Robust control method for aircraft engine input saturation based on transfer function

Abstract

The present invention discloses a robust control method for aircraft engine input saturation based on a transformation function, comprising the following steps: step 1), establishing an uncertain nonlinear system with noise and saturated input constraints, and using a hyperbolic tangent function to approximate the saturation constraint model; step 2), calculating the bound value of uncertainty in the nonlinear system; step 3), designing a robust controller; step 4), verifying the global consistent boundedness of the system. The present invention transforms the sudden saturation constraint model into a continuously differentiable limiting model, and then designs a controller by using the Lagrange multiplier method, which effectively solves the problem of decreased control effect and even instability of the closed-loop system in the aircraft engine nonlinear system due to the simultaneous existence of uncertainties such as input saturation and unknown bounded external disturbances.

Description

Technical Field

[0001] The invention belongs to the technical field of aircraft engine control, and in particular relates to an aircraft engine input saturation robust control method based on a transformation function. Background Art

[0002] The working process of aircraft engines is subject to various constraints, including mechanical constraints such as speed, thermal constraints such as pressure and temperature, and safe and stable working process constraints such as surge margin and combustion chamber flameout limit. Aircraft engine control mainly includes steady-state control (also called design point control), transient process control and safety limit protection control, so that aircraft engines can maximize their potential as power devices under various flight conditions and working conditions, respond quickly during the transition process of acceleration and deceleration, and ensure that various safety boundaries are not violated.

[0003] Since the control system will be affected by external disturbances, the controlled object has physical limitations, or due to system safety and environmental protection considerations, the actual control system is inevitably subject to some constraints. For a complex system such as an aircraft engine with strong nonlinearity and a wide range of operating point changes, the fuel supply in the aircraft engine oil pump is limited and the change of the tail nozzle area can only be within a limited range due to physical condition limitations and energy-saving considerations. This also means that modern aircraft engine control is a typical control problem with input saturation constraints.

[0004] When input saturation occurs, if the controller is not redesigned and saturation is ignored, the input and output of the controller will not match. If the system cannot get correct feedback, chaos will occur inside the system, which will greatly reduce the system performance. For a system with strict control requirements such as an aircraft engine, it may cause system instability and even system collapse in severe cases. Therefore, the present invention proposes an aircraft engine input saturation robust control method based on a transformation function. Summary of the invention

[0005] Purpose of the invention: In order to solve the problem of weakened control effect and even instability of the closed-loop system caused by input saturation and unknown bounded external disturbances in the uncertain nonlinear system of aircraft engines, the present invention proposes an aircraft engine input saturation robust control method based on a transformation function. The saturation constraint model is continuously differentiable by introducing a limit function to approximate saturation. In view of the nonlinearity caused by approximate errors, external disturbances and saturation constraint models, a continuous robust controller is proposed based on the Lagrange multiplier method to make the uncertain system consistently bounded and consistently ultimately bounded.

[0006] Technical solution: To achieve the above purpose, the technical solution adopted by the present invention is:

[0007] A method for robust control of aircraft engine input saturation based on a transformation function comprises the following steps:

[0008] Step 1), establish an uncertain nonlinear system with noise and saturation input constraints, and use the hyperbolic tangent function to approximate the saturation constraint model;

[0009] Step 2), calculate the bounds of uncertainty in the nonlinear system;

[0010] Step 3), design a robust controller;

[0011] Step 4), verify the global consistent boundedness of the system.

[0012] Furthermore, the specific steps in step 1) are as follows:

[0013] Step 1.1), consider a class of uncertain nonlinear systems with noise and saturation input constraints

[0014]

[0015] where x(t)=[x1(t),…,x n (t)] T represents the state, u(v(t))=[u1(t),…,u m (t)] T represents the input subject to saturation limitation, v(t) = [v1(t), v2(t), …, v m (t)] T represents control, g(t)=[g1(t),…,g p (t)] T represents noise, is a known constant matrix, n, m, and p represent the dimensions of state, input, and noise, respectively;

[0016] Step 1.2), construct the state space model of the aircraft engine, and based on the component level model of the aircraft engine, obtain the coefficient matrices A, B and B0 of the system (1) respectively by the small perturbation method;

[0017] Step 1.3), for the saturated input model

[0018]

