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A Dead Time Compensator Based on Linear Algebra (DTCLA)
Proceedings of the 20th World The International Federation of Congress Automatic Control The International Federation of Congress Automatic Control Toulouse, France, July 2017 Proceedings of the 20th9-14, World Proceedings of theJuly 20th9-14, World Congress Toulouse, France, 2017 Available online at www.sciencedirect.com The International Federation of Automatic Control The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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A Dead Time Compensator Based on Linear Algebra (DTCLA) IFAC PapersOnLine 50-1 (2017) 3075–3080 A Dead Time Compensator Based on Linear Algebra (DTCLA) 1,2 3 A Dead Time Compensator Based on Algebra Camacho , Gustavo , O. Lucia Quintero4 (DTCLA) A Dead Time Oscar Compensator BasedScaglia on Linear Linear Algebra (DTCLA)
3 Oscar Camacho1,2, Gustavo Scaglia , O. Lucia Quintero4 1,2 3 4 1,2 3 4 Oscar Camacho 1,2, Gustavo Scaglia3, O. Lucia Quintero4 Oscar Camacho , Gustavo Scaglia , O. Lucia Quintero 1 Universidad de Los Andes, Mérida, Venezuela, (e-mail: [email protected]) 1 2 Universidad de Nacional, Los Andes,Quito, Mérida, Venezuela, (e-mail: [email protected]) Escuela Politécnica Ecuador, (e-mail: [email protected]) 1 2 3 1 Escuela Politécnica Nacional, Quito, Ecuador, (e-mail: [email protected]) Universidad de Los Andes, Mérida, Venezuela, (e-mail: [email protected]) 1 Universidad Nacional de SanMérida, Juan, CONICET, Universidad de Los Andes, Venezuela,(e-mail: (e-mail:[email protected]) [email protected]) 22 3 Universidad 4 (e-mail: Nacional de San Juan, CONICET, (e-mail: [email protected]) Politécnica Nacional, Quito, Ecuador, [email protected]) 2 Escuela Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Escuela Politécnica Nacional, Quito, Ecuador, (e-mail: [email protected]) 3 4 3 Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Nacional 3 Universidad Universidad Nacional de de San San Juan, Juan, CONICET, CONICET, (e-mail: (e-mail: [email protected]) [email protected]) 4 4 4 Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Abstract: This paper proposes a dead time compensator based on a combination of the linear algebra control Abstract: This(proposed paper proposes a dead time compensator based on a combination of the linear algebratocontrol methodology by Scaglia, 2009) and an internal model structure (Smith predictor), solve Abstract: This paper proposes a dead time compensator based on a combination of the linear algebratocontrol methodology (proposed by Scaglia, 2009) and an internal model structure (Smith predictor), solve elevated dead time system problems. Moreover, the process transfer function of the nonlinear system Abstract: This paper proposes a dead time compensator based on a combination of the linear algebra control methodology (proposed by problems. Scaglia, 2009) and anthe internal model structure predictor), tosystem solve elevated dead timeoperation system Moreover, transfer function of transfer the nonlinear linearized at some point is approximated aprocess second order plus dead(Smith time function. The methodology (proposed by Scaglia, 2009) and anbyinternal model structure (Smith predictor), to solve linearized at some operation point is approximated by a second order plus dead time transfer function. The elevated dead time system problems. Moreover, the process transfer function of the nonlinear system controller performance is judged by simulations and it is evaluated using the ISE performance index. We elevated dead time system problems. Moreover, the process transfer function of the nonlinear system controller performance isagainst judged by simulations and ita is evaluated using the results. ISE performance index. The We linearized at some operation point is approximated by second order plus dead time transfer function. compared this approach a typical DTC-PID controller with improved linearized at some operation point is approximated by a second order plus dead time transfer function. The compared performance this approachisagainst a by typical DTC-PID controller with improved results. controller judged simulations and it is evaluated using the ISE performance index. controller performance is judged by simulations andControl) it is evaluated using the ISELtd. performance index. We We Keywords: dead time systems, linear algebra, chemical processes, internal model, and linear systems. © 2017, IFAC (International Federation of DTC-PID Automatic Hosting by Elsevier All rights reserved. compared this approach against a typical controller with improved results. Keywords:this dead time systems, algebra, chemical processes, model, and linear systems. compared approach againstlinear a typical DTC-PID controller withinternal improved results. Keywords: time processes, internal model, and systems. Keywords:1.dead dead time systems, systems, linear linear algebra, algebra, chemical chemical processes, internal model,damped and linear linear systems. INTRODUCTION very oscillatory or highly when the process has a large 1. INTRODUCTION very oscillatory or highly damped when thewith process a large time delay (Astrom et al, 1994). To deal this has additional very oscillatory or highly damped when the process has a large time delay (Astrom et al, 1994). To deal with this additional 1. INTRODUCTION new or structures were when proposed, decoupling the The presence of time1.delays in many industrial processes is a problem INTRODUCTION very oscillatory highly damped the process has a large time delay (Astrom et al, 1994). To deal with this additional problem new structures were proposed, decoupling the The presence of time delays in many industrial processes is a disturbance from the et setal, point response (Zhang 1996) well-recognized problem. Time lag, transportation lag, time time delay (Astrom 1994). To deal with and thisSun, additional problem new structures were proposed, decoupling the disturbance from the set point response (Zhang and Sun, 1996) The presence of time delays in many industrial processes is a well-recognized problem. Time lag, transportation lag, time (Astrom etnew al, 1994). In general, approaches have some delaypresence and dead time delays are common in industrial structures were these proposed, decoupling the The of time in manyphenomena industrial processes is a problem (Astrom et al, 1994). In point general, approaches have some from the set response (Zhang and Sun, 1996) well-recognized problem. Time lag,phenomena transportation lag, time delay and Time