A Distributed Cooperative Guidance Law with Prescribed-Time Consensus Performance

3.1. Design Strategy

Inspired by the mathematical expression of the total flight time in Equation (15), this paper proposes a two-step control strategy to achieve the distributed cooperative guidance law.

First, it is assumed that the aircraft design is based on Equation (7), with

c_{i} = 1

. Once the initial conditions are given, the total flight time can be calculated using Equation (15). Next, each time step is treated as the initial time, and the instantaneous state is considered as the initial state to recalculate the flight time using Equation (15). The equation is updated as follows:

$t_{i}^{'} = \frac{σ_{i} R_{i}}{V_{i} \sin σ_{i}}, i = 1, 2, \dots, n .$

(16)

Combining with Equation (16), the relative impact time error of the i-th aircraft is defined as follows:

$ϵ_{i} = \sum_{j = 1}^{n} a_{i j} (t_{i} - t_{j}), i = 1, 2, \dots, n .$

(17)

where

$t_{i} = t_{i}^{'} + t = \frac{σ_{i} R_{i}}{V_{i} \sin σ_{i}} + t$

(18)

Let

ϵ = {[ϵ_{1}, \dots, ϵ_{n}]}^{T}

and

t_{i} = {[t_{1}, \dots, t_{n}]}^{T}

; then, the compact vector form can be expressed as:

Second, the control parameters are adjusted to achieve consensus on the total flight time. From the previous discussion, the consensus on the total flight time allows for simultaneous arrival. Once the consensus error calculated using Equation (17) becomes zero, all aircraft will set their control parameters to $c_{i} = 1$ and maintain that value.

3.2. Undirected Topology Cooperative Guidance Law

To accomplish the task of cooperative arrival, each aircraft must estimate its own remaining flight time while also obtaining the remaining flight times of other aircraft. By analyzing this information, they can continuously adjust their flight speeds to achieve cooperative guidance. During the flight, if aircraft can exchange information with one another, they can maximize the use of flight data, enhancing cooperative efficiency, which represents the optimal communication state. Based on these conditions, this subsection considers the ideal scenario of designing cooperative guidance under an undirected topology communication state.

Lemma 3.

A first-order multi-agent system with

n

agents is described by the following dynamic equation:

${\dot{x}}_{i} (t) = u_{i} (t), i = 1, 2, \dots, n .$

(20)

where $u_{i} (t) \in R$ denotes the designed protocol, and ${\dot{x}}_{i} (t) \in R$ represents the state of the i-th agent. If the graph $Ω$ connecting the agents is connected and undirected, each agent follows the protocol defined as:

$u_{i} (t) = - (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α) \sum_{j = 1}^{n} a_{i j} (x_{i} (t) - x_{j} (t))$

(21)

where $λ_{2} (L)$ is defined by Lemma 1, $0 < δ ≪ 1$ , $α > 0$ are constant parameters, and $ξ (t)$ is a time-varying function with the following properties:

1.: $ξ (t)$ is twice continuously differentiable on the interval $[0, \infty)$ .
2.: $ξ (t)$ is monotonically increasing from $ξ (0) = 0$ to $ξ (t_{p r e}) = 1$ , where $t_{p r e} < \infty$ represents the prescribed time.
3.: $ξ^{'} (0) = ξ^{'} (t_{p r e}) = 0$ .
4.: $ξ^{'} (t) = 0$ and $ξ (t) = 1$ when $t > t_{p r e}$ .

Thus, the system will achieve consensus within the prescribed time, expressed as:

$\{\begin{array}{l} \lim_{t \to t_{pre}} | x_{i} (t) - x_{j} (t) | \leq (n - 1) \sqrt{\frac{2 δ}{1 + δ} V_{1} (0)} \\ | x_{i} (t) - x_{j} (t) | \leq (n - 1) \sqrt{\frac{2 δ}{1 + δ} V_{1} (0)}, \forall t > t_{pre} \\ \lim_{t \to + \infty} | x_{i} (t) - x_{j} (t) | = 0 \end{array} i = 1, 2, \dots, n .$

(22)

where $V_{1} (0)$ is the initial value of a non-negative function $V_{1} (t) = \frac{1}{2} x^{T} (t) L x (t)$ .

For the cooperative arrival problem of multiple aircraft under an undirected topology, this paper makes the following assumptions:

Assumption 1.

The communication topology between the aircraft is undirected and connected.

Assumption 2.

During the flight, the changes are small-angle variations; thus,
$\sin σ_{i} \approx σ_{i}$ , $\cos σ_{i} \approx 1 - σ_{i}^{2} / 2$ .

