Robust Control E-Book

Farhad Assadian

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley-ASME Press Series
• Sprache: Englisch

Seitenzahl: 654

Veröffentlichungsjahr: 2022

Beschreibung

Robust Control Robust Control Youla Parameterization Approach Discover efficient methods for designing robust control systems In Robust Control: Youla Parameterization Approach, accomplished engineers Dr. Farhad Assadian and Kevin R. Mallon deliver an insightful treatment of robust control system design that does not require a theoretical background in controls. The authors connect classical control theory to modern control concepts using the Youla method and offer practical examples from the automotive industry for designing control systems with the Youla method. The book demonstrates that feedback control can be elegantly designed in the frequency domain using the Youla parameterization approach. It offers deep insights into the many practical applications from utilizing this technique in both Single Input Single Output (SISO) and Multiple Input Multiple Output (MIMO) design. Finally, the book provides an estimation technique using Youla parameterization and controller output observer for the first time. Robust Control offers readers: * A thorough introduction to a review of the Laplace Transform, including singularity functions and transfer functions * Comprehensive explorations of the response of linear, time-invariant, and dynamic systems, as well as feedback principles and feedback design for SISO * Practical discussions of norms and feedback systems, feedback design by the optimization of closed-loop norms, and estimation design for SISO using the parameterization approach * In-depth examinations of MIMO control and multivariable transfer function properties Perfect for industrial researchers and engineers working with control systems, Robust Control: Youla Parameterization Approach is also an indispensable resource for graduate students in mechanical, aerospace, electrical, and chemical engineering.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Acknowledgments

Introduction

I.1 Why Feedback Control?

I.2 Why Youla Parameterization?

About the Companion Website

Part I: Control Design Using Youla Parameterization: Single Input Single Output (SISO)

1 Review of the Laplace Transform

1.1 The Laplace Transform Concept

1.2 Singularity Functions

1.3 The Laplace Transform

1.4 Inverse Laplace Transform

1.5 The Transfer Function and the State Space Representations (State Equations)

1.6 Problems

Notes

2 The Response of Linear, Time‐Invariant Dynamic Systems

2.1 The Time Response of Dynamic Systems

2.2 Frequency Response of Dynamic Systems

2.3 Frequency Response Plotting

2.4 Problems

Note

3 Feedback Principals

3.1 The Value of Feedback Control

3.2 Closed‐Loop Transfer Functions

3.3 Well‐Posedness and Internal Stability

3.4 The Youla Parameterization of all Internally Stabilizing Compensators

3.5 Interpolation Conditions

3.6 Steady‐State Error

3.7 Feedback Design, and Frequency Methods: Input Attenuation and Robustness

3.8 The Saturation Constraints

3.9 Problems

Notes

4 Feedback Design For SISO: Shaping and Parameterization

4.1 Closed‐Loop Stability Under Uncertain Conditions

4.2 Mathematical Design Constraints

4.3 The Neoclassical Approach to Internal Stability

4.4 Feedback Design And Parameterization: Stable Objects

4.5 Loop Shaping Using Youla Parameterization

4.6 Design Guidelines

4.7 Design Examples

4.8 Problems

Note

5 Norms of Feedback Systems

5.1 The Laplace and Fourier Transform

5.2 Norms of Signals and Systems

5.3 Quantifying Uncertainty

5.4 Problems

Notes

6 Feedback Design By the Optimization of Closed‐Loop Norms

6.1 Introduction

6.2 Optimization Design Objectives and Constraints

6.3 The Linear Fractional Transformation

6.4 Setup for Loop‐Shaping Optimization

6.5

‐norm Optimization Problem

6.6

Design

6.7

Solutions Using Matlab Robust Control Toolbox for SISO Systems

6.8 Problems

7 Estimation Design for SISO Using Parameterization Approach

7.1 Introduction

7.2 Youla Controller Output Observer Concept

7.3 The SISO Case

7.4 Final Remarks

8 Practical Applications

8.1 Yaw Stability Control with Active Limited Slip Differential

8.2 Vehicle Yaw Rate and Side‐Slip Estimation

Notes

Part II: Control Design Using Youla Parametrization: Multi Input Multi Output (MIMO)

