# Reaction Control System Optimization for Maneuverable Reentry Vehicles Based on Particle Swarm Optimization

Authors: Hang Gui; Ruisheng Sun; Wei Chen; Bin Zhu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

This paper presents a new parametric optimization design to solve a class of reaction control system (RCS) problem with discrete switching state, flexible working time, and finite-energy control for maneuverable reentry vehicles. Based on basic particle swarm optimization (PSO) method, an exponentially decreasing inertia weight function is introduced to improve convergence performance of the PSO algorithm. Considering the PSO algorithm spends long calculation time, a suboptimal control and guidance scheme is developed for online practical design. By tuning the control parameters, we try to acquire efficacy as close as possible to that of the PSO-based solution which provides a reference. Finally, comparative simulations are conducted to verify the proposed optimization approach. The results indicate that the proposed optimization and control algorithm has good performance for such RCS of maneuverable reentry vehicles.


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## Body

## 1. Introduction

Nowadays, hypersonic maneuverable reentry vehicles have received considerable attention from various countries, owing to its promising application in civilian and military aspects [1]. In the military, high mobility and large-scale hypersonic vehicles are characterized by their ability to cope with future near-space operations, break through ballistic defence systems, and achieve rapid global precision strikes. Usually, the initial reentry flight is with high altitude and thin atmosphere density, which results in low aerodynamic control efficiency. Reaction control system (RCS) is a kind of spacecraft system that uses thrusters arranged around the side direction of aircraft to provide attitude control torque [2], and sometimes translation force. Consequently, RCS is a better solution scheme to provide efficient control performance. Currently, a lot of research studies on RCS are focused on RCS thruster configuration [2], RCS nozzle selection logic [3, 4], and RCS combined with pneumatic rudder surface control [5, 6]. Liu provides a new method to obtain necessary and sufficient conditions for impulse controllability in continuous linear dynamics, which performs discrete-time actions for spacecraft maneuvering [7]. However, due to the discontinuity and limited working time of RCS, it is different from conventional continuous servo control.At present, mathematical optimal control is divided into two categories, the direct method [8, 9] and indirect method [10, 11]. The direct method uses the original discrete optimal control problem as the parameter optimization problem and then finds the optimal solution through nonlinear programming [12, 13]. Generally, the direct method is more popular than the indirect method because the direct analytical solution of the complex nonlinear system is easier to solve [14, 15]. HP adaptive pseudospectral method is a combination of Legendre pseudospectral method and HP adaptive method. The HP adaptive pseudospectral method discretizes state variables and control variables into a series of Legendre–Gauss–Lombarto (LGL) points, which is one of the most popular and effective direct methods at present [16, 17]. Due to the nondifferentiability of RCS variables and flexible time interval, the application of pseudospectral method in RCS is limited, and the Karush–Kuhn–Tucker (KKT) condition is not satisfied.In order to effectively solve such problems, some intelligent algorithms such as PSO algorithm, genetic algorithm (GA), and neural network algorithm are becoming more and more attractive due to their high speed and high precision. These intelligent algorithms are inspired by natural phenomena, such as the association between ants, birds, and even human social behaviour [18, 19]. PSO was proposed by Eberhart and Kennedy [20] in 1995 and later promoted by other researchers [21]. Due to its advantages of simple theory, easy programming, and high computational efficiency, PSO has achieved excellent results in solving discontinuous problems. For example, Pires et al. proposed a new method to control the convergence rate of PSO algorithm in 2010 by using the concept of fractional calculus [22]. Unlike traditional optimization methods, PSO algorithm does not depend on the strict mathematical characteristics (derivability, continuity) and constraints of the optimization problem. In 2010, Yang et al. proposed a new method for solving missile fuel impulse optimal control problem using improved PSO technology, but this method did not consider the optimal operating mode and elastic interval [23]. Rahimi et al. proposed a novel PSO method to solve the optimal control problem of spacecraft reentry orbit, which avoided the computational burden of common analytical methods [24]. Zhou et al. using the particle swarm algorithm for the design of hypersonic vehicle guidance [25].Therefore, a new method of solving RCS problem of maneuverable reentry vehicles with discrete and flexible time intervals by using improved particle swarm optimization (PSO) method is proposed in this paper. This discrete and flexible time optimization problem cannot be solved by HP adaptive pseudospectral method. The rest of this paper is organized as follows. In Section2, the mathematical model of RCS control for maneuverable reentry vehicle is deduced, and the optimal RCS problem is given. Section 3 details the modifications based on basic PSO and RCS design. Aiming at the optimal setting of RCS control for maneuverable reentry vehicles, the structure and parameter design of the controller are discussed. On this basis, a suboptimal control guidance law based on PSO is proposed, which can be used in real-time designs. In Section 4, comparative simulations are conducted to verify the proposed optimization approach. The results indicate that the proposed optimization and control algorithm has good performance for such RCS of maneuverable reentry vehicles. Finally, Section 5 gives some conclusions of this paper.


