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Nonlinear Model Predictive Control for Guidance Law with Target Input Estimation

Minho Jang, Minjeong Kim, Sungsu Park

arXiv:2608.26551v1eess.SY

TL;DR

Strapdown seekers provide look angles rather than direct LOS-rate measurements, limiting conventional PNG and existing guidance approaches under measurement and target-maneuver uncertainties. The paper proposes look angle-based nonlinear MPCG with LGRPM transcription, explicit constraints, and AEKF–IMM target-acceleration estimation. Across pitch-weaving, yaw-weaving, and barrel-roll scenarios, MPCG intercepted targets while satisfying practical constraints, although brief FOV exceedances and transient estimation errors occurred during aggressive maneuver transitions.

  • Problem

    Strapdown seekers do not directly provide LOS rates, while existing guidance methods remain limited in handling both seeker measurement constraints and maneuvering-target uncertainties.

  • Method

    The paper combines look angle-based nonlinear MPCG, LGRPM transcription, explicit FOV and acceleration constraints, and AEKF–IMM target-acceleration estimation.

  • Results

    MPCG successfully guided the missile to interception in pitch-weaving, yaw-weaving, and barrel-roll scenarios while satisfying practical constraints.

  • Takeaways & Limitations

    MPCG provides a constraint-aware guidance solution for strapdown seeker systems with stable terminal control and acceleration commands within 25G through a first-order lag autopilot.

  • Takeaways & Limitations

    During aggressive endgame maneuvers, look angles briefly exceeded FOV and AEKF–IMM estimation errors appeared at maneuver transitions before recovering within the 1 −σ bound.

Abstract

from arXiv · show

This paper presents a look angle-based nonlinear model predictive control guidance (MPCG) method for missiles equipped with strapdown seekers. Conventional proportional navigation guidance (PNG) requires line-of-sight (LOS) rate measurements, which are not directly available in strapdown systems. MPCG instead employs look angles and their derivatives as state variables, eliminating body-rate coupling and associated parasitic feedback. The guidance problem is formulated as a continuous-time optimal control problem (OCP), discretized via the Legendre-Gauss-Radau pseudo-spectral method (LGRPM), and solved as a nonlinear program (NLP) incorporating explicit field-of-view (FOV) and acceleration constraints. Target acceleration at the first step of the prediction horizon is estimated using an adaptive extended Kalman filter (AEKF) integrated with an interacting multiple model (IMM) framework. Simulation results under single-maneuver scenarios, which include pitch and yaw plane weaving as well as barrel-roll maneuvers, demonstrate that MPCG achieves reliable interception while satisfying operational constraints, outperforming pure PNG (PPNG) in stability and resilience. This indicates that MPCG offers a practical and effective solution for modern missile guidance systems constrained by seeker measurement limitations.

I. INTRODUCTION

Strapdown seekers measure look angles rather than LOS rates, creating phase-mismatch and parasitic-loop issues for conventional PNG. The paper addresses these constraints with a look angle-based MPCG architecture combining constraint-aware optimization and adaptive target-acceleration estimation.

  • I. INTRODUCTION: Strapdown seekers provide look angles instead of directly measured LOS rates, complicating conventional PNG guidance.PNG typically reconstructs LOS rates by differentiating look angles and combining them with body rates.
  • I. INTRODUCTION: Latency in look-angle measurements combined with real-time body rates creates phase mismatch and parasitic loops that degrade GNC performance.The paper identifies estimation delay and phase mismatch as the source of these feedback effects.
  • I. INTRODUCTION: Existing look-angle guidance methods avoid explicit LOS-rate reconstruction but often use simple control schemes without advanced predictive optimization.Related MPCG approaches provide nonlinear modeling and constraint handling but do not jointly address strapdown measurement constraints and maneuvering-target uncertainty.
  • A. Research Contributions and Overview: The proposed MPCG uses look-angle dynamics, nonlinear predictive optimization, and AEKF–IMM target-acceleration estimation for strapdown-seeker guidance.The architecture combines an IMM-based estimator with an NMPC law whose prediction model incorporates estimated target acceleration.
  • A. Research Contributions and Overview: The nonlinear OCP is transcribed with LGRPM, while the guidance law explicitly incorporates FOV and maneuver constraints.The formulation is intended to generate optimal acceleration commands under seeker and maneuver limits.
  • A. Research Contributions and Overview: Validation covers pitch-plane weaving, yaw-plane weaving, and barrel-roll engagements using miss distance, interception time, constraint satisfaction, and estimation accuracy.The reported results confirm stability and adaptability against agile targets.