9 Introduction to Multivariable Feedback Control

9.1 Nonoptimal, Optimal, and Robust Control

9.2 Review of the SISO Transfer Function

9.3 Basic Aspects of Transfer Function Matrices

9.4 Problems

10 Matrix Fractional Description

10.1 Transfer Function Matrices

10.2 Polynomial Matrix Properties

10.3 Equivalency of Polynomial Matrices

10.4 Smith Canonical Form

10.5 Smith–McMillan Form

10.6 MIMO Controllability and Observability

10.7 Straightforward Computational Procedures

10.8 Problems

11 Eigenvalues and Singular Values

11.1 Eigenvalues and Eigenvectors

11.2 Matrix Diagonalization

11.3 Singular Value Decomposition

11.4 Singular Value Decomposition Properties

11.5 Comparison of Eigenvalue and Singular Value Decompositions

11.6 Generalized Singular Value Decomposition

11.7 Norms

11.8 Problems

Note

12 MIMO Feedback Principals

12.1 Mutlivariable Closed‐Loop Transfer Functions

12.2 Well‐Posedness of MIMO Systems

12.3 State Variable Compositions

12.4 Nyquist Criterion for MIMO Systems

12.5 MIMO Performance and Robustness Criteria

12.6 Open‐Loop Singular Values

12.7 Condition Number and its Role in MIMO Control Design

12.8 Summary of Requirements

12.9 Problems

Notes

13 Youla Parameterization for Feedback Systems

13.1 Neoclassical Control for MIMO Systems

13.2 MIMO Feedback Control Design for Stable Plants

13.3 MIMO Feedback Control Design Examples

13.4 MIMO Feedback Control Design: Unstable Plants

13.5 Problems

14 Norms of Feedback Systems

14.1 Norms

14.2 Linear Fractional Transformations (LFT)

14.3 Linear Fractional Transformation Explained

14.4 Modeling Uncertainties

14.5 General Robust Stability Theorem

14.6 Problems

15 Optimal Control in MIMO Systems

15.1 Output Feedback Control

15.2

Control Design

15.3

‐ Robust Optimal Control

15.4 Problems

16 Estimation Design for MIMO Using Parameterization Approach

16.1 YCOO Concept for MIMO

16.2 MIMO Estimator Design

16.3 State Estimation

16.4 Applications

16.5 Final Remarks

17 Practical Applications

17.1 Active Suspension

17.2 Advanced Engine Speed Control for Hybrid Vehicles

17.3 Robust Control for the Powered Descent of a Multibody Lunar Landing System

17.4 Vehicle Yaw Rate and Sideslip Estimation

A: Cauchy Integral

A.1 Contour Definitions

A.2 Contour Integrals

A.3 Complex Analysis Definitions

A.4 Cauchy–Riemann Conditions

A.5 Cauchy Integral Theorem

A.6 Maximum Modulus Theorem

A.7 Poisson Integral Formula

A.8 Cauchy's Argument Principle

A.9 Nyquist Stability Criterion

B: Singular Value Properties

B.1 Spectral Norm Proof

B.2 Proof of Bounded Eigenvalues

B.3 Proof of Matrix Inequality

B.4 Triangle Inequality

C: Bandwidth

C.1 Introduction

C.2 Information as a Precise Measure of Bandwidth

C.3 Examples

C.4 Summary

D: Example Matlab Code

D.1 Example 1

D.2 Example 2

D.3 Example 3

D.4 Example 4

References

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Matrix form parameters.

Chapter 2

Table 2.1 Stability definitions.

Table 2.2 The denominator of the

th‐order Butterworth transfer function,

. ...

Chapter 3

Table 3.1 Important rational functions for characterizing closed‐loop perfor...

Table 3.2 Transfer functions for internal stability.

Chapter 4

Table 4.1 Requirements.

Chapter 5

Table 5.1 Robust stability conditions.

Chapter 7

Table 7.1 RMS of unsprung mass velocity results for the actual, the Youla an...

Chapter 16

Table 16.1 Heave acceleration results for both cases, with and without the e...

Chapter 17

Table 17.1 Nomenclature.

Table 17.2 Nomenclature.

Table 17.3 Nomenclature.

Table 17.4 Guidance parameter values and boundary conditions.

Table 17.5 Parameter values for Lander model.

Table 17.6 Parameter values for Lander model.

Table 17.7 Closed‐loop requirements for

,

, and

.

List of Illustrations

Introduction

Figure I.1 There is more uncertainty about the book's location with (a) clos...

Figure I.2 Robustness and performance.

Figure I.3 Brief history of feedback control.

Figure I.4 System model.

Figure I.5 System model with output disturbance.

Figure I.6 System model.

Figure I.7 Closed‐loop feedback system.

Figure I.8 Block diagram manipulation of system.

Figure I.9 Closed‐loop feedback system.

Figure I.10 Plant

‐domain.

Figure I.11 Frequency response of

for various gains

.

Chapter 1

Figure 1.1 Laplace transform process.

Figure 1.2 Riemann integral lower sum,

.

Figure 1.3 Riemann integral upper sum,

.

Figure 1.4 Mean value theorem development of singularity functions.

Figure 1.5 An mass with an applied force.

Figure 1.6 Impulse approximate response.

Figure 1.7 Doublet approximate response.

Figure 1.8 Impulse exact response (

).

Figure 1.9 Doublet exact response (

).

Figure 1.10 Unit step time domain.

Figure 1.11 Unit step

‐domain.

Figure 1.12 Decaying exponential time response.

Figure 1.13 Decaying exponential time domain.

Figure 1.14 Cosine function time response.

Figure 1.15 Cosine function

‐domain.

Figure 1.16 Decaying cosine time response.

Figure 1.17 Decaying cosine

‐domain.

Figure 1.18 Ramp input.

Figure 1.19 A mass with an applied force.

Figure 1.20 Forced mass block diagram.

Figure 1.21 Poles of the forced mass.

Figure 1.22 Impulse function.

Figure 1.23 Block diagram of a dynamic system.

Figure 1.24 State‐space block diagram.

Figure 1.25 An improper system.

Figure 1.26 An expansion of

into proper and improper components.

Figure 1.27 Derivative common limit.

Figure 1.28 Integral of

.

Figure 1.29 The integral is the common limit of the upper and lower sum.

Figure 1.30 Integration block diagram.

Figure 1.31 Time domain functions for Problem 1.1. (a) Truncated ramp. (b) D...

Figure 1.32 Controlled metal‐rolling device.

Chapter 2

Figure 2.1 Dynamic system.

Figure 2.2 Response of a dynamic system.

Figure 2.3 Response to step input.

Figure 2.4 Convolution.

Figure 2.5 Transmission blocking response.

Figure 2.6 Transmission blocking of a unit step input.

Figure 2.7

roots.

Figure 2.8 Unbounded response.

Figure 2.9 Step input.

Figure 2.10 Step response.

Figure 2.11 Minimum phase.

Figure 2.12 Non‐minimum phase.

Figure 2.13 Minimum phase response.

Figure 2.14 Non‐minimum phase response.

Figure 2.15 Pole–zero diagram of a first order system.

Figure 2.16 Single pole step response.

Figure 2.17 Second‐order system

‐domain.

Figure 2.18 Quadratic pole step response.

Figure 2.19 Polar form of the factor.

Figure 2.20 Effect of a zero on the second‐order response (

).

Figure 2.21 Step response of filters in the Butterworth pattern.

Figure 2.22 Frequency response.

Figure 2.23 Frequency response magnitude and phase.

Figure 2.24 Frequency response magnitude and phase in the time domain.

Figure 2.25 Bode magnitude plot, single pole.

Figure 2.26 Bode phase plot, single pole.

Figure 2.27 Magnitude response for a zero.

Figure 2.28 Minimum (solid) and non‐minimum (dashed) phase response for a ze...

Figure 2.29 (a) Minimum and (b) non‐minimum phase zeros.

