Force analysis and system equations

Below are the free-body diagrams of the two elements of the inverted pendulum system.

Summing the forces in the free-body diagram of the cart in the horizontal direction, you get the following equation of motion.

(1)

Note that you can also sum the forces in the vertical direction for the cart, but no useful information would be gained.

force Summing the forces in the free-body diagram of the pendulum in the horizontal direction, you get the following expression for the reaction .

(2)

If you substitute this equation into the first equation, you get one of the two governing equations for this system.

(3)

To get the second equation of motion for this system, sum the forces perpendicular to the pendulum. Solving the system along this axis greatly simplifies the mathematics. You should get the following equation.

(4)

To get rid of the and terms in the equation above, sum the moments about the centroid of the pendulum to get the following equation.

(5)

Combining these last two expressions, you get the second governing equation.

(6)

Since the analysis and control design techniques we will be employing in this example apply only to linear systems, this set of equations needs to be linearized. Specifically, we will linearize the equations about the vertically upward equillibrium position, = , and will assume that the system stays within a small neighborhood of this equillbrium. This assumption should be reasonably valid since under control we desire that the pendulum not deviate more than 20 degrees from the vertically upward position. Let represent the deviation of the pedulum's position from equilibrium, that is, = + . Again presuming a small deviation ( ) from equilibrium, we can use the following small angle approximations of the nonlinear functions in our system equations:

(7)

(8)

(9)

After substiting the above approximations into our nonlinear governing equations, we arrive at the two linearized equations of motion. Note has been substituted for the input .

(10)

(11)

Car Automatic cruise control Physical setup

Automatic cruise control is an excellent example of a feedback control system found in many modern vehicles. The purpose of the cruise control system is to maintain a constant vehicle speed despite external disturbances, such as changes in wind or road grade. This is accomplished by measuring the vehicle speed, comparing it to the desired or reference speed, and automatically adjusting the throttle according to a control law.

We consider here a simple model of the vehicle dynamics, shown in the free-body diagram (FBD) above. The vehicle, of mass m, is acted on by a control force, u. The force u represents the force generated at the road/tire interface. For this simplified model we will assume that we can control this force directly and will neglect the dynamics of the powertrain, tires, etc., that go into generating the force. The resistive forces, bv, due to rolling resistance and wind drag, are assumed to vary linearly with the vehicle velocity, v, and act in the direction opposite the vehicle's motion.

System equations

With these assumptions we are left with a first-order mass-damper system. Summing forces in the x-direction and applying Newton's 2nd law, we arrive at the following system equation:

(1)

Since we are interested in controlling the speed of the vehicle, the output equation is chosen as follows

(2)

System parameters

For this example, let's assume that the parameters of the system are:

(m) vehicle mass 1000 kg

(b) damping coefficient 50 N.s/m

State-space model

First-order systems have only has a single energy storage mode, in this case the kinetic energy of the car, and therefore only one state variable is needed, the velocity. The state-space representation is therefore:

(3)

(4)

We enter this state-space model into MATLAB using the following commands:

m = 1000;

b = 50;

A = -b/m;

B = 1/m;

C = 1;

D = 0;

cruise_ss = ss(A,B,C,D);

DC Motor Speed

Physical setup

A common actuator in control systems is the DC motor. It directly provides rotary motion and, coupled with wheels or drums and cables, can provide translational motion. The electric equivalent circuit of the armature and the free-body diagram of the rotor are shown in the following figure.

For this example, we will assume that the input of the system is the voltage source (V) applied to the motor's armature, while the output is the rotational speed of the shaft d(theta)/dt. The rotor and shaft are assumed to be rigid. We further assume a viscous friction model, that is, the friction torque is proportional to shaft angular velocity.

The physical parameters for our example are:

(J) moment of inertia of the rotor 0.01 kg.m^2

(b) motor viscous friction constant 0.1 N.m.s

(Ke) electromotive force constant 0.01 V/rad/sec

(Kt) motor torque constant 0.01 N.m/Amp

(R) electric resistance 1 Ohm

(L) electric inductance 0.5 H