## 2. Problem Description

### 2.1. Dynamics of the Vehicle System

The mathematical model of the maneuverable reentry vehicle includes the body dynamic model, the RCS thrust model, and the aerodynamic model, as well as the path constraint conditions during the reentry process of the vehicle [26, 27]. In order to facilitate discussion, only the motion of maneuverable reentry vehicle in longitudinal plane is considered, and the earth is assumed to be flat and stationary. Thus the dynamic equations are(1)dVdt=Pcosα−D−mgsinθ+uRCS⋅FRCSsinαm,dθdt=Psinα+L−mgcosθ−uRCS⋅FRCScosαmV,dxdt=Vcosθ,dydt=Vsinθ,dωzdt=uRCS⋅FRCS⋅Xcg+MzJz,dϑdt=ωz,dmdt=−mc,dIRCSdt=uRCS⋅FRCS,α=ϑ−θ,where the state variables include flight velocity V, the horizontal position x, trajectory inclination angle θ, height position y, pitch angular ratio ωz, pitch angle ϑ, gravity acceleration g, mass of the reentry vehicle m, and RCS thrust total impulse IRCS. Engine thrust P, RCS thrust FRCS, RCS thrust to centroid distance Xcg. uRCS is the switching quantity of the RCS nozzle, which has three states: 1, 0, −1. Jz is the moment of inertia of the z-axis. mc is mass flow. α means the attack of angle (Figure 1). The lift force L, drag force D, and aerodynamic pitch moment Mz are all the functions of reference area S, dynamic pressure q, and reference length l. The lift, drag, and pitch moment coefficients CL, CD, and mz are shown in equation (2).Figure 1
Motion coordinate systems of the maneuverable reentry vehicle. This is the basic shape of the maneuverable reentry, with the tail of the main engine and the RCS nozzle.(2)
M
a
=
V
Sonic
,
L
=
q
S
C
L
α
,
M
a
,
D
=
q
S
C
D
α
,
M
a
,
M
z
=
q
S
l
m
z
α
,
M
a
.The Mach numberMa here is a function of velocity V and local sonic speed.The RCS can rapidly provide thrust force and control torque to maneuverable reentry vehicle by a nozzle switch. Since the nozzle switch between ON and OFF is very fast, the switching time delay here can be almost ignored. Thus, the RCS thrust model is(3)Fi=FRCS,U=1,0,U=0,where Fi is the thrust force generated by the nozzle i. U is switch instruction of the nozzle, 0 means OFF, 1 means ON. RCS nozzles are generally symmetrically arranged, and the resulting control torque is shown in Figure 2.Figure 2
Schematic of RCS control torque. The horizontal axis is the working time, and the vertical axis is the control torque generated by the RCS.


### 2.2. Constraints during Flight Process

#### 2.2.1. Path Constraints

During the reentry phase, to ensure the structural safety of the maneuverable reentry vehicle, the constraints of dynamic pressure and angle of attack must be strictly satisfied, and the reentry trajectory should be as smooth as possible [28].(a)
Dynamic pressure: dynamic pressure is an important characteristic variable in aerodynamics. The dynamic pressure at the reentry stage strictly meets the following constraints:(4)q=ρV22<qmax.(b)
Angle of attack: the angle of attack is an important state variable in flight dynamics, which determines the heat rate and overload experienced by the reentry vehicle. The angle of attack during the reentry phase strictly meets the following constraints:(5)αmin<α<αmax.


#### 2.2.2. Terminal Constraint

In order to ensure the terminal guidance, the terminal guidance status during the reentry phase must satisfy the terminal constraints [28]. The error of the trajectory inclination angle θ inclination should be less than a certain value, i.e.(6)θmin<θ<θmax.


#### 2.2.3. Control Constraints

Due to the fuel limitation and nozzle switch delay of RCS, we consider total working time and minimum working time limitation [29], i.e.(7)∑i=1nTi≤TTotal_Max,Ti≥TMin.


### 2.3. Objective of the System Design

For maneuverable reentry vehicles, the accuracy of control and the fuel consumption of RCS are a contradictory relationship. If high control accuracy is required, the nozzle must be opened frequently to consume fuel. Considering that the RCS fuel of the maneuverable reentry vehicle is limited, the purpose of optimizing the RCS thrust control is to ensure the control accuracy while reducing the amount of fuel consumed. Therefore, the objective functionJ for minimizing control energy and control errors is written as(8)minJ=λ∫t0tfuFRCSdtIRCS+∫t0tfet2dt,where λ is the weight coefficient, u means the switching quantity of the RCS nozzle, and the control error et=x1−xc=θ−θc. Here t0 and tf, respectively, represent the initial and end times.Obviously, the above design is optimization problem with various constraints, so how to transform a constrained optimization problem to an unconstrained optimization problem will make the design easier. Penalty function method is a good way to do this [26]. In order to enhance the computational efficiency of trajectory optimization, penalty function P(t) can be introduced to rewrite the objective function [30, 31]. When the constraints in Section 2.2 are not satisfied, P(t) will take a very large value to penalize the objective function. So equation (8) can be modified as follows:(9)minJ=λ∫t0tfuFRCSdtIRCS+∫t0tfet2dt+μPt,where μ is the penalty factor which is positive.


