A fixed-time sliding mode control method for aero turbofan engine based on polytopic LPV system

Abstract

The application belongs to the technical field of aero-engine control, and proposes a kind of fixed time sliding mode control method of aero turbofan engine based on polytopic LPV system.Through input-output data, small deviation system modeling is carried out on turbofan engine, on the basis of considering input uncertainty, interpolation fitting is carried out, the LPV system model of turbofan engine system is obtained, sliding surface is designed and equivalent control law is calculated, stability analysis is carried out, control law is designed, reachability analysis of sliding surface is carried out, and maximum reaching time is solved.The application establishes polytopic LPV system model on turbofan engine, at polytopic vertex, infinite-dimensional inequality constraint is converted into a finite number of linear matrix inequality, and the stability condition of aero-engine control process is obtained.In addition, the method ensures the reachability of sliding surface, and gives the upper bound of reaching time, and then effectively enhances the robustness of engine control system.

Description

Technical Field

[0001] This invention relates to the field of aero-engine control technology, and in particular to a fixed-time sliding mode control method for aero-engine turbofan engines based on a multi-cell LPV system. Background Technology

[0002] Turbofan engines are widely used in civil aircraft, offering advantages such as low fuel consumption and high economic efficiency. However, these engines also possess complex aerodynamic and thermodynamic models and strong nonlinear characteristics. LPV (Liquid Mode Controller) systems are widely used in aero-engine system modeling due to their wide modeling range and ability to directly reflect system nonlinearity. Among these, multi-cell LPV models exhibit convexity and vertex advantages, with a finite number of vertices effectively describing the entire time-varying system. Controllers designed based on multi-cell vertices are applicable to the entire aero-engine LPV system control, significantly reducing the number of solutions and computational load. Furthermore, due to the complexity and diversity of flight environments, engine systems require strong robustness and anti-interference capabilities to ensure the accuracy of the control system. When an aircraft is in maneuvering flight, the engine thrust changes in real time with changes in flight attitude; therefore, the engine control system needs a fast response speed to meet real-time flight requirements. Sliding mode controllers, as an advanced robust control technology, offer advantages such as fast response speed and strong anti-interference capabilities, and are widely used in the aerospace field. The design process of a sliding mode controller typically includes two parts: 1) designing a sliding surface with uncertainty suppression effects. When the system state trajectory moves on the sliding surface, the system is not affected by disturbances and uncertainties; 2) Design an effective control law. A well-designed control law can effectively guide the system state to the designed sliding surface. Therefore, in the design of a sliding controller, the accessibility of the sliding surface is the basic guarantee of the robustness of the sliding control. Most existing sliding controller designs for aero-engines neglect the accessibility of the sliding surface and the solution of the arrival time. To further ensure the accessibility of the sliding surface in the engine control process, this invention proposes a fixed-time sliding control method for aero-turbofan engines. Through fixed-time arrival theory analysis, the upper bound of the arrival time of the sliding surface is effectively calculated. Applying it to the high-pressure rotor speed control of a turbofan engine ensures the robustness of the engine system and improves the engine's rapid response capability. Summary of the Invention

[0003] To ensure the robustness of the aero-engine operation and improve its rotor speed rapid response capability, this invention proposes a fixed-time sliding mode control method for aero-turbofan engines based on a multi-cell LPV system. Based on the multi-cell LPV system model of the aero-turbofan engine, fixed-time sliding mode control is performed and applied to the high-pressure rotor speed control of the aero-turbofan engine in takeoff state.

[0004] The technical solution of this invention is as follows: A fixed-time sliding mode control method for aero-engines based on a multi-cell LPV system, the specific steps of which are as follows:

[0005] Step 1: Divide the flight envelope based on the aircraft's flight altitude and Mach number. In the divided working area, perform small deviation system modeling of the turbofan engine at multiple design points using input-output data. Considering input uncertainties, obtain the LPV system model of the turbofan engine system through interpolation fitting.

[0006]

[0007] Where, x = [ΔN1 ΔN2] T u=ΔW f ΔN1 represents the per-unit error of the engine's low-pressure rotor speed relative to the design point steady-state value; ΔN2 represents the per-unit error of the engine's high-pressure rotor speed relative to the design point steady-state value; ΔW f The per-unit error of the engine's air-fuel ratio relative to the steady-state value at the design point is represented by θ; A(θ) and B(θ) are time-varying system matrices, where θ represents the scheduling parameters, and f(x,t) is used to represent the input uncertainty of the turbofan engine system. satisfy

[0008] For the established LPV system, the polytope is designed as follows:

[0009]

[0010] Where m represents the number of vertices of the polytope, β i The parameters are convex decomposition parameters; for the i-th vertex, the parameter matrix of the LPV system is represented as M. i =(A i B i Therefore, the following multicellular LPV system is obtained.

