  # Feedback Control of Mechanical Systems

This book covers the fundamentals of feedback control of dynamic systems. Appropriate for upper-division engineering undergraduate students with some knowledge of dynamic systems.

# System Modeling

### Overview

By now, you should be comfortable with the idea of feedback control and how the unity-feedback block diagram (Fig. 1.1) applies to many different systems. You should also understand what each element in the feedback control diagram represents, but for this chapter, we are going to specifically study what the plant P(s) represents.

Before we can begin discussing controller design techniques, we must have a deep understanding of what the plant P(s) actually is. It turns out we generally have an intuitive idea of what a plant is, but it is very difficult to define what P(s) is in so many words. For example, in the stick balancing example from Chapter 1, the plant was the inverted stick with mass m, inertia J, length l, all in a gravitational field defined by the acceleration g. We simply lumped all of those elements together and called it the plant. To be a little more specific, you should start thinking about the plant P(s) as a dynamic system. For the remainder of the chapter, we are going to take a look at what system dynamics really are. We begin with a general definition:

DEFINITION: A dynamic system is any set of connected things or parts forming a complex whole, in which the current output is a function of all past and current inputs.

It's true that this definition is very general, but it must be this way because there are so many different types of dynamic systems. Essentially, as long as the above definition applies, the system can be considered dynamic. Additionally, although we will primarily study mechanical dynamic systems, there are also electrical, biological, and economical systems, to name a few, that also fall under the general descriptor of a dynamic system. Let's get a bit more specific and discuss system dynamics within the confines of a real world example.

Consider getting into your car and depressing the gas pedal all the way down (you are late for class). This is considered a constant input to the system. The output of this dynamic system is the velocity v(t) (see Fig. 2.1).

The question is, if you floor the gas pedal, what does the output look like? Does the velocity profile follow the input as in Fig. 2.1, output a? Or does the velocity profile look something more like Fig. 2.1 output b? Intuitively, we all know that the velocity of the car behaves more like response b. That is, the vehicle accelerates, gradually picking up speed until a terminal velocity is reached, at which point the force due to wind resistance cancels out the forward thrust generated by the car's engine. This example should give you a sense of what the dynamics of a system are. In so many words, the dynamics of system refer to how the input is mapped or translated to the output.

KEY POINT: The dynamics of a system refer to HOW the input translates to the output. In the context of the car example in Fig. 2.1, the WAY the vehicle reaches its final speed is described the the dynamics of the system.

The goal of the previous example was to give you an intuitive sense for what the dynamics of a system, and hence the plant P(s), really refer to. The question then becomes: Can we capture the dynamics of a system mathematically? After all, we are engineers and we would like to be able to somehow quantify the dynamics of a system. The answer is yes. The dynamics of a system can be represented mathematically through the process of modeling.

KEY POINT: We can capture the dynamics of a system mathematically through the process of modeling.

It was mentioned that there are several types of dynamic systems; mechanical, electrical, biological, economical, to name a few. For each type of system, the process of modeling requires a different set of tools, but the general process is to use some known relationships in order to describe the system. For mechanical systems, we generally use free-body-diagrams, Newton's laws, and rigid body dynamics to generate a model for the system. More specifically, for mechanical systems, the modeling process boils down to following these steps:

• Draw an accurate free-body-diagram with all relevant external forces
• Apply Newton's 2nd law for translation or rotation
• Write the equation(s) in terms of the input and output variables you are concerned with
• The resulting differential equation(s) are the model

The remainder of this chapter will illustrate how this modeling process applies to several real life mechanical systems.

### Modeling an automobile with wind resistance

Generate a model (i.e. do the modeling process) for the automobile shown in the figure below:

The input to the system is the force u(t), which is provided by the engine. The output is taken to be the car's velocity, denoted v(t).

Since this is a mechanical system, the modeling process begins with an accurate free-body-diagram (an FBD). We are to consider wind resistance in this example, and we will use a first-order linear drag force, proportional to the velocity. Once an FBD is created, we apply newton's law for translation and rearrange the equation.

