Modeling Systems

Systems are most easily modeled using systems of linear constant coefficient differential equations. They can be represented either as a set of state-space equations or as a transfer function in the Laplace domain.

Electrical and Mechanical Systems

Eletrical Systems

Mechanical Systems

Electro-Mechanical Equivalence

It turns out that electrical and mechanical systems are analogous to each other. In other words, given an electrical system, we can convert it into a mechanical system and vice versa. Capacitors act like springs as energy storage, resistors act like dampers which dissipate energy, and inductors act like inertial masses which resist movement. These are clear from their force/voltage differential equations (in the Laplace domain) in Table 1. Under these analogies, forces are like voltages, currents are like velocities, and charge is like position.

Linearization

Because non-linear systems often have dynamics which are complicated to analyze, a standard trick to make them simpler is to linearize them.

Definition 5

Using Definition 5, we can see that around our operating point, we have

State-Space Equations

Definition 6

System variables are variables which depend on either the input or the system's internal state.

Definition 7

We can easily go from State-Space Equations to a transfer function via the Unilateral Laplace transform. After taking the Laplace Transform of both sides of Equation 2, Equation 3,

Phase Variable Form

We can also derive state space equations from their transfer functions. First, we assume that the transfer function comes from the LCCDE

meaning our transfer function will be of the form

Using this intermediary variable, we can now let

Converting this back to the time-domain,

When we do control in State-Space Control, this makes it easier to place the system poles where we want them to be.

Time Domain Solution

Notice that

Combining these two equations, we see that

Notice that Equation 7 is broken into two pieces.

Definition 8

The zero-input response is how the system will behave when no input is supplied.

Definition 9

Controllability

Definition 10

By the Cayley-Hamilton Theorem (see Cayley-Hamilton),

Definition 11

The controllability matrix is

Theorem 1

Observability

Definition 12

Definition 13

The observability matrix is

A theorem analogous to Theorem 1 exists for observability.

Theorem 2

Time Delays

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