[0011]

[0012] Step 2: Design the sliding surface and calculate the equivalent control law;

[0013] To achieve high-pressure rotor speed control of a turbofan engine, the controlled output variable is defined.

[0014] y = Cx = ΔN²,

[0015] Where C = [0 1];

[0016] The sliding mode function is defined as follows:

[0017] S(t)=ε(y-ΔN 2cmd)

[0018] Where, ΔN 2cmd It is the expected increment of the high-voltage rotor speed, ε>0, which is a given constant;

[0019] By defining the ideal sliding surface S(t) = 0, from Obtain the equivalent control law.

[0020] u eq (t)=-[CB(θ)] -1 CA(θ)x(t)-f(x,t)

[0021] Step 3: Perform stability analysis; substitute the equivalent control law into the LPV system to obtain the sliding mode system equations as follows:

[0022]

[0023] in,

[0024] For sliding mode systems, construct Lyapunov functions.

[0025] V1(x)=x T Px.

[0026] Given a parameter α > 0, if there exists a matrix P > 0 such that the following matrix inequalities hold,

[0027]

[0028] Then the sliding mode system is stable;

[0029] Based on the theory of multicell LPV systems, the above infinite-dimensional matrix inequalities are transformed into the following finite-dimensional linear matrix inequalities at the vertices of the multicell.

[0030]

[0031] in,

[0032] Step 4: Design the control law, perform reachability analysis of the sliding surface, and solve for the maximum arrival time; the designed control law is as follows:

[0033] u(t) = -[CB(θ)] -1 CA(θ)x(t)-g(θ)sign(S(t)),

[0034] Among them, g(θ)=ξ+ρ(θ)||S(t)|| p +κ(θ)||S(t)|| q 0 < p < 1, q > 1 ρ1>0, κ1>0,

[0035] Construct a Lyapunov function for the control law.

[0036]

[0037] Further differentiation yields,

[0038]

[0039] The maximum arrival time of the sliding surface is then obtained as follows:

[0040]

[0041] Finally, a turbofan engine speed simulation experiment was conducted to analyze the control performance.

[0042] The beneficial effects of this invention are as follows: This invention designs a fixed-time sliding mode controller for aero-turbofan engines. By establishing a multi-celled LPV system model of the turbofan engine, the infinite-dimensional inequality constraints are transformed into a finite number of linear matrix inequalities at the vertices of the cell, thus obtaining the stability conditions for the aero-engine control process. The reachability of the sliding surface is ensured through the design of the control law. This invention is beneficial in ensuring the robustness of the aero-turbofan engine's operation and the speed response of the rotational speed process. Attached Figure Description

[0043] Figure 1 The high-pressure rotor speed curve of a turbofan engine;

[0044] Figure 2 This is an incremental fuel flow curve;

[0045] Figure 3 This represents the trajectory curve of the sliding mode function. Detailed Implementation

[0046] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0047] Specific implementation steps:

[0048] 1) System modeling of aero-turbofan engines;

[0049] The actual operation of turbofan engine systems is complex, with severe parameter coupling, making the study of mechanistic models within the entire flight envelope extremely difficult. To better utilize engine performance and reduce the conservatism of the control process, controller design needs to be integrated with the actual model. Therefore, for modeling aero-turbofan engines, this invention considers using the input-output data of the turbofan engine's operation process and employing system identification methods to perform LPV system modeling of the aero-engine, resulting in the following model:

[0050]

[0051] Where, x = [ΔN1 ΔN2] T u=ΔW f ΔN1, ΔN2 and ΔW f These represent the per-unit errors of the engine's low-pressure rotor speed, high-pressure rotor speed, and air-fuel ratio relative to the steady-state value at the design point, respectively. A(θ) and B(θ) are time-varying system matrices, where θ represents the scheduling parameters, and f(x,t) represents the system's input uncertainty. satisfy

[0052] Next, for the established LPV system, the polytope is designed as follows:

[0053]

[0054] Where m represents the number of vertices of the polytope, β i Let M represent the convex decomposition parameters. For the i-th vertex, the parameter matrix of the LPV system is represented as M. i =(A i B i Therefore, the following multicellular LPV system is obtained.

[0055]

[0056] As shown in equation (1), this invention controls the engine rotor speed by controlling the fuel quantity, thereby controlling the engine thrust. Furthermore, this invention considers the influence of uncertainties on the control effect, making the established model more general and practical in real-world applications.

[0057] 2) Sliding surface design

[0058] The basic idea of ​​sliding mode control consists of two parts. First, a sliding surface is designed. When the system state trajectory runs on the sliding surface, the system is insensitive to external disturbances, noise, and uncertainties. Second, a control law is designed to guide the system state trajectory to the sliding surface. Therefore, to ensure system robustness, it is necessary to ensure that the system state can reach the designed sliding surface, i.e., to ensure the reachability of the sliding surface. This invention focuses on controlling the high-pressure rotor speed of a turbofan engine; therefore, the controlled output variable is defined as follows:

[0059] y = Cx = ΔN²,

[0060] Where C =

[01] .