C. Engagement Geometry

The engagement geometry represents missile–target motion across navigation, LOS, body-fixed, and body-LOS frames. Look angles describe the target direction relative to the missile body axis, while LOS and look-angle relationships support the guidance formulation.

  • C. Engagement Geometry: The engagement geometry uses navigation, LOS, body-fixed, and body-LOS coordinate frames to represent missile–target motion.The formulation is described in both navigation and LOS frames.
  • C. Engagement Geometry: Relative position and velocity are expressed in navigation and body-fixed frames to derive the engagement kinematics.Relative velocity is obtained by differentiating relative position and incorporating LOS-frame angular velocity.
  • C. Engagement Geometry: The conventional seeker cannot measure the LOS-frame angular velocity directly, motivating a separate LOS angular-velocity definition.The unavailable component includes angular rate along the LOS direction.
  • C. Engagement Geometry: Look angles specify the target’s angular deviation from the missile body axis through the body-LOS frame.Pitch and yaw look angles define the orientation of the body-LOS frame relative to the body-fixed frame.
  • C. Engagement Geometry: The seeker FOV angle is obtained from relative-position and velocity vectors and approximated using pitch and yaw look angles under a small-angle assumption.The total look angle is linked to the seeker FOV through the dot product of unit vectors.
  • C. Engagement Geometry: The roll look angle compensates for missile roll about the body longitudinal axis and, under narrow FOV, has the opposite sign to the missile bank angle.The paper derives simplified LOS-angle relations as λθ ≈ γ + σθ and λψ ≈ χ + σψ.

D. Optimal Control Problem

The guidance problem is posed as a continuous-time optimal control problem with dynamics, path, and boundary constraints. LGRPM discretizes this problem into a finite-dimensional nonlinear program for numerical optimization.

  • OCP formulation: The OCP minimizes a performance index over control inputs subject to system dynamics and operational constraints.The formulation includes path and boundary conditions in addition to the dynamics.
  • LGRPM transcription: LGRPM transforms the continuous-time OCP into an NLP by enforcing dynamics at collocation points and interpolating states and controls with polynomials.This provides an accurate and efficient finite-dimensional transcription.
  • LGRPM transcription: LGR points are generated from roots of PN(τ) + PN−1(τ), excluding the endpoint τN+1 = 1.The endpoint is reserved for state interpolation rather than differential-equation enforcement.
  • LGRPM transcription: States are interpolated at N +1 points, while control inputs are discretized at N collocation points using Lagrange basis polynomials.The differentiation matrix enforces the equations of motion at the LGR collocation points.
  • NLP solution: The discretized objective uses Gaussian quadrature, and the resulting NLP combines dynamic constraints, path constraints, boundary conditions, and optimization variables.Solvers such as IPOPT, SQP methods, and SNOPT can solve the resulting NLP.

E. A Look Angle-Based Nonlinear Model Predictive Control for Guidance Law

MPCG predicts missile-target behavior over a finite horizon and applies only the first optimized acceleration command. Its look angle-based formulation incorporates seeker and vehicle constraints directly into nonlinear predictive guidance.

  • MPCG framework: MPC predicts future behavior over a finite horizon, applies only the first control input, and repeats this process as the state is updated.This is the receding horizon strategy used by MPCG.
  • Guidance formulation: The LGRPM-based MPC formulation minimizes a quadratic cost whose weighting matrices regulate state and control contributions.Quadrature weights correspond to the LGR collocation points.
  • Guidance formulation: MPCG uses relative position and velocity, look angles and rates, and simplified LOS angular velocities as state variables, with missile body-frame acceleration as the control input.This state representation avoids including LOS rates directly in the state vector.
  • Operational constraints: MPCG explicitly constrains total look angle, look angle rates, relative range, and missile acceleration within admissible bounds.The missile’s body x-axis acceleration is fixed to zero, and target acceleration is applied uniformly across the prediction horizon.
  • Guidance objectives: Minimizing look angles, their rates, and simplified LOS angular velocities supports target pointing, reduced oscillation, and stable tracking within the guidance framework.The formulation also relies on narrow-FOV and near-zero-roll assumptions for the STT missile.

