Lesson 3: First Order Open Loop Systems

A first-order system is one whose output (___) is modeled by a first-order differential equation

y(t)

What is f(t) in first order systems?

Input (forcing function)

What is 𝜏_p? Formula and Name

𝜏_p - time constant

𝜏_p = a1/a0

What is K_p? Formula and Name

K_p - Steady state gain, static gain, gain of the process

K_p = b/a0

What are the initial conditions for input and output for steady state?

y(0) = 0

f(0) = 0

G(s) is the

Transfer function

What is the general formula for the transfer function?

What is the transfer function for steady state conditions?

First order processes with steady state initial conditions are also called

• First order lag

• Linear lag

• Exponential transfer lag

If a_o = 0, what is the transfer function

What are the processes called if a_o = 0?

Purely capacitive or pure integrator

First-order processes are characterized by:

1. Their capacity to store material, energy, or momentum

2. The resistance associated with the flow of mass, energy, or momentum in reaching the capacity

Typical first order systems based on how they are characterized:

• Tanks that store liquids and gases

  • Resistance - pumps, valves, weirs, pipes

• SLG systems that can store thermal energy

  • Resistance - heat transfer

most common class of dynamic components in a chemical plant, with the capacity to store primarily mass and energy

First-order lags

What is the input at s for a pure capacitive process?

1/s

What happens to the output for a pure capacitive process?

As time approaches infinity, y(t) also approaches infinity

A __________________ will cause serious control problems, because it cannot balance itself.

pure capacitive process

A _______________________ is a system that cannot reach a new steady state on its own after a disturbance.

non-self-regulating system

We can adjust manually the speed of the constant-displacement pump, so as to balance the flow coming in and thus keep the level constant. But any small change in the flow rate of the inlet stream will make the tank flood or run dry (empty). This attribute is known as

non-self-regulation

A first-order lag process is ________________

self-regulating

Unlike a purely capacitive process, it reaches a new steady state.

First-order lag process

The smaller the value of the time constant τp, what is the effect on the system response?

The smaller the value of the time constant τp, the steeper/faster the initial response of the system.

The ____________ of a process is a measure of the time necessary for the process to adjust to a change in its input

time constant τ_p

The value of the response y(t) reaches _____ of its final value when the time elapsed is equal to one time constant, τ_p

63.2%

Value of the response y(t) when time elapsed different multiples of the time constant

1τ_p = 63.2%

2τ_p = 86.5

3τ_p = 95

4τ_p = 98

Thus, after ____ time constants, the response has essentially reached its ultimate value.

four

The ultimate value of the response (i.e., its value at the new steady state) is equal to ____ for a unit step change in the input,

K_p

The ultimate value of the response (i.e., its value at the new steady state) is equal to K_p for a unit step change in the input, or ____ for step size of A

AK_p

Relationship of static gain to the input and output

Δ(output) = K_pΔ(input)

A small change in the input if

K_p is large (for sensitve systems)

A large change in the input if

K_p is small

The larger the static gain of a process, the ______ the steady-state value of its output for the same input change

larger

Possible solutions on finding the dynamic response of first order systems with variable time constraints and static gains:

• Analytical solutions with variable coefficients

• Assume time constant and static gain are constant for a limited period of time

e can assume that such systems possess constant time constants and static gains for a certain limited period of time only. At the end of such a period we will change the values of t, and K, and consider that we have a _______________

new first-order system