MEM255Su06-07.7.Intro2sSpace.studs

MEM255Su06-07.7.Intro2sSpace.studs - M EM 255: I nt r oduct...

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Unformatted text preview: M EM 255: I nt r oduct ion t o Cont r ols (I ntr o to M atr i x Eqns.) [3] Dr. Ajmal Yousuff Dept . M EM Dr exel Univer sit y An algebr aic example Solve: 2 x1 + 3 x2 = 4 x1 - 4 x2 = 5 for x 1 and x 2. Rewr it e: Yousuff MEM255: Intro2Control 2 Solut ion t o Ax=y Solut ion exist s iff eqns. ar e consist ant . rank [ A y ] = rank [ A] r ank = # of independent columns (r ows) Examples: x1 5 0 5 3 0 x = 3 has solutions 2 5 0 x1 5 3 0 x = 4 does not have a solution 2 Yousuff MEM255: Intro2Control 3 A differ ent ial eqn. example Consider : x1 + 2 x1 + 2 x2 = u1 + u 2 x2 + x1 - 3 x2 = u1 ; (u1 and u 2 : inputs) Rewr it e: Yousuff MEM255: Intro2Control 4 St at e Space Repr esent at ion (L inear ) x = Ax + Bu y = Cx + Du Yousuff MEM255: Intro2Control 5 M echanical syst em m1q1 + kq1 - kq2 = 0 m2 q2 + kq2 - kq1 = f Put in state space form with output y = q1 and input u = f: Yousuff MEM255: Intro2Control 6 Set up St at es Define: Yousuff MEM255: Intro2Control 7 M ech. Cont inued Express xi in terms of only x j and u Yousuff MEM255: Intro2Control 8 St at e Space M odel (cont d) x 1 0 0 x 2 = 3 k / m1 x - x k 4 / m2 0 0 k / m1 -k / m2 1 0 x1 0 0 1 x2 0 + u 0 0 x3 0 0 0 x4 m2 1/ y = [ 1 0 0 0] x + [0]u Yousuff MEM255: Intro2Control 9 Simpler appr oach m 1 0 (k -k2 x1 0 1 1 + k2 ) x 0 x -k + = 0 (k2 + k3 ) 2 m2 x2 2 M + Cx + K x = B u x M : mass matrix C: damping matrix K: stiffness matrix zero, in this example 10 MEM 423 Yousuff St at e-space model M q+ Cq+ K q = Bu Define states as: (1) q x = R2 n , with q Rn q From (1): q = -M -1Cq - M -1K q+ M -1Bu In 0 x + -1 u -1 -M C B M Yousuff q 0 x = -1 = q -M K 11 MEM423 Vibrations St at e Equat ions x = Ax + Bu; y = Cx + Du. x e n , u m , y k Analysis: u (t ) = ? x(t ) or y (t ) = Y acceptable " Cont r ol Pr oblem: (t ) Y x(t ), y (t ) = ? Given u A.Yousuff MEM 255 Control 12 Tr ansfer funct ion fr om st at e space x = Ax + Bu; y = Cx; x0 = 0 L{ } sX = AX + BU ; Y = CX ( sI - A) X = BU X = ( sI - A) -1 B Y(s) = C(sI-A)-1B U(s) adj ( sI - A) G(s) = C B | sI - A | Poles of G = eigenvalues of A A.Yousuff MEM 255 Control 13 Tr ansfer funct ion t o st at e space Example (det ails in r ecit at ions) U(s) b2 s 2 + b1s + b0 1s 3 + a2 s 2 + a1s + a0 Y(s) 14 MEM255: Intro2Control Yousuff ...
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