Model Predictive Static Programming Research Articles

A model predictive Stackelberg solution to orbital pursuit-evasion game

In this paper, we investigate analytical numerical iterative strategies for the pursuit-evasion game involving spacecraft with leader–follower information. In the proposed problem, the interplay between two spacecraft gives rise to a dynamic and real-time game, complicated further by the presence of J2 perturbation. The primary challenge lies in crafting control strategies that are both efficient and applicable to real-time game problems within a nonlinear system. To overcome this challenge, we introduce the model prediction and iterative correction technique proposed in model predictive static programming, enabling the generation of strategies in analytical iterative form for nonlinear systems. Subsequently, we proceed by integrating this model predictive framework into a simplified Stackelberg equilibrium formulation, tailored to address the practical complexities of leader–follower pursuit-evasion scenarios. Simulation results validate the effectiveness and exceptional efficiency of the proposed solution within a receding horizon framework.

An improved MPSP‐based path‐following control method for USV with input disturbances

AbstractThis study proposes an improved model predictive static programming (MPSP) based path‐following control method for unmanned surface vessel (USV) subject to input disturbances. The method addresses the challenges of accurate USV dynamics modeling, unpredictable maritime environments, and limited power and energy systems. A trajectory generator is designed to construct smooth reference trajectories, and the MPSP algorithm is adapted to handle path‐following problems while considering state and input constraints. An event‐triggered mechanism is introduced to reduce computational burden and conserve energy. Comparative simulations demonstrate the superiority of the proposed method over open‐loop tracking and the original MPSP approach in terms of tracking accuracy, disturbance rejection, and overall control performance. The improved MPSP‐based control method offers a robust and efficient solution for USV path‐following tasks, ensuring accurate tracking even in the presence of environmental disturbances and system uncertainties.

Adaptive Terminal Time and Impact Angle Constraint Cooperative Guidance Strategy for Multiple Vehicles

This paper addresses the guidance of various flight vehicles under multiple constraints in three-dimensional space. A cooperative guidance strategy that satisfies both time and angle constraints is designed to reach a moving target. The strategy is organized into two parts: modeling and programming calculations. First, a nonlinear motion model for guidance is established and normalized, including both the vehicle and the target. Later, the arrival method is automatically determined according to the strategy and depending on the type of target. The cooperative terminal time is determined based on an augmented proportional navigation method. An improved model predictive static programming (MPSP) algorithm was designed as a means of adjusting the adaptive terminal time. Then, the algorithm was used to update the control quantity iteratively until the off-target quantity and the angle of constraints were satisfied. The simulation results showed that the strategy could enable multiple flight vehicles at different initial positions to reach the target accurately at the same time and with the ideal impact angle. The strategy boasts a high computational efficiency and is capable of being implemented in real time.

Approach and landing guidance using constrained model predictive static programming

The paper proposes a guidance scheme during the approach and landing phase of an unpowered vehicle to achieve high precision at touchdown by limiting the control inputs within allowable bounds using constrained Model Predictive Static Programming. The advantages of a single flight phase are incorporated into the design since the guess control history is obtained from a single segment based guidance approach. The requirement of ensuring continuity in states and guidance commands are hence avoided leading to minimization of control deviations using a simple cost function. Constrained input commands are used for determining control surface deflection based on dynamic inversion considering the longitudinal pitch rate dynamics. Comparison is carried out between the results obtained for constrained as well as unconstrained schemes for the nominal trajectory. Comparisons of load factor, dynamic pressure and flight path angle for three segment, two segment, single segment guidance strategies have been carried out with MPSP based guidance to evaluate the relative advantages offered by the scheme. Performance analysis of the proposed scheme is done based on numerous simulations with simultaneous variations in initial states, vehicle parameters and aerodynamic parameter variations. Simulation results indicate that the proposed scheme is suitable for landing on runways of shorter lengths due to minimal touchdown errors.

Optimal control of mineral processing plants using constrained model predictive static programming

The model predictive static programming (MPSP) technique, which is extended recently to incorporate applicable state and control constraints, operates on the philosophy of nonlinear model predictive control (NMPC). However, it reduces the problem into a lower-dimensional problem of control variables alone, thereby enhancing computational efficiency significantly. Because of this, problems with larger dimensions and/or increased complexity can be solved using MPSP without changing the computational infrastructure. In this paper, the MPSP technique with applicable constraints is applied to two challenging control problems in the mineral processing industry: (i) a single-stage grinding mill circuit model, and (ii) a four-cell flotation circuit model. The results are compared with a conventional nonlinear MPC approach. Comparison studies show that constrained MPSP executes much faster than constrained MPC with similar/improved performance. Therefore, it can be considered a potential optimal control candidate for mineral processing plants.