dead delay time are in industrial problems: sensitive to these modelling since the processes. can common be produced by measurement lag, disturbance disturbancethey fromare the set point response (Zhangerrors and Sun, 1996) well-recognized problem. Time lag, transportation lag, time (Astrom et they al, 1994). In general, these approaches have some problems: are sensitive to modelling errors since the delay andand dead time are phenomena inlag industrial processes. Time delay can common be produced by measurement lag, design requires the use of a process model, which can be analysis computation time, communication or the (Astrom et al, 1994). In general, these approaches have some delay and dead time are common phenomena in industrial problems: they are sensitive to modelling errors since requires the use of aModelling process model, which can the be processes. Time delay can be produced by measurement lag, analysis and computation time, communication lag orAlso, the design difficult to obtain in practice. errors are unavoidable transport time required for a fluid to flow through a pipe. problems: they are sensitive to modelling errors since the processes. Time delay can be produced by measurement lag, design requires the use of a process model, which can be difficult to obtain in practice. Modelling errors are unavoidable analysis and computation time, communication lag or the transport time required for a fluid to flow through a pipe. Also, and theyrequires result inthe a mismatch modelwhich and thecan actual several concerntime, with dynamics and certain delays design use of abetween processthe model, be analysisbioprocesses and computation communication lag or the difficult to obtain practice.designed Modelling errors areand unavoidable and they result incontrollers ainmismatch between the model themodels actual transport time required for a fluid flow through a pipe.adelays Also, several bioprocesses concern withto dynamics and certain plant. Thus, the using particular related to the capability of microorganisms of reaching good difficult to obtain in practice. Modelling errors are unavoidable transport time required for a fluid to flow through a pipe. Also, and they result a mismatch between the model and themodels actual Thus, theincontrollers designed using several bioprocesses concern with dynamics and certain related to theorcapability ofdue microorganisms of reaching adelays good plant. may perform differently when they areparticular implemented on growth rate inhibition to the concentration of product and they resultquite in a mismatch between the model and the actual several bioprocesses concern with dynamics and certain delays plant. may perform quite differently when they are implemented on Thus, the controllers designed using particular models related to the capability of microorganisms of reaching a good growth rate or inhibition due to the concentration of product the actual process (Quintero et al, 2009). and substrate (Quintero et al, 2008) (Amicarelli et al, 2016). plant. Thus, the controllers designed using particular models related to the capability of microorganisms of reaching a good may perform quite differently when they are implemented on the actual process (Quintero et al, 2009). growth rate or inhibition due to the concentration of product and substrate (Quintero et al, 2008) (Amicarelli et al, 2016). Certainly, of some of the the concentration variables for these kinds may perform quite differently when they are implemented on growth ratetheorcontrol inhibition due to of product In addition, the methodology based on linear algebra and actual process (Quintero et al, 2009). and substrate (Quintero al, 2008) etthese al, 2016). Certainly, the control of et some of the (Amicarelli variables kinds of systems must be performed Nonlinearfor Point view. the In addition, the methodology based on linear algebra and the actual process (Quintero ettoal, 2009). and substrate (Quintero et al, from 2008)a (Amicarelli et al,of2016). numerical methods concepts design control algorithms is a Certainly, the control of we some of thea variables for these kinds of systems must be2009 performed from Nonlinear Point of view. In Quintero al, demonstrate that Linear Algebra addition, the methodology based on linear algebra and numerical methods concepts to design control algorithms isofa Certainly, theetcontrol of some of the variables for these kinds In simple and relatively new method that allows the control of systems must be2009 performed from a Nonlinear Point of view. In addition, the methodology based on linear algebra and In Quintero et al, that Linear Algebra based controllers can, we in demonstrate an easily and understandable methods concepts to design control algorithms is simple and relatively new method that allows the control ofa of systems must be performed from a Nonlinear Point of view. numerical highly nonlinear systems. It has been applied the designisof In Quintero et al, 2009 demonstrate that tracking Linear Algebra based controllers can, in antheeasily and understandable numerical methods concepts to design controltoalgorithms a formulation, perform notwe only trajectory control simple and relatively new method that allows the control highly nonlinear systems. It has been applied to the design of In Quintero et al, 2009 we demonstrate that Linear Algebra differentand class of control systems such asallows trajectory tracking of based controllers can, in an easily and understandable formulation, perform not only the trajectory tracking control simple relatively new method that the control of but alsocontrollers the positioning point.and Butunderstandable the controller highly different class and of control such as2016), trajectory nonlinear systems. It has been to thetracking design of based can, at in certain an easily mobile robots UAV’ssystems (Capito et al,applied chemical plants formulation, not at only the atrajectory control but alsomust the perform positioning certain point. Buttracking the controller highly nonlinear systems. It has been applied to the design of design be performed under zero error model of the different class and of control such trajectory tracking of mobile robots UAV’ssystems (Capito et al,as 2016), chemical plants formulation, perform not only the trajectory tracking control and bioprocesses (Quintero, et al 2009, Scaglia et al, 2015) just but also the positioning at certain point. But the controller design must be performed under a zero error model of the different class of control systems such as trajectory tracking of nonlinear system, in