Data-driven model predictive control design and stability analysis method thereof

Abstract

The invention belongs to the field of control algorithm design, and discloses a data-driven model predictive control design and a stability analysis method thereof. For a linear time-invariant system, a prediction model is constructed by using input and output data in a data driving mode, and system parameter identification is not needed. Wherein iterative updating of model parameters is realized in combination with a recursive least square technology, and adaptive control of a complex nonlinear system is realized in combination with subspace identification and model prediction control. Then, by constructing a Lyapunov function, the closed-loop exponential stability of the data-driven model prediction control method is theoretically proved. Compared with standard subspace prediction control, the method has smaller tracking errors and overshoot, and stability and accuracy are remarkably improved. Therefore, the prediction model can be updated online without a prior model, and the adaptability is high; stability is strictly proved, and the method is suitable for systems with high safety requirements. And the aero-engine verifies that the method shows excellent engineering practicability.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of control algorithm design, and relates to a data-driven model predictive control design and a stability analysis method thereof. BACKGROUND

[0002] Model predictive control (MPC) algorithm can include multiple control objectives and multiple constraint conditions, and can be used to predict future behavior and develop optimal control strategies, making it very suitable for solving control challenges in complex systems. Another key feature of MPC is its ability to manage various constraints in complex multi-input multi-output (MIMO) systems, thereby ensuring optimal control performance. These capabilities have led to the widespread adoption of MPC in various modern industrial sectors.

[0003] However, system complexity poses a major challenge to model-based control strategies, with obtaining an accurate system model being the most critical but time-consuming step. A variety of identification methods have been developed, including first-principle-based methods and data-driven methods. Compared with first-principle-based controllers, data-driven methods have the advantage of not requiring system identification, but directly utilizing measured input-output data to design controllers. Among them, subspace model predictive control (SPC) integrates subspace identification and model predictive control, and has a smaller amount of calculation. However, standard SPC is designed specifically for linear systems, which makes it less suitable for nonlinear systems, especially complex nonlinear models such as aircraft engines. This deficiency stems from the fact that the dynamic characteristics of aircraft engines change with operating conditions. Therefore, the predictive model needs to be iteratively updated to enhance model adaptability and minimize tracking errors. Therefore, the present application proposes a DD-RSPC (Data-Driven Model Predictive Control based on Recursive Subspace Least Squares Estimation) algorithm to more accurately describe the control dynamics of the engine.

[0004] DD-RSPC is a method that can iteratively update the predictive model, thereby enhancing the adaptability of the model and minimizing tracking errors. This method was proposed by Peter Verheijey, combining recursive least squares with data-driven model predictive control. In existing research, it is found that compared with traditional SPC, this method is suitable for nonlinear systems and meets more complex control systems. It has advantages such as adaptability, stability and accuracy.

[0005] The research field of DD-RSPC is gradually expanding, for example, the method can be applied to high-speed motor integral sliding mode control, large-aperture telescope driving control, vehicle trajectory tracking, etc. It can be seen that it has certain universality. The previous aero-engine control is generally based on model-based control. Therefore, it is of great significance to extend DD-RSPC to the field of aero-engine control. It is well known that stability is crucial for a controller, and in the field of actual control, a stable aero-engine control system can ensure flight safety and improve fuel efficiency. Therefore, the stability analysis of the DD-RSPC method in the aero-engine has a certain degree of safety guarantee. SUMMARY

[0006] In view of the shortcomings of the existing model predictive control method in the application of complex nonlinear systems, the present application provides a data-driven model predictive control design and its stability analysis method, which is suitable for the field of control system design and application, and helps to improve its performance, mainly solving the problems of model dependence, large amount of calculation and linear system in the model predictive control method.

[0007] The technical scheme of the present application is as follows:

[0008] A design and control method of an aero-engine control system, comprising the following steps:

[0009] Step A: aero-engine control system modeling and aero-engine data-driven model predictive controller construction;

[0010] 1) aero-engine control system structure and control target definition;

[0011] 2) linear state space model definition for predicting aero-engine dynamics;

[0012] 3) aero-engine data-driven model identification method;

[0013] 4) aero-engine data-driven model predictive controller construction;

[0014] 5) model recursive update method;

[0015] 6) optimal control input solving problem;

[0016] Step B: stability guarantee mechanism design;

[0017] Step C: aero-engine system operation and control execution mechanism.

[0018] The beneficial effects of the present application are as follows:

[0019] (1) DD-RSPC algorithm uses the input-output (I / O) data measured online by each sampling instance to update the prediction model. It is continuously updated to match the dynamics of the real system, avoiding the mismatch problem.

[0020] (2) By constructing Lyapunov function, the stability theorem of DD-RSPC is derived, solving the control safety problem.

