic arm.

3. The secure iterative learning control method based on homomorphic encryption technology according to claim 2, characterized in that: In step S3, the preset initial control signal μ 0,i (t) and the preset reference output signal x d,j (t) After direct encryption processing, the initial encryption control signal is obtained. and encrypted reference output signal The details are as follows: Among them, μ 0,i (t) represents the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, that is, the preset initial control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the initial stage; represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, i = 1, 2, that is, the preset encrypted control signal of the torque at each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the initial stage; x d,j (t) represents the j-th preset reference output signal of the planar two-degree-of-freedom manipulator at time t, j = 1, 2, that is, the reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t; r 3,0,i represents the encrypted parameter of the i-th preset initial control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage, r 2,j represents the encrypted parameter of the jth preset reference output signal of the planar two-degree-of-freedom manipulator at time t, r 3,0,i and represents the set of integers, Represents a set of integers Elements in It represents the j-th encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t, that is, the encrypted reference angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t.

4. The secure iterative learning control method based on homomorphic encryption technology according to claim 2, characterized in that: In step S6, the initial encrypted control signal in the cloud platform is Encrypted reference output signal and the initial encrypted output signal Input into the secure iterative learning controller of the cloud platform, which then outputs the encrypted input signal for the next iteration. The security iterative learning controller is as follows: in, It represents the encrypted input signal i applied to the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, k = 0, 1, 2, ..., that is, the encrypted torque applied to each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t in the k+1th iteration. When k = 0, represents the i-th encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k-th iteration. When k = 0, It represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; represents the jth initial encrypted reference output signal of the planar two-degree-of-freedom manipulator at time t+1; It represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in ciphertext form at time t during the kth iteration. When k = 0, It represents the jth initial encrypted output signal of the planar two-degree-of-freedom manipulator at time t in the initial stage; Represents the encrypted element in the i-th row and j-th column of the learning gain matrix L; The learning gain matrix L satisfies the following conditions: Where I represents the identity matrix; G(x k ) represents the jth encrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration The input dynamic matrix.

