1. Introduction
When evaluating the performance of a missile system, its guidance and control system is the main factor to be considered. Traditional guidance laws, such as proportional navigation guidance law, generate acceleration commands under the given target-missile kinematics. The commands are followed by an autopilot system that generates actuator commands to achieve the desired acceleration. In general, guidance and control are designed separately without considering interactions between the two systems. Although a separated design principle has proven to be reliable and effective over the decades, the method shows degradation in combined response compared to the separated conditions. This is because traditional guidance laws cannot guarantee optimal characteristics under autopilot lags and dynamic constraints. In [
1], for example, the circular navigation guidance law that theoretically promises zero miss-distance was proposed. Although their method showed a robust performance even under uncertain autopilot models, its performance has been inevitably constrained by autopilot and dynamic response.
Integrated guidance and control (IGC) design concept has been considered as an alternative to solve the afore-mentioned issues. An IGC design is a single system that performs the role of both guidance and control, generating fin commands based on missile and target states. Since control commands are designed considering the interaction between guidance and control loop, IGC has the potential to enhance missile performance. Additionally, it is helpful to reduce the number of iterations and costs for the entire design process. To perform IGC, various control techniques are introduced. From classical control techniques to various nonlinear control techniques, including feedback linearization [
2], sliding mode control [
3,
4], backstepping control [
5], dynamic surface technique [
6,
7,
8], and optimization-based methods [
9] have been applied to solve the problem. Shima et al. [
3] proposed a sliding mode control (SMC)-based IGC, enhancing the robustness of overall systems. Shtessel and Tournes [
4] extended the research to a higher-order SMC to attenuate the chattering problem for dual control missiles. In [
6], Hou and Duan utilized dynamic surface control (DSC) for the IGC problem under unmatched uncertainties. The DSC-based IGC technique is expanded by Liang et al. [
7], with the additional consideration of input saturation. Kim et al. [
9] proposed an explicit solution of finite time-varying state feedback control with a feedforward term.
Of many optimization-based methods, the Model Predictive Control (MPC) technique has been thought to be a powerful solution for IGC [
10,
11,
12,
13,
14,
15]. Compared to the other control techniques utilized in the previous research [
2,
3,
4,
5,
6,
7,
8,
9], online MPC provides the optimal solution within certain state constraints. MPC produces a control input that minimizes the objective function specified on the receding prediction horizon. The technique repeatedly solves the finite horizon open-loop optimal control problem and implements it in the form of closed-loop control. It can be applied not only to linear time-invariant systems, but also to multivariable, time-varying nonlinear systems [
16]. In general, the optimal control problem is a quadratic programming (QP) problem. As it is an explicit solution to the time-varying state feedback form, it is easy to be adopted on-board. Additionally, MPC has the advantage of being able to set state and output constraints. Especially for the missile terminal guidance phase, the acceleration limit and seeker field-of-view (FOV) are crucial constraints caused by the finite maneuver capacity and seeker’s image plane. It is essential to consider these limits as inequality constraints in the optimization problem. Considering the advantages, MPC is a suitable control technique for terminal homing guidance.
Despite its outstanding performance, applications of online MPC have been limited to slow dynamic systems because of computational bottlenecks. The issue is mainly caused by the optimization process that requires excessive computational capacities. To ease the problem, various studies on optimization algorithms and acceleration methods are conducted. In particular, convex optimization algorithms have been considered as a conductive solution for their computational efficiency and parallelizable characteristic. Gradient-based convex optimization techniques, such as the alternating direction method of multipliers (ADMM) [
17,
18,
19], primal-dual interior point method (PD-IPM), parallel quadratic programming (PQP) [
20,
21], and active set method (ASM) [
22,
23,
24], are employed. In this work, PD-IPM [
24,
25], which is the most commonly used technique for convex optimization, is applied. PD-IPM is developed using the Newton direction of the optimality conditions for the logarithmic barrier problem. The method simultaneously updates primal and dual variables by setting a residual function. Compared to ASM and PQP, PD-IPM requires a smaller number of iterations to reach the desired convergence level [
26,
27]. Additionally, the PD-IPM technique satisfies strict interior point feasibility by adopting a backtracking line search. This eases the constraint that the initial point must be feasible.
However, despite the high efficiency of PD-IPM, MPC for the IGC problem needs further improvements for real-time implementation. As the dynamics of the missile and target show fast responses, the update rate-of-control command should be large enough for stability and to yield a smaller miss distance [
28]. Furthermore, the large size of the prediction horizon is required for precise interception performance. Consequentially, the optimization process in MPC for IGC demands frequent operations of multiplication and inversion for large-sized matrix. For this reason, we adopt the parallel design for real-time GPU implementation. Research on accelerating the PD-IPM is conducted, as shown in
Table 1. Even though there is limited research [
29,
30,
31,
32,
33,
34,
35,
36] that deals with the real-time problem of PD-IPM, it focuses on the acceleration of the linear equation solver part of PD-IPM. However, except for the linear equation solver part, we found that the KKT condition construction part also requires considerable computation time. Moreover, there is no related work that applies the PD-IPM to IGC systems.
