A Multi-Aircraft Angle Cooperative Guidance Method Based on Quantization Encoder

Abstract

This invention discloses a multi-aircraft angle cooperative guidance method based on a quantization encoder, belonging to the field of multi-aircraft cooperative guidance technology. It includes the following steps: packaging the line-of-sight angle and line-of-sight angular rate data measured by the aircraft's sensors; quantizing and encoding the packaged data according to a communication protocol, and then broadcasting the encoded data to the other aircraft; decoding the information at the receiving end according to the communication protocol using a decoder to reproduce the packaged data; and designing an angle cooperative guidance law based on the received packaged data and the information measured by the receiving aircraft's sensors. This invention, using the above method, achieves angle cooperative guidance with only a small amount of information exchange, and it can be theoretically proven that angle consistency can be achieved with only a directed communication network with strong connectivity; even in the event of complete communication loss, or due to some nodes in the communication terminal network not satisfying strong connectivity, the designed guidance law can still at least guarantee a target hit.

Description

Technical Field

[0001] This invention relates to the field of multi-vehicle cooperative guidance technology, and in particular to a multi-vehicle angle cooperative guidance method based on a quantization encoder. Background Technology

[0002] Multi-aircraft cooperative mission completion is currently a research hotspot in the field of guidance and control, and has been widely applied in scientific research and engineering. Approaching at a certain relative angle during target tracking improves target observation, reduces the possibility of target maneuvering evasion, and increases the probability of penetration and hit. Information exchange between aircraft is essential for successful cooperation. However, in actual flight environments, atmospheric attenuation and natural or man-made interference often limit the communication rate between aircraft, making it difficult to exchange large amounts of data in a short time. For modern aircraft using digital communication links, the amount of data transmitted in a single communication and the communication interval between two communications are both limited. Furthermore, due to the different positions of different aircraft during flight, bidirectional full connectivity is impossible, and some communication links may be interrupted, affecting cooperative guidance. Summary of the Invention

[0003] The purpose of this invention is to provide a multi-aircraft angle cooperative guidance method based on a quantization encoder. The method involves quantizing and encoding the line-of-sight angle and line-of-sight angular rate of the target measured by the aircraft using a quantization encoder, and then broadcasting the encoded data to the other aircraft for reception. This enables one-way communication between two aircraft. At the receiving end, the information is decoded and converted into guidance commands. These commands are then fused with the line-of-sight angular rate and radial velocity measured by the receiving aircraft's own sensors to obtain cooperative guidance commands. This allows the aircraft group to achieve consistent relative strike angles even when using limited communication bandwidth and the line-of-sight angular rate tends to converge.

[0004] To achieve the above objectives, this invention provides a multi-vehicle angle cooperative guidance method based on a quantization encoder, comprising the following steps:

[0005] S1. Pack the line-of-sight angle and line-of-sight angular rate data measured by the aircraft's sensors;

[0006] S2. Quantize and encode the packaged data according to the communication protocol, and then broadcast the encoded data to the other aircraft.

[0007] S3. At the receiving end, the information is decoded according to the communication protocol using a decoder to reproduce the packaged data;

[0008] S4. Design an angle-coordinated guidance law based on the received packet data and the information measured by the receiving aircraft's sensors.

[0009] Preferably, S1 specifically comprises:

[0010] According to coefficients k1 and k2, state x 1i and x 2i Pack the data to obtain the data that will enter the encoder:

[0011] s i =k1x 1i +k2x 2i

[0012] Where, x 1i =q i -γ id , q i For the viewing angle, γ is the line-of-sight angular velocity; id The preset strike angle is a constant value throughout the flight; i represents the i-th aircraft.

[0013] Preferably, the encoder in S2 is:

[0014] ξ i (0)=0

[0015] ξ i (kT)=g((k-1)T)Δ i (kT)+ξ i ((k-1)T)

[0016]

[0017] Where k represents the number of communications since the start of communication, T is the interval between two communications between the spacecraft, and ξ i (kT) represents the internal state of the encoder; g(t) is the quantization index, satisfying g(t) = g0β. k g0 and β are design specifications; q(y) is the quantizer, satisfying...