[0019] Among them, u M =[u 1M ,…,u mM ] T Represents the saturation limit of the input. Using the range characteristics of the hyperbolic tangent function, a smooth function h(v(t)) is designed to approximate the saturation function, that is:

[0020] h(v)=[h1(v),h2(v),…,h n (v)] T (3)

[0021]

[0022] Let d(v) be the error due to approximation, and

[0023] d(v):=sat(v)-h(v)=[d1(v),d2(v),…,d n (v)] T (5)

[0024] Then the saturation function can be expressed as u(v)=h(v)+d(v), and system (1) can be written as

[0025]

[0026] Furthermore, the specific steps in step 2) are as follows:

[0027] Step 2.1), let the diagonal matrix matrix and vector They are

[0028]

[0029] Then system (6) can be expressed as

[0030]

[0031] Step 2.2), let the matrix Make it meet the matching conditions The total uncertainty in the system is defined as

[0032] k(t):=L -1 d(v)+Eg(t) (9)

[0033] Then the system (8) can be written as

[0034]

[0035] Step 2.3), prove that the system uncertainty k(t) has a bound, and define its bound as ρ(t).

[0036] Further, the specific steps in step 2.3) are as follows:

[0037] Step 2.3.1), for the similarity error defined in (5), we can get

[0038] |d(v i)|=|sat(v i )-h(v i )|≤|sat(v i * )-h(v * i )|=d i (11)

[0039]

[0040] When 0≤|v i |≤u iM When |v i |Increase from 0 to v i * ,|d(v i )|Increases from 0 to d i , with |v i |From v i * Increase to u iM ,|d(v i )|From d i Reduce to |d(u iM )|; when|v i |>u iM When |d(v i )|With|v i |Increase from |d(u iM )|decrease to 0;

[0041] d max =max(d1,d2,…d m ) (13)

[0042] Then we can get that the similarity error has a maximum value, and

[0043] ||d(v)||≤d max (14)

[0044] Step 2.3.2), let ρ g (t)=||g(t)||,ρ gmax =||g(t)|| max Combining (9) and (17), we can obtain the upper bound of uncertainty k(t):

[0045] k(t)≤||L -1 ||d max +||E||ρ g (t)=ρ(t) (15)

[0046] and in is a given positive constant.

[0047] Furthermore, the specific steps in step 3) are as follows:

[0048] Step 3.1), design the continuous function γ(·): when hour, For any satisfy

[0049] γ(||v||)≤v T φ(v) (16)

[0050] Design continuous function ψ(·): when hour, For any satisfy

[0051]

[0052] Step 3.2), design the controller v(t) = p(x(t), t), where p(x, t) is a Carathedory function and p(·): For a given positive constant ε,

[0053]

[0054] in, μ(x,t):=α(x,t)ρ(t), V(x,t) is the Lyapunov equation, and V(·):

[0055] Furthermore, the specific steps in step 3.1) are as follows:

[0056] Step 3.1.1), select the state variable as the low pressure rotor speed n L and high pressure rotor speed n H , the control variable is the fuel flow rate W in the main combustion chamber f and tail nozzle throat area A8, then in system (1) m = n = p = 2,

[0057]

[0058] Combining (11) and (19) we can get

[0059] γ(||v||)≤v1 tanh(v1)+v2 tanh(v2) (20)

[0060] in Convert (20) into a conditional extreme value problem, that is,

[0061] θ(v1,v2):=v1 tanh(v1)+v2 tanh(v2) (21)

[0062] For any ξ ≥ 0, find all v1 and v2 that satisfy The minimum value of θ(v1,v2); therefore, for any v1 and v2, solve to obtain γ(ξ)=min{θ(v1,v2)|ξ=||v||};

[0063] Step 3.1.2), use the Lagrange multiplier method to solve the conditional extreme value problem in step 3.1.1), let

[0064]

[0065] where λ is the Lagrange multiplier; taking partial derivatives of equation (22) yields

[0066]

[0067] Solving equation (23), we can obtain the solution of the optimization problem: Will and Substituting into (22), we can get the minimum value of θ(v1,v2) is and then

[0068]

[0069] Step 3.1.3), combining (24) and (12) yields ψ(·), which is

[0070]

[0071] Furthermore, the specific steps in step 4) are as follows:

[0072] Step 4.1) for autonomous systems

[0073]