Cubature Kalman Filters Model Predictive Static Programming Guidance Method with Impact Time and Angle Constraints Considering Modeling Errors

This paper proposes a CKF-MPSP guidance method for hitting stationary targets with impact time and angle constraints for missiles in the presence of modeling errors. This innovative guidance scheme is composed of three parts: First, the model predictive static programming (MPSP) algorithm is used to design a nominal guidance method that simultaneously satisfies impact time and angle constraints. Second, the cubature Kalman filter (CKF) is introduced to estimate values of the influence of the inevitable modeling errors. Finally, a one-step compensation scheme is proposed to eliminate the modeling errors’ influence. The proposed method uses a real missile dynamics model, instead of a simplified one with a constant-velocity assumption, and eliminates the effects of modeling errors with the compensation scheme; thus, it is more practical. Simulations in the presence of modeling errors are conducted, and the results illustrate that the CKF-MPSP guidance method can reach the target with a high accuracy of impact time and angles, which demonstrates the high precision and strong robustness of the method.

Fault-Tolerant Attitude Control for Hypersonic Flight Vehicle Subject to Actuators Constraint: A Model Predictive Static Programming Approach

In this article, an improved model predictive static programming (MPSP) based fault-tolerant control (FTC) scheme is proposed to solve the attitude tracking control problem of the hypersonic vehicle (HSV). In the field of HSV, the MPSP technique has been applied successfully to solve guidance problems for its high computational efficiency. While we try to address the attitude control problem directly using it. The attitude model of HSV with uncertainty and disturbance, together with the fault model of aircraft body injury, are constructed firstly. The actuator of HSV is suffering from input constraints. Then, a feasible attitude control trajectory is generated by the improved MPSP method. The methodological innovation in this paper extends MPSP technique to the direct control of the attitude of HSV both in fixed and flexible final time. By utilizing the improved MPSP technique, the complexity of processing multiple constraints and the computation are reduced. The effectiveness of the designed FTC scheme is demonstrated through simulation under different cases with actuator constraints.

Pseudospectral Convex Optimization Based Model Predictive Static Programming for Constrained Guidance

This paper presents a pseudospectral convex optimization-based model predictive static programming (PCMPSP) for the constrained guidance problem. First, the sensitivity relation between the state increment and control correction is reformulated using Legendre-Gauss (LG) and Legendre-Gauss-Radau (LGR) pseudospectral transcriptions. Second, the convex optimal control problem associated with the trajectory optimization is defined by introducing the quadratic performance index. Third, modifications to the initial guess solution and reference trajectory update are introduced to enhance the accuracy and robustness of the algorithm. Finally, a model predictive guidance law is designed based on the proposed PCMPSP algorithm for the air-to-surface missile guidance with impact angle constraint. The simulation results show that the PCMPSP has lower sensitivity to the initial guess trajectory, higher accuracy, as well as faster convergence speed than existing convex programming methods. Moreover, the robustness of the proposed guidance law to uncertainties is demonstrated through the Monte Carlo campaign.

Adaptive Trajectory Tracking Algorithm for the Aerospace Vehicle Based on Improved T-MPSP

To deal with the uncertainty and disturbance that exist in the tracking system of an aerospace vehicle, an adaptive trajectory-tracking method based on a novel tracking model predictive static programming (T-MPSP) is proposed. Firstly, to make the proposed method more adaptive to uncertain parameter deviations, an extended Kalman filter (EKF) parameter correction strategy is designed. Then, the control constraints are considered to form a novel T-MPSP algorithm. By combining the parameter correction strategy with the improved T-MPSP algorithm, a novel adaptive tracking guidance scheme is presented. Finally, simulations are carried out to demonstrate the effectiveness of the proposed method.

Ascent adaptive energy management method for solid rocket‐powered hypersonic vehicle under uncertainties

SummaryThe solid rocket booster is widely used in the launch mission for hypersonic vehicles nowadays, whose ascent energy management is still a challenging problem due to the inherent dynamic nonlinearity. This paper casts the problem into an equivalent ascent guidance one and proposes a model predictive static programming (MPSP) based solution. The MPSP‐based method is a computationally efficient solver that adapts to different energy demands while guaranteeing the satisfaction of terminal constraints. To further improve the performance of MPSP in this problem, a Bézier spline guidance method is proposed to generate high‐quality reference trajectories such that the MPSP can converge with high accuracy while satisfying the real‐time requirement. Additionally, an online parameter estimation based on the extended Kalman filter is employed to avoid large terminal state deviations incurred by parametric uncertainties during the Bézier curve computation and linearization in MPSP. Numerical simulations with different hard terminal constraints and uncertainties are conducted to demonstrate the effectiveness and robustness of the proposed algorithm. The simulation results show that our proposed algorithm can guide the hypersonic vehicle to a given terminal state under uncertainties and outperforms several existing ascent energy management methods.