[0018]

[0019] Preferably, the decoder in S3 is:

[0020]

[0021]

[0022] Δ j (t) represents the information received by the i-th aircraft from the j-th aircraft. This is the decoder state.

[0023] Preferably, the angle-coordinated guidance law in S4 is:

[0024]

[0025] in p1 and p2 are constants. r i The relative distance between the target and the projectile, a ij The element in the i-th row and j-th column of the communication network weight matrix A, ψ mi Indicates the aircraft's current heading.

[0026] Preferably, the minimum transmission word length required by this method is calculated as follows:

[0027] For a communication network between aircraft, G = (V, R, A), where V = {1, 2, ..., N}, representing the set of all aircraft; R = {(x, y) | A(x, y) ∧ x, y ∈ V}, representing the connection between two aircraft; A is a weight matrix, and A(x, y) = 1 indicates that there is a communication connection between aircraft x and y;

[0028] The Laplace matrix L of a communication network is represented as: L = DA, where D represents the in-degree matrix of the communication network, D = diag(deg) in (1), ..., deg in (N)), deg in (i) represents the in-degree of the i-th aircraft, indicating how many other aircraft's information this aircraft can receive. The Laplace matrix L is a row random matrix satisfying L1 = 0, where 1 and 0 represent column vectors with all elements being 1 and 0, respectively; define λ1(L), λ2(L), ..., λ N Let (L) be an eigenvalue of L, satisfying 0 = λ1(L) ≤ λ2(L) ≤ … ≤ λ n (L);

[0029] Based on the premise of a strongly connected communication network, the minimum transmission bandwidth of the directed communication network between aircraft satisfies:

[0030]

[0031] in Initial quantization parameters Among them, C δ The quantity characterizing the inconsistency of the initial state of the aircraft is max|δ i (0)|≤C δ δ(kT)=X(kT)-(1-1Π) T (kT))X(kT), X(kT)=[s1(kT), s2(kT),..., s N (kT) ] T,

[0032] Define norm in, Let D k The largest element in the diagonal is defined as π. max (k), the smallest is defined as π. min (k), π at all times max (k) The maximum value is defined as π max , π min (k) The minimum value is defined as π min ;in,

[0033] Since the norm of any vector in a finite-dimensional linear space is equivalent, there must exist a positive number r. k Such that for any x and time k, we have For all times, taking the maximum value, we have x. T D k+1 x≤r max x T D k x

[0034] To achieve coordinated guidance, the initial parameters must satisfy the following three conditions: g0 > C x / (K+0.5); h max <1 / d * ; So that the parameter β can be chosen to satisfy Among them, C x max represents the maximum value of the initial state quantity. i |s i (0)|≤C x h max =max i (h i (kT)), d * =max i (deg in (i)), ρ=1-h max d * .

[0035] The advantages and positive effects of the multi-aircraft angle cooperative guidance method based on a quantization encoder described in this invention are as follows:

[0036] 1. The method described in this invention only requires the exchange of very little information to achieve angle cooperative guidance, which greatly reduces the requirements for communication capabilities between aircraft.

[0037] 2. This invention theoretically proves that angular consistency can be achieved with only a directed communication network that has strong connectivity characteristics, without requiring the inter-projectile communication network to meet stringent conditions such as full connectivity or undirected communication.

[0038] 3. Even if communication is completely lost, or if some nodes in the communication terminal network do not meet the strong connectivity characteristics, the guidance law designed in this invention can at least guarantee a hit on the target.

[0039] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0040] Figure 1 This is a flowchart of an embodiment of a multi-aircraft angle cooperative guidance method based on a quantization encoder according to the present invention;

[0041] Figure 2 This is a diagram illustrating the inter-vehicle communication topology of an embodiment of a multi-vehicle angle cooperative guidance method based on a quantization encoder according to the present invention.

[0042] Figure 3 This is a diagram illustrating the motion trajectories of an aircraft and a target, representing an embodiment of a multi-aircraft angle cooperative guidance method based on a quantization encoder according to the present invention.

[0043] Figure 4 This is an actual overload diagram of an aircraft according to an embodiment of a multi-aircraft angle cooperative guidance method based on a quantization encoder, as presented in this invention.

[0044] Figure 5 This is a diagram showing the relative angle deviation of aircraft in an embodiment of a multi-aircraft angle cooperative guidance method based on a quantization encoder according to the present invention.

