Guidance and control of flying vehicle by explicit model predictive controller

Subject Areas : Mechanical Engineering

mohammad mahdi soori ¹ , Hosein Sadati ²

1 -
2 -

Received: 2025-01-03 Accepted : 2025-01-27 Published : 2025-04-07

Keywords: flying vehicle, target, integrated guidance and control, model predictive control, optimization,

Abstract :

The integrated guidance and control (IGC) system offers a significant advantage by leveraging the synergy between guidance and control subsystems to enhance the overall performance of flying vehicles. In the context of air defense missile systems, where speed is critical for intercepting fast-moving targets, this paper introduces a novel approach for designing and implementing an explicit linear model predictive controller (MPC) specifically tailored for three-dimensional scenarios. The proposed controller is developed to minimize the time to collision and miss distance by fully exploiting the interactions within the IGC framework, significantly enhancing the response time and speed, making it suitable for high-speed air defense applications. A key innovation of this work lies in the adoption of the explicit MPC approach, where the optimization problem is solved offline for all potential state vector values. The optimal control commands are formulated as explicit functions of the state variables and stored in memory. During real-time operation, the controller rapidly evaluates these precomputed functions to generate control commands, eliminating the need for computationally expensive online optimizations. This design significantly reduces the computational load, making it particularly suitable for hardware with limited processing capacity. Simulation results validate the superior performance of the proposed explicit MPC compared to conventional PID and LQR controllers. Specifically, the IGC system employing the proposed controller demonstrated a marked reduction in both miss distance and time to collision. These findings underscore the effectiveness and practicality of the explicit MPC in improving the precision, speed, and efficiency of guidance and control for advanced flying vehicles, particularly in air defense missile systems.

References:

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‏]‏‎8‎‏[‏ Jiayi Tian, Neng Xiong, Shifeng Zhang, Huabo Yang, ‎Zhenyu Jiang,Integrated guidance and control for missile ‎with narrow field-of-view strapdown seeker, ISA ‎Transactions, vol. 106,2020, Pages 124-137. https: //doi. ‎org/10. 1016/j. isatra. 2020. 06. 012.‎
‏]‏‎9‎‏[‏ Sinha, Abhinav & Kumar, Shashi & Mukherjee, ‎Dwaipayan, Integrated Guidance and Control For Dual ‎Control Interceptors Under Impact Time Constraint, AIAA ‎scitech Forum Conference, January 2021. doi. org/10. ‎‎2514/6. 2021-1463.‎
‏]‏‎10‎‏[‏ Li Z. , Dong Q. , Zhang X. , Zhang H. , Zhang F. ‎Field-to-View Constrained Integrated Guidance and ‎Control for Hypersonic Homing Missiles Intercepting ‎Supersonic Maneuvering Targets, Aerospace. 2022; 9(11): ‎‎640. https: //doi. org/10. 3390/aerospace9110640.‎
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Full-Text:



Journal of Applied Dynamic Systems and Control,Vol.8, No.1, 2025: 25-43

25

 

Guidance and Control of Flying Vehicle by Explicit Model Predictive Controller

 

Mohammad Mahdi Soori1, Hosein Sadati2*

 

 

1 Department of Mechanical Engineering K. N. Toosi University of Technology Iran. Email: [e-mail]

2* Corresponding Author : Department of Mechanical Engineering K. N. Toosi University of Technology, Iran. Email: [e-mail]

Received: 2025.01.03; Accepted: 2025.01.27

Abstract–The integrated guidance and control (IGC) system offers a significant advantage by leveraging the synergy between guidance and control subsystems to enhance the overall performance of flying vehicles. In the context of air defense missile systems, where speed is critical for intercepting fast-moving targets, this paper introduces a novel approach for designing and implementing an explicit linear model predictive controller (MPC) specifically tailored for three-dimensional scenarios. The proposed controller is developed to minimize the time to collision and miss distance by fully exploiting the interactions within the IGC framework, significantly enhancing the response time and speed, making it suitable for high-speed air defense applications. A key innovation of this work lies in the adoption of the explicit MPC approach, where the optimization problem is solved offline for all potential state vector values. The optimal control commands are formulated as explicit functions of the state variables and stored in memory. During real-time operation, the controller rapidly evaluates these precomputed functions to generate control commands, eliminating the need for computationally expensive online optimizations. This design significantly reduces the computational load, making it particularly suitable for hardware with limited processing capacity. Simulation results validate the superior performance of the proposed explicit MPC compared to conventional PID and LQR controllers. Specifically, the IGC system employing the proposed controller demonstrated a marked reduction in both miss distance and time to collision. These findings underscore the effectiveness and practicality of the explicit MPC in improving the precision, speed, and efficiency of guidance and control for advanced flying vehicles, particularly in air defense missile systems.

