Finite-Time Fault-Tolerant Control Method for Fixed-Wing UAV Formations Based on Neural Networks

Abstract

The present invention discloses a finite-time formation fault-tolerant control method for fixed-wing unmanned aerial vehicles based on neural networks: First, consider using a virtual leader-follower structure for formation control, and adopt a distributed formation method for information interaction. Based on this, design a corresponding communication topology graph and calculate the corresponding adjacency matrix and Laplacian matrix; Then, for the longitudinal model of the fixed-wing unmanned aerial vehicle, simplify the dynamic model to a strict-feedback form based on reasonable assumptions, and consider the failure fault of elevator deflection; Based on the backstepping method and combined with sliding mode control, use a radial basis function neural network to process the unknown nonlinear functions of the model, and adopt an adaptive estimation of the failure fault of the elevator, so as to achieve finite-time fault-tolerant control. The present invention overcomes the problem that it is difficult to perform formation fault-tolerant control due to the longitudinal model of the fixed-wing unmanned aerial vehicle being a high-order nonlinear system, making the system have strong robustness and being able to stabilize within a finite time.

Description

Technical Field

[0001] The present invention relates to a finite-time formation fault-tolerant control method for fixed-wing unmanned aerial vehicles based on neural networks, belonging to the technical field of multi-agent systems based on intelligent control methods. Background Art

[0002] In recent years, unmanned aerial vehicles (UAVs) have been widely used due to their unique advantages such as low cost, high mobility, light weight, and autonomous flight function. Fixed-wing UAVs have advantages such as faster speed, larger payload, higher flight altitude, longer endurance, and longer range compared to rotary-wing UAVs. The formation system composed of multiple UAVs is more superior than a single UAV not only in terms of endurance, coverage, and task execution efficiency, but also can complete more and more complex tasks. Therefore, the research on fixed-wing UAV formation systems has broad application prospects.

[0003] Since fixed-wing UAVs are underactuated, strongly coupled, and non-linear high-speed moving bodies, their complex dynamic characteristics increase the difficulty of control design for the formation system. Although the present invention decouples the fixed-wing UAV model and only considers the longitudinal model, as a high-order non-linear model, the design of the controller for the fixed-wing longitudinal model is still difficult. The UAV formation system is an interconnected system, and there is information interaction between the UAVs in the formation. Therefore, adjacent UAVs will affect each other, further increasing the design difficulty of the formation control system. Currently, there are few research results on the formation control of high-order non-linear multi-agents. Therefore, the research on the formation control of the fixed-wing longitudinal model is in the forefront. Considering the actual situation, there will also be unknown non-linear terms with model uncertainties, which greatly increases the difficulty of controller design. The existing related research has low control accuracy and poor reliability.

[0004] Although the UAV formation system has stronger fault-tolerant ability and robustness than a single UAV, when some UAVs fail, through the information interaction between UAVs, it may lead to the failure of adjacent healthy UAVs' tasks, and also makes the UAV formation system lose the redundancy advantage of multiple UAVs. Through fault-tolerant control, the system performance can be maintained within an acceptable range when the system fails, which is crucial for ensuring the safety and reliability of the flight control system. In applications, the convergence time is an important evaluation criterion. The ability to control the system to be stable within a finite time has great application value in engineering practice. Therefore, the finite-time fault-tolerant control for the fixed-wing longitudinal model formation system has important practical significance. Summary of the Invention

[0005] To solve the technical problems raised in the above background art, the present invention aims to provide a finite-time formation fault-tolerant control method for fixed-wing UAVs based on neural networks, overcome the problem that it is difficult to achieve formation fault-tolerant control due to the complexity of high-order nonlinear models, and strive to achieve finite-time convergence to achieve a faster and more reliable control effect.

[0006] The present invention adopts the following technical solutions to solve the above technical problems:

[0007] A finite-time formation fault-tolerant control method for fixed-wing UAVs based on neural networks includes the following steps:

[0008] (1) Consider using a virtual leader-follower structure for formation control, and adopt a distributed formation method for information interaction. Based on this, design a corresponding communication topology graph and calculate the corresponding Laplacian matrix L;

[0009] (2) For the longitudinal model of the fixed-wing UAV, simplify the dynamic model to a strict-feedback form based on reasonable assumptions, deal with the difficult-to-handle non-linear terms and model uncertainties in the model, and consider the failure fault of elevator deflection to form a unified strict-feedback dynamic model with faults;

[0010] (3) Based on the radial basis function neural network to handle the unknown non-linear problems of the system, combine with adaptive control to achieve fault estimation; based on the backstepping method and sliding mode control, achieve distributed finite-time fault-tolerant control of the formation system.

