⚡ electrical engineering

Adaptive MPC for Constrained Trajectory Tracking of Uncertain LTI System with Input-Rate Limits

This paper proposes a systematic adaptive Model Predictive Control (MPC) framework that ensures recursive feasibility and closed-loop stability for discrete-time uncertain linear systems tracking trajectories under simultaneous hard constraints on states, inputs, and input rates by integrating parameter estimation with a reformulated optimization routine to overcome temporal coupling challenges.

Original authors: Bishal Dey, Abhishek Dhar, Sumit kr. Pandey, Anindita Sengupta

Published 2026-05-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bishal Dey, Abhishek Dhar, Sumit kr. Pandey, Anindita Sengupta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to drive a car that you've never seen before, down a winding road you've never traveled. The car is a bit "wobbly" because you don't know exactly how heavy it is or how responsive its engine is (this is the uncertainty). Furthermore, you have strict rules: you can't drive faster than a certain speed (state constraints), you can't press the gas pedal harder than a certain limit (input constraints), and you can't slam the pedal from zero to full instantly—you have to ease into it gradually (input-rate constraints).

This paper presents a smart driving system called Adaptive Model Predictive Control (MPC) designed to handle this exact scenario. Here is how it works, broken down into simple concepts:

1. The Problem: The "Wobbly" Car and the Strict Rules

Most standard driving systems assume they know the car perfectly. If the car turns out to be heavier than expected, the system might crash or break the rules.

The Challenge: The authors are dealing with a system where the "engine" and "weight" (the math parameters) are unknown and can change.
The Difficulty: Usually, if you have to follow a specific path (trajectory) while obeying strict rules about how fast you can accelerate or decelerate, the math gets incredibly messy. It's like trying to solve a puzzle where the pieces keep changing shape, and you have to make sure you never touch the edges of the table.

2. The Solution: A "Learning" GPS

The authors created a system that acts like a GPS that learns the car's behavior while driving.

The "Error" Map: Instead of trying to drive the car from point A to point B directly, the system rewrites the problem. It asks, "How far off are we from the ideal path?" and focuses entirely on fixing that gap. This turns a complex "follow the road" problem into a simpler "stay in the center of the lane" problem.
The Adaptive Learner: Since the system doesn't know the car's true specs, it makes a guess. As it drives, it watches how the car actually reacts. If the car moves slower than the guess predicted, the system updates its internal model. It's like a student taking a test, seeing which answers were wrong, and immediately studying those specific topics to get better next time.
The "Tube" of Safety: To handle the fact that the system is still learning, it builds a safety buffer (a "tube") around the ideal path. Even if the car wobbles because the model isn't perfect yet, the system ensures the car stays inside this safe tube, never hitting the walls (the constraints).

3. The "Input-Rate" Twist

A unique part of this paper is the focus on input-rate limits.

The Analogy: Imagine you are steering a boat. You can't just spin the wheel 90 degrees instantly; the water resistance and the boat's momentum mean you have to turn the wheel smoothly.
The Innovation: Most previous methods ignored this "smoothness" rule or only handled it for simple cases. This paper figures out how to plan a series of smooth turns while the system is still learning the boat's weight. It ensures that the plan for the next turn is always possible, even if the previous turn was slightly different than expected.

4. The Guarantee: "Recursive Feasibility"

This is a fancy term for a very practical promise: "If we can solve the puzzle right now, we guarantee we can solve it again in the next second, and the one after that."

Because the system is learning and the rules are strict, there was a fear that the system might get stuck in a situation where no valid move exists (a dead end).
The authors proved mathematically that their "learning GPS" never gets stuck. It always finds a valid, safe path forward, no matter how the car's unknown parameters behave.

5. The Results: The Simulation

The authors tested this on a computer simulation of a two-wheeled vehicle (like a drone or a robot).

They gave the system a winding path to follow.
They made the vehicle's physics unknown and slightly different from the model.
The Outcome: The vehicle successfully followed the path, never broke the speed limits, never slammed the controls, and the "learning" part quickly figured out the vehicle's true behavior. The errors (the distance between where the car was and where it should be) shrank to zero over time.

Summary

In short, this paper teaches a robot how to drive a car it doesn't fully understand, while obeying strict rules about speed and smoothness. It does this by constantly updating its understanding of the car, planning several steps ahead, and mathematically guaranteeing that it will never run out of safe moves, ensuring a smooth and safe journey to the destination.

Technical Summary: Adaptive MPC for Constrained Trajectory Tracking of Uncertain LTI Systems with Input-Rate Limits

Problem Statement
This paper addresses the trajectory-tracking problem for discrete-time linear time-invariant (LTI) systems subject to bounded parametric uncertainty. The control design must simultaneously satisfy hard constraints on system states, control inputs, and, crucially, input rates (increments). While Model Predictive Control (MPC) is a standard framework for handling constraints, classical MPC relies on accurate plant models, which are often unavailable due to modeling errors and parameter variations. Existing adaptive MPC approaches often focus on regulation problems, omit input-rate constraints, or fail to address the temporal coupling introduced by input-rate limits under full parametric uncertainty. The specific challenge lies in determining the control input required for zero tracking error when system parameters are unknown, while ensuring that the time-varying admissible control set induced by input-rate constraints does not compromise recursive feasibility or closed-loop stability.

