The claims to novelty in the thesis are
• The Matlab implementation of the DI method originally proposed by Helton
et al. [46] is presented for the purpose of comparison and analysis with other algorithms. The nonparametric spectral factorization methods by Wilson [99] and Harris and Davis [36] are also coded in Matlab and incorporated as part of the optimization code. Although the solution of the DI method is not optimal to the general H∞ optimization problem, the optimization
problem can be approached by iterating to the closest circular form and solved in terms of the Nehari problem. Thus, its sup-optimal solution is still useful for nonparametric control and, in the problems with circular performance function, an optimal solution is obtained. Since the original programs that implement the DI method were coded in Mathematica [41], for use in engineering applications, it is practically useful to integrate a Matlab version of the DI method as part of a controller design toolbox.
• An alternative nonparametric method known as Q-parameterization or You-
la-Kucera parameterization of sensitivity functions by the sensitivity func- tion Qcan be used in order to meet the internal stability conditions in the
feedback H∞ problem and to replace the parametric interpolation method
proposed by Helton et al. [47, 106]. The interpolation method with the DI method has previously been implemented on an automotive engine control problem by using the FrobeniusH∞ norm [108], however, this interpolation
method required the parametric identication of the RHP pole and zero of the plant. The Q-parameterization method skips this parametric identica- tion process and thereby allows the approach to become fully nonparamet- ric method. In this thesis, the Q-parameterization method is successfully applied to an automotive engine control problem with nonparametric opti- mization algorithms.
method, the NI method is capable of dealing with multiple function op- timization with non-circular type objective functions. The solution by the NI method is potentially optimal to the general optimization problem. Fur- thermore, the NI method should in principle converge quickly to the optimal solution due to its second order convergence rate. These properties of the NI method potentially save cost in computing eort during the optimization process as well as improving the accuracy of the solution. The NI method is therefore more suitable for general nonparametric control problems. The NI method is also implemented in Matlab in this thesis for the purpose of analysis and practical use. This implementation is the rst published implementation of this important algorithm.
• The comparative analysis of the DI method and the NI method is presented.
This is the rst independent comparison of these two methods. The analytic solutions by both methods are investigated by applying to several numerical examples.
• A nonparametric approach by Streit's [86] linear programming (LP) method
forH∞control problems is proposed. Compared to that the DI method and
the NI method which are both based on the computation of the Fourier co- ecients of frequency response function, the LP method is formed by the approximation of the analytic solution directly on the basis of the frequency response itself. The central optimization approach originally studied by Helton and Sideris [40] using Streit's method [86] is described in this the- sis. The implementation of Streit's algorithm is also coded in Matlab for practical use and for comparison purpose. An example of a SISO control problem with multiplicative uncertainty by this nonparametric approach is demonstrated.
• The nonparametric approach using the dierent analytic function methods,
i.e. the DI method, the NI method and the LP method, is applied to an automotive engine control problem. The analysis of the dierent controllers from the dierent analytic function methods is presented.
Chapter 2
Disk Iteration Method for
H
∞
Optimization
The standardH∞control problem is often viewed as a mathematical optimization
problem. The optimization problem mentioned in Chapter 1 can be solved by the Disk Iteration (DI) method proposed by Helton and Merino [46, 47]. Most parts of this chapter are originated from the work of Helton and Merino[46, 47] and the programs to implement the DI method are also available in [41]. However, the Matlab implementation of the DI method is rstly accomplished for the purpose of comparison and practical use.
In this chapter, starting with the statement of the optimization problem and the transformation from the OP T eproblem to the OP T dproblem in
Section 2.1, the underlying theory and the implementation of the Disk Iteration (DI) algorithm are presented in more details in Section 2.2.
2.1 Optimization Problem
This section is the summary of the work on H∞ optimization in the frequency
domain by Helton et al in [37, 47] and in other related publications [38, 39, 40, 42, 43, 45, 46, 48]. In this section, the H∞ control problem in the frequency
domain approach is formulated by using Equation 1.24 . Equation 1.24 species a mathematical minimax problem :
Given a continuous positive-valued functionΓinCN whereN is the dimension
of the problem, and a set of all the stable functions on the jω axis ( denotes as RH∞ , the space of all functions whose poles have negative parts. ), nd the
optimal function f∗(jω)∈RH∞ such that
γ = inf
f∈RH∞
sup
ω
Γ (ω, f(jω)) (2.1)
where ω represents the frequency, and f(jω) is the frequency response function This problem is known as the OP T problem in [48]. It can be translated into
an optimization problem that searches the stabilizing controller in terms of the measureγ in the worst performance case.
