Preliminaries: linear algebra State-space models System interconnections Control Theory (035188) lecture no. 6 Leonid Mirkin Faculty of Mechanical Engineering Technion | IIT Preliminaries: linear algebra State-space models System interconnections Outline Preliminaries: linear algebra State-space models System interconnections in terms of state-space realizations Preliminaries: linear algebra State-space models System interconnections Outline Preliminaries: linear algebra State-space models System interconnections in terms of state-space realizations Preliminaries: linear algebra State-space models System interconnections Linear algebra, notation and terminology A matrix A Rn×m is an n m table of real numbers, 2 × 2 3 a a a m 11 12 ··· 1 6 a21 a22 a1m 7 A = 6 . ··· . 7 = [aij ]; 4 . 5 · an an anm 1 2 ··· accompanied by their manipulation rules (addition, multiplication, etc). The number of linearly independent rows (or columns) in A is called its rank and denoted rank A. A matrix A Rn×m is said to 2 have full row (column) rank if rank A = n (rank A = m) − be square if n = m, tall if n > m, and fat if n < m − be diagonal (A = diag ai ) if it is square and aij = 0 whenever i = j − f g 6 be lower (upper) triangular if aij = 0 whenever i > j (i < j) − 0 0 be symmetric if A = A , where the transpose A = [aji ] − · The identity matrix I = diag 1 Rn×n (if the dimension is clear, just I ). n · It is a convention that ·A0 = I fforg every2 nonzero square A. Preliminaries: linear algebra State-space models System interconnections Block matrices n×m P P Let A R and let i=1 ni = n and i=1 mi = n for some ni ; mi N. Then A2 can be formally split as 2 2 3 A A A 11 12 ··· 1 6 A21 A22 A1 7 A = 6 . ··· . 7 = [Aij ] 4 . 5 A A A 1 2 ··· ni ×mj Pi−1 for Aij = [aij;kl ] R with entries aij;kl = apq, where p = r=1 nr + k Pj−1 2 and q = r=1 mr + l. For example, we may present 2 1 2 3 4 3 A A A A = 5 6 7 8 = 11 12 13 4 5 A A A 9 10 11 12 21 22 23 1 2 3 4 for A11 = 5 6 , A12 = 7 , A13 = 8 , A21 = 9 10 , A22 = 11, A23 = 12. This split is not unique and is normally done merely for convenience. Preliminaries: linear algebra State-space models System interconnections Block matrices (contd) We say that a block matrix A = [Aij ] is block-diagonal (A = diag Ai ) if = and Aij = 0 whenever i = j − f g 6 block lower (upper) triangular if Aij = 0 whenever i > j (i < j) − If A is square, i.e. n = m, a natural split is with ni = mi for all i. Then all Aii are square too. Operations on block matrices are performed in terms of their blocks, like A A B B A + B A + B 11 12 + 11 12 = 11 11 12 12 ; A21 A22 B21 B22 A21 + B21 A22 + B22 A A B B A B + A B A B + A B 11 12 11 12 = 11 11 12 21 11 12 12 22 : A21 A22 B21 B22 A21B11 + A22B21 A21B12 + A22B22 Sum, product, inverse of block-diagonal (block-triangular) matrices remain block-diagonal (block-triangular). Preliminaries: linear algebra State-space models System interconnections Eigenvalues & eigenvectors Given a square matrix A Rn×n, its eigenvalues are the solutions C to 2 2 n n−1 A() = det(I A) = + n−1 + + 1 + 0 = 0 · − ··· (characteristic equation). There are n (not necessarily different) eigenvalues of A and spec(A) denotes their set. If i spec(A) is such that Im i = 0, ¯ 2 6 then i spec(A) too. 2 Right and left eigenvectors associated with i spec(A) are nonzero vectors such that 2 0 (i I A)Ái = 0 and Á (i I A) = 0; − i − respectively. Note that k k 0 k k 0 A Ái = Ái and Á A = Á ; k N i i i i 8 2 k k so i spec(A) = spec(A ), keeping left and right eigenvectors. 