The boxed ordinary differential equation (ODE) above is the model for the system and it is important to understand that this equation represents the dynamics of the system. In fact, we can reconcile our intuition from before, by actually solving this equation with a specific input u(t). It turns out that, if the input force u(t) is a constant value (i.e. flooring the gas pedal), then the general form of the solution to the ODE is

Without actually specifying initial conditions and solving for a particular solution, we know that this response looks like:

which should agree with our intuition from back in the example in Fig. 2.1

### Modeling a swinging pendulum

Another very commonly studied mechanical dynamic system is the swinging pendulum. Again, it's a mechanical system, so the modeling process is exactly the same as before: produce an accurate FBD, then apply Newton's law to generate an ODE (model) representing the system.

There are a couple of differences between this example and the automobile example. First, the FBD was drawn with all of the forces broken down into radial and tangential forces. This is because it is a rotational system, and thus Newton's law for rotation should be used. Drawing the FBD as such simplifies the sum-of-torques part of the equation (the left-hand side). The rotational moment of inertia I for this problem is equal to mL2 which just means we are assuming the mass is a point mass at the end of a massless rod of length L. The variable α is the angular acceleration, and is denoted by

KEY POINT: Despite these slight differences, remember to keep in mind that the modeling process was exactly the same as in the automobile example.

Now, let's do a bit of analysis on our pendulum model. Often, engineers like to linearize our models as an approximation in order to simplify some of the math required to do analysis. In this case, we have an ODE with a sin(θ) term in it. This is a non-linear term, and we might want to linearize the model. One method to linearize this system is to invoke the small angle approximation, which states that sin(θ) ≈ θ, and cos(θ) ≈ 1, for small values of θ. If we linearize (apply the small angle approximation) our pendulum model, we get

The general solution to this 2nd-order homogeneous ODE is

where a and b will depend on the initial conditions. Let's suppose we pull back the pendulum to a specific angle, rest, then release it. Mathematically, this scenario is equivalent to applying the following initial conditions:

With these initial conditions, the particular solution to the ODE is

which looks like:

This response is the solution to the ODE, but more importantly, it represents the actual output of the system, i.e. it describes the system response under the given initial conditions.

At this point, we should ask ourselves whether or not this seems logical. On one hand, we do expect that the system should oscillate back and forth as a pendulum does. On the other hand, we intuitively know that a typical pendulum in our earth's atmosphere does not oscillate forever, as indicated by this model. So, is this really a good model?  ...not really.

Let's look at the same system, with a slight modification.

### Modeling a swinging pendulum with wind resistance

The only difference between this modified version and the previous version is now we are going to include a drag force that scales linearly (via the drag coefficient b) with the tangential velocity of the mass.

The resulting ODE is similar to the previous example, but now there is a first-order term, commonly referred to as the damping term. Now, if you can, recall the general form for this 2nd-order homogenous ODE with a damping term. If you can't, that's ok; it looks like:

Now, without actually applying initial conditions and muddying up the math, let's instead focus on the form of this equation. It's the product of an exponential term and a sinusoidal term. Again, relating this back math back to the real system, it represents the actual time function for the angle. Qualitatively, it looks like this:

which is a much more realistic response to expect from a normal, everyday pendulum. The main point here is that we have choices as engineers in the modeling process, and our assumptions will carry through to the final model. The pendulum with drag model is clearly more realistic than the pendulum without drag, which should serve to illustrate that your model will differ greatly depending on your assumptions.

KEY POINT: We have choices when modeling dynamic systems, and those choices will affect the accuracy of the final model.

### Recap

We've done some examples of modeling dynamic, mechanical systems, and hopefully the process is clear. It is imperative, however, to remember where this process fits in with the overall scheme of feedback control systems. Remember that the models we have been deriving serve to represent the dynamic systems that we wish to control. The model is a representation of the plant, which is synonymous with the dynamic system.

### Concept check questions:

Q2.1

A model of a system in the context of this course can be thought of as

A

a physical representation of the system in 3D space

B

a mathematical representation of the system that captures its dynamics

C

a sketch of the system output vs time, under specific initial conditions

D

a small-scale representation of the real life system

Q2.2

The process of modeling is necessary because we need a mathematical representation of the plant P(s). Through studying the examples in this chapter, you have learned how to model mechanical systems. What is generally the result of the modeling process? i.e. What form the does the model generally take?

A

An ordinary differential equation (ODE)

B

A closed-form solution to an initial-value problem

C

A plot of a specific output

D

The system response

Q2.3

Describe the modeling process for a mechanical system by putting the following steps in chronological order.

A

Define all underlying assumptions

B

Apply an appropriate form of Newton's law

C

Produce a free-body-diagram (FBD)

D

Obtain a model (ODE) for the system