Interarea Oscillations Starrett Mini-Lecture #5

Interarea Oscillations - Linear or Nonlinear? l Mostly studied as a linear phenomenon l More evidence of nonlinear or stressed system problem

Why do We Like Linear Systems? l Easy to solve differential equations l Can calculate frequencies and damping l Design control systems easily l Pretty good approximation

Small-Signal Stability -> Linear System Analysis State Space representation  x = A  x + B  u  y = C  x + D  u A =  f 1 /  x 1...  f 1 /  x n B = {  f/  u}  f 2 /  x 1...  f 2 /  x n...  f n /  x 1...  f n /  x n

Linear System Terms l Eigenvalues l Eigenvectors l Jordan Canonical Form l System Trajectories l Measures of system performance

Eigenvalues l Roots of characteristic equation l Tell stability properties of linear system (Hartman-Grobman Theorem) Eigenvalues =>  j 

Linear System Solution x(t) = C 1 e 1 t + C 2 e 2 t... + C n e n t x(t) = C 1 e (  1+ j  1)t + C 2 e (  2+ j  2)t … + C n e (  n+ j  n)t x(t) = D 1 e  1t cos(  1t) + D 2 e  2t cos(  2t) … l Constants are dependent on initial conditions

Calculating Eigenvalues A r i = i r i l A = system plant matrix  = eigenvalue l r = an nX1 vector (right eigenvector) l Rearranging …  (A - I )r = 0 => det(A - I ) = 0

Solving for Right Eigenvectors A r i = i r i l Solve system of linear algebraic equations for components of r i, (r 1i, r 2i, r 3i, etc.) l Right Modal Matrix, l R = square matrix with r i 's as columns

Left Modal Matrix & Left Eigenvectors l L = R -1 l left eigenvectors = l i 's = rows of L l i A = i l i

Free Response of a System of Linear Differential Equations  x = A  x l Define a variable transformation   x = R z l Substitute into diff. eq. yR z’ = A R z l Pre-multiply both sides by R -1 = L  L R z’ = R -1 R z = R -1 A R z = L A R z =  z

Jordan Canonical Form z’ =  z  = diagonalized matrix with i 's on diagonal  = n

The Jordan Form System is Decoupled z 1 = 1 z 1 =>z 1 (t) = z 1o e 1t z 2 =  2 z 2 => z 2 (t) = z 2o e 2t … z n = n z n => z n (t) = z no e nt

Now Transform Solutions Back to x-Space  x = R z=>  x(t) = R z(t)  x 1 (t) = r 11 z 1 (t) + r 12 z 2 (t) +... R 1n z n (t)  x 1 (t) = r 11 z 1o e 2t + r 12 z 2o e 2t … + r 1n z no e nt

Initial Conditions l z io 's are the initial conditions in z-space x o is the vector of initial conditions in x-space  x = Rz => L  x = L R z => L  x = z l so... z o = L x o and z io = l 11 x 1o + l 12 x 2o l 1n x no

Visualize the Linear Systems Analysis x2x2 x1x1 z1z1 z2z2 Jordan State Space Angle-Speed Space

In-Class Exercise A = R = L = Eigenvalues = ? 2. What values of x 1o and x 2o correspond to z 1o = 1, z 2o = 0?

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