An Adaptive Robust MPC Method for Systems with Parameter Uncertainties and Input Delays

Abstract

The present invention discloses an adaptive robust MPC method for processing a system containing parameter uncertainty and input delay, comprising the following steps: establishing a discrete linear system containing parameter uncertainty and input delay; establishing an optimization control problem for infinite time domain; based on an estimation system and an error transmission system, establishing an adaptive update law based on a time-varying update rate and updating the estimated parameters of the system online; based on the estimated parameters of the system updated online, for the infinite time domain optimization control problem, using an ellipsoid parameterization method to simplify the optimization process, using Tube‑MPC to design a controller, obtaining an offline optimization problem and an online optimization problem; solving the offline optimization problem and the online optimization problem, and obtaining an optimization result. The control method proposed by the present invention has the advantages of high computational efficiency, strong adaptability and good robustness, and can be widely used in fields such as automatic control and industrial process control, and is particularly suitable for complex systems with parameter uncertainty and input delay.

Description

Technical Field

[0001] The invention belongs to the field of automation technology, and in particular relates to an adaptive robust MPC method for processing a system containing parameter uncertainty and input delay. Background Art

[0002] In practical applications, there are always uncertainties in the system caused by various factors, such as random noise, uncertainty in model parameters, including additive or multiplicative uncertainty. These uncertainties will affect the performance of the system and even cause system instability. In addition, in some real-time response systems, input delays such as sensor delays, actuator delays, and signal transmission delays will also affect system performance and even cause safety hazards. At present, MPC research for uncertain system models mainly focuses on two aspects: robust MPC and adaptive MPC.

[0003] Robust MPC is a common and effective method to solve system uncertainty, mainly including min-max MPC and Tube-MPC. Traditional robust MPC usually considers the worst case of parameter uncertainty, and the results are relatively conservative. Therefore, combining adaptive control is a promising technical solution while considering both system parameter uncertainty and input delay.

[0004] The core idea of ​​adaptive MPC is to design an adaptive algorithm to update uncertain parameters online and calculate the optimal control input based on the updated system model while ensuring system stability. Currently, a variety of parameter identification strategies for adaptive MPC have been studied, including membership identification, recursive least squares, neural networks, and adaptive update laws.

[0005] However, in practical systems, time input delays often occur, and there are relatively few studies on MPC for systems with both time delays and model uncertainty. Therefore, how to handle input delays and parameter uncertainties in the system in a stable and efficient way remains a difficult point in current research. Summary of the invention

[0006] In view of the deficiencies of the above-mentioned prior art, the present invention provides an adaptive robust MPC method for processing systems with parameter uncertainty and input delay. Based on the Tube model predictive control (Tube-MPC) framework, the system parameters are estimated adaptively online and the input delay problem is processed in combination with the state expansion method, so as to achieve robust control of the system and solve the problem of parameter uncertainty and input delay in linear systems.

[0007] In order to achieve the above technical objectives, the technical solution adopted by the present invention is:

[0008] An adaptive robust MPC method for processing a system with parameter uncertainty and input delay comprises the following steps:

[0009] Step 1: Establish a discrete linear system with parameter uncertainty and input delay;

[0010] Step 2: Using the state variable expansion method, the discrete linear system is converted into an augmented system and an estimation system and an error transfer system of the augmented system are established, and an optimization control problem is established based on the estimation system;

[0011] Step 3: Based on the estimation system and the error transfer system, an adaptive update law based on a time-varying update rate is established and the estimation parameters of the system are updated online to ensure that the error of the estimation parameters is bounded;

[0012] Step 4: Based on the estimated parameters of the system updated online, for the infinite time domain optimization control problem, the ellipsoid parameterization method is used to simplify the optimization process, and the controller is designed using the Tube-MPC concept to obtain the offline optimization problem and the online optimization problem;

[0013] Step 5: Adopting the adaptive robust MPC algorithm to solve the offline optimization problem and the online optimization problem to obtain the optimization result.

