**3. Benchmark**

## *3.1. Benchmark Description*

The popular three-tank benchmark problem is used to exemplify the usefulness of marks in the context of uncertainty, vagueness, and indiscernibility [20,21]. It consists of three cylindrical tanks of liquid connected by pipes of circular section, as depicted in Figure 1. The first tank has an incoming flow, which can be controlled using a pump (actuator) and the outflow is located in the last tank.

**Figure 1.** Schematic representation of the three-tank system.

The model in form of difference equations for this systems is

$$\begin{aligned} h\_1(t+1) &= h\_1(t) + \Delta t \cdot (q\_1(t) - c\_{12} \cdot \operatorname{sch}\_{12} \cdot \operatorname{cc}\_1) / s\_1 \\ h\_2(t+1) &= h\_2(t) + \Delta t \cdot (q\_2(t) + c\_{12} \cdot \operatorname{sch}\_{12} \cdot \operatorname{cc}\_1 - c\_{23} \cdot \operatorname{sch}\_{23} \cdot \operatorname{cc}\_2) / s\_1 \\ h\_3(t+1) &= h\_3(t) + \Delta t \cdot (q\_3(t) + c\_{23} \cdot \operatorname{sch}\_{23} \cdot \operatorname{cc}\_2 - c\_{30} \cdot \operatorname{sch}\_{30} \cdot \operatorname{cc}\_3) / s\_3 \\ &\text{with} \\ s \text{r} h\_{12} &= \operatorname{sign}(h\_1(t) - h\_2(t)) \cdot \sqrt{|h\_1(t) - h\_2(t)|} \\ s \text{r} h\_{23} &= \operatorname{sign}(h\_2(t) - h\_3(t)) \cdot \sqrt{|h\_2(t) - h\_3(t)|} \\ s \text{r} h\_{30} &= \operatorname{sign}(h\_3(t) - 0) \cdot \sqrt{|h\_3(t) - 0|} \end{aligned} \tag{17}$$

where the state variables *h*1, *h*2, *h*3 are the level of liquid in the tanks, *s*1, *s*2, *s*3 their respective areas, *q*1, *q*2, *q*3 the incoming flows, *c*12, *c*23, *c*30 the valves constants which represent the flux between the tanks and Δ*t* is the simulation step, in seconds.

The following values were considered: the three tanks are the same heights *h* = 2 m, areas *s*1 = *s*2 = *s*3 = 1 m2, and intermittent inputs of the maximum incoming flows (in m3/s) are the following:

$$q\_1 = 0.01 \quad , \ q\_2 = q\_3 = 0. \tag{18}$$

For the valves' constants (in m5/2/s) the values are

$$c\_{12} = 0.009 \quad , \ c\_{23} = 0.008 \quad , \ c\_{30} = 0.007 \tag{19}$$

and the initial liquid levels (in m) are

$$h\_1(0) = 0.1 \quad h\_2(0) = 1.5 \quad h\_3(0) = 0.6 \,\text{.}\tag{20}$$

As an example of the application of marks for this benchmark model, we present three general problems related to many mathematical models: simulation, fault detection, and control to show the suitability of the marks for dealing with mathematical models with uncertainty and indiscernibility.
