Warm Start

The core mechanism of a warm start is using the optimal solution vector (e.g., predicted state and input trajectories) from the previous MPC iteration as the initial guess for the current online optimization. This is in contrast to a cold start, which uses a generic or zero initial guess. For iterative solvers like Sequential Quadratic Programming (SQP), starting near the optimum significantly reduces the number of iterations required for convergence, which is essential for meeting real-time optimization (RTO) deadlines.

Warm starting is most impactful in Nonlinear MPC (NMPC), where the underlying Optimal Control Problem (OCP) is a Nonlinear Programming (NLP) problem. NLP solvers are iterative and sensitive to initial conditions. A good warm start:

• Prevents convergence to local minima far from the global optimum.
• Reduces solver iterations from tens or hundreds to just a few.
• Makes solving complex NMPC problems within a control system's sampling period (e.g., < 100 ms) feasible.

A simple warm start uses the previous solution directly, but a more effective technique involves shifting the trajectory. Since MPC uses a receding horizon, the optimal trajectory from time step k-1 is shifted forward by one step. The state/input values for the new time step at the end of the horizon must be extrapolated, often by assuming a steady-state or repeating the final predicted control input. This provides a physically meaningful initial guess that respects the system's dynamics.

Warm starting is a defining feature of real-time iteration schemes used in NMPC. Solvers like those based on Real-Time Iteration (RTI) or acados are designed to perform only one (or a few) SQP iterations per control step. They rely entirely on a high-quality warm start from the previous step's solution to provide a near-optimal control input within the strict time budget. This makes warm start not just an optimization but a fundamental requirement for stability.

• Cold Start: Initializes the solver with a default guess (often zeros). Leads to longer, variable solve times, making real-time performance unpredictable. Used primarily for the very first step or after a major system reset.
• Explicit MPC: This method pre-computes the optimal control law offline as a piecewise affine function of the state, eliminating the online optimization problem entirely. Therefore, Explicit MPC does not require a warm start, as the control action is found via a simple lookup table evaluation.

A consistent warm start contributes to nominal stability by ensuring the optimizer converges to a consistent region of the solution space. It also aids in maintaining feasibility. If the previous solution satisfied all state constraints and input constraints, a properly shifted warm start is likely to be a feasible initial point for the next problem, helping the solver find a feasible solution quickly. This is critical for safety-critical systems where constraint violation is unacceptable.

In Nonlinear MPC (NMPC) for legged robots or manipulators, solving the optimization within a tight sampling period (e.g., 1-10 ms) is paramount. A warm start provides the Sequential Quadratic Programming (SQP) solver with an initial guess very close to the new optimum, as robot states and optimal controls evolve smoothly between time steps. This is essential for real-time optimization (RTO) on embedded hardware.

In chemical plants or refineries, Economic MPC controllers continuously optimize for profit or efficiency. Process dynamics are often slow-moving. Using the prior solution as a warm start allows the Quadratic Programming (QP) or Nonlinear Programming (NLP) solver to converge in fewer iterations, enabling faster response to market price changes or feedstock variations while maintaining hard and soft constraints on temperatures and pressures.

Model Predictive Control (MPC) is used for local path planning and vehicle stabilization. As the vehicle moves along a planned trajectory, the optimal control problem at time k+1 is a slight perturbation of the problem at time k. A warm start, often using a shifted initialization of the previous control sequence, ensures the online optimizer finds a feasible, optimal trajectory within the perception-planning-control loop latency budget, crucial for safe navigation.

Finite Control Set MPC (FCS-MPC) for inverters and drives involves solving mixed-integer problems at very high frequencies (kHz). While the discrete switching states change, the underlying current and voltage trajectories are continuous. A warm start from the previous optimal switching sequence or a relaxed solution provides a hot starting point for the solver, reducing computational jitter and enabling higher switching frequencies or more complex cost functions.

For satellite orientation or drone flight control, MPC must handle nonlinear dynamics and actuator constraints. The optimal control problem (OCP) is solved repeatedly as sensor data updates the state estimate. A warm start is highly effective because the spacecraft's angular velocity and control torques are continuous between samples. This allows the use of more accurate nonlinear models without violating the real-time compute budget.

Warm start is not a panacea. Its effectiveness depends on:

• Solver Type: Interior-point methods benefit greatly, while active-set methods may see less gain.
• Problem Changes: Large, abrupt changes in references, constraints, or model parameters (e.g., a robot picking up a heavy object) can make the previous solution a poor initial guess.
• Numerical Stability: In Nonlinear MPC, the warm-started point must still be feasible or near-feasible for the new problem to aid convergence. Poorly managed, it can sometimes lead to slower convergence or numerical issues.