 

Keywords: Flying vehicle, target, integrated guidance and control, model predictive control, optimization

 

 

 

 

1. Introduction

 

Guidance, navigation and control functions are critical to all forms of air and space vehicles, including missiles. In practice, these functions work together in series to maneuver a vehicle. It is now common to develop guidance completely separate from control (autopilot) and vice versa. Almost all textbooks and technical articles on this topic have dealt with it [1].

Some more advanced guidance algorithms not only achieve interception, but also control the interception angle of the missile upon impact. However, all these algorithms are rooted in the collision triangle concept, which minimizes the change of line of sight between the interceptor and the target, and may suffer from instability at the end of the task. In a multi-loop structure, steering is generated using engagement

kinematics while the autopilot stabilizes the body dynamics and follows the acceleration provided by the steering.

 

 

1.1        Integrated guidance and control

Unlike the conventional three-loop autopilot structure, Integrated Guidance and Control (IGC) is an integrated framework in which guidance and control are considered to be integrated within rather than independent of each other.

The advantage of IGC is their ability to use interactions between command and control subsystems. IGC intends to increase the performance of the missile by taking advantage of the synergy between the guidance and control processes. Depending on the structure of the IGC, some provide additional feedback paths in the flight control system, while others require less. Putting G&C into a single IGC system improves its optimization potential. Because optimization of parameters can be done directly. Cost functions include key performance parameters such as missile and target relative speed of approach, line-of-sight angle, impact angle, and many parameters not readily available to autopilot are now directly available. In the conventional approach, the guidance law has no knowledge of the amount of spin or acceleration applied to the missile, instead, guidance only knows the relative position and speed of engagement. As the range-target decreases, small changes in geometry result in large acceleration commands that can exceed the performance range of the autopilot. In addition, the autopilot cannot adjust itself based on relative engagement kinematics, as it does not receive this information. As a result, conventional G&C systems rely on making the autopilot time constant as small as possible to improve stability. The autopilot time constant designs the distance from miss to target in conventional G&C systems[2].

One of the main approaches to IGC using SMC was presented in 2010 by Harrell and Balakrishnan using terminal second-order sliding mode control [3]. by Harrell and Balakrishnan using terminal second-order sliding mode (TSM) control. In 2019, Wang et al proposed an integrated guidance and control method with limited impact angle for the missile to achieve unidirectional attack capability[4]. To improve the ability to damage the target, He et al.[5] designed an integrated guidance and control law with impact angle constraint to deal with the problem of tracking unknown maneuvering targets. To deal with the limitations of stimulus saturation in real systems, Ma et al in [6] investigated an integrated control law using dynamic level control, feedforward control and adaptive neural network. And Michel and Stechel thoroughly investigated the sliding mode control for the integrated plane model[7]. In 2020, Tian et al. presented a unified model to avoid practical problems such as the field of view limitation, and solved the field limitation by converting output to input saturation[8]. In 2021, Sinha et al presented an integrated guidance and control model with limited time. In this research, due to the simplicity of the design, sliding mode control is used, while a non-linear finite time disturbance observer is used to estimate the target maneuver [9]. In 2022, Lee proposed a unified model for hypersonic homing missiles. In this design, high-speed targets are hit with proper accuracy by using the sliding model controller, and by using the Monte Carlo method, the non-hit distance was reduced to the minimum[10]. In 2023 Xiaohui Liang et al investigated the nonlinear integrated missile guidance and control system with external uncertainties and disturbances and proposed a new adaptive neural network (NN) control scheme with the help of estimates obtained by NN and disturbance observer (DOB). In this paper, the weight learning rule NN and DOB are updated according to the tracking and estimation errors. Under the operation of the proposed adaptive NN rules, a good tracking characteristic and guidance effects can be obtained for the integrated missile guidance and control (IGC) system. Finally, the simulation results of two different scenarios show the correctness of the designs. It is worth mentioning that the missile tracking process shows a smoother trajectory and a shorter distance can be achieved with the proposed NN adaptive control approach[11]. Xiangyu et al. 2024 investigates the integrated guidance and control (IGC) law design problem with impact angle and general field of view (FOV) constraints. First, the IGC model for non-maneuvering moving target tracking is parameterized by state-dependent coefficient matrices. The nominal IGC law for target interception with the desired impact angle is obtained by solving the state-dependent Riccati equation. Second, since the relative degrees of general FOV constraints exceed one according to the IGC model, high-order control barrier functions are constructed. Satisfying the FOV constraints is equivalent to ensuring that the hypersurface sets defined by the barrier functions are constant, which translate into dependent constraints on the control input. The nominal IGC law is modified in a minimally invasive way by quadratic programming. Then, the proposed method is extended to the case of maneuvering target tracking using a relative coordinate framework. Finally, numerical simulations are performed to confirm the effectiveness of the proposed method. [12]

