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Robust Model Predictive Control Design for Autonomous Vehicles with Perception-based Observers

Nariman Niknejad, Gokul S. Sankar, Bahare Kiumarsi, Hamidreza Modares

arXiv:2509.05201v1cs.ROcs.CVcs.LGcs.SYeess.SY

TL;DR

Gaussian, zero-mean noise assumptions may not capture perception errors in real-world settings, motivating robust control that can account for bounded, non-Gaussian uncertainty. The paper combines a zonotopic observer with LP-based MPC, and reports bounded estimation under heavy-tailed noise and improved performance relative to Gaussian-noise-based designs.

  • Problem

    Gaussian, zero-mean perception-noise assumptions may fail to capture the complex and structured noise encountered in real-world applications.

    Source: §I · p. 1

  • Method

    The framework combines a robust zonotopic observer with LP-based tube MPC that explicitly accounts for process and measurement noise with nonzero means.

    Source: §I · p. 2

  • Results

    Under heavy-tailed noise, the zonotopic observer provides bounded estimates while the Kalman filter degrades under incorrect Gaussian assumptions, and Perception-MPC is reported to improve overall performance over Gaussian-MPC.

    Source: A. Observer Design Simulation Example · p. 9, 4) MPC-Perception Performance Comparison · p. 11, §VII · p. 11

  • Takeaways & Limitations

    Constrained zonotopes represent uncertainty and invariant sets in a control formulation that the paper reports as more reliable and computationally efficient.

    Source: §VII · p. 11

    The paper identifies nonlinear systems, adaptive process-noise estimation, and validation on more diverse platforms and conditions as future research directions.

    Source: §VII · p. 11

Abstract

from arXiv · show

This paper presents a robust model predictive control (MPC) framework that explicitly addresses the non-Gaussian noise inherent in deep learning-based perception modules used for state estimation. Recognizing that accurate uncertainty quantification of the perception module is essential for safe feedback control, our approach departs from the conventional assumption of zero-mean noise quantification of the perception error. Instead, it employs set-based state estimation with constrained zonotopes to capture biased, heavy-tailed uncertainties while maintaining bounded estimation errors. To improve computational efficiency, the robust MPC is reformulated as a linear program (LP), using a Minkowski-Lyapunov-based cost function with an added slack variable to prevent degenerate solutions. Closed-loop stability is ensured through Minkowski-Lyapunov inequalities and contractive zonotopic invariant sets. The largest stabilizing terminal set and its corresponding feedback gain are then derived via an ellipsoidal approximation of the zonotopes. The proposed framework is validated through both simulations and hardware experiments on an omnidirectional mobile robot along with a camera and a convolutional neural network-based perception module implemented within a ROS2 framework. The results demonstrate that the perception-aware MPC provides stable and accurate control performance under heavy-tailed noise conditions, significantly outperforming traditional Gaussian-noise-based designs in terms of both state estimation error bounding and overall control performance.

I. INTRODUCTION

The introduction motivates perception-aware robust control because Gaussian, zero-mean noise models can miss structured perception errors, while efficient tube-based MPC formulations remain needed. The paper combines constrained-zonotope estimation and control with an LP-based MPC, and introduces the set representations used to formulate it.

I. INTRODUCTION

  • Conventional Gaussian, zero-mean models may fail to capture real-world perception noise, while computationally efficient tube-based MPC for perception-induced uncertainty remains an open need.The introduction also notes that quadratic MPC can be computationally intensive and LP-based MPC can exhibit idle or deadbeat behavior.Source: §I · p. 1
  • The proposed perception-driven tube MPC uses linear programming and accounts for process and measurement noise with nonzero means.It uses constrained zonotopes for uncertainty and invariant control sets, with a Minkowski-function cost and an ∞/∞ formulation to address idle or deadbeat behavior.Source: §I · p. 2
  • Simulations and omnidirectional-robot experiments compare Perception-MPC favorably with approaches relying on Gaussian noise assumptions.The experiments use CNN-based state estimation with a robust zonotopic observer and a lightweight model trained for a Husarion ROSbot XL.Source: §I · p. 2