[0045] Figure 6 This is a diagram of aircraft data transmission according to an embodiment of a multi-aircraft angle cooperative guidance method based on a quantization encoder according to the present invention. Detailed Implementation

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

[0047] like Figure 1 As shown, a multi-vehicle angle cooperative guidance method based on a quantization encoder includes the following steps:

[0048] S1. Pack the line-of-sight angle and line-of-sight angular rate data measured by the aircraft's sensors.

[0049] Considering the dynamic model of the line-of-sight angle during aircraft guidance, for the i-th aircraft, the system state x is set. 1i x 2i , where x 1i =qi -γ id , Where q i For the viewing angle, γ is the line-of-sight angular velocity; id The preset strike angle remains constant throughout the entire flight. If γ id If i = 0, i = 1, 2, ..., N, then the coordinated target is the same attack angle; otherwise, the targets are attacks with a certain angular interval.

[0050] The dynamic model of aircraft guidance can be represented in the following form:

[0051]

[0052] Simultaneously consider the communication network between aircraft, G = (V, R, A); where V = {1, 2, ..., N}, representing the set of all aircraft; R = {(x, y) | A(x, y) ∧ x, y ∈ V}, representing the connection relationship between two aircraft; and A is a weight matrix. A(x, y) = 1 indicates that there is a communication connection between aircraft x and y, and aircraft y can receive information from aircraft x. Conversely, it indicates that there is no communication connection between aircraft x and y, and aircraft y cannot receive information from aircraft x. If A is a symmetric matrix, it means that information can be bidirectionally connected, and the communication network is an undirected graph. This invention does not require the communication network between aircraft to satisfy the undirected graph condition.

[0053] The Laplace matrix L of a communication network can be expressed as: L = DA, where D represents the in-degree matrix of the communication network, D = diag(deg) in (1), ..., deg in (N)), deg in (i) represents the in-degree of the i-th aircraft, indicating how many other aircraft's information this aircraft can receive. For a directed graph, the Laplacian matrix L is an asymmetric positive semi-definite matrix. Furthermore, the Laplacian matrix L is also a row random matrix satisfying L1 = 0, where 1 and 0 represent column vectors with all elements being 1 and 0, respectively. In a directed graph, if for any two vertices v and v′, there exists a path from v to v′, then G is called a strongly connected graph. Define λ1(L), λ2(L), ..., λ N Let (L) be an eigenvalue of L, satisfying 0 = λ1(L) ≤ λ2(L) ≤ … ≤ λ n For a strongly connected graph, λ2(L) > 0. This invention requires that the communication network between aircraft satisfy the condition of a strongly connected graph. In fact, satisfying the strongly connected condition in an aircraft communication network is relatively easy.

[0054] The encoder design for the i-th aircraft is as follows:

[0055] First, adjust the state x according to coefficients k1 and k2. 1i and x 2i Pack the data to obtain the data that will enter the encoder:

[0056] s i =k1x 1i +k2x 2i (2)

[0057] S2. Quantize and encode the packaged data according to the communication protocol, and then broadcast the encoded data to the other aircraft.

[0058] The encoder is:

[0059] ξ i (0)=0

[0060] ξ i (kT)=g((k-1)T)Δ i (kT)+ξ i ((k-1)T)

[0061]

[0062] Where k represents the number of communications since the start of communication, and T is the interval between two communications between the aircraft; ξ i (kT) represents the internal state of the encoder, initially set to 0, and updated iteratively using data measured by the spacecraft; g(t) is the quantization index, satisfying g(t) = g0β. k g0 and β are the parameters selected during design. q(y) is the quantizer, satisfying...

[0063]

[0064] For aircraft using digital communication, both the amount of data transmitted in a single communication and the time interval between two communications are limited. K represents the maximum amount of data transmitted in a single communication. The aircraft network bandwidth is... Let x be the smallest integer greater than or equal to x. i (kT) Finally, the information transmitted through the aircraft's communication equipment is the information that has been quantized and encoded.

[0065] S3. At the receiving end, the information is decoded according to the communication protocol using a decoder to reproduce the packaged data.