In this article, the development of an explicit linear model predictive controller (MPC) for an integrated guidance and control (IGC) model is introduced and explored in depth. The proposed approach leverages the explicit formulation of the controller to enhance real-time performance in control systems. Unlike traditional MPCs, where optimization is performed online at every time step, the explicit controller precomputes the solution to the optimization problem offline for all feasible values of the state vector. This process involves determining the optimal control input as an explicit function of the state variables, which is subsequently stored in the controller's memory. During real-time operation, the controller simply evaluates this precomputed function at each time step, using the current values of the state vector to generate the control commands. This method significantly reduces the computational burden during runtime, as the need for solving optimization problems repeatedly in real time is eliminated. Consequently, this approach is particularly advantageous for applications where computational resources are limited, such as hardware with constrained processing capabilities or systems requiring high-speed control actions. The benefits of this explicit MPC design were validated through simulations, which demonstrated its superior performance compared to traditional controllers like proportional-integral-derivative (PID) and linear quadratic regulator (LQR) controllers. Specifically, the results highlighted that the proposed controller, when implemented alongside the integrated guidance and control model for air defense missile systems, achieved a notable reduction in both the miss distance and the time to collision. These improvements underscore the effectiveness of the proposed method in achieving more precise, efficient, and high-speed control outcomes, making it a promising solution for advanced missile defense systems, particularly in time-critical and resource-constrained environments. The results from these simulations demonstrate that the proposed approach yields the best performance in terms of speed and accuracy for air defense missile systems.

 

2. Mathematical modeling

A missile-target engagement scenario involves a missile trying to intercept a target by changing course. All through homing guidance, sensors onboard the missile are used to guide the missile to impact. The initial conditions of this scenario include three main assumptions. (a) Midcourse guidance is successful, (b) The speed of the missile and the target are close to each other in the collision course, (c) In order to intercept and completely destroy the target, the impact angle of the missile and the target is limited. The engagement geometry of this conflict scenario is shown in Fig. 1.

 

Fig. 1 -Kinematics of conflict

 

The general purpose of this article will be to design a proper controller for accurate target tracking. Therefore, the problem of missile-target engagement is discussed, which includes all the topics required for accurate modeling, including engagement kinematics, missile dynamics, and integrated guidance and control model. The o-xyz in Fig. 1 represents the inertial coordinate system. The coordinate system in which the missile and target velocities are described is shown in Fig. 1 as () , and  is the relative missile/target range. In respective orders, vectors  are the velocities of the missile and target, () and () are the angles of the missile and the target relative to the velocity coordinates and line of sight coordinate system, () is the elevation angle and the side angle[13]. The kinematics of the engagement is described by the equations

 

 

In equations 1 to 3, () and , () represent the acceleration components for the missile and target, respectively. Also, θ is the angle of the flight path. Since the acceleration for the missile is usually provided by the aerodynamic force, in particular in the final phase of the guidance, closer attention must be given to the relationship between the acceleration of the missile and the aerodynamic force. The aerodynamic force on the missile is calculated in the inertial coordinate system as equations 4 to 7.