## 2.1. Dynamics of the Vehicle System

The mathematical model of the maneuverable reentry vehicle includes the body dynamic model, the RCS thrust model, and the aerodynamic model, as well as the path constraint conditions during the reentry process of the vehicle [26, 27]. In order to facilitate discussion, only the motion of maneuverable reentry vehicle in longitudinal plane is considered, and the earth is assumed to be flat and stationary. Thus the dynamic equations are(1)dVdt=Pcosα−D−mgsinθ+uRCS⋅FRCSsinαm,dθdt=Psinα+L−mgcosθ−uRCS⋅FRCScosαmV,dxdt=Vcosθ,dydt=Vsinθ,dωzdt=uRCS⋅FRCS⋅Xcg+MzJz,dϑdt=ωz,dmdt=−mc,dIRCSdt=uRCS⋅FRCS,α=ϑ−θ,where the state variables include flight velocity V, the horizontal position x, trajectory inclination angle θ, height position y, pitch angular ratio ωz, pitch angle ϑ, gravity acceleration g, mass of the reentry vehicle m, and RCS thrust total impulse IRCS. Engine thrust P, RCS thrust FRCS, RCS thrust to centroid distance Xcg. uRCS is the switching quantity of the RCS nozzle, which has three states: 1, 0, −1. Jz is the moment of inertia of the z-axis. mc is mass flow. α means the attack of angle (Figure 1). The lift force L, drag force D, and aerodynamic pitch moment Mz are all the functions of reference area S, dynamic pressure q, and reference length l. The lift, drag, and pitch moment coefficients CL, CD, and mz are shown in equation (2).Figure 1
Motion coordinate systems of the maneuverable reentry vehicle. This is the basic shape of the maneuverable reentry, with the tail of the main engine and the RCS nozzle.(2)
M
a
=
V
Sonic
,
L
=
q
S
C
L
α
,
M
a
,
D
=
q
S
C
D
α
,
M
a
,
M
z
=
q
S
l
m
z
α
,
M
a
.The Mach numberMa here is a function of velocity V and local sonic speed.The RCS can rapidly provide thrust force and control torque to maneuverable reentry vehicle by a nozzle switch. Since the nozzle switch between ON and OFF is very fast, the switching time delay here can be almost ignored. Thus, the RCS thrust model is(3)Fi=FRCS,U=1,0,U=0,where Fi is the thrust force generated by the nozzle i. U is switch instruction of the nozzle, 0 means OFF, 1 means ON. RCS nozzles are generally symmetrically arranged, and the resulting control torque is shown in Figure 2.Figure 2
Schematic of RCS control torque. The horizontal axis is the working time, and the vertical axis is the control torque generated by the RCS.


## 2.2. Constraints during Flight Process

### 2.2.1. Path Constraints

During the reentry phase, to ensure the structural safety of the maneuverable reentry vehicle, the constraints of dynamic pressure and angle of attack must be strictly satisfied, and the reentry trajectory should be as smooth as possible [28].(a)
Dynamic pressure: dynamic pressure is an important characteristic variable in aerodynamics. The dynamic pressure at the reentry stage strictly meets the following constraints:(4)q=ρV22<qmax.(b)
Angle of attack: the angle of attack is an important state variable in flight dynamics, which determines the heat rate and overload experienced by the reentry vehicle. The angle of attack during the reentry phase strictly meets the following constraints:(5)αmin<α<αmax.


### 2.2.2. Terminal Constraint

In order to ensure the terminal guidance, the terminal guidance status during the reentry phase must satisfy the terminal constraints [28]. The error of the trajectory inclination angle θ inclination should be less than a certain value, i.e.(6)θmin<θ<θmax.


### 2.2.3. Control Constraints

Due to the fuel limitation and nozzle switch delay of RCS, we consider total working time and minimum working time limitation [29], i.e.(7)∑i=1nTi≤TTotal_Max,Ti≥TMin.


## 2.2.1. Path Constraints

During the reentry phase, to ensure the structural safety of the maneuverable reentry vehicle, the constraints of dynamic pressure and angle of attack must be strictly satisfied, and the reentry trajectory should be as smooth as possible [28].(a)
Dynamic pressure: dynamic pressure is an important characteristic variable in aerodynamics. The dynamic pressure at the reentry stage strictly meets the following constraints:(4)q=ρV22<qmax.(b)
Angle of attack: the angle of attack is an important state variable in flight dynamics, which determines the heat rate and overload experienced by the reentry vehicle. The angle of attack during the reentry phase strictly meets the following constraints:(5)αmin<α<αmax.


## 2.2.2. Terminal Constraint

In order to ensure the terminal guidance, the terminal guidance status during the reentry phase must satisfy the terminal constraints [28]. The error of the trajectory inclination angle θ inclination should be less than a certain value, i.e.(6)θmin<θ<θmax.


## 2.2.3. Control Constraints

Due to the fuel limitation and nozzle switch delay of RCS, we consider total working time and minimum working time limitation [29], i.e.(7)∑i=1nTi≤TTotal_Max,Ti≥TMin.


