Hybrid PI controller constructed with paraconsistent annotated logic

Control Engineering Practice 84 (2019) 112–124

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Hybrid PI controller constructed with paraconsistent annotated logic Marcelo Saraiva Coelho a,b , João Inácio da Silva Filho a ,∗, Hyghor Miranda Côrtes a , Arnaldo de Carvalho Jr. a , Maurício Fontoura Blos a , Mauricio Conceição Mario a , Alexandre Rocco a a b

Laboratory of Applied Paraconsistent Logic, Santa Cecilia University - UNISANTA, Oswaldo Cruz Street, 288, Santos city, SP, 11045-000, Brazil SENAI College of Santos, Brás Cuba Street, 344, Santos city, SP, 11013-162, Brazil

ARTICLE Keywords: Level control mesh Process control Paraconsistent logic Hybrid PI control Algorithm

INFO

ABSTRACT In this work a new type of hybrid PI controller is presented using the Paraconsistent Logic (PL) as the basis for the mathematical and logical treatment of the signals corresponding to the control variables. This control has the conventional actions PI (Proportional and Integral) together with the logical actions structured in the concepts of Paraconsistent Logic. Compared to conventional techniques, this new type of Hybrid PI Controller with Paraconsistent Logic responded well to the various tests performed in the control loop. The tests and the technical validations were done with evidences that prove the efficiency of the paraconsistent concepts.

1. Introduction The evolution of the technologies used in the systems of instrumentation and automatic control of industrial processes brought the controller as main equipment. Therefore, the controller is the main equipment responsible for keeping the variables of a process in set values, through real-time application with control actions usually of Proportional, Integral and/or Derivative type. A controller can be adjusted with these three actions being applied separately or applied together. Controllers with the joint action of the type Proportional–Integral (PI) and Proportional–Integral–Derivative (PID) have found a wide range of applications in industrial controls (Astrom & Haglund, 1988; Marlin, 2000; Ogata, 2014). With technological development, controllers received electronic and computational resources around the 1980s, and became more and more efficient. The inclusion of chips (microcontroller integrated circuits) in the controllers provided new possibilities for industrial process controls, allowing the incorporation of algorithms with P, PI and PID actions, and the development of systems with auto-tuning, as well as an enormous ease in control parameter settings (Astrom & Haglund, 1988; Ogata, 2014; Zhao, Chai, & Wang H. Fu, 2014). At the end of the 1990s, with the advent of industrial networks, the so-called ‘‘Fieldbus’’ systems came with strong characteristics such as the large reduction of electrical cables and the possibility of systems with real-time diagnostics, thus allowing greater ease of maintenance and a enormous agility to reconfigure the system in a simple way. Even with these new technologies, the algorithm with the joint actions of PI and PID control, with some small variations in its modeling, remained in the most modern control systems (Chai, Ding, & Wu, 2011; Kuphaldt, 2011; Zhao et al., 2014). In parallel to the technological development of control equipment,

industrial processes always seek higher production with the highest quality index of the manufactured product and with low energy wastage in production. In order to respond well to this demand, process control systems must act efficiently and reliable so that the plant responsible for production has its processes under control at all stages of production. To achieve this goal, all controller actions are directed so that the process control assets remain reliable and efficient. This required condition must be maintained through controllers and efficient systems, which must be active from the adequate capture of the data originated from sensors, through the correct treatment of these data by means of special equations until the final presentation with reliable results that will act on the element end of the mesh (Chai et al., 2011; Murrill, 2000). The main objective of this work is to present the results of the application of a new technique using the Paraconsistent Logic in the level control of the pressurized vessels, by implementing a real pilot plant with confirmation of a good response by comparing it with the controllers based on the conventional methods Ziegler–Nichols and IMC (Internal Model Control). 1.1. Proportional and integral control (PI) In industrial process control meshes, the PI control formed by Proportional and Integral joint actions is the most used, mainly in the control of level, flow, pressure and other variables that do not present very large delays. The great use of this type of control is due to the advantage of not presenting the offset error associated with the pure proportional control, besides increasing the speed of response in relation to the integral action alone (Marlin, 2000; Murrill, 2000). Nevertheless, when it comes to problems in industrial plants, the conventional PI

∗ Corresponding author. E-mail address: [email protected] (J.I. da Silva Filho).

https://doi.org/10.1016/j.conengprac.2018.11.007 Received 23 April 2018; Received in revised form 13 October 2018; Accepted 15 November 2018 Available online xxxx 0967-0661/© 2018 Elsevier Ltd. All rights reserved.

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Control Engineering Practice 84 (2019) 112–124

controller still has some problems inherent in their own actions. When faced with a larger load demand, the PI controller responds with high initial value and with long and excessive response time. Besides that, the transient response of the conventional PI controller presents high overshoot and long stabilization time for high variations in the process, which in general has a detrimental impact on the control plant (Chai et al., 2011; Murrill, 2000). The joint action PI presents these problems precisely due to the presence of the integral action, because the stability of the control mesh decreases and the control result is impaired. In these same conditions, there will also be the risk of saturation by the integral mode, in which case the controller continues to integrate the error even without resulting in effective correction in the process (Kuphaldt, 2011; Marlin, 2000; Murrill, 2000).

the level control test using a conventional controller. Also in Section 3 the configurations of the PANs and the constructed modules that make up the Paraconsistent Hybrid PI controller with the description of their operation in the control process are presented. In Section 4 the results of the Paraconsistent Hybrid PI controller applied to the control mesh are presented that show its efficiency, as well as the comparative discussions between the results obtained through this new controller in relation to the conventional controller. Section 5 concludes with the considerations about the Paraconsistent Hybrid PI controller based on results of Section 4. 2. Paraconsistent logic The classical or Aristotelian logic, which supports our current technology, was based on rigid binary laws and, therefore, does not admit situations of redundancy, inconsistencies or those that are expressed by incompleteness (Da Costa, 1974; Da Costa & de Ronde, 2013). In classical logic there is no contradiction, that is, something cannot be both true and not true at the same time, when dealing with the same context. However, with the limits of standards of control and quality currently required, increasingly refined, make the principles of Classical Logic impossible to provide efficient control models so optimized for this demand (Da Costa, 1974). In order to act in situations where binary logic is impossible to be applied, other types of logic, called non-classical, such as, multivalued, Fuzzy, paraconsistent, etc., have recently been created (Da Costa, 1974; Da Costa & de Ronde, 2013). Non-classical logics have as their main objective to oppose the binary principles of classical logic and thus capture different aspects of informal arguments. For example, in classical logic the principle of noncontradiction (PNC) states that contradictory statements cannot both be true in the same sense at the same time. However, in the case of signals with respect to a physical quantity originating from two sensors, they may be contradictory, which provides inconsistent information for decision making. Thus, control systems will be more efficient if they are able to act in situations where information may be inconsistent. Paraconsistent (PL) logic is a type of non-classical logic that has the property of accepting the contradiction in its foundations and is therefore capable of dealing with contradictory signals. The PL has as its extension the Paraconsistent Annotated Logic (PAL), which has a Hasse Lattice (Lattice FOUR) associated with logical states represented at its vertices. In this way, sentences can be obtained where propositions can be analyzed on the basis of evidences (Da Costa, 1974; Da Costa & de Ronde, 2013; Da Silva Filho, Lambert-Torres, & Abe, 2010). In this PAL representation, the four extreme logic states represented at the vertices of the PAL lattice are: True (t ), False (f ) Paracomplete (⊥) and Inconsistent (⊺)) (Baptista, Da Silva Filho, & Morilla, 2013; Da Silva Filho et al., 2010).

