Exam date: Friday 07 June (5 hour written exam)
Textbook:
"Multivariable feedback control" av Skogestad og Postlethwaite.
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Required material from the book (syllabus) (Pensum)
The numbers refer to the March/April version of the book.
The complete table of contents is given below.
Ch. 1
Ch. 2 (assumed known from previous courses with exception of
2.7 on H-infinity etc.)
Ch. 3
Ch. 4.4, 4.5, 4.6, 4.7, 4.9.4, 4.9.5, 4.10 (the rest is for orientation or
is assumed known)
Ch. 5: The most importants parts are: 5.1, 5.3.4 (but not Thm. 5.5), 5.5,
5.6.4, 5.8, 5.14 (summary). You must be able to derive the rules
given in the summary.
Ch. 6: 6.1, 6.2.3, 6.3, 6.5 (need not remember details of Thm 6.4), 6.6,
6.7 (need not remember details), 6.8, 6.9.1, 6.10.2, 6.10.4 (only
last page with Conclusions on input uncertainty and feedback control),
6.10, 6.11
Ch. 7: 7.1, 7.2, 7.4, 7.5, 7.6
Ch. 8: 8.1, 8.2.4, 8.3, 8.4, 8.5, 8.6, 8.7, know the most important results
in 8.8 (definition of mu, eq. 8.82, 8.86, 8.87, 8.98), 8.9, 8.10
8.11, 8.14.
Ch. 9: 9.1 (the rest is not required for the exam, but it is assumed
that you are familiar with most of it during the project)
Ch. 10: 10.1, 10.4, 10.6 is assumed known from previous courses, 10.8
(most important results are in 10.8.1 and 10.8.5 (summary)).
Ch. 11: For orientation only.
Ch. 12: For orientation only
Appendix A: It is recommended that you read though this!
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TABLE OF CONTENTS (final version):