[0014] To optimize the above technical solutions, the specific measures taken also include:

[0015] The above step 1 establishes the following discrete linear system with parameter uncertainty and input delay:

[0016] x θ (k+1)=A θ x θ (k)+B θ u(kd) (1)

[0017] where x θ (k+1) is the system state measured at time k+1;

[0018] and are the state variables and input variables of the system, is a known time delay, and is the system matrix with uncertainty;

[0019] The state and input of the system are in a set containing the origin, defined as:

[0020] C:={(x θ ,u)|Fx θ +Gu≤1} (2)

[0021] in

[0022] The above step 2 converts the discrete linear system into an augmented system and establishes an estimation system of the augmented system, including:

[0023] Step 201, using the state variable expansion method, the original system is converted into the following augmented system:

[0024] x(k+1)=Ax(k)+Bu(k) (3)

[0025] Where x(k+1) represents the state vector after expansion at time k+1;

[0026] represents the state vector after expansion at time k;

[0027] and Represents the expanded system matrix in the following form:

[0028] x(k)=[u(k-1) T ,…,u(kd) T ,x θ (k) T ] T

[0029]

[0030] Step 202, establishing an estimation system of the augmented system:

[0031]

[0032] in and are the estimated values ​​of A and state x(k) respectively;

[0033] Step 203, subtracting (4) from (3) to obtain the following error transfer system:

[0034]

[0035] in is the state error;

[0036] is the estimated error of matrix A;

[0037] Step 204, define the current state vector δ(k) = [u(kd) T x θ (k) T ] T , referring to formula (5), the state estimation error is defined as follows:

[0038]

[0039] in is the state estimation error;

[0040] is the parameter estimation error, represents the domain of real numbers, n represents the dimension of the system's state variables, and m represents the dimension of the system's input variables;

[0041] a and They are:

[0042]

[0043] a=[B θ A θ ]

[0044]

[0045] Where a is the real parameter vector of the system;

[0046] is the estimate of the system state matrix at time k;

[0047] is the estimate of the system state matrix at time k.

[0048] The above step 2 described in establishing the optimization control problem based on the estimation system includes:

[0049] The parameter uncertainty is expressed in the form of additive disturbance, and the estimation system shown in formula (4) is written as follows:

[0050]

[0051] in represents parameter uncertainty in the form of perturbations, satisfying the set Satisfy the quadratic constraint w T P w w≤1;

[0052] is a positive definite matrix, which determines the shape and range of the perturbation set and ensures that the quadratic form is non-negative;

[0053] D=I md+n is the perturbation matrix;

[0054] Construct a set of collection sequences using the Tube-MPC algorithm It is called the state Tube, which contains the input sequence All the trajectories of the system that can be generated, the infinite time domain optimization control problem is as follows:

[0055]

[0056] Fx l|k +Gu l|k ≤1 (11)

[0057]

[0058] Where l represents the time step index in the prediction time domain, and N represents the length of the prediction time domain;

[0059] J(·,·) and J T (·) represent the stage and terminal cost functions defined based on the state Tube, respectively, and the form is as follows:

[0060]

[0061] in and is a positive definite matrix, is the positive definite terminal cost matrix calculated offline,

[0062] The above step 3, based on the estimation system and the error transfer system, establishes an adaptive update law based on a time-varying update rate and updates the estimation parameters of the system online, including:

[0063] First, define the state error The cost function form is as follows:

[0064]

[0065] J is obtained by the following formula x Relative to The gradient is:

[0066]

[0067] Then an adaptive update law is used to minimize the cost function, introducing a time-varying update rate λ(k)>0 as follows:

[0068]

[0069] λ(k) is obtained by the following cost function:

[0070]

[0071] To J x Taking the derivative with respect to λ(k) and setting it equal to zero:

[0072]

[0073] get:

[0074]

[0075] Substituting formula (18) into (15), we get:

[0076]

[0077] in At time k+1, by calculating Get, x θ (k+1) is the system state measured at time k+1.

[0078] The above step 4 uses the ellipsoid parameterization method to simplify the optimization process, including:

[0079] Status Tube Parameterized using a predefined ellipsoid in the following form:

[0080]

[0081] in represents the state set, and x represents the system state vector;

[0082] is a positive definite matrix, L is given by P = L T Obtained by Cholesky decomposition of L;

[0083] ||·||2 represents the 2-norm;

[0084] Using Scaling Variables and shift variables Parameterize the state Tube as follows:

[0085]

[0086] where e l|k =x l|k -z l|k ;

[0087] It represents the scaling factor at the lth moment when predicting the time k, which is used to scale the size of the state Tube;

[0088] represents the state offset vector at the lth moment when predicting time k;

[0089] ⊕ represents the Minkowski sum;

[0090] The control input is parameterized as follows:

[0091]

[0092] Where K is the feedback gain of the offline design, For online optimization of variables;

[0093] The terminal set is defined using an ellipsoid as follows:

[0094]

[0095] in Represents a terminal set.