certain cases too muchBut complicated and mobile robots and UAV’s (Capito et al, Scaglia 2016), chemical plants and bioprocesses (Quintero, etIts al 2009, et al,is2015) just but also the positioning at certain point. the controller to name some applications. main advantage that the design must be performed under a zero error model of the nonlinear system, in certain cases too much complicated and mobile robots and UAV’s (Capito et al, 2016), chemical plants highly must nonlinear. Some approaches wereerror alsomodel explored to and to name some applications. Its main advantage is2015) that the bioprocesses (Quintero, al 2009, et al, just design be performed under a zero of the for the trackinget tendScaglia to zero and control nonlinear certain cases too much also complicated highly Some approaches were to conditions and bioprocesses (Quintero, eterror al 2009, Scaglia et al, 2015) just control nonlinear. thissystem, kind ofin Sciascio et al,explored 2004). and conditions for the tracking error tend to zero and control name some applications. Its main advantage is that the nonlinear system, inbioprocesses certain cases(ditoo much complicated and to obtained with low computational cost which makes highly approaches were also control nonlinear. this kind ofSome bioprocesses (di Sciascio et al,explored 2004). to actions to namearesome applications. Its main advantage is that the actions are obtained with low computational cost which makes for the tracking error tend to zeroconditions and control highly nonlinear. Some approaches were also explored to conditions it easy to implement in a microcontroller. These are On the other hand, the achievable performance of typical control this kind of bioprocesses (di Sciascio et al, 2004). conditions for the tracking error tend to zero and control it easyby to implement in a microcontroller. These are are obtained with low computational cost which makes On the this other hand, the achievable performance of typical control kind ofsystems bioprocesses Sciascio al, 2004). found solving a system of linear equations andconditions conditioning feedback control can (di decline if aetprocess has a actions actions are obtained with low computational cost which makes found by solving a system of linear equations and conditioning it easy to implement in a microcontroller. These conditions are On the other hand, the achievable performance of typical feedback control systems can decline if a process has a theeasy system to havein exact solution. For all conditions the previous relatively largehand, time the delay comparedperformance to the dominant time it to implement a microcontroller. These are On the other achievable of typical by solving a system of linear andby conditioning the system tothehave exact solution. For all the previous feedback control systems decline process has relatively large time delay can compared toifthea dominant timea found applications, controllers wereequations designed using the constant (Smith Corripio, 1997). Predictive structures found by solving a system of linear equations and conditioning feedback controlandsystems can decline if a process hasanda the controllers were designed by system tothehave solution. For all(Quintero the using previous relatively large timeCorripio, delay compared to the(Camacho, dominant constant (Smith and 1997). Predictive structurestime and complete nonlinear andexact discrete process model etthe al, sliding mode controllers (Camacho, al, applications, the system to have exact solution. For all the previous relatively large time delay compared2002), to the dominant ettime complete nonlinear and discrete process model (Quintero al, the controllers were designed by usingetthe constant (Smith andused Corripio, 1997). Predictive structures sliding mode controllers (Camacho, 2002), (Camacho, etand al, applications, 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the 2007) have been to solve such problems. Primarily, applications, the controllers were designed by using the constant (Smith and Corripio, 1997). Predictive structures and complete 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the nonlinear and discrete process model (Quintero et al, sliding mode controllers (Camacho, 2002), (Camacho, et al, 2007) have been used to solve such problems. Primarily, controller algorithm applied toetthe internal modelcontrollers control (IMC) and the 2002), Smith predictor (SP) complete mathematical nonlinear and discrete process modelis (Quintero al, sliding mode (Camacho, (Camacho, et are al, obtained obtained mathematical controller algorithm is Therefore, applied to the Scaglia et al, 2009, Serrano et al, 2016). internal model control and the Smith predictor (SP) are 2009; 2007) have been used(IMC) to solve such problems. Primarily, specific process. Hence, for each different process a different the most popular predictive structures used for time delay 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the 2007) have been used to solve such problems. Primarily, obtained process. for each different process a different mathematical controller algorithm isit applied to thea internal control (IMC) the Smith predictor (SP) are specific the mostmodel popular predictive structures used for time delay controller should Hence, be obtained, in other words, is generated compensation (Marlin, 1991,and and Corripio, 1997). obtained mathematical controller algorithm is applied to the internal model control (IMC) andSmith the Smith predictor (SP) are specific controller should be obtained, in other words, it is generated process. Hence, for each different process a different the most popular predictive structures used for time delay compensation (Marlin, 1991, Smith and Corripio, 1997). differentprocess. control law in each case. different process a differenta Furthermore, whenpredictive the processstructures presents an integral behaviour specific Hence, for each the most popular used for time delay controller control in each case. shouldlaw be obtained, in other words, it is generated a compensation (Marlin, 1991, be Smith Corripio, 1997). Furthermore, when the process presents an integral behaviour the original structures used and since a constant load different controller should be obtained, in with otherthe words, generated compensation (Marlin,cannot 1991, Smith and Corripio, 1997). Besides, there arelaw two problems use ofit aismodel as fara different control in each case. Furthermore, when the process presents an integral behaviour the original structures cannot be used since a constant load disturbance results