## 2.3. Objective of the System Design

For maneuverable reentry vehicles, the accuracy of control and the fuel consumption of RCS are a contradictory relationship. If high control accuracy is required, the nozzle must be opened frequently to consume fuel. Considering that the RCS fuel of the maneuverable reentry vehicle is limited, the purpose of optimizing the RCS thrust control is to ensure the control accuracy while reducing the amount of fuel consumed. Therefore, the objective functionJ for minimizing control energy and control errors is written as(8)minJ=λ∫t0tfuFRCSdtIRCS+∫t0tfet2dt,where λ is the weight coefficient, u means the switching quantity of the RCS nozzle, and the control error et=x1−xc=θ−θc. Here t0 and tf, respectively, represent the initial and end times.Obviously, the above design is optimization problem with various constraints, so how to transform a constrained optimization problem to an unconstrained optimization problem will make the design easier. Penalty function method is a good way to do this [26]. In order to enhance the computational efficiency of trajectory optimization, penalty function P(t) can be introduced to rewrite the objective function [30, 31]. When the constraints in Section 2.2 are not satisfied, P(t) will take a very large value to penalize the objective function. So equation (8) can be modified as follows:(9)minJ=λ∫t0tfuFRCSdtIRCS+∫t0tfet2dt+μPt,where μ is the penalty factor which is positive.


## 3. Optimization and Control Design

### 3.1. Optimal Design Based on Improved PSO

PSO algorithm is one of popular optimization techniques, which was originally developed to graphically simulate the bird flock. The birds form groups and move in a flock, and each bird is a particle. Each particle is assumed to have two characteristics: position and velocity. Each particle wanders around in the design space and remembers the best position (in terms of objective function value) it has discovered. The particles communicate information or good positions to each other and adjust their individual positions and velocities based on the information received on the good positions [32]. Several variants of the PSO technique have been proposed so far, following Eberhart and Kennedy [33, 34].Here we define the notation: assuming the search space isd-dimensional, the i-th particle of the group is represented by a d-dimensional vector xi=xi1,xi2,…,xid, and the experienced best position is expressed as pi=pi1,pi2,…,pid. The particle the best position that has been experienced is gi=gn,gi2,…,gid and the position change (velocity) of the i-th particle is vi=vi1,vi2,…,vi3 [21]. During the iteration, particle i updates its position and velocity according to the following formulas:(10)vidk+1=w⋅vidk+c1⋅rand⋅pidk−xidk+c2⋅rand⋅gidk−xidk,(11)xidk+1=xidk+vidk+1,where w is the inertia weight, c1, c2 are the acceleration factors, k is the iteration counter, and rand() is a random number between (0, 1). In addition, the velocity of each dimension of the particle is limited by the maximum velocity vmax. When the value of vmax is large, the particle can fly fast, which is conducive to global search, but it is possible to fly over the optimal solution. When vmax is small, the particle can be searched in the feature area, but it is easy to fall into the local optimal solution.The inertia weightw was originally introduced by Eberhart and Shi [35]. When using PSO found that usually the particle velocities build up too fast and the minimum of the objective function is easily skipped [32]. Hence, an inertia weight w is added to reduce the velocity. Generally, the value of w is assumed to be a constant. A large value of w promotes global exploration and a smaller value promoted a local search. Therefore, to achieve a balance between global and local exploration to speed up convergence to the true optimum, we design the inertia weight as an exponentially decreasing function. The exponentially changing inertia weight coefficient w can achieve better global optimization capabilities in the initial stage of optimization and better local optimization capabilities in the later stages of optimization, i.e.,(12)w=w0−w∞e−k+w∞.Simultaneously, RCS is a class of discontinuous actuators with limitation of fuel consumption. Consequently, we select the switch stateui and working time Ti as design variables, where ui is discrete, and Ti is continuous. Since the movement of the particle position is a continuous process, the particle position vector is corresponded to the design variable by a sign function.Therefore, we define the structure of the particle’s position vectorxi and sign function signx. u1,u2,…,uid/2 and T1,T2,…,Tid/2 form a complete RCS nozzle assignment.(13)signx=+1x>0.5,0−0.5<x<0.5,−1x<−0.5,ui1Ti1ui2Ti2⋮uid/2Tid/2=signxi1xi2signxi3xi4⋮signxid−1xid=xi.The upper and lower bounds of the RCS nozzle switch stateui and working time Ti are defined as(14)umax=+1,umin=−1,Tmax=5s,Tmin=0.1s.


