Chapter 3 System model
3.2 Proposed system
We can consider the water treatment process as the process that water transport from one tank to another tank, and proceed different physical or chemical reaction in each tank. The factors that we measure as the standard for water quality would vary according to different treatments in each tanks and the change of water volume. Namely, water quality is impacted by water processing action in those tanks before current tank where tank quality is measured as well. Hence, the two tank system is proposed to meet the dynamic of the water hydraulic properties and the situation of the interaction between any connected two tanks. The two-tank water system as Figure 3.2.
3.2.1 Two-tank system
Figure 3.2: Two-tank System
There are two tanks in the proposed model, tank 1 and tank 2. There is also one pipe to connect these two tanks. External water resource enter the system from the input pipe of tank 1, denoted as Qin. Qinis the only input node that import water from outside of the system. The value of Qinis the maximum input flow rate of the external resource. Cin q1refers to the inflow
water quality of a specific substance. We assume that the concentration of the substance, which is denoted as q1, will affect the water quality. There is only one outlet of the two tank system, which is at tank 2. The output pipe of the tank 2 is the only one place for water leaving the two tank system.
3.2.2 Proposed system with controller
To control the water level and quality issue of the two tank system, we modify the above two tank system with controller to command the control signal to the valves. With the controller, we expect the system will achieve the states that we desire. The two tank system with controller is shown as Figure 3.3.
Figure 3.3: general architecture of control systems
In Figure 3.3, there are numerous sensors to monitor system states. Sensors for monitoring water level in tanks are denoted as ShTi, where i = 1, 2 refers to tank 1 and tank 2. For example, ShT 1represent to the information about the water level of tank 1. Sensors for sensing the water quality could be represented as SqjTi, where i = 1, 2 refers to tank 1 and tank 2 and j = 1, 2...n
refers to the substances that we concern about. For instance, Sq1T1 represent to the information that sensor read about the concentration of solute q1 in tank 1. The water quality at the output node will be sensed by SqjT2b, where j = 1, 2...n refers to the different water quality factors. The sensor will read the concentration of qj at the output pipe of tank 2 and send the information back to the controller.
There are also lots of valves for the controller to control the system to modify the system states. Valves for adding chemicals could be represented as vqjT i, where i = 1, 2 refers to tank 1 and tank 2 and j = 1, 2...n refers to the substances that we concern about. For instance, vq1T 1 represents the valve to control the dosage of q1 injected to tank 1. Valve vin is for controlling the input flow rate of importing resource. The value of vindetermine the open ratio of the pipe;
the value to vin proportions to the inflow rate. The valve, v12, is to control the water flow from tank 1 to tank 2. Still, the value of v12proportions to the opened cross sectional area of the pipe between two tanks. And there is a valve, vd, to control the output flow from tank 2. The control value of vddepends on the consumer water demand. If the water consumption is heavy, then vd should be more open relatively, and vise versa.
For the proposed two tank system, we aim to maintain the water level and the water quality.
The water level should lower the hight of the tank and also higher than a minimum hight. The minimum value of hight is designed to meet the consumer water need, since we have to preserve certain amount of water in the tanks to handle the unpredictable changes of consumer demands.
3.2.3 Dynamic model
We apply the mathematical equations to characterize the features of the proposed model.
We can derive the dynamic model of the proposed system. The dynamic model is discussed separately as the followings, the water hydraulics and water quality .
• Water hydraulics
The dynamic equations could be obtained as follows,
hT 1˙ = uvinQin− uv12Ap√
2g(hT 1− hT 2)
AT 1 (3.6)
hT 2˙ = uv12Ap
√2g(hT 1− hT 2)− udAp
√2ghT 2
AT 2 (3.7)
hT 1and hT 2denote the water level of tank 1 and tank 2, respectively. Qin (L/sec) is the maximum inflow rate of water entering into tank 1. Ap is the cross sectional area of the pipe. AT 1and AT 2are the cross sectional area of the tank 1 and tank 2.
uvinis the control value suggesting the status of valve vinof the input pipe of tank 1. The value of uvin is 1 indicating the valve is completely opened and 0 meaning the valve is fully closed. The value of valve will affect the flow rate of the input pipe of tank 1. uv12is the control value suggesting the status of valve v12between tank 1 and tank 2. Similarly, value 1 for u12indicates open and zero means close. The value of valve will affect the flow rate from tank 1 to tank 2.
• Water quality
The dynamic model of water quality is described as follows, is the concentration of q1 of the inflow resource to the tank 1. Vp is the volume of water in the pipe between tank 1 and tank 2.
We attempt to add more q1 in water to maintain the water quality.uT 1q1 and uT 2q1 are two control variables for controller to control the injection of q1 to tank 1 and tank 2. uT 1q1 and uT 2q1 denote the value for the mass of q1 increasing to tank 1 and tank 2, respectively. uT 1q1 and uT 2q1 should be greater than zero, since the mass of additional q1 would always greater or equal to zero.
Chapter 4 Implementation
In this chapter, we discuss the method to design the controller for the proposed model in chapter 3. We will consider to compute the linear model of the water system first to design the controller. Then we will utilize MPC to design an appropriate controller and consider the constraints to the control signal into designing.