In this section, the control variable is selected as

{\dot{σ}}_{i}

, and a prescribed-time guidance law for stationary targets under an undirected graph is proposed, taking the following form:

${\dot{σ}}_{i} = - (1 + \frac{2 ϵ_{i} (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}}) \frac{V_{i}}{R_{i}} \sin σ_{i}, i = 1, 2, \dots, n .$

(23)

where $λ_{2} (L)$ satisfies the definition in Lemma 1, $δ$ , $ξ (t)$ , and $α$ meet the conditions outlined in Lemma 3, and $ϵ_{i}$ follows the definition in Equation (17).

Theorem 1.

If Assumption 1 and Assumption 2 are satisfied, the guidance law defined by Equation (23) guarantees that the actual flight time reaches prescribed-time consensus. Moreover, when $t > t_{p r e}$ , due to $ϵ_{i} \approx 0$ , it can be assumed that ${\dot{σ}}_{i} = \frac{V_{t_{p r e}, i}}{R_{t_{p r e}, i}} \sin σ_{t_{p r e}, i} = C_{i}$ , where $C_{i}$ is a constant.

Proof 1.

Taking the derivative with respect to time yields:

${\dot{t}}_{i} = \frac{{\dot{R}}_{i} σ_{i} \sin σ_{i} + {\dot{σ}}_{i} (R_{i} \sin σ_{i} - R_{i} σ_{i} \cos σ_{i})}{V_{i} \sin^{2} σ_{i}} + 1$

(24)

Combining Assumption 2 and substituting into Equations (2), (4), and (23) yields:

$\begin{array}{l} {\dot{t}}_{i} = \frac{{\dot{R}}_{i} σ_{i}}{V_{i} \sin σ_{i}} - (1 + \frac{2 ϵ_{i} (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}}) (\frac{\sin σ_{i} - σ_{i} \cos σ_{i}}{\sin σ_{i}}) + 1 \\ = \frac{- σ_{i} \cos σ_{i}}{\sin σ_{i}} + \frac{σ_{i} \cos σ_{i}}{\sin σ_{i}} - \frac{2 ϵ_{i} (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}} (1 - \frac{σ_{i} \cos σ_{i}}{\sin σ_{i}}) \\ = - \frac{2 ϵ_{i} (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}} (1 - 1 + σ_{i}^{2} / 2) \\ = - \sum_{j = 1}^{n} a_{i j} (t_{i} - t_{j}) (\frac{\dot{ξ} (t)}{2 λ_{2} (L) (1 - ξ (t) + δ)} + α) \end{array}$

(25)

Equation (25) satisfies the form of Equation (21), which confirms that the total flight time can reach consensus within the prescribed time.

Consider the Lyapunov function of the form:

$V_{1} (t) = \frac{1}{4} \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i j} {(t_{i} - t_{j})}^{2} = \frac{1}{2} t^{T} L t$

(26)

It is clear that

V_{1} (t)

is positive and semi-definite, and

V_{1} (t) = 0

if and only if

t_{1} = t_{2} = \dots = t_{n}

. This indicates that once

V_{1} (t) = 0

, consensus on the remaining flight time is achieved. Combining Property 3 and Lemma 3 allows for the derivation of the following:

$\lim_{t \to t_{pre}} | t_{i} - t_{j} | \leq \frac{(n - 1)}{2 δ (1 + δ)} V_{1} (0) i, j = 1, 2, \dots, n .$

(27)

where $V_{1} (0)$ is the initial value of $V_{1} (t)$ in Equation (26). Now, consider the case when $t > t_{p r e}$ . By applying Property 3 from Lemma 3 and Equation (25), it can be concluded that:

${\dot{t}}_{i} = - α ϵ_{i}, t > t_{pre}$

(28)

Using Equations (19) and (28), the first-order time derivative of

V_{1} (t)

can be expressed as:

$\dot{V_{1}} (t) = t^{T} L t = - α \sum_{i = 1}^{n} ϵ_{i}^{2} \leq 0, t > t_{pre}$

(29)

This indicates that

V_{1} (t)

monotonically decreases for

t > t_{p r e}

. Combining with Equation (27), it follows that:

$| t_{i} - t_{f} | \leq \frac{(n - 1)}{2 δ (1 + δ)} V_{1} (0), t > t_{pre} i, j = 1, 2, \dots, n .$

(30)

This means that when $δ$ is chosen to be sufficiently small, $| t_{i} - t_{f} |$ ensures that for all $i, j \in V$ , the values will approach to a sufficiently small region within the prescribed time and remain in that neighborhood until interception, thereby ensuring the accuracy of the impact timing. □

3.3. Directed Topology Cooperative Guidance Law

In practical applications, the communication channels between aircraft are often unreliable due to the varying performance of communication devices. A more general scenario is examined to further lessen the communication burden, with the communication topology between the aircraft being directed. It is assumed that in addition to the $n$ aircraft, there is one leader aircraft (indexed as 0). The communication topology among the n + 1 aircraft is characterized by a directed graph $\tilde{Ω} (\tilde{ν}, \tilde{ε})$ , where $\tilde{ε}$ denotes the communication links and $\tilde{ν}$ denotes the set of aircraft.