[0096] The above step 4 uses the ellipsoid parameterization method to simplify the optimization process, and also includes:

[0097] Re-stating the optimization problem based on variables, the initial condition of equation (9) is expressed as the following quadratic constraint:

[0098]

[0099] The conditions for formula (10) to be valid are as follows:

[0100]

[0101] in For a given scalar, d l|k =Az l|k +Bv l|k -z l+1|k , Formula (25) is about the variable z l|k ,α l|k ,α l+1|k ,τ 1,l|k and τ 2,l|k Linear matrix inequality LMI;

[0102] The conditions for formula (11) to be valid are as follows:

[0103]

[0104] in is the maximum value vector of offline calculation, which provides the worst case of the state space of the state Tube in all directions to ensure the satisfaction of the state constraints. Each element is Among them [F+GK] i Denotes the i-th row of the matrix (F+GK), then the parameterized form of the control input and state Tube of equation (11) is obtained, and the state and input constraints can be restated as linear inequalities;

[0105] The conditions for formula (12) to be valid are as follows:

[0106]

[0107] where τ 1,T For a given scalar, Equation (27) is about the variable z N|k ,α N|k and τ 1,T Linear matrix inequality LMI in .

[0108] The controller is designed using the Tube-MPC concept as described in step 4 above, and the offline optimization problem is finally obtained, including:

[0109] The optimization problem is formulated offline to calculate the ellipsoid shape matrix P and feedback gain K. First, in order to ensure the recursive feasibility, the terminal set of the MPC controller is designed to remain unchanged under the terminal controller, that is, the terminal set The following conditions must be met:

[0110]

[0111] where x + represents the system state at the next time step;

[0112] Written in the framework of the ellipsoid Tube:

[0113]

[0114] If there is a positive scalar So that the following is true:

[0115]

[0116] Where Y = KP -1 , τ 1,O +τ 2,O ≤1, then the terminal set Satisfy condition (29);

[0117] In addition, the terminal set is designed to satisfy the state and input constraints under the terminal control law, which is in the following form:

[0118]

[0119] in Used together with Schur's complement lemma, we get the above result;

[0120] Finally, an offline optimization problem needs to be formulated to calculate the ellipsoid shape P and the feedback gain K in order to maximize the controller’s attraction area and thus the size of the terminal set, which is achieved by minimizing the determinant of P. Therefore, the offline optimization problem is expressed as:

[0121]

[0122] Terminal cost matrix P C The solution conditions are as follows:

[0123]

[0124] Where Q x is the state weight matrix, Q u Input weight matrix for control.

[0125] The online optimization problem obtained in step 4 above is as follows:

[0126]

[0127] in For online optimization variables.

[0128] The adaptive robust MPC algorithm of step 5 above is as follows:

[0129] First, the initial state of the system is given Initial estimate matrix Control and prediction time domain N p , cost matrix Q;

[0130] Then, solve the offline optimization problem, including:

[0131] Step 501: k←1;

[0132] Step 502: Setting the grid τ 1,O ∈(0,1);

[0133] Step 503: Through each τ 1,O Solve the offline optimization problem (32) to calculate P and K;

[0134] Step 504: Select P, K to minimize det(P);

[0135] Step 505: Calculate P C Satisfy equation (33) and minimize trace(P C );

[0136] Solve online optimization problems, including:

[0137] Step 511: Solve the online optimization problem (34);

[0138] Step 512: Apply control input u k =K(x k -z 0|k )+v 0|k ;

[0139] Step 513: Measure the state x(k+1) of the actual uncertain system;

[0140] Step 514: Use the adaptive update law to estimate the system renew;

[0141] Step 515: Setup

[0142] Step 516: k←k+1.

[0143] The present invention has the following beneficial effects:

[0144] The present invention first expands the original system state to compensate for the influence of input delay, and then designs an adaptive update law to estimate the system parameters in real time. Subsequently, the updated uncertain parameters are expressed in the form of additive interference, and a robust controller is designed through the Tube-MPC algorithm to ensure the stability and performance optimization of the system in the presence of disturbances.

[0145] The control method proposed in the present invention has the advantages of high computational efficiency, strong adaptability and good robustness, and can be widely used in the fields of automatic control, industrial process control, etc., and is particularly suitable for complex systems with parameter uncertainty and input delay.