erroran (Camacho and De la different Besides, there are two problems with the use of a model as far control law in each case. Furthermore, when in thea steady-state process presents integral behaviour the original structures used a constant load disturbance in a cannot steady-state errorsince (Camacho and De la as industrial processes are concerned. First, the development Cruz, 2004).results To overcome thisbe obstacle different approaches as industrial processes are concerned. First, the development there are two problems with the use of a model as far the original structures cannot be used since a constant load Besides, a complete difficult due complexity Cruz, 2004).proposed To overcome this obstacle approaches disturbance results in a steady-state error different (Camacho andstudies De la of Besides, there model are twoisproblems withmainly the usetoofthe a model as far have been Simulation of a complete model is difficult due mainly to the complexity industrial processes are concerned. First, the development disturbance results in a(Watanabe, steady-state1981). error (Camacho and De la as of industrial the process itself, and the lack of knowledge of some Cruz, 2004).proposed To the overcome this obstacle different approaches have been 1981). Simulation as processes aretoconcerned. First, the development have set(Watanabe, point and load disturbances arestudies either of processmodel itself, isand to thedue lackmainly of knowledge of some athe complete difficult to the complexity Cruz,shown 2004).that To overcome this obstacle different approaches have shown been proposed 1981). Simulationarestudies that the set(Watanabe, point and load disturbances either of a complete model is difficult due mainly to the complexity have been proposed (Watanabe, 1981). Simulation studies of the process itself, and to the lack of knowledge of some have shown that the set point and load disturbances are either of the process itself, and to the lack of knowledge of some have shown thatIFAC the set point and load disturbances are either3130 Copyright © 2017
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Proceedings of the 20th IFAC World Congress 3076 Oscar Camacho et al. / IFAC PapersOnLine 50-1 (2017) 3075–3080 Toulouse, France, July 9-14, 2017
process parameters. Second, most process models relating the controlled and the manipulated variables are of higher-order. Therefore, the traditional numerical controller procedure can produce more complex controllers and also their application is just for the process of analysis. An efficient alternative modelling method for process control is the use of empirical models, which use low-order linear models with dead time (Marlin, 1991; Smith and Corripio, 1997). Most times, first-order-plus dead-time (FOPDT) models are adequate for process control analysis and design. In many cases, mobile robotics as well as chemical processes, can be represented by first-order plus dead time (FOPDT) models (Capito, et al 2016). The internal model control design is based on the idea of forming an ideal and simple controller based on the transfer function of the process model, and it has presented satisfactory results, as in (Camacho et al 2002). When controlling chemical processes, it is important to simplify the process model, because these processes are often represented by several state space equations, and then the controllers must be resistant to modelling errors and uncertainties. Guevara et al, 2016 designed a Linear Algebra (and Numerical Methods) based controller (NMCr) designed on a first order plus dead time (FOPDT) model of the actual process. The overall idea is to develop a general NMCr, which can be used for different self-regulating industrial processes, if they have a similar open loop response, such as an FOPDT model. Hence, this work summarizes the easily and simplicity of numerical methods procedures along to a reduced order model of the process to obtain a simple and versatile controller. The performance of the proposed controller in this article is tested by simulations, but the performance degrades when the 𝑡𝑡 controllability relationship ( 0) increases. 𝜏𝜏
This paper proposes from a Smith Predictor structure the synthesis of a Deadtime compensator that uses the linear algebra methodology. This new approach uses a second order plus dead-time model obtained from the reaction curve of the nonlinear process. This novel proposal has the capability of solving elevated dead time problems. The proposed scheme is tested and compared against the previous approach built from reduced first order plus dead time model of the process (Guevara et al, 2016). This work is organized as follows: section 2 presents the fundamentals of the linear algebra based controllers and IMC control and section 3 explains the Dead Time Compensator Based on Linear Algebra (DTCLA). Then section 4, presents some study cases where the DTCLA scheme is presented and compared with a regular DTC-PID. Finally, conclusions are presented in section 5. 2. BASIC CONCEPTS This section presents briefly some basics of the technique developed by Scaglia in 2009 and then basic ideas of Internal model control (Francis and Wonham, 1976). These concepts are the foundation for the approach that is presented in section 3. It is important to mention that the internal model structure is used as a Smith predictor.
2.1 Linear Algebra methodology The numerical methods and linear algebra tracking control methodology has been proved over highly nonlinear systems for bioprocesses (Quintero et al, 2009) (Scaglia et al, 2009, 2014), and also in under-actuated mechanical systems, dynamic nonlinear systems controlled with a kinematic model for tracking and positioning control tasks for fast and slow dynamics systems such as mobile robots (Scaglia et al, 2009, 2010, 2015; Serrano et al 2014, Rosales et al 2015). Stability and robustness demonstrations can be found in (Scaglia et al, 2010, 2015) and parameters optimization through Monte Carlo methods (Serrano et al, 2016). The methodology for calculating the control actions can be summarized in the following steps: 1.
To describe the system using state equations and then approximate them using a numerical method.
2.
To set out the control actions calculation as a problem of solving a system of linear equations.
3.
To choose the state variables to be followed and under what conditions the system of linear equations, of the previous point, has exact solution, and then to define the reference trajectory of the remaining state variables.
4.
To solve the system of equations.