(1)
(2)
(3)

 

 

In equations 4 to 7, () are the acceleration components of the missile along the inertial coordinate system, m indicates the mass of the missile,  is the air density, () indicate the upward and lateral forces, () is the dynamic pressure,  represents the aerodynamic reference area of the missile,  is the angle of attack ,  is the lateral slip angle, () denotes the deflection angles of the missile wings, () are partial derivatives of the lift force coefficient, and () partial derivatives are the lateral force coefficients.

By combining equations 2, 3, 6, and 7, the equations for the line of sight angle appear as equations 8 and 9, as follows.

(4)
(5)
(6)
(7)

 

 

In equations 8 and 9, parameters and  are defined as relations 10 and 11.

(8)
(9)

 

In equations 8 to 11,  is the ballistic angle,  the rotation angle, () are the approximate error for the and . Kinematic equations of flight path angle and ballistic angle are shown as equations 12 and 13, as follows.

(10)
(11)

 

 

The kinematic and dynamic equations of the missile rotating around the center of mass in three-dimensional space are given as equations 14 through 16 as follows.

(12)
(13)

 

In equations 14 to 16, parameters  ,  , and   are the angular velocities in the roll, yaw, and pitch channels, respectively.

The dynamic equations of the state of the missile are obtained as equation 17.

(14)
(15)
(16)

(17)

 

In equation 17, denote the moments of inertia for the roll, yaw, and pitch channels, respectively. Also, the three moments ),( are defined as in equation 18.

(18)

 

In equation 18,  is the reference length, (,) are the moment coefficient of the partial derivatives related to the roll channel, () are those related to the yaw channel, and () are those related to the pitch channel.

By combining relations 8, 9, and 14 with 18, the three-dimensional equation of integrated missile guidance and control can be written as equation 19 [13].In these equations, x the state variables of the system are

(19)

 

 

 

 

, where

،

,

and  shows the approximate system error. Moreover, the matrices  and are shown in relations (20) to (25).

3- Controller design

The missile hits the target when R (the distance between the missile and the target) approaches  (the minimum distance between the missile and the target). The goal of the controller design in this article is to the make the angles  and  converge to their reference values  and , respectively, and also make the distance between the missile and approach .

Backtracking method is used to solve the problem. And the control input (u) will be calculated by the Higher order continuous sliding mode control. The backsteping approach states that a virtual control signal named  is designed at first. This signal is the desired behavior of the state variable  and its design is such that the state variable  tends to zero. So, in fact, the desired behavior of  is obtained in such a way that the first goal of the control problem is established. In the next step, we design the virtual control signal . This signal is the reference signal of state variable  and it is designed in such a way that if  follows it,  also follows  and as a result  will tend to zero. In the last stage, the real control input u is designed in such a way that it converges  to , and as a result, the control loop is completed. By doing this, the second goal of the control problem (zeroing the distance between the missile and the target (R) will be established.

Two S virtual control signals are obtained by the backtracking method, but the real control signal, which is the main goal of this article, will be calculated from the explicit linear model predictive controller. To calculate virtual signals, we will have relations (26) and (27).

(20)
(21)
(22)
(23)
(24)
(25)

 

In equations (26) and (27), the coefficients c, , and  are fixed and will be obtained with the help of optimization methods of PSO along with the variables related to the real controller.

 

3-1 PID controller design

PID controller is introduced as a standard control structure in classical control theory. The performance of the system is improved by accurately adjusting the values of the three proportional gains , integral gain , and derivative gain . By adjusting these parameters, the steady state error and output fluctuations can be controlled in response to the step input. In order to evaluate the proposed controller of this article, first a simple PID controller is designed. According to the system equations, the output of the missile system is its angular velocities. These speeds are to tend to zero. As a result, tracking error is defined as equation 28.

