1: State-Space Models

Objective: Derive the state-space equations of electromechanical systems from first principles. Use Matlab/Oc- tave to represent state-space systems numerically and simulate their response to impulses and steps.

Problem 1 Permanent Magnet DC Motor with Gearbox

In this problem you will extend the model of the permanent magnet DC motor presented in class to include a gearbox as shown in the diagram below. The motor load in this case is modeled by the inertia J_L and the damping coefficient BL instead of an external torque Τ_L (t ). Using the Firestone analogy (see Reference Notes),

where torques and forces correspond to currents and velocities correspond to voltages, the above diagram can be turned into an electric circuit with linear elements and dependent sources whose dynamic equations are identi- cal to the equations of the original electromechanical system. I will refer to these analog circuits as generalized circuit diagrams. The generalized circuit diagram shown below corresponds to a permanent magnet DC motor driving a load through a gear box with a rigid axis. Note that capacitances represent moments of inertias, of the rotor and load, and conductances represent viscous-dampers.

This a consequence of choosing currents (Τ_m and Τ_L ) to represent torques and voltages ω_m and ω_L ) to represent angular velocities. Dependent sources are used to model the magnetic coupling and ideal transforms to model the mechanical gear box as shown below

In the generalized circuit diagram, v_a(t ) is the voltage input to the motor, i_a(t ) the motor (armature) current, ω_m, ω_L the angular speed of the motor and load, respectively, L_a[H] the motor (armature) inductance, R_a [?] the motor (armature) resistance, K_i [N-m/A] the torque constant, K_e [V/rad/sec] the back-electromotive force constant, B_m [N-m sec/rad] the motor damping, J_m [Kg m²] the motor inertia, JL [Kg m²] the load inertia and BL [N-m sec/rad] the load damping. (Note: in SI units K_i = Ke = K_m.) The (inverse) gear ratio is N : 1 (That is N = N₁/N₂, where Ni is the number of teth in the i^th gaer)

Note: To obtain more compact state-space equations use the "derived parameters" B_e = B_m + B_L/N² and J_e = J_m + J_L/N² where appropriate. Electrical Engineering students should recognize B_e and J_e as equivalent impedances (after reflecting capacitances and conductances from the secondary to primary and combining)

a) Choosing ωm and ωL as outputs and (i_a , θ_m , ω_m , θ_L , ω_L ) as states write the state-space equations of the system. Give your final answer in matrix form. (Hint: add the state equations for θ_m and θ_L once you are done with the derivation of the dynamic equations.)

Note: if your background is in Electrical or Computer Engineering you may find it easier to write the dynamic equations using nodal analysis. If your background is in Mechanical Engineering you can use free body dia- grams and assume "ideal gears" ( the gear inertias can be lumped in JL and its losses in BL )

b) I_s the state ω_L required? That is, if we remove the state ωL from state-equations, is it still possible to obtain ω_L from the remaining states ? Explain.

c) Write a Matlab function called DCmotor.m that takes as input argument, pars, a structure with the parameters of the motor and their values and returns a state-space LTI object. Use the following prototype for your Matlab/Octave function

1 function G=DCmotor(pars);

2 DCmotor: Generates an LTI object of the state-space representation of a permanent

3 magnet DC motor with a gear box driving a load through a rigid shaft.
4
5 Usage: G=DCmotor(pars);
6
7 inputs: pars - a structure with the motor parameters as follows:
8 pars.Ra = Armature Resistance
9 pars.La = Armature Inductance
10 pars.Ki = torque constant
11 pars.Ke = bemf constant
12 pars.Bm = Motor damping
13 pars.N = reciprocal of gear ratio
14 pars.JL = Load inertia
15 pars.BL = Load damping
16 outputs: G - a state-space LTI object
17
18 Author:
19 Revised:
20
21 Check for correct number of input arguments
22 if nargin ~= 1
23 help DCmotor
24 return
25 end

d) Write a Matlab/Octave script, hw1_plots.m to generate one figure with the following three plots:

(a) The impulse response of the system when the output is the load position θ_L (t ); e.g., use [yimp,t] = impulse(G(1,1)).

(b) The step response when the output is the load velocity ω_L (t ),e.g.; use ystp = step(G(2,1),t). (c)The sinusoidal response when when the output is the load position and the voltage input is va (t ) = 10 sin(5t ), t ≥ 0 ; use ysin = lsim(G(1,1), u, t).

To generate a single figure with the three plots use the subplot function. Make sure to label the plots properly,
e.g., using xlabel, ylabel and title).

For this part use the following parameter values (all in SI units): Ra = 3 ?, La = 0.03 H, Ki = 0.2 [Nm/A],

K_e = 0.2 Vs/rad, J_m = 0.02 Kgm2, B_m = 0.002 Nms/rad, J_L = 0.003 Kgm², B_L = 1 Nms/rad and N = 12.

e) How is the impulse response plot related to the step response plot ? Why do they look similiar ? Explain.

f) In the impulse and step simulations you will observe that the output does not oscillate.

Is it possible to have an oscillatory impulse or step response by changing the parameters of the motor ? Explain

(Hint: Do not change the parameters blindly, instead analyze the problem using your understanding of the physics of the system and or of the equivalent generalized circuit that describes it)