[0074] There exists a Lyapunov function V(x,t) and a positive constant λ1 such that

[0075]

[0076] Step 4.2), take the derivative of the Lyapunov equation of the system and get

[0077]

[0078] When ||μ||>ε, take τ=ψ(ρ)ρ / ||μ||, p(x,t)=-τα, combined with (11) we can get

[0079]

[0080] α≠0 and p(x,t)≠0; combined with (28), we can get

[0081]

[0082] When ||μ||≤ε, if α≠0 and p(x,t)≠0, combining (13) and (11) we can obtain

[0083]

[0084]

[0085] Combining (11) we can get

[0086]

[0087] If p(x,t)=0, then we can get α T φ(p(x,t))=0,

[0088]

[0089] If α=0, then we can get α T φ(p(x,t))=α T k(t)=0,

[0090]

[0091] In summary, combining (30), (33), (34) and (35), we can obtain that for any All

[0092]

[0093] Therefore, when ||x|| is large enough, For must be negative;

[0094] Step 4.3), combined with (36), it is easy to prove that it is ultimately bounded and always ultimately bounded.

[0095] The beneficial effects of the present invention are:

[0096] (1) Most existing technical methods based on function transformation have special requirements on the form of the system and are difficult to be used in aircraft engines. The robust control proposed in this paper is suitable for uncertain systems such as aircraft engines, filling the technical gap.

[0097] (2) Make full use of the filtering characteristics of the hyperbolic tangent function for noise;

[0098] (3) The proposed method solves the problems of saturation nonlinearity and noise nonlinearity in aero-engines and has universality, which helps solve other saturation nonlinear problems in engineering and society. BRIEF DESCRIPTION OF THE DRAWINGS

[0099] Figure 1 Schematic diagram of the process of the present invention.

[0100] Figure 2 It is a schematic diagram of the control system structure of the present invention.

[0101] Figure 3 Schematic diagram of the approximation between the limiting function and the saturation function.

[0102] Figure 4 It is a schematic diagram of the structure and cross-section identification of the aircraft engine in the present invention.

[0103] Figure 5 It is a schematic diagram comparing the x1 response results of the control method of the present invention and the LQR control method.

[0104] Figure 6 It is a schematic diagram comparing the response results of the control method of the present invention and the LQR control method x2.

[0105] Figure 7 It is a schematic diagram comparing the u1 response results of the control method of the present invention and the LQR control method.

[0106] Figure 8 It is a schematic diagram comparing the u2 response results of the control method of the present invention and the LQR control method. DETAILED DESCRIPTION

[0107] The present invention discloses a robust control method for aircraft engine input saturation based on a transformation function, comprising the following steps: step 1), establishing an uncertain nonlinear system with noise and saturated input constraints, and using a hyperbolic tangent function to approximate the saturation constraint model; step 2), calculating the bound value of uncertainty in the nonlinear system; step 3), designing a robust controller; step 4), verifying the global consistent boundedness of the system. The present invention transforms the sudden saturation constraint model into a continuously differentiable limiting model, and then designs a controller through the Lagrange multiplier method, effectively solving the problem of weakened control effect and even instability of the closed-loop system in the aircraft engine nonlinear system due to the simultaneous existence of uncertainties such as input saturation and unknown bounded external disturbances.

[0108] A robust control method for aircraft engine input saturation based on transformation function, such as Figure 1 The steps include:

[0109] Step 1), using the hyperbolic tangent function to approximate the saturation constraint model;

[0110] Step 1.1), consider a class of uncertain nonlinear systems with noise and saturation input constraints

[0111]

[0112] where x(t)=[x1(t),…,x n (t)] T represents the state, u(v(t))=[u1(t),…,u m (t)] T represents the input subject to saturation limitation, v(t) = [v1(t), v2(t), …, v m (t)] T represents control, g(t)=[g1(t),…,g p (t)] T represents noise, is a known constant matrix, n, m, and p represent the dimensions of state, input, and noise, respectively;

[0113] Step 1.2), construct the state space model of the aircraft engine, and based on the component level model of the aircraft engine, obtain the coefficient matrices A, B and B0 of the system (1) respectively by the small perturbation method;

[0114] Step 1.3), for the saturated input model

[0115]