[0146] The present invention aims at linear systems with parameter uncertainty and input delay, and adopts an extended state method to effectively deal with the input delay problem, ensuring the effectiveness and accuracy of the control strategy. The proposed adaptive update law dynamically adjusts the system parameters and enhances the robustness of the system in the face of uncertainty. BRIEF DESCRIPTION OF THE DRAWINGS

[0147] Figure 1 The schematic diagram of the adaptive robust MPC method for processing a system with parameter uncertainty and input delay according to the present invention;

[0148] Figure 2 It is the control input diagram of the system at different delay steps in the example shown in the present invention;

[0149] Figure 3 It is the state trajectory diagram of the system under different delay steps in the example shown in the present invention. DETAILED DESCRIPTION

[0150] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0151] Although the steps in the present invention are arranged with numbers, they are not used to limit the order of the steps. Unless the order of the steps is clearly stated or the execution of a certain step requires other steps as a basis, the relative order of the steps can be adjusted. It is understood that the term "and / or" used in this article involves and covers any and all possible combinations of one or more of the associated listed items.

[0152] The invention discloses an adaptive robust MPC method for processing a system with parameter uncertainty and input delay, an adaptive robust model predictive control method (Model Predictive Control, MPC), which is used to process a constrained linear system with constant parameter uncertainty and input delay to achieve system robust stability and performance optimization, and realizes robust control of the system by online adaptive estimation of system parameters and combining a state expansion method to process the input delay problem, such as Figure 1 As shown, the following steps are included:

[0153] Step 1: Establish a discrete linear system with parameter uncertainty and input delay;

[0154] Step 2: Use the state variable expansion method to convert the discrete linear system into an augmented system and establish an estimation system and an error transmission system of the augmented system, and establish an optimization control problem based on the estimation system;

[0155] Step 3: Based on the estimation system and the error transfer system, an adaptive update law based on a time-varying update rate is established and the estimation parameters of the system are updated online to ensure that the error of the estimation parameters is bounded;

[0156] Step 4: Based on the estimated parameters of the system updated online, for the infinite time domain optimization control problem, the ellipsoid parameterization method is used to simplify the optimization process, and the controller is designed using the Tube-MPC concept to obtain the offline optimization problem and the online optimization problem;

[0157] Step 5: Adopt the adaptive robust MPC algorithm to solve the offline optimization problem and the online optimization problem and obtain the optimization result.

[0158] In the embodiment, step 1 establishes a discrete linear system with parameter uncertainty and input delay, which is expressed as follows:

[0159] x θ (k+1)=A θ x θ (k)+B θ u(kd) (1)

[0160] in and are the state variables and input variables of the system, is a known time delay, and is the system matrix with uncertainty. The state and input of the system are located in a set containing the origin and are defined as:

[0161] C:={(x θ ,u)|Fx θ +Gu≤1} (2)

[0162] in

[0163] In the embodiment, in step 2, since the system contains input delay, it is necessary to consider the impact of historical input data on the system. Here, a state variable expansion method is used to convert the original system into an augmented system by adding a state vector to incorporate the input delay, and an optimization control problem for the model is proposed;

[0164] Using the state variable expansion method, the original system is converted into an augmented system to compensate for the input delay as follows:

[0165] x(k+1)=Ax(k)+Bu(k) (3)

[0166] in represents the expanded state vector, the matrix and Represents the expanded system matrix in the following form:

[0167] x(k)=[u(k-1) T ,…,u(kd) T ,x θ (k) T ] T

[0168]

[0169] Since the above system matrix A contains uncertainty, in order to deal with this problem, the estimation system of the augmented system (3) is established as follows:

[0170]

[0171] in and are the estimated values ​​of the system expansion matrix A and state x respectively; subtracting (4) from (3) yields the following error transfer system:

[0172]

[0173] in is the state error, is the estimation error of the state matrix A of the estimated system (3);

[0174] Define the current state vector δ(k) = [u(kd) T x θ (k) T ] T ,in Referring to formula (5), the state estimation error is defined as follows:

[0175]

[0176] in is the state estimation error, is the parameter estimation error, represents the domain of real numbers, n represents the dimension of the system's state variables, and m represents the dimension of the system's input variables;

[0177] Formula (6) expresses the error of parameter estimation The prediction error state calculated by the current state vector δ(k) is used to describe the evolution of the prediction error at the next moment. It is part of the model prediction and ensures that the control strategy can be adjusted dynamically.

