Stability of linear systems Daniele Carnevale Dipartimento di Ing. - PowerPoint PPT Presentation

Matrix properties Stability Stability of linear systems Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome “Tor Vergata” Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015 1 / 8

Matrix properties Cayley-Hamilton Stability Minimal characteristic polynomial Cayley-Hamilton A generic square matrix A renders zero its characteristic polynomial p A ( λ ) such as p A ( A ) = A n + α n − 1 A n − 1 + · · · + α 1 A + α 0 I = 0 , this also implies that the matrix A n can be written as a linear combination of lower order power of A such as − A n = α n − 1 A n − 1 + · · · + α 1 A + α 0 I, (1) that is named the Cayley-Hamilton Theorem. Consequently A n + h for any h ≥ 0 can be rewritten as a linear combination of A j with j ∈ { 0 , 1 , . . . , n − 1 } . . The matrix A nullify its characteristic polynomial by definition, in fact p A ( λ ) = det( λI − A ) ⇒ p A ( A ) = det( A − A ) = 0 . (2) 2 / 8

Matrix properties Cayley-Hamilton Stability Minimal characteristic polynomial Minimal characteristic polynomial The minimal characteristic polynomial q A ( λ ) is defined as the minimum order polynomial such that q A ( A ) = 0 . Note that in general the degree of q A ( λ ) is smaller than p A ( λ ) . The polynomial q A ( λ ) can be obtained as the least common multiple (LCM) denominator of the rational matrix ( sI − A ) − 1 . Note that the roots of q A ( λ ) are the roots of p A ( λ ) , i.e. the eigenvalues of the matrix A . As an example, if ν are the distinct eigenvalues of the matrix A , then ν � ( λ − λ i ) m i . q A ( λ ) = (3) i =1 Note that m i = n i − µ i + 1 , (4) than m i > 1 iif the matrix A is not diagonalizable. m i is the dimension of the Jordan block associated to the eigenvalue λ i . Then, the free state response of a linear systems has polynomial terms only if m i > 1 . 3 / 8

Matrix properties Cayley-Hamilton Stability Minimal characteristic polynomial Minimal characteristic polynomial: examples Consider   1 0 0  → p A ( s ) = ( s − 1) 2 ( s + 1) . A = 0 − 1 2  0 0 1 In this case the minimal polynomial q A ( s ) is the LCM denominator of ( sI − A ) − 1 that is   1 0 0 s − 1 ( sI − A ) − 1 = 1 2  0   → q A ( s ) = ( s − 1)( s + 1) ,  s +1 ( s − 1)( s +1) 1 0 0 s − 1 in fact defining λ 1 = − 1 , λ 2 = 1 , then µ 1 = 2 = n 1 → m 1 = 1 , µ 2 = 1 = n 2 → m 2 = 1 . 4 / 8

Matrix properties Cayley-Hamilton Stability Minimal characteristic polynomial Minimal characteristic polynomial: examples cont’d Consider now   1 0 1  → p A ( s ) = ( s − 1) 2 ( s + 1) . A = 0 − 1 2  0 0 1 yielding   1 1 0 s − 1 ( s − 1) 2 ( sI − A ) − 1 =  1 2   → q A ( s ) = ( s − 1) 2 ( s + 1) , 0  s +1 ( s − 1)( s +1) 1 0 0 s − 1 since λ 1 = − 1 , λ 2 = 1 , then µ 1 = 1 < n 1 → m 1 = 2 , µ 2 = 1 = n 2 → m 2 = 1 . In this case the degree of p A ( s ) is equal to q A ( s ) , i.e. p A ( s ) = q A ( s ) . 5 / 8

Matrix properties LTI stability condition Stability Examples The stability property do not depend on the system input To retrieve if the equilibrium (set) x e associated to u e is stable, than consider the solution of a linear continuous time (in discrete time is similar) such as � t x ( t ) = ϕ ( t, t 0 , x 0 , u e ) = e A ( t − t 0 ) x 0 + e A ( t − τ ) B dτu e (5) t 0 = e A ( t − t 0 ) x 0 + L − 1 � ( sI − A ) − 1 B u e � ( t − t 0 ) . (6) s By linearity, the stability property of x e can be equivalently derived by considering directly the stability property of the origin. Then || x ( t ) || = || e A ( t − t 0 ) x 0 || ≤ || e A ( t − t 0 ) |||| x 0 || ≤ || e A ( t − t 0 ) || δ ǫ , (7) that if Re { λ i } < 0 for all λ i ∈ σ ( A ) , implies that || e A ( t − t 0 ) || ≤ m and then selecting δ ǫ < ǫ/m yield || x ( t ) || < ǫ, and t → + ∞ || x ( t ) || = 0 , lim so that the origin is Globally Asimptotically Stable (GAS). 6 / 8

Matrix properties LTI stability condition Stability Examples Stabiity conditions of LTI A continuous time LTI system (all the equilibria have the same properties) is stable iif Re { λ i } ≤ 0 for all λ i ∈ σ ( A ) and µ i = n i for all λ i such that Re { λ i } = 0 globally asymptotically stable iif Re { λ i } < 0 for all λ i ∈ σ ( A ) unstable otherwise. A discrete time LTI system (all the equilibria have the same properties) is stable iif | λ i | ≤ 1 for all λ i ∈ σ ( A ) and µ i = n i for all λ i such that | λ i | = 1 globally asymptotically stable iif | λ i | < 1 for all λ i ∈ σ ( A ) unstable otherwise. 7 / 8

Matrix properties LTI stability condition Stability Examples Economic linear model Let d ( k ) be the demand of a given product at month k − th , s ( k ) the production offer and p ( k ) the product cost. At market equilibrium it has to hold that d = s . We are interested in understanding if such equilibrium can be reached in the market. Assume that the dynamic relation between s , p and d is given by d ( k + 1) = D − αp ( k + 1) (8) s ( k + 1) = S + βp ( k ) , (9) with D > 0 , S > 0 , α > 0 and β > 0 . Then p ( k + 1) = D − d ( k + 1) D − s ( k + 1) = − β α p ( k ) + D − S = . α α α ���� equilibrium s = d Does exist a price p e that is an equilibrium and is it globally stable, unstable or globally asymptotically stable? Υ . There is a bracket missing in the use of Υ function up to Definition 5. 8 / 8