Figure 2.30 Bode magnitude plot, second‐order pole.

Figure 2.31 Block diagram of time delay.

Figure 2.32 Bode magnitude plot, time delay.

Figure 2.33 Bode phase plot, time delay.

Figure 2.34 Time day in a block diagram.

Figure 2.35 Various delay approximations.

Figure 2.36 Bode Plot.

Figure 2.37 Nyquist Plot. The dotted lines show the closed‐loop frequency re...

Figure 2.38 Polar Plot.

Figure 2.39 Three‐tank hydraulic system.

Chapter 3

Figure 3.1 Feedback controller block diagram.

Figure 3.2 Open‐loop synthesis.

Figure 3.3 Single‐loop feedback with prefilter

, controller

, plant

, and...

Figure 3.4 Single‐loop feedback with disturbances.

Figure 3.5 Single‐loop feedback with unity feedback and with disturbances an...

Figure 3.6 Block diagram for determining return ratio.

Figure 3.7 Bode magnitude plot for

,

, and

. Controller commands

operat...

Figure 3.8

and

relationship.

Figure 3.9 A mass with a disturbance and damper force.

Figure 3.10 Block diagram of the mass and damper system.

Figure 3.11 SISO block diagram.

Figure 3.12 Block diagram of system.

Figure 3.13 Ill‐posed feedback.

Figure 3.14 Feedback system with an unstable pole/zero cancellation.

Figure 3.15 The internal model control scheme.

Figure 3.16 Open‐loop synthesis for a system with plant

.

Figure 3.17 Dynamically equivalent single‐loop feedback system.

Figure 3.18 Schematic for velocity control.

Figure 3.19 Schematic for position control.

Figure 3.20 Single‐loop system with unity feedback.

Figure 3.21 Block diagram.

Figure 3.22 Graphical interpretation of the return ratio and return differen...

Figure 3.23 Potential

and

yielding

.

Figure 3.24 Potential

and

where

.

Figure 3.25 Youla design with high‐gain plant.

Chapter 4

Figure 4.1 Marginally stable system.

Figure 4.2 Block diagram for harmonic consistency.

Figure 4.3 An intuitive justification of the Nyquist stability criterion. Im...

Figure 4.4 Nyquist plot with stability margins.

Figure 4.5 The phase margin from the intersection of

and

(valid only if

Figure 4.6 Crossover frequency on Nyquist plot.

Figure 4.7 Nyquist plot.

Figure 4.8 Crossover frequency on magnitude plot of

and

.

Figure 4.9 Block diagram of a controlled system,

,

.

Figure 4.10

‐Plane location of LHP zero.

Figure 4.11

and

for a LHP without cancellation in black and with cancell...

Figure 4.12

when canceling a zero.

Figure 4.13

when canceling a zero.

Figure 4.14 Magnitude plot of

for Example 4.2.

Figure 4.15 Magnitude plot of

for Example 4.2.

Figure 4.16 Magnitude plot of

for Example 4.2.

Figure 4.17 Magnitude plot of

for Example 4.2.

Figure 4.18

and

with pole placement.

Figure 4.19

and

without pole placement.

Figure 4.20 Magnitude plot of

for Example 4.3.

Figure 4.21 Magnitude plot of

for Example 4.3.

Figure 4.22 Magnitude plot of

for Example 4.3.

Figure 4.23 Magnitude plot of

for Example 4.3.

Figure 4.24

when canceling a pole.

Figure 4.25

when canceling a pole.

Figure 4.26 Magnitude plot of

for Example 4.4.

Figure 4.27 Magnitude plot of

for Example 4.4.

Figure 4.28 Magnitude plot of

for Example 4.4.

Figure 4.29 Magnitude plot of

for Example 4.4.

Figure 4.30

and

for a typical system.

Figure 4.31 Return ratio

for a typical system.

Figure 4.32 Youla design with high‐gain plant.

Figure 4.33 Stability margins of the controlled system.

Figure 4.34 Magnitude plot of

and

.

Figure 4.35 Magnitude plot of

and

.

is now at 2 rad/s.

Figure 4.36 Magnitude plot of

.

Figure 4.37 Magnitude plot of

,

, and

.

Figure 4.38 Closed‐loop step response. The controller produces unacceptably ...

Figure 4.39

when adding a pole.

Figure 4.40 Magnitude response of

.

Figure 4.41 Block diagram for Problem 4.5.

Figure 4.42 Block diagram of the active suspension with sprung mass velocity...

Chapter 5

Figure 5.1 Impulse response for Example 5.1.

Figure 5.2 Two disturbances.

Figure 5.3 An energy signal.

Figure 5.4 A power signal.

Figure 5.5 Impulse response for an arbitrary system.

Figure 5.6 An arbitrary complex dynamic system.

Figure 5.7 A simplified dynamic system.

Figure 5.8 A dynamic system with modeled uncertainty

.

Figure 5.9 True feedback.

Figure 5.10 Impact of uncertainty on closed‐loop.

Figure 5.11 Types of uncertainty.

Figure 5.12 Bode plot of

and appropriate filter.

Chapter 6

Figure 6.1 Frequency response constraint.

Figure 6.2 A dynamic system with upper and lower LFTs.

Figure 6.3 LFT with matrix representation of

.

Figure 6.4 Block diagram of a controlled system.

Figure 6.5 Block diagram organized in the LFT framework.

Figure 6.6 The dynamic system manipulated to match the LFT form.

Figure 6.7 Controlled system in form for

synthesis.

Figure 6.8 Simplified block diagram for internal model control.

Figure 6.9 Direct loop shaping with Youla parameter.

Figure 6.10 Model matching with Youla parameter.

Figure 6.11 Sensitivity transfer function weighting function.

Figure 6.12 Sensitivity and complementary sensitivity transfer functions upp...

Figure 6.13 Frequency division of the sensitivity transfer function for meet...

Figure 6.14 Frequency division of the complementary sensitivity transfer fun...

Figure 6.15 The

weighting function.

Figure 6.16 The

weighting function.

