National Instruments 370753C-01

Chapter 1 Introduction

The linearized state-space equations, including the actuator and sensor

dynamics, are as follows:

θ is the angle (in radians) the wedge makes with the vertical axis, x is the

position of the sliding mass, and u is the control input voltage. The outputs

are scaled to give the measured angle in degrees and the measured position

in meters.

A = [0,0,1,0;0,0,0,1;

15.54,-10.93,0,0;

-5.31,0,0,-16.24];

B = [0,0,0,1.96]';

C = [57.29,0,0,0;0,29.9,0,0];

D = [0;0];

states = ["Angle", "Mass Position",

"Angular Velocity","Mass Velocity"];

wsys = system(A,B,C,D,

{inputNames= "Voltage",

stateNames = states,

outputNames=["Measured Angle","Measured

Position"]});

You need to ensure that you have no uncontrollable or unobservable modes

of the system:

[,,nuco] = minimal(wsys)

nuco (a scalar) = 0

Because there are no uncontrollable or unobservable states, you can

proceed with the design of a regulator and estimator. The weighting matrix

used here in designing the regulator reflects the desire to bring the value of

the first state, the angle with the vertical, to zero as quickly as possible.

··

0010

0001

15.54 10.93–00

5.31–0 0 16.24–

1.96

u+=

y57.29000

0 29.9 0 0