[0178] a and They are:

[0179]

[0180] a=[B θ A θ ]

[0181]

[0182] Where a is the real parameter vector of the system, is the estimate of the system state matrix at time k, is the estimate of the system state matrix at time k;

[0183] The parameter uncertainty is expressed in the form of additive disturbance, and the estimation system shown in formula (4) is written as follows:

[0184]

[0185] in represents parameter uncertainty in the form of perturbations, satisfying the set w satisfies the quadratic constraint w T P w w≤1; is a positive definite matrix, which determines the shape and range of the perturbation set and ensures that the quadratic form is non-negative; D = I md+n is the perturbation matrix;

[0186] In order to deal with disturbances, a set of set sequences is constructed using the Tube-MPC algorithm idea It is called the state Tube, which contains the input sequence All trajectories of the system that can be generated. The optimization problem is as follows:

[0187]

[0188] Fx l|k +Gu l|k ≤1(11)

[0189]

[0190] Where l represents the time step index in the prediction time domain, N represents the length of the prediction time domain; J(·,·) and J T (·) represent the stage and terminal cost functions defined based on the state Tube, respectively, and the form is as follows:

[0191]

[0192] in and is a positive definite matrix, is the positive terminal cost matrix calculated offline, and the parameters in formula (11) are and

[0193] The above formula (8) is an optimization problem, the goal is to minimize the sum of the stage cost and the terminal cost in the prediction time domain;

[0194] Formula (9) is the initial condition, indicating that the initial state must be within the initial state set, ensuring that the optimization problem starts from the correct initial state, indicating that the state update formula (7) takes effect at the initial moment of the optimization problem;

[0195] Formula (10) shows the recursive relationship of the state Tube. It uses the dynamic characteristics of the state update formula to describe the evolution of the state set in the prediction time domain, that is, the state Tube at the next moment must contain the result of the state Tube at that time after the system dynamics, input and disturbance;

[0196] Formula (11) is the state and control input constraint, indicating that the result calculated in the state update formula (7) satisfies this constraint, ensuring that the state and control input at each moment meet the constraint conditions;

[0197] Formula (12) is the terminal state constraint, which means that the state update formula (7) updates the state at each prediction step, and the final state must be within the terminal set to satisfy the terminal constraint, ensuring that at the end of the prediction time domain, the system state is within the expected terminal state set.

[0198] In the embodiment, step 3 designs an adaptive update law based on a time-varying update rate to update the estimated parameters of the system online. (i.e. the above );

[0199] Design an adaptive update law based on time-varying update rate. First, define the state error The cost function form is as follows:

[0200]

[0201] J is obtained by the following formula x Relative to The gradient is:

[0202]

[0203] Then an adaptive update law is used to minimize the cost function, introducing the update rate λ(k)>0 as follows:

[0204]

[0205] λ(k) can be obtained by the following cost function:

[0206]

[0207] For the objective function J x Taking the derivative with respect to λ(k) and setting it equal to zero:

[0208]

[0209] get:

[0210]

[0211] Substituting formula (18) into (15), we can obtain:

[0212]

[0213] in At time k+1, we can calculate Get, x θ (k+1) is the system state measured at time k+1. It is used to actually calculate the prediction error state. By subtracting the estimated influence term from the actual measured system state, a more accurate prediction error can be obtained.

[0214] In the embodiment, step 4 is based on the estimated parameters of the online updating system, the uncertain parameters of the system are expressed in the form of additive disturbance, the controller is designed using the Tube-MPC concept, the ellipsoid parameterization method is used to replace the traditional polyhedron parameterization method to simplify the optimization process, and the state Tube containing all possible future trajectories is constructed, and the constraints are ensured to ensure the robustness of the system;

[0215] Parameterized control input and state Tube method, first set Use a predefined ellipsoid for parameterization in the following form:

[0216]

[0217] in represents the state set; x represents the system state vector; is a positive definite matrix, L is given by P = L T Obtained by Cholesky decomposition of L; ||·||2 represents the 2-norm (Euclidean norm);

[0218] Using Scaling Variables and shift variables The state Tube is parameterized as follows:

[0219]

[0220] where e is defined l|k =x l|k -z l|k ; It represents the scaling factor at the lth moment when predicting the time k, which is used to scale the size of the state Tube; represents the state offset vector at the lth moment when predicting time k; ⊕ represents the Minkowski sum, that is, the point-by-point addition of two sets;

[0221] The control input is parameterized as follows:

[0222]

[0223] Where K is the feedback gain of the offline design, For online optimization of variables;

[0224] The terminal set is defined using an ellipsoid as follows:

[0225]

[0226] in Represents a terminal set.