[0021] (3) Tests are carried out on aero-engine to prove the adaptability, stability and superiority of the method compared with standard SPC. BRIEF DESCRIPTION OF DRAWINGS

[0022] Figure 1 is the framework diagram of direct performance adaptive prediction control of aero-engine.

[0023] Figure 2 is the design principle diagram of DD-RSPC.

[0024] Figure 3 is the closed-loop response diagram of comparing DD-RSPC and SPC to π T .

[0025] Figure 4 is the closed-loop response diagram of comparing DD-RSPC and SPC to N2.

[0026] Figure 5 is the error diagram between predicted value and actual value by using DD-RSPC prediction method and SPC prediction method respectively. DETAILED DESCRIPTION

[0027] The specific embodiments of the present application are further illustrated below in combination with the drawings and technical solutions.

[0028] The improved control algorithm structure diagram of the present application is shown in Figure 2 The DD-RSPC controller mainly includes recursive least squares and data driving to achieve the effect of model predictive control, that is, adaptability, stability and effectiveness can be realized for very complex nonlinear model.

[0029] The specific composition of each part of the controller is as follows:

[0030] (1) Recursive least squares are used to extract the dynamic model of the system from the input-output data. It uses the recursive structure of input-output data and updates the model parameters through iteration to adapt to the dynamic changes of the system. It is suitable for processing time series data and has good real-time performance and computational efficiency. It effectively captures the dynamic characteristics of the system and still maintains good modeling accuracy when the data volume is large.

[0031] (2) Data-driven, can use a large number of actual data to build system model, avoid the dependence on prior knowledge and assumptions of the system. With strong flexibility and adaptability, can cope with system complexity and nonlinear characteristics.

[0032] (3) Model predictive control, can effectively deal with multivariable system and multi-objective control problem, has good stability and robustness. It can maximize system performance through optimization control input, and can control under the condition of considering constraints.

[0033] The basic standard for measuring the control algorithm is its accuracy, stability and adaptability, the present application meets the above standards while also has practicality, the data-driven recursive subspace model predictive control method based on the present application mainly has the following advantages:

[0034] (1) Good accuracy. It can be known from Figure 5 that the prediction error obtained by the control algorithm described in the present application is smaller than that of the standard SPC algorithm, which shows that the algorithm has good accuracy.

[0035] (2) Good stability. It can be known from Figure 3 that under the same conditions, the DD-RSPC is compared with the standard SPC, the DD-RSPC algorithm proposed in the present application has good trajectory tracking effect, which reflects excellent stability, and the stability is also proved by the constructed Lyapunov function.

[0036] (3) Strong adaptability. It can be known from Figure 3 that compared with the standard SPC, the DD-RSPC has better trajectory tracking effect on WF and A8, that is, in the complex turbofan engine system, trajectory tracking can also be well realized, and good control effect is obtained.

[0037] The data-driven recursive subspace model predictive control method proposed in the present application is as follows:

[0038] Step A: modeling of the aero-engine control system and construction of the aero-engine data-driven model predictive controller;

[0039] 1) Definition of the aero-engine control system structure and control target:

[0040] The aero-engine control system takes fuel injection flow W f and nozzle area A8 as control input variables, takes high-pressure rotor speed N2 and total pressure ratio π t of the aero-engine, that is, the ratio of the total pressure at the outlet of the compressor to the total pressure at the inlet, as the output vector, and realizes closed-loop control through the aero-engine data-driven model predictive controller;

[0041] 2) Linear state-space model definition for predicting aero-engine dynamics:

[0042] A simplified dynamic behavior of an aero-engine is represented by a discrete-time linear time-invariant system:

[0043] x k+1 = Ax k + Bu k

[0044] y k = Cx k (1)

[0045] where k denotes the discrete time instant of the current control period; x k ∈ R n represents the aero-engine control system state vector, which represents the aero-engine internal dynamic variables that cannot be directly measured, and the state vector dimension n = 4; u k ∈ R m represents the control input variable, including fuel injection flow W f and nozzle area A8, and the control input dimension m = 2; y k ∈ R l represents the output vector, including high-pressure rotor speed N2 and total pressure ratio π t , and the output vector dimension l = 2; A ∈ R n×n , B ∈ R n×m , C ∈ R l×n are the state transition matrix, control input matrix, and output mapping matrix, respectively;

[0046] 3) Aero-engine data-driven model identification method:

[0047] Using the recursive subspace identification method and Willems' basic lemma, the behavior matrix of the aero-engine is constructed through historical input and output data without the need for aero-engine physical structure modeling, and the Hankel matrix of input and output data is constructed as follows:

[0048]

[0049] where u d represents the input data sequence of the aero-engine control system; N p represents the historical length of the aero-engine control system, used to extract the past aero-engine response behavior; N f represents the future prediction step of the aero-engine control system, corresponding to the number of steps in the prediction control time domain;