2.2 Internal Model Structure as Smith Predictor This controller structure can be seen in Francis and Wonham, 1976. In this structure 𝐺𝐺𝑃𝑃 (𝑠𝑠) represents the process, 𝐺𝐺𝐶𝐶 (𝑠𝑠) is the controller transfer function. 𝐺𝐺𝑚𝑚 (𝑠𝑠) is the process model, em represents the modelling error, 𝑦𝑦𝑦𝑦 is the process output, 𝑦𝑦𝑦𝑦 is − is the output of the non-invertible the model output and 𝑦𝑦𝑚𝑚 part of the model. To implement it, it is necessary to obtain the process model and separate it into two parts: a non-invertible 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ and an invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠)− , as shown in (1). 𝐺𝐺𝑚𝑚 (𝑠𝑠) = 𝐺𝐺𝑚𝑚 (𝑠𝑠)− 𝐺𝐺𝑚𝑚 (𝑠𝑠)+
(1)
To obtain this controller, only the invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠) is used, because the non-invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ can present problems such as being unstable, not causal or cannot be found. −
3. A DEAD TIME COMPENSATOR BASED ON LINEAR ALGEBRA (DTCLA) In this section is shown the development of the proposed Dead Time Compensator Based on Linear Algebra (DTCLA). This proposal is based on the improvement of the control scheme by (Guevara et al, 2016) in which the linear algebra controller design is based on a first order plus dead time model of the system. For this new approach, it is assumed that the process presents an overdamped dynamics and the parameters of a transfer function are obtained from the reaction curve method identification (Smith and Corripio, 1997). The process model is represented as a second order system plus delay transfer function, which is divided into two components to be added to the control scheme (Astrom et al, 1994).
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Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Oscar Camacho et al. / IFAC PapersOnLine 50-1 (2017) 3075–3080
In figure 1, it is represented the proposed scheme, it can be seen that the controller based on linear algebra needs as the signal to its calculations the output of the invertible part of the transfer function from the internal model structure in a similar way as in the Smith predictor architecture. Therefore, the Linear Algebra method is applied to the part of the reduced model without time delay.
3077
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 = 𝜒𝜒̇ 2 + 𝐾𝐾𝐴𝐴 𝜒𝜒2 + 𝐾𝐾𝐵𝐵 𝜒𝜒1
(10)
Applying Euler approximation to discretise the derivative of the state variables, the following equations are obtained:
𝜒𝜒2 𝑛𝑛 =
𝜒𝜒1
𝑛𝑛+1 −𝜒𝜒1 𝑛𝑛
𝑇𝑇
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 𝑛𝑛 =
𝜒𝜒2
(11)
𝑛𝑛+1 −𝜒𝜒2 𝑛𝑛
𝑇𝑇
+ 𝐾𝐾𝐴𝐴 𝜒𝜒2 𝑛𝑛 + 𝐾𝐾𝐵𝐵 𝜒𝜒1 𝑛𝑛
(12)
Where, 𝑇𝑇 is the sampling time. The sampling time for the discrete controller will be in terms of the time constants of 𝜏𝜏𝑚𝑚1 and 𝜏𝜏𝑚𝑚2 . min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 )/15 < 𝑇𝑇 < min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 )/4
Fig.1 Proposed scheme for the DTCLA. First of all, let us show the model of the process, the system model to approximate the process is described by (2): 𝐺𝐺𝑚𝑚 (𝑠𝑠) = 𝑘𝑘𝑚𝑚
𝑒𝑒 −𝑡𝑡0 𝑠𝑠
(2)
(𝜏𝜏𝑚𝑚1 𝑠𝑠+1)(𝜏𝜏𝑚𝑚2 𝑠𝑠+1)
Where: − (s) = (𝜏𝜏 𝐺𝐺𝑚𝑚
𝑘𝑘𝑚𝑚
𝑚𝑚1 𝑠𝑠+1)(𝜏𝜏𝑚𝑚2 𝑠𝑠+1)
; 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ = 𝑒𝑒 −𝑡𝑡0𝑠𝑠
(3)
Where: 𝑘𝑘𝑚𝑚 : Static gain. 𝜏𝜏𝑚𝑚1 : Time constant 1. 𝜏𝜏𝑚𝑚2 : Time constant 2. 𝑡𝑡0 : Dead time. − Rearranging some terms in Eq. (2), the invertible part 𝐺𝐺𝑚𝑚 (s), can be represented as: − (s) = 𝐺𝐺𝑚𝑚
− (𝑠𝑠) 𝑦𝑦𝑚𝑚
𝑢𝑢(𝑠𝑠)
=
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑠𝑠 2 +𝐾𝐾𝐴𝐴 𝑠𝑠+𝐾𝐾𝐵𝐵
(4)
Where: 𝐾𝐾𝐴𝐴 =
𝐾𝐾𝐵𝐵 =
𝜏𝜏𝑚𝑚1 +𝜏𝜏𝑚𝑚2 𝜏𝜏𝑚𝑚1 𝜏𝜏𝑚𝑚2
(5)
𝜏𝜏𝑚𝑚1 𝜏𝜏𝑚𝑚2
1
(6)
𝜒𝜒̈ + 𝐾𝐾𝐴𝐴 𝜒𝜒̇ + 𝐾𝐾𝐵𝐵 𝜒𝜒 = 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢
(7)
Representing this second order equation in state variables form, the following expression is obtained: 0 𝜒𝜒̇ [ 1 ] = [−𝐾𝐾 𝐵𝐵 𝜒𝜒̇2
1 𝜒𝜒1 0 −𝐾𝐾𝐴𝐴 ] [ 𝜒𝜒2 ] + [ 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 ] [𝑢𝑢]
(8)
If a faster response is required, the value of 𝐾𝐾𝑥𝑥 should be close to 0 and if a slower response is required, the value should be close to 1. To introduce this parameter into the algorithm is required change the nomenclature of (11) to have a variation of 𝐾𝐾𝑥𝑥 related with the difference between the reference and output value (error). Having the next resulting expression with reference terms:
𝜒𝜒2 𝑛𝑛 =
(9)
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 −𝜒𝜒1 𝑛𝑛 −𝐾𝐾𝑥𝑥 (𝜒𝜒1 𝑇𝑇
𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 −𝜒𝜒 𝑛𝑛 )
(14)
Where:
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 : is the reference. 𝜒𝜒1 𝑛𝑛 : is the measured output.