[0227] The ellipsoid parameterization method is used to simplify the optimization process, which also includes reformulating the conditions of the optimization problem based on variables. The initial condition of equation (9) is expressed as the following quadratic constraint:

[0228]

[0229] The above formula (24) obtained from formula (21);

[0230] The conditions for formula (10) to be valid are as follows:

[0231]

[0232] in For a given scalar, d l|k =Az l|k +Bv l|k -z l+1|k , formula (25) is about the variable z l|k ,α l|k ,α l+1|k ,τ 1,l|k and τ 2,l|k The linear matrix inequality (LMI) of .

[0233] The conditions for formula (11) to be valid are as follows:

[0234]

[0235] in is the maximum value vector of offline calculation, which provides the worst case of the state space of the state Tube in all directions to ensure the satisfaction of the state constraints. Each element is Among them [F+GK] i Denotes the i-th row of the matrix (F+GK), then the parameterized form of the control input and state Tube in equation (11) is obtained. Therefore, the state and input constraints can be restated as linear inequalities.

[0236] The conditions for formula (12) to be valid are as follows:

[0237]

[0238] where τ 1,T For a given scalar, Equation (27) is about the variable z N|k ,α N|k and τ 1,T Linear matrix inequalities (LMI) in .

[0239] The optimization problem is formulated offline to calculate the ellipsoid shape matrix P and feedback gain K. First, in order to ensure the recursive feasibility, the terminal set of the MPC controller is designed to remain unchanged under the terminal controller, that is, the terminal set The following conditions must be met to ensure that the system state always remains within the terminal set under the action of the terminal controller, thereby ensuring the recursive feasibility and stability of the system:

[0240]

[0241] where x + represents the system state at the next time step;

[0242] In the framework of the ellipsoid Tube, it can be written as:

[0243]

[0244] If there is a positive scalar So that the following is true:

[0245]

[0246] Where Y = KP -1 , τ 1,O +τ 2,O ≤1, then the terminal set Satisfies condition (29).

[0247] In addition, the terminal set is designed to satisfy the state and input constraints under the terminal control law, which is in the following form:

[0248]

[0249] in Used together with Schur's complement lemma, we get the above result.

[0250] Finally, an offline optimization problem needs to be formulated to calculate the ellipsoid shape P and the feedback gain K. In order to maximize the attraction area of ​​the controller, and thus the size of the terminal set, this can be achieved by minimizing the determinant of P. Therefore, the offline optimization problem can be formulated as:

[0251]

[0252] Terminal cost matrix P C The solution conditions are as follows:

[0253]

[0254] Where Q x is the state weight matrix, Q u Input weight matrix for control.

[0255] The online optimization problem is as follows:

[0256]

[0257] in For online optimization variables.

[0258] In the embodiment, step 5 proposes an adaptive robust MPC algorithm step for processing a system with parameter uncertainty and input delay.

[0259] The specific steps of the algorithm include offline part and online part. First, the initial state of the system is given Initial estimate matrix Control and prediction time domain N p , cost matrix Q;

[0260] Then, solve the offline optimization problem, including:

[0261] Step 501: k←1;

[0262] Step 502: Setting the grid τ 1,O ∈(0,1);

[0263] Step 503: Through each τ 1,O Solve the offline optimization problem (32) to calculate P and K;

[0264] Step 504: Select P, K to minimize det(P);

[0265] Step 505: Calculate P C Satisfies (33) and minimizes trace(P C );

[0266] Solve online optimization problems, including:

[0267] Step 511: Solve the online optimization problem (34);

[0268] Step 512: Apply control input u k =K(x k -z 0|k )+v 0|k ;

[0269] Step 513: Measure the state x(k+1) of the actual uncertain system;

[0270] Step 514: Use the adaptive update law (Formula (19)) to estimate the system renew;

[0271] It is understandable that based on the above definition at the same time Include These two items, because is the estimated form of A(k+1), so the adaptive update law updates Further update

[0272] Step 515: Setup

[0273] Step 516: k←k+1.

[0274] In the embodiment, in order to make the technical solution and advantages of the present invention more clear, the technical solution in the embodiment of the present invention is clearly and completely described below in conjunction with a calculation example.