[0050] 4) Construction of aero-engine data-driven model predictive controller:

[0051] The historical feature vector sp future input, i.e. the future time instant engine input sequence u f , the engine subspace prediction model is obtained as:

[0052]

[0053] where, represents the future N f step prediction of the high pressure rotor speed and total pressure ratio; s p represents the historical feature vector constructed from past input-output data; u f ∈R 2Nf represents the future N f step fuel flow and nozzle opening input sequence; L s , L u represents the mapping matrix obtained from the subspace method, representing the influence of the historical vector s p and the future input u f on the prediction result;

[0054] 5) Model recursive updating method:

[0055] To adapt to the dynamic changes of the engine in real time, the recursive least square algorithm is used to update the engine dynamic model parameters:

[0056]

[0057] where, represents the estimated engine dynamic model parameter vector at the kth time instant, used to describe how the control input variables affect the high pressure rotor speed N2and total pressure ratio π t ; represents the engine dynamic model parameter estimate at the previous time instant; y k represents the output vector of the engine control system at the current time instant N2and π t ; Φ k represents the regression vector, composed of the current control input variable u k = [W f,k , A 8,k ] T and the state estimate of the previous data at the current time instant, used to calculate the next output of the engine data-driven model; K k represents the gain matrix; P k represents the covariance matrix; P k-1 represents the covariance matrix at the previous time instant, recording the confidence of the previous model estimate; I represents the identity matrix, with the same dimension as P k ; λ represents the forgetting factor, 0 < λ ≤ 1, used to balance the influence of historical data and new data;

[0058] 6) Optimal control input solving problem:

[0059] The quadratic optimal problem of the aero-engine is established, and the following is solved in each control period:

[0060]

[0061] Where i represents the current time, y ref is the target value of the aero-engine, Q and R are weighting matrices, u and y are constraint sets of control input and output respectively;

[0062] Step B: Stability guarantee mechanism design;

[0063] To ensure the closed-loop stability of the aero-engine control system under the model predictive control framework, the following design method is proposed:

[0064] 1) Terminal equality constraint setting: Introduce terminal state constraint in the aero-engine predictive control optimization problem, that is, let the state at the end of the prediction time domain satisfy x k+Nf = x s , where x s represents the stable operating state of the engine corresponding to the target equilibrium point;

[0065] 2) Lyapunov function construction: To verify the stability of the closed-loop system of the aero-engine, define the following quadratic candidate Lyapunov function:

[0066] V(x k ) = (x k -x s ) T P(x k -x s ) (6)

[0067] Where P is a symmetric positive definite matrix;

[0068] 3) Stability criterion:

[0069] If there exists α>0, such that:

[0070] V(x k+1 )-V(x k )≤-α||x k -x s || 2 (7)

[0071] It is shown that the closed-loop system of the aero-engine has exponential stability under the action of the aero-engine data-driven controller, and α is a constant;

[0072] 4) Performance function boundedness analysis:

[0073] Simultaneously guarantee the optimal value of the aero-engine rolling optimization objective function There is a quadratic upper bound as follows:

[0074]

[0075] Where: β, γ are constants related to aero-engine performance and constraints, further illustrating the robustness and convergence of the aero-engine control system;

[0076] Step C: Aero-engine system operation and control execution mechanism;

[0077] 1) Closed-loop control process: The aero-engine control system periodically collects the current time fuel flow W f , nozzle area A8, high-pressure rotor speed N2 and total pressure ratio π t , predicts the future output trajectory through recursive updating of the subspace model, and solves the optimal control input sequence;

[0078] 2) Control strategy execution: Only the first step control input u k * of the current period is executed in the prediction time domain, and the rest is recalculated in the next period, so as to realize the aero-engine rolling optimization control;

[0079] 3) Constraint processing: All optimization solving is carried out under the condition of meeting the engine structure allowed range, ensuring that the aero-engine controller will not output unimplementable instructions.

