[Paper Review] Asymptotic Tracking Control of Uncertain MIMO Nonlinear Systems with Less Conservative Controllability Conditions

B. Zhou, Xiucai Huang+1 more|arXiv (Cornell University)|Aug 3, 2022

Stability and Control of Uncertain Systems4 citations

TL;DR

This paper proposes a robust adaptive asymptotic tracking control scheme for uncertain MIMO nonlinear systems with unknown time-varying control gain and direction. By integrating a novel Nussbaum gain technique with a positive integrable function and a feasible auxiliary matrix, the method relaxes controllability conditions, achieves global asymptotic tracking without linearization, and handles intermittent actuator faults without fault detection, validated via robotic system simulations with zero steady-state error.

ABSTRACT

For uncertain multiple inputs multi-outputs (MIMO) nonlinear systems, it is nontrivial to achieve asymptotic tracking, and most existing methods normally demand certain controllability conditions that are rather restrictive or even impractical if unexpected actuator faults are involved. In this note, we present a method capable of achieving zero-error steady-state tracking with less conservative (more practical) controllability condition. By incorporating a novel Nussbaum gain technique and some positive integrable function into the control design, we develop a robust adaptive asymptotic tracking control scheme for the system with time-varying control gain being unknown its magnitude and direction. By resorting to the existence of some feasible auxiliary matrix, the current state-of-art controllability condition is further relaxed, which enlarges the class of systems that can be considered in the proposed control scheme. All the closed-loop signals are ensured to be globally ultimately uniformly bounded. Moreover, such control methodology is further extended to the case involving intermittent actuator faults, with application to robotic systems. Finally, simulation studies are carried out to demonstrate the effectiveness and flexibility of this method.

Motivation & Objective

• To address the challenge of asymptotic tracking in uncertain MIMO nonlinear systems with unknown time-varying control gain and direction.
• To relax the stringent controllability conditions typically required in MIMO systems, especially under actuator faults.
• To eliminate the need for fault detection and diagnosis modules by enabling automatic actuator failure compensation.
• To achieve global asymptotic tracking without linearization or approximation techniques.
• To extend the method to robotic systems with intermittent actuator faults, demonstrating robustness and effectiveness.

Proposed method

• Introduces a novel Nussbaum gain function to handle unknown control direction (sign) in time-varying control coefficients.
• Incorporates a positive integrable function ν(t) = 0.5e^(-0.5t) in the control law to ensure convergence and handle time-varying uncertainties.
• Employs a feasible auxiliary matrix α(·) to relax the controllability condition, allowing broader system applicability beyond prior works.
• Designs a robust adaptive control law that does not require linearization or neural network approximation, reducing computational complexity.
• Uses a Lyapunov-based stability analysis to prove global uniform ultimate boundedness of all closed-loop signals.
• Extends the scheme to handle intermittent actuator faults by modeling effectiveness via a time-varying matrix ρ(t), without requiring fault detection.

Experimental results

Research questions

• RQ1 Can asymptotic tracking be achieved for uncertain MIMO nonlinear systems with unknown time-varying control gain and direction?
• RQ2 How can the controllability condition be relaxed to include more practical and less conservative scenarios?
• RQ3 Can the control scheme compensate for intermittent actuator faults without relying on fault detection and diagnosis?
• RQ4 What is the impact of using a non-diagonal auxiliary matrix α(·) on system controllability and stability?
• RQ5 How does the proposed method compare in performance and robustness to existing methods under undetectable faults?

Key findings

• The proposed control scheme achieves asymptotic tracking with zero steady-state error, as confirmed by simulation results showing tracking error e converging to zero.
• The control input signals remain bounded, as illustrated in Fig. 6, ensuring practical implementability.
• The method successfully handles intermittent actuator faults with ρ(t) non-differentiable at t=5s and not definitively positive/negative, demonstrating robustness.
• Compared to [11], the proposed method exhibits stronger robustness under undetectable faults, though control chattering increases when ν(t) is smaller.
• The auxiliary matrix α(·) chosen as in Eq. (51) satisfies Assumption 4, validating the feasibility of the relaxed controllability condition.
• The control structure is low-complexity and computationally efficient, avoiding linearization and approximation techniques.