Temperature control of water-bath system in presence of constraints by using MPC

© 2016 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).


Introduction
*Temperature is one of the key variables to be controlled in industrial processes. Temperature regulation of fluids to transport heat for particular use has leaded the industrial scenario for both power generation and process heating. Thermal fluids have numerous applications in industries of molding Processes, lamination, furniture, automotive manufacturing etc. Water-Bath temperature control is one of the major control process and has numerous applications in both food process industries and medical centers i-e Nestle, F&N, Yeoh Hiop Seng, General district hospital Kifisia Greece etc. (Om et al., 2012;Lochaitis et al., 1992). Aqua therapy is the internal or external use of water in any of its forms (water, ice, steam) for health improvement or treatment of numerous diseases with specific temperatures, pressure, duration and it is one of the ways of treatment mostly used in domain of natural medicine (Moventhan and Nivethitha, 2014). It is the naturopathic treatment used on a huge scale in earlier cultures of China, India and Egypt for maintaining health, preventing and curing different diseases (Fleming and Gutknechet, 2010). Effect of warm water aqua therapy directly depends on the accuracy with which temperature of Water-Bath is controlled. Traditionally Proportional Integral Derivative (PID) control technique is used for temperature control systems. PID controller is one of the most commonly used controllers in industrial processes for its simplicity of usage it accounts for more than 90% of the installed automatic control systems. Although majority of control processes operate at constraints, but PID does not has the ability to handle input and output constraints effectively. Moreover, gains of PID have to be tuned again and again for added disturbance or change in system dynamics (Holkar and Waghmere, 2010). Solution to these issues is Model Predictive Control (MPC). MPC is an optimization based technique and its performance depends on selection of two important parameters prediction horizon (Np) and control horizon (Nc). It optimizes the control sequence by minimizing the cost function at every sampling instant of time to reduce the difference between desired value and actual value of the system (Sohaib et al., 2016). In this research work simulations of Water-Bath temperature controller using MPC in three cases without constraints, with input constraints and with output constraints have analyzed by first modeling the water tank then implementing MPC and finally simulated in MATLAB. In section 2 Implementation of MPC is done. Section 3 and 4 presents simulation results and conclusion respectively

Implementation of MPC
MPC is a model based control technique and it uses process model to optimize the control vector without violating input or output constraints by minimizing the cost function. The basic structure of MPC is shown in Fig. 1 (Darandale et al., 2013). The implementation of MPC is based on system modeling and optimizer. Optimizer is a mathematical function which optimizes the control signal by minimizing the cost function in presence of constraints. Outputs or past inputs in terms of temperature are presented at system model to get predicted output. The differences between the predicted output and reference value in form of errors are sent to optimizer which optimizes the future input and this procedure will remain continuous.

System modeling
The performance of heating process depends on the induction, maintenance and control. There are several heating methods used now a days in domestic and industrial processes i.e. direct heating, indirect heating, electric arc heating etc. In this research work direct resistance heating method is used in which immersion rod is dipped in the water tank. The heating capacity of immersion rod depends on its wattage power and the required wattage power for the heating rod is calculated by the Eq. 1: (1) where, is the heater wattage, is the weight of water, is the specific heat of water, T is the change in temperature, and T is the time required to raise temperature at desired level . In this case, is 33 lb, is 1 Btu/lb/F, T is 140 F, and T is 1 hours. By substituting the values in Eq. 1 the required calculated wattage is given below: = 88 * 1 * 140 3412 * 1 = 1.8 As required power of immersion rod is 1.8KW so 2KW heating rod is selected. This immersion rod is used to regulate the temperature for Water-Bath system by using MPC control technique. Performance of MPC depends on how well the dynamics of system have been captured. The system model of water tank is shown in Fig. 2.

Fig. 2: System model
Where is the input power, C 1 is the energy in the tank, ρ is the density of water, V is the volume, T1 is the water temperature in the tank, Tin is the inlet temperature, U is the thermal conductivity, and W is the mass flow rate of water.
The parameter values of liquid tank are shown in Table 1. Specific Heat Capacity C = 4200J/Kg-K 5 Thermal Conductivity The thermodynamic equation for water tank is Eq. 2: by taking Laplace of Eq. 2 it becomes: assuming that = solving for T1(s) Eq. 4 becomes: τ is the time delay between an excitation of heating element and the response of temperature sensor.
applying Laplace transformation to Eq. 6 it becomes: by putting 1 ( ) in Eq. 7, the transfer function becomes: function H(s) becomes: By putting the values of parameters from Table 1 to Eq. 9, transfer function becomes: The Eqs. 11 and 12 represent the State Space Model (SSM) of the liquid tank that will be used in Fig. 1 to implement MPC temperature control of Water-Bath System.