[0011] Furthermore, in step (1), consider using a directed communication topology graph, which is represented by to represent the communication topology of the multi-agent system; where, represents the set of vertices of the communication graph, v i represents the i-th vertex, that is, the i-th agent, i = 1, 2,..., N, and N represents the number of agents; E = e ij represents the set of edges of the communication graph, e ij =(v i , v j ) means that v i can directly obtain the information of the j-th agent v j , j = 1, 2,..., N; A = [a ij ∈ R N×N is the adjacency matrix. If (v i , v j ) ∈ E, then a ij = 1, otherwise a ij = 0; define the in-degree matrix D of G = diag(d1, d2,..., d N ), define the Laplacian matrix L = [lij ∈R N×N , L = D - A; Define matrix B = diag(b1, b2, …, b N ) represents the communication topology between the leader and the followers. If the i-th follower can communicate with the leader, then b i = 1, otherwise b i = 0.

[0012] Assumption 1: There is at least one communication path from the leader to each follower.

[0013] Furthermore, the longitudinal model of the i-th follower UAV in step (2) is expressed as follows:

[0014]

[0015] where i = 1, 2, …, N, and N represents the number of agents; V i , h i , γ i , α i , q i represent the speed, altitude, flight path angle, angle of attack, and pitch rate of the i-th follower UAV, respectively; T i , D i , L i , M i represent the thrust, drag, lift, and pitch moment of the i-th follower UAV, respectively; m i , I iy represent the mass and pitch moment of inertia of the i-th follower UAV; g represents the acceleration due to gravity.

[0016] Assumption 2: Since γ i is very small during flight, it can be considered that sinγ i ≈ γ i ; Since T i sinα i << L i , T i sinα i can be neglected.

[0017] The strict feedback dynamic model with faults of the i-th follower UAV is obtained through simplification and is expressed as follows:

[0018]

[0019] where, represents the true control input quantity with multiplicative faults of the i-th follower UAV, δ ei is the control input quantity to be designed for the i-th follower UAV, 0 < i(t) < 1 is the multiplicative fault of the i-th follower UAV; f Vi = -D i / m i - gsinγ i and are the known non - linear terms in the i-th follower model, where is the static pressure, s = 1.463m 2 is the reference area, c = 0.451m is the mean aerodynamic chord length, C M0 = -0.0954, C Mα = -2.8206, C Mq = -13.8189 are the aerodynamic coefficients; f γi (V i , γ i ), f αi (V i , γ i , α i ) are the unknown non - linear terms in the i-th follower model; are all the known functions in the i-th follower model, where C Lα = 4.6333, is the aerodynamic coefficient.

[0020] Furthermore, the design method of the fault - tolerant controller combining backstepping and sliding - mode control in step (3) is as follows:

[0021] First, define the sliding surface as:

[0022]

[0023] where, e Vi = V i - V id , e hi = h i - h id , e γi = γ i - γ id , e αi = α i - α id , e qi = q i - q id are respectively the tracking errors of the speed, altitude, flight path angle, angle of attack and pitch - angle rate of the i-th follower; V id and h id are respectively the desired speed signal and desired altitude signal of the i-th follower: and V0 and h0 respectively represent the speed and altitude of the virtual leader, d i and d jrespectively represent the relative expected heights between the \(i\)-th follower and the \(j\)-th follower and the leader;

[0024] γ id ,α id ,q id are the estimated values of the virtual control inputs to be designed; the parameters \(a\) Vi ,b Vi ,c Vi ,a hi ,b hi ,c hi are positive real numbers; sgn() represents the sign function.