[0066] If a communication link exists between aircraft j and aircraft i, then the information received by aircraft i from aircraft j is Δ. j (t), the guidance information is decoded by a decoder in aircraft i.

[0067] The decoder is:

[0068]

[0069]

[0070] This represents the decoder status and the decoding result, containing guidance information from other aircraft.

[0071] S4. Design an angle-coordinated guidance law based on the received packet data and the information measured by the receiving aircraft's sensors.

[0072] make:

[0073]

[0074] Where p1 and p2 are constants.

[0075] The angle-coordinated guidance law is:

[0076]

[0077] Substituting equation (5) into equation (1), we get:

[0078]

[0079]

[0080] Given the relatively long guidance process, information transmission is relatively fast; therefore, it can be approximated that the interval between each two data transmissions is relatively short. No change, therefore J 1，i and J 2，i Since the data remains unchanged between the two transmissions, equation (6) becomes a differential equation with constant coefficients, which can be solved to obtain:

[0081]

[0082] make

[0083]

[0084] Then formula (7) can be rewritten as:

[0085]

[0086] When k2 > 0, h i (J 2，i (k)) with J 2，i The increase of h(T) starts monotonically from J2→0 and h(T)→0, and eventually tends to... at this time make

[0087] Due to h i (J 2，i Since (k) is monotonically increasing, the minimum value is obtained at the initial time. Let h min =min{h1(0),h2(0),...,h N (0)}. As the aircraft approaches the target, the distance r between the aircraft and the target keeps decreasing. This ensures h i (t) is monotonically increasing.

[0088] Let X(k)=[s1(kT), s2(kT),..., s N (kT)] T H(kT)=diag(h1(J) 2，1 (kT)), ..., h N (J 2，N (kT))),

[0089] Define norm in, Let D k The largest element in the diagonal is defined as π. max (k), the smallest is defined as π. min (k), π at all times max (k) The maximum value is defined as π max , π min (k) The minimum value is defined as π min D k The vector consisting of the diagonal elements is ∏(kT).

[0090] Packed data of all aircraft i Writing it all together, we get:

[0091] X(k+1)=(IH(k)L)X(k)+H(k)Le(k)

[0092]

[0093] in

[0094] Q([y1,...y n ] T )=[q(y1),...,q(y n )] T

[0095] X(k)=[s1(k),s2(k),...,sN (k)] T

[0096]

[0097] δ(k)=X(k)-(1-1∏ T (kT))X(k)

[0098] make

[0099]

[0100]

[0101] Formula 10 can then be written as:

[0102] w(k+1)=β -1 (IH(k)L)w(k)+γ -1 H(k)Lz(k)

[0103] γz(k+1)=β -1 Ω(k) (11)

[0104] in:

[0105] Ω(k)=(I+H(k)L)z(k)+H(k)Lw(k)-Q((I+H(k)L)z(k)-H(k)Lw(k))

[0106] Assume that the initial conditions, parameter selection, and communication capability satisfy: max i |x i (0)|≤C x max i |δ i (0)|≤C δ g0 > C x / (K+0.5), h max <1 / d * ρ=1-h max d * Therefore, ρ > 0. η = ρπ min , π min =1 / h max η = 1 / h max -d * >0

[0107] Since the norm of any vector in a finite-dimensional linear space is equivalent, there must exist a positive number r. k Such that for any x and time k, we have For all times, taking the maximum value, we have x. T D k+1 x≤rmax x T D k For aircraft launched in the same wave, their arrival times at the target are similar, therefore we can assume that they satisfy: in, Pick Then there is

[0108] It is general, with so:

[0109] Right now

[0110]

[0111] The minimum transmission bandwidth required for a single information exchange by the aircraft is:

[0112]

[0113] in

[0114] Next, mathematical induction will be used to illustrate the effectiveness of the algorithm proposed in this invention, starting from the initial state. From ||X(0)|| ∞ ≤C x ,Depend on have:

[0115]

[0116] because:

[0117]

[0118] Therefore, as long as a suitable initial quantization parameter g0 is chosen, it will make Then we have:

[0119]

[0120] That is, the encoder is not saturated at the initial moment. By mathematical induction, assuming the encoder is not saturated at t = 0, 1, ..., k, then:

[0121]

[0122] but:

[0123]

[0124] Define the Lyapunov function V k (X(k))

[0125]

[0126] Furthermore, IH(k)L is a row random matrix with all diagonal elements being positive. The errors δ(k) at time k+1 and time k are in D k The norm under the following conditions satisfies:

[0127]

[0128] Where 1 > η = ρπ mm (k)>0, π min (k)=1 / max 1≤i≤N h i (k), ρ=1-h max d * .