[0061] For the LPV system, the sliding mode function is designed as follows:

[0062] S(t)=ε(y-ΔN 2cmd (3)

[0063] Where, ΔN 2cmd It is the expected incremental value of the high-voltage rotor speed, and ε>0 is a given constant.

[0064] The ideal sliding surface is defined as S(t) = 0. The control law for the state trajectory of the LPV system (1) running on the sliding surface is called the equivalent control law. To obtain the specific expression of the equivalent control law, we can let Therefore, the following equation holds true.

[0065]

[0066] From equation (4), we can obtain the specific manifestation of the equivalent control law.

[0067] u eq (t)=-[CB(θ)] -1 CA(θ)x(t)-f(x,t). (5)

[0068] 3) Stability analysis

[0069] After deriving the equivalent control law, the stability of the control system is further analyzed. First, when the state trajectory of the LPV system (1) runs on the sliding surface, the closed-loop control system at this time is called the sliding mode. From equations (1) and (5), the sliding mode equation can be obtained as follows:

[0070]

[0071] in,

[0072] Secondly, to derive the stability conditions for the system state operating on the sliding surface, this invention performs a stability analysis on the sliding mode equation (6). Firstly, based on stability analysis in modern control theory, for a sliding mode system, when there exists a Lyapunov function V1(x) and a real number α > 0, satisfying...

[0073]

[0074] Then, from equation (7), we can obtain

[0075] V1(x(t))≤exp{-α(t-t0)}V1(x(t0)). (8)

[0076] Therefore, as time t→∞, V1(x) approaches 0, thus ensuring the stability of the sliding mode (6).

[0077] Based on the proposed stability condition (7), the Lyapunov function is constructed as follows for the sliding mode system:

[0078] V1(x)=x T Px,

[0079] Where P > 0 is the positive definite symmetric matrix to be solved.

[0080] By differentiating the Lyapunov function, we can obtain

[0081]

[0082] Substituting the sliding mode equation (6) into equation (9), we can obtain

[0083]

[0084] Based on equation (10), the stability condition (7) is equivalent to

[0085]

[0086] Since inequality (11) is related to the scheduling parameter θ and has infinite dimensions, it suffers from high computational complexity and difficulty in solving. To simplify the computation, according to the theory of multi-cell LPV systems, it is only necessary to solve for the controller gain of a finite number of linear matrix inequalities at the vertices to ensure the effectiveness of the controller throughout the entire operating range. Therefore, if there exists a matrix P > 0 such that the following linear matrix inequalities hold,

[0087]

[0088] in, This ensures that stability condition (7) holds, thereby guaranteeing the stability of the sliding mode system and the stability of the engine control system.

[0089] 4) Control law design and reachability analysis

[0090] This invention not only considers the stability of the engine control process but also focuses on the robustness of the engine control system to uncertainties. Therefore, while ensuring the stability of the engine control system, this invention needs to further explore the reachability of the designed sliding surface and provide the maximum arrival time. First, to guide the state trajectory of the LPV system to the sliding surface, the control law is designed as follows:

[0091] u(t) = -[CB(θ)] -1 CA(θ)x(t)-g(θ)sign(S(t)), (13)

[0092] Among them, g(θ)=ξ+ρ(θ)||S(t)|| p+κ(θ)||S(t)|| q 0 < p < 1, q > 1 ρ1>0, κ1>0,

[0093] From the sliding mode function (3) and the control law (13), we can obtain

[0094]

[0095] Next, the accessibility of the sliding surface is analyzed using the fixed-time theory. To analyze the accessibility of the sliding surface and obtain the arrival time, the Lyapunov function for equation (14) is constructed as follows:

[0096]

[0097] Further, we can obtain

[0098]

[0099] Substituting equation (14) into equation (15), we can obtain

[0100]

[0101] As can be seen from the definition of uncertainty,

[0102] ξ||S(t)||≥S T f(e,t), (17)

[0103] From equation (17), it can be seen that the designed control law (13) can effectively eliminate the input uncertainty in the system. Combining equations (16) and (17), we can obtain...

[0104]

[0105] According to equation (18) and the fixed-time theory, the designed sliding surface is accessible, and the maximum arrival time does not exceed [a certain value].

[0106]

[0107] 5) Simulation experiment on speed control of turbofan engine

[0108] This invention targets the maximum thrust state of an aircraft during takeoff, corresponding to the altitude and Mach number: H = 0 km, Ma = 0. The high-pressure rotor speed N2 of the aero-turbofan engine is selected as the scheduling parameter, where N2 ∈ [N...]. 2min N 2max ] = [0.88 1]. Simultaneously, according to the following transformation...