[0025] Design the virtual control inputs and the actual control inputs as follows:

[0026]

[0027] The adaptation laws are as follows:

[0028]

[0029] where, \(k\) mi ,p mi (m = V, h, γ, α, q, i = 1, 2, …, N) are adjustable positive real numbers; and are the first-order derivatives of \(V\) id and \(h\) id respectively; represents the adaptive weight vector of the RBF neural network in the \(i\)-th agent; represents the basis function vector of the RBF neural network in the \(i\)-th agent; represents the estimated value of the multiplicative fault \(\rho\) i in the controller of the \(i\)-th agent; \(\Gamma\) i1 ,\(\Gamma\) i2 are positive definite diagonal matrices to be designed; \(\eta\) i1 and \(\eta\) i2 are positive real numbers to be designed.

[0030] Considering that the derivatives of multiple virtual control input signals are required, design a first-order low-pass filter as follows:

[0031]

[0032] where, is a positive real number, and \(\gamma\) id ,\(\alpha\) id ,q id are the estimated values of \(\gamma\) ic ,\(\alpha\) ic ,q ic respectively.

[0033] The fault-tolerant control method designed above can make the state error of the formation system of the fixed-wing longitudinal model converge in finite time.

[0034] Furthermore, for the convenience of analysis and problem-solving, the error dynamic equation of the $i$-th follower is obtained as follows:

[0035] Define the filtering error $y$ iγ $=\gamma$ id $-\gamma$ ic , $y$ iα $=\alpha$ id $-\alpha$ ic , $y$ iq $=q$ id $-q$ ic , we get

[0036]

[0037] Among them, the characteristics of the radial basis function neural network are used to approximate the non-linear function to deal with $f$ γi $(V$ i , $\gamma$ i ) and $f$ αi $(V$ i , $\gamma$ i , $\alpha$ i ), and the specific formula is as follows:

[0038]

[0039] Among them, represents the optimal weight vector of the RBF neural network in the $i$-th agent controller, $m$ represents the number of hidden neurons of the RBF neural network; is the basis function vector, $j = 1, 2, \ldots, m$, is the input vector of the RBF neural network, and respectively represent the center point vector and width of the high-order basis function of the RBF neural network; is the approximation error of the RBF neural network in the $i$-th agent controller, and satisfies is a very small positive real number.

[0040] The finite-time stability of the formation system is verified by using the Lyapunov function method:

[0041] Lemma 1: For the system $f(0)=0$, assuming that there exists a continuously differentiable positive definite function $H(x)$, if there exist real numbers $a\gt0$, $b\gt0$, $0\lt p\lt1$ and $0\lt\varepsilon\lt+\infty$ satisfying then the system is finite-time stable.

[0042] Consider the Lyapunov function as follows:

[0043]

[0044] where \(i = 1, 2, \ldots, N\) represents the \(i\)-th follower UAV.

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051] where is a positive real number;

[0052] Taking the derivative, we can get the following form:

[0053]

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] Simplify using the following inequalities:

[0060] 1.

[0061] where

[0062] 2.

[0063] 3.

[0064] 4.

[0065] By further simplification, we can get

[0066]

[0067] Continue to simplify using the following inequality:

[0068] 1. (|x1| + |x2| + … + |x n |) p ≤|x1| p +|x2| p +…|x n | p , 0 < p ≤ 1

[0069] Through simplification, we can obtain:

[0070]

[0071] Among them,

[0072]

[0073]

[0074]

[0075] Through the final simplification, we can obtain:

[0076]

[0077] Among them,

[0078] l1 = min{2a1, …, 2a N},

[0079] Proof completed.

[0080] Compared with the prior art, the present invention adopts the above technical solutions and has the following technical effects: The present invention designs a distributed finite-time fault-tolerant controller for the longitudinal model of a fixed-wing UAV based on the theoretical knowledge of neural network adaptation; by using the designed controller, not only the problem of difficult formation tracking control of nonlinear high-order multi-agent systems is overcome, but also the finite-time fault-tolerant control of nonlinear high-order multi-agent systems is achieved; at the same time, when the agents in the system have multiplicative faults in the actuators and the system model is inaccurate, compared with the prior art that lumps and approximates the actuator faults and uncertainties, the present invention separately approximates and estimates the faults and nonlinear terms, uses the adaptive principle to separately estimate the fault information of the actuators, and uses neural networks to approximate the nonlinear uncertainties; finally, fast formation tracking within a finite time is achieved. The controller designed by the present invention has strong robustness and rapidity. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is the flowchart of the present invention;

[0082] Figure 2 is the communication topology graph G;