[0116] Among them, u M =[u 1M ,…,u mM ] T Represents the saturation limit of the input. Using the range characteristics of the hyperbolic tangent function, a smooth function h(v(t)) is designed to approximate the saturation function, that is:

[0117] h(v)=[h1(v),h2(v),…,h n (v)] T (3)

[0118]

[0119] Let d(v) be the error due to approximation, and

[0120] d(v):=sat(v)-h(v)=[d1(v),d2(v),…,d n (v)] T (5)

[0121] Then the saturation function can be expressed as u(v)=h(v)+d(v), and system (1) can be written as

[0122]

[0123] Step 2), calculate the bounds of uncertainty in the nonlinear system;

[0124] Step 2.1), let the diagonal matrix matrix and vector They are

[0125]

[0126] Then system (6) can be expressed as

[0127]

[0128] Step 2.2), let the matrix Make it meet the matching conditions The total uncertainty in the system is defined as

[0129] k(t):=L -1 d(v)+Eg(t) (9)

[0130] Then the system (8) can be written as

[0131]

[0132] Step 2.3), prove that the system uncertainty k(t) has a bound, and define its bound as ρ(t);

[0133] Step 2.3.1), for the similarity error defined in (5), we can get

[0134] |d(v i )|=|sat(v i )-h(v i )|≤|sat(v i * )-h(v * i )|=d i (11)

[0135]

[0136] When 0≤|v i |≤u iM When |v i |Increase from 0 to v i * ,|d(v i )|Increases from 0 to d i , with |v i |From v i *Increase to u iM ,|d(v i )|From d i Reduce to |d(u iM )|; when|v i |>u iM When |d(v i )|With|v i |Increase from |d(u iM )|decrease to 0;

[0137] d max =max(d1,d2,…d m ) (13)

[0138] Then we can get that the similarity error has a maximum value, and

[0139] ||d(v)||≤d max (14)

[0140] Step 2.3.2), let ρ g (t)=||g(t)||,ρ gmax =||g(t)|| max Combining (9) and (17), we can obtain the upper bound of uncertainty k(t):

[0141] k(t)≤||L -1 ||d max +||E||ρ g (t)=ρ(t) (15)

[0142] and in is a given positive constant;

[0143] Step 3), design a robust controller;

[0144] Step 3.1), design the continuous function γ(·): when hour, For any satisfy

[0145] γ(||v||)≤v T φ(v) (16)

[0146] Design continuous function ψ(·): when hour, For any satisfy

[0147]

[0148] Step 3.1.1), select the state variable as the low pressure rotor speed n L and high pressure rotor speed n H , the control variable is the fuel flow rate W in the main combustion chamber f and the tail nozzle throat area A8, in system (1), m = n = p = 2, we have

[0149]

[0150] Combining (11) and (19) we can get

[0151] γ(||v||)≤v1 tanh(v1)+v2 tanh(v2) (19)

[0152] in Thus (19) can be transformed into a conditional extreme value problem, that is,

[0153] θ(v1,v2):=v1 tanh(v1)+v2 tanh(v2) (20)

[0154] For any ξ ≥ 0, find all v1 and v2 that satisfy The minimum value of θ(v1,v2). Therefore, for any v1 and v2, we can solve for γ(ξ)=min{θ(v1,v2)|ξ=||v||}.

[0155] Step 3.1.2), use the Lagrange multiplier method to solve the conditional extreme value problem in step 3.1.1), let

[0156]

[0157] where λ is the Lagrange multiplier; taking partial derivatives of equation (21) yields

[0158]

[0159] Solving equation (22), we can obtain the solution of the optimization problem: Will and Substituting into (21), we can get the minimum value of θ(v1,v2) is and then

[0160]

[0161] Step 3.1.3), combining (12) and (23) yields ψ(·), which is

[0162]

[0163] Step 3.2), design the controller v(t) = p(x(t), t), where p(x, t) is a Carathedory function and p(·): For a given positive constant ε,

[0164]

[0165] in, μ(x,t):=α(x,t)ρ(t), V(x,t) is the Lyapunov equation, and V(·):

[0166] Step 4), verify the global consistent boundedness of the system;

[0167] Step 4.1) for autonomous systems

[0168]

[0169] There exists a Lyapunov function V(x,t) and a positive constant λ1 such that

[0170]

[0171] Step 4.2), take the derivative of the Lyapunov equation of the system and get

