Under the terms of the US/German MoU (Helicopter Aeromechanics) the Task IX - Modeling and Simulation for Rotorcraft Systems - is defined:
”The overall objective of this task is to improve the modeling accuracy and understanding of helicopter dynamics and control. Improved modeling and understanding of the important issues can be used to increase the fidelity of ground-based simulations, thus allowing early pilot evaluation during the development of new control systems, compatibility checks for improved safety, decreases in experimental flight testing, and hence a reduction in costs and risks.”
One of the recent subtasks under Task IX has been a disturbance rejection study [1], resulting in a UH-60 Black Hawk control equivalent turbulence simulation model [2].
Figure 1: Turbulence model extraction
As illustrated in Figure 1 the basic idea is to have a pilot loosely stabilize a helicopter in a turbulent (input T ) environment (e. g. hovering on the leeward side of a high building), and measure the pilot control inputs (P ) and the reaction (rates, velocities, . . . ) of the helicopter (x). In the off-line extraction phase the measured reaction x (which includes the reaction of the helicopter to both the turbulence and the pilot input) is fed into an inverse model of the helicopter, resulting in the corresponding control input P+T that would be necessary to produce the measured reaction. Again, P+T includes turbulence and pilot input. If the measured pilot input P is subtracted, an equivalent turbulence input Teq remains.
This equivalent turbulence input can then directly be used as an additional control input in any kind of simulator; without the need to gain and implement more complex turbulence models.
During the actual model extraction approach [1] it became clear that
”Ideally, an exact numerical inverse of the coupled MIMO model would be used.”
This paper will therefore present and discuss different approaches to invert dynamical systems.
At a glance
- Compares inversion methods using helicopter models and turbulent flight-test data
- Includes the Matlab source code for the sym2tf conversion function
- Uses a two-input, two-output, three-state MIMO example
- Applies feedback-based inversion to a forced mathematical pendulum
- 25 figures
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
Objectives and Topics
This work examines how dynamic systems can be inverted to reconstruct their inputs from their outputs. Motivated by the extraction of helicopter turbulence models, it compares analytical and feedback-based approaches and investigates their implementation for linear, multi-input multi-output, unstable and nonlinear systems.
- Transfer-function and state-space inversion
- Improper inverses, propering filters and numerical solvers
- Feedback-based inversion of MIMO systems
- Instability and right-half-plane transmission zeros
- Nonlinear pendulum and EC-135 helicopter examples
Excerpt from the Book
1 Introduction
Under the terms of the US/German MoU (Helicopter Aeromechanics) the Task IX - Modeling and Simulation for Rotorcraft Systems - is defined:
”The overall objective of this task is to improve the modeling accuracy and understanding of helicopter dynamics and control. Improved modeling and understanding of the important issues can be used to increase the fidelity of ground-based simulations, thus allowing early pilot evaluation during the development of new control systems, compatibility checks for improved safety, decreases in experimental flight testing, and hence a reduction in costs and risks.”
One of the recent subtasks under Task IX has been a disturbance rejection study [1], resulting in a UH-60 Black Hawk control equivalent turbulence simulation model [2].
As illustrated in Figure 1 the basic idea is to have a pilot loosely stabilize a helicopter in a turbulent (input δT) environment (e. g. hovering on the leeward side of a high building), and measure the pilot control inputs (δP) and the reaction (rates, velocities, . . . ) of the helicopter (x).
In the off-line extraction phase the measured reaction x (which includes the reaction of the helicopter to both the turbulence and the pilot input) is fed into an inverse model of the helicopter, resulting in the corresponding control input δP+T that would be necessary to produce the measured reaction. Again, δP+T includes turbulence and pilot input. If the measured pilot input δP is subtracted, an equivalent turbulence input δTeq remains.
This equivalent turbulence input can then directly be used as an additional control input in any kind of simulator; without the need to gain and implement more complex turbulence models.
During the actual model extraction approach [1] it became clear that
”Ideally, an exact numerical inverse of the coupled MIMO model would be used.”
This paper will therefore present and discuss different approaches to invert dynamical systems.
Chapter Summaries
1 Introduction: The authors introduce helicopter turbulence-model extraction and explain why a dynamic inverse is useful for recovering equivalent control inputs.
2 Transfer Function Inversion: A single-input, single-output example illustrates inversion by exchanging the numerator and denominator of a transfer function.
3 State-Space Inversion: The inverse state-space matrices are derived for a system with an invertible feedthrough matrix and demonstrated in simulation.
4 Improper Inverse: The chapter explains why strictly proper systems produce improper inverses and considers propering poles and solver selection.
5 Proper Inversion: A high-gain feedback loop provides an alternative way to implement an approximate inverse, with attention to gain and numerical tradeoffs.
6 The MIMO Case: A two-input, two-output example is used to examine numerical and symbolic inversion, filtering, feedback implementation and root-locus behavior.
7 Unstable Systems: An unstable example illustrates numerical difficulties and the use of feedback stabilization when evaluating an inverse.
8 Transmission Zeros: The authors consider right-half-plane transmission zeros and an all-pass filtering approach, including its time-delay implications.
9 Inversion of a Nonlinear System: A forced mathematical pendulum serves as an example of feedback-based inversion for a nonlinear system.
10 EC-135 Inversion: Analytical inversion with propering filters and direct proper inversion are applied to an EC-135 model using flight-test inputs.
Keywords
dynamic system inversion, transfer functions, state-space models, MIMO systems, propering filters, feedback control, transmission zeros, numerical simulation, nonlinear systems, helicopter dynamics, EC-135, turbulence modeling, Matlab, Simulink
Frequently Asked Questions
What is this work about?
It examines methods for constructing and simulating inverses of dynamic systems, particularly for reconstructing helicopter control inputs.
What are the central topics?
The work covers transfer functions, state-space models, proper and improper inverses, MIMO systems, instability, transmission zeros and nonlinear inversion.
What is the main objective?
The authors aim to present and discuss approaches to dynamic-system inversion motivated by helicopter turbulence-model extraction.
What methods are used?
The authors combine mathematical derivations and symbolic calculations with numerical examples and Matlab and Simulink simulations.
What does the main discussion cover?
It develops inversion methods from simple linear systems to MIMO, unstable and nonlinear examples, before applying them to a helicopter model.
Which keywords characterize the work?
Key terms include dynamic inversion, state-space models, propering filters, feedback, transmission zeros and helicopter dynamics.
How is the EC-135 model used?
It provides an application in which analytical inversion with propering filters is compared with direct proper inversion using flight-test input data.
Why are propering filters introduced?
They add fast poles to improper inverses so that the resulting systems can be implemented in state-space simulation.
What nonlinear example is examined?
The authors use a forced mathematical pendulum to demonstrate simulation of an inverse based on high-gain feedback.
- Arbeit zitieren
- Prof. Dr.-Ing. Jörg Buchholz (Autor:in), Wolfgang v. Grünhagen (Autor:in), 2007, Inversion Impossible?, München, GRIN Verlag, https://www.hausarbeiten.de/document/85745