[0083] Figures 3 to 5 is the actual flight speed V of the fixed-wing UAVs 1-3 within 100 s i (t) compared with the leader V0(t);

[0084] Figures 6 to 8 is the state error e between the flight speed of the fixed-wing UAVs 1-3 and the desired speed command Vi (t) graph;

[0085] Figures 9 to 11 is the altitude h of the fixed-wing UAVs 1-3 within 100 s i (t) compared with the desired altitude command h id (t);

[0086] Figures 12 to 14 is the state error e between the altitude of the fixed-wing UAVs 1-3 and the desired altitude command hi (t) graph;

[0087] Figure 15 and Figure 16 is the actuator fault estimation of the fixed-wing UAV 1 and the fixed-wing UAV 2 compared with the true fault ρ i (t). Detailed implementation manners

[0088] The present invention takes the following 3 fixed-wing UAVs as the implementation objects, where V i , h i , γ i , α i , q i are the state variables of the system, respectively representing the speed, altitude, flight path angle, angle of attack and pitch angle rate of the UAV; T i and δ ei are the control inputs of the system, respectively representing the thrust and elevator deflection of the UAV.

[0089] Consider the following longitudinal dynamic equation of the fixed-wing UAV:

[0090]

[0091] Among them, let the elevator control input with a fault be expressed as i (t) represents the unknown efficiency factor of the actuator of the i-th agent; f γi (V i , γ i ) and f αi (V i , γ i,α i ) represents the unknown nonlinear term in the i-th agent model; f Vi and f qi is a known nonlinear term in the i-th follower model; g Vi ,g γi ,g αi ,g qi are all known functions in the i-th follower model.

[0092] First, construct the communication link diagram of the multi-agent system and represent it as a directed graph G. Figure 2 As shown, 0 represents the virtual leader; 1-3 represent the three fixed-wing drone followers in the formation system. The Laplace matrix L and the adjacency matrix B are as follows:

[0093]

[0094] The expected formation of the agent is d = d1, d2, d3, where d i (i=1,2,3) represents the expected height difference between the ith follower UAV and the virtual leader, and d1=0m, d2=5m, d3=-5m are set; the flight speed of the follower UAV is consistent with that of the leader; the virtual leader is used to give a reference speed signal V0 and a reference height signal h0.

[0095] 1. Reference signals given by virtual leaders:

[0096] V0=40m / s,h0=1000m.

[0097] 2. The initial state and related parameters of the first drone:

[0098] V1(0)=36.5m / s; h1(0)=975m; γ1(0)=0rad; α1(0)=0rad; q1(0)=0rad / s; k V1 =8; p V1 =0.1; a V1 =1; b V1 =0.05; c V1 =0.01; k h1 =3; p h1 =0.1; a h1 =1; b h1 =0.05; c h1 =0.01; k γ1 =3; p γ1 =1;Γ 11 =10*I 13 ; η 11 =0.008; b 11 =0.5; kα1 = 5; p α1 = 0.01; η 12 = 0.009; Γ 12 = 10*I 13 ; b 12 = 8; k q1 = 15; p q1 = 0.1; The number of neurons m = 13.

[0099] 3. Initial state and related parameters of the second drone:

[0100] V2(0) = 36 m / s; h2(0) = 974.6 m; γ2(0) = 0 rad; α2(0) = 0 rad; q2(0) = 0 rad / s; k V2 = 9; p V2 = 0.1; a V2 = 2; b V2 = 0.05; c V2 = 0.01; k h2 = 3; p h2 = 0.1; a h2 = 2; b h2 = 0.05; c h2 = 0.01; k γ2 = 4; p γ2 = 1; Γ 21 = 10*I 13 ; η 21 = 0.008; b 21 = 0.6; k α2 = 6; p α2 = 0.2; η 22 0.008; Γ 22 = 10*I 13 ; b 22 = 8; k q2 = 15; p q2 = 1; The number of neurons m = 13.

