Eigenvalues and eigenvectors System Theory: electricmotor

Transcription

1 Eigenvalues and eigenvectors System Theory: electricmotor Alireza Abouhossein Ph.D. 1

2 Example-1 Can you calculate A^2, A -1 -I? without multiplying A or finding the inverse of A? Give it try We know: Av= v=> A 2 v=a( v)= (Av)= 2 v A -1 v=a -1 (Av/ )=A -1 A(v/ )=1/ 0 2

3 Example-1 Let s find eigenvalues and eigenvectors of A matrix: Try to see if you get the above equation and how you wish to solve this problem. 3

4 Example-1 Remember we have the equation Ax= x to solve: 4

5 Example-1 Sub-back 2 (the distinct eigenvalue) to the A, for the first one we have: The matrix is singular and the rank is 2 5

6 Example-1 Matrix was singular: 6

7 Example-2 Second eigenvalues (1), linearly dependent eigenvectors. Singular matrix determinant =0 show that det. is zero 7

8 Example-1 8

9 Example-2 Recall that this eigenvector has a bases in the same plane as it is multiple of the β, pay attention to the equation, the sign has changed: Try to solve the equation by hand and get the above eigenvector. 9

10 Simulink How does Simulink work? System description through block diagrams Simulink provides a large library of components that describe basic algebraic and dynamical system (e.g. transfer function, gain, saturation and mathematical function, etc.) The user must set up the complete the block diagram of the system via block interconnections. Step #2: System simulation Simulink uses the Matlab core to numerically integrate the differential equations that describe the dynamics of the system over a time interval. 10

11 Electromotor in direct current Write the transfer function for the armature-controlled DC motor system In an armature-current controlled DC motor, the field current i f is held constant, and the armature current is controlled through the armature voltage V a. Therefore, we can conclude that the motor torque increases Linearly with the armature current. T m =K ma i a K ma is a constant that depends on the chosen motor. The transfer function from the input armature current to the resulting motor torque is: T m /I a (s)=k ma (1) 11

12 Electromotor cont s Now, let s take a look at the voltage/current relationship for the armature side of the motor: V a =V R +V L +V b= R a i a +L a (di a /dt)+v b (2) V b Represents the back counterelectromotive force (EMF) induced by the rotation of the armature windings in a magnetic field. V b is proportional to the rotational speed, V b = K b ω(s) Taking Laplace form of Eq. 1: V a (s)-v b (s)=(r a +L a s)i a (s) or V a (s)-k b ω(s)=(r a +L a s)i a (s) (3) 12

13 Electromotor cont s What else is missing? Applying Newton s law (By summing the moments) for the rotational motion of the motor gives: (positive CCW) M=T m -cω=jω T C(dθ/dt)=c ; friction m torque constant (c) Jω +cω=t m (4) θ Thus, the transfer function from the input motor torque to rotational speed changes is ω/t m (s)=(1/j)/(s+(c/j)) (5) 13

14 Combining the equations 1,3,5 can be represented by the closed loop block diagram. Try to implement it in the Simulink. 14

15 Transfer function Write the transfer function from the input armature voltage to the resulting speed change 15

16 Where to pick up the slides 16