HOW NONCOMMUTING ALGEBRA ARISES IN SYSTEMS THEORY Bill at UC San Diego helton@ucsd.edu
� dx ( t ) = Ax ( t ) + Bv ( t ) � y v dt G � ✲ ✲ x -state y ( t ) = Cx ( t ) + Dv ( t ) � � A, B, C, D are matrices � � x, v, y are vectors Re(eigvals( A )) ≺ 0 ⇐ ⇒ Asymptotically stable � � A T E + E A ≺ 0 E ≻ 0 E = E T � 0 Energy dissipating ∃ � G : L 2 → L 2 � H := A T E + E A + � � T � T � + E BB T E + C T C = 0 | v | 2 dt ≥ | Gv | 2 dt � � 0 0 x (0) = 0 � E is called a storage function Two minimal systems ∃ M invertible, so that � � M A M − 1 = a [ A, B, C, D ] and [ a, b, c, d ] � � M B = b with the same input � C M − 1 = c to output map. � ( B AB A 2 B · · · ) : ℓ 2 → X Every state is reachable � � is onto from x = 0 �
H ∞ Control Problem w out Given ✲ ✲ A , B 1 , B 2 , C 1 , C 2 y u D 12 , D 21 ✲ ✲ Find ✛ ✛ K dx dt = Ax + B 1 w + B 2 u out = C 1 x + D 12 u + D 11 w y = C 2 x + D 21 w D 21 = I D 12 D ′ 12 = I D ′ 12 D 12 = I D 11 = 0 PROBLEM: Find a control law K : y → u which makes the system dissipative over every finite horizon: T � T � | out ( t ) | 2 dt ≤ | w ( t ) | 2 dt 0 0 The unknown K is the system dξ dt = a ξ + b u = c ξ So a , b , c are the critical unknowns.
CONVERSION TO ALGEBRA Engineering Problem: Make a given sys- tem dissipative by designing a feedback law. Given D ✲ ✲ ✲ A , B 1 , C 1 , || � 0 1 � B 2 C 2 1 0 ✲ ✲ Find ✛ ✛ a b c DYNAMICS of “closed loop” system: BLOCK matrices A B C D ENERGY DISSIPATION : H := A T E + E A + E BB T E + C T C = 0 � E 11 E 12 � T E = E 12 = E 21 E 21 E 22 � H xx H xy � H xy = H T H = yx H yx E yy
H ∞ Control Problem ALGEBRA PROBLEM: Given the polynomials: H xx = E 11 A + A T E 11 + C T 1 C 1 + E 12 T b C 2 + C T 2 b T E 12 T + E 11 B 1 b T E 12 T + E 11 B 1 B T 1 E 11 + E 12 b b T E 12 T + E 12 b B T 1 E 11 H xz = E 21 A + a T ( E 21 + E 12 T ) + c T C 1 + E 22 b C 2 + c T B T 2 E 11 T + 2 E 21 B 1 b T ( E 21 + E 12 T ) 1 E 11 T + E 22 b b T ( E 21 + E 12 T ) + E 21 B 1 B T + E 22 b B T 1 E 11 T 2 2 1 c + ( E 12 + E 21 T ) a H zx = A T E 21 T + C T 2 b T E 22 T + + E 11 B 2 c + C T 2 + ( E 12 + E 21 T ) b B T 1 E 21 T + ( E 12 + E 21 T ) b b T E 22 T 1 E 21 T E 11 B 1 b T E 22 T + E 11 B 1 B T 2 2 H zz = E 22 a + a T E 22 T + c T c + E 21 B 2 c + c T B T 2 E 21 T + E 21 B 1 b T E 22 T + 1 E 21 T + E 22 b b T E 22 T + E 22 b B T E 21 B 1 B T 1 E 21 T (HGRAIL) A , B 1 , B 2 , C 1 , C 2 are knowns. � H xx H xz � Solve the inequality � 0 for un- H zx H zz knowns a , b , c and for E 11 , E 12 , E 21 and E 22 When can they be solved? If these equations can be solved, find formulas for the solution. 5
TEXTBOOK SOLUTION TO THE H ∞ PROB DGKF = Doyle-Glover Kargonekar - Francis 1989 ish KEY Riccatis DGKF X := ( A − B 2 C 1 ) ′ X + X ( A − B 2 C 1 ) + X ( γ − 2 B 1 B ′ 1 − B − 1 2 B ′ 2 ) X DGKF Y := A × Y + Y A ×′ + Y ( γ − 2 C ′ 1 C 1 − C ′ 2 C 2 ) Y here A × := A − B 1 C 2 . THM DGKF There is a system K solving the control problem if there exist solutions X � 0 Y ≻ 0 and to inequalities the DGKF Y � 0 and DGKF X � 0 which satisfy the coupling condition X − Y − 1 ≺ 0 . This is iff provided Y � 0 and Y − 1 is inter- preted correctly.
ALL THE RAGE Riccati Inequalities A ′ 1 X + X A 1 + X Q 1 X + R 1 � 0 A ′ 2 X + X A 2 + X Q 2 X + R 2 � 0 X � 0 These are “matrix convex” in the unknown X provided Q 1 , Q 2 are positive semidefinite ma- trices. If such an X exists, then can simultane- ously control or stablize several systems. Numerical Solution Can solve convex (es- pecially linear) matrix inequalities numerically with X smaller than 150 × 150 matrices us- ing interior point optimization methods - called semidefinite programming . Main Algebra Problem ”Convert” your engineering problem to a set of equiv- alent‘convex matrix inequalities” .
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