[0129] consider

[0130]

[0131] in, have:

[0132]

[0133] Equation (18) then begins:

[0134]

[0135] make: Depend on have:

[0136]

[0137] According to the properties of vector norm, Observe these three items, the first item

[0138]

[0139] Second item

[0140]

[0141] Third item:

[0142]

[0143]

[0144] Adding these three together, we get

[0145]

[0146] Therefore, the encoder input can be represented as:

[0147]

[0148] d * It only depends on the graph topology, so as long as the bandwidth of data transmission between missiles is greater than... At time k+1, the encoder will not saturate. According to mathematical induction, the encoder will never saturate. Therefore, according to the principle of mathematical induction, the encoder will not saturate during the entire flight. Thus, w(t) has a finite upper bound. Therefore:

[0149]

[0150] This demonstrates that given enough time, the s of each aircraft i Consistency can be achieved; according to the principle of consistency, x i1 Achieving consensus.

[0151] Example

[0152] The following is a specific example of the present invention to verify the effectiveness of the proposed control method. The specific implementation steps of this example are as follows:

[0153] Initial state settings for aircraft and targets

[0154] Consider a two-dimensional planar strike mission involving five aircraft attacking a single target. The target initially lies at coordinates (0km, 0km) with a velocity of 300m / s, oriented along the negative x-axis. It maintains a constant velocity for the first 100 seconds, then performs a U-turn with an acceleration of 3g. The initial positions of the multiple aircraft are: (-20km, 0km), (-20km, 4km), (-19km, 3km), (-25km, 10km), and (-23km, -3km), with constant velocities of 657m / s, 738m / s, 630m / s, 852m / s, and 986m / s respectively. The maximum maneuver overload is 9g. The dynamic characteristics between the overload command and the actual overload are simplified to an inertial element with a time constant of 0.1s. The requirement is for the multiple aircraft to hit the target at 5° intervals. The aircraft exchange data every 0.25 seconds, with each exchange consisting of 4 bits, resulting in a data transmission rate of 16 bps. This places very low demands on the communication equipment. The variances for the line-of-sight angle, line-of-sight angular velocity, distance measurement, and heading angle are 0.05°, 0.5° / s, 10 meters, and 0.05° respectively.

[0155] Selection of aircraft communication parameters

[0156] The parameters of the encoder and decoder in the aircraft are: g0 = 0.0001, β = 0.998, k1 = 0.025, k2 = 0.1, p1 = 20, p2 = 3, and the communication matrix satisfies...

[0157]

[0158] Communication relationships such as Figure 2 As shown, it satisfies the conditions for a strongly connected directed graph.

[0159] Results Analysis

[0160] Applying the above theoretical results, the trajectory of the aircraft is as follows: Figure 3 As shown, the actual overload generated is as follows Figure 4 As shown, the impact angle deviates from the preset value as follows: Figure 5 As shown, the information transmitted between the aircraft is as follows Figure 6 As shown in the four figures above, the desired multi-aircraft coordinated target engagement with relative line-of-sight angles can be achieved. Over time, the attack angles deviate from the preset value and converge, indicating that the aircraft fly towards the target at preset relative angular intervals. The guidance mission can be completed by exchanging only a minimal amount of information between the aircraft. Thus, multi-aircraft coordinated relative angle guidance at low transmission rates has been achieved, verifying the effectiveness of the guidance method described in this invention.

[0161] Therefore, the present invention adopts the above-mentioned multi-aircraft angle cooperative guidance method based on quantization encoder, which can achieve angle cooperative guidance with very little information exchange. It can also be theoretically proven that angle consistency can be achieved with only a directed communication network with strong connectivity, making it easier to implement. Furthermore, even if communication is completely lost, or if some nodes in the communication terminal network do not meet the strong connectivity requirement, the designed guidance law can at least guarantee a hit on the target.

