- Background
- Linear State Space Models
- Introduction to Laplace Transforms
- Transfer Functions
- First-Order Behavior
- Integrating System
- Second-Order Behavior
- Lead-Lag Behavior
- Poles and Zeros
- Processes with Dead Time
- Padé Approximation for Dead Time
- Converting State Space Models to Transfer Functions
- Matlab and Simulink
- Summary
- References
- Student Exercises

## 3.6 Integrating System

We found in the previous chapter that material balances on liquid surge vessels or gas drums often yielded models with the following form:

In the Laplace domain, this is

Consider an integrating process initially at steady state, with *y*(0) = 0.

#### Step Response

If a step input change of D*u* is made at *t* = 0,

and we find the time-domain value

That is, the output ramps with a constant slope of *k*D*u*.

#### Impulse Response

If an impulse input of magnitude *P* is made at *t* = 0,

then the output immediately changes to a new steady-state value of

#### Example 3.5: Tank-Height Problem

The mathematical model for a liquid surge tank is (see Example 1.3)

where *h* is the liquid height, *A* is the constant cross-sectional area of the tank, *F*_{1} is the inlet flow rate, and *F*_{2} is the outlet flow rate. Assume that the outlet flow rate remains constant at a steady-state value of *F*_{2}* _{s}*. Defining the output and input in deviation variable form as

For a constant cross-sectional area of 10 m^{2}, the model is

#### Step Response

For a step input change of 0.25 m^{3}/min, the output response is

which is shown in Figure 3-7. If the steady-state height is 2 meters, then the height as a function of time is

**Figure 3-7. Step response of a liquid surge tank. Deviation variables.**

#### Impulse Response

For an impulse input of 1 m^{3}, the output response is

which makes sense, because the cross-sectional area is 10 m^{3}.