1.2. Hybrid controllers Hybrid controllers are the result of the investigation of new forms of actions that can offer better means for tuning meshes in industrial processes with more efficient control (Miki et al., 1991; Murrill, 2000; Venayagamoorthy et al., 2003). In the search for control systems that better process information from sensors, based on the constant demand of industrial processes for more efficient controllers, some controllers have begun to introduce non-classical logic into their algorithms (Lee, Kim, & Park, 1998; Venayagamoorthy & others, 2003). The new forms of logical–mathematical treatment of variables using different types of non-classical logics led to the creation of controllers that fit other different techniques of PID actions (Astrom & Haglund, 1988). From the end of the 1990s onwards, controllers arose bringing this option to the control of processes, basically a block containing Fuzzy Logic, being this type of non-classic logic most used in this period. Several works have been published with hybrid PI controllers using Fuzzy logic, however, the complexity of the different process control meshes currently in the industry allows us to consider investigations of new techniques based on other types of non-classical logics (Lee et al., 1998; Renwal & Kumar, 2015; Venayagamoorthy & others, 2003). 1.3. Construction of a paraconsistent hybrid PI control Despite the evolution in the study of alternative applications of nonclassical logics in controllers of industrial processes, it can be observed that their actual use in industrial systems is still limited to a small number of cases when compared to the use of conventional control systems. In some works as in Brock (2014), Chai et al. (2011), Kuphaldt (2011), Lee et al. (1998), Miki and others (1991), Mudi, ChanchalDey, and Lee (2008), Murrill (2000), Renwal and Kumar (2015), Venayagamoorthy and others (2003), and especially those that deal with the concepts of level control in pressurized vessels, where there is no need for derivative anticipatory action, PI control is considered ideal. The hybrid PI control constructed with Paraconsistent Logic consists of an algorithmic computational structure that unites the data processing techniques of Paraconsistent Logic (Ayob, Salam, & Azli, 2010; Bousserhanel, Hazzabl, Rahli, Kamli, & Mazari, 2006) to the mathematical processes of PI actions (Chai et al., 2011; Zhao et al., 2014). In this work are presented details of the construction of the Hybrid Paraconsistent PI Controller and the results of its application to a pressurized vessel level control mesh. In addition to this introduction, the text is organized as follows: Section 2 presents the details of Paraconsistent Logic (PL) and its main properties. Section 2 still presents the PL extension called of Paraconsistent Annotated Logic with Annotation of Two Values (PAL2v) and its algorithm called of Paraconsistent Analysis Node (PAN) that will compose the module of the Paraconsistent Hybrid PI controller. In Section 3 (Materials and Methods) it is presented the pressurized vessel level control mesh, its tuning parameters and the results obtained in

2.1. Paraconsistent annotated logic with annotation of two values From a Lattice of four vertices it is possible to associate a type of Paraconsistent Annotated Logic in which two evidential values related to a particular proposition are considered. By interpreting the evidential values by means of annotation or normalized degrees of evidence we can obtain equations involving the logical states that are represented with degrees of evidence from measurements. This type of logic is called Paraconsistent Annotated Logic with annotation of two values (PAL2v) (Da Costa & de Ronde, 2013; Da Silva Filho et al., 2010). Fig. 1(a) shows the Lattice FOUR associated at PAL2v. In PAL2v an evidential atomic formula of the form P(𝜇, 𝜆) can be seen as an annotation for the proposition p, where 𝜇, 𝜆 ∈ [0,1] (real unit interval) (Da Costa & de Ronde, 2013; Da Silva Filho et al., 2010). Thus, the Evidence degree (𝜇) is a value that represents the evidence favorable to the proposition p, and the Evidence degree (𝜆) is a value that represents the evidence unfavorable to the proposition p. The association of a pair (𝜇, 𝜆) with a proposition p means that the 113

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Control Engineering Practice 84 (2019) 112–124

Fig. 1. (a) Lattice associated at Paraconsistent Annotated Logic with annotation of two values PAL2v. (b) PAL2v - Hasse finite lattice with annotations.

degree of favorable evidence at p is 𝜇, and the degree of unfavorable evidence at p is 𝜆. According to the annotations in the associated lattice shown in Fig. 1(b) (Da Silva Filho et al., 2010): (1,1) → indicates the existence of both, favorable and unfavorable evidence totals, assigning a logical connotation of inconsistency to the proposition p. (1,0) → indicates the existence of total favorable evidence and unfavorable evidence equal to zero, signaling a connotation of logical truth for the proposition p. (0,0) → indicates the existence of favorable and unfavorable evidence, but both equals zero, assigning a logical connotation of indetermination (paracomplete) for the proposition p. (0,1) → indicates the existence of favorable evidence equal to zero and total unfavorable evidence, signaling a connotation of logical falsehood for the proposition p. The equations of the PAL2v are obtained through a transformation where, initially, are considered to be represented in a Unit Square in the Cartesian Plane (USCP) the degree of favorable Evidence 𝜇 in the 𝑦-axis, and the degree of Unfavorable Evidence 𝜆 in the 𝑥-axis, according to Fig. 2 (Da Silva Filho, 2011; Da Silva Filho et al., 2010). A transformation that allows the degrees of evidence of normalized values represented in the x, y axes of the (USCP) to be located on the X and Y axes of a four vertex PAL2v Lattice, given by: 𝑇 (𝑋, 𝑌 ) = (𝑥 − 𝑦, 𝑥 + 𝑦 − 1)

The Normalized Degree of Contradiction–𝜇ctr , where a variation between 0 and 1 is obtained, can be calculated through equation (Da Silva Filho, 2011; Da Silva Filho et al., 2010): 𝜇𝑐𝑡𝑟 =

(5)

The Real Certainty degree (𝐷𝐶𝑅 ) is obtained by determining the distance d in the PAL2v lattice (Fig. 2) √ 𝑑 = (1 − |𝐷𝑐|)2 + 𝐷𝑐𝑡2 (6) The (𝐷𝐶𝑅 ) values are calculated according to the conditions shown below (Da Silva Filho, 2011; Da Silva Filho et al., 2010): √ (7) If 𝐷𝑐 > 0 𝐷𝐶𝑅 = 1 − (1 − |𝐷𝑐|)2 + 𝐷𝑐𝑡2

If 𝐷𝑐 < 0

𝐷𝐶𝑅 =

√ (1 − |𝐷𝑐|)2 + 𝐷𝑐𝑡2 − 1

(8)

The logical states and the location of the axes and values found in the equations can be seen in Fig. 2. In order to have a reference on how much the two signals in the input 𝜇 and 𝜆 are bringing information of the proximity to the true logical state, one can use the value of the Resultant Real Evidence degree (𝜇𝐸𝑅 ), that is obtained by normalized value of the 𝐷𝐶𝑅 (Da Silva Filho, 2011; Da Silva Filho et al., 2010). Therefore, calculated from Eqs. (7) and (8) and equated by:

(1)

Relating the components of the transformation T(X, Y ) according to the usual nomenclature of PAL2v, where (Bousserhanel et al., 2006; Da Silva Filho, 2011): 𝑥 = 𝜇→ Favorable Evidence degree, with 0 ≤ 𝜇 ≤ 1 and 𝑦 = 𝜆→ degree of Unfavorable Evidence, with 0 ≤ 𝜆 ≤ 1, where: (a) the first term obtained in the ordered pair of the transformation equation is: 𝑋 = 𝑥 − 𝑦 = 𝜇 − 𝜆→ which will be called the certainty degree-Dc (Da Silva Filho, 2011; Da Silva Filho et al., 2010). Therefore, the certainty degree is obtained by: 𝐷𝑐 = 𝜇 − 𝜆

𝜇+𝜆 2

𝜇𝐸𝑅 =

𝐷𝐶𝑅 + 1 2

(9)

The equations obtained by the interpretations made in the lattice associated to PAL2v allow the creation of algorithms used in analysis and logical treatment of information signals. The algorithm Paraconsistent Analysis Node (PAN) can be used in several fields of knowledge where incomplete and contradictory information is treated appropriately through the PAL2v equations. In this work, the PAN will be used to build a network of signal analysis that represents the control variables in an industrial mesh (Da Silva Filho, 2011; Da Silva Filho et al., 2010).