Figure 6.17 The

weighting function.

Figure 6.18 The

weighting function.

Figure 6.19 Bode magnitude plot of a typical

weighting function.

Figure 6.20 Block diagram for Problem 6.1.

Figure 6.21 Active suspension for eighth‐car model.

Figure 6.22 Active suspension for quarter‐car model.

Figure 6.23 Block diagram for Problem 6.8.

Figure 6.24 LFT.

Figure 6.25 Robust performance.

Chapter 7

Figure 7.1 Automotive application for controller output observer.

Figure 7.2 Classical observer design. (a) Classical observers. (b) Observer ...

Figure 7.3 General controller output observer.

Figure 7.4 Youla controller output observer.

Figure 7.5 Comparison of classical feedback loop and Youla feedforward trans...

Figure 7.6 Schematic.

Figure 7.7 Damper.

Figure 7.8 Closed‐loop frequency response.

Figure 7.9 SISO state estimation with D‐road PSD input (a) wheel stroke (b) ...

Figure 7.10 SISO state estimation with a speed bump input (a) Wheel stroke (...

Chapter 8

Figure 8.1 E‐locker.

Figure 8.2 A schematic of the torque transfer capability and limitation of t...

Figure 8.3 Bicycle model.

Figure 8.4 Controller structure.

Figure 8.5 Feedback control loop.

Figure 8.6 A third order plant model.

Figure 8.7 Actuator model.

Figure 8.8 Yaw rate response to the locking torque step input request.

Figure 8.9 Actuator effort to the locking torque step input request.

Figure 8.10 Bode plot of

and

(

) filter.

Figure 8.11 Yaw rate response to the locking torque step input request with ...

Figure 8.12 Actuator effort to the locking torque step input request compari...

Figure 8.13 Actuator effort to the locking torque step input request compari...

Figure 8.14 Actuator effort response to the locking torque request compariso...

Figure 8.15 Yaw rate response to the locking torque request comparison of al...

Figure 8.16 Block diagram of controller output observer (COO) using accelera...

Figure 8.17 Front steering bicycle model system diagram.

Figure 8.18 Bode plot of linear plant.

Figure 8.19 Bode plot for SISO case with large bandwidth.

Figure 8.20 Bode plot of closed‐loop transfer function of yaw rate estimatio...

Figure 8.21 Lateral acceleration estimation using different strategies with

Figure 8.22 Yaw rate estimation using different strategies with

road wheel...

Figure 8.23 Yaw rate estimation using different strategies with

road wheel...

Figure 8.24 Yaw rate estimation using different strategies with

road wheel...

Figure 8.25 Yaw rate estimation using different strategies with

mass.

Figure 8.26 Yaw rate estimation using different strategies with

.

Chapter 9

Figure 9.1 General Input–Output System.

Figure 9.2 LTI Input–Output System.

Figure 9.3 Full LTI Input‐Output System.

Figure 9.4 LTI Input–Output System.

Figure 9.5 General plant configuration with structured uncertainty.

Figure 9.6 Example 1 – system configuration 1.

Figure 9.7 Example 1 – system configuration 2.

Chapter 10

Figure 10.1 RLC circuit.

Chapter 11

Figure 11.1 SVD orthonormal basis.

Figure 11.2 Row and column spaces of

.

Figure 11.3 Multidimensional input/output map.

Figure 11.4 Projection of a vector onto a plane.

Figure 11.5 Gain as a function of input direction.

Figure 11.6 Altered view of the input space.

Figure 11.7 Input and output spaces of a dynamic system.

Figure 11.8 A system with 3 inputs and 2 outputs.

Figure 11.9 A system with 2 inputs and 3 outputs.

Figure 11.10 A multivariable dynamic system.

Chapter 12

Figure 12.1 Block diagram of a multivariable control system.

Figure 12.2 State feedback control system.

Figure 12.3 Open‐loop gain from the plant output.

Figure 12.4 Open‐loop gain from the plant input.

Figure 12.5 Block diagram of a multivariable control system.

Figure 12.6 Nyquist plot for a MIMO system.

Figure 12.7 MIMO feedback system without noise and output disturbance inject...

Figure 12.8 MIMO feedback system.

Figure 12.9 Frequency plot of

Figure 12.10 Frequency plot of

and

.

Figure 12.11 Frequency plot of

.

Figure 12.12 Comparison of plot of

and

for poorly conditioned

.

Figure 12.13 Comparison of plot of

and

for well‐conditioned

.

Figure 12.14 Frequency plot of

.

Figure 12.15 Frequency plot of

.

Chapter 13

Figure 13.1 Block diagram of a multi‐variable control system.

Figure 13.2 Interpretation of a neoclassical control system.

Figure 13.3 Open‐loop Youla control.

Figure 13.4 Internal model control.

Figure 13.5 Internal model control.

Figure 13.6 Block diagram of a multi‐variable control system.

Figure 13.7 Closed‐loop and return ratio singular values.

Figure 13.8 MIMO controller, plant and Youla transfer function matrices sing...

Figure 13.9 Step input responses.

Figure 13.10 MIMO feedback system without noise injection.

Chapter 14

Figure 14.1

norm of a TFM

in terms of its singular values.

Figure 14.2 Standard I/O relationship.

Figure 14.3

‐Norm of MIMO systems.

Figure 14.4 General linear control feedback design.

Figure 14.5 Disturbance decomposition.

Figure 14.6 Degrees of freedom for control design: (a) 1 degree of freedom c...

Figure 14.7 Lower LFT on systems.

Figure 14.8 Lower LFT.

Figure 14.9 Lower MIMO LFT.

Figure 14.10 Upper system LFT.

Figure 14.11 MIMO feedback system.

Figure 14.12 LFT block diagram.

Figure 14.13 Augmented closed‐loop feedback system.

Figure 14.14 Additive uncertainty topology.

Figure 14.15 Multiplicative uncertainty topology.

Figure 14.16 Inverse multiplicative uncertainty topology.

Figure 14.17 Multiplicative uncertainty topology.