Lemma 4.

For a multi-agent system with a stationary leader (indexed as 0), let the dynamics of the leader be represented by ${\dot{x}}_{0} (t) = u_{0} (t) = 0$ , where $u_{0} (t) \in R$ and $x_{0} (t) \in R$ represent the control input and state of the leader, respectively. It is assumed that the communication topology $\tilde{Ω}$ contains a directed spanning tree with the leader as the root. The rule established for each follower is:

$u_{i} (t) = - (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α) \sum_{j = 0}^{n} a_{i j} (x_{i} (t) - x_{j} (t))$

(31)

For Equation (31), if

δ

and

α

satisfy the conditions of Lemma 3,

h_{m a x}

and

λ_{m i n} (U)

meet the definitions of Lemma 2; then:

$\{\begin{matrix} \lim_{t \to t_{pre}} | x_{i} (t) - x_{0} (t) | \leq \sqrt{\frac{2 δ}{h_{\min} (1 + δ)} V_{2} (0)} \\ | x_{i} (t) - x_{0} (t) | \leq \sqrt{\frac{2 δ}{h_{\min} (1 + δ)} V_{2} (0)} \\ \lim_{t \to + \infty} | x_{i} (t) - x_{0} (t) | = 0 \end{matrix} t_{i} > t_{pre} i = 1, 2, \dots, n .$

(32)

where $h_{\min}$ satisfies the definition of Lemma 2, and $V_{2} (0)$ denotes the initial value of the non-negative function $V_{2} (t)$ , defined as follows:

$V_{2} (t) = \frac{1}{2} e^{T} (t) H e (t)$

(33)

For the functions $V_{2} (t)$ , $H$ satisfies the definitions in Lemma 2 and has the following properties: $e (t) = {[e_{1} (t), \dots, e_{n} (t)]}^{T} = {[x_{1} (t) - x_{0} (t), \dots, x_{n} (t) - x_{0} (t)]}^{T} .$

Assumption 3.

Among the n + 1 aircraft, the communication topology is structured as a leader–follower graph $\tilde{Ω} (\tilde{ν}, \tilde{ε})$ , where the leader aircraft acts as the root of a directed spanning tree. The communication network among the other aircraft, excluding the leader, is represented by $Ω (ν, ε)$ .

Remark 1.

It should be emphasized that the leader aircraft does not receive any information from other aircraft. Consequently, it adheres to the guidance law specified in Equation (14) to navigate towards the target.

The relative impact time error of the i-th aircraft (as a follower) in the leader–follower setup is defined as:

$ϵ_{i} = \sum_{j = 0}^{n} a_{i j} (t_{i} - t_{j}), i = 1, 2, \dots, n .$

(34)

where $t_{i} = t_{i}^{'} + t = \frac{σ_{i} R_{i}}{V_{i} \sin σ_{i}} + t$ . Additionally, $e (t) = {[e_{1} (t), \dots, e_{n} (t)]}^{T}$ and $e_{i} = t_{i} - t_{0}$ are defined.

Then, for a stationary target, the actual prescribed-time guidance law designed under the directed graph is expressed as:

${\dot{σ}}_{i} = - (1 + \frac{2 ϵ_{i} (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}}) \frac{V_{i}}{R_{i}} \sin σ_{i}, i = 1, 2, \dots, n .$

(35)

where $δ$ , $ξ (t)$ , and $α$ satisfy the conditions outlined in Lemma 3, while $h_{\max}$ and $λ_{\min} (U)$ meet the conditions specified in Lemma 2.

Theorem 2.

If Assumption 2 and Assumption 3 hold, the guidance law defined by Equation (35) ensures that the actual flight time achieves prescribed-time consistency. Additionally, $| e_{i} |, i = 1, 2, \dots, n .$ is bounded for $t > t_{p r e}$ .

Proof 2.