(2)

(b) the second term obtained in the ordered pair of the transformation equation is: 𝑌 = 𝑥 + 𝑦 − 1 = 𝜇 + 𝜆 − 1→ which will be called the Contradiction Degree-Dct (Da Silva Filho, 2011; Da Silva Filho et al., 2010). Therefore, the degree of contradiction is obtained by:

2.2. Algorithm Paraconsistent Analysis Node (PAN) 𝐷𝑐𝑡 = 𝜇 + 𝜆 − 1

(3) In its complete version, the Paraconsistent Analysis Node algorithm (PAN) receives two information signals represented by degrees of evidence, equates its values according to the previous equations and presents as a result a single value of the resulting degree of evidence (Da Silva Filho et al., 2010). This output, called the Resultant Real Evidence degree, is a value that expresses a unique representation of the paraconsistent analysis of the two values of input, with nullification of the effect of the contradiction. The description of a generic PAN

The equations of PAL2v allow the paraconsistent logical states to be found within the lattice (Bousserhanel et al., 2006; Da Silva Filho et al., 2010), these are points of interpolation given by: 𝜀𝜏 = (𝐷𝑐, 𝐷𝑐𝑡)

(4)

One can therefore make decisions based on the proximity of the logical state 𝜀𝜏 to the extreme logic states True (t ) or False (f ), located at the vertices of the PAL2v lattice. 114

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Fig. 2. Unit Square in the Cartesian Plane (USCP) and Lattice FOUR associated at PAL2v with values.

algorithm is given below (Baptista et al., 2013; Da Silva Filho, 2011; Da Silva Filho et al., 2010; Ziegler & Nichols:, 1942): Define: E1 = Input 1; E2 = Input 2 S1 = Output 1; S2 = Output 2; S3 = Output 3 1. Present two input values: E𝟏 = 𝜇1 */ favorable Evidence degree , where, 𝜇 1 ∈[0, 1]⊂ℜ E𝟐 = 1 − 𝜇2 = 𝜆 */ unfavorable Evidence degree, where, 𝜆 ∈ [0, 1]⊂ℜ 2. Calculate the Certainty Degree: 𝐷𝑐 = 𝜇1 − 𝜆 3. Calculate the Contradiction Degree: 𝐷𝑐𝑡 = 𝜇1 + 𝜆 − 1 4. Calculate the Normalized Degree of Contradiction: 𝜇𝑐𝑡𝑟 =

Fig. 3. Symbol of a typical paraconsistent analysis node PAN.

𝜇1 +𝜆 2

network (PANnet). Two PANs will be networked together to compose the hybrid PI controller module. The first PAN will have a complete structure, so its output will be the value of the resultant real evidence ( ) degree 𝜇𝐸𝑅 according to Eq. (9), and the other PAN will have the partial result value and will present in the output the value of the Contradiction degree (𝐷𝑐𝑡) according to Eq. (3), and the value of the ( ) Normalized Degree of Contradiction 𝜇𝑐𝑡𝑟 according to Eq. (5). The initial treatment given to the signals of the variables by the PANnet is complemented by the conventional actions, Proportional and Integral. In this way the hybrid PI control with paraconsistent logic is performed. The validation tests of the Paraconsistent Hybrid PI controller were made in a pilot plant of pressurized vessels, as described below.

5. Calculate the distance d (projection on the axis (horizontal) √ of the degrees of certainty on the PAL2v-lattice): 𝑑 = (1 − |𝐷𝑐|)2 + 𝐷𝑐𝑡2 6. Determine the output signals: If d > 1, then do: 𝑆1 = 0.5; 𝑆2 = 𝜇𝑐𝑡𝑟 , 𝑆3 = 𝐷𝑐𝑡 Consider Undefinition and go to end. Otherwise go to the next item. 7. Determine the Real Certainty Degree: If 𝐷𝑐 > 0 then calculate 𝐷𝐶𝑅 = 1 − 𝑑 If 𝐷𝑐 < 0 then calculate 𝐷𝐶𝑅 = 𝑑 − 1 𝐷 +1 8. Calculate the resultant real evidence degree: 𝜇𝐸𝑅 = 𝐶𝑅2 9. Present the results in the outputs: Do 𝑆1 = 𝜇𝐸𝑅 ; 𝑆2 = 𝜇𝑐𝑡𝑟 , 𝑆3 = 𝐷𝑐𝑡 10. End

3.1. Industrial process pilot plant The pilot plant used in the tests and studies of the Paraconsistent Hybrid PI controller was designed for researches involving various instrumentation and process control technologies for a water transfer system. In this plant two of the main variables of the industrial instrumentation can be controlled: level and flow. Fig. 4 shows the pilot plant used. The diagram of the pilot plant used in the validation tests of the Paraconsistent Hybrid PI controller and Conventional controller is detailed in Fig. 5. In this work both controllers were used to control the level of the T-102 vessel.

The paraconsistent uncertainty treatment system defined as PAN can be used in several fields of knowledge. With its application, incomplete and contradictory information will receive adequate treatment through the equations of PAL2v (Da Costa, 1974; Da Silva Filho et al., 2010). As its output has normalized value, its application can be forming different configurations of networks of analysis and logical treatments of signals according to the purpose that it is proposed. The symbol of the generic PAN algorithm (Baptista et al., 2013; Ziegler & Nichols:, 1942) is shown in Fig. 3. The procedures analysis of PAN enables for accessions to the resulting stream of values through their internal equations, thus a PAN can be used in full, or only the values of partial results, that may be representative for the purpose of its application (Baptista et al., 2013; Da Silva Filho et al., 2010; Ziegler & Nichols:, 1942).

Level control of the T-102 pressurized vessel by the conventional controller To control the level in the T-102 vessel, the flow of the liquid inlet of the T-102 vessel is manipulated through the mesh formed by: 1. LIT-102 level transmitter; 2. LIC-103 level controller and; 3. two control valves FV-101 and FV-102. In this control strategy, the output signal from the controller will be applied simultaneously to two valves, the FV-101 and the FV-102.

3. Materials and methods In this work the Paraconsistent Hybrid PI controller was constructed using the PAL2v algorithms interconnected in a Paraconsistent Analysis 115

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Fig. 6. S-shaped process variable response.

3.2. Determining the characteristics of the mesh and tuning of the pi controller

Fig. 4. Overview of the Industrial Process Pilot Plant used in the validation tests of the Paraconsistent Hybrid PI controller.

The control system requires adjustment and tuning, which consists in determining the parameters to be used in the PID controller related to the process and the control mesh in which it will be applied. These parameters will define the optimization of the process through the equations of the control actions P, PI or PID used in the controller that will act in mesh (Antelo, Exler, Banga, & Alonso, 2008; Lopes, Miller, Murrill, & Smith, 1967). The tests to obtain the plant parameters were developed following the procedures described below.

The FV-101 valve operates inversely proportional to the FV-102 valve, so that by reducing the opening of the FV-102 valve, the FV101 opening is increased so as not to overload the operation of the P101 pump with a possible pressure increase in its discharge. The valve FV-101 of P-101 pump recirculation operates in its full range when the signal coming from the controller varies between 0 and 100%, that is, when the controller output signal is 0%, the FV-101 is fully open and when the controller output signal is 100%, the FV-101 is fully closed. The FV-102 valve operates within its full range when the signal from the controller varies between 0 and 100%, that is, when the controller output signal is 0%, the FV-102 is fully closed, and when the signal controller output is 100%, the FV-102 is fully open. Therefore, when the controller output is at its lowest value, we will have the lowest liquid flow in the T-102 inlet and when its output is at the maximum value, will occur the maximum liquid flow in the T-102 inlet. The LIC-103 controller, called in this work of conventional PI controller, must be configured to operate in reverse action, in such a way that operating in automatic mode, increasing the PV value in relation to SP, the action decrease the output signal, decreasing the liquid flow rate in the T-102 vessel causing the return of the level to the value equal to SP. As this level control in a pressurized vessel, the PI actions are sufficient. Thus, all the tests were performed considering only the adjustments of 𝐾𝑝 proportional action) and 𝑇𝑖 (integral action).