[0275] Consider a constrained system with input delays and uncertain parameters, whose dynamics is modeled as:

[0276] x θ (k+1)=A θ x θ (k)+B θ u(kd) (35)

[0277] Where d = 1, 2, 3,

[0278] In formula (35), A θ and B θ are unknown, their nominal parameters are known and used as initial values ​​for the estimation matrix as follows:

[0279]

[0280] System (35) contains input delay, and the extended state method is used to transform the system to eliminate the delay. In this example, the initial state of the system is set to x θ (0) = [0.8, 1], the prediction time domain and control time domain are N p =10, the sampling interval is 0.1, and the weight matrix Q x =diag[0.1,0.1],Q u =diag[0.1,0.1], input constraint ||u||≤2, state constraint ||x||≤2, the above parameters need to be expanded according to different delay step sizes;

[0281] Figure 2 and Figure 3 The performance of the system with uncertain parameters under different input delays is compared, so it can be concluded that the present invention is effective.

[0282] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0283] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. An adaptive robust MPC method for processing systems with parameter uncertainty and input delay, characterized in that: The following steps are involved: Step 1: Establish a discrete linear system with parameter uncertainty and input delay; Step 2: Using the state variable expansion method, the discrete linear system is converted into an augmented system and an estimation system and an error transfer system of the augmented system are established, and an optimization control problem is established based on the estimation system; Step 3: Based on the estimation system and the error transfer system, an adaptive update law based on a time-varying update rate is established and the estimation parameters of the system are updated online to ensure that the error of the estimation parameters is bounded; Step 4: Based on the estimated parameters of the system updated online, for the infinite time domain optimization control problem, the ellipsoid parameterization method is used to simplify the optimization process, and the controller is designed using the Tube-MPC concept to obtain the offline optimization problem and the online optimization problem; Step 5: using an adaptive robust MPC algorithm to solve the offline optimization problem and the online optimization problem to obtain an optimization result; The ellipsoid parameterization method described in step 4 simplifies the optimization process, including: Status Parameterized using a predefined ellipsoid in the following form: in represents the state set, and x represents the system state vector; is a positive definite matrix, L is given by P = L T Obtained by Cholesky decomposition of L; ‖·‖2 represents the 2-norm; Using Scaling Variables and shift variables Parameterize the state Tube as follows: where e l|k =x l|k -z l|k ; It represents the scaling factor at the lth moment when predicting the time k, which is used to scale the size of the state Tube; represents the state offset vector at the lth moment when predicting time k; represents the Minkowski sum; The control input is parameterized as follows: Where K is the feedback gain of the offline design, For online optimization of variables; The terminal set is defined using an ellipsoid as follows: in Represents a terminal set; The ellipsoid parameterization method is used to simplify the optimization process in step 4, and further includes: Re-stating the optimization problem based on variables, the initial condition of equation (9) is expressed as the following quadratic constraint: The conditions for formula (10) to be valid are as follows: in For a given scalar, d l|k =Az l|k +Bv l|k -z l+1|k , formula (25) is about the variable z l|k ,α l|k ,α l+1|k ,τ 1,l|k and τ 2,l|k Linear matrix inequality LMI; The conditions for formula (11) to be valid are as follows: in is the maximum value vector of offline calculation, which provides the worst case of the state space of the state Tube in all directions to ensure the satisfaction of the state constraints. Each element is Among them [F+GK] i Denotes the i-th row of the matrix (F+GK), then the parameterized form of the control input and state Tube of equation (11) is obtained, and the state and input constraints can be restated as linear inequalities; The conditions for formula (12) to be valid are as follows: where τ 1,T For a given scalar, Equation (27) is about the variable z N|k ,α N|k and τ 1,T Linear matrix inequality LMI in; Step 4 describes the design of the controller using the Tube-MPC concept, and ultimately obtains the offline optimization problem, including: The optimization problem is formulated offline to calculate the ellipsoid shape matrix P and feedback gain K. First, in order to ensure the recursive feasibility, the terminal set of the MPC controller is designed to remain unchanged under the terminal controller, that is, the terminal set The following conditions must be met: where x + represents the system state at the next time step; Written in the framework of the ellipsoid Tube: If there is a positive scalar So that the following is true: Where Y = KP -1 , τ 1,O +τ 2,O ≤1, then the terminal set Satisfy condition (29); In addition, the terminal set is designed to satisfy the state and input constraints under the terminal control law, which is in the following form: in Used together with Schur's complement lemma, we get the above result; Finally, an offline optimization problem needs to be formulated to calculate the ellipsoid shape P and the feedback gain K in order to maximize the controller’s attraction area and thus the size of the terminal set, which is achieved by minimizing the determinant of P. Therefore, the offline optimization problem is expressed as: Terminal cost matrix P C The solution conditions are as follows: Where Q x is the state weight matrix, Q u Input weight matrix for control; The online optimization problem obtained in step 4 is as follows: in For online optimization variables.