5. The secure iterative learning control method based on homomorphic encryption technology according to claim 2, characterized in that: In step S6, the encrypted input signal of the next iteration is The input executor performs decryption and saturation processing in sequence to obtain the unencrypted input signal u for the next iteration. 1,i (t) and encrypted control signal The details are as follows: Among them, u k+1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration, i.e., the torque applied to each joint of the planar two-degree-of-freedom manipulator in plain text at time t during the k+1th iteration. When k = 0, u 1,i (t) represents the i-th unencrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage, u 0,i (t) and They represent the i-th initial unencrypted input signal and the preset initial encrypted input signal applied to the planar two-degree-of-freedom manipulator at time t in the initial stage; s and l represent the first quantization parameter and the second quantization parameter, respectively; represents a first mapping function based on the first quantization parameter s and the second quantization parameter l; Represents a decryption operation; It represents the i-th encrypted control signal of the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. When k=0, represents the i-th initial encrypted control signal of the planar two-degree-of-freedom manipulator at time t in the next iteration after the initial stage; ε Λ () indicates encryption operation; represents a second mapping function based on the first quantization parameter s and the second quantization parameter l; sat() represents a saturation function; b u Represents the saturation parameter of the saturation function sat(); r 3,k+1,i It represents the i-th unencrypted input signal u applied to the planar two-degree-of-freedom manipulator at time t during the k+1th iteration. k+1,i (t) encryption parameters, 6. The secure iterative learning control method based on homomorphic encryption technology according to claim 5, characterized in that: The first quantization parameter s and the second quantization parameter l specifically satisfy the following quantization constraints: -2 l-s-1 ≤u k+1,i (t)≤2 l-s-1 -2 -s -2 l-s-1 ≤x k+1,j (t)≤2 l-s-1 -2 -s (2 l-1 -1)(2n x (2 l-1 -1)+2 s )+1≤2 l-1 Among them, x k+1,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t in the k+1th iteration, that is, the angular displacement of each joint of the planar two-degree-of-freedom manipulator in plain text at time t in the k+1th iteration; the first quantization parameter s is the first quantization parameter that satisfies the quantization constraint. -s The smallest integer value, the second quantization parameter l is the second quantization parameter that satisfies the quantization constraint. l The largest integer value; n x Represents the dimension of the output signal of a planar two-degree-of-freedom manipulator; The mapping function based on the first quantization parameter s and the second quantization parameter l The details are as follows: The second mapping function based on the first quantization parameter s and the second quantization parameter l is specifically as follows 7. The secure iterative learning control method based on homomorphic encryption technology according to claim 5, characterized in that: The decryption operation The details are as follows: λ=lcm(p1-1,p2-1) Λ=p1p2 Wherein, λ and Λ represent the private key and public key respectively, η represents the modular inverse of the private key λ, ηλ = 1 mod Λ; p1 and p2 represent the first prime number and the second prime number respectively, the first prime number p1 and the second prime number p2 are the two largest prime numbers in the solution of gcd(p1p2, (p1-1)(p2-1)) = 1; gcd() and lcm() represent the greatest common divisor and the least common multiple respectively; The public key Λ satisfies the following conditions: <h2 style=";text-align:left;direction:ltr">2<h2 style=";text-align:left;direction:ltr"> l-1 <h2 style=";text-align:left;direction:ltr"> +2n<h2 style=";text-align:left;direction:ltr"> x <h2 style=";text-align:left;direction:ltr"> (2<h2 style=";text-align:left;direction:ltr"> l-1 <h2 style=";text-align:left;direction:ltr"> -1)<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> ≤Λ Among them, n x Represents the dimension of the output signal of a planar two-degree-of-freedom manipulator; The encryption operation ε Λ () The details are as follows: Among them, r 1,k+1,j Indicates that at the k+1th iteration, the encryption operation ε Λ () encryption parameters, 8. The secure iterative learning control method based on homomorphic encryption technology according to claim 2, characterized in that: In step S6, the unencrypted input signal and the encrypted control signal of the next iteration are repeatedly subjected to the same operations as those of the initial unencrypted input signal and the initial encrypted control signal in steps S5-S6. When repeating step S5, the unencrypted output signal obtained in the current iteration is sequentially quantized and encrypted by the sensor in each iteration to obtain the encrypted output signal of the current iteration, as follows: Among them, x k,j (t) represents the jth unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration; represents a quantization function based on a first quantization parameter s and a second quantization parameter l; It represents the jth unencrypted output signal x of the planar two-degree-of-freedom manipulator at time t during the kth iteration. k,j (t) quantized unencrypted output signal after quantization processing; It represents the jth quantized unencrypted output signal of the planar two-degree-of-freedom manipulator at time t during the kth iteration. Encrypted output signal after encryption processing; r 1,k,j Indicates that at the kth iteration, the encryption operation ε Λ () encryption parameters, 9. The secure iterative learning control method based on homomorphic encryption technology according to claim 8, characterized in that: The quantization function based on the first quantization parameter s and the second quantization parameter l The details are as follows: in, represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l; q represents a set of rational numbers based on the first quantization parameter s and the second quantization parameter l Elements in A set of rational numbers based on the first quantization parameter s and the second quantization parameter l The details are as follows: Among them, a l and a v All are 0-1 variables; v represents an element in the set {1,2,…,l-1}.

10. The secure iterative learning control method based on homomorphic encryption technology according to claim 2, characterized in that: In step S6, the unencrypted input signal obtained in the iteration is input into the controlled object to control the controlled object, specifically, the torque applied to each joint of the planar two-degree-of-freedom manipulator obtained in the iteration is used as the torque at each joint of the planar two-degree-of-freedom manipulator to control the planar two-degree-of-freedom manipulator; The tracking error between the unencrypted output signal of the controlled object and the preset reference output signal reaches the preset index, as follows: |x d,j (t)-x k,j (t)|≤ε Where ε represents the preset tracking error.

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