Figure 14.18 Multiplicative uncertainty topology with factored

.

Figure 14.19 Upper LFT.

Figure 14.20 Frequency plot of MIMO mixed sensitivity loop‐shaping technique...

Figure 14.21 Augmented plant block diagram.

Figure 14.22 Linear Fractional Transformation (LFT).

Figure 14.23 Augmented classical control feedback loop.

Figure 14.24 A two‐mass system.

Chapter 15

Figure 15.1 PSD I/O relationship.

Figure 15.2 LQG using LFT framework.

Figure 15.3 Kalman filter topology.

Figure 15.4 Full LQG feedback diagram.

Figure 15.5

.

Figure 15.6 Augmented plant.

Figure 15.7 Lower LFT.

Figure 15.8 Standard I/O relationship.

Figure 15.9 Duality in optimization. (a) Weak duality. (b) Strong duality.

Figure 15.10

and

LFT.

Figure 15.11 LQG/

block diagram.

Figure 15.12

block diagram.

Chapter 16

Figure 16.1 Youla Controller Output Observer.

Figure 16.2 Comparison of classical feedback loop and Youla feedforward tran...

Figure 16.3 Bode for decoupled system.

Figure 16.4 Bode for the coupled system.

Figure 16.5 MIMO state estimation with D‐Road PSD input – sprung mass veloci...

Figure 16.6 MIMO state estimation with D‐Road PSD input – unsprung mass velo...

Figure 16.7 MIMO state estimation with D‐Road PSD input – wheel stroke.

Figure 16.8 MIMO state estimation with D‐Road PSD input – tire deflection.

Figure 16.9 Damping force feedback control structure.

Figure 16.10 Actual versus estimated damping force over a motorway at 100 kp...

Figure 16.11 Power spectral density of road profile for a motorway (bottom) ...

Chapter 17

Figure 17.1 Quarter‐car model with two active control forces.

Figure 17.2 General MIMO feedback control loop.

Figure 17.3 Sensitivity and complementary sensitivity closed‐loop transfer f...

Figure 17.4 Disturbance rejection formulation.

Figure 17.5 Frequency gain from road disturbance and actuator forces to susp...

Figure 17.6 Step responses of the nominal actuator dynamics.

Figure 17.7 Step responses of the nominal actuator dynamics including the un...

Figure 17.8 Suspension deflection and body acceleration frequency responses ...

Figure 17.9

, right, and MIMO Youla, left, comparison for a smooth bump roa...

Figure 17.10

, right, and scaled MIMO Youla with actuator dynamics, left, c...

Figure 17.11 Body acceleration, suspension deflection, and body travel frequ...

Figure 17.12 Actuator effort frequency responses to road disturbance input....

Figure 17.13 Time domain robustness test to a road bump for Youla, left, and...

Figure 17.14 Schematic of the HEV powertrain with a BISG and a CISG.

Figure 17.15 Block diagram of the simplified parallel diesel hybrid electric...

Figure 17.16 MISO feedback control loop.

Figure 17.17 MISO feedforward/feedback control loop.

Figure 17.18 Plant augmentation.

Figure 17.19

plant augmentation.

Figure 17.20

, (b), and Youla first method, (a), time response comparison f...

Figure 17.21

, (b), and Youla first method, (a), actuator effort comparison...

Figure 17.22

, MISO and SISO control on‐vehicle results, (a), simulation re...

Figure 17.23 Youla second method, time response for an engine reference spee...

Figure 17.24 Time domain robustness to a step reference input for Youla firs...

Figure 17.25 A high‐level guidance, navigation, and control block diagram.

Figure 17.26 A multibody dynamics model of a lunar landing system.

Figure 17.27 Translation trajectory traced by the mass center of the nonline...

Figure 17.28 Pitch trajectory traced by the body of the nonlinear lander mod...

Figure 17.29 Singular values of

,

, and

.

Figure 17.30 Tracking trajectory of the reference altitude (dash‐black), tru...

Figure 17.31 Tracking trajectory of the reference downrange (dash‐black), tr...

Figure 17.32 Tracking trajectory of the reference pitch angle (dash‐black), ...

Figure 17.33 The block diagram of the proposed control method for unstable u...

Figure 17.34 Stabilizing a plant with double pole at zero with root locus.

Figure 17.35 Stabilizing a plant with double pole at zero and a RHP pole wit...

Figure 17.36 Tracking trajectory of the reference pitch angle (dash‐black), ...

Figure 17.37 General controller output observer.

Figure 17.38 Youla controller output observer.

Figure 17.39 Front and rear steering bicycle car model system diagram with t...

Figure 17.40 Singular values of

,

, and

.

Figure 17.41 The yaw rate estimation results for different strategies.

Figure 17.42 The vehicle sideslip estimation for different strategies.

Figure 17.43 The yaw rate estimation results for different strategies with

Figure 17.44 The vehicle sideslip estimation results for different strategie...

Figure 17.45 The yaw rate estimations for different strategies with 30% addi...

Figure 17.46 The vehicle sideslip estimations for different strategies with ...

Figure 17.47 The yaw rate estimations for different strategies with

.

Figure 17.48 The vehicle sideslip estimations for different strategies with

Figure 17.49 The yaw rate estimation of different strategies with a sensor b...

Figure 17.50 The vehicle sideslip estimations for different strategies with ...

Figure 17.51 The vechile sideslip estimation of YCOO including sensor bias e...

Appendix A

Figure A.1 Functions and nonfunctions.

Figure A.2 Parameterization of arc length.

Figure A.3 Jordan and non‐Jordan arcs.

Figure A.4 Jordan curve.

Figure A.5 Smooth and non‐smooth curves.

Figure A.6 Basic Jordan contour.

Figure A.7 Generic Jordan contour.

Figure A.8 Contour

and enclosed region

.

Figure A.9 A simple contour to prove Greene's theorem.

Figure A.10 A point in the complex plane.

Figure A.11 The limit

.

Figure A.12 Line and surface integral in the complex plane.

Figure A.13 Mapping the

and

planes.

