First Order Dynamics

How many poles does a first order system have?

1

First order lag has

1 real pole on the negative axis

A transfer function is defined by

Effect variable divided by cause variable

Generic transfer function for first order system

g(s)=\frac{\overline{x}(s)}{\overline{u}(s)}=\frac{K_{p}}{s\tau+1}

What is Kp?

Process gain

Indicates how much effect there is for a given cause

At steady state, Kp = ?

K_{p}=\frac{x^{SS}}{u^{SS}}

Units of Kp

Units of x divided by units of u

What is \tau ?

Time constant

What is u(t)?

Independent variable (cause)

What is x(t)?

Dependent variable (effect)

What are the cause and effect for a first order step response?

Step input (cause) \overline{\Delta F_{IN}}(s)^{}

Step response (effect) \overline{\Delta h}(s)

\overline{\Delta h}(s) for a first order step response inherits…

The pole of the step input

The pole of the transfer function

\overline{\Delta F^{IN}}(s)=

\overline{\Delta F^{IN}}(s)=\frac{\Delta F^{IN,SS}}{s}

\overline{\Delta h}(s)=

\overline{\Delta h}(s)=\Delta F^{in,SS}\cdot\frac{Kp}{s(s\tau+1)}

How do we determine poles?

Partial fraction expansion

The step response can then be expressed as

A sum of exponentials

How do we actually determine steady state behaviour?

Set t to infinity

What is the physical meaning of the time constant?

At one time constant, the first order step response reaches 63% of its final steady state value

What do we use to approximate steady state / system settling?

Four time constants

How is an impulse signal represented

Delta function

Example of a pulse signal

Inflow into a tank changes for a short time \Delta T

Then changes back, creates an impulse

Laplace of a unit impulse is equal to

Its area

What is the area of an impulse signal for first order impulse response

\Delta F^{IN,SS}\cdot\Delta T

Impulse response \overline{\Delta h}(s)=

\overline{\Delta h}(s)=\Delta F^{IN,SS}\Delta T\cdot\frac{Kp}{s\tau+1}

What is the physical meaning of the transfer function? (Impulse response)

x(s) = g(s) * u(s)

g(s) is the Laplace transform of the system’s impulse response

For first order response of a non linear system, what control implications are there?

Process gain and time constant depend on the steady state operating point

More difficult to control because dynamics differ at different operating points

Complex numbers can be represented as

z=\left\vert z\right\vert e^{j\theta} in polar coordinates

z=a+ib in Cartesian

What is |z|?

\left\vert z\right\vert=\sqrt{a^2+b^2}

What is \theta

arctan(b/a)

Also called the argument

Frequency response: input

Unit amplitude sine wave

Frequency response: output

Sine wave with amplitude ratio A(\omega) and phase angle \phi(\omega)

If there is a phase angle, then sin is…

Not in pure polar form

Due to 1/2j multiplier

Frequency response input/output can also be a

Cosine wave

Still uses amplitude ratio and phase angle

What is \omega

Angular frequency

Units of rads^{-1}

What is f

Frequency in Hz

What is \phi

Phase angle in radians

One cycle of oscillations is 2\pi radians, therefore

\omega=2\pi f

What is T_{\phi}

Time interval between peaks

Corresponds to the unknown phase angle

What is T_{p}

Time period of the oscillation

How do we find phase angle?

1. Measure T_{\phi}

2. Use T_{p} of the oscillation to determine phase angle

Relation between phase angle and T_{\phi}

\frac{T_{\phi}}{T_{p}}=\frac{\phi}{2\pi}

So equation for phase angle in terms of time periods

Effect of lag on peaks

The peaks in the effect signal are later than peaks in the cause signal