[0172]

[0173] When ||μ||>ε, take τ=ψ(ρ)ρ / ||μ||, p(x,t)=-τα, combined with (11) we can get

[0174]

[0175] α≠0 and p(x,t)≠0. Combining (28) we can get

[0176]

[0177] When ||μ||≤ε, if α≠0 and p(x,t)≠0, combining (13) and (11) we can obtain

[0178]

[0179]

[0180] Combining (11) we can get

[0181]

[0182] If p(x,t)=0, then we can get α T φ(p(x,t))=0,

[0183]

[0184] If α=0, then we can get α T φ(p(x,t))=α T k(t)=0,

[0185]

[0186] In summary, combining (30), (33), (34) and (35), we can obtain that for any All

[0187]

[0188] Therefore, when ||x|| is large enough, For must be negative;

[0189] Step 4.3), combined with (36), it is easy to prove that it is ultimately bounded and always ultimately bounded.

[0190] Example

[0191] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0192] The present invention can be better understood according to the following examples. However, it is easy for those skilled in the art to understand that the specific material ratios, process conditions and results described in the examples are only used to illustrate the present invention, and should not and will not limit the present invention described in detail in the claims.

[0193] by Figure 4 As an example of a turbofan engine of this type, based on the component-level model of turbofan engine aerodynamic thermodynamics, the state variable is selected as the low-pressure rotor speed n L and high pressure rotor speed n H , the control variable is the fuel flow rate W in the main combustion chamber f and the tail nozzle throat area A8, establish its state space model at the working state point of height h = 0km and Mach number Ma = 0.3, and the coefficient matrix obtained is:

[0194]

[0195] The state space model is

[0196]

[0197] Where g(t) is the noise and ρ g (t)=||g(t)||≤0.015. Select ε=0.01, and the control limits are u 1M =1.5,u2M =1.2. Combined Figure 3 The similarity between the limiting function and the saturation function model can be obtained by max =0.2075,

[0198]

[0199] Figure 5-Figure 8 It shows that when [0.02 0.02] is applied to the system T The response curves of the state variables and the control variables after the initial disturbance are calculated, and the traditional LQR controller is selected for comparison with the controller designed by the present invention. The simulation results show that when the system is disturbed, the controller designed by the present invention can return the system to a steady state, and its control variables meet the constraint requirements (the slight oscillation in the figure is caused by noise). Compared with the traditional LQR control, both control methods can return the system to a steady state within 0.2 seconds, but in the LQR control, the minimum value of u1 is less than -1.5, which exceeds the constraint boundary. Therefore, it can be concluded that for aircraft engine nonlinear systems with input saturation uncertainty, the performance of the controller designed by the present invention is better than that of the traditional LQR controller.