Figure A.14 Contour

and enclosed region

for Example A.2.

Figure A.15 Open disc about

with inner contour

.

Figure A.16 The modulus

and and argument

.

Figure A.17

contour.

Figure A.18 Polar coordinates of contour

.

Figure A.19 Polar coordinates for an application of the Cauchy Integral Theo...

Figure A.20 Contour encompassing the RHP.

Figure A.21 Poles and zeros of

within the RHP contour.

Figure A.22 A point

within the RHP contour.

Figure A.23 An arbitrary line and surface integral in the complex plane.

Figure A.24 Mapping of contour

.

Figure A.25 A Nyquist locus.

Figure A.26 Nyquist locus for a first‐order transfer function. Negative freq...

Figure A.27 Nyquist locus for small

.

Figure A.28 Nyquist locus for large

.

Figure A.29 Nyquist locus with poles on the

axis.

Figure A.30 Nyquist contour with poles on the

axis.

Appendix C

Figure C.1 Feedback Control Loop.

Figure C.2 Feedback control loop with

block.

Figure C.3 Youla in feedback control loop with

block.

Figure C.4 Single mass system with force input and velocity output.

Figure C.5 Closed‐Loop and Return Ratio Transfer Functions for Example C.1....

Figure C.6 Controller and Youla transfer functions for Example C.1.

Figure C.7 Single mass system with force input and position output.

Figure C.8 Closed‐Loop and Return Ratio Transfer Functions for Example C.2....

Figure C.9 Controller and Youla transfer functions for Example C.2.

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

Acknowledgments

Introduction

Begin Reading

A: Cauchy Integral

B: Singular Value Properties

C: Bandwidth

D: Example Matlab Code

References

Index

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Wiley‐ASME Press Series

Robust Control: Youla Parameterization ApproachFarhad Assadian, Kevin R. Mallon

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Robust Control

Youla Parameterization Approach

Farhad Assadian and Kevin R. MallonUniversity of California, Davis, USA

This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Library of Congress Cataloging‐in‐Publication Data

Names: Assadian, Farhad, 1960‐ author. | Mallon, Kevin R., 1990‐ author.

Title: Robust control : Youla parameterization approach / Farhad Assadian and Kevin R. Mallon, University of California, Davis, USA.

Description: Hoboken, NJ : John Wiley & Sons Ltd., 2022. | Series: Wiley‐ASME press series

Identifiers: LCCN 2021058893 (print) | LCCN 2021058894 (ebook) | ISBN 9781119500360 (cloth) | ISBN 9781119500353 (adobe pdf) | ISBN 9781119500308 (epub)

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LC record available at https://lccn.loc.gov/2021058893

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Cover Design: WileyCover Image: © VikaSuh/Getty Images; Image by Farhad Assadian and Kevin R. Mallon

To Professor John Brewer

Preface

In the past several decades, there have been many published control system design books in classical (PID), modern (state‐space), and neoclassical and optimal robust control. Then, why is it necessary to publish yet another control system design book?

To clearly answer this question, I need to say a few words about my own background. Before becoming a Professor in System Dynamics and Control, I have been a control engineer with both energy and automotive industries for over 30 years. In these industries, feedback is still a novelty and majority of feedback control technique is around designing PID controllers. Control theory has drastically advanced and yet with these advancements, the gap between industrial application of controls and theoretical controls has been increased. Majority of the current control publications are very much math heavy and make the assumptions that all practitioners and students have the right background to fully appreciate these theoretical development. Most importantly, in all these developments, the beauty of control design at times is not clearly addressed and hence, large number of the students and the practitioners are not still have a good understanding of why feedback control is needed.

In this book, we try to close this gap by illustrating that feedback control could be elegantly designed in the frequency domain using Youla parameterization approach. It is the system bandwidth which elegantly defines a trade‐off between performance and robustness. The father of quantitative feedback theory (QFT), Issac Horowitz, said that the bandwidth is like an open window during a hot summer night. If the window is kept widely open, then one will let the flies in and if the window is kept shut, then one will limit the summer night breeze. This simple and elegant metaphor defines the trade‐off or balance between performance (speed) and noise rejection capability in feedback control design.

Youla parameterization technique is not new; the technique was first developed in the 1970s [1] and addressed in several other existing textbooks, for instance [2, 3]. However, in this book, we will provide deeper insights with many practical applications in utilizing this technique in both single input single output (SISO) and multiple input multiple output (MIMO) frequency domain feedback control design. In addition, an estimation technique using Youla parameterization and controller output observer both for SISO and MIMO plants, for the first time, is introduced. Although, the emphasis of these two parts is on frequency domain, we will discuss optimal robust control approaches such as and using state‐space approaches in Chapter 5 of this book. We will provide comparative results through practical examples between the optimal robust control methods and the Youla parameterization approach.

We have to apologize for the level of mathematical rigor provided in this book. It goes without saying that control system design is a subject which requires mastery of basic mathematical concepts. We have tried hard to avoid theorem and proof format when introducing mathematical concepts and provided the level mathematics to give the readers the necessary background for understanding and appreciating the control theoretical concepts discussed in this book.

This book is divided into two parts: Part I covers SISO control and estimation design using the Youla parameterization approach, while Part II covers MIMO control and estimation design using Youla parameterization approach.

Part I of this book includes eight chapters:

In Chapter 1, a review of the Laplace transform is given. In this chapter, we tried to avoid many details in introducing the Laplace transform and concentrated on providing the properties of the Laplace transform. It is these properties will be utilized throughout this book for developing the main concepts.

In Chapter 2, the response of linear time‐invariant (LTI) dynamic systems is provided. A review of stability of LTI dynamic systems is given and frequency response including Bode and Nyquist diagrams are discussed.

In Chapter 3, we provide a discussion of feedback principals, a review of well‐posedness and internal stability is given, and the role of Youla parameter and interpolation conditions in assuring internal stability is provided. Complementary sensitivity transfer function and sensitivity transfer function are derived, and modeling uncertainty is discussed.