[0101] 4. Initial state and related parameters of the third drone:

[0102] V3(0) = 36.5 m / s; h3(0) = 975 m; γ3(0) = 0 rad; α3(0) = 0 rad; q3(0) = 0 rad / s; k V3 = 10; p V3 = 0.1; a V3 = 1; b V3 = 0.05; c V3 = 0.01; k h3 = 3; ph3 = 0.1; a h3 = 1; b h3 = 0.05; c h3 = 0.01; k γ3 = 4; p γ3 = 1; Γ 31 = 10*I 13 ; η 31 = 0.01; b 31 = 0.6; k α3 = 5; p α3 = 0.2; η 32 = 0.008; Γ 32 = 10*I 13 ; b 32 = 8; k q3 = 15; p q3 = 0.1; The number of neurons m = 13.

[0103] UAV 1 and UAV 2 have faults, and UAV 3 has no fault. The specific fault parameters are as follows:

[0104] ρ1(t) = 1, t ≤ 5s; ρ1(t) = 0.6 + 0.4exp(-0.8(t - 5)), t > 5s;

[0105] ρ2(t) = 1, t ≤ 10s; ρ2(t) = 0.7 + 0.3exp(-0.8(t - 10)), t > 10s.

[0106] To verify the effect of the distributed finite-time fault-tolerant control method of the present invention, the simulink template in matlab is used for simulation verification. Figures 3 to 5 The curves in represent the comparison diagrams of the actual flight speeds V i (t) of fixed-wing UAVs 1 to 3 within 100s and the leader speed V0(t); Figures 6 to 8 The curves in represent the state error e Vi (t) diagrams between the flight speeds of fixed-wing UAVs 1 - 3 and the desired speed commands; Figures 9 to 11 The curves in represent the actual heights h i (t) of fixed-wing UAVs 1 to 3 within 100s and the given desired altitude formation h id (t); Figures 12 to 14 The curves in represent the state error e hi (t) diagrams between fixed-wing UAVs 1 - 3 and the desired commands; Figure 15 and Figure 16 are the actuator fault estimations of fixed-wing UAV 1 and fixed-wing UAV 2 respectively and the true fault ρi Comparison diagram of (t).

[0107] It can be seen from the simulation results that by using the designed distributed finite-time fault-tolerant controller, agents at arbitrary initial positions can move according to the predetermined formation instructions within a short time to achieve the desired formation, and when one or more agents in the system have actuator failures, stability can still be ensured. The present invention has important reference value and practical application value in the related research on finite-time fault-tolerant control of nonlinear high-order multi-agent systems.

[0108] The embodiments are only for explaining the technical idea of the present invention, and the protection scope of the present invention cannot be limited thereby. Any modification made on the basis of the technical solution according to the technical idea proposed by the present invention falls within the protection scope of the present invention.