1 INTRODUCTION
1.1 The process of control system design
1.2 The control problem
1.3 Transfer functions
1.4 Scaling
1.5 Deriving linear models
1.6 Notation
2 CLASSICAL FEEDBACK CONTROL
2.1 Frequency response
o Frequency-by-frequency sinusoids
2.2 Feedback control
2.2.1 One degree-of-freedom controller
2.2.2 Closed-loop transfer functions
2.2.3 Why feedback?
2.3 Closed-loop stability
2.4 Evaluating closed-loop performance
2.4.1 Typical closed-loop responses
2.4.2 Time domain performance
2.4.3 Frequency domain performance
o Gain and phase margins
o Maximum peak criteria
2.4.4 Relationship between time and frequency domain peaks
2.4.5 Bandwidth and crossover frequency
2.5 Controller design
2.6 Loop shaping
2.6.1 Trade-offs in terms of $L$
2.6.2 Fundamentals of loop-shaping design
o Limitations imposed by RHP-zeros and time delays
2.6.3 Inverse-based controller design
2.6.4 Loop shaping for disturbance rejection
2.6.5 Two degrees-of-freedom design
o Loop shaping applied to a flexible structure
2.6.6 Conclusions on loop shaping
2.7 Shaping closed-loop transfer functions
2.7.1 The terms ${\cal H}_\infty$ and ${\cal H_2}$
2.7.2 Weighted sensitivity
2.7.3 Stacked requirements: mixed sensitivity
2.8 Conclusion
3 INTRODUCTION TO MULTIVARIABLE CONTROL
3.1 Introduction
3.2 Transfer functions for MIMO systems
o Negative feedback control systems
3.3 Multivariable frequency response analysis
3.3.1 Obtaining the frequency response from $G(s)$
3.3.2 Directions in multivariable systems
3.3.3 Eigenvalues are a poor measure of gain
3.3.4 Singular value decomposition
o Non-Square plants
o Use of the minimum singular value of the plant
3.3.5 Singular values for performance
3.4 Control of multivariable plants
3.4.1 Decoupling
3.4.2 Pre- and post-compensators and the SVD-controller
3.4.3 Diagonal controller (decentralized control)
3.4.4 What is the shape of the ``best'' feedback controller?
3.4.5 Multivariable controller synthesis
3.4.6 Summary of mixed-sensitivity ${\cal H}_\infty $ design
($S/KS$)
3.5 Introduction to multivariable RHP-zeros
3.6 Condition number and RGA
3.6.1 Condition number
3.6.2 Relative Gain Array (RGA)
3.7 Introduction to MIMO robustness
3.7.1 Motivating robustness example no. 1: Spinning Satellite
3.7.2 Motivating robustness example no. 2: Distillation Process
3.7.3 Robustness conclusions
3.8 General control problem formulation
3.8.1 Obtaining the generalized plant $P$
3.8.2 Controller design: Including weights in $P$
3.8.3 Partitioning the generalized plant $P$
3.8.4 Analysis: Closing the loop to get $N$
3.8.5 Generalized plant $P$: Further examples
3.8.6 Deriving $P$ from $N$
3.8.7 Problems not covered by the general formulation
3.8.8 A general control configuration including model
uncertainty
3.9 Additional exercises
3.10 Conclusion
4 ELEMENTS OF LINEAR SYSTEM THEORY
4.1 System descriptions
4.1.1 State-space representation
4.1.2 Impulse response representation
4.1.3 Transfer function representation - Laplace transforms
4.1.4 Frequency response
4.1.5 Coprime factorization
4.1.6 More on state-space realizations
4.2 State controllability and state observability
4.3 Stability
4.4 Poles
4.4.1 Poles and stability
4.4.2 Poles from state-space realizations
4.4.3 Poles from transfer functions
4.5 Zeros
4.5.1 Zeros from state-space realizations
4.5.2 Zeros from transfer functions
4.6 More on poles and zeros
4.6.1 Directions of poles and zeros
4.6.2 Remarks on poles and zeros
4.7 Internal stability of feedback systems
4.7.1 Implications of the internal stability requirement
4.8 Stabilizing controllers
4.8.1 Stable plants
4.8.2 Unstable plants
4.9 Stability analysis in the frequency domain
4.9.1 Open and closed-loop characteristic polynomials
o Relationship between characteristic polynomials
4.9.2 MIMO Nyquist stability criteria
4.9.3 Eigenvalue loci
4.9.4 Small gain theorem
4.10 System norms
4.10.1 ${\cal H}_2$\ norm
4.10.2 ${\cal H}_\infty $\ norm
4.10.3 Difference between the ${\cal H}_2$\ and
${\cal H}_\infty $\ norms
4.10.4 Hankel norm
4.11 Conclusion
5 LIMITATIONS ON PERFORMANCE IN SISO SYSTEMS
5.1 Input-Output Controllability
5.1.1 Input-output controllability analysis
5.1.2 Scaling and performance
5.1.3 Remarks on the term controllability
5.2 Perfect control and plant inversion
5.3 Constraints on $S$ and $T$
5.3.1 $S$ plus $T$ is one
5.3.2 The waterbed effects (sensitivity integrals)
o Pole excess of two: First waterbed formula
o RHP-zeros: Second waterbed formula
5.3.3 Interpolation constraints
5.3.4 Sensitivity peaks
5.4 Ideal ISE optimal control
5.5 Limitations imposed by time delays
5.6 Limitations imposed by RHP-zeros
5.6.1 Inverse response
5.6.2 High-gain instability
5.6.3 Bandwidth limitation I
5.6.4 Bandwidth limitation II