Claims

1. An aeroengine input saturation robust control method based on a transformation function, characterized in that: The following steps are involved: Step 1), establish an uncertain nonlinear system with noise and saturation input constraints, and use the hyperbolic tangent function to approximate the saturation constraint model; Step 2), calculate the bounds of uncertainty in the nonlinear system; Step 3), design a robust controller; Step 4), verify the global consistent boundedness of the system; The specific steps of step 1) are as follows: Step 1.1), consider a class of uncertain nonlinear systems with noise and saturation input constraints where x(t)=[x1(t),…,x n (t)] T represents the state, u(v(t))=[u1(t),…,u m (t)] T represents the input subject to saturation limitation, v(t) = [v1(t), v2(t), …, v m (t)] T represents control, g(t)=[g1(t),…,g p (t)] T represents noise, is a known constant matrix, n, m, and p represent the dimensions of state, input, and noise, respectively; Step 1.2), construct the state space model of the aircraft engine, and based on the component level model of the aircraft engine, obtain the coefficient matrices A, B and B0 of the system (1) respectively by the small perturbation method; Step 1.3), for the saturated input model Among them, u M =[u 1M ,…,u mM ] T Represents the saturation limit of the input; using the range characteristics of the hyperbolic tangent function, a smooth function h(v(t)) is designed to approximate the saturation function, that is: h(v)=[h1(v),h2(v),…,h n (in)] T (3) Let d(v) be the error caused by the approximation, and d(v):=sat(v)-h(v)=[d1(v),d2(v),…,d n (v)] T (5) Then the saturation function can be expressed as u(v)=h(v)+d(v), and system (1) can be written as The specific steps of calculating the boundary value of uncertainty of the uncertain nonlinear system in step 2) are as follows: Step 2.1), let the diagonal matrix matrix and vector They are Then system (6) can be expressed as Step 2.2), let the matrix Make it meet the matching conditions The total uncertainty in the system is defined as k(t):=L -1 d(v)+Eg(t) (9) Then the system (8) can be written as Step 2.3), prove that the system uncertainty k(t) has a bound, and define its bound as ρ(t); The specific steps of designing a robust controller in step 3) are as follows: Step 3.1), design continuous function γ(0)=0, when hour, For any satisfy Designing continuous functions ψ(0)=0, when hour, For any satisfy Step 3.2), design the controller v(t) = p(x(t), t), where p(x, t) is a Carathedory function, and For a given positive constant ε, in, μ(x,t):=α(x,t)ρ(t), V(x,t) is the Lyapunov equation, and The specific steps of step 4) to verify the global consistent boundedness of the system are as follows: Step 4.1) for autonomous systems There exists a Lyapunov function V(x,t) and a positive constant λ1 such that Step 4.2), take the derivative of the Lyapunov equation of the system and get When ||μ||>ε, take τ=ψ(ρ)ρ / ||μ||, p(x,t)=-τα, combined with (11) we can get α≠0 and p(x,t)≠0; combined with (16), we can get When ||μ||≤ε, if α≠0 and p(x,t)≠0, combining (13) and (11) we can obtain Combining (11) we can get If p(x,t)=0, then we can get α T φ(p(x,t))=0, If α=0, then we can get α T φ(p(x,t))=α T k(t)=0, In summary, combining (18), (21), (22) and (23), we can obtain that for any All Therefore, when ||x|| is large enough, For must be negative; Step 4.3), combined with (24), it is easy to prove that it is ultimately bounded and always ultimately bounded.

2. The method for robust control of aircraft engine input saturation based on a transformation function according to claim 1, characterized in that: The specific steps of proving that the system uncertainty k(t) has a bounded value and defining its bounded value as ρ(t) in step 2.3) are as follows: Step 2.3.1), for the similarity error defined in (5), we can get |d(v i )|=|sat(v i )-h(v i )|≤|sat(v i * )-h(v * i )|=d i (25) When 0≤|v i |≤u iM When |v i |Increase from 0 to v i * ,|d(v i )|Increases from 0 to d i , with |v i |From v i * Increase to u iM ,|d(v i )|From d i Reduce to |d(u iM )|; when|v i |>u iM When |d(v i )|With|v i |Increase from |d(u iM )|decrease to 0; d max =max(d1,d2,…d m ) (27) Then we can get that the similarity error has a maximum value, and ||d(v)||≤d max (28) Step 2.3.2), let ρ g (t)=||g(t)||,ρ gmax =||g(t)|| max Combining (9) and (28), we can obtain the upper bound of uncertainty k(t): k(t)≤||L -1 ||d max +||E||r g (t)=ρ(t) (29) and in is a given positive constant.

3. The method for robust control of aircraft engine input saturation based on a transformation function according to claim 1, characterized in that: The specific steps of step 3.1) are as follows: Step 3.1.1), select the state variable as the low pressure rotor speed n L and high pressure rotor speed n H , the control variable is the fuel flow rate W in the main combustion chamber f and tail nozzle throat area A8, then in system (1) m = n = p = 2, Combining (11) and (30), we can get γ(||v||)≤v1tanh(v1)+v2tanh(v2) (31) in Convert (31) into a conditional extreme value problem, that is, θ(v1,v2):=v1tanh(v1)+v2tanh(v2) (32) For any ξ ≥ 0, find all v1 and v2 that satisfy The minimum value of θ(v1,v2); therefore, for any v1 and v2, solve to obtain γ(ξ)=min{θ(v1,v2)ξ=||v||}; Step 3.1.2), use the Lagrange multiplier method to solve the conditional extreme value problem in step 3.1.1), let where λ is the Lagrange multiplier; taking partial derivatives of equation (33) yields Solving equation (23), we can obtain the solution of the optimization problem: Will and Substituting into (33), we can get the minimum value of θ(v1,v2) is and then Step 3.1.3), combining (35) and (12) yields ψ(·), which is

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