To reduce the resulting expressions, the difference between the reference 𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 and the actual output 𝜒𝜒 𝑛𝑛 will be represented by 𝑒𝑒 𝑛𝑛 . Assuming that the current value of variable 𝜒𝜒2 𝑛𝑛 is the same required for a next discrete time 𝜒𝜒2 𝑛𝑛+1 ( see remark 1). Then it can replace (14) in (12) to obtain: 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 𝑛𝑛 =
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 − 𝜒𝜒1 𝑛𝑛 − 𝐾𝐾𝑥𝑥 𝑒𝑒 𝑛𝑛 𝑇𝑇 2
𝐾𝐾𝐵𝐵 𝜒𝜒1 𝑛𝑛
−
𝜒𝜒2 𝑛𝑛 𝑇𝑇
+ 𝐾𝐾𝐴𝐴 𝜒𝜒2 𝑛𝑛 +
(15)
Finally, from previous equation, the control law 𝑢𝑢 𝑛𝑛 , is obtained: 1 [𝜒𝜒 − 𝜒𝜒1 𝑛𝑛 + 𝐾𝐾𝑥𝑥 𝑒𝑒 𝑛𝑛 − 𝑇𝑇 𝜒𝜒2 𝑛𝑛 ] + 𝑢𝑢 𝑛𝑛 = 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑇𝑇 2 1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 𝜒𝜒1 𝑛𝑛 𝑘𝑘𝑚𝑚
Where:
Where 𝜒𝜒1 and 𝜒𝜒2 are the state variables of the system, being the derivate of the controlled variable and u is the controller output. From the state variables representation, the following equations are obtained:
𝜒𝜒̇ 1 = 𝜒𝜒2
Where min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 ) means the minimum between the time constants. In order to improve the algorithm and makes decrease slowly the variation of error, a constant 𝐾𝐾𝑥𝑥 is added and it takes values between 0 and 1 depending on how fast the control action is needed. This becomes the first tuning parameter to calibrate the response of the controller.
− (𝑠𝑠) 𝑦𝑦𝑚𝑚
= 𝜒𝜒(𝑠𝑠). To facilitate the next calculations, let us call Now for controller design (4) can be represented in differential equation form as follows:
(13)
𝜒𝜒2
𝑛𝑛
=
𝜒𝜒1 𝑛𝑛 −𝜒𝜒1 𝑛𝑛−1 𝑇𝑇
+
𝐾𝐾𝐴𝐴 𝜒𝜒 2 𝑛𝑛 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵
(16)
(17)
One of the main advantages of the proposed scheme is that it presents a fixed architecture facilitating its implementation, therefore the proposed controller that can be easily used for nonlinear and different kinds of processes that fit the premise
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of second order model plus dead time model; which is a good approximation to the main percentage of chemical processes. Remark 1: 𝑥𝑥2(𝑛𝑛+1) can be estimated using the Taylor’s formula: 𝑥𝑥2(𝑛𝑛+1) = 𝑥𝑥2(𝑛𝑛) +
𝑑𝑑𝑑𝑑2 𝑑𝑑𝑑𝑑
𝑇𝑇 +
𝑑𝑑 2 𝑥𝑥2 𝑇𝑇 2 𝑑𝑑𝑡𝑡 2
2
+ ⋯ + 𝐶𝐶
(18)
Where, 𝐶𝐶 is the complementary term (Hildebrand, 1987). So, if the sampling time is small, 𝑥𝑥2(𝑛𝑛+1) can be approximated in one of following ways: 𝑥𝑥2(𝑛𝑛+1) ≈ 𝑥𝑥2(𝑛𝑛)
(19)
𝑥𝑥2(𝑛𝑛+1) = 𝑥𝑥2(𝑛𝑛) + 𝑥𝑥2(𝑛𝑛+1) ≈ 𝑥𝑥2(𝑛𝑛) +
𝑑𝑑𝑑𝑑2 𝑑𝑑𝑑𝑑
𝑇𝑇 ≈ 2𝑥𝑥2(𝑛𝑛) − 𝑥𝑥2 (𝑛𝑛−1)
𝑥𝑥2(𝑛𝑛) −𝑥𝑥2(𝑛𝑛−1) 𝑇𝑇
𝑇𝑇 +
(20)
𝑥𝑥2(𝑛𝑛) −2𝑥𝑥2(𝑛𝑛−1) +𝑥𝑥2(𝑛𝑛−2) 𝑇𝑇 2 𝑇𝑇
The previous model can be decomposed in two parts 1 𝐺𝐺𝑚𝑚3 − (𝑠𝑠) = (0.9173𝑠𝑠 + 1)(0.7120𝑠𝑠 + 1)
𝐺𝐺𝑚𝑚3 + (𝑠𝑠) = 𝑒𝑒 −9.95𝑠𝑠 The controller design was based on 𝐺𝐺𝑚𝑚3 − (𝑠𝑠) as process model and the procedure was the same as indicated in (17). In Fig. 2 can be seen the response of the system when the set point goes from 1 to 1.5 in time equal to 175 seconds. It is also necessary to establish certain boundaries for stability and robustness against disturbances. For this reason, the system was perturbed with magnitude 0.3 in time 100 seconds until the end of the simulation, from Fig. 2 can be seen that the controller can lead the system to the reference very fast.