### 3.2. Improved PSO Algorithm Process

Combined with the algorithm described in Section3.1, the entire calculation process is depicted in Figure 3. The specific steps are as follows:(1)
The initial trajectory parameters include the initial velocity, position, and attitude of the maneuvering reentry vehicle. PSO initialization includes setting group size, neighborhood size, iteration times, acceleration coefficient, inertia weight, random group position, and speed.(2)
The inertia weight coefficientw and the particle velocity vidk+1 and position xidk+1 information are updated according to equations (10)–(12).(3)
Getui and Ti as control variables based on the updated particle position.(4)
Input control variablesui and Ti into trajectory program for calculation. Since the RCS nozzle switch status is only (1, 0, −1), so ui = sign(ui).(5)
The value of objective functionJ was calculated to update the individual optimal fitness and population optimal fitness.(6)
Determine whether the new particles generated by iteration reach the minimum objective value or whether the number of iterations reaches the preset maximum value. If one of these two conditions is satisfied, the optimal solution of the problem corresponds to the global optimal solution of the particle and stops iteration. Otherwise, go to step (2) to update the position and velocity of particles in the next iteration.Figure 3
Process of propose PSO algorithm. After the optimization program starts, the particle is initialized first and then iteratively calculated according to the algorithm until the program is terminated after the stop condition is satisfied.


### 3.3. Suboptimal Online Controller Design

Through the particle swarm optimization method, the optimal solution of the maneuverable reentry vehicle is obtained, which minimizes the RCS consumption and control error. Obviously, the particle swarm algorithm spent long time to converge, which cannot satisfy real-time control requirements [14]. Therefore, it is very important to find an online control scheme to obtain a suboptimal control design. By tuning the suboptimal control parameters, we here try to acquire efficacy as close as possible to that of the PSO-based solution. Figure 4 shows the guidance and control process of maneuverable reentry vehicle.Figure 4
Guidance and control structure of maneuverable reentry vehicle. A guidance command is generated according to the guidance law by the current location information and the location information of the target. The control system controls the nozzle according to the guidance command.For the optimal flight control problem consideration of equation (1) and Figure 4, the mathematical model based on disturbance linearization motion can be used as a reference model for the controlled object [14].(15)x˙=Ax+Bu,where x=x1x2x3=θθ˙θ¨, A=0100010−1/TM2−2ξM/TM, B=00KM. Here KM, ξM, and TM, respectively, represent the transfer coefficient, damping, and time constant of the vehicle.It can be seen from the schematic diagram of the guidance control system of the maneuverable reentry vehicle shown in Figure4. The detailed control structure is shown in Figure 5, where f(e(t)) is the RCS designed with reference model. The work of this paper is mainly divided into two steps: (1) In the previous section, the PSO numerical optimization method was used to obtain the optimal RCS numerical solution; (2) in this section, the suboptimal online controller will be determined by establishing the mathematical relationship between the error e(t) and the control uRCS.Figure 5
Structure of RCS.Pulse width pulse frequency (PWPF) modulator is a popular technique used in RCS nozzle control [36, 37], which is designed by combining a first-order filter and a Schmitt trigger in the feedforward loop (Figure 5) [5, 38]. PWPF modulator is widely used in spacecraft control systems [6] because it works with an almost linear input/output relationship which makes the design easier. The first-order filter has the transfer function f(s):(16)fs=KTs+1.The Schmitt trigger [5, 39] is a switching relay with hysteresis and dead zone, as shown in Figure 5. The mathematical description of the Schmitt trigger is formulated as [37, 39](17)u=+1,e≥Uon,+1,Uoff≤e<Uon,e˙<0,0,−Uon≤e<Uoff,e˙<0,0,−Uoff≤e<Uon,e˙>0,−1,−Uon≤e−Uoff,e˙>0,−1,e<−Uon.Subject to the following restrictions:(1)
Due to the fuel limitations carried by the RCS system, the total working time of the nozzle is shown in equation (7)(2)
According to the discrete characteristics of the RCS nozzle, the generated control torque is as follows:(18)Mc=MRCS,u=1,0,u=0,−MRCS,u=−1,where Mc is the control torque generated by the RCS nozzle.By selecting suitable parametersK, T, Uon, and Uoff, the online control effect is as close as possible to the trajectory optimization result of PSO [14]. Similarly, the online controller parameters can also be optimized by using the PSO method.


### 3.4. Guidance Law Design

Guidance can be defined as a method of guiding and controlling the flight of a vehicle toward a target. We need to design an appropriate guidance law to enable the maneuvering reentry vehicle along the predetermined trajectory, that is, to express guidance law in terms of the relative relationship between the vehicle and the target. Proportional navigation (PN) is one of the most popular guidance methods [40, 41]. Here, considering the influence of gravity, a modified PN law is expressed as(19)θ˙c=K1q˙+K2gVcosθ,where θ˙c is guidance command, and K1, K2 are guidance gains.According to equation (19), the guidance law can be written in an integral formation:(20)θct−θ0t0=K1qt−qt0+K2∫totgVcosθdt.