In mathematical analysis, the functions are often treated on the unit disk rather than in the complex plane because of the use of the Fourier Transform, the ease of implementation in the computer codes and other properties in complex analysis. The transformation from the imaginary axis to the unit circle ∂D is
possible via a Linear Fractional Transform (LFT). A LFT is a one-to-one, linear, and bounded mapping in this case from the imaginary axis ( s = jω ) onto
the unit circle ∂D. It transforms the poles and zeros of the function in the left
half plane onto a point inside the unit cirle and maps the poles and zeros of the function in the right half plane onto the point outside the unit circle. The sampling points on the imaginary axis ( e.g.. jω ) are thus mapped to points on
the unit circle∂D by the linear fractional transform, as illustrated in Figure 2.1.
Figure 2.1: Linear fractional transform
Thus by means of the linear fractional transform, the functionf(jω)inRH∞ in theOP T problem is mapped to the functionf ejθ
of all analytic functions f ejθthat are bounded and continuous in the unit disk D. The OP T problem is therefore transformed to the so-called OP T e problem
given by
OPTe Given a continuous, real, positive valued function Γ in D, nd the
optimal function f∗ ejθ such that γ∗ = inf f∗∈ A sup ω Γ ejθ, f∗ ejθ= inf f∗∈ A Γ ejθ, f∗ ejθ ∞ (2.2)
where ejθ are the points spaced around the unit circle, f ejθ is con-
tinuous in H∞ and ∗ denotes the function value at the optimum.
For simplicity, we will denote z = ejθ in the rest of the thesis. It is stated in
[45] that the OP T e problem is closely related to the sub-optimization problem OP T cproblem in terms of the sublevel setSθ(c) = {Γ (z, f (z))≤c, c∈R+}with
a prescribed performance function c∈R+ :
OPTc Find f∗(z)∈A with Γ (z, f(z))≤c, ∀θ,and c∈R+such that
γ∗ = inf f∗∈ A kΓ (z, f∗(z))k∞= inf f∗∈ A kk(z)−f∗(z)k2∞ ≤c (2.3)
where k(z) is any complex-valued function
It is also shown in [45] that the sublevel sets Sθ have the shape of disk and
their properties of boundness, simple connectness, smoothness and convexity are closely related to the properties of solutions to the OP T e problem. Therefore,
the solution to the OP T c problem can be viewed as the solution to the OP T
problem.
Furthermore, in [46, 47], the generalization of the above OP T c problem for N ≥1 is considered :
OPTd Given the positive continuous functions w(z) and k(z), nd f(z) ∈ H∞N such that γ = inf f∈H∞ N sup θ Γ (z, f(z)) = inf f∈H∞sup θ N X i=1 wj(z)|ki(z)−fi(z)| 2 (2.4)
Two examples to illustrate the relationship between the OP Tdand the OP T c
Example1 [47] Given a scalar-valued function ( i.e. theOP T dproblem forN = 1) Γ (z, f) = w(z)|k(z)−f(z)|2 (2.5) = (2 + cosθ) 1 z+ 0.5−f 2 (2.6) ≤ γ
it is seen that the solution must lie in the disk centred at 1
z+0.5 with the radius
p
γ/(2 + cosθ). This can be related to theOP T cproblem by observing that the
sublevel sets of the solutions f are
f =f ∈H∞;|k−f|2 ≤r2,for r∈R+
where the centre function k = z+01.5 and the radius function r=pγ/(2 + cosθ). Consequently, the equivalent OP T c problem for this objective function Γ (z, f) can be reformed as inf f∈H∞ sup θ Γ (z, f) = inf f∈H∞ sup θ (2 + cosθ) 1 z+ 0.5 −f 2 (2.7) = inf f∈H∞sup θ k− ˜ f 2 (2.8) where k =pγ/(2 + cosθ) −1/2 · 1 z+0.5 and f˜= p γ/(2 + cosθ) −1/2 f
As the OP T cproblem is solvable by solving the Nehari problem, we immedi-
ately have the solution to the optimization for this type of Γ (z, f).
Example2 [47] An objective function that has more than one term of the form : Γ (z, f) = w1|f|2+w2|1−f|2 ≤γ (2.9)
it can be considered a quasi-circular problem. The sublevel sets of the function Γ (z, f) are fomulated as the disks centred at w2
w1+w2 with the radius
1 (w1+w2) γ− w1w2 w1+w2 1/2
. Furthermore, the problem can be written in terms of
f = ( f ∈H∞; w2 w1+w2 −f 2 ≤ 1 (w1 +w2) γ− w1w2 w1+w2 )
This implies that the problem can be viewed as a circular problem with the centre function w2
w1+w2 and the radius function
1 (w1+w2) γ− w1w2 w1+w2 1/2 . There-
fore, it is concluded that any function in the form of Γ (z, f) = N X i=1 wi(z)|ki(z)−fi(z)|2
is said to be a quasi-circular function, which is solvable by means of its equivalent circular form.
In summary, whether the function is circular or quasi-circular, the OP T d
problem can be solved by solving the equivalent OP T c problem in terms of the
solution to the Nehari problem. To deal with other functions that are not of circular or quasi-circular form, the approximation to the nearest circular function is performed. In the ANOPT package [41], the approximation is coded in terms of using an interpolation method. The interpolation method is discussed in depth in Section 2.2.