带输入饱和限制的LQR与基础MPC在实际中是否等效？

Great question—this is a super common point of confusion when moving between unconstrained and constrained optimal control frameworks. Let’s break down the key practical differences, even though there’s a mathematical overlap between the two:

1. Horizon: Infinite vs. Finite & Receding #

• LQR (even with saturation) is fundamentally an infinite-horizon controller. It computes a fixed feedback gain (or a set of gains for constrained cases) offline to minimize a cost function that stretches from the current moment to infinity. Once tuned, it runs as a simple state feedback loop online.
• Basic MPC uses a finite, receding horizon. At every time step, it optimizes inputs for the next N steps, but only executes the first input. The next iteration repeats the process with the latest measured state. This rolling optimization lets you adapt to real-time changes in references, constraints, or system behavior that LQR can’t handle easily.

2. Constraint Flexibility #

• Saturated LQR is typically limited to handling input magnitude constraints (upper/lower bounds on u). Most implementations use workarounds like anti-windup compensation to mitigate issues from saturation, rather than enforcing the constraint directly during optimization.
• Basic MPC excels here: you can directly encode state constraints (e.g., maximum pressure in a reactor, minimum position for a robot arm), input rate constraints (how fast u can change), and output constraints into the online optimization problem. These are strictly enforced at every step, which is critical for industrial safety and performance.

3. Computational Load & Real-Time Performance #

• Saturated LQR has near-zero online computation. The control law (gain matrix or lookup table) is computed offline, so online operation is just a simple matrix multiplication or lookup. This makes it perfect for high-speed systems where even microsecond delays matter (e.g., aircraft attitude control).
• Basic MPC requires solving a quadratic program (QP) online at every time step. While modern embedded hardware and fast QP solvers make this feasible for most industrial systems, high-dimensional models or long prediction horizons can push computational limits. That said, this online optimization is what lets MPC adapt to unmodeled disturbances or system drift.

4. Trajectory Tracking Capabilities #

• Saturated LQR is designed primarily for setpoint regulation—pulling the system back to a fixed point after a disturbance. Tracking time-varying reference trajectories requires adding complex feedforward loops, and results can be suboptimal under constraints.
• Basic MPC is built for trajectory tracking. You can directly incorporate a time-varying reference trajectory into the finite-horizon cost function, and the controller will generate smooth, constraint-compliant inputs to track it. This is a huge win for applications like automated assembly lines or autonomous vehicle path following.

5. Robustness to Disturbances & Unmodeled Dynamics #

• Saturated LQR relies on offline robustness analysis (e.g., pole placement, H∞ methods). If the system experiences unmodeled disturbances or drift, anti-windup can help, but it’s a reactive fix rather than a proactive one.
• Basic MPC gains robustness from its receding horizon: at every step, it uses the latest measured state to re-plan inputs, effectively correcting for disturbances in real time. Many MPC variants even let you model disturbance bounds directly in the optimization to build in safety margins.

Quick Rule of Thumb #

Use saturated LQR if you have a low-dimensional system, only need input saturation constraints, prioritize maximum real-time performance, and focus on setpoint regulation. Use basic MPC if you need to handle complex constraints, track time-varying trajectories, or adapt to real-world disturbances.

内容的提问来源于stack exchange，提问作者euraad