[0109]

[0110] Normalizing the scheduling parameter θ to the range [-1, 1], the LPV system matrix of the turbofan engine is obtained as follows:

[0111]

[0112]

[0113] The system matrix is ​​obtained at the multicell vertices θ = -1 and θ = 1 of the aero-engine LPV system.

[0114]

[0115]

[0116] Given the parameter α = 0.1, by solving the linear matrix inequality (12) at the vertex, the positive definite symmetric matrix is ​​obtained as follows:

[0117]

[0118] Therefore, the engine control system is stable. Furthermore, the control objective of this invention is to control the high-pressure rotor speed of the turbofan engine from 0.88 to 1.0 during takeoff. The input uncertainty is set to f(x,t) = 0.1sin(ΔN1 + ΔN2), and the controller parameters are selected as p = 0.5, q = 2, ρ1 = 2, κ1 = 1.2, and ε = 0.1. Figure 1 and Figure 2 The high-pressure rotor speed curve and incremental fuel curve of the turbofan engine are shown respectively. Under the control of the designed sliding mode controller, the high-pressure rotor speed of the engine can track the target value well. The engine control system has high control accuracy and response speed, and strong robustness under the influence of input uncertainty. Figure 3 The trajectory of the sliding mode function is shown. Based on the sliding mode function curve and the maximum arrival time calculation formula (19), it can be seen that the system trajectory can reach the preset maximum arrival time. The internal structure reaches the designed sliding surface, thereby ensuring the accessibility of the sliding surface and the robustness of the engine control system.

[0119] This invention proposes a fixed-time sliding mode control method for aero-engines based on a multi-cell LPV system. Results show that the proposed controller exhibits high control accuracy and fast response speed. Furthermore, the proposed method ensures the reachability of the sliding surface and provides an upper bound on the reach time, thereby effectively enhancing the robustness of the engine control system.

Claims

1. A fixed-time sliding mode control method for aero-turbofan engines based on a multi-cell LPV system, characterized in that, The specific steps are as follows: Step 1: Divide the flight envelope based on the aircraft's flight altitude and Mach number. In the divided working area, perform small deviation system modeling of the turbofan engine at multiple design points using input-output data. Considering input uncertainties, obtain the LPV system model of the turbofan engine system through interpolation fitting. Where, x = [ΔN1 ΔN2] T u=ΔW f ΔN1 represents the per-unit error of the engine's low-pressure rotor speed relative to the design point steady-state value; ΔN2 represents the per-unit error of the engine's high-pressure rotor speed relative to the design point steady-state value; ΔW f The per-unit error of the engine's air-fuel ratio relative to the steady-state value at the design point is represented; A(θ) and B(θ) are time-varying system matrices, where θ represents the scheduling parameters, and f(x,t) is used to represent the input uncertainty of the turbofan engine system. Satisfy ||f(x,t)||≤ξ, For the established LPV system, the polytope is designed as follows: Where m represents the number of vertices of the polytope, β i The parameters are convex decomposition parameters; for the i-th vertex, the parameter matrix of the LPV system is represented as M. i =(A i B i Therefore, the following multicellular LPV system is obtained. Step 2: Design the sliding surface and calculate the equivalent control law; To achieve high-pressure rotor speed control of a turbofan engine, the controlled output variable is defined. y = Cx = ΔN², Where C = [0 1]; The sliding mode function is defined as follows: S(t)=ε(y-ΔN 2cmd ) Where, ΔN 2cmd It is the expected increment of the high-voltage rotor speed, ε>0, which is a given constant; By defining the ideal sliding surface S(t) = 0, from Obtain the equivalent control law. u eq (t)=-[CB(θ)] -1 CA(θ)x(t)-f(x,t) Step 3: Perform stability analysis; substitute the equivalent control law into the LPV system to obtain the sliding mode system equations as follows: in, For sliding mode systems, construct Lyapunov functions. V1(x)=x T Px Given a parameter α > 0, is there a matrix P > 0 such that the following matrix inequalities hold? Then the sliding mode system is stable; Based on the theory of multicell LPV systems, the above infinite-dimensional matrix inequalities are transformed into the following finite-dimensional linear matrix inequalities at the vertices of the multicell. in, Step 4: Design the control law, perform reachability analysis of the sliding surface, and solve for the maximum arrival time; the designed control law is as follows: u(t)=-[CB(θ)] -1 CA(θ)x(t)-g(θ)sign(S(t)) Where, g(θ)=ξ+ρ(θ)||S(t)|| p +κ(θ)||S(t)|| q ,0<p<1,q>1, p1>0,κ1>0, Construct a Lyapunov function for the control law. Further differentiation yields, The maximum arrival time of the sliding surface is then obtained as follows:

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