DC Motor Driving an Inertial Load

w(t): angular rate of the load, output vapp(t): applied voltage, the input i(t) armature current vemf(t) back emf voltage generated by the motor rotation vemf(t) = constant * motor velocity t(t): mechanical torque generated by the motor t(t) = constant * armature current

State Space model

Matlab R= 2.0; % Ohms L= 0.5; % Henrys Km = .015; % torque constant Kb = .015; % emf constant Kf = 0.2; % Nms J= 0.02 % kg.m^2; A = [-R/L -Kb/L; Km/J -Kf/J]; B = [1/L; 0]; C = [0 1]; D = [0]; sys_dc = ss(A,B,C,D)

Matlab output a = x1 x2 x1 -4 -0.03 x2 0.75 -10 b = u1 x1 2 x2 0 c = y1 0 1 d = y1 0

SS to TF or ZPK representation >> sys_tf = tf(sys_dc) Transfer function: 1.5 ------------------------ s^2 + 14 s + 40.02 >> sys_zpk = zpk(sys_dc) Zero/pole/gain: ------------------------- (s+4.004) (s+9.996)

Note: The state-space representation is best suited for numerical computations. For highest accuracy, convert to state space prior to combining models and avoid the transfer function and zero/pole/gain representations, except for model specification and inspection.

4 ways to enter system model sys = tf(num,den) % Transfer function sys = zpk(z,p,k) % Zero/pole/gain sys = ss(a,b,c,d) % State-space sys = frd(response,frequencies) % Frequency response data s = tf('s'); sys_tf = 1.5/(s^2+14*s+40.02) Transfer function: 1.5 ------------------------ s^2 + 14 s + 40.02 sys_tf = tf(1.5,[1 14 40.02])

4 ways to enter system model sys_zpk = zpk([],[-9.996 -4.004], 1.5) Zero/pole/gain: 1.5 ------------------------- (s+9.996) (s+4.004)

Liquid Level System Qi H Qo Qi = input flow rate Qo = output flow rate H = liquid level in tank A = cross section of tank V = volume of liquid in tank V = AH

Conservation of matter: Q Conservation of matter: Qo is dependent on the “head” H const. coeff. H

∴ This is nonlinear. To find eq. points, set derivative=0 To linearize: let where

Substitute into eq on top: use

Output flow: The quantity = R is the called the resistance of the valve and A is also denoted as C & is called the capacitance of the tank. Then: Note:

Two tank system:

In eq pt: all flow=same 1 3 2 4

Op Amp circuit example Note: ip1=0, ∴vp1=vo=vA & vB=vp2=0 Let vC1 & vC2 be s.v., vo output.

KCL at A: vo is not s.v. nor input, use vo=vC2

KCL at B: vo1 not s.v. nor input, vo1=vA+vC1=vn1+vC1 =vp1+vC1=vo+vC1 =vC2+vC1

Output eq:

Modeling Types of systems electric mechanical electromechanical Types of models I/O o.d.e. models Transfer Function state space models

I/O o.d.e. model: o.d.e. involving input/output only. linear: where u: input y: output

State space model: linear: or in some text: where: u: input y: output x: state vector A,B,C,D, or F,G,H,J are const matrices

Other types of models: Transfer function model (This is I/O model) from I/O o.d.e. model, take Laplace transform:

Then I/O model in L.T. domain becomes: This is the T.F. model of the system. ∴T.F. or i.e. output L.T. is eq. to input L.T. with gain H(s) denote

State space model to T.F. / block diagram: s.s. Take L.T. : From sX(s)-AX(s)=BU(s) sIX(s)-AX(s)=BU(s) (sI-A)X(s)=BU(s) X(s)=(sI-A)-1BU(s) 1 2 1

into : Y(s)=C(sI-A)-1BU(s)+DU(s) Y(s)=[C(sI-A)-1B+D] U(s) H(s)= D+C(sI-A)-1B is the T.F. from u to y from 2 1

Example

>> [n,d]=ss2tf(A,B,C,D) n = 0 3.0000 1.0000 d = 1 3 2 >> [A,B,C,D]=tf2ss(n,d) A = -4 -5 -6 1 0 0 0 1 0 B = 1 C = 1 2 3 D = >> tf(n,d) Transfer function: s^2 + 2 s + 3 --------------------- s^3 + 4 s^2 + 5 s + 6 In Matlab: >> A=[0 1;-2 -3]; >> B=[0;1]; >> C=[1 3]; >> D=[0]; >> [n,d]=ss2tf(A,B,C,D) n = 0 3.0000 1.0000 d = 1 3 2