## 3.1. Optimal Design Based on Improved PSO

PSO algorithm is one of popular optimization techniques, which was originally developed to graphically simulate the bird flock. The birds form groups and move in a flock, and each bird is a particle. Each particle is assumed to have two characteristics: position and velocity. Each particle wanders around in the design space and remembers the best position (in terms of objective function value) it has discovered. The particles communicate information or good positions to each other and adjust their individual positions and velocities based on the information received on the good positions [32]. Several variants of the PSO technique have been proposed so far, following Eberhart and Kennedy [33, 34].Here we define the notation: assuming the search space isd-dimensional, the i-th particle of the group is represented by a d-dimensional vector xi=xi1,xi2,…,xid, and the experienced best position is expressed as pi=pi1,pi2,…,pid. The particle the best position that has been experienced is gi=gn,gi2,…,gid and the position change (velocity) of the i-th particle is vi=vi1,vi2,…,vi3 [21]. During the iteration, particle i updates its position and velocity according to the following formulas:(10)vidk+1=w⋅vidk+c1⋅rand⋅pidk−xidk+c2⋅rand⋅gidk−xidk,(11)xidk+1=xidk+vidk+1,where w is the inertia weight, c1, c2 are the acceleration factors, k is the iteration counter, and rand() is a random number between (0, 1). In addition, the velocity of each dimension of the particle is limited by the maximum velocity vmax. When the value of vmax is large, the particle can fly fast, which is conducive to global search, but it is possible to fly over the optimal solution. When vmax is small, the particle can be searched in the feature area, but it is easy to fall into the local optimal solution.The inertia weightw was originally introduced by Eberhart and Shi [35]. When using PSO found that usually the particle velocities build up too fast and the minimum of the objective function is easily skipped [32]. Hence, an inertia weight w is added to reduce the velocity. Generally, the value of w is assumed to be a constant. A large value of w promotes global exploration and a smaller value promoted a local search. Therefore, to achieve a balance between global and local exploration to speed up convergence to the true optimum, we design the inertia weight as an exponentially decreasing function. The exponentially changing inertia weight coefficient w can achieve better global optimization capabilities in the initial stage of optimization and better local optimization capabilities in the later stages of optimization, i.e.,(12)w=w0−w∞e−k+w∞.Simultaneously, RCS is a class of discontinuous actuators with limitation of fuel consumption. Consequently, we select the switch stateui and working time Ti as design variables, where ui is discrete, and Ti is continuous. Since the movement of the particle position is a continuous process, the particle position vector is corresponded to the design variable by a sign function.Therefore, we define the structure of the particle’s position vectorxi and sign function signx. u1,u2,…,uid/2 and T1,T2,…,Tid/2 form a complete RCS nozzle assignment.(13)signx=+1x>0.5,0−0.5<x<0.5,−1x<−0.5,ui1Ti1ui2Ti2⋮uid/2Tid/2=signxi1xi2signxi3xi4⋮signxid−1xid=xi.The upper and lower bounds of the RCS nozzle switch stateui and working time Ti are defined as(14)umax=+1,umin=−1,Tmax=5s,Tmin=0.1s.


## 3.2. Improved PSO Algorithm Process

Combined with the algorithm described in Section3.1, the entire calculation process is depicted in Figure 3. The specific steps are as follows:(1)
The initial trajectory parameters include the initial velocity, position, and attitude of the maneuvering reentry vehicle. PSO initialization includes setting group size, neighborhood size, iteration times, acceleration coefficient, inertia weight, random group position, and speed.(2)
The inertia weight coefficientw and the particle velocity vidk+1 and position xidk+1 information are updated according to equations (10)–(12).(3)
Getui and Ti as control variables based on the updated particle position.(4)
Input control variablesui and Ti into trajectory program for calculation. Since the RCS nozzle switch status is only (1, 0, −1), so ui = sign(ui).(5)
The value of objective functionJ was calculated to update the individual optimal fitness and population optimal fitness.(6)
Determine whether the new particles generated by iteration reach the minimum objective value or whether the number of iterations reaches the preset maximum value. If one of these two conditions is satisfied, the optimal solution of the problem corresponds to the global optimal solution of the particle and stops iteration. Otherwise, go to step (2) to update the position and velocity of particles in the next iteration.Figure 3
Process of propose PSO algorithm. After the optimization program starts, the particle is initialized first and then iteratively calculated according to the algorithm until the program is terminated after the stop condition is satisfied.