In Chapter 4, we provide the fundamentals of feedback design using Youla parameterization. We discuss algebraic and analytical constraints in feedback design and we reveal how Youla parameter could be utilized in designing a feedback control by considering the constraints and trade‐offs in SISO control design.

In Chapter 5, a review of signal and system size (norms) is provided. linear fractional transformation (LFT) platform and its use in control design are discussed including uncertainty modeling with the use of this platform. A brief review of state‐space method including concepts such as controllability and observability are discussed.

In Chapter 6, we discuss the Mixed Sensitivity Method and method for SISO robust control design. In this chapter, we provide insights to the frequency domain solution of control and selection of the weighting filters in shaping complementary sensitivity, sensitivity, and Youla transfer functions.

In Chapter 7, we discuss how Youla parameterization could be utilized in deriving a state and input estimation method. In this chapter, an introduction to Luenberger observer and Kalman filtering is provided including a discussion about the Controlled Output Observer approach. A step‐by‐step transformation approach from Luenberger observer to Youla parameterization estimation using the Controller Output Observer framework is developed.

In Chapter 8, we present two automotive examples, one for control example and another for estimation. The control example demonstrates yaw stability control with an active differential. The estimation example consists of vehicle yaw rate and sideslip estimation using the Youla controller output observer (YCOO) method. In the control example, we reveal the differences between using SISO robust control design using the standard Matlab toolbox and Youla parameterization hand computation technique. In the estimation example, we compare the result of YCOO to both linear and nonlinear Kalman filter method.

Part II of this book includes nine chapters.

In Chapter 9, we provide an overview of different multivariable feedback control methods and the importance of Youla parameterization approach in designing multivariable feedback control system. In the same chapter, we introduce multivariable transfer function matrix, Schur complement, and Rosenbrock's system matrix including its application for gaining insight into poles and zeros of a transfer function matrix.

In Chapter 10, Matrix Fractional Description, unimodular matrices, and the concept of polynomial matrices' equivalency with several insightful examples are introduced. A discussion on Smith canonical form and Smith‐McMillan form of a transfer function matrix including several examples is provided. The idea of coprimeness and several approaches for transforming a transfer function matrix to its equivalent stat‐space form are discussed.

In Chapter 11, we discuss the eigenvalue problem and matrix diagonalization using eigenvectors and eigenvalues. The diagonalization issues using eigenvalues and eigenvectors are discussed, and the use of singular value decomposition as a general method for transfer function matrices diagonalization is reviewed.

In Chapter 12, we discuss MIMO feedback principals. We introduce MIMO return ratio and return difference, complementary sensitivity and sensitivity transfer function matrices, and Youla transfer function matrix. Concepts such as Nyquist stability, well‐posedness, and internal stability for MIMO feedback systems are discussed. Some of the properties of singular value decomposition are reviewed and their implications on the limit of MIMO feedback performance are revealed.

In Chapter 13, we introduce the concept of MIMO feedback control design using Youla parameterization. We illustrate the simplicity of this technique by showing that for low‐order transfer function matrices, the MIMO control design could be easily performed by hand.

In Chapter 14, a review of vector signal and MIMO system size (norms) is provided. Linear fractional transformation (LFT) platform is revisited, and the uncertainty modeling for MIMO feedback systems with the use of this platform is discussed.

In Chapter 15, we utilize the norms discussed in the previous chapter to derive MIMO controllers. We discuss optimal control methods. We review the derivation of linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) and discuss the equivalency between and optimal control. We provide the state‐space solutions to and optimal robust control.

In Chapter 16, we revisit the system from Chapter 7 and discuss how Youla parameterization could be utilized in deriving state and input estimation method for MIMO systems. A step‐by‐step design approach for MIMO Youla parameterization estimation using the Controller Output Observer framework is developed.

In Chapter 17, we present three MIMO control examples and one MIMO estimation example. The control examples include active suspension control, advanced engine speed control for hybrid vehicles, and robust control for the powered descent of a multibody lunar landing system. The estimation example revisits the vehicle yaw rate and sideslip estimation example of Chapter 8, transforming the SISO problem into a MIMO one through the introduction of rear wheel steering. In the control examples, we reveal the differences between using MIMO robust control design using the standard Matlab toolbox and MIMO YCOO hand computation technique. In the estimation example, we compare the result of MIMO YCOO with both linear and nonlinear Kalman filter method.

University of California, DavisJuly, 2017

Farhad Assadian

Kevin R. Mallon

Acknowledgments

My sincere thanks and appreciation to Professor John Brewer who introduced me to the concept of Youla parameterization in 1990s. Unfortunately, Professor Brewer passed way on 17 October 2003 at the age of 65. His enthusiasms and passion for teaching control system design to his students was exemplary. He shall always be remembered as one of the best teachers by his students.

F.A.

Introduction

“It seems that without electronic, controls nothing works.”

I.1 Why Feedback Control?

Control systems are increasingly becoming an integral part of every piece of equipment that we humans use. Without control systems our lives would drastically change. From aircrafts to household appliances, control systems play a vital role in the operation of these machines. In the automotive domain, nowadays, the majority of vehicle attributes are software and control‐system based. It is therefore important for everyone, especially machine operators and technical managers, to have a basic understanding of control system operation and their impact. It is also becoming significantly important that all engineers, from every specialty, should be required to take at least one good control system design class throughout their academic education.

When it comes to control system design, our emphasis in this book is on feedback control system design, particularly model‐based feedback control design. We will describe what we mean by model based in the next few paragraphs. But, first let's clarify the need for feedback controls. It is interesting, especially in industry, that there is still not a good understanding about why feedback control is necessary.

Let's take a look at Figure I.1a, where a person without eyesight is trying to pick up a book, which is located on a table. From this person's point of view, there is an uncertainty about the exact position of this book. To reduce this uncertainty, the person would have to find the book by examining different locations on the table with their hands. Hence, it normally takes more effort or energy to overcome the uncertainty and locate this book. Now, imagine a person with eyesight, as in Figure I.1b. For this person, the exact location of the book is found by making a measurement using the person's eyesight and feeding back this information to the brain. Therefore, uncertainty about the position of the book is reduced by this feedback measurement. Interestingly, the effort and energy required to locate the book is also reduced when compared to the person without eyesight. We can say then that the fundamental reason for feedback in control system design is simply to reduce uncertainty. We will see that this uncertainty exist in many sources including the act of the measurement itself. Amazingly, the same uncertainty that gives rise to the use of feedback control itself and can result in its demise!