Claims

1. A finite-time formation fault-tolerant control method for fixed-wing UAVs based on neural networks, characterized in that, It includes the following steps: (1) Conduct formation control through a virtual leader-follower structure, and adopt a distributed formation method for information interaction. Based on this, design the corresponding communication topology graph and calculate the corresponding Laplacian matrix L; (2) For the longitudinal dynamic model of a fixed-wing UAV, simplify the dynamic model into a strict feedback form based on reasonable assumptions. Deal with the difficult-to-handle nonlinear terms and model uncertainties in the model, and consider the failure fault of elevator deflection to form a unified strict feedback dynamic model with faults; (3) Based on a radial basis function neural network to handle the unknown nonlinear problems of the system, and combine with adaptive control to achieve fault estimation; Based on backstepping and sliding mode control, achieve distributed finite-time fault-tolerant control of the formation system; In step (1), a directed communication topology graph is adopted, and represents the communication topology of the multi-agent system; among them, represents the set of vertices of the communication graph, and v i represents the i-th vertex, that is, the i-th agent, i = 1, 2, …, N, and N represents the number of agents; E = e ij represents the set of edges of the communication graph, represents can directly obtain the information of the j-th agent , j = 1, 2, …, N; A = [a ij ∈ R N×N is the adjacency matrix. If (v i , v j ) ∈ E, then a ij = 1, otherwise a ij = 0; define the in-degree matrix D = diag(d1, d2, …, d N ) of G, define the Laplacian matrix L = [l ij ∈ R N×N , L = D - A; define the matrix B = diag(b1, b2, …, b N ) to represent the communication topology between the leader and the followers. If the i-th follower can communicate with the leader, then b i = 1, otherwise b i = 0; Step (2) specifically includes: Assumption 1: There is at least one communication path from the leader to each follower; The longitudinal dynamic model of the i-th follower UAV in step (2) is expressed as follows: where \(i = 1, 2, \cdots, N\), and \(N\) represents the number of agents; \(V\) i , \(h\) i , \(\gamma\) i , \(\alpha\) i , \(q\) i represent the velocity, altitude, flight path angle, angle of attack, and pitch rate of the \(i\)-th follower UAV, respectively; \(T\) i , \(D\) i , \(L\) i , \(M\) i represent the thrust, drag, lift, and pitch moment of the \(i\)-th follower UAV, respectively; \(m\) i , \(I\) iy represent the mass and pitch moment of inertia of the \(i\)-th follower UAV, respectively; \(g\) represents the acceleration due to gravity; Hypothesis 2: Since γ i is very small during flight, sinγ i can be considered approximately equal to γ i ; Since T i sinα i << L i , T i sinα i can be neglected; The strict feedback dynamic model with faults of the i-th follower UAV obtained through simplification is expressed as follows: wherein, represents the true control input quantity with multiplicative fault of the i-th follower UAV, δ ei is the control input quantity to be designed for the i-th follower UAV, 0 < ρ i (t) < 1 is the multiplicative fault of the i-th follower UAV; f Vi = -D i / m i - gsinγi and are the known non-linear terms in the i-th follower model, where is the static pressure, s = 1.463m 2 is the reference area, c = 0.451m is the mean aerodynamic chord, C M0 = -0.0954, C Mα = -2.8206, C Mq = -13.8189 are the aerodynamic coefficients; f γi (V i , γ i ), f αi (V i , γ i , α i ) are the unknown non-linear terms in the i-th follower model; are all known functions in the i-th follower model, where is the aerodynamic coefficient; In step (3), based on a radial basis function neural network to handle the unknown nonlinear problems of the system, and combine with adaptive control to achieve fault estimation specifically includes: Obtain the error dynamic equation of the i-th follower according to the strict feedback dynamic model obtained in step (2), specifically: Define the filtering error y iγ = γ id - γ ic , y iα = α id - α ic , y iq = q id - q ic , we get Among them, the characteristics of the radial basis function neural network for approximating nonlinear functions are used to process f γi (V i ,γ i ) and f αi (V i ,γ i ,α i ), and the specific formula is as follows: Among them, W iι * (ι = 1, 2) ∈ R m×1 represents the optimal weight vector of the RBF neural network in the i-th agent controller, and m represents the number of hidden neurons of the RBF neural network; is the basis function vector, X iι is the input vector of the RBF neural network, C iι and b iι respectively represent the center point vector and width of the high-order basis function of the RBF neural network; σ iι * is the approximation error of the RBF neural network in the i-th agent controller, and satisfies is a positive real number; The design method of the fault-tolerant controller combining backstepping and sliding mode control in step (3) is as follows: First, define the sliding mode surface as: where, e Vi = V i - V id , e hi = h i - h id , e γi = γ i - γ id , e αi = α i - α id , e qi = q i - q id are the tracking errors of the speed, altitude, flight path angle, angle of attack, and pitch rate of the i-th follower respectively; V id and h id are the desired speed signal and desired altitude signal of the i-th follower respectively: and V0 and h0 represent the speed and altitude of the virtual leader respectively, and d i and d j represent the relative desired altitudes between the i-th follower and the j-th follower and the leader respectively; γ id , α id , q id are the estimated values of the virtual control inputs to be designed; the parameters a Vi , b Vi , c Vi , a hi , b hi , c hi are positive real numbers; sgn(·) represents the sign function; Design the virtual control input and the actual control input as follows: The adaptive law is as follows: where k mi , p mi are adjustable positive real numbers, m = V, h, γ, α, q, i = 1, 2, …, N; and are the first-order derivatives of V id and h id respectively; represents the adaptive weight vector of the RBF neural network in the i-th agent; represents the basis function vector of the RBF neural network in the i-th agent; represents the estimated value of the multiplicative fault ρ i in the i-th agent controller; Γ i1 , Γ i2 are positive definite diagonal matrices to be designed; η i1 and η i2 are positive real numbers to be designed; Design a first-order low-pass filter as follows: Among them, is a positive real number, γ id , α id , q id are the estimated values of γ ic , α ic , q ic respectively.

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