## 3.3. Suboptimal Online Controller Design

Through the particle swarm optimization method, the optimal solution of the maneuverable reentry vehicle is obtained, which minimizes the RCS consumption and control error. Obviously, the particle swarm algorithm spent long time to converge, which cannot satisfy real-time control requirements [14]. Therefore, it is very important to find an online control scheme to obtain a suboptimal control design. By tuning the suboptimal control parameters, we here try to acquire efficacy as close as possible to that of the PSO-based solution. Figure 4 shows the guidance and control process of maneuverable reentry vehicle.Figure 4
Guidance and control structure of maneuverable reentry vehicle. A guidance command is generated according to the guidance law by the current location information and the location information of the target. The control system controls the nozzle according to the guidance command.For the optimal flight control problem consideration of equation (1) and Figure 4, the mathematical model based on disturbance linearization motion can be used as a reference model for the controlled object [14].(15)x˙=Ax+Bu,where x=x1x2x3=θθ˙θ¨, A=0100010−1/TM2−2ξM/TM, B=00KM. Here KM, ξM, and TM, respectively, represent the transfer coefficient, damping, and time constant of the vehicle.It can be seen from the schematic diagram of the guidance control system of the maneuverable reentry vehicle shown in Figure4. The detailed control structure is shown in Figure 5, where f(e(t)) is the RCS designed with reference model. The work of this paper is mainly divided into two steps: (1) In the previous section, the PSO numerical optimization method was used to obtain the optimal RCS numerical solution; (2) in this section, the suboptimal online controller will be determined by establishing the mathematical relationship between the error e(t) and the control uRCS.Figure 5
Structure of RCS.Pulse width pulse frequency (PWPF) modulator is a popular technique used in RCS nozzle control [36, 37], which is designed by combining a first-order filter and a Schmitt trigger in the feedforward loop (Figure 5) [5, 38]. PWPF modulator is widely used in spacecraft control systems [6] because it works with an almost linear input/output relationship which makes the design easier. The first-order filter has the transfer function f(s):(16)fs=KTs+1.The Schmitt trigger [5, 39] is a switching relay with hysteresis and dead zone, as shown in Figure 5. The mathematical description of the Schmitt trigger is formulated as [37, 39](17)u=+1,e≥Uon,+1,Uoff≤e<Uon,e˙<0,0,−Uon≤e<Uoff,e˙<0,0,−Uoff≤e<Uon,e˙>0,−1,−Uon≤e−Uoff,e˙>0,−1,e<−Uon.Subject to the following restrictions:(1)
Due to the fuel limitations carried by the RCS system, the total working time of the nozzle is shown in equation (7)(2)
According to the discrete characteristics of the RCS nozzle, the generated control torque is as follows:(18)Mc=MRCS,u=1,0,u=0,−MRCS,u=−1,where Mc is the control torque generated by the RCS nozzle.By selecting suitable parametersK, T, Uon, and Uoff, the online control effect is as close as possible to the trajectory optimization result of PSO [14]. Similarly, the online controller parameters can also be optimized by using the PSO method.


## 3.4. Guidance Law Design

Guidance can be defined as a method of guiding and controlling the flight of a vehicle toward a target. We need to design an appropriate guidance law to enable the maneuvering reentry vehicle along the predetermined trajectory, that is, to express guidance law in terms of the relative relationship between the vehicle and the target. Proportional navigation (PN) is one of the most popular guidance methods [40, 41]. Here, considering the influence of gravity, a modified PN law is expressed as(19)θ˙c=K1q˙+K2gVcosθ,where θ˙c is guidance command, and K1, K2 are guidance gains.According to equation (19), the guidance law can be written in an integral formation:(20)θct−θ0t0=K1qt−qt0+K2∫totgVcosθdt.


## 4. Numerical Simulations

In this section, trajectory simulations are presented to verify the efficacy of nonlinear RCS suboptimal control algorithm in detail. The fourth-order Runge–Kutta method is used to integrate the dynamic model. The simulation parameters are as follows: theRCS control force FRCS = 150 N, minimum switching time of nozzle Tmin = 0.1 s, and maximum working time Ttotal_Max = 80 s. Initial state values of the maneuverable reentry vehicle: t0 = 0 s, m0 = 1248.95 kg, v0 = 2500 m/s, x0 = 0 m, y0 = 150000 m, ωz0 = 0 rad/s, = 0 deg, ϑ0=θ0=0deg, l = 2.14 m. Main engine thrust P=40kN, mass flow mc = 10 kg/s, working time 60 s. Target position x = 300 km and the end condition of the trajectory calculation is that the height reaches 70 km reentry height.


### 4.1. Efficacy of Improved PSO Algorithm

Here in order to validate the performance of the improved PSO, comparative simulations are conducted between improved PSO with exponential decreasing inertia weight and basic PSO with constant inertia weight. The initial conditions for the optimization design are set as follows. The control parameters of PSO are particle numberm = 10, dimension n = 100, c1 = c2 = 1.8. The constant inertia weight w=0.8, exponential decreasing inertial weight w=1−0.8e−0.03k+0.8, and the number of iteration reaches 200. The objective function J results are shown in Figure 6.Figure 6
Curve of objective function J with iteration counter.As shown in Figure6, the value of objective function with improved PSO method drops faster than that with basic PSO in the initial stage, which indicates its better global optimization ability. At the later stage of optimization, the value of objective function with improved PSO is smaller, which indicates that it has better local optimization capabilities. It is seen that the objective function value remains at the same value for a long iteration period, which indicates that the particles have reached the global optimum point in the feasible area.


