[Paper Review] Observer-based correct-by-design controller synthesis

Sofie Haesaert, Paul M.J. Van den Hof+1 more|arXiv (Cornell University)|Sep 11, 2015

Formal Methods in Verification12 references3 citations

TL;DR

This paper extends correct-by-design controller synthesis to partially observable linear time-invariant (LTI) systems using an observer-based output-feedback approach, ensuring quantifiable robustness to stochastic disturbances. By combining a state observer with a formally synthesized controller, the method achieves bounded trajectory deviation and significantly improves performance over feedforward designs in a smart building case study.

ABSTRACT

Current state-of-the-art correct-by-design controllers are designed for full-state measurable systems. This work first extends the applicability of correct-by-design controllers to partially observable LTI systems. Leveraging 2nd order bounds we give a design method that has a quantifiable robustness to probabilistic disturbances on state transitions and on output measurements. In a case study from smart buildings we evaluate the new output-based correct-by-design controller on a physical system with limited sensor information.

Motivation & Objective

• To extend correct-by-design controller synthesis from full-state measurable systems to partially observable LTI systems.
• To ensure quantifiable robustness against stochastic disturbances in state transitions and output measurements.
• To develop an output-feedback control architecture that maintains formal guarantees on system behavior.
• To validate the approach on a real-world smart building model with limited sensor data.
• To provide a formal interface between discrete abstractions and continuous stochastic systems using approximate bisimulation.

Proposed method

• Introduces a state observer to estimate unmeasurable states from noisy output measurements, enabling output-based control.
• Employs a two-stage design: first synthesizing a correct-by-design controller on a discrete abstraction of the system, then interfacing it with the continuous system via observer-based feedback.
• Uses second-order bounds to quantify the expected deviation between the abstracted and concrete system trajectories under stochastic disturbances.
• Applies optimal LQ and Kalman gain designs for the observer and controller, respectively, to minimize estimation and control errors.
• Utilizes approximate bisimulation relations to formally relate the abstracted (discrete) model to the concrete (continuous) system with bounded error.
• Employs PESSOA for discrete controller synthesis on a quantized state and action space, followed by interface construction with the physical system.

Experimental results

Research questions

• RQ1 Can correct-by-design controller synthesis be extended to systems with only partial state observations?
• RQ2 How can robustness to stochastic disturbances in state dynamics and sensor measurements be formally quantified in output-feedback control?
• RQ3 What is the achievable accuracy of the closed-loop system when using an observer-based controller compared to a feedforward interface?
• RQ4 How do observer gains and controller gains jointly affect the deviation between abstract and concrete system trajectories?
• RQ5 Can formal guarantees on system behavior be preserved when transitioning from full-state to output-feedback control in stochastic LTI systems?

Key findings

• The observer-based controller reduces the steady-state error bound from 0.4845 (feedforward) to 0.1240, representing a 74.4% improvement in accuracy.
• The initial state error is reduced from 3.9618 to 2.1194, indicating better transient performance under observer feedback.
• The state estimation error converges to zero over time, confirming the observer's effectiveness in tracking the true system state.
• The ambient temperature deviation from its mean is effectively bounded, demonstrating robustness to external stochastic disturbances.
• The proposed interface achieves a 74.4% reduction in long-term error compared to the feedforward approach, validating the effectiveness of observer-based feedback.
• The method successfully enables correct-by-design synthesis for partially observable LTI systems while maintaining formal robustness certificates under stochastic uncertainty.