Linear Matrix Inequalities in Control

Optimisation approach (11)
when the synthesis is well-posed?
A
C
cl
cl
A B2D f
C2
B C
C
1
f
12
2
D D C
f
2
B2C
f
A
f
D C ; D
12
; B
f
cl
cl
B1
Df
D
B D
D
11
f
21
12
21
D D D
1. Simultanouos stabilisability of the pair (A,B 2 ) and observability of the pair
(C 2 ,A).
2. D-matrix conditions:
1. D 22 =0,
2. D cl =0 (if H 2 norm is used),
3. often D 11 =0 required (Robust Toolbox),
4. D 12 full column rank matrix,
5. D 21 full row rank matrix.
f
21
KSO materiały do wykładu 2008/09 16
Optimisation approach (warning)
• Many (all ?) control analysis and synthesis problems can be formulated
as optimisation problems.
• Only some of them can be transformed into LMI (not BMI), solvable
problems (e.g. there exists no general output controller synthesis for
uncertain plant – like LFT framework).
• As LMI – the price is paid in large number of slack (Lapunov)
variables.
• Many interesting papers on BMI solvers and non-smooth, non-convex
optimisation have been published recently. P.Apkarian ONERA-CERT,
D. Noll Uni Toulouse, M.Overton, Uni of New York.
KSO materiały do wykładu 2008/09 17
Scope of the lecture
• Control – classical vs. modern
• Optimisation approach
• Computer tools support
• LMI primer
• Basic LMI tools for control
• Analysis
• Synthesis
• Conclusions
KSO materiały do wykładu 2008/09 18
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