Unifying Sequential Quadratic Programming and Linear-Parameter-Varying Algorithms for Real-Time Model Predictive Control

1. Problem Statement

Model Predictive Control (MPC) is a standard technique for controlling nonlinear dynamical systems with constraints. However, solving the underlying Nonlinear Programming (NLP) problem in real-time is computationally demanding. Two dominant approaches exist:

• Sequential Quadratic Programming (SQP): Iteratively solves Quadratic Programs (QPs) by linearizing the system dynamics and constraints. It offers strong local convergence properties but relies on Jacobian evaluations.
• Iterative Linear-Parameter-Varying MPC (LPV-MPC): Embeds the nonlinear system into an LPV form using the Second Fundamental Theorem of Calculus (FTC). This allows for an exact embedding of nonlinearities without linearization approximation, solved via successive QPs.

The Gap: While both methods share similarities (inexact Newton-type methods), their theoretical relationship has not been fully unified. Furthermore, recent advancements in zero-order approximations (decoupling uncertainty propagation to reduce computational scaling) have been applied separately to SQP and LPV-MPC, but the connection between these specific implementations remains unexplored. The paper aims to unify these frameworks to enhance computational efficiency for robust and stochastic MPC, particularly in learning-based control scenarios like autonomous racing.

2. Methodology

The authors propose a unified differential formulation that encompasses both SQP and FTC-based LPV-MPC as special cases.

A. Unified Framework via Differential Formulation

The core of the unification lies in reformulating the LPV-MPC problem using differential variables ($\Delta x, \Delta u$) similar to SQP.

• FTC Embedding: The nonlinear dynamics $f(x, u)$ are represented exactly using the FTC:
  $$f(x, u) = \int_0^1 \frac{\partial f}{\partial \xi} d\lambda \cdot (x - \tilde{x}) + \dots + f(\tilde{x}, \tilde{u})$$
  Here, $\tilde{x}, \tilde{u}$ are anchor points.
• The Unification: The authors demonstrate that the LPV-MPC iteration becomes mathematically identical to the SQP iteration if the anchor points are chosen as the previous iteration's solution ($\tilde{x} = \hat{x}, \tilde{u} = \hat{u}$).
  • In this specific case, the integrated Jacobians in the LPV formulation reduce to the standard Jacobians used in SQP linearization.
  • This proves that SQP is a sub-case of the FTC-based LPV-MPC.

B. Zero-Order Approximation Integration

The paper integrates zero-order approximations into this unified framework.

• Concept: A subset of states (e.g., uncertainty propagation states) is decoupled from the optimization variables. These states are propagated outside the QP solver using a simplified dynamics model (often linear or based on the previous iterate).
• Unification: The authors show that the zero-order approximation in SQP (where a tailored Jacobian structure decouples states) is equivalent to using an extended scheduling variable in the LPV-MPC variant. This allows the unified framework to handle robust and stochastic MPC problems (like GP-MPC) efficiently by eliminating the quadratic scaling of optimization variables with respect to state dimensions.

3. Key Contributions

1. C1: Unified Solution Method: Introduced a general framework where both SQP and FTC-based LPV-MPC are sub-cases. Proved that LPV-MPC recovers SQP exactly when anchor points match the previous iterate.
2. C2: Zero-Order Unification: Demonstrated how zero-order Jacobian approximations can be integrated into the unified scheme, interpreting them as using extended scheduling variables in the LPV context.
3. C3: Comparative Analysis: Provided a rigorous comparison of computational complexity and convergence properties between the schemes.
4. C4: Real-World Validation: Applied the unified zero-order scheme to Gaussian Process (GP)-based MPC for autonomous racing, validating performance on real hardware.

4. Results

Simulation Studies (Cart-Pendulum & Autonomous Racing)

• Convergence:
  • When anchor points are set to the previous solution, LPV-MPC behaves identically to SQP, converging to a KKT point.
  • When using fixed anchor points (e.g., origin), LPV-MPC converges faster (fewer iterations) but to a suboptimal solution compared to SQP.
• Computational Cost:
  • SQP: Faster preparation time (evaluating Jacobians is cheaper than numerical integration).
  • LPV-MPC: Requires more preparation time due to numerical integration of Jacobians (FTC embedding). However, this integration can be parallelized.
  • Trade-off: For high-dimensional systems, LPV-MPC may be advantageous if the reduction in QP iterations outweighs the integration cost.

Real-World Experiments (Autonomous Racing)

The authors tested Nominal MPC and GP-based MPC (learning residual dynamics) on a 1/28-scale electric race car.

• Safety: Nominal controllers (SQP and LPV) required manually tightened track constraints to avoid collisions due to model mismatch. GP-based controllers (both GP-SQP and GP-LPV) operated safely without constraint tightening.
• Performance:
  • GP-LPV achieved the lowest average cost (best lap performance) due to superior model approximation via the FTC embedding.
  • SQP was computationally faster in the Real-Time Iteration (RTI) loop because it avoids the heavy numerical integration burden of the LPV method.
  • Zero-order GP-LPV successfully handled the stochastic nature of the problem, proving the feasibility of the unified approach on real hardware.

5. Significance

• Theoretical Insight: The paper bridges a significant gap in control theory by formally proving the equivalence between SQP and FTC-based LPV-MPC under specific conditions. This clarifies the relationship between "linearization" and "exact embedding."
• Practical Efficiency: By unifying these methods, the authors provide a flexible toolkit. Engineers can choose between SQP (for speed in low-dimensional problems) or LPV-MPC (for better global approximation in complex, uncertain systems) within the same software architecture.
• Advancement in Learning-Based Control: The successful application of the zero-order GP-MPC to autonomous racing demonstrates that complex stochastic control problems can be solved in real-time on embedded hardware. The ability to decouple uncertainty propagation while maintaining the benefits of exact nonlinear embedding is a major step forward for safe, high-performance autonomous systems.
• Open Source: The authors released the implementation via L4ACADOS, facilitating reproducibility and further research in real-time nonlinear control.