II. PRELIMINARIES

  • Constrained zonotopes are presented as a computationally efficient representation of polytopes, with containment methods relevant to error-bounded observers.Source: §II · p. 2

A. Notations

  • The notation section defines symbols including I, N, R, and the vector of ones, and defines a ball centered at xd with radius ρ.Source: A. Notations · p. 2

B. Set Representations and Properties

  • The set-representation preliminaries describe polytopes as intersections of half-spaces and zonotopes as centrally symmetric polytopes; constrained zonotopes add generator constraints.Constrained zonotopes retain low computational complexity without requiring central symmetry.Source: B. Set Representations and Properties · p. 2

C. Containing Constrained Zonotopes

The section presents a linear-programming formulation for checking whether one constrained zonotope is contained in another.

C. Containing Constrained Zonotopes

D. Minkowski Function

The section defines set functions used in the formulation, then describes a bounded-state and bounded-input control problem with process noise represented by a constrained zonotope.

D. Minkowski Function

  • The Minkowski function gives the smallest nonnegative scale of a proper C-set containing a point, while the support function takes the supremum of inner products over a set.Source: D. Minkowski Function · p. 3
  • A polar set is defined by inequalities against every point in the original set; the section also lists support-function relations for inclusion, Minkowski sums, and linear transformations.Source: D. Minkowski Function · p. 3

III. PROBLEM SETTING AND FORMULATION

  • The problem formulation assumes convex constrained-zonotope bounds on state, input, and process noise, and uses bounded uncertainty to keep trajectories within safety constraints.The system matrices are known, while the observation process may be generated by an unknown nonlinear, high-dimensional model.Source: §III · pp. 3–4

1) Perception Map:

The section models the perception module as a noisy sensor, bounds its error near the operating point, and develops an observer and nominal-state deviation model for robust control.

1) Perception Map:

  • The perception map converts module output into an imperfect prediction of partial state information, with state-dependent noise that is smaller near the training data.The error is bounded within a ball around the operating point, and sensor noise is represented by a constrained zonotope.Source: 1) Perception Map · p. 4

2) State Estimator:

  • A static-gain observer estimates the system state from measurements, and its estimation error is defined as the difference between the true and estimated states.The state estimator assumes the pair (A, C) is observable.Source: 2) State Estimator · p. 4

3) Nominal System and State Deviation:

  • The control input combines a nominal input with feedback on the estimation deviation, and the resulting deviation dynamics depend on the feedback gain and observer error.Bounding both estimation error and state deviation supports choosing nominal states and inputs that keep the actual system within state and input constraints.Source: 3) Nominal System and State Deviation · p. 4

4) Robust Invariant Set Principles:

Bounded estimation errors and state deviations let the nominal trajectory be chosen so the actual system remains within state and input constraints. The formulation uses zonotopic invariant sets and expresses robust MPC as a linear program.

4) Robust Invariant Set Principles:

  • Bounded estimation error and state deviation sets let nominal states and inputs be selected so the actual system adheres to state and input constraints.The estimation-error and state-deviation sets are design parameters that affect overall system performance.Source: 4) Robust Invariant Set Principles · pp. 4–5
  • λ-contractivity of the error and deviation sets prevents those quantities from growing unbounded and guarantees convergence to the actual state at a rate of at least λ.A λ-contractive set maps the next state into λ times the set.Source: 4) Robust Invariant Set Principles · p. 4
  • Tightening nominal input and state constraints by the deviation and estimation-error sets preserves true-state and control constraint satisfaction.The stated nominal constraints are ¯u ∈ Su ⊖ K S˜x and ¯x ∈ Sx ⊖ (Se ⊕ S˜x).Source: 4) Robust Invariant Set Principles · p. 5

B. Problem Formulation

  • The robust MPC uses Minkowski-function stage and terminal costs to cast the finite-horizon optimization as a linear program.The formulation includes a prediction horizon, terminal constraint set, estimation-error set, state-deviation set, and state and input constraints.Source: B. Problem Formulation · p. 5

IV. ROBUSTLY BOUNDING THE ESTIMATION ERROR AND STATE DEVIATION

The observer design relaxes Gaussian, white, and zero-mean noise assumptions and instead bounds estimation error in a constrained zonotope. Its gain is selected to provide bounded error with a prescribed convergence rate.