### 4.2. Verification of Suboptimal Algorithm

Controller gains areK = 1, T = 0.05. The parameters of the Schmitt trigger Uon = 1, Uoff = 0.1. Additionally, α is chosen as a state constraint of αlower = −15 deg, αupper = +15 deg. Comparative simulation results between optimal and suboptimal algorithms are shown in Figures 7–13.Figure 7
Trajectory curves under different conditions.Figure 8
Curve ofMa, V and t.Figure 9
Curve ofθ and t.Figure 10
Curve ofα and t.Figure 11
Curve ofϑ and t.Figure 12
Curve ofFRCS and t.Figure 13
Curve ofIRCS and t.From Figures7–13, it can be concluded that(1)
Figures7 and 8 show that the trajectory and velocity curves with the optimal PSO algorithm are similar to those with the suboptimal online algorithm, which indicates that the online design has good performance.(2)
Figures9–11 indicate that the velocity inclination θ of the optimal trajectory is smoother than that of the suboptimal trajectory. From the angle of attack and pitch curve (Figures 10 and 11), it can be seen that the oscillation amplitude of the optimal trajectory is smaller. In addition, there is a high maneuverable with large angle of attack at the initial stage of optimal scheme, and then the angle of attack gradually converges to zero. Because the direction of the velocity is easier to change at lower speed, maneuverable at this time is beneficial to the overall trajectory performance. Also we find that the changing trend of angle of attack with suboptimal online control is similar to that of PSO scheme.(3)
As shown in Figures12 and 13, it can be indicated that the RCS nozzle with suboptimal scheme has very close opening time and total impulse consumption to that of the PSO algorithm, which verifies the effectiveness of the propose algorithm.


## 4.1. Efficacy of Improved PSO Algorithm

Here in order to validate the performance of the improved PSO, comparative simulations are conducted between improved PSO with exponential decreasing inertia weight and basic PSO with constant inertia weight. The initial conditions for the optimization design are set as follows. The control parameters of PSO are particle numberm = 10, dimension n = 100, c1 = c2 = 1.8. The constant inertia weight w=0.8, exponential decreasing inertial weight w=1−0.8e−0.03k+0.8, and the number of iteration reaches 200. The objective function J results are shown in Figure 6.Figure 6
Curve of objective function J with iteration counter.As shown in Figure6, the value of objective function with improved PSO method drops faster than that with basic PSO in the initial stage, which indicates its better global optimization ability. At the later stage of optimization, the value of objective function with improved PSO is smaller, which indicates that it has better local optimization capabilities. It is seen that the objective function value remains at the same value for a long iteration period, which indicates that the particles have reached the global optimum point in the feasible area.


## 4.2. Verification of Suboptimal Algorithm

Controller gains areK = 1, T = 0.05. The parameters of the Schmitt trigger Uon = 1, Uoff = 0.1. Additionally, α is chosen as a state constraint of αlower = −15 deg, αupper = +15 deg. Comparative simulation results between optimal and suboptimal algorithms are shown in Figures 7–13.Figure 7
Trajectory curves under different conditions.Figure 8
Curve ofMa, V and t.Figure 9
Curve ofθ and t.Figure 10
Curve ofα and t.Figure 11
Curve ofϑ and t.Figure 12
Curve ofFRCS and t.Figure 13
Curve ofIRCS and t.From Figures7–13, it can be concluded that(1)
Figures7 and 8 show that the trajectory and velocity curves with the optimal PSO algorithm are similar to those with the suboptimal online algorithm, which indicates that the online design has good performance.(2)
Figures9–11 indicate that the velocity inclination θ of the optimal trajectory is smoother than that of the suboptimal trajectory. From the angle of attack and pitch curve (Figures 10 and 11), it can be seen that the oscillation amplitude of the optimal trajectory is smaller. In addition, there is a high maneuverable with large angle of attack at the initial stage of optimal scheme, and then the angle of attack gradually converges to zero. Because the direction of the velocity is easier to change at lower speed, maneuverable at this time is beneficial to the overall trajectory performance. Also we find that the changing trend of angle of attack with suboptimal online control is similar to that of PSO scheme.(3)
As shown in Figures12 and 13, it can be indicated that the RCS nozzle with suboptimal scheme has very close opening time and total impulse consumption to that of the PSO algorithm, which verifies the effectiveness of the propose algorithm.


## 5. Conclusions

Based on the basic particle swarm optimization (PSO) method, an exponentially decreasing inertia weight function is introduced to improve convergence performance of the PSO algorithm. Considering PSO algorithm spends long calculation time, a suboptimal control and guidance scheme is developed for online practical design. By tuning the control parameters, we try to acquire efficacy as close as possible to that of the PSO-based solution which provides a reference. Finally, comparative simulations are conducted to verify the proposed optimization approach. The results indicate that the proposed optimization and control algorithm has good performance for such RCS of maneuverable reentry vehicles. This paper proposes a new parametric optimization design to solve a class of reaction control system (RCS) problem with discrete switching state, flexible working time, and finite-energy control for maneuverable reentry vehicles. In order to enhance the nonlinear global optimization capacity, an improved PSO algorithm is used to find the optimal setting of the RCS nozzle control variable by selecting the switch state and working time as the RCS nozzle. For practical application requirements, a suboptimal guidance scheme is online designed. The simulation results demonstrate that the online control algorithm has good performance, which is very close to the optimal scheme. The closed-loop system has good performance, satisfy the process constraints and limitation of fuel consumption.Meanwhile, the proposed PSO control method can be extended to the composite control with RCS and pneumatic control for reentry vehicles. The real-time online programming of the PSO algorithm has attracted more and more attention, but the further improvement of the convergence speed of the PSO has yet to be resolved.



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*Source: 6518531-2020-03-12.xml*