Controller tuning by the Ziegler–Nichols method The tuning methods defined by Ziegler and Nichols (Desborough & Miller, 2002; Economou & Morari, 1986; Lopes et al., 1967; Morari & Zafiriou, 1989) are widely used and are developed based on the determination of some dynamic characteristics of the process. For the open mesh, the Ziegler & Nichols method has as principle to obtain the characteristics of the process with the analysis of the response from a variation by step. Initially with the controller in manual, a step variation is stimulated in its output and the response curve is observed. If the response curve presents an ‘‘S’’ as a result, according to Fig. 6, two constants are defined, the delay ‘‘L’’ and the time constant ‘‘T’’ (Antelo et al., 2008; Desborough & Miller, 2002; Economou & Morari, 1986; Morari & Zafiriou, 1989; Ogata, 2014; Skogestad, 2003; Skogestad S. Postlethwaite, 1996). Exactly at the inflection point of the curve, a tangent line must be drawn and at the intersection of that tangent line with the time axis and

Fig. 5. Diagram of the pilot plant used in the validation tests.

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Table 1 ISA-PID parameters according to Ziegler & Nichols for open-loop testing (Lopes et al., 1967). Controller

𝐾𝑝

𝑇𝑖

𝑇𝑑

P PI PID

T/L 0.9 × T/L 1.2 × T/L

– L/0.3 2×L

– – 0.5 × L

Table 2 PID parameters according to IMC method. Process First order 𝐻(𝑠) =

𝐾𝑝 𝐺𝑝 𝑥𝑒−𝜃𝑠 1+𝜏𝑠

Second order 𝐻(𝑠) = Integrator 𝐻(𝑠) =

the line relating to the stabilization of the process variable, the delay and the time constant are defined. From these values and using Table 1, the ideal parameters for the ISA – Instrument of Society of America – PID are defined (Economou & Morari, 1986; Lopes et al., 1967). The test to survey the characteristic of the level control process was carried out with the plant operating in open loop for this method, with the following conditions:

𝐺𝑝 𝑠

𝑇𝑖

𝐺𝑝 𝑥𝑒−𝜃𝑠

(1+𝜏1 𝑠)𝑥(1+𝜏2 𝑠) 𝑥𝑒−𝜃𝑠

𝑇𝑑 {

𝜏 𝐺𝑝 𝑥(𝜆𝐼𝑀𝐶 +𝜃) 𝜏1 𝐺𝑝 𝑥(𝜆𝐼𝑀𝐶 +𝜃) 1 𝐺𝑝 𝑥(𝜆𝐼𝑀𝐶 +𝜃)

𝑚𝑖𝑛 𝜏, 4𝑥(𝜆𝐼𝑀𝐶 { 𝑚𝑖𝑛 𝜏, 4𝑥(𝜆𝐼𝑀𝐶 4𝑥(𝜆𝐼𝑀𝐶 + 𝜃)

} + 𝜃) } + 𝜃)

– 𝜏2 –

Tuning the controller by IMC (Internal Model Control) The method of the Internal Model Control (Economou & Morari, 1986; Morari & Zafiriou, 1989) was proposed for the tuning of PID controllers in a process with first order dynamics with dead time. In Skogestad S. Postlethwaite (1996), a modification of the initially developed IMC method was proposed, presenting a new set of rules to tune the controllers according to Table 2 (Skogestad, 2003). These parameters were used in this work. Using the IMC tuning method developed in Economou and Morari (1986) and presented in Table 2, and adopting as parameter 𝜆IMC = 20 s, the calculated values of the proportional gain (𝐾𝑝 ) and integral time 93 𝑇𝑖 for the PI controller were: 𝐾𝑝 = 𝐺𝑝×(𝜆 𝜏 +𝜃) = 2.142×(20+5) → 𝐾𝑝 = 1.73

• Pressure on top of vessel T-102 in 100 mbar; • Output of the LIC-103 controller by 50%; The fully opened HV-105 valve, thus allowing maximum liquid transfer between the T-102 vessel and the T-101 vessel. Thus, after a few minutes, the level in the T-102 vessel stabilized at 30.40%, as presented in Fig. 7, which illustrates the response of the monitoring and supervision system screen of the control system studied. Exactly at 13 h18 min00 s a +15% step was manually applied at the output of the LIC-103 controller, corresponding to the blue line, on curve A of Fig. 7, which from 50% changed to 65% at that time. Considering that the step response curve behaved as a first order process with dead time, it was initially determined that the dead time of this process was 𝜃 = 5 s. After 6 min, it was noticed that the level in the T-102 stabilized in 62.53%, therefore with 𝛥𝑃 𝑉 = 32.13%. For a first-order process, the value of PV corresponding to the time constant was calculated, which corresponded to 63.2% of 𝛥PV. Therefore:

𝐼𝑀𝐶

𝑇𝑖 = min{𝜏, 4 × (𝜆𝐼𝑀𝐶 + 𝜃)} = min{93.4 × (20 + 5)} = min{93, 100} → 𝑇𝑖 = 93 s = 1.55 min The results achieved with this PI controller configuration applied to the level control mesh of the pressurized vessels are shown in Fig. 9, where adopting 𝜆IMC = 9 s by the IMC method, we have: proportional gain (𝐾𝑝 = 3.10), integral time (𝑇i = 0.93 min), time for PV equal to 35.30% (𝑡PV = 35.30% = 14 s), rise time (𝑡r = 35 s), settling time (ts = 90 s), error with margin of 5% (error = ∓5%), and maximum overshoot (MP = 1.01%). In Fig. 9 it is seen that the Process Variable (PV) had a rise time of 108 s (tr = 108 s) with an overshoot of approximately 0.1%. The time in which the Process Variable (PV) reaches approximately 50% of the path to reach the Reference (SP) was 27 s. It can also be observed that in the variation of the Reference (SP) from 30.5 to 40.5%, the Manipulated Variable did not have a great variation and, after, it remained close to the Process Variable (PV). Considering the error of 5% in relation to reference (SP), a settling time (ts) of 144 s was obtained.

𝛥𝑃 𝑉0.63.2% = 32.13 × 0.632 = 20.31% In analysis to the graph of response by step entry, it was determined the instant that the PV = 50.71% corresponded exactly to 3 h19 min38 s Therefore, the time constant of this first-order process was equal to the difference of this instant in relation to the end of dead time. 𝜏 = 3 h19 min38 s − 3𝐻18 min0.5 s = 1 min33 s = 93 s By means of the graph of Fig. 7, the gain of the process ‘‘Gp’’ was calculated as the relationship between the variation of the level and the variation of the step: 𝐺𝑝 = 𝛥𝑃 𝑉 ∕𝛥𝑀𝑉 = 32.13%∕15% = 2.142. Based on the tuning method developed by Ziegler & Nichols (Da Costa & de Ronde, 2013), previously presented in Table 1, the calculated values for the ISA type PI controller were:

3.3. Hybrid PI Paraconsistent controller applied to level control in the pilot plant To verify the validation of the control technique using PAL2v, the conventional PI controller, which acts in the plant control process, has been replaced by a Paraconsistent Hybrid PI controller. For the insertion of the Hybrid Paraconsistent Control in the conventional plant that was shown in Fig. 5 and had its detailed description in Section 3.1, some adaptations were made. The LIT-102 (Transmitter and level indicator) was replaced by two components LT1 and LT2 (Level Transmitters) as shown in Fig. 10, both connected directly to the Controller. Valves FV-101 and FV-102 of Fig. 5 were set to a working configuration with Paraconsistent Hybrid PI controller and, in Fig. 10, they were given the designations FV-001A and FV-001B. Therefore, as shown in Fig. 10, two LT1 and LT2 transmitters are used in level monitoring and the control resulting from the analysis will act on the valves FV-001A and FV-001B.