A. Bounding the Estimation Error

  • The observer gain is designed to bound estimation error with a prescribed convergence rate despite colored, biased, non-Gaussian, or non-zero-mean disturbances.The paper notes that Kalman-gain optimality assumes white, zero-mean Gaussian noise and that a Luenberger observer can introduce bias or instability under non-ideal noise.Source: A. Bounding the Estimation Error · pp. 5–6
  • Under the stated assumptions, the error dynamics propagate a constrained-zonotope estimation-error set using the sensor and process noise sets.Theorem 1 gives conditions on observer gain L for the estimation-error set to be contractive with ratio λL.Source: A. Bounding the Estimation Error · p. 6
  • Zonotope propagation through the error dynamics, followed by containment in a λ-contracted initial set, establishes a robust invariant estimation-error bound.The construction uses constrained zonotopes and the contraction result to keep the estimation error within its predefined boundary.Source: A. Bounding the Estimation Error · p. 6

B. Bounding the State Deviation

The state-deviation bound is built from constrained zonotopes and the error dynamics, with a feedback gain chosen to make the deviation set contractive. This keeps deviations within a predefined boundary over time.

B. Bounding the State Deviation

  • Given an initial deviation bound and an observer whose error set is robust invariant, the deviation dynamics propagate the state deviation within a constrained zonotope.The propagation accounts for sensor and process noise sets.Source: B. Bounding the State Deviation · p. 6
  • The resulting bounded deviation set keeps the state deviation within its predefined constrained-zonotopic boundary for all time steps.The result is derived from the contraction construction and the deviation propagation lemma.Source: B. Bounding the State Deviation · p. 6
  • A feedback gain K can make the state-deviation set contractive with ratio λ˜x under the stated assumptions.The theorem specifies the gain as part of the state-deviation dynamics.Source: B. Bounding the State Deviation · pp. 6–7

V. ROBUST MPC DESIGN FOR SYSTEMS WITH PERCEPTION-BASED OBSERVERS

The design combines an online MPC optimization with offline observer, feedback, and terminal-set design. Minkowski-function costs support the terminal stability conditions, while a shared slack variable balances cumulative state and input costs.

V. ROBUST MPC DESIGN FOR SYSTEMS WITH PERCEPTION-BASED OBSERVERS

  • The terminal gain Kf is designed to satisfy the MPC terminal cost condition, with the terminal cost expressed as a Minkowski function.The gain depends on the stage and terminal costs; the terminal-set choice sets an aspiration level, with a larger set relaxing the inequality.Source: §V · p. 7
  • The maximum-volume ellipsoidal terminal set is constrained to be contractively invariant under A + BKf and to satisfy tightened state and input constraints.A zonotopic approximation can preserve a purely linear constraint formulation for the terminal-set inclusion.Source: §V · pp. 7–8
  • Minimizing slack variable β jointly bounds cumulative state-tracking error and actuator effort, preventing persistent idle or dead-beat behavior while preserving the stated stability results.The optimizer tightens whichever cumulative norm currently dominates.Source: §V · p. 8
  • Offline design computes observer, feedback, and terminal gains and tightened constraints; online, the controller solves MPC, applies its control, and refreshes the estimate from perception measurements.The algorithm obtains a measurement through image capture and the perception map, then updates the estimate with the observer.Source: §V · pp. 7–8

VI. EXPERIMENTAL RESULTS

The experiments assess the state estimator from Theorem 1 against a standard Kalman filter and compare the perception-based MPC with MPC using Kalman-based estimates. The evaluation includes cumulative costs and estimated and ground-truth trajectories.

