System equations

In general, the torque generated by a DC motor is proportional to the armature current and the strength of the magnetic field. In this example we will assume that the magnetic field is constant and, therefore, that the motor torque is proportional to only the armature current i by a constant factor Kt as shown in the equation below. This is referred to as an armature-controlled motor.

(1)

The back emf, e, is proportional to the angular velocity of the shaft by a constant factor Kb.

(2)

In SI units, the motor torque and back emf constants are equal, that is, Kt = Ke; therefore, we will use K to represent both the motor torque constant and the back emf constant.

From the figure above, we can derive the following governing equations based on Newton's 2nd law and Kirchhoff's voltage law.

(3)

(4)

1. Transfer Function

Applying the Laplace transform, the above modeling equations can be expressed in terms of the Laplace variable s.

(5)

(6)

We arrive at the following open-loop transfer function by eliminating I(s) between the two above equations, where the rotational speed is considered the output and the armature voltage is considered the input.

(7)

However, during this example we will be looking at the position as the output. We can obtain the position by integrating the speed, therefore, we just need to divide the above transfer function by s.

(8)

2. State-Space

The differential equations from above can also be expressed in state-space form by choosing the motor position, motor speed and armature current as the state variables. Again the armature voltage is treated as the input and the rotational position is chosen as the output.

(9)

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Bus Suspension Physical setup

Designing an automotive suspension system is an interesting and challenging control problem. When the suspension system is designed, a 1/4 model (one of the four wheels) is used to simplify the problem to a 1D multiple spring-damper system. A diagram of this system is shown below. This model is for an active suspension system where an actuator is included that is able to generate the control force U to control the motion of the bus body.

System parameters

(M1) 1/4 bus body mass 2500 kg

(M2) suspension mass 320 kg

(K1) spring constant of suspension system 80,000 N/m

(K2) spring constant of wheel and tire 500,000 N/m

(b1) damping constant of suspension system 350 N.s/m

(b2) damping constant of wheel and tire 15,020 N.s/m

(U) control force

Equations of motion

From the picture above and Newton's law, we can obtain the dynamic equations as the following:

(1)

(2)

Aircraft Pitch

Physical setup and system equations

The equations governing the motion of an aircraft are a very complicated set of six nonlinear coupled differential equations. However, under certain assumptions, they can be decoupled and linearized into longitudinal and lateral equations. Aircraft pitch is governed by the longitudinal dynamics. In this example we will design an autopilot that controls the pitch of an aircraft.

The basic coordinate axes and forces acting on an aircraft are shown in the figure given below.

We will assume that the aircraft is in steady-cruise at constant altitude and velocity; thus, the thrust, drag, weight and lift forces balance each other in the x- and y-directions. We will also assume that a change in pitch angle will not change the speed of the aircraft under any circumstance (unrealistic but simplifies the problem a bit). Under these assumptions, the longitudinal equations of motion for the aircraft can be written as follows.

(1)

(2)

Please refer to any aircraft-related textbooks for the explanation of how to derive these equations. You may also refer to the Extras: Aircraft Pitch System Variables page to see a further explanation of what each variable represents.