Min-Max Control Over Transfer Characteristics in Models of Technical and Non-Technical Systems

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2015 17th UKSIM-AMSS International Conference on Modelling and Simulation Min-Max Control Over Transfer Characteristics in Models of Technical and Non-Technical Systems Darko Kovacevic, Mijo Zujic

1 th UKSIM-AMSS International Conference on Modelling and Simulation Min-Max Control Over Transfer Characteristics in Models of Technical and Non-Technical Systems Darko Kovacevic, Mijo Zujic Faculty of Maritime Studies Split University of Split Split, Croatia Asja Kovacevic Information Centre Clinical Hospital Centre Split Split, Croatia Abstract Several types of Min-Max Circuits including Singleton Changeable Min-Max Circuit, D-Operator Circuits and Changeable Mid Circuit have been introduced in this paper. Inherent functional changeability of Min, Max and Mid Circuits is related to outside (environmental) stimuli (information) what makes these circuits suitable for different control purposes in complex min-max based electronic circuits and in hardware (min-max) modeling of non-technical systems and processes. Controllable double ended clipper, full wave rectifier (absolute value circuit) and peak detector are the circuits that are used to explain the notion of min-max control. As a non-technical system, a newspaper distribution and sale process has been chosen as a modeling environment. Simple model of multi-agent coordination system in which an operator plays the main control role is modeled using Changeable Mid Circuit. The operator s decision making bias (un-bias) problem, expressed in the change of its evolving transfer characteristic is presented in a few words. Keywords-Min-Max circuits, min-max control, D-Operator circuits, Mid circuits, changeable circuits, min-max modeling, decision-making bias modeling. I. INTRODUCTION In our previous works Min and Max Circuits were discussed on an individual basis and in very simple combinations. When Min, Max and Compl(ement) circuits are connected together to produce a specified output for certain specified combinations of input variables, with no storage involved, the resulting Min-Max Circuit is in the category of combinational Min-Max Circuits. In Min-Max combinational circuits, the output is at all times dependent on the combination of input signals (voltage levels in the case when Min-Max circuitry supports Boolean operations). We will initiate this paper with the brief description of a Singleton Changeable Min-Max Circuit and its operation. Then, in a row, D-Operator Circuit and Changeable Mid Circuit are described along with some simple application; controllable double ended limiting circuits, voltage controlled full wave rectifier, a model of a multiagent environment etc. The last of the mentioned applications is related to the problem of decision making bias (de-bias) in a multiagent environment. The problem was observed in a work of a human operator(s) controlling the newspaper distribution and the sale process (NDSP) along the Adriatic coast during a tourist season [1]. In this paper, inherent device s min-max controllability is related to geometric transformation of device s transfer characteristic which is made up of a set of manipulative straight lines in min-max environment. II. COMBINATIONS OF MIN AND MAX CIRCUITS Min and Max Circuits as elementary building blocks can be used repeatedly in various topological configurations to realize specific circuits capable to perform different functions and to form more complex circuits in much diversified areas of electronics. For example, a simple organic Mid Circuit, with limited capabilities, can be designed using organic Min and Max Circuits [1], i.e. mid(x, Y, Z) = min(z, max(x, Y)), where X, Y and Z are circuit inputs. The resulting circuit is in the category of combinational logic. Another origin of combinational circuits is originated in the fact that all min-max expressions can be converted into either of two standard forms (just as Boolean expressions): the min of Maxs (mom) or Max of mins (Mom). When two or more min terms consisting of Mined variables (sets) or their complements (complement sets) are Maxed the resulting expression is a Max of mins (Mom), or sum of products (SOP) if Min and Max Circuits operate as Boolean devices. In both chases the output can be expressed as Y = max(min(a, B), min(c, D)). Even more, the logic of min-max design (approach) can be introduced in various scientific disciplines [1]. Min and Max Circuits (chips, sub-circuits) are easy to use and required no special handling precautions. They are designed in different transistor and/or Op-Amp techniques [2]. The design of a family of changeable Min, Max and Mid Circuits introduces information (input voltage) as a kind of trigger on a time or event scale over circuits operations, giving new combinational and controlled dimension to minmax circuitry in general sense. III. SINGLETON CHANGEABLE MIN-MAX CIRCUIT A compact electronic circuit with inherent capability to perform min or max operation, over two or three input variables (sets) is called Singleton Changeable Min-Max Circuit (see Fig. 1 and Fig. 2). Which one of two inherent circuit operations will be executed depending on incoming instruction /15 $ IEEE DOI /UKSim

Figure 1. Switch C (control input) is drawn out of the circuit to emphasize the changeable character of the circuit transfer characteristic(s).

