MatricesAn m by n matrix is a set of numbers which are arranged in m rows and n columns. We denotematrices by capital letters, and the element of i-th row and j-th column is denoted by ai j.A...

Matrices An m by n matrix is a set of numbers which are arranged in m rows and n columns. We denote matrices by capital letters, and the element of i-th row and j-th column is denoted by ai j. A = ? ???? a11 a12 ··· a1n a21 a22 ··· a2n . . . am1 am2 ··· amn ? ???? = [ai j]. (1.1) The summation of two given m×n matrices are defined entry-wise: A B = ? ???? a11 b11 a12 b12 ··· a1n b1n a21 b21 a22 b22 ··· a2n b2n . . . am1 bm1 am2 bm2 ··· . amn bmn ? ???? . (1.2) For given two m×n matrices A and B, we have A B = B A, (commutative property) (1.3) A (B C) = (A B) C, (Associativity of addition).


1 Linear Systems Theory Prof. Nader Motee Copyright c© 2015 Lehigh University HTTP://WWW.DCDS-LAB.COM/ Fall Semester 2015 Contents 1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Matrices 5 1.1.1 Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.4 Principle subspaces of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 State Space Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Transfer Function 17 2.2 Canonical Realizations of Transfer Function 18 2.2.1 Controllable Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 Observer Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Algebra operations on Systems 23 2.3.1 Cascade interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 Parallel interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.3 Feedback interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Similarity Transformation 28 3 Solution to Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Solution to LTI systems 29 3.2 The state transition matrix calculation 30 3.2.1 First method (Laplace Transform): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2 Second method (Taylor expansion): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2.3 Third method (Cayley-Hamilton) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.4 Forth method (diagonalization) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Parallel realization 34 4 Solution to different initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.1 Controllability 43 5.2 Observability 49 5.2.1 Observability for LTI systems: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.2 x0 calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Duality of Controllability and Observability for LTI Systems 51 5.3.1 PBH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Discrete-time systems: 53 5.4.1 Hankel Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4.2 Transfer function for Discrete-time systems