Singularity avoidance controller design for spacecraft attitude control using double-gimbal variable-speed control moment gyro

A sub-optimal singularity avoidance large angle rest-to-rest attitude maneuver control of a rigid spacecraft actuated using a double-gimbal variable-speed control moment gyro (DGVSCMG) is proposed for avoiding the control torque saturation and the gimbal lock singularity. To achieve this objective, a modified model predictive static programming (MPSP) algorithm subject to control and state constraints is proposed and developed. A cuboid rigid spacecraft equipped with a single DGVSCMG device for the attitude control is considered to validate the proposed control algorithm. A new MPSP based control architecture for the attitude control of a spacecraft using a DGVSCMG is developed, wherein the desired gimbal angle rates and rotor spin acceleration are taken as the control input and computed directly using the MPSP algorithm without requiring the steering law. The control law ensures attitude stabilization of the spacecraft-DGVSCMG system while successfully avoiding the control input saturation and the gimbal lock singularity of the DGVSCMG, thereby enabling large angle attitude maneuvers. Multiple test cases of rest-to-rest attitude maneuvers are simulated to show the efficacy of the proposed control.

Active-set pseudospectral model predictive static programming for midcourse guidance

This paper focuses on the midcourse guidance law for the interceptor to engage the hypersonic gliding vehicles (HGVs). Firstly, an active-set pseudospectral model predictive static programming (APS-MPSP) algorithm is presented, which entails two major improvements: the application of pseudospectral discretion and handling of control constraint using active-set iteration strategy. Except inheriting the high efficiency of the MPSP, this new algorithm can address the control constraint directly. Secondly, the optimal predictive guidance design is established, which embeds the target prediction into the guidance framework by formulating a simultaneous nonlinear optimal problem. Employing the APS-MPSP algorithm to solve this optimal problem efficiently, the predictive guidance command that ensures interceptor reach an appropriate predicted interception point (PIP) can be obtained directly. This strategy has the merits of no need to precompute the PIP or time-to-go and high prediction accuracy. The reliability and efficiency of the proposed methods are verified by numerical examples and comparisons with already methods.

High-precision computational guidance in terminal phase with impact angle, lead angle and lateral acceleration constraints

This paper proposes a high-precision three-dimensional nonlinear optimal computational guidance law in the terminal phase of an interceptor that ensures near-zero miss-distance as well as the desired impact angle. Additionally, it achieves these ambitious objectives while ensuring that the lead angle and lateral acceleration constraints are not violated throughout its trajectory. This ensures (i) the target does not escape the field of view of its seeker at any point in time (a state constraint) and (ii) it does not demand unreasonable lateral acceleration that cannot be generated (a control constraint). The guidance problem is formulated and solved using newly proposed Path-constrained Model Predictive Static Programming (PC-MPSP) framework. All constraints, both equality and inequality, are equivalently represented as linear constraints in terms of the errors in the control history, thereby reducing the complexity and dimensionality of the problem significantly. Coupled with a quadratic cost function in control, the problem is then reduced to a standard quadratic optimization problem with linear constraints, which is then solved using the computationally efficient interior-point method. Results clearly demonstrate the advantage of the proposed guidance scheme over the conventional Biased PN as well as the recently proposed GENeralized EXplicit (GENEX) guidance techniques. Numerical simulations with variation in initial conditions and Monte–Carlo simulations with parametric uncertainty demonstrate the robustness of the proposed guidance scheme.

Three-dimensional field of view and impact angle constrained guidance with terminal speed maximization

A three-dimensional strap-down seeker based multi-constrained guidance law for an interceptor is proposed in this paper to achieve near zero miss distance with minimal control effort (lateral acceleration) and interception time. When the interceptor is guided based on strap-down seeker measurements, the field of view limit throughout the flight trajectory is one of the major concerns to guarantee tracking of the target. To enhance the capability of warhead while approaching a target, the interceptor needs to ensure the desired impact angles and maximum speed at terminal time as well as various path inequality constraints throughout the trajectory, such as lateral acceleration limit and field of view limit. To meet these requirements, the proposed guidance is designed using a new extended version of the computationally efficient, suboptimal generalized model predictive static programming (GMPSP) technique in an unspecified final time framework. In addition, to make all the states available, to estimate parameter uncertainties and to filter out the sensor measurement noises, an extended Kalman filter based state and parameter estimation is incorporated with this proposed guidance law. Using simulation studies, the efficacy of the proposed GMPSP based suboptimal guidance method is shown and compared with that of the Chebyshev pseudo-spectral method as well. The extensive simulation results also demonstrate that the proposed robust suboptimal guidance design can perform well, in the presence of aerodynamic parameter uncertainties and sensor noises.