𝐾𝑝 = 0.9 × 𝜏∕(𝐺𝑝 × 𝜃) = 0.9 × 93∕(2.142 × 5) → 𝐾𝑝 = 7.81 𝑇𝑖 = 𝜃∕0.3 = 5∕0.3 = 16.67 s → 𝑇𝑖 = 0.56 min The parameters calculated by this method were implemented in the LIC-103 level controller and the graphical results are shown in Fig. 8. After stabilizing the level in the T-102 vessel, with the output of the controller manually positioned at 50%, it was observed that the level was around 30%, specifically with 30.22%, at the time the LIC102 controller was changed to automatic mode and at exactly 17:14:00 PM, SP varied in step from 30.0% to 40.0%. In Fig. 8 it is verified that the Process Variable (PV) presented a rise time of 17 s (tr = 17 s) with an overshoot of approximately 1.65%. It can also be observed that in the variation of the Reference (SP) from 30 to 40% the Manipulated Variable (MV) had an abrupt variation remaining for a time in the maximum limit so that the Process Variable (PV) responded with the rise time of 17 s. Considering the error of 5% in relation to reference (SP) a settling time (ts) of 48 s was obtained.

Hybrid controller architecture The architecture of the Paraconsistent Hybrid PI Controller was obtained after a preliminary analysis of how the configurations of the PANs and their respective representations of the physical variables obtained in the pilot control mesh could be used. These procedures are set out below. 117

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Control Engineering Practice 84 (2019) 112–124

Fig. 7. Level in vessel T-201 after step of + 10% in LV-201. (A) Step, (B) Process variable, (C) Additional signal without measurement, (1) Dead time 𝜃 of 5 s, (2) 𝛥𝑃 𝑉0.63.2% = 20.31%, (3) 𝛥𝑡0.63.2% = 93 s, (4) 𝛥𝑃 𝑉 = 32.13%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Level in T-201 after step in SP from 30% to 40% with LIC-103 in automatic mode with PI adjustments by Ziegler & Nichols. (A) Manipulated Variable, (B) Process Variable, (C) Step, (1) 𝑇 𝑟 = 17 s (2) ) 𝑒𝑟𝑟𝑜𝑟 = ∓5%, (3) 𝑀𝑃1.65% = 41.70%, (4) 𝑡𝑠 = 48 s.

Fig. 9. Level in T-201 after step in SP from 30.5% to 40.5% with LIC-103 in automatic mode with PI adjustments by IMC (𝜆IMC = 20 s). (A) Variable Manipulated, (B) Process Variable, (C) Step, (1) tPV=36.80% = 27 s, (2) tr = 108 s, (3) ts = 144 s (4) 𝑒𝑟𝑟𝑜𝑟 = ∓5%, (5) 𝑀𝑃 = 0.1%.

degree 𝜇𝐸𝑅 . The second PAN is a partial algorithm that receives in its input the Resultant Real Evidence degree from PAN1 and in its other input receives the third signal from an external source in the representative form 𝜆3 = 1 − 𝜇3 . This PAN2 has two outputs, the first being ( ) the normalized degree of Contradiction 𝜇𝑐𝑡𝑟 found through Eq. (4) and the second the Contradiction degree (𝐷𝑐𝑡) found through Eq. (3). In the control strategy, according to Fig. 11, two level transmitters are used in PAN network. For the paraconsistent analysis node PAN1, the LT1 transmitter was used as the first source of evidence and the LT2

3.4. Paraconsistent analysis network for logical processing of variable signals For the initial analysis of the signals of the variables in the Hybrid PI Paraconsistent controller applied to the level control mesh, a network configuration composed by two algorithms of the PAL2v (PANs) interconnected was used as shown in Fig. 11. The first PAN named PAN 1 is a complete algorithm that receives in its inputs the two representative values of 𝜇1 and 𝜆2 = 1 − 𝜇2 from external sources and presents in its output the Resultant Real Evidence 118

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Control Engineering Practice 84 (2019) 112–124

Fig. 10. Representation of the Paraconsistent Hybrid PI Controller installed in the general synoptic of the Industrial Process Pilot Plant.

If LT2 = 4 mA → 𝜇2 = 0→ Level for pressurized vessel 1 = 0% (0) If LT2 = 20 mA → 𝜇2 = 1→ Level for pressurized vessel 1 = 100% (1) If 4 mA ≤ LT2 ≤ 20 mA → 0 ≤ 𝜇2 ≤ 1 → 𝜇1 = (LT2-4)/16 and 𝜆3 = 1 − 𝜇2 The first paraconsistent analysis node will therefore receive the evidence from the two level transmitters to answer the following proposition: p: The level of Pressurized Vessel 1 is at its maximum value (P (𝜇1,𝜆1) ). Therefore, the transmitter LT1 will be the source of favorable evidence (𝜇1 ) and the LT2 transmitter the source of unfavorable evidence of the analysis 𝜆 = 1 − 𝜇2 , according to Fig. 11.

Fig. 11. Network of PANs to be used in the Paraconsistent Hybrid PI controller.

transmitter as the second source of evidence, its output corresponding ( ) to the representation of the Resultant Real Evidence degree 𝜇𝐸𝑅 . For the second PAN2, as it is part of the network of paraconsistent analysis (PANnet), it has as a source of Favorable evidence the output of the ( ) Resultant Real Evidence degree 𝜇𝐸𝑅 of the PAN 1 and as a source of Unfavorable Evidence the desired value of the level in the vessel (SP) ( ) producing as outputs the Normalized degree of Contradiction 𝜇𝑐𝑡𝑟 and the Contradiction degree (𝐷𝑐𝑡).

Initial trials with PAN 1 For verification in this first phase three tests were done in PAN 1, in which were considered: 1. Error of 0% for both transmitters; 2. LT1 with −5% error and LT-2 with 0% error; 3. LT1 with 0% error and LT2 with error of + 2.5%. The results of this test, represented by values of Resultant Real Evidence degree obtained at PAN 1 output, are shown in Tables 3–5.

3.5. Modeling of the paraconsistent signals for PAN Network

Modeling of paraconsistent signals for PAN 2 In the configuration proposed in this paper, the output signal of the Resultant Real Evidence degree (𝜇𝐸𝑅 ) from of the first paraconsistent analysis node PAN 1 will be applied to the degree of favorable Evidence of the second paraconsistent analysis node PAN 2, and still on the second node, the degree of Unfavorable Evidence will receipt a value corresponding to Set Point for the level control. By convention the Set Point values vary between +1VDC and +6VDC in the instrumentation, and to be used by PAL2v they were normalized by line equations of proportional form, such that:

The algorithms of PAL2v only work with degrees of evidence with values between 0 and 1, so for the application of the Paraconsistent Hybrid PI controller in the control loop, the PAN network must receive normalized signals. In this way the signals of the physical quantities are transformed into degrees of evidence through a modeling. This is presented below. Modeling of paraconsistent signals for PAN 1 In this work the degrees of evidence were generated through the following relations: For the signal coming from the LT1 ranging from 4 to 20 mA proportional of tank level variation of the 0 and 100%, and this applied to the first analog input, is obtained:

𝑆𝑒𝑡 Point − 1 . 5 For the PAN 2: 𝑆𝑃 =

• The contradiction degree (𝐷𝑐𝑡) will be at zero whenever PV = SP; • The contradiction degree (𝐷𝑐𝑡) will be negative whenever the error (SP-PV) is negative; • The contradiction degree (𝐷𝑐𝑡) will be positive whenever the error (SP-PV) is positive; ( ) • The normalized degree of contradiction 𝜇𝑐𝑡𝑟 will be 0.5, whenever PV = SP; ( ) • The normalized degree of contradiction 𝜇𝑐𝑡𝑟 will decrease, whenever the error (SP-PV) becomes more negative; ( ) • The normalized degree of contradiction 𝜇𝑐𝑡𝑟 will increase, whenever the error (SP-PV) becomes more positive.