Under Remark 1, it is known that the leader’s acceleration variation satisfies $\dot{a} \equiv 0$ . According to Equation (15), the leader’s total flight time can be expressed as:

$t_{0} = \frac{σ_{0} R_{0}}{V_{0} \sin σ_{0}}$

(36)

Then, the flight time

t_{i}

is differentiated for the followers, and Equations (2), (4), and (35) are substituted into it to obtain:

$\begin{array}{l} {\dot{t}}_{i} = \frac{{\dot{R}}_{i} σ_{i}}{V_{i} \sin σ_{i}} - (1 + \frac{2 ϵ_{i} (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}}) (\frac{\sin σ_{i} - σ_{i} \cos σ_{i}}{\sin σ_{i}}) + 1 \\ = \frac{- σ_{i} \cos σ_{i}}{\sin σ_{i}} + \frac{σ_{i} \cos σ_{i}}{\sin σ_{i}} - \frac{2 ϵ_{i} (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}} (1 - \frac{σ_{i} \cos σ_{i}}{\sin σ_{i}}) \\ = - \frac{2 ϵ_{i} (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α)}{σ_{i}^{2}} (1 - 1 + σ_{i}^{2} / 2) \\ = - (\frac{h_{\max} \dot{ξ} (t)}{λ_{\min} (U) (1 - ξ (t) + δ)} + α) ϵ_{i} \end{array}$

(37)

Equation (37) satisfies the form of Equation (31), which indicates that the flight times $t_{i}$ can achieve consensus within the prescribed time.

Consider the Lyapunov function of the form:

$V_{2} (t) = \frac{1}{2} e^{T} (t) H e (t)$

(38)

where H is a diagonal matrix with positive entries and satisfies the conditions of Lemma 2. Clearly, $V_{2} (t)$ is positive and definite, and for Equation (38), $V_{2} (t) = 0$ if and only if all $e_{i} = 0$ .

This indicates that once $V_{2} (t) = 0$ , the total flight times of all aircraft can be expressed as $t_{i} = t_{0} = \frac{σ_{0} R_{0}}{V_{0} \sin σ_{0}}$ , achieving consensus in flight times.

According to Lemma 4 and Equation (37), it can be concluded that when

t \to t_{p r e}

$\lim_{t \to t_{pre}} | e_{i} | \leq \sqrt{\frac{2 δ}{h_{\min} (1 + δ)} V_{2} (0)}, i = 1, 2, \dots, n .$

(39)

where $h_{\min}$ satisfies the definition in Lemma 2, and $V_{2} (0)$ is the initial value of $V_{2} (t)$ in Equation (38). Now consider the case when $t > t_{p r e}$ . According to Property 3 in Lemma 3 and Equation (37), the time derivative of $e$ can be expressed as:

$\dot{e} = - α M e, t > t_{p r e}$

(40)

where $M$ is defined by Lemma 2. Differentiating $V_{2} (t)$ and substituting Equation (40) yields:

$\begin{array}{l} {\dot{V}}_{2} (t) = - α e^{T} H M e \\ = - \frac{1}{2} α e^{T} (H M + M^{T} H) e \\ = - \frac{1}{2} α e^{T} U e \\ \leq - \frac{1}{2} α λ_{\min} (U) e^{T} e \\ \leq 0 \end{array}$

(41)

where from Lemma 2, $U = H M + M^{T} H > 0$ . Therefore, $V_{2} (t)$ is monotonically decreasing for $t > t_{p r e}$ , and for Equation (39), it follows that:

$| e_{i} | \leq \sqrt{\frac{2 δ}{h_{\min} (1 + δ)} V_{2} (0)}, t > t_{pre}, i, j = 1, 2, \dots, n .$

(42)

This means that within the prescribed time, the errors of all aircraft can converge to a sufficiently small neighborhood around the origin and remain within that neighborhood before arrival occurs, thereby ensuring the accuracy of cooperative arrival timing. □

Source link

Chao Ou www.mdpi.com

Greenberg News

A Distributed Cooperative Guidance Law with Prescribed-Time Consensus Performance

3.1. Design Strategy

3.2. Undirected Topology Cooperative Guidance Law

3.3. Directed Topology Cooperative Guidance Law

Greenberg

3.1. Design Strategy

3.2. Undirected Topology Cooperative Guidance Law

3.3. Directed Topology Cooperative Guidance Law

Related Posts

Environmental Factor – August 2020: NIEHS newsletter editor-in-chief wins writing award

Cannabis sativa L. Leaf Oil Displays Cardiovascular Protective Effects in Hypertensive Rats

A Project 2025 advisor takes the reins at EPA Region 6

Greenberg