6CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
several studies involving process control applications, particularly high purity distillation
columns. These are detailed by Skogestad and Morari in [15, 16, 17, 18]
Section 2.2 introduces robust control perturbation models and linear fractional
transformations. WeightedH∞design is covered in Section 2.3. Theanalysis of closed
loop systems with the structured singular value (µ) is overviewedin Section 2.4.
Section 2.5 discusses µsynthesis and the D-Kiteration. Model reduction is often used
to reduce the controller order prior to implementation and this is covered in Section 2.6.
2.1.1 Notation
Wewill use some fairly standard notation and this is given here for reference.
Rset of real numbers
Cset of complex numbers
Rnset of real valued vectorsof dimension n×1
Cnset of complex valued vectorsof dimension n×1
Rn×mset of real valued matrices of dimension n×m
Cn×mset of complex valued matricesof dimension n×m
Inidentity matrix of dimension n×n
0 matrix (or vector orscalar) of zeros of appropriate dimension
The following apply to a matrix, M∈C
n×m
.
M
Ttranspose of M
M∗complex conjugate transpose of M
|M|absolute value of each element of M(also applies if Mis a vector or scalar)
Re{M}real part of M
Im{M}imaginary part of M
dim(M) dimensions of M
σmax(M) maximum singular value of M
σmin(M) minimum singular value of M
Mij element of Min row i, column j. (also used for the i,jpartition of a previously defined
partition of M)
λi(M) an eigenvalue of M
ρ(M) spectral radius (maxi|λi(M)|)
kMknorm of M(see section 2.1.2 for more details)