∙ LT1 ∈ℜ | 0 < LT1 ≤ 1 and proportional at 4 mA ≤ LT1 ≤ 20 mA If LT1 = 4 mA → 𝜇1 = 0→ Level for pressurized vessel 1 = 0% (0) If LT1 = 20 mA → 𝜇1 = 1→ Level for pressurized vessel 1 = 100% (1) If 4 mA ≤ LT1 ≤ 20 mA → 0 ≤ 𝜇1 ≤ 1 → 𝜇1 = (LT1- 4)/16 and 𝜆2 = 1 − 𝜇2 For the signal from the LT-2 ranging from 4 to 20 mA proportional of tank level variation of the 0 and 100%, and this applied to the second analog input, is obtained: ∙ LT2 ∈ℜ | 0 < LT2 ≤ 1 and proportional at 4 mA ≤ LT2 ≤ 20 mA 119

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Table 3 Resultant real evidence degree 𝜇𝐸𝑅 for LT1 and LT2 with 0% error. LEVEL T-102 (%)

VALUE LT1 (%)

ERROR LT1 (%)

VALUE LT2 (%)

ERROR LT2 (%)

𝜇1

𝜇2

𝜆2

𝐷𝑐

𝐷𝑐𝑡

d

𝐷𝐶𝑅

𝜇𝐸𝑅

0,0 20,0 40,0 50,0 60,0 80,0 100,0

0.0 20.0 40.0 50.0 60.0 80.0 100.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 20.0 40.0 50.0 60.0 80.0 100.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.000 0.200 0.400 0.500 0.600 0.800 1.000

0.000 0.200 0.400 0.500 0.600 0.800 1.000

1.000 0.800 0.600 0.500 0.400 0.200 0.000

−1.000 −0.600 −0.200 0.000 0.200 0.600 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.400 0.800 1.000 0.800 0.400 0.000

−1.000 −0.600 −0.200 0.000 0.200 0.600 1.000

0.000 0.200 0.400 0.500 0.600 0.800 1.000

Table 4 Resultant real evidence degree 𝜇𝐸𝑅 for LT1 with 5% and LT2 without error. LEVEL T-102 (%)

VALUE LT1 (%)

ERROR LT1 (%)

VALUE LT2 (%)

ERROR LT2 (%)

𝜇1

𝜇2

𝜆2

𝐷𝑐

𝐷𝑐𝑡

d

𝐷𝐶𝑅

𝜇𝐸𝑅

0.0 20.0 40.0 50.0 60.0 80.0 100.0

−5.0 15.0 35.0 45.0 55.0 75.0 95.0

−5.0 −5.0 −5.0 −5.0 −5.0 −5.0 −5.0

0.0 20.0 40.0 50.0 60.0 80.0 100.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.000 0.150 0.350 0.450 0.550 0.750 0.950

0.000 0.200 0.400 0.500 0.600 0.800 1.000

1.000 0.800 0.600 0.500 0.400 0.200 0.000

−1.000 −0.650 −0.250 −0.050 0.150 0.550 0.950

0.000 −0.050 −0.050 −0.050 −0.050 −0.050 −0.050

0.000 0.350 0.750 0.950 0.850 0.450 0.070

−1.000 −0.640 −0.240 −0.040 0.140 0.540 0.920

0.000 0.170 0.370 0.470 0.570 0.770 0.96

Table 5 Resultant real evidence degree 𝜇𝐸𝑅 for LT1 without error and LT2 with + 2.5% error. LEVEL T-102 (%)

VALUE LT1 (%)

ERROR LT1 (%)

VALUE LT2 (%)

ERROR LT2 (%)

𝜇1

𝜇2

𝜆2

𝐷𝑐

𝐷𝑐𝑡

d

𝐷𝐶𝑅

𝜇𝐸𝑅

0.0 20.0 40.0 50.0 60.0 80.0 100.0

0.0 20.0 40.0 50.0 60.0 80.0 100.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

2.5 22.5 42.5 52.5 62.5 82.5 102.5

2.5 2.5 2.5 2.5 2.5 2.5 2.5

0.000 0.150 0.350 0.450 0.550 0.750 0.950

0.025 0.225 0.425 0.525 0.625 0.825 1.000

0.975 0.775 0.575 0.475 0.375 0.175 0.000

−0.975 −0.575 −0.175 0.025 0.225 0.625 1.000

−0.025 −0.025 −0.025 −0.025 −0.025 −0.025 −0.025

0.035 0.426 0.825 0.975 0.775 0.376 0.000

−0.965 −0.574 −0.175 0.025 0.225 0.624 1.000

0.018 0.213 0.413 0.512 0.612 0.812 1.000

Table 7 Normalized degree of contradiction considering PV fixed at 50%.

Table 6 Normalized degree of contradiction considering SP fixed at 50%. 𝜇𝐸𝑅 (PV)

𝜇3 (SP)

𝜆3

𝐷𝑐

𝐷𝑐𝑡

𝑑

𝐷𝑐𝑟

𝜇𝐸𝑅

𝜇𝑐𝑡𝑟

𝜇𝐸𝑅 (PV)

𝜇3 (SP)

𝜆3

𝐷𝑐

𝐷𝑐𝑡

𝑑

𝐷𝑐𝑟

𝜇𝐸𝑅

𝜇𝑐𝑡𝑟

0.00 0.20 0.40 0.50 0.60 0.80 1.00

0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.50 0.50 0.50 0.50 0.50 0.50 0.50

−0.50 −0.30 −0.10 0.00 0.10 0.30 0.50

−0.50 −0.30 −0.10 0.00 0.10 0.30 0.50

0.70 0.76 0.90 1.00 0.90 0.76 0.70

−0.29 −0.23 −0.09 0.00 0.09 0.23 0.29

0.35 0.38 0.45 0.50 0.54 0.61 0.64

0.25 0.35 0.45 0.50 0.55 0.65 0.75

0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.00 0.20 0.40 0.50 0.60 0.80 1.00

1.00 0.80 0.60 0.50 0.40 0.20 0.00

−0.50 −0.30 −0.10 0.00 0.10 0.30 0.50

0.50 0.30 0.10 0.00 −0.10 −0.30 −0.50

0.70 0.76 0.90 1.00 0.90 0.76 0.70

−0.29 −0.23 −0.09 0.00 0.09 0.23 0.29

0.35 0.38 0.45 0.50 0.54 0.61 0.64

0.75 0.65 0.55 0.50 0.45 0.35 0.25

Table 8 Normalized degree of contradiction considering SP fixed at 40%.

Such considerations were obtained by the analysis of the behaviors of the degrees of Certainty and Normalized Contraction 𝐷𝑐𝑡 and 𝜇𝑐𝑡𝑟 through the variations of Favorable and Unfavorable evidences (𝜇1 and 𝜇2 ). Initial trials with PAN 2 Preliminary tests were performed in PAN 2 and the results of four conditions are shown in Tables 6–9 where the behavior of its two outputs 𝐷𝑐𝑡 and 𝜇𝑐𝑡𝑟 can be observed with the variations of the output of PAN 1 and Set Point. 3.6. Paraconsistent Hybrid PI Controller configuration

𝜇𝐸𝑅 (PV)

𝜇3 (SP)

𝜆3

𝐷𝑐

𝐷𝑐𝑡

𝑑

𝐷𝑐𝑟

𝜇𝐸𝑅

𝜇𝑐𝑡𝑟

0.00 0.10 0.30 0.40 0.50 0.70 0.90 1.00

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

−0.60 −0.50 −0.30 −0.20 −0.10 0.10 0.30 0.40

−0.40 −0.30 −0.10 0.00 0.10 0.30 0.50 0.60

0.56 0.58 0.70 0.80 0.90 0.94 0.86 0.84

−0.43 −0.41 −0.29 −0.20 −0.09 0.05 0.14 0.15

0.28 0.29 0.35 0.40 0.45 0.52 0.56 0.57

0.30 0.35 0.45 0.50 0.55 0.65 0.75 0.80

reference. The result of this analysis in PAN 2 (𝜇𝑐𝑡𝑟 and 𝐷𝑐𝑡) are applied directly to inputs P (proportional) and I (integral) of the equation of the conventional controller. The analysis of the values obtained in the tests of the network constructed with the PANs and the modeling of the physical variables allow the configuration of the Paraconsistent Hybrid PI Controller. As can be seen in Tables 6 to 9, whenever the values of PV and SP are ( ) equal, the output of the normalized degree of contradiction 𝜇𝑐𝑡𝑟 will

The Paraconsistent Hybrid PI Controller (Hybrid-PI) works by treating the contradiction. Firstly, the treatment of contradiction is made through PAN 1 between the measurements values of LT1 and LT2, generating the 𝜇𝐸𝑅 values, according to Fig. 2 and its algorithm. Then, it obtains the values referring to the contradiction (divergence) between the result obtained by the paraconsistent analysis in PAN 1 and SP 120

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Table 9 Normalized degree of contradiction considering PV fixed at 40%.