o Performance at low frequencies
o Performance at high frequencies
5.6.5 Limitations at low or high frequencies
5.6.6 Remarks on the effects of RHP-zeros
5.7 Non-causal controllers
5.8 Limitations imposed by RHP-poles
5.9 Combined RHP-poles and RHP-zeros
5.10 Performance requirements imposed by disturbances and commands
5.11 Limitations imposed by input constraints
5.11.1 Inputs for perfect control
5.11.2 Inputs for acceptable control
5.11.3 Unstable plant and input constraints
5.12 Limitations imposed by phase lag
5.13 Limitations imposed by uncertainty
5.13.1 Feedforward control
5.13.2 Feedback control
5.14 Controllability analysis with feedback control
5.15 Controllability analysis with feedforward control
5.16 Applications of controllability analysis
5.16.1 First-order delay process
5.16.2 Application: Room heating
5.16.3 Application: Neutralization process
5.16.4 Additional exercises
5.17 Conclusion
6 LIMITATIONS ON PERFORMANCE IN MIMO SYSTEMS
6.1 Introduction
6.2 Constraints on $S$ and $T$
6.2.1 $S$ plus $T$ is the identity matrix
6.2.2 Sensitivity integrals
6.2.3 Interpolation constraints
6.2.4 Sensitivity peaks
6.3 Functional controllability
6.4 Limitations imposed by time delays
6.5 Limitations imposed by RHP-zeros
6.5.1 Moving the effect of a RHP-zero to a specific output
6.6 Limitations imposed by RHP-poles
6.7 RHP-poles combined with RHP-zeros
6.8 Performance requirements imposed by disturbances
6.9 Limitations imposed by input constraints
6.9.1 Inputs for perfect control
6.9.2 Inputs for acceptable control
o Exact conditions
o Approximate conditions in terms of the SVD
6.10 Limitations imposed by uncertainty
6.10.1 Input and output uncertainty
6.10.2 Effect of uncertainty on feedforward control
6.10.3 Uncertainty and the benefits of feedback
6.10.4 Uncertainty and the sensitivity peak
o Upper bound on $\sigma (S')$ for output uncertainty
o Upper bounds on $\sigma (S')$ for input uncertainty
o Lower bound on $\sigma (S')$ for input uncertainty
o Conclusions on input uncertainty and feedback control
6.10.5 Element-by-element uncertainty
6.10.6 Steady-state condition for integral control
6.11 Input-output controllability
6.11.1 Controllability analysis procedure
6.11.2 Plant design changes
6.11.3 Additional exercises
6.12 Conclusion
7 UNCERTAINTY AND ROBUSTNESS FOR SISO SYSTEMS
7.1 Introduction to robustness
7.2 Representing uncertainty
7.3 Parametric uncertainty
7.4 Representing uncertainty in the frequency domain
7.4.1 Uncertainty regions
7.4.2 Representing uncertainty regions by complex perturbations
7.4.3 Obtaining the weight for complex uncertainty
7.4.4 Choice of nominal model
7.4.5 Neglected dynamics represented as uncertainty
7.4.6 Unmodelled dynamics uncertainty
7.5 SISO Robust stability
7.5.1 RS with multiplicative uncertainty
7.5.2 Comparison with gain margin
7.5.3 RS with inverse multiplicative uncertainty
7.6 SISO Robust performance
7.6.1 SISO nominal performance in the Nyquist plot
7.6.2 Robust performance
7.6.3 The relationship between NP, RS and RP
7.6.4 The similarity between RS and RP
7.7 Examples of parametric uncertainty
7.7.1 Parametric pole uncertainty
7.7.2 Parametric zero uncertainty
7.7.3 Parametric state-space uncertainty
7.8 Additional exercises
7.9 Conclusion
8 ROBUST STABILITY AND PERFORMANCE ANALYSIS
8.1 General control configuration with uncertainty
8.2 Representing uncertainty
8.2.1 Differences between SISO and MIMO systems
8.2.2 Parametric uncertainty
8.2.3 Unstructured uncertainty
o Lumping uncertainty into a single perturbation
o Moving uncertainty from the input to the output
8.2.4 Diagonal uncertainty
8.3 Obtaining $P$, $N$ and $M$
8.4 Definitions of robust stability and robust performance
8.5 Robust stability of the $M\Delta $-structure
8.6 RS for complex unstructured uncertainty
8.6.1 Application of the unstructured RS-condition
8.6.2 RS for coprime factor uncertainty
8.7 RS with structured uncertainty: Motivation
8.8 The structured singular value
8.8.1 Remarks on the definition of $\mu $
8.8.2 Properties of $\mu $ for real and complex $\Delta $
8.8.3 $\mu $ for complex $\Delta $
o Properties of $\mu $ for complex perturbations
8.9 Robust stability with structured uncertainty
8.9.1 What do $\mu \not =1$ and skewed-$\mu $ mean?
8.10 Robust performance
8.10.1 Testing RP using $\mu $
o Block diagram proof of Theorem 8.7
o Algebraic proof of Theorem 8.7
8.10.2 Summary of $\mu $-conditions for NP, RS and RP
8.10.3 Worst-case performance and skewed-$\mu $
8.11 Application: RP with input uncertainty
8.11.1 Interconnection matrix
8.11.2 RP with input uncertainty for SISO system
8.11.3 Robust performance for $2\times 2$ distillation process
8.11.4 $\mu $ and the condition number
o Worst-case performance (any controller)
