[Paper Review] Robust Stabilization of Nonlinear Systems by Quantized and Ternary Control

Claudio De Persis|University of Twente Research Information|Oct 5, 2008

Stability and Control of Uncertain Systems12 references4 citations

TL;DR

This paper presents a robust semi-global practical stabilization method for nonlinear systems using quantized and ternary controllers, leveraging a discontinuous semi-global backstepping lemma with logarithmic quantization. It establishes that minimum-phase uncertain systems can be stabilized with finite control values, providing bandwidth estimates and enabling implementation via simple switched ternary control.

ABSTRACT

Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems.

Motivation & Objective

• Address the challenge of stabilizing nonlinear continuous-time systems when feedback information is constrained by limited bandwidth.
• Develop a robust control strategy using a finite number of control values (quantized or ternary) to handle parametric uncertainties.
• Provide a theoretical framework based on a discontinuous version of the semi-global backstepping lemma for systems with partial-state measurements.
• Estimate an upper bound on the required communication bandwidth for the proposed quantized control scheme.
• Demonstrate the feasibility of semi-global practical stabilization using a simple ternary controller, avoiding complex adaptive or dense quantization schemes.

Proposed method

• Adapt the semi-global backstepping lemma to handle discontinuous control laws under logarithmic quantization of the state or measurement.
• Apply a discontinuous backstepping tool to systems with uncertain parameters, ensuring robustness via Lyapunov-based analysis.
• Use a logarithmic quantizer to discretize the feedback signal, enabling finite-data transmission while maintaining stability.
• Introduce a hysteresis-based quantization scheme to prevent chattering and ensure robustness in the presence of measurement errors.
• Propose a ternary controller (on/off/on) as a simplified alternative to quantized control, reducing implementation complexity.
• Estimate the required bandwidth by analyzing the data rate needed to transmit quantized or ternary control signals over a communication channel.

Experimental results

Research questions

• RQ1 Can minimum-phase nonlinear systems with parametric uncertainty be semi-globally practically stabilized using only a finite number of control values?
• RQ2 How can a discontinuous backstepping approach be adapted to work under state quantization while preserving robustness?
• RQ3 What is the upper bound on the required communication bandwidth for implementing the proposed quantized control scheme?
• RQ4 Can a simple ternary controller achieve semi-global practical stabilization without requiring dense quantization or complex adaptive mechanisms?
• RQ5 How does the proposed method compare in complexity and robustness to existing approaches relying on input-to-state stability or system copies?

Key findings

• A discontinuous version of the semi-global backstepping lemma enables robust practical stabilization of uncertain minimum-phase nonlinear systems using quantized feedback.
• The proposed quantized control scheme ensures that trajectories converge to a residual set of size proportional to the quantization density, with the set size bounded independently of initial conditions.
• An upper bound on the required communication bandwidth is derived, depending on the system's Lyapunov function and quantization parameters.
• Semi-global practical stabilization is achieved using a simple ternary controller, which switches between three discrete control levels based on state regions.
• The method avoids the need for dense quantization or input-to-state stability assumptions, offering a computationally lighter alternative to prior approaches.
• The framework is applicable to output-feedback control problems, including systems with relative degree ≥2 and globally stable zero dynamics, via dynamic output feedback with quantized/ternary control.