4.1. Controller test with 𝐾𝑝 = 3 and 𝐾𝑖 = 1 rpm

𝜇𝐸𝑅 (PV)

𝜇3 (SP)

𝜆3

𝐷𝑐

𝐷𝑐𝑡

𝑑

𝐷𝑐𝑟

𝜇𝐸𝑅

𝜇𝑐𝑡𝑟

0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40

0.00 0.10 0.30 0.40 0.50 0.70 0.90 1.00

1.00 0.90 0.70 0.60 0.50 0.30 0.10 0.00

−0.60 −0.50 −0.30 −0.20 −0.10 0.10 0.30 0.40

0.40 0.30 −0.10 0.00 −0.10 −0.30 −0.50 −0.60

0.56 0.58 0.70 0.80 0.90 0.94 0.86 0.84

−0.43 −0.41 −0.29 −0.20 −0.09 0.05 0.14 0.15

0.28 0.29 0.35 0.40 0.45 0.52 0.56 0.57

0.70 0.65 0.55 0.50 0.45 0.35 0.25 0.20

This first essay considered: • • • •

Initial value of controller output in manual mode = 50%; Hybrid PI Controller with 𝐾𝑝 = 3.0; Hybrid PI Controller with 𝐾𝑖 = 1.0 rpm; Value of the level variable when stabilized (PV) = 31.00%, before the application of the step; • Value of the Set Point variable (SP) = 30.5%, before the application of the step; • Value of the Set Point variable (SP) = 40.5%, after application of the step;

be 0.5, that is, in the middle of the range, therefore, this value is the initial optimum output value of the control in which PV = SP for any SP condition. Thus, a good configuration could be made with the PI controller having the corresponding Normalized degree of contradiction ) ( 𝜇𝑐𝑡𝑟 of the PAN-2 as input of the proportional function P, and the Contradiction degree (𝐷𝑐𝑡) as input of the integral function I. As shown in Tables 6 to 9, with the input of the integral function of the PI controller connected to the output corresponding to the Contradiction degree (𝐷𝑐𝑡), this will be with the value 0.0 whenever the values of PV and SP are equal. As Contradiction degree (𝐷𝑐𝑡) varies between −1.0 and +1.0, it is verified that in the condition of positive difference between PV and SP, the value of the Contradiction degree (𝐷𝑐𝑡), will also be positive and in the condition of negative difference, the value of the Contradiction degree (𝐷𝑐𝑡) will also be negative. Therefore, with the degree of Contradiction (𝐷𝑐𝑡) being the input of the integral function, it can be stated that in the absence of error (PV-SP), the integral function will not act. And, in the positive or negative error conditions, Will have the sense of correction according to the difference, as long as the error exists. Fig. 12 shows the diagram of the Paraconsistent Hybrid PI controller installed in the control circuit with the proportional and integral actions. In the outputs of the PAN 2 the Normalized degree of Contradiction ( ) 𝜇𝑐𝑡𝑟 is the signal of the variable for the proportional action P and the Contradiction degree (𝐷𝑐𝑡) is the signal of the variable for the integral action I, which together will act on the PI controller. When the controller is in MANUAL mode, the output of the PI block will be receiving signal from the supervisory system on a scale of 0 to 100% and internally will be scaled to the working range of the paraconsistent logic, being transcript from 0.0 to 1.0.

In this first test, for 𝐾𝑝 and 𝐾𝑖 parameters, the same test values of the conventional controller with IMC tuning with 𝜆IMC = 9 s were used. Fig. 13 shows the graphical response of this first test, where it is possible to verify the existence of a divergence between the two sources of evidence: the level transmitter LT1 and LT2. 4.2. Controller test with 𝐾𝑝 = 1.8 and 𝐾𝑖 = 0.67 rpm This second essay considered: • • • •

Initial value of controller output in manual mode = 50%; Hybrid PI Controller with 𝐾𝑝 = 1.8; Hybrid PI Controller with 𝐾𝑖 = 0.67 rpm; Value of the level variable when stabilized (PV) = 31.00%, before the application of the step; • Value of the SET POINT variable (SP) = 30.5%, before the application of the step; • Value of the SET POINT variable (SP) = 40.5%, after application of the step; In this second test, for 𝐾𝑝 and 𝐾𝑖 parameters, the same test values of the conventional controller with IMC tuning with 𝜆𝐼𝑀𝐶 = 20 s were used. Fig. 14 shows the graphical response of this second test. 4.3. Comparisons of the results To verify the efficiency of the Paraconsistent Hybrid PI Controller compared to conventional, a comparison was made with some characteristics that demonstrate the best performance of the level control mesh. In Table 10 are the results of accommodation time (ts), maximum Overshoot MP and rise time (tr), for each previously performed assay. The first row of the table, defined as ‘‘OPEN LOOP’’, shows the results of the test performed to define the dynamic response characteristic of the open loop process, with a 15% step in the valve opening. The second line, defined as ‘‘PI (Z & N)’’, shows the results of the test performed for a +10% step in SP, and the PI adjustments with 𝐾𝑝 = 7.81 and 𝑇𝑖 = 0.56 min, these values calculated by the Ziegler & Nichols method. The third line, defined as ‘‘PI (IMC) 𝜆IMC = 20’’, presents the results of the test performed for a +10% step in SP, with PI adjustments with the values calculated by the IMC method, considering the factor (𝜆IMC ) in 20 s. The fourth line defined as ‘‘PI (IMC) 𝜆IMC = 9’’, presents the results of the test performed for a step of +10% in the SP, with PI adjustments with the values calculated by the IMC method, considering the factor (𝜆IMC ) in 9 s. The last two rows of Table 10, defined as ‘‘HYBRID PI (1st Test)’’ and ‘‘HYPHID PI (2nd Test)’’, show the results of the Paraconsistent Hybrid PI controller with emphasis on the accommodation time (ts), maximum Overshot (MP) and rise time (tr), considering the adjustments of 𝐾𝑝 and 𝐾𝑖 similar to those used in the conventional controller with 𝜆IMC = 9 and 𝜆IMC = 20 respectively. It is worth stressing that to consider the similarity between integral time and integral gain, the relation 𝐾𝑖 = 1/𝑇𝑖 was used.

4. Results and discussion The operation of the Paraconsistent Hybrid PI controller was performed using as a test condition the following sequence: • Control mode selection of the Hybrid PI controller for the ‘‘MANUAL’’ condition; • Adjustment of the output value randomly by the supervisory system. • Wait of the stabilization of the process, from the analysis of PV variation; • Adjustment by supervisory system the value of the SP to the same value of the PV variable after its stabilization; • Adjustments of the 𝐾𝑝 and 𝐾𝑖 values of the Hybrid PI controller; • Switching the control mode of the Hybrid PI controller from ‘‘MANUAL’’ to ‘‘AUTOMATIC’’; • Application of the step in the value of SP; • Analysis of the graphical response of the PV and MV values with performance of the Hybrid PI controller. The simulations performed to test the controller response in the industrial process pilot plant presented the following results: 121

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Control Engineering Practice 84 (2019) 112–124

Fig. 12. Final configuration of the Paraconsistent Hybrid PI controller.