Methodology
The proposed solution is an Adaptive MPC framework that transforms the trajectory-tracking problem into a regulation problem using error coordinates relative to a time-varying reference trajectory. The methodology proceeds through several key steps:

System Reformulation: The system dynamics are expressed in terms of error states ( $x^e_k = x_k - x^r_k$ ) and error inputs ( $u^e_k = u_k - u^r_k$ ). This transformation allows the tracking problem to be treated as a regulation problem where the objective is to drive the error states to the origin.
Indirect Adaptive Law: A projection-based gradient descent adaptive law is employed to estimate the lumped uncertain parameters ( $\Theta = [A, B]$ ) online. The estimates are constrained to lie within a known convex hull ( $\Psi$ ) of possible parameter values to ensure boundedness.
Error Set Characterization: The paper rigorously defines sets bounding the errors induced by parameter adaptation ( $\tilde{x}^e_k$ ) and the variation in parameter estimates ( $\Delta \hat{\Theta}$ ). These sets are used to tighten the constraints in the MPC optimization problem.
Constraint Reformulation: State, input, and input-rate constraints are reformulated in the error-dynamics domain. A significant contribution is the handling of input-rate constraints, which introduce temporal coupling between successive control inputs. The admissible control set is shown to be time-varying due to its dependency on the previous control input, yet it remains a convex polytope.
Robust MPC Formulation: The control input is decomposed into a stabilizing state feedback term ( $Kx^e_k$ ) and a free optimization variable ( $v^e_k$ ). The MPC optimization problem is formulated using a nominal predicted model based on current parameter estimates. To ensure robustness against estimation errors, the constraints on the nominal predicted states are tightened by subtracting the calculated error sets ( $Z^\Theta_i$ and $Z^{\tilde{x}^e}$ ).
Stability and Feasibility Analysis:
- Recursive Feasibility: The paper establishes that despite the time-varying nature of the admissible control set, the optimization problem remains recursively feasible. This is proven by constructing a feasible candidate solution at the next time step based on the optimal solution of the current step, accounting for the shift in the input-rate constraint.
- Closed-Loop Stability: Using a Lyapunov-based analysis, the paper proves that the closed-loop system is bounded and that the tracking error asymptotically converges to zero. The analysis accounts for the boundedness of the parameter estimation errors and the disturbance terms arising from model mismatch.

Key Contributions

Systematic Adaptive MPC for Tracking: Unlike prior works focusing on regulation or partial constraints, this work provides a systematic solution for constrained trajectory tracking under full parametric uncertainty.
Handling Input-Rate Constraints: The paper explicitly addresses the complexity introduced by input-rate constraints, which couple successive control inputs and create a time-varying admissible set. It demonstrates how to maintain recursive feasibility in this specific context.
Rigorous Feasibility and Stability Proofs: The authors provide rigorous mathematical proofs for both recursive feasibility and closed-loop stability (convergence of tracking error and boundedness of physical states) despite the presence of uncertainty and time-varying constraints.
Constraint Tightening Strategy: The method utilizes adaptively estimated parameters and derived error bounds to tighten constraints in the optimization routine, ensuring that the actual system states satisfy the original physical constraints.

Simulation Results
The effectiveness of the proposed framework is validated through simulations on a second-order discrete-time plant with uncertain parameters. The system is tasked with tracking a time-varying reference trajectory while adhering to constraints on states ( $\|x\|_\infty \leq 2.25$ ), inputs ( $\|u\|_\infty \leq 3$ ), and input rates ( $\|\Delta u\|_\infty \leq 2.5$ ).

The simulations demonstrate that the system states successfully track the reference trajectory while remaining within the prescribed bounds.
Control inputs and their rates satisfy the imposed constraints throughout the simulation.
Parameter estimation errors are shown to remain bounded. A notable observation is that a sudden change in the reference trajectory at $t=30$ s induces a transient in the parameter estimation error, which subsequently decays as the adaptation law re-engages to track the new reference.
The error states converge to the origin, confirming the stability claims.

Significance and Claims
The paper claims to offer a computationally tractable and theoretically sound control framework for safety-critical applications (such as aerospace, autonomous vehicles, and robotics) where accurate trajectory tracking is required despite model uncertainties and strict physical limitations on actuators. The primary significance lies in overcoming the difficulty of guaranteeing constraint satisfaction and performance in the presence of both parametric uncertainty and input-rate constraints, a combination that renders standard recursive feasibility arguments inapplicable. The work asserts that by systematically utilizing estimated parameters and reformulating the MPC problem with appropriate constraint tightening, it is possible to achieve robust constrained trajectory tracking with guaranteed stability and recursive feasibility.