Inversion Impossible?

Table of Contents

1 Introduction

2 Transfer Function Inversion

3 State-Space Inversion

4 Improper Inverse

5 Proper Inversion

6 The MIMO Case

6.1 Example

6.2 Numerical Inversion

6.3 Analytical Inversion

6.4 Propering Filter

6.5 Proper Inversion

6.5.1 Analytical Proper Inversion

6.5.2 Inverse via Limit

6.5.3 Root Locus

6.5.4 Implementation of the Proper Inverse

7 Unstable Systems

8 Transmission Zeros

9 Inversion of a Nonlinear System

10 EC-135 Inversion

A Matlab Function sym2tf

Objectives and Research Scope

This paper explores various computational and analytical methods for inverting dynamical systems, particularly focusing on the challenges presented by multi-input-multi-output (MIMO) systems, improper transfer functions, and nonlinear models. The primary goal is to provide robust techniques for accurate system inversion that can be implemented within simulation environments like MATLAB and Simulink.

• Inversion of linear time-invariant (LTI) and nonlinear systems.
• Handling of improper transfer functions using "propering" filters.
• Implementation of robust MIMO inversion techniques, including feedback-based implicit inversion.
• Mitigation of instabilities in system inversion via transmission zero mirroring.
• Practical application of inversion techniques to rotorcraft flight dynamics, specifically the EC-135 model.

Excerpt from the Book

Summary of Chapters

1 Introduction: This chapter defines the research task under the US/German MoU and introduces the concept of helicopter disturbance rejection and turbulence model extraction.

2 Transfer Function Inversion: This section covers the fundamental approach to inverting SISO LTI systems by exchanging the numerator and denominator of a transfer function.

3 State-Space Inversion: The chapter details the matrix-based inversion process for state-space representations of dynamical systems.

4 Improper Inverse: This chapter defines properness in transfer functions and presents solutions for non-realizable improper systems using high-frequency "propering" poles.

5 Proper Inversion: An alternative, feedback-based approach for inversion is introduced, proving its effectiveness for complex systems by placing the model into a closed loop.

6 The MIMO Case: This extensive chapter applies inversion techniques to multi-input-multi-output systems, covering numerical and analytical methods, propering filters, and specific implementation strategies.

7 Unstable Systems: The authors discuss the impact of system instabilities and how numerical computation errors can arise when inverting unstable models.

8 Transmission Zeros: This chapter examines the problem of right-half-plane transmission zeros and proposes a "mirror and compensate" method using all-pass filters.

9 Inversion of a Nonlinear System: The chapter demonstrates how the previously discussed feedback-based inversion method can be successfully applied to nonlinear dynamical systems like a forced pendulum.

10 EC-135 Inversion: The book concludes with a comprehensive real-world application of the described inversion methods to a 10-state linear model of an EC-135 helicopter.

A Matlab Function sym2tf: Appendix A provides the source code for the custom MATLAB function required to convert symbolic transfer function matrices into numerical representations.

Keywords

System Inversion, MIMO, Transfer Function, State-Space, Improper Systems, Propering Filters, Helicopter Dynamics, Disturbance Rejection, Turbulence Modeling, MATLAB, Simulink, Nonlinear Systems, Transmission Zeros, Feedback Control, EC-135.

Frequently Asked Questions

What is the core focus of this work?

The work focuses on the methodologies and challenges of inverting dynamical systems, providing a framework for transforming system outputs back into their original inputs.

What types of systems are addressed?

The book covers Linear Time-Invariant (LTI) systems, specifically Single-Input-Single-Output (SISO) and Multi-Input-Multi-Output (MIMO) configurations, as well as nonlinear dynamic systems.

What is the primary goal of the research?

The objective is to achieve accurate model inversion, enabling improved simulation fidelity for rotorcraft control systems and disturbance rejection analysis.

Which scientific methods are primarily employed?

The methods include mathematical inversion of transfer functions, state-space matrix operations, symbolic computation in MATLAB, and the use of feedback loops to implicitly approximate inverses.

What is covered in the main body of the text?

The main body systematically progresses from basic SISO inversion to complex MIMO inversion, addressing issues like system properness, numerical stability, unstable systems, and non-linearities.

How is the effectiveness of the proposed methods verified?

Verification is performed through simulations in MATLAB/Simulink and the application of these techniques to real-world identified flight models of the UH-60 and EC-135 helicopters.

How does the "propering" filter work?

A propering filter appends high-frequency poles to an otherwise improper transfer function, making the system physically realizable in simulation environments without significantly altering the system dynamics within the frequency range of interest.

How does the feedback-based inversion method differ from explicit inversion?

Instead of calculating a mathematical inverse directly, the system is placed into a closed-loop structure with a high gain. This implicitly creates an inverse system that is inherently proper and stable.

How does the book handle right-half-plane transmission zeros?

The authors propose using an all-pass filter to "mirror" the transmission zeros from the right half-plane to the left half-plane, thereby stabilizing the inverse, followed by a time-delay compensation to restore the original system's phase characteristics.