Fig. 13. Response of Paraconsistent Hybrid PI controller after step in SP from 30.50% to 40.50% with adjustments of 𝐾𝑝 = 3.0 and 𝐾𝑖 = 1.0 rpm. (A) Manipulated Variable, (B) Process Variable (𝜇𝐸𝑅 of PAN 1), (C) Step, (D) LT1, (E) LT2, (1) tr = 37 s (2) ts = 78 s 3) 𝑒𝑟𝑟𝑜𝑟 = ∓5%, (4) MP0.59% .

Fig. 14. Response of Paraconsistent Hybrid PI controller after step in SP from 30.50% to 40.50% with adjustments of 𝐾𝑝 = 1.8 and 𝐾𝑖 = 0.67 rpm. (A) Manipulated Variable, (B) Process Variable (𝜇𝐸𝑅 of PAN1), (C) Step, (D) LT1, (E) LT2, (1) ts = 61 s (2) tr = 62 s (3) 𝑒𝑟𝑟𝑜𝑟 = ∓5%, (4) 𝑀𝑃 0.18% .

The identification ‘‘Ref.’’ is characterized by the reference selected as the best value obtained among all the tests. It is verified that the results of Table 10 show that: (a) For the first test, the Paraconsistent Hybrid PI Controller (HybridPI) showed a rise time tr with better performance than the PI (IMC) 𝜆IMC = 20, and with PI (IMC) 𝜆IMC = 9 but showed a worse result than PI (Z & N), with a value of 61 s. In relation to overshoot, the Paraconsistent Hybrid PI Controller (Hybrid-PI) showed better performance than PI (Z & N) and PI (IMC) 𝜆IMC = 9 and it was worse than PI (IMC) 𝜆IMC = 20, with value of MP = 0.18%. In relation to settling time ts Paraconsistent Hybrid PI Controller (Hybrid-PI), it showed better performance than PI

(IMC) 𝜆IMC = 20 and worse than PI (IMC) 𝜆IMC = 9 and PI (Z & N) with a value of 62 s. (b) For the second test, the Paraconsistent Hybrid PI Controller (Hybrid-PI) showed a rise time tr with better performance than PI (IMC) 𝜆IMC = 20 and PI (IMC) 𝜆IMC = 9 but showed a worse result than PI (Z & N), with a value of 78 s In relation to overshoot, the Paraconsistent Hybrid PI Controller (Hybrid-PI) showed better performance than PI (Z&N) and PI (IMC) 𝝀𝑰𝑴𝑪 = 9 and it was worse than PI (IMC) 𝝀𝑰𝑴𝑪 = 20, with MP = 0.59%. In relation to the settling time ts, the Paraconsistent Hybrid PI Controller (Hybrid-PI) showed better performance than the PI (IMC) 122

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Control Engineering Practice 84 (2019) 112–124

Table 10 Comparison of controller results.

– –

10% 7.81 0.56 – 10% 1.73 1.55 –

48 s Ref. 1.65% 1550% 17 s Ref. 108 s 125% 0.10% Ref. 144 s 747.1%

support logic. The results obtained with the functional structure in the form of a network with two nodes of paraconsistent analyzes are very important for the development of new research projects that may encompass other configurations, especially those where, due to the inherent responsibility of risks of various nature present the need for multiple variables for control.

–

10% 3.10 0.93 –

90 s

5.1. Future works

– –

TEST

𝛥MV 𝛥SP 𝐾𝑃

𝑇𝑖

𝐾𝑖

𝑡𝑠

OPEN LOOP PI (Z & N) PI (IMC) 𝜆𝐼𝑀𝐶 = 20 PI (IMC) 𝜆𝐼𝑀𝐶 = 9 Hybrid-PI (1st) Hybrid-PI (2nd)

15% –

–

–

360 s –

–

𝑡𝑠 %

𝑀𝑃

𝑀𝑃 % 𝑡𝑟

–

–

𝑡𝑟 %

–

87.5% 1.01% 910%

35 s

105.9%

10% 1.80 1.49 0.67 61 s

27.1% 0.18% 80%

62 s

264.7%

10% 3.00 1.00 1.00 78 s

62.5% 0.59% 490%

37 s

117.6%

In this work, it was demonstrated the possibility of applying this new technique using Paraconsistent Logic when dealing with level control in pressurized vessels. For this, the original pilot plant of the article was considered, where, in the case of a conventional controller, it would work with single input information from a single sensor. And in this case if you had any problem in measuring, it would impact directly on the output. With the application of the Paraconsistent Control, two sensors are used in PAN 1 and it was found that, without major divergences in the measurements of LT1 and LT2 in relation to the actual value of the Variable, a greater system security is provided. We know that when there were two or more sensors, instead of the result of the analysis performed by the PANs, the average of the values in the conventional controller would be used. However, because of the variance, the average may not optimally express the measurement related to the process. In future works, it will be done a better investigation to know about in which control situations the average values would be sufficient to act on the control, and also under what conditions the intensities of contradiction that can be measured by the paraconsistent controller in real time would present for the better operation in the control of the mesh. Regarding the control behavior comparisons, in this work, the references of the controllers responses of the Ziegler–Nichols and IMC (Internal Model Control) methods were used to demonstrate a satisfactory response of the algorithms that make up the Paraconsistent Hybrid Controller. We did not make a comparison with another type of hybrid controller; however, we strong believe that the two techniques used were sufficient to support the satisfactory answers for this new technique in the proposed scenario, which is the control level of the pressurized vessel. In future works, we intend to expand the research by comparing the Paraconsistent Controller PI hybrid with other hybrid controllers, such as those that use the Fuzzy Logic in its foundations.

𝝀𝑰𝑴𝑪 = 20, and worse than the PI (IMC) 𝝀𝑰𝑴𝑪 = 9 and PI (Z&N) with a value of 37 s. In this work it was used only the Proportional and Integral actions (PI controller) for the implementations, since these two actions are sufficient for most of the dynamics of the processes that involve level controls in pressurized vessels. The tests applied with the paraconsistent logical control in the mesh confirmed these considerations. In the analysis of the values shown in Table 10, it is seen that the Paraconsistent Hybrid PI Controller (Hybrid-PI) responds well in different control situations. The Hybrid Paraconsistent PI control may also present the possibility of using more input variables to control the mesh, making it more reliable, keeping the measurements of the inputs of PAN 1 (in this paper LT-1 and LT-2) on acceptable error values and, respectively, their contradiction. This is advantageous over conventional control, which generally uses a single input and output. In this way, the Hybrid PI Paraconsistent control can better fit into a real control mesh, leaving it with the good control response. 5. Conclusions In this paper we present a hybrid controller, whose results are very promising and therefore presents itself as a good control alternative in situations where a higher level of complexity is required for an optimized response. This new type of control, which has the conventional PI actions (Proportional and Integral) coupled with the logical treatment actions of the variables structured in the concepts of Paraconsistent Logic, allowed the use of more than one input signal as a process variable adding more reliability to the actual information, and thus inducing a more aligned response to reality. In this research the PAL2v algorithms received the signals from two sensors, made the treatment with paraconsistent logic in the data and acting on the final control element in hybrid mode together with the PI actions controlling the level plant with pressurized vessel. In this analysis the results of this configuration allow us to conclude that the Paraconsistent Hybrid PI controller behaved with a damped but non-slow response and with an excellent maximum overshoot, almost non-existent, thus showing excellent control efficiency. Although the PI algorithm is consolidated in industrial process control systems, with this work, it can be verified that it is possible, with the use of non-classical Logics, to present other ways to improve control efficiency. The Paraconsistent Logic used here is quite effective in dealing with the uncertainties and contradictions of the measurement systems of the variables to be controlled, especially in processes that add complexity to achieve equilibrium in their controls due to incomplete and/or contradictory information. The fundamentals of the Paraconsistent Logic that support the PAN algorithms were of fundamental importance in the elaboration of the controller, working together with the conventional techniques, forming a reliable hybrid control (working with input sensors with an acceptable error range). It is worth noting that this work has shown alternative routes, in the area of uncertainty treatment with non-classical logics, where the Paraconsistent Annotated Logic with its algorithms is used as the control

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