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A Course in Robust
Con trol Theory
a convex approach
Geir E. Dullerud
F ernando G. Paganini
University of Illinois
Univ ersity of California
Urbana-Champaign
Los Angeles
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Contents
0 In tro duction 1
System representations . . . . . . . . . . . . . . . . . . . 2
Blo ck diagrams . . . . . . . . . . . . . . . . . . . 2
Nonlinear equations and linear p ositions . . 4
Robust control problems and uncertainty . . . . . . . . . 9
Stabilization . . . . . . . . . . . . . . . . . . . . . 9
Disturbances mands . . . . . . . . . . . . 12
Unmo deled dynamics . . . . . . . . . . . . . . . . 15
1 Preliminaries in Finite Dimensional Space 18
Linear spaces and mappings . . . . . . . . . . . . . . . . 18
V ector spaces . . . . . . . . . . . . . . . . . . . . 19
Subspaces . . . . . . . . . . . . . . . . . . . . . . 21
Bases, spans, and linear indep endence . . . . . . 22
Mappings and matrix representations . . . . . . 24
Change of basis and invariance . . . . . . . . . . 28
Subsets and Conv exit y . . . . . . . . . . . . . . . . . . . 30
Some basic top ology . . . . . . . . . . . . . . . . 31
Convex sets . . . . . . . . . . . . . . . . . . . . . 32
Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 38
Eigenvalues and Jordan form . . . . . . . . . . . 39
Self-adjoint, unitary and p ositive de nite matrices 41
Singular value p osition . . . . . . . . . . . 45
Linear Matrix Inequalities . . . . . . . . . . . . . . . . . 47
ii Con tents
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2 State Space System Theory 57
The autonomous system . . . . . . . . . . . . . . . . . . 58
Con trollability . . . . . . . . . . . . . . . . . . . . . . . . 61
Reachability . . . . . . . . . . . . . . . . .