[Paper Review] Control of nonlinear underactuated systems

Dave Auckly, Lev Kapitanski+1 more|ArXiv.org|Jan 29, 1999

Dynamics and Control of Mechanical Systems4 citations

TL;DR

This paper introduces a novel control design method for nonlinear underactuated systems that generates an infinite-dimensional family of control laws, each associated with a natural Lyapunov function. The approach successfully stabilizes systems that are open-loop unstable and cannot be stabilized by linear control, as demonstrated on the inverted pendulum cart and an abstract nonlinear system.

ABSTRACT

In this paper we introduce a new method to design control laws for non-linear underactuated systems. Our method produces an infinite dimensional family of control laws, whereas most control techniques only produce a finite dimensional family. These control laws each come with a natural Lyapunov function. The inverted pendulum cart is used as an example. In addition, we construct an abstract system which is open loop unstable and cannot be stabilized using any linear control law, and demonstrate that our method produces a stabilizing control law.

Motivation & Objective

• Address the challenge of stabilizing nonlinear underactuated systems, which are common in robotics and mechanical systems but difficult to control with standard methods.
• Overcome the limitation of existing control techniques that yield only finite-dimensional families of control laws.
• Develop a systematic method that guarantees stability via naturally associated Lyapunov functions.
• Demonstrate the method's effectiveness on systems that are inherently unstable and unresponsive to linear control strategies.

Proposed method

• Propose a control design framework based on a constructive method that generates an infinite-dimensional family of control laws for nonlinear underactuated systems.
• Utilize a Lyapunov-based approach where each control law is intrinsically linked to a Lyapunov function, ensuring stability by construction.
• Apply the method to the inverted pendulum on a cart as a canonical example to illustrate its practical implementation.
• Construct an abstract nonlinear system that is open-loop unstable and provably non-stabilizable by any linear control law to test the method's robustness.
• Use differential geometric and control theoretic tools to derive control laws that satisfy the necessary stability conditions.
• Ensure the control laws are explicitly computable and depend on a continuous family of parameters, enabling flexibility in design.

Experimental results

Research questions

• RQ1 Can a systematic method be developed to generate an infinite-dimensional family of stabilizing control laws for nonlinear underactuated systems?
• RQ2 Can such a method guarantee stability through a naturally associated Lyapunov function for each control law?
• RQ3 Can the method stabilize systems that are open-loop unstable and cannot be stabilized by linear control laws?
• RQ4 How does the method perform on benchmark nonlinear systems such as the inverted pendulum cart?
• RQ5 What structural properties of the system allow this method to succeed where linear control fails?

Key findings

• The proposed method produces an infinite-dimensional family of control laws, significantly expanding the design space compared to traditional finite-dimensional approaches.
• Each control law in the family is associated with a natural Lyapunov function, ensuring asymptotic stability by construction.
• The method successfully stabilizes the inverted pendulum cart system, a classic example of a nonlinear underactuated system.
• An abstract nonlinear system was constructed that is open-loop unstable and non-stabilizable by any linear control law, yet the method produces a stabilizing control law for it.
• The results demonstrate the method's capability to handle systems where linear control techniques fail, highlighting its robustness and generality.
• The framework provides a constructive and systematic way to design stabilizing controllers for complex nonlinear systems without requiring linearization or restrictive assumptions.