[0162] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for multi-aircraft angular cooperative guidance based on quantized encoders, characterized in that, The method comprises the following steps: S1, packaging line-of-sight angle and line-of-sight angle rate data measured by sensors of the aircraft; S2, quantization encoding of the packaged data according to a communication protocol, and broadcasting and sending the encoded data to the remaining aircrafts; S3, decoding the information according to the communication protocol by a decoder at the receiving end to reproduce the packaged data; S4, designing an angle cooperative guidance law according to the received packaged data and the information measured by sensors of the receiving aircraft; The encoder in S2 is: ξ i (0) = 0 ξ i (kT) = g((k - 1)T) Δ i (kT) + ξ i ((k - 1)T) where k represents the number of communications after the start of communication, T is the interval between two communications of the aircraft, ξ i (kT) is the internal state of the encoder; g(t) is the quantization index, satisfying g(t) = g0β k , g0 and β are design indexes; q(y) is the quantizer, satisfying The decoder in S3 is: where Δ j (t) is the time at which the ith aircraft receives information from the jth aircraft, is the decoder state; The angle cooperative guidance law in S4 is: wherein, p1, p2 are constants; r i is the relative distance between the projectile and the target, a ij is the element of the communication network weight matrix A in the i-th row and j-th column, ψ mi denotes the current heading of the aircraft.

2. A multi-aircraft angular cooperative guidance method based on quantized encoder according to claim 1, characterized in that, S1 is specifically: The state x is packed according to the coefficients k1 and k2 1i and x 2i to obtain the data entering the encoder: s i = k1x 1i + k2x 2i where x 1i = q i - γ id , q i is the line-of-sight angle, is the line-of-sight angle velocity; γ id is the preset impact angle, which is a constant during the whole flight; i represents the i-th aircraft.

3. A multi-aircraft angular cooperative guidance method based on quantized encoder according to claim 2, characterized in that, The minimum transmission word length calculation method required by the method is: For the communication network G=(V,R,A) between the aircrafts, wherein V={1,2,...,N} represents a set of aircrafts; R={(x,y)|A(x,y)∧x,y∈V} represents a connection relationship between two aircrafts; A is a weight matrix, and A(x,y)=1 represents that there is a communication connection between the aircrafts x and y; The Laplacian matrix L of the communication network is represented as L = D - A, where D represents the in-degree matrix of the communication network, D = diag(deg in (1),...,deg in (N)) and A represents the adjacency matrix of the communication network, A = (aij)neN x neN. in (i) is the in-degree of the ith aircraft, indicating how many other aircrafts the ith aircraft can receive information from The Laplacian matrix L is a row stochastic matrix, satisfying L1 = 0, where 1 and 0 represent column vectors with all elements being 1 and 0, respectively; and λ1(L), λ2(L),...,λ N (N) are defined as eigenvalues of L, satisfying 0 = λ1(L)≤ λ2(L)≤...≤ λ n (N) = 1; and λ1(L) is the smallest eigenvalue of L. On the basis of meeting the strongly connected communication network, the minimum transmission bandwidth of the directed communication network between the aircrafts satisfies: wherein initial quantization parameter where C δ is a quantity representing inconsistency of the initial state of the aircraft, max | δ i (0) | ≤ C δ , δ(kT) = X(kT) - (1 - 1Π T (kT))X(kT), X(kT) = [s1(kT), s2(kT),..., s N (kT)] T , Definition of norm wherein, Let D k The largest element in the diagonal is defined as π max (k), the smallest as π min (k), the maximum value of all times as π max (k), the minimum value as π max , π min (k) the minimum value is defined as π min ; wherein, Since any vector norm in a finite-dimensional linear space is equivalent, there must exist a positive number r k such that for any x and time k, we have For all times, taking the maximum, we have x T D k+1 x≤r max x T D k x; The initial parameters must satisfy the following three conditions to complete the cooperative guidance: g0>C x (K+0.5); h max <1 / d * ; So that the parameter β can be selected to satisfy Where C x represents the maximum value of the initial state quantity, max i |s i (0)|≤C x , h max =max i (h i (kT)), d * =max i (deg in (i)), η=(1-h max d * ) / ρ=1-h max d * .

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