Energy Management of a Parallel Hybrid Electric Vehicle using Model Predictive Static Programming

Energy management of Parallel Hybrid Electric Vehicles (PHEVs) involves computation of energy efficient torque-speed set-points for the engine and the motor. Application of Nonlinear Model Predictive Control (NMPC) for this problem has been reported in the recent literature. In this work, a fast iterative implementation of the NMPC, known as Model Predictive Static Programming (MPSP) is deployed for the first time for energy management of PHEVs, with an appropriate multi-objective cost function and the applicable equality and inequality constraints. Additionally, a systematic approach based on the Theory of Dominance is used to tune parameters of the controller efficiently. Performance is evaluated with a validated PHEV model on the ADVISOR simulation environment, in terms of various performance metrics on standard drive cycles. The results show that the performance of the MPSP-based technique is close to the ideal performance achieved with Dynamic Programming (DP) and improved over a standard implementation using the Linear Time-Varying MPC (LTV-MPC) and Sequential Quadratic Programming MPC (SQP-MPC) algorithm. Moreover, its execution time on an industry standard micro-controller board is seen to be significantly less than that of LTV-MPC and SQP-MPC. This algorithm therefore appears to be one of the best candidates for on-board vehicle implementation of an optimal energy management strategy for a PHEV.

Mars Atmospheric Entry Guidance using MPSP with State and Control Constraints

An optimal Mars entry guidance scheme is presented in this paper using the Model Predictive Static Programming (MPSP) technique accounting for the applicable state and control constraints. The guidance scheme is designed to maximize the terminal parachute deployment altitude while applying minimum control effort and satisfying hard constraints on desired terminal conditions such as final velocity and downrange. The proposed guidance computes the optimal bank angle profile to shape the trajectory of the spacecraft. Path constraints on heat rate, dynamic pressure, aerodynamic load, and bounds on bank angle are considered to guide the vehicle safely through the martian atmosphere. Moreover, in order to generate practically realizable bank angle profiles, an additional constraint on the bank angle rate is also applied. Next, using the MPSP technique, the nonlinear constrained optimal control problem is converted into a static quadratic optimization problem with linear equality and inequality constraints to solve it in a computationally efficient manner. The concept of flexible final time MPSP is incorporated to update the final time in an optimal fashion. Numerical simulations illustrate the ability of the proposed method to solve the guidance problem efficiently while satisfying the path and terminal constraints within the desired accuracy.

Controlling a Grinding Mill Circuit using Constrained Model Predictive Static Programming

A constrained Model Predictive Static Programming (MPSP) method is implemented in simulation to a single-stage grinding mill circuit model. The results are compared to a constrained Nonlinear Model Predictive Control (NMPC) method. Both the constrained MPSP and NMPC controllers were able to track the desired output set-points without exceeding any constraints. The comparison shows that the constrained MPSP has a faster computational time than that of the NMPC controller with similar performance. Therefore, constrained MPSP shows promise as a model-based controller for large processes where computational time limits the use of NMPC.

Model Predictive Static Programming Guidance Method with Trajectory Way-points Constraints

Model Predictive Static Programming Guidance Method with Trajectory Way-points Constraints

Sub-optimal fixed-finite-horizon spacecraft configuration control on SE(3)

For achieving the desired configuration of spacecraft at the desired fixed time, a sub-optimal fixed-finite-horizon configuration control method on the Lie group SE(3) is developed based on the Model Predictive Static Programming (MPSP). The MPSP technique has been widely used to solve finite-horizon optimal control problems and is known for its high computational efficiency thanks to the closed-form solution, but it cannot be directly applied to systems on SE(3). The methodological innovation in this paper enables that the MPSP technique is extended to the geometric control on SE(3), using the variational principle, the left-invariant properties of Lie groups, and the topology structure of Lie algebra space. Moreover, the energy consumption, which is crucial for spacecraft operations, is considered as the objective function to be optimized in the optimal control formulation. The effectiveness of the designed sub-optimal control method is demonstrated through an online simulation under disturbances and state measurement errors.

An impulsive model predictive static programming based station-keeping guidance for quasi-halo orbits

In this paper, a control effort minimizing optimal station-keeping guidance is designed and implemented to regulate a spacecraft around an L1 quasi-halo orbit in the Sun–Earth–Moon elliptic four-body problem. The station-keeping guidance is formulated as a finite time, non-linear optimal control problem with hard terminal output constraints and Impulsive Model Predictive Static Programming (I-MPSP) is used to obtain station-keeping maneuvers. The algorithm is iterative, where the guessed station-keeping maneuvers are optimally updated by a simple closed-form equation until an output terminal constraint is satisfied and an optimal cost function is obtained. The technique involves the calculation of sensitivity matrices that is done in a computationally efficient manner owing to their recursive nature. Through extensive simulations, in the presence of disturbance forces and uncertainties, a closed-loop application of I-MPSP guidance results in a trajectory that remains tightly bound to the reference orbit over a long-duration mission.