Two menus define the circuit configuration and in the consequence, a corresponding circuit transfer characteristic (G Di). Figure 2. Changeable Mid Circuit transfer characteristic.

z min( x, ) 2 into standard Op-Amp hardware[4]. The designed circuit can be seen as an analog computing electronic device having outputs W 2 and W 3.(see Fig. 1).

2 Figure 1. Switch C (control input) is drawn out of the circuit to emphasize the changeable character of the circuit transfer characteristic(s). Note that two circuit outputs W 2 =min/ c max(x, and W 3 =min/ c max(x,y,z) are allowable for further processing. Figure 3. Four basic D i-operators (D i, i = 1, 2, 3, 4). Two menus define the circuit configuration and in the consequence, a corresponding circuit transfer characteristic (G Di). Figure 2. Changeable Mid Circuit transfer characteristic. Positive part of the shown characteristic y =max(y,0) can be used to model wholesaler attitudes (event filter or shaper) to incoming orders; not less then 2, or not greater then 10 (linguistic modeling). The design of the Singleton Changeable Min-Max Circuit is based on the mapping a correlation of two terms [3]: 1 max( z, max( x, ) = ( z + max( x, + z max( x, ) 2 1 min( z, min( x, ) = ( z + min( x, z min( x, ) 2 into standard Op-Amp hardware[4]. The designed circuit can be seen as an analog computing electronic device having outputs W 2 and W 3.(see Fig. 1). If Singleton Changeable Min-Max Circuits (3-in-m-M) are used instead of transistor Min and Max Circuits in an electronic scheme shown in Fig. 2, then emerging circuit transfer characteristic can be broken into two standalone pieces: Y =max(y,0) and Y =min(y,0). IV. D-OPERATOR CIRCUITS Min/ c max clipping circuits, are modified to obtain a class of transfer characteristics called D-operators, i.e. ideal diode operators [3]. For switching purposes the ideal diode is one way switch that is open when the imposed voltage is of one polarity and closed when the polarity is opposite. The D-Operator Circuit is a voltage-to-voltage min (max) circuit with the same response as a circuit which has ideal diode as a switching element. D-operators along with other min-max circuits can be used in the design of more complex electronic circuits, in synthesis of different non-linear transfer characteristics, in modeling discrete event systems, etc [5]. For example, if two D-operators D 1 and D 2 are tied together in a summing amplifier and if Changeable Min and Max Circuits are used to generate both D 1 and D 2 oprators, a voltage controlled absolute value circuit, i.e. full wave rectifier, will be designed (along with voltage controlled D 1 and D 2 operators). Note that the output of this circuit (Y) can be clamped to zero voltage level if control A calls min operation and control B calls max operation (or vice versa) (see oscillograph in Fig. 4). Anyway, D i -operators can be easily translated and transformed around the electronic phase plain in an effort to design circuits with quite different operating characteristics (see Fig. 5). Figure 4. Transformable (voltage controlled) absolute value circuit (full wave rectifier). Controlled circuit output Y is expressed as: A B min max voltage control A B min max voltage control Y = v in 0 v, or in a straight manner as A B min max voltage control Y = v in v in. in 589

3 Figure 5. D 1 transfer characteristic translation along with D 1 transfer function transformation can be used to design a rudimental signal compression (concentration) or expansion (dilatation) circuits. If D 2 -operator is shifted along x-axis (a D2 = 1) and D 4 - operator is shifted on the same axis (a D4 = 2), too and if a controlled DC voltage level (y DC = c) is added to the sum of shifted D-operators, then a transfer characteristic ( O G full ) suitable in a certain AI modeling and simulation will emerge (see next section for more details)[1], O G full =D 2 +D 4 +c, i.e. O G full = max(-x + 1, 0) + min(-x + 2, 0) + c. V. CHANGEABLE MID CIRCUIT Mid circuit have three inputs and a single output and operates in accordance with the following definition: The output of Mid Circuit assumes the middle voltage (set) among three different voltages (sets) applied to its inputs. Research related to the development of Min, Max and Mid Circuits has showed that there are several kinds of mid circuits capable to perform action in the accordance to the given definition [1]. In this paper we are going to deal with a version of so called algebraic Changeable Mid Circuit [6]. The circuit operates according the algorithm showed in Fig. 6 and is applied in the part of hardware model of an NDSP agent. The model is used to simulate behavior (only a dedicated part in a decision making domain) of the wholesaler agent operating in NDSP and dealing with three newspaper orders ( O n(k), I n(k) and P n(k)) received from field agents in the same time window (see Fig. 6). Other modes of wholesaler agent operation are available through his (her) simulated control menu (Zinputs shown in Fig.7). Agent s control menu allows, for example, modeling of seasonal biases over wholesaler agent decision making; on the raising edge of tourist season the agent will, as a rule, select the greatest among received orders (max operation will be performed), while the minimal order among applied orders will be chosen at the end of tourist season (min operation will be executed). Note that the voltage (information) changes over inputs in a control menu will induce corresponding changes of the transfer characteristic of Changeable Mid Circuit (see Fig. 8). One new application of Mid Circuits, presented in this paper, is related to AI modeling. Changeable Mid Circuit can be used to embody, in hardware environment, a small control (event evolving) rule base that is used by local operator controlling NDSP (see Fig. 9). On the other hand, the same rule base can be seen as a model of AI operator that controls distribution and sale of foreign newspapers in NDSP simulation. In the NDSP computer simulation, an operator (agent) is presented (activated) in, so called, 4-part multiagent diagram shown in Fig. 10 [1]. Note that this paper thematic digression was needed for pure presentation reasons, because only one figure is necessary to show a set of correlated entities: time evolution of TCO, TCO biases and geometric transformation of TCO reaching quasi-stable state from pure chaotic behavior. Such complex process (agents) state presentation can be done neither in EWB simulation program nor in hardware environment due to our poor laboratory equipment. Also note that our previous NDSP work was based on Min, Max and Op Amp circuits as modeling tools [1]. In this paper Mid Circuit is used for rule-base-to-hardware mapping and consequently in modeling some operator s seasonal biases. In that case, TCO that is shown in Fig. 9 as full TCO be expressed in mid terms as: O + 0 G full ( k ) = mid ( mid ( p ( k ), p ( k ), p ( k )), p c1 ( k ), p c 2 ( k )). Control over current TCO, representing operator s desires and capabilities in NDSP, is established through two control menus serving inputs of two Changeable Mid Circuits. A combination of applied straight lines (p +, p -, p c1, p c2 ) will define (changes of) current TCO (k). Our research showed that TCO was changing occasionally, representing the bias state of managerial environment, bias state of local operator or their combination [7]. In situations when biases were strong (sometimes almost irrational), a degraded TCOs ( o G deg ), or biased TCOs were emerging [8]. Simple hardware model presented in Fig. 11 can be used to represent (illustrate) complexity of operator s decision making bias over his (her) TCO during the tourist season, but does not include explicit biases over operator s dead zone [1]. Operator s dead zone (the part of p 0 straight line that lies between straight lines p + and p - ), i.e. operator s quasistable state, or a process evolutionary termination criteria is not the subject of our interest in this paper, but could be in the focus of the next paper, as the dead zone width and position changes affect operator s control behavior [1]. It is worth to emphasize that all 2-in-1 switches, presented in these hardware models can be designed in min-max technique, min-max controlled multiplexers, and they can be a subject of information control [6]. Mentioned transformability of Mid Circuit can have a pure electronic application (see Fig. 12). Mid Circuit with a feedback from its output can be used as signal minimum or maximum (peak) value detector over a certain input frequency and amplitude ranges. 590

Figure 6. Wholesaler agent operating algorithm in AI hardware model of NDSP [1].

(see Fig 7). In an electronic sense Mid Circuit can be set up to preform AND/ c OR Boolean logic operations. Figure 7.

..), among three applied field orders, for further processing in a day k.

4 Figure 6. Wholesaler agent operating algorithm in AI hardware model of NDSP [1]. In a multi agent preset coordination process, mid operation of wholesaler agent is chosen as a problem solving tool when three different orders O n(k), I n(k) and P n(k) are question (see Fig 7). In an electronic sense Mid Circuit can be set up to preform AND/ c OR Boolean logic operations. Figure 7. Mid Circuit in AI model of a human operator controlling ordering in NDSP. W In the preset mid mode the wholesaler agent will pick up an order o(k) = P n(k) =(2, 20, ), among three applied field orders, for further processing in a day k. Selection (preset) of the mid mode operation of the wholesaler agent can be seen as act of cooperation among agents in NDSP. Figure 8. Input control (Z) over the transfer characteristic(s) of Changeable Mid Circuit; min, max and mid operations are available depending on the state of input control menu (switches M and C). A set of batteries (Min and Max) represents different straight lines that are used to (de)compose Mid Circuit transfer characteristic into its organic Min and Max parts. 591

Figure 9. Straight lines representing operator s actions are Maxed and Mined to design transfer characteristic of an operator controlling NDSP; TCO O G full =min(p - (k), max(p + (k), p 0 (k))).

Geometric manipulations over TCO will affect operator s control responses (new orders) in a computer simulation of NDSP (see Fig. 10). Figure 10.

11), is introduced at a certain stage of evolution to reach termination criterion as soon as possible. Injected bias means transformation of TCO in geometric sense.

5 Figure 9. Straight lines representing operator s actions are Maxed and Mined to design transfer characteristic of an operator controlling NDSP; TCO O G full =min(p - (k), max(p + (k), p 0 (k))). In min-max TCO synthesis operational amplifiers as well, as batteries are seen as straight lines generators. Geometric manipulations over TCO will affect operator s control responses (new orders) in a computer simulation of NDSP (see Fig. 10). Figure 10. Simplified computer simulation of operator s behavior in NDSP (I quadrant). Injected TCO bias, meaning min-max modification of existing TCO (see Fig. 11), is introduced at a certain stage of evolution to reach termination criterion as soon as possible. Injected bias means transformation of TCO in geometric sense. Note that sale process is simulated in IV quadrant of 4-part multiagent diagram. Second quadrant is dedicated to wholesaler agent control and bias over incoming operator s order while the third quadrant represent distribution agent better say distribution loses, i.e. noises that affect operator s control actions. Figure 11. Geometric control over a seasonal TCO. In existing circuit (model) different scenarios over TCO are possible depending of the state of the mind of an operator that can be influenced by strong managerial biases [1]. For example, TCO* (G O deg) can be expressed as G O deg(k)= mid(mid(p + (k),p + (k),p - (k)),p c2,0) and can present operator's attitudes: There will be no order increase in September. Only decreases will be executed! 592

Figure 12. MID positive/negative peak detector can be used in modeling NDSP agent s attitudes in a coordination processes (agent s oscillations from one extreme attitude to the other) [1]. VI.

6 Figure 12. MID positive/negative peak detector can be used in modeling NDSP agent s attitudes in a coordination processes (agent s oscillations from one extreme attitude to the other) [1]. VI. CONCLUSION Different kinds of processing and modeling capabilities of min-max circuitry have been shown in this paper. Those capabilities are based on hardware combinational logic of Min, Max and Mid Circuits, on their functional changeability and ability to process different kinds of inputs (from variables to sets). Inherent functional changeability of these circuits can be effectuated in, at least, three ways: by means of dedicated control signals applied to the control input ( MIN Signal MAX ) through input voltage connections (combinations) over input terminals Connections Connections ( MIN MID MAX ) by choice of input signal set (universe of discourse) Choice{ 1,0} Choice{ 1,0} ( MIN AND, MAX OR) We have shown how distinctive features of these circuits can be used in the design of hardware circuits and in (hardware) modeling. For example, Changeable Min and Max Circuits were used to design controllable rectifier (absolute value circuit) and Mid Circuit was used to model decision making bias (control behavior of TCO) in the (AI) model of the human operator in NDSP. Almost all aspects of operator s seasonal biases over NSDS can be modeled through geometric transformations of TCO. The design stage for min-max circuitry applications is the (electronic) phase plain (v out versus v in ) in which geometry plays an important role. Productive min-max geometric manipulations over input variables and sets can be seen as new circuit design procedure. We can talk about min-max circuit computational geometry.in that sense min-max circuit computational geometry can be regarded as min-max circuit design and description tool that studies ways (algorithms) for solving geometric circuit design problems related to hardware design, modeling and control problems in min-max environment. The input to a min-max circuit computational geometry problem can be a set of geometric objects (straight lines), a set of line segments, or even polygons. The output is often a new geometric object (geometric portrait of a circuit) that can represent a transfer characteristic or dedicated controls efforts. In sum, min-max environment can be considered as a friendly environment when modeling and circuit control design are in questions. The further modeling work in min-max environment would include introduction of dead zone (evolutionary termination criterion of TCO) bias problem in the design of current TCO and its influence on the evolution of TCO. At the very end, note that associative and self-explanatory figures (illustrations), with incorporated paper s red line were used to avoid text pitfalls. REFERENCES [1] D. Kovacevic, Hardware Modeling of Newspaper Distribution and Sell Process Control Based on Intelligent Agent Network Realization, Doctoral Thesis, FESB, Split, Croatia, [2] D. Kova evi, A. Kova evi, Fuzzy Logic Controller Based on Standard Operational Amplifiers, Edited by J.F. Baldwin, John Wiley & Sons, Chichester, England, pp [3] R.Yager, P.Filev, Essentials of Fuzzy Modeling and Control, John Wiley and Sons, New York, USA, 1994, pp [4] D. Kovacevic, Fuzzy Gates Interchangeability, Proc. MIPRO. Symp MEET, May 2014, pp [5] The Engineering Staff of Analog Devices, Inc., Edited by D.H. Sheingold, Nonlinear Circuits Handbook, Analog Devices, Inc.,Norwood Massachusetts, USA, [6] D. Kovacevic, MIN-MAX Circuits as a Problem Solving Tool(s), Proc. MIPRO Symp. MEET, June 2013, pp [7] D. Kovacevic, A. Kovacevic, M. Burazer, ESTAN-project, Proc. IAESTED Symp. International Conference, May 1994, pp [8] D. Kovacevic, A. Kovacevic, An Opinion Survey; Useful Tool on a Way to Design an Information System, Proc. KOI'96. Symp. Operational Research, September 1996, pp

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