F. Summary

The MPCG loop converts seeker observations into an augmented optimization state and computes constrained acceleration commands. Its objective emphasizes alignment-related variables to support stable target tracking.

  • MPCG architecture: Seeker observations are processed into measured and augmented state vectors for the MPCG optimization.The augmented state adds look angles, their rates, and simplified body-frame LOS angular velocities.
  • MPCG architecture: The control problem is a quadratic optimization over the prediction horizon subject to dynamic and path constraints.Weighting matrices determine the relative priority of states and control inputs.
  • Guidance objective: Minimizing look angles, their rates, and simplified LOS angular velocities keeps the missile pointed toward the target and reduces excessive oscillations.The paper connects reduced LOS angular velocities with body alignment to the LOS vector.

III. TARGET INPUT ESTIMATION

The AEKF models nonlinear missile-target measurements and adapts its noise covariances for changing engagement conditions. Within MPCG, it estimates target-related states using seeker-derived measurements.

  • Adaptive filtering: The AEKF adapts process and measurement noise covariances online to improve estimation under noisy and uncertain measurements.Its adaptive updates are intended to support abrupt target maneuvers and time-varying noise conditions.
  • Filter model: The AEKF state-space model uses state, input, and measurement vectors with linear dynamics and a nonlinear measurement function.Process and measurement noises are modeled as zero-mean Gaussian variables with covariances Qk and Rk.
  • MPCG integration: In MPCG, the filter state includes relative position, velocity, target acceleration, and jerk in navigation coordinates.The input is missile acceleration, while measurements include range, range rate, and pitch/yaw look angles.
  • Filter operation: EKF prediction and correction use prior estimates, measurement innovations, and residuals to update the state estimate.The innovation determines the Kalman gain, whereas the residual supports adaptive measurement-noise updating.

B. Interacting Multiple Model

The IMM framework estimates maneuvering target inputs by running multiple motion models, interacting through probabilistic mixing, filtering, model updates, and estimate fusion.

  • Target acceleration is estimated because maneuvering intensity cannot be directly measured and is generally unknown in practical engagements.
  • The MM framework classifies model structures as AMM, CMM, or VSMM according to model interaction and configuration changes.CMM is described as balancing computational efficiency and estimation accuracy for online applications.
  • IMM runs multiple dynamic models concurrently and combines their estimates using mode transition probabilities and interactive reweighting.
  • The IMM procedure comprises model-conditioned reinitialization, model-conditioned filtering, model probability update, and estimate fusion.
  • AEKF-IMM adaptively updates process and measurement noise covariances, weights model estimates by updated probabilities, and fuses them into an overall estimate.

C. Modeling of Target Maneuvers within the IMM Framework

The IMM represents target maneuvering with multiple motion models, including CV, CA, Singer, and Markov oscillatory dynamics, using discretized state-space formulations.

  • Four motion models are considered within the IMM: Constant Velocity, Constant Acceleration, Singer, and a Markov oscillatory model.Their dynamic matrices are derived continuously, discretized for implementation, and paired with a common seeker measurement model.
  • Constant Velocity: The Constant Velocity model describes three-dimensional target motion with Gaussian white process noise representing external disturbances.The residual acceleration noise determines maneuver intensity, so the model is also called nearly constant velocity.
  • Constant Velocity: The continuous-time Constant Velocity system is converted to a discrete-time model using the state-transition matrix exponential and sampling interval ∆t.
  • Constant Acceleration: The Constant Acceleration model differs from the Constant Velocity model through an identity matrix in the third block of its process-noise covariance.Its process noise induces target jerk, motivating the term nearly constant acceleration model.
  • Constant Acceleration: The Constant Acceleration continuous-time system is discretized with sampling interval ∆t, with discretized process noise characterized by its mean and covariance.

3) Singer Acceleration Model

The Singer model represents target acceleration as a zero-mean first-order stationary Markov process, with its correlation time controlling maneuver persistence and behavior.

  • The Singer model treats target acceleration as a zero-mean first-order stationary Markov stochastic process.
  • The correlation time constant τ controls acceleration persistence and maneuverability, with larger τ approximating CV and smaller τ approximating CA behavior.
  • By combining temporal correlation with process noise, the Singer model captures more realistic maneuvering behaviors.
  • The model permits flexible acceleration-variance modeling using probabilistic distributions such as ternary-uniform mixtures.
  • The Singer dynamics are discretized using the sampling interval ∆t, with a corresponding discretized process-noise covariance Qk.