Another important takeaway from this discussion is that reducing uncertainty requires effort and energy and the source of this effort and energy is the actuator. We will relate the actuator effort and energy to system robustness when discussing robust control development in the following chapters.

There are several different methods that could be utilized in designing feedback controllers. Our intent in this book is to illustrate that the use of Youla parameterization technique is one of the most straightforward methods in designing model‐based feedback control system, especially for single‐input, single‐output (SISO) control system design. We will show that control system design becomes procedural with minimal calibration effort when implementing these controllers on the actual systems. Reducing controller calibration is significantly important to industry, as reducing calibration effort directly reduces development costs. We will argue that the elegance of Youla parameterization is its simplicity and in reducing controller calibration overhead.

Imagine designing a control system for a very complex system which could be mathematically described by nonlinear partial differential equations (PDEs). What are the choices to design a controller for this complex system?

Figure I.1 There is more uncertainty about the book's location with (a) closed eyes, compared to (b) open eyes.

Direct Control of Nonlinear PDEs

: As far as the authors of this book are aware, there are no methods which directly address the control design of nonlinear PDEs, especially the open loop unstable kinds. In addition, if there were control methodologies to address nonlinear PDEs directly, quantifying model errors and uncertainties would have been an extremely difficult task. The quote by George Box “Essentially, all models are wrong, but some are useful” should be kept in mind when complex systems are modeled using nonlinear PDEs.

Indirect Control by Model Reduction of the Nonlinear PDEs to

linear time‐invariant

(

LTI

) PDEs

: There are methods which address the control of LTI PDEs, such as boundary control of LTI PDEs using backstepping control method

[4]

, or robust control of infinite dimensional LTI using

control

[5]

.

It should be noted that at times, linearization of particular nonlinear systems results in loss of controllability. We will not address these systems in this book.

We will use model reduction techniques tp go one step further than indirect control by reducing the original complex model to LTI ordinary differential equations (ODEs). Here stands the trade‐off between complexity and uncertainty: As the complex model is reduced to a simpler model, naturally the complexity is reduced. However, uncertainty is increased due to loss of information. Therefore, the entire idea of robust control is to quantify these uncertainties in the control design phase.

There are generally two objectives in designing a feedback control system: performance and robustness. “Performance” indicates the ability of the control system to respond quickly and accurately under ideal conditions. “Robustness” indicates the ability of the control system to respond accurately in the face of disturbances, noise, or other sources of uncertainty. Generally (but not always), there is a trade‐off between robustness and performance: Speeding up the response requires sacrificing robustness, while increasing robustness requires sacrificing the nominal performance. This concept – the trade‐off between robustness and performance – is illustrated in Figure I.2.

More specifically, the conflicting measures are:

Figure I.2 Robustness and performance.

Figure I.3 Brief history of feedback control.

Stability (is the controller robustly stable?)

Command tracking and disturbance rejection

Sensor noise rejection

Actuator effort and saturation.

The history of control systems theory and the path to robust control is briefly outlined in Figure I.3.

I.2 Why Youla Parameterization?

In the prior section, we briefly discussed the benefits of feedback control and various options for model‐based control development. In the next few paragraph, we will elaborate more on the benefits of feedback control and the utility of Youla parameterization in feedback control design. This section presumes some knowledge of the Laplace transform, transfer functions, and block diagram algebra.

When dealing with model‐based controls, the starting point is a mathematical description of the system. Consider the simple system shown in Figure I.4, where is the plant model and is the control.,

In order to enforce perfect target following, where and is the Laplace operator, we have to select the controller . This very simple example illustrates that plan invertability is very important in control system design. We will elaborate on this point when designing feedback controllers using Youla parameterization method. Let's assume for a moment that this plant is invertible. What happens when we implement this simple controller on an actual plant?

In order to capture the reality a bit better, we added a disturbance signal at the output of the plant, as illustrated in Figure I.5. This disturbance signal could, for example, model a wind gust acting on a vehicle while driving. Based on the previous feedforward controller, we can write the following equations.

Figure I.4 System model.

Figure I.5 System model with output disturbance.

(1)

but

(2)

so therefore,

(3)

So, we are here stating the obvious point that if there is a disturbance or a source of uncertainty, the feedforward controller is not capable of coping with this. As a result, the target following in a feedforward loop deteriorates when compared to the situation when perfect knowledge of the plant was available. The output no longer follows the commanded input .

Let's now add a feedback to Figure I.5 as illustrated in Figure I.6.

The output can be written as a function of the target signal, , and disturbance signal, , as follows.

(4)

Two important points should be evident from this equation. First, we cannot simply set , as we did in the feedforward case, if we are designing for good target following and disturbance rejection. Second, if we set , then we will recover the target following of the feedforward controller by rejecting or minimizing the disturbance or uncertainty effect. There are other possible disturbances and uncertainties to consider, but the value of feedback controls in the presence of uncertainties has been made clear.

In the next few paragraphs, we provide a very basic introduction to Youla parameterization. The feedback control loop is revisited in Figure I.7. However, this time, we replaced the dynamic model of the plant with a generalized symbol and replaced the feedforward controller, with a generalized symbol for feedback controller .

The closed‐loop transfer function or the complementary sensitivity transfer function, , can be written as a function of and as follows:

(5)

Figure I.6 System model.

Figure I.7 Closed‐loop feedback system.

Figure I.8 Block diagram manipulation of system.

Equivalently, we define the Youla transfer function to be the transfer function from the input target, to the actuator input, ,

(6)

At the first glance, defining the Youla transfer function seems to be trivial and not useful. However, as we discuss below and in the following chapters, this simple mapping will have important consequences both in SISO and