The Algorithm635 proposed by Streit [87] for solving the systems of complex linear equations in terms of L∞ norm, which is summarized below.
Given a n×m matrix A∈C, an×r matrixB ∈C, and the vectorsf ∈Cm, g ∈Cr, a∈
Cn, d∈Rn and c∈Rr, the optimization problem
min
z∈CkzA−fk∞ (5.14)
subject to
|z−a| ≤ d
|zB−g| ≤ c (5.15)
where k·k∞ denotes the L∞ norm, can be discretized to the problem
min ε∈R,z∈Cn ε (5.16) subject to |zAi−fi|D ≤ε , i= 1,· · · , m |zBi−gi|D ≤ci , i= 1,· · · , r (5.17) |z−ai|D ≤di , i= 1,· · · , n
where Ai and Bi are the i-th columns of the matrices A and B.
This algorithm solving the discretized problem is originally published in [86] and implemented in the FORTRAN language in [87]. Several applications of the Algorithm635 are also studied in [40, 72, 85, 88]. It is seen from the above derivation that the algorithm is based on the extension of the solution to the complex semi-innite program (SIP) formulation of the unconstrained problem to the constrained optimization problem.
It is shown in [87] that the norm |u|∞ can be discretised as a p-th polygon
: |u|D = max
θ∈D u
Rcosθ+uIsinθ
where D ={θ1, θ2, . . . , θp} is the subset of the
interval[0,2π)anduR,uI are the real and imaginary parts ofurespectively . The
discretised problem 5.16 can then be transformed to the standard constrained linear programming problem [87] :
min zR zI ε ∈ R h zR zI ε i h 0 n 0n 1 iT (5.18)
subject to ε≥0and for each θ∈[0,2π]
h zR zI ε i
ARcosθ+AIsinθ BRcosθ+BIsinθ Icosθ ARsinθ−AIcosθ BRsinθ−BIcosθ Isinθ
−1m 0r 0n
≤h fRcosθ+fIsinθ c+gRcosθ+gIsinθ d+aRcosθ+aIsinθ i (5.19)
It is reported [15] that the linear programming problem is usually solved by the simplex method. The simplex method, however, requires large memory ca- pacity to store the huge matrices in this problem [87]. A revised simplex method is presented by Streit [87] to reduce the memory storage for solving the corre- sponding dual problem. In the Matlab programs to implement Streit's algorithm, the interior point method is used by calling the standard code linprog [105] for solving the large scale linear programming problem.
It is also noted in [87] that since the discretised problem is an approximation to the original problem, there may be no feasible solution to the original problem but we can still calculate a solution to the discrete problem. It is discussed in [87] that this is because the region of the constraints in the Inequalities 5.17 is larger than those in the Inequalities 5.15 . However, as the discretisation number of p increases, the discrete problem grows closer to the original problem and the
false solution to the discrete problem can be excluded. In [87], the discretisation number p is suggested to be 1024. However, it is known that increasing the
discretisation number p increases the computing eorts dramatically as shown
in Figure 5.1. It is observed in the gure that the computing time starts to increase potentially from the point p = 16. In the experience of running many similar Matlab programmes, if a feasible solution is found, the value of p is not
neccessary to be 1024 and p = 16 is enough to provide a satisfactorily accurate solution.
Figure 5.1: The inuence of the discretisation number p to the computing time • Application of the Linear Programming Method to the Mixed
Sensitivity Control Problem
To solve the mixed sensitivity problem in Equation 5.8, we restate the problem to be a constrained optimization problem as
γ∗ = min
θ∈D kWS ?(1−GQ)k∞ (5.20)
subject to
kWT ?(GQ)k∞≤1 (5.21)
We rst focus on this relaxed mixed sensitivity problem for a SISO system as follows and extend it to MIMO systems afterwards :
γ∗ = min
θ∈D kWS(1−GQ)k∞ (5.22)
subject to
kWT (GQ)k∞ ≤1 (5.23)
Due to Q∈H∞, the power expasion of Q up to n-the order can be expressed in
Q eiθ = z0+z1ejθ+z2e2jθ+· · ·+znenjθ (5.24) = h z0 z1 z2 · · · zn i ·h e0 ejθ e2jθ · · · enjθ iT (5.25) = z·A ejθ (5.26) where z=h z0 z1 z2 · · · zn i and A =h e0 ejθ e2jθ · · · ejnθ iT.
Therefore, the mixed sensitivity optimization problem becomes
γ∗ = min θ∈D WS ejθ 1−G ejθzA ejθ ∞ (5.27) subject to WT ejθ G ejθ·zA ejθ∞≤1 (5.28)
In the spirit of Streit's algorithm, the problem can then be discretized as
min ε∈R,z∈Cnε subject to z ˜ Ak−WS,k D ≤ε , k= 1,· · ·, n z ˜ Bk−0 D ≤1 , k= 1,· · ·, n where ˜ Ak =WS ejkθ G ejkθA ejkθ WS,k =WS ejkθ ˜ Bk=WT ejkθ G ejkθA ejkθ
and n ∈Z+ is the number of discretizing points. Such problem is closely related
to the linear programming problem proposed by Streit [86] and is thus solvable by the standard Matlab algorithm:linprog [105].
• Extension of the Linear Programming Method for MIMO Systems
The above derivation may also extend for the mixed sensitivity problem for MIMO systems. For instance, for a 2×2 system, the optimization problem stated in Inequality 5.11 may be re-formulated to the constrained optimization problem :
γ∗ = min θ∈DkWS,11(1−G11Q11−G12Q21)k∞ (5.29) subject to kWT ,11(G11Q11+G12Q21)k∞ ≤ 1 (5.30) kWS,12(0−G12Q22−G11Q12)k∞ ≤ 1 (5.31) kWT ,12(G12Q22+G11Q12)k∞ ≤ 1 (5.32) kWS,21(0−G21Q11−G22Q21)k∞ ≤ 1 (5.33) kWT ,21(G21Q11+G22Q21)k∞ ≤ 1 (5.34) kWS,22(1−G22Q22−G21Q12)k∞ ≤ 1 (5.35) kWT ,22(G22Q22+G21Q12)k∞ ≤ 1 (5.36)
In principle, the problem is possibly solved by the LP method in the spirit of the LP method for SISO systems. It may be solvable by other techniques of convex optimization in [14, 15]. In the thesis, the software CVX [33, 34] providing the solution to convex problems will be used in the next chapter to speed up the LP