In this paper, we propose a GPU-accelerated PD-IPM method, which is conducted in MPC for real-time IGC systems, which parallelizes the KKT condition construction part to reduce the computation time of the PD-IPM. A series of complex matrix operations are performed on the KKT condition construction. The proposed method transforms these complex matrix operations into easier forms in the context of parallelization. Then, the transformed matrices are reformed to sparse matrices. Finally, parallelization is conducted with the sparse matrices through both built-in and customized CUDA kernels. The contributions of this paper are as follows.
This is the first approach to accelerate missile MPC on GPU.
The problem of considerable computation time in the KKT condition construction part of PD-IPM is firstly addressed and analyzed.
A new parallelization method is developed for the KKT condition construction part of PD-IPM.
The computation time for PD-IPM is significantly reduced, even considering the overhead time for the CUDA (Compute Unified Device Architecture) initialization on a widely-used embedded system.
The remainder of this paper is organized as follows.
Section 2 describes the optimization problem of the IGC system and MPC with PD-IPM to solve it. Additionally, the real-time problem of PD-IPM is addressed. In
Section 3, after computation times for PD-IPM are profiled in a block-wise manner, a new parallelization method for the KKT condition construction part of PD-IPM is proposed. In
Section 4, the evaluation results of the proposed method are shown and quantitatively compared with other methods on a widely-used embedded system. Finally,
Section 5 presents the conclusions.
2. Problem Description
For the IGC problem, we considered missile terminal homing phase geometry in a two-dimensional plane.
Figure 1a depicts planar homing engagement geometry, where the subscripts
and
denote the missile and target. Reference coordinate system X–Z is centered at the missile’s center of gravity; initial target position
and deviated target position
are defined on the reference coordinate. Initial line-of-sight (LOS) angle
, LOS angle displacement from initial LOS frame
, and range-to-go
are also represented. Missile acceleration, velocity, and flight-path angle are denoted by
, respectively. Relative displacement
is defined as a normal distance between the target position and initial LOS. In
Figure 1b, the seeker look angle
, angle of attack
, and body-fixed coordinate system
are denoted. Reference coordinate frame X
L0–Z
L0 is the initial LOS frame whose origin is also located at the missile’s center of gravity. It is assumed that, in the terminal homing phase, the distance between the missile and target is small enough so that linearization can be performed on the initial LOS frame. Additionally, missile velocity is assumed to be constant.
The main objective of terminal homing is actuating the missile to intercept the target under the finite maneuver capacity and seeker look-angle limit. In addition, based on the previous study [
13], the look-angle rate is limited in bound to prevent image distortion and signal intensity reduction problems. With the acceleration limit
, look-angle limit
, and look-angle rate limit
, the constraints can be expressed as follows:
2.1. Augmented Model for Integrated Guidance and Control
Considering the missile short-period dynamics, kinematics, and actuator dynamics, augmented continuous equations for IGC are given by [
9,
12,
13]
where
is pitch rate,
is control fin command, and
is actuator response.
and
are aerodynamic dimensional derivatives. The actuator dynamics are modeled as a 1st-order lag system with time constant
.
Equation (2) is characterized by its input and state variables. Compared to conventional guidance and control design, augmented equations for IGC simultaneously consider target-missile kinematics and dynamics. For simplicity, state variable vector and input are represented as
System and input matrices are denoted as
. Equation (2) is discretized with sampling interval
.
System and input matrices of the discretized equation are ,, respectively. The notation represents sampling time step.
As mentioned above, control input
should be generated within the extent that it does not violate the restrictions. Inequality constraints defined in Equation (1) are linearized and expressed in matrix form [
12,
13]. As shown below, linearized matrix
is time-varying.
is range-to-go in
th time step.
Overall, the optimization problem for MPC-based IGC is formulated as follows:
where
is the size of finite horizon, and
and
are weightings for state variable and input. As all the equations of dynamics and constraints in Equations (3) and (4) are linear, a basic linear MPC framework is adopted. Equation (4) can be transferred into a single-term quadratic problem by introducing new variable
.
2.2. The Problem of PD-IPM for Real-Time MPC
MPC relies on the real-time solution of a convex optimization problem. The optimization problem in Equation (8) should be defined and computed at every calculation time. To solve the problem, the Primal-Dual Interior Point Method (PD-IPM) was applied, as shown in Algorithm 1. According to the expressions defined in Equations (7) and (8), Algorithm 1 presents a flow chart of PD-IPM for solving a given optimization problem. PD-IPM is one of the most famous algorithm to solve the convex optimization problem. The algorithm alleviates inequality constraints using barrier function and minimize residuals based on perturbed KKT conditions. It attains Newton step computations in every iteration.
In the whole IGC process, the PD-IPM in MPC requires the most computation time because the PD-IPM repeatedly conducts matrix operations until the cost converges. Therefore, the PD-IPM needs to be highly accelerated to be applied to real-time MPC in the IGC process.
Algorithm 1. Primal-Dual Interior Point Method [24]. |
Choose K: maximum iteration, |
while
or
|
| // Find update direction by solving Newton Step |
| |
| |
| |
|
// Backtracking Line Search to find θ |
| |
| | |
| | |
| | |
| // Primal-Dual Update |
| |