Optimization
MPC optimizes the control vector ∆ by minimizing the cost function with predetermined constraints as shown in Eq. 13: where, the first term in Eq. 13 minimizes the difference between the set-point signal and the predicted output while, second term reflects the importance given to change in control signal ∆U. R ̅ is a diagonal matrix and R ̅ = r w I N c ×N c . r w ≥ 0 is a user defined tuning parameter for desired closed loop performance. When rw = 0 is used in cost function no importance is given to change in control signal and for large value of rw results in careful attention to change in control signal. The optimization of control vector is shown in Eq. 14 (Liuping, 2009).
where, F and are the vectors shown below: There are three major types of constraints two of them are applied at the control signal and change in control signal respectively while, the third one is applied at the output of the system. These constraints are of the form given below (Liuping, 2009).
Traditionally, the constraints are imposed on the whole control vector and all constraints are expressed in terms of vector∆ as shown below: rearranging the Eq. 16 in a compact matrix form by relating C1 and C2 to appropriate matrices and subject to constraints for control signal as shown in equation below: −( 1 ( − 1) + 2 ∆ ) ≤ − ( 1 ( − 1) + 2 ∆ ) ≤ where, and are column vectors with Nc elements of u min and u max similarly for change in control signal constraints are: After minimizing the cost function for optimization of control vector, ∆ is subjected to the inequality constraints by following equation: transforming the above Eq. in more compact form as shown in equation below and use it to find an optimized control without violating the constraints.

∆ ≤
where, M is a vector of constraints with its rows equal to number of constraints and columns equal to dimension of ∆ (Liuping, 2009).

Simulations based analysis
The simulation results of Water-Bath temperature controller are analyzed in three cases ie without constraints, with input constraints and with output constraints. Finally, MPC is further evaluated under dynamic environment by varying inlet flow rates.

MPC implementation flow chart
Simulations of MPC are developed by first acquiring the state-space model (SSM) of water tank then include the design parameters Nc, Np, reference point and constraints. Finally, the cost function subject to constraints is minimized to optimize control vector which is further applied to SSM to get predicted output and this procedure continues until the desired set point is achieved. MPC implementation hierarchy is shown in Fig. 3.

Simulation results
The simulation results of water-Bath temperature control for different cases are presented below:

Water-bath temperature control without constraints
The water-Bath temperature control is developed to be used in medical centers for the purpose of aqua therapy at different temperatures. Aqua therapy at 28°C not only provides treatment for bacterial coldwater diseases but also gives immunity against causative agent Flavobacterium psychrophilum (Sughara and Equichi, 2012). Head-out body immersion in Water-Bath at 38°C for 30 min results in decrease in blood viscosity which improves the blood pressure (Digeisi et al., 1986). Water immersion at 48 °C up to shoulder level results in increased oxygen (O2) consumption level. Aqua therapy at 60 °C for 15 min improves cardiac function (Choukron and Varene, 1990). The simulation results of MPC with set-points of 28°C, 38°C, 48°C and 60°C at different times are shown in Fig. 4. In the Fig. 4, MPC effectively regulates the temperature at different set-points for use in aqua therapy.

Water-bath temperature control with input constraints
The immersion heating rod is normally operated with 10% to 35% of safety factor. The 2kW heating rod used in Water-Bath temperature controller is allowed to operate maximum at 1.7kW with 15% safety factor, so the constraint on control signal is U max = 1.7kW with the desired set-point is 130 °C. The simulations have been presented without input constraints and with input constraints in Figs. 5 and 6 to compare the effect of constrained control signal. 2 minutes and the control signal first gets to the peak value of 1.9kW and finally gets off. After input constraint is applied on the amplitude of control signal temperature still reaches the desired value in 15.36 minutes so from Figs. 5 and 6 it is evident that MPC successfully regulates the temperature of Water-bath system without violating the input constraints but with only little time delay.

Water-bath temperature control with output constraints
In aqua therapy different temperature for Waterbath is required to cure different diseases but Water-Bath at temperature above 80°C becomes dangerous and causes skin burns immediately so temperature of water-Bath system should not exceed 80°C. The output constraint is imposed at upper limit of Y max = 80°C with set-point of 120°C. The simulation results are shown in Fig. 7. Fig. 7 elaborates that output constraint is not violated.

Water-bath temperature control in dynamic environment
Robustness of MPC is evaluated in dynamic environment by varying the inlet flow rate to the water tank. Initially inlet flow rate is fixed at 20kg/min for all above cases. Here firstly, MPC regulates the desired temperature with flow rate of 20kg/min then flow rate is varied at random to 30kg/min and then 40kg/min so the results have shown that MPC efficiently regulated the desired temperature but with little increase in time to reach the steady state at the desired temperature. The simulation results are shown in Fig. 8 From Fig. 8 it is clear that with inlet flow rates of 20kg/min, 30kg/min and 40kg/min controlled variable reaches the desired value of 60 °C in 15.87 min, 16.03 min and 16.2 min. MPC performs better in dynamic environment by regulating the desired temperature of Water-Bath system with different flow rates as an added disturbance randomly.

Conclusion
In this research work MPC based Water-Bath temperature controller is developed for purpose of aqua therapy in medical centers for treatment of different diseases. As PID is a simple structure control technique but it cannot effectively deal with input and output constraints therefore, simulation results of MPC have been analyzed in three modes temperature control without constraints, with input constraints and with output constraints. In all three modes MPC effectively regulates the temperature without violating the constraints. Finally, MPC effectively regulates the desired temperature of Water-Bath system in dynamic environment of variable inlet flow rate and proves to be a robust control technique. In future prospect MPC can be real time implemented on embedded device for Water-Bath temperature control system to test its real performance and moreover, research can be made to make this system more robust and energy efficient.