4) Markov Models for Oscillatory Target

The oscillatory Markov target model represents periodic lateral acceleration through a second-order pre-whitening system driven by zero-mean white noise.

  • The oscillatory target model is designed for periodic acceleration oscillations in a specific lateral direction, unlike the non-periodic Singer model.
  • It uses a second-order pre-whitening system that produces target acceleration in response to zero-mean white-noise inputs.
  • The model’s parameters include damping ratio ζ, undamped natural frequency ωn, actual damped frequency ωc, and damping coefficient α.
  • Compared with the Singer model, the oscillatory model captures high-frequency oscillatory behavior more effectively and suits targets exhibiting bending motion.
  • The frequency-domain representation is converted into the time domain, with acceleration drift expressed through jerk as a Kalman-filter state variable.
  • The derivative of the white-noise process is treated as zero because white noise is non-differentiable, and the resulting dynamics are discretized for implementation.

IV. NUMERICAL SIMULATION

The simulations evaluate MPCG and its integrated AEKF-IMM framework across pitch weaving, yaw weaving, and barrel-roll target maneuvers. They assess guidance, seeker-constraint compliance, interception distance, timing, estimation accuracy, and model-probability consistency.

  • Scenario design: Three target maneuvers—pitch-plane weaving, yaw-plane weaving, and barrel-roll—test adaptability across vertical, lateral, and full three-dimensional engagements.The scenarios represent distinct engagement geometries and dynamic target behaviors.
  • Scenario design: Pitch-plane weaving creates rapid vertical oscillations that can temporarily exceed the missile’s pitch look-angle limit, challenging vertical guidance authority.Yaw-plane weaving instead produces periodic horizontal oscillations and an S-shaped target path that stresses lateral tracking.
  • Scenario design: High-G barrel-roll motion combines circular movement perpendicular to flight with forward velocity while using the target’s maximum allowable acceleration.Its minimum turning radius depends on both maximum acceleration and angular roll rate ωbr.
  • Implementation: The numerical implementation discretizes the continuous-time OCP with LGRPM and solves the resulting NLP in Julia using JuMP.jl, IPOPT.jl, MA97, and warm starts.The warm-start strategy is used to enhance computational efficiency.
  • Evaluation: Evaluation covers trajectory behavior, FOV compliance, bounded missile acceleration, final MD, TTI, estimation errors, and model-probability transitions.Together, these metrics assess guidance and estimation feasibility and precision under maneuvering-target conditions.

B. Single Maneuver Scenarios

Across pitch-weaving, yaw-weaving, and barrel-roll engagements, MPCG achieved interception while improving or matching PPNG performance and enforcing practical look-angle and acceleration constraints. The AEKF-IMM supported target-motion estimation, while brief FOV exceedances remained during aggressive terminal maneuvers.

  • Performance comparison: In pitch weaving, MPCG reached interception in 23.66 seconds, approximately 0.3 seconds faster than PPNG, while reducing final miss distance by approximately 0.267 meters.The reported comparison indicates improved interception time and accuracy for this scenario.
  • Performance comparison: Across the three scenarios, MPCG provided comparable or improved miss distance and time-to-interception performance relative to PPNG.MPCG outperformed PPNG in both metrics during barrel roll, while yaw weaving showed a time-to-interception advantage but not a miss-distance advantage.
  • Constraint compliance: Unlike PPNG, MPCG kept look angles within allowable FOV limits for most of each engagement, with slight pre-interception violations in yaw weaving and barrel roll.The exceedances were attributed to aggressive terminal maneuvering near the engagement-envelope boundary.
  • Constraint compliance: MPCG acceleration commands remained within the 25G limit, and first-order lag autopilot dynamics attenuated command spikes for smoother control responses.The resulting commands were described as practical and physically feasible across the scenarios.
  • Target estimation: The AEKF-IMM reduced estimation errors over time, adaptively adjusted noise covariances, and identified representative target-motion models during maneuvering.CV2 was dominant during strong curved motion, while Singer models generally had low probabilities; transient estimation errors occurred at maneuver transitions before recovery within the 1σ bound.
  • Scenario evaluation: MPCG achieved interception across pitch-weaving, yaw-weaving, and barrel-roll scenarios while satisfying practical guidance constraints.The simulations assessed final miss distance, time-to-interception, seeker FOV compliance, and missile acceleration limits.
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