VI. EXPERIMENTAL RESULTS

  • The experimental section compares the Perception-MPC framework with an MPC scheme using Kalman-based state estimates.The comparison includes cumulative costs and estimated and ground-truth trajectories.Source: §VI · p. 8
  • The state-estimation assessment compares the estimator derived from Theorem 1 with a standard Kalman filter that presumes Gaussian sensor noise.Source: §VI · p. 8
  • Fig. 2 presents a state-estimation comparison between the zonotopic observer and the Kalman gain.Source: Figure 2 · p. 9

A. Observer Design Simulation Example

The simulation tests a zonotopic observer under biased, heavy-tailed Laplace noise and compares it with a Kalman gain designed under a Gaussian-noise assumption. The zonotopic observer maintains bounded estimates but converges more slowly to the reference trajectory.

A. Observer Design Simulation Example

  • The perception noise is modeled as a Laplace distribution with deterministic bias c_v = 0.5 and spread G_v = 0.5.Source: A. Observer Design Simulation Example · p. 9
  • Under heavy-tailed noise, the zonotopic observer provides bounded estimates, while the Kalman filter degrades under incorrect Gaussian-noise assumptions.The zonotopic observer converges more slowly to the reference trajectory because of its conservative gain.Source: A. Observer Design Simulation Example · p. 9

B. Real-world Implementation

The real-world implementation combines a ROS2-controlled omnidirectional robot with camera-based position estimation from RobotPerceptionNet. It constructs zonotopic error and deviation bounds and derives associated gains and a terminal set for the MPC.

B. Real-world Implementation

  • The ROS2 implementation runs Algorithm 1 on a Husarion ROSbot XL with a Jetson Orin Nano, using an Astra Pro Plus camera and Wi-Fi command communication.The robot operates in a bounded area with an overhead camera and a master control system.Source: 1) Hardware and Setup · p. 9, Figure 3 · p. 9

1) Hardware and Setup:

  • The robot’s x and y positions come from perception, while θ is measured by a SLAM-based algorithm; images are processed at 30 Hz with a 12-step prediction horizon.The experiment assumes an exact θ reading at each step.Source: 1) Hardware and Setup · p. 9

2) Perception Data and Convolutional Neural Network Map:

  • RobotPerceptionNet regresses robot coordinates from images using convolutional feature extraction and fully connected layers; its perception-error set uses 95% of the noise data points.The dataset maps 300×300 pixel images to physical coordinates in a 2×2 m² space.Source: 2) Perception Data and Convolutional… · p. 10, Figure 5 · p. 10

3) Computing the Invariant Sets and Associated Gains:

  • The observer gain and compensation feedback term are selected to keep estimation-error and state-deviation sets bounded, and the terminal set is a maximum-size ellipsoidal approximation with a three-generator inscribed zonotope.The inscribed zonotope keeps the MPC as a linear program.Source: 3) Computing the Invariant Sets and… · pp. 10–11

4) MPC-Perception Performance Comparison:

Perception-MPC keeps estimated states within prescribed bounds where Gaussian-MPC occasionally has larger estimation errors, and its control inputs remain within constraints. The conclusion presents constrained zonotopes and invariant sets as supporting more reliable, computationally efficient control under disturbances.

4) MPC-Perception Performance Comparison:

VII. CONCLUSION

  • The conclusion attributes more reliable performance to set-based approaches and says constrained zonotopes and invariant sets support accurate, computationally efficient control with bounded true-state errors despite significant disturbances.The framework integrates CNN-based perception, a robust zonotopic observer, and LP-based MPC.Source: §VII · p. 11
  • Future work includes nonlinear systems, adaptive process-noise estimation, and further validation across diverse robotic platforms and conditions.The authors describe these as promising directions that could enhance robustness and scalability assessment.Source: §VII · p. 11

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