2. The adaptive robust MPC method for processing systems with parameter uncertainty and input delay according to claim 1, characterized in that: The step 1 establishes the following discrete linear system with parameter uncertainty and input delay: x θ (k+1)=A θ x θ (k)+B θ u(k-d) (1) where x θ (k+1) is the system state measured at time k+1; and are the state variables and input variables of the system, is a known time delay, and is the system matrix with uncertainty; The state and input of the system are in a set containing the origin, defined as: C:={(x θ ,u)|Fx θ +Gu≤1} (2) in 3. The adaptive robust MPC method for processing systems with parameter uncertainty and input delay according to claim 1, characterized in that: The step 2 of converting the discrete linear system into an augmented system and establishing an estimation system of the augmented system includes: Step 201, using the state variable expansion method, the original system is converted into the following augmented system: x(k+1)=Ax(k)+Bu(k) (3) Where x(k+1) represents the state vector after expansion at time k+1; represents the state vector after expansion at time k; matrix and Represents the expanded system matrix in the following form: x(k)=[u(k-1) T ,…,u(k-d) T ,x θ (k) T ] T Step 202, establishing an estimation system of the augmented system: in and are the estimated values ​​of A and state x(k) respectively; Step 203, subtracting (4) from (3) to obtain the following error transfer system: in is the state error; is the estimated error of matrix A; Step 204, define the current state vector δ(k) = [u(kd) T x θ (k) T ] T , referring to formula (5), the state estimation error is defined as follows: in is the state estimation error; is the parameter estimation error, represents the domain of real numbers, n represents the dimension of the system's state variables, and m represents the dimension of the system's input variables; a and They are: a=[B θ A θ ] Where a is the real parameter vector of the system; is the estimate of the system state matrix at time k; is the estimate of the system input matrix at time k.

4. The adaptive robust MPC method for processing systems with parameter uncertainty and input delay according to claim 3, characterized in that: The step 2 of establishing an optimization control problem for infinite time domain based on the estimation system includes: The parameter uncertainty is expressed in the form of additive disturbance, and the estimation system shown in formula (4) is written as follows: in represents parameter uncertainty in the form of perturbations, satisfying the set w satisfies the quadratic constraint w T P w w≤1; is a positive definite matrix, d is the input delay; D=I md+n is the perturbation matrix; Construct a set of collection sequences using the Tube-MPC algorithm It is called the state Tube, which contains the input sequence All the trajectories of the system that can be generated, the infinite time domain optimization control problem is as follows: Where l represents the time step index in the prediction time domain, and N represents the length of the prediction time domain; J(·,·) and J T (·) represent the stage and terminal cost functions defined based on the state Tube, respectively, and the form is as follows: in and is a positive definite matrix, is the positive definite terminal cost matrix calculated offline, 5. The adaptive robust MPC method for processing systems with parameter uncertainty and input delay according to claim 1, characterized in that: Step 3: Based on the estimation system and error transfer system, establish an adaptive update law based on the time-varying update rate and update the estimated parameters of the system online. include: First, define the state error The cost function form is as follows: J is obtained by the following formula x Relative to The gradient is: Then an adaptive update law is used to minimize the cost function, introducing the update rate λ(k)>0 as follows: λ(k) is obtained by the following cost function: To J x Taking the derivative with respect to λ(k) and setting it equal to zero: get: Substituting formula (18) into (15), we get: in At time k+1, by calculating Get, x θ (k+1) is the system state measured at time k+1.

6. The adaptive robust MPC method for processing systems with parameter uncertainty and input delay according to claim 1, characterized in that: The adaptive robust MPC algorithm of step 5 is specifically as follows: First, the initial state of the system is given Initial estimate matrix Control and prediction time domain N p , cost matrix Q; Then, solve the offline optimization problem, including: Step 501: k←1; Step 502: Setting the grid τ 1,O ∈(0,1); Step 503: Through each τ 1,O Solve the offline optimization problem (32) to calculate P and K; Step 504: Select P, K to minimize det(P); Step 505: Calculate P C Satisfy equation (33) and minimize trace(P C ); Solve online optimization problems, including: Step 511: Solve the online optimization problem (34); Step 512: Apply control input u k =K(x k -z 0|k )+v 0|k ; Step 513: Measure the state x(k+1) of the actual uncertain system; Step 514: Use the adaptive update law to estimate the system renew; Step 515: Setup Step 516: k←k+1.

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