8.11.5 Comparison with output uncertainty
8.12 $\mu $-synthesis and $DK$-iteration
8.12.1 $DK$-iteration
8.12.2 Adjusting the performance weight
8.12.3 Fixed structure controller
8.12.4 Example: $\mu $-synthesis with $DK$-iteration
o Analysis of $\mu $-``optimal'' controller $K_3$
8.13 Further remarks on $\mu $
8.13.1 Further justification for the upper bound on $\mu $
8.13.2 Real perturbations and the mixed $\mu $ problem
8.13.3 Computational complexity
8.13.4 Discrete case
8.13.5 Relationship to linear matrix inequalities (LMIs)
8.14 Conclusion
o Practical $\mu $-analysis
9 CONTROLLER DESIGN
9.1 Trade-offs in MIMO feedback design
9.2 LQG control
9.2.1 Traditional LQG and LQR problems
9.2.2 Robustness properties
9.2.3 Loop transfer recovery (LTR) procedures
9.3 ${\cal H}_2$ and ${\cal H}_\infty$ control
9.3.1 General control problem formulation
9.3.2 ${\cal H}_2$ optimal control
9.3.3 LQG: a special ${\cal H}_2$ optimal controller
9.3.4 ${\cal H}_\infty $ optimal control
9.3.5 Mixed-sensitivity ${\cal H}_\infty $ control
9.3.6 Signal-based ${\cal H}_\infty $ control
9.4 ${\cal H}_\infty $ loop-shaping design
9.4.1 Robust stabilization
9.4.2 A systematic ${\cal H}_\infty $ loop-shaping design
procedure
9.4.3 Two degrees-of-freedom controllers
9.4.4 Observer-based structure for ${\cal H}_\infty $
loop-shaping controllers
9.4.5 Implementation issues
9.5 Conclusion
10 CONTROL STRUCTURE DESIGN
10.1 Introduction
10.2 Optimization and control
10.3 Selection of controlled outputs
o Measurement selection for indirect control
10.4 Selection of manipulations and measurements
10.5 RGA for non-square plant
10.6 Control configuration elements
10.6.1 Cascade control systems
10.6.2 Cascade control: Extra measurements
10.6.3 Cascade control: Extra inputs
10.6.4 Extra inputs and outputs (local feedback)
10.6.5 Selectors
10.6.6 Why use cascade and decentralized control?
10.7 Hierarchical and partial control
10.7.1 Partial control
10.7.2 Hierarchical control and sequential design
o Sequential design of cascade control systems
10.7.3 ``True'' partial control
10.8 Decentralized feedback control
10.8.1 RGA as interaction measure for decentralized control
10.8.2 Stability of decentralized control systems
o Sufficient conditions for stability
o Necessary steady-state conditions for stability
10.8.3 The RGA and right-half plane zeros
10.8.4 Performance of decentralized control systems
10.8.5 Summary: Controllability analysis for decentralized
control
10.8.6 Sequential design of decentralized controllers
10.8.7 Conclusions on decentralized control
10.9 Conclusion
11 MODEL REDUCTION
11.1 Introduction
11.2 Truncation and residualization
11.2.1 Truncation
11.2.2 Residualization
11.3 Balanced realizations
11.4 Balanced truncation and balanced residualization
11.5 Optimal Hankel norm approximation
11.6 Two practical examples
11.6.1 Reduction of a gas turbine aero-engine model
11.6.2 Reduction of an aero-engine controller
11.7 Reduction of unstable models
11.7.1 Stable part model reduction
11.7.2 Coprime factor model reduction
11.8 Model reduction using MATLAB
11.9 Conclusion
12 CASE STUDIES
12.1 Introduction
12.2 Helicopter control
12.2.1 Problem description
12.2.2 The helicopter model
12.2.3 ${\cal H}_\infty $ mixed-sensitivity design
12.2.4 Disturbance rejection design
12.2.5 Comparison of disturbance rejection properties of the two
designs
12.2.6 Conclusions
12.3 Aero-engine control
12.3.1 Problem description
12.3.2 Control structure design: output selection
12.3.3 A two degrees-of-freedom ${\cal H}_\infty $ loop-shaping
design
12.3.4 Analysis and simulation results
12.3.5 Conclusions
12.4 Distillation process
12.4.1 Idealized $LV$-model
12.4.2 Detailed $LV$-model
12.4.3 Idealized $DV$-model
12.4.4 Further distillation case studies
12.5 Conclusion
A MATRIX THEORY AND NORMS
A.1 Basics
A.1.1 Some useful matrix identities
A.1.2 Some determinant identities
A.2 Eigenvalues and eigenvectors
A.2.1 Eigenvalue properties
A.2.2 Eigenvalues of the state matrix
A.2.3 Eigenvalues of transfer functions
A.3 Singular Value Decomposition
A.3.1 Rank
A.3.2 Singular values of a $2\times 2$ matrix
A.3.3 SVD of a matrix inverse
A.3.4 Singular value inequalities
A.3.5 SVD as a sum of rank $1$ matrices
A.3.6 Singularity of matrix $A+E$
A.3.7 Economy-size SVD
A.3.8 Pseudo-inverse (Generalized inverse)
o Principal component regression (PCR)
A.3.9 Condition number
A.4 Relative Gain Array
A.4.1 Properties of the RGA
A.4.2 RGA of a non-square matrix
A.4.3 Computing the RGA with MATLAB
A.5 Norms
A.5.1 Vector norms
A.5.2 Matrix norms
o Induced matrix norms
o Implications of the multiplicative property
A.5.3 The spectral radius
A.5.4 Some matrix norm relationships
A.5.5 Matrix and vector norms in MATLAB
A.5.6 Signal norms
A.5.7 Signal interpretation of various system norms
A.6 Factorization of the sensitivity function
A.6.1 Output perturbations
A.6.2 Input perturbations
A.6.3 Stability conditions
A.7 Linear fractional transformations
A.7.1 Interconnection of LFTs
A.7.2 Relationship between $F_l$ and $F_u$.
A.7.3 Inverse of LFTs
A.7.4 LFT in terms of the inverse parameter
A.7.5 Generalized LFT: The matrix star product
B PROJECT WORK and SAMPLE EXAM
B.1 Project work
B.2 Sample exam
BIBLIOGRAPHY
INDEX