(26)
(27)

The general form of the PID controller is of the form given in Eq. (29) as:

(28)

 

ZieglerNichols method is used to obtain the ,  , and   gains in relation (29). For this purpose, the system should be linearized first. By a frequency analysis then, the PID controller gains are obtained from Eq. (30) [14].

(29)

The values of  and  are calculated from Eq. (31).

(30)

(31)

In relation (31), the parameter  is the phase margin and  is the frequency where the phase margin is measured and the system phase will be -180.

3-2 LQR controller design

 

The 2nd-order linear regulator or LQR controller attempts to minimize the objective function (32).

(32)

 

In Eq. (32), the parameter  denotes the final time. Also, Q and R are two weight matrices that are obtained through optimization. The equation for the control input associated with this objective function is obtained as given in Eq. (33).

 

(33)

 

where B is the matrix multiplier for the control input in the linearized system, and   is obtained from Eq. (34) below.

 

(34)

 

where  is the matrix multiplier for the state vector in the linearized system.

3-3 Design of predictive model controller

In most control projects where the use of predictive model control is considered a necessity, the linear type predictive model controller option is preferred. In particular, in industrial projects, due to the need for high speed and reliability along with cost considerations, this type of controller has been almost the only option for designers [15]. In general, the pre-linear model controller can be implemented in two ways: online (implicit) and offline (explicit). In the implementation of the online type, the optimization problem of the control cost function is solved online in each time step, and the answer to this problem will determine the optimal control order. But in the offline method, the optimization problem is solved once for all possible values of the state vector and the optimal answer is calculated as an explicit function of the variables of the state vector and loaded into the controller memory. At each time step then, the controller will determine the value of this function based on the values of the state variables to issue the control commands.

As such, the time required to perform control calculations is reduced, hence making it possible to implement the devised controller on the control hardware with limited processing volume.

3-3-1 Designing an explicit linear predictive model controller

 

In the design of the predictive model controller, the calculation of the optimal control command vector requires solving a quadratic optimization problem at each time step. Although efficient quadratic programming solvers based on active set and interior point methods are available today, it is not possible to use these solvers in control hardware with limited processing capacity.

To reduce the need for online processing power to increase the speed of calculating the control command in the constrained linear predictive model controller used for real applications, the explicit predictive model controller is usually used. This controller has all the stability and performance features of the equivalent online predictive model controller, and only the need for calculations related to solving the online optimization problem has been eliminated. For this purpose, multi-parameter programming is used to solve the optimization problem of the predictive model controller.

Parametric programming is a term used to solve the optimization problem for a range of values of a parameter. The difference between parametric programming and multi-parameter programming is that in the latter we deal with a vector of parameters. We show that in this method, the control law can be calculated offline in the form of a continuous piecewise affine function of the state variables and the reference signal vector such as . This control law can be set in the form of a reference table and loaded in the memory of the control hardware, thus reducing the online processing effort to finding the i related to the state vectors and the reference signal.

The main idea of the multi-parameter algorithm is to use the necessary and sufficient conditions of optimality to construct the critical region in the neighborhood of a given parameter and search the space outside this region in a recursive method.

The predictive model controller is usually implemented as discrete time in control systems. Therefore, we convert the state equations into the discrete time state space equation form using the zeroth order retainer and the sampling time .

(35)

Therefore, the output equations of the discrete-time system will be in the following form.

(36)

Relations (35) and (36) express the model based on which the predictive model controller is developed. Therefore, this model is called control-oriented model. With the aim of including the defined constraint on the rate of acceleration changes in the control equations and also creating the integration behavior in the controller to reduce the steady state error, the control input by relating its increase value at each sampling time as  in the form of new state variables, we add to the equations of the previous control-oriented model[30] .

(37)

This new control-oriented model is called the augmented model. The output equations of the augmented model can be expressed as follows.

(38)

 

By defining  and  as prediction and control horizons, the input and output equations of the prediction model can be calculated using the following equation.

(39)

In the equation of the above forecasting model, Y vector can be taken from equation (40),

(40)

ΔU vector from equation (41),

(41)

matrix A from equation (42),

 

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