2 ) i 2 Preliminaries: linear algebra State-space models System interconnections Eigenvalues & eigenvectors Given a square matrix A Rn×n, its eigenvalues are the solutions C to 2 2 n n−1 A() = det(I A) = + n−1 + + 1 + 0 = 0 · − ··· (characteristic equation). There are n (not necessarily different) eigenvalues of A and spec(A) denotes their set. If i spec(A) is such that Im i = 0, ¯ 2 6 then i spec(A) too. 2 Right and left eigenvectors associated with i spec(A) are nonzero vectors such that 2 0 (i I A)Ái = 0 and Á (i I A) = 0; − i − respectively. Note that k k 0 k k 0 A Ái = Ái and Á A = Á ; k N i i i i 8 2 k k so i spec(A) = spec(A ), keeping left and right eigenvectors. 2 ) i 2 Preliminaries: linear algebra State-space models System interconnections Eigenvalues & eigenvectors of block-triangular matrices Let A = A11 A12 . Because 0 A22 I A A Á (I A )Á 11 12 = 11 ; −0 I− A 0 −0 − 22 Ái every eigenvalue of A11 is that of A (its right eigenvector is 0 ). Because 0 I A11 A12 0 0 Á − − = 0 Á (I A22) ; 0 I A22 − − 0 every eigenvalue of A22 is that of A (its left eigenvector is 0 Ái ). Thus i is an eigenvalue of A iff it is an eigenvalue of A or A . − 11 22 The same arguments work for block-lower-triangular and block-diagonal matrices too. − Preliminaries: linear algebra State-space models System interconnections Cayley{Hamilton In essence: each square matrix satisfies its own characteristic equation: n n−1 A(A) = A + n−1A + + 1A + 0In = 0: · ··· Important consequences: Ak for all k n is a linear combination of Ai , i = 0;:::; n 1, like − ≥ − n n−1 A = n− A A In − 1 − · · · − 1 − 0 n+1 n 2 A = n− A A A − 1 − · · · − 1 − 0 n−1 2 = n− (n− A + + A + In) A A 1 1 ··· 1 0 − · · · − 1 − 0 2 n−1 = ( n− )A + + (n− )A + n− In n−1 − 2 ··· 1 1 − 0 1 0 . A−1, if exists, is also a linear combination of Ai , i = 0;:::; n 1: − − −1 1 n−2 n−1 A = 1I + + n−1A + A : −0 ··· Preliminaries: linear algebra State-space models System interconnections Matrix functions P1 i Let f (x) = i=0 fi x be analytic. Its matrix version f (A) is defined as 1 X f (A) = f Ai : · i · i=0 It is readily seen that f (A)Ái = Ái f (i ) for all eigenvalue-eigenvector pairs. By Cayley{Hamilton, we know that gi , i = 0;:::; n 1 such that 9 − n−1 X i f (A) = gi A : i=0 Pn−1 i This does not imply that f (x) = g(x) = i gi x for all x, but still · =0 dj f (x) dj g(x) f ( ) = g( ) and = ; j = 1; : : : ; 1 i i j j i dx x=i dx x=i 8 − for each eigenvalue i of A having multiplicity i . These equations can be used to calculate all n coefficients gi and, hence, f (A). Preliminaries: linear algebra State-space models System interconnections Matrix functions (contd) If all n eigenvalues of A are different, 2 1 3 n−1 X j 6 i 7 f (i ) = gi = g0 g1 gn−1 6 . 7 : i ··· 4 . 5 j =0 n−1 i Thus, g g gn− V = f ( ) f ( ) f (n) , where 0 1 ··· 1 1 2 ··· 2 1 1 1 3 ··· 6 7 6 1 2 n 7 V = 6 . ··· . 7 · 4 . 5 n−1 n−1 n−1 1 2 ··· n Q is the Vandermonde matrix (with det V = (j i ) = 0). Hence, 1≤i<j≤n − 6 −1 g g gn− = f ( ) f ( ) f (n) V : 0 1 ··· 1 1 2 ··· Preliminaries: linear algebra State-space models System interconnections Matrix exponential The matrix exponential is defined (here t R, we shall need it later on) as 2 1 1 exp(At) = eAt = I + At + (At)2 + (At)3 + · 2! 3! ··· Example Let A = −1 1 with eigenvalues = 1 and = 2. Then 0 2 1 − 2 −1 (g (t) = 2 e−t + 1 e2t −t 2t 1 1 0 3 3 g0(t) g1(t) = e e 1 2 () g (t) = 1 e−t + 1 e2t : − 1 − 3 3 and hence 1 1 1 0 1 1 e−t 1 (e2t e−t ) exp t = g (t) + g (t) = 3 : −0 2 0 0 1 1 −0 2 0 e−2t Note, the exponential of this triangular matrix is triangular too (generic). Preliminaries: linear algebra State-space models System interconnections Matrix exponential The matrix exponential is defined (here t R, we shall need it later on) as 2 1 1 exp(At) = eAt = I + At + (At)2 + (At)3 + · 2! 3! ··· Example Let A = −1 1 with eigenvalues = 1 and = 2. Then 0 2 1 − 2 −1 (g (t) = 2 e−t + 1 e2t −t 2t 1 1 0 3 3 g0(t) g1(t) = e e 1 2 () g (t) = 1 e−t + 1 e2t : − 1 − 3 3 and hence 1 1 1 0 1 1 e−t 1 (e2t e−t ) exp t = g (t) + g (t) = 3 : −0 2 0 0 1 1 −0 2 0 e−2t Note, the exponential of this triangular matrix is triangular too (generic).