Claims

1. A design and control method for an aircraft engine control system, comprising the following steps: Step A: Modeling of aircraft engine control systems and construction of aircraft engine data-driven model predictive controllers; 1) Aeroengine control system structure and control target definition: The control system of aircraft engine is based on the fuel injection flow rate W f With the tail nozzle area A8 as the control input variable, the high pressure rotor speed N2 and the total pressure ratio π of the aircraft engine t , that is, the ratio of the total pressure at the compressor outlet to the total pressure at the inlet is used as the output vector, and closed-loop control is achieved through the aircraft engine data-driven model predictive controller; 2) Definition of the linear state-space model for predicting aero-engine dynamics: The simplified dynamic behavior of an aerospace engine is represented by a discrete-time linear time-invariant system: x k+1 =Ax k +Bu k y k =Cx k (1) Where k represents the discrete moment of the current control cycle; x k ∈R n represents the state vector of the aircraft engine control system, which represents the dynamic variables inside the aircraft engine that cannot be directly measured. The state vector dimension is n = 4; u k ∈R m Represents the control input variable, including the fuel injection flow W f With the tail nozzle area A8, the control input dimension m=2; y k ∈R l Represents the output vector, including the high-pressure rotor speed N2 and the total pressure ratio π t , output vector dimension l = 2; A∈R n×n , B∈R n×m , C∈R l×n They are state transfer matrix, control input matrix, and output mapping matrix respectively; 3) Aero-engine data-driven model identification method: Using the recursive subspace identification method and Willems' fundamental lemma, without the need for physical structure modeling of the aircraft engine, the aircraft engine behavior matrix is ​​constructed using historical input and output data. The Hankel matrix of the input and output data is constructed as follows: Among them, u d Represents the input data sequence of the aircraft engine control system; N p Represents the history length of the aircraft engine control system, which is used to extract the past aircraft engine response behavior; N f represents the future prediction step size of the aircraft engine control system, corresponding to the number of steps in the predictive control time domain; 4) Construction of a data-driven model predictive controller for aircraft engines: Construct the historical feature vector s through the subspace method p and future input, that is, the future aircraft engine input sequence u f , we get the aircraft engine subspace prediction model: in, Indicates the future N f The high-pressure rotor speed and total pressure ratio predicted by the first step; p Represents the historical feature vector constructed by past input and output data; u f ∈R 2Nf Indicates the future N f The fuel flow rate and tail nozzle opening input sequence of the step; L s , L u Represents the mapping matrix obtained by the subspace method, representing the history vector s p and future input u f Impact on forecast results; 5) Model recursive update method: In order to adapt to the dynamic changes of the aircraft engine in real time, the recursive least squares algorithm is used to update the aircraft engine dynamic model parameters: in, represents the aircraft engine dynamic model parameter vector estimated at the kth moment, which is used to describe how the control input variables affect the high-pressure rotor speed N2 and the total pressure ratio π t ; represents the estimated value of the aircraft engine dynamic model parameters at the previous moment; k Represents the output vector N2 and π of the aircraft engine control system at the current moment t Φ k Represents the regression vector, which is controlled by the current input variable u k =[W f,k ,A 8,k ] T and the state estimation value of the previous data at the current moment, which is used to calculate the output of the aircraft engine data-driven model at the next moment; K k represents the gain matrix; P k represents the covariance matrix; P k-1 Represents the covariance matrix of the previous moment, recording the confidence of the previous model estimate; I represents the identity matrix, with the same dimension as P k Same; λ represents the forgetting factor, 0<λ≤1, which is used to weigh the impact of historical data and new data; 6) Solving the problem of optimal control input: Establish a quadratic optimization problem for an aircraft engine and solve it in each control cycle: Among them, i represents the current moment, y ref is the target setting value of the aircraft engine, Q and R are weighted matrices, are the sets of constraints that control input and output respectively; Step B: Design of stability guarantee mechanism; To ensure the closed-loop stability of the aircraft engine control system under the model predictive control framework, the following design method is proposed: 1) Terminal equality constraint setting: Introduce terminal state constraints in the aircraft engine predictive control optimization problem, that is, make the state at the end of the prediction time domain satisfy x k+Nf =x s , where x s Indicates the stable operating state of the engine corresponding to the target balance point; 2) Lyapunov function construction: To verify the stability of the closed-loop system of an aero-engine, the following quadratic candidate Lyapunov function is defined: V(x k )=(x k -x s ) T P(x k -x s )(6) Where P is a symmetric positive definite matrix; 3) Stability criteria: If there exists α>0 such that: V(x k+1 )-V(x k )≤-α||x k -x s || 2 (7) It is shown that the closed-loop system of the aero-engine has exponential stability under the action of this aero-engine data-driven controller, and α is a constant; 4) Analysis of the boundedness of performance function: At the same time, the optimal value of the objective function of the aircraft engine rolling optimization is guaranteed There exists the following quadratic upper bound: Where: β and γ are constants related to the performance and constraints of the aircraft engine, which further illustrate the robustness and convergence of the aircraft engine control system; Step C: Aero-engine system operation and control execution mechanism; 1) Closed-loop control process: The aircraft engine control system periodically collects the current fuel flow rate W f , tail nozzle area A8, high pressure rotor speed N2 and total pressure ratio π t , predict the future output trajectory through the recursively updated subspace model and solve the optimal control input sequence; 2) Control strategy execution: only the first step of the control input of the current cycle, i.e. u, is executed in the prediction time domain. k * , the rest is recalculated in the next cycle, thus realizing the rolling optimization control of the aircraft engine; 3) Constraint processing: All optimization solutions are performed under the conditions that meet the allowable range of the engine structure to ensure that the aircraft engine controller does not output unfeasible instructions.

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