2
4. SIMULATION RESULTS AND DISCUSSION
(21)
System response without Error
1.6 1.4
Magnitud
1.2
The process model will be described by the following transfer function: 1 𝑒𝑒 −9.7𝑠𝑠 𝐺𝐺𝑝𝑝 (𝑠𝑠) = (𝑠𝑠 + 1)(0.5𝑠𝑠 + 1)(0.25𝑠𝑠 + 1)(0.125𝑠𝑠 + 1)
1
Reference System Response
0.8 0.6 0.4
4.1 Guevara et al, approach
0.2
From reaction curve procedure a FOPDT model is obtained. 𝑒𝑒 −10.38𝑠𝑠 𝐺𝐺𝑚𝑚1 (𝑠𝑠) = (1.305𝑠𝑠 + 1) 𝑡𝑡 10.38 The controllability relationship is given by 0 = = 𝜏𝜏𝑚𝑚
1.305
7.95. The time delay can be replaced by a pole located in 1/10.38. Thus, the former transfer function can be modified as indicated in (Camacho et al, 2003), 1 𝐺𝐺𝑚𝑚2 − (𝑠𝑠) = (1.305𝑠𝑠 + 1)(10.38𝑠𝑠 + 1) Following the procedure as was described in (Guevara et al, 2016), the best results obtained can be seen in the following figure: Original Controller
2 1.8 1.6 1.4
0 0
50
100
150
Time (s)
200
250
300
Fig 2. System response without error. It also can be seen that the system reacts to the events in a time instant equal to the delay, following the setpoint changes and rejecting the constant disturbances with zero error. To test the robustness of the system 1000 tests were calculated through Monte Carlo simulations as in (Serrano et al, 2016), varying the parameters kA, kB y km with a uniform distribution. Results can be seen in Fig. 3. From Fig. 3 can be seen that system follows the reference signal even with a constant disturbance in presence of modeling errors. The proposed control structure allows following different continuous trajectories for each change in the set point.
Magnitude
1.2 1
MC Set Point Changes
1.8
0.8
1.6
System Response Reference
0.6 0.4
1.4
0.2
50
100
150
Time (s)
200
250
1.2
300
Magnitud
0 0
Fig.1 Original Controller (Guevara et al, 2016)
1 0.8 0.6
4.2 Proposed approach
0.4
For this part, the system is described, as was mentioned in the previous section, by a second order model plus time delay and using in the scheme shown in figure 1. The second order plus time delay representation of the process is: 𝑒𝑒 −9.95𝑠𝑠 𝐺𝐺𝑚𝑚3 (𝑠𝑠) = (0.9173𝑠𝑠 + 1)(0.7120𝑠𝑠 + 1)
0.2 0 0
50
100
150
Time (s)
200
250
300
Fig.3 System responses against set point changes and disturbances considering parameter errors.
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1.4
Constant Set point - Advanced reference
1.6 1.4 1.2
Magnitude
Fig. 4 shows the system response when the set point is a continuous signal variant in time and also it is present a load disturbance, the system is capable of following the reference signal. If the reference is known each time instant, it is possible to advance the reference. The system response with an advanced reference can be seen in Fig. 5 and Fig. 6, with variable and constant set points respectively, also the system follows the reference signal in a precise way without the delay shown in Fig. 2 and 4. The closed loop system reacts to the disturbance because of neither the magnitude nor the time instant are known.
3079
1
Advanced Reference
0.8
System response
0.6 0.4 0.2
Variable reference - without advance
0 0
50
100
150
Time (s)
200
250
300
1.2
Fig.6 System response advancing the reference signal and constant disturbance.
Magnitud
1 0.8
1.8 0.6
Reference System response
1.4
0.2
50
100
150
200
250
Magnitude
1.2 300
Time (s)
Fig.4 System response against a variable reference and constant disturbance.
1 0.8 0.6 0.4
Variable Reference - Advanced Reference
1.4
0.2 0 0
1.2 1
Magnitude
Linear Algebra Controller Variable reference PID controller
1.6
0.4
0 0
Closed loop with Linear Algebra and PID Controllers
100
200
300
400 500 Time (s)
600
700
800
900
Fig.7 Comparison of the DTC with a PID controller and a LA controller following a variable reference.
0.8 0.6
System response Reference
Fig. 7 presents the results of the DTC with a PID controller, tuned with Dahlin formula, and the DTC with a LA controller following a variable reference. It can be seen that the DTC-LA follows properly the desired trajectory. In order to compare the performance of the proposed architecture, we used the ISE criterion: ISE_DTC-PID = 52.9709 and ISE_DTC-LA controller = 26.016. This result shows that the proposed scheme overcome the DTC-PID over 100%.
which leads to an easy design with better and faster control actions. The scheme showed accurate results and presented a good performance against disturbances of big magnitude. Also, the robustness of the proposed controller was achieved through Monte Carlo simulations to find the accurate boundary for the parameters. This controller is a good option for elevated dead time systems. Therefore, it can be used for processes with a controllability relationship greater than two. The proposed approach can track variable time references, therefore it can be applied in robotics, teleoperation systems, bioprocesses, chemical processes and so on. The proposed approach was compared against a typical deadtime compensator for the same kind of systems and presented better results. The deadtime compensator based on linear algebra presents an easy algorithm which does it very suitable for industrial implementation.
5. CONCLUSIONS
ACKNOWLEDGMENTS
In this work was proposed a dead time compensator that combines the Smith Predictor control architecture and the Linear algebra controller methodology. For designing purposes, a second order plus dead time model was used. To get the controller only the invertible part of the model was used
Authors want to acknowledge the Research Office from Universidad EAFIT for their support to join the conference. OC thanks to PROMETEO Project of SENESCYT, Republic of Ecuador, for its support for the realization of this work. This work was partially funded by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET),
0.4 0.2 0 0
50
100
150
200
250
300
Time (s)
Fig.5 System response advancing the reference signal and constant disturbance.
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Argentina. OLQ wants to acknowledge Hèlida for her incomparable support during her academic career. REFERENCES Amicarelli, A, Quintero Montoya O. L and Di Sciascio, F. (2016). “Substrate Feeding Strategy Integrated with a Biomass Bayesian Estimator for a Biotechnological Process. Int. J. Chem. React. Eng.; aop. DOI 10.1515/ijcre2015-0182 Amicarelli A., Quintero O., Sciascio F., (2014). Behavior comparison for biomass observers in batch processes. Asia‐ Pacific Journal of Chemical Engineering 9, 81–92 Astrom KJ., Hang CC., Lim, BC. (1994). A new Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control; 39(2):343–5. Camacho, O. (2002). “A Predictive Approach Based-Sliding Mode Control”. 15th Triennial IFAC World Congress, Barcelona, Spain Camacho, O., Smith, C., Moreno, W. (2003). “Development of an Internal Model Sliding Mode Controller”. Industrial and Engineering Chemistry Research 42 (3), 568-573 Camacho O., De la Cruz, F. (2004). “Smith Predictor basedsliding mode Controller for Integrating Processes with Elevated Deadtime” ISA transactions 43 (2), 257-270 Camacho, O. Rojas, R., García-Gabín, W. (2007). “Some long time delay Sliding Mode Control Approaches”. ISA Transactions 46 (1), 95-101 Capito L., Proaño P., Camacho O., Rosales A., Scaglia G. (2016). “Experimental comparison of control strategies for trajectory tracking for mobile robots,” International Journal of Control and Automation (in press), 2016. di Sciascio, F., Quintero, O.L., Alvarez, H. (2004). Estructura de Dos Grados de Libertad y Sensor Virtual para Controlar un Bioproceso. Conference Paper . Conference: XIX Congreso Argentino de Control Automático, AADECA'2004, At Buenos Aires, Argentina Francis, B. A. and Wonham, W. M., (1976). The internal model principle of control theory, Automatica, Vol 12, pp. 457-465 Pergamon press 1976. Guevara, L., Guevara, J., Scaglia, G., Camacho, O. and Rosales, A. (2016). "A New Approach of a Numerical Methods Controller for Self-Regulating Processes," DOI: 10.1109/ARGENCON.2016.7585282. Marlin T. E. (1995). Process control. New York: McGrawHill. Quintero, O.L., Scaglia, G., Amicarelli, A., di Sciascio F. (2008). “Bio process control strategy based on numerical methods and linear algebra: second approach,” in Proc. 27th IASTED International Conference on Modelling, Identification, and Control, MIC At Innsbruck, Austria Quintero, O.L., Scaglia, G., di Sciascio F., Mut, V. (2008). Numerical methods based strategy and particle filter state estimation for bio process control. Conference Industrial Technology, ICIT 2008. IEEE International Conference on May 2008 DOI: 10.1109/ICIT.2008.4608611 · Source: IEEE Xplore. Quintero O., Amicarelli A., di Sciascio F. (2012). Biomass Estimation in Batch Process through Particle Filters. XV
Congreso Latinoamericano de Control Automático – CLCA 2012, At Perú-Lima Quintero O., Nieto J., Amicarelli A., Scaglia G., Luna T., di Sciascio F., (2008). Control engineering perspective of fermen- tation process from zymomonas mobilis: modeling, state estimation and control. XIII Latin American conference on automatic control, Venezuela. Quintero O.L., Amicarelli A.A., Di Sciascio F., Scaglia G., (2008). State estimation in alcoholic continuous fermentation of Zymomonas mobilis using recursive bayesian filtering: A simulation approach. BioResources 3, 316–34. Quintero O.L., Amicarelli A.A., Scaglia G., di Sciascio F., (2009). Control based on numerical methods and recursive Bayesian estimation in a continuous alcoholic fermentation process. BioResources 4, 1372–95. Rosales, Claudio., Gandolfo, Daniel., Scaglia, Gustavo., Jordan, Mario., Carelli, Ricardo. Trajectory tracking of a mini four-rotor helicopter in dynamic environments - a linear algebra approach. Robotica. doi:10.1017/S0263574714000952. ISSN:0263-5747. Robotica / Volume 33 / Issue 08 / October 2015, pp 16281652 Serrano, Mario E., Scaglia, Gustavo., Godoy, Sebastian., Mut Vicente., and Ortiz, Oscar (2014), Trajectory Tracking of Underactuated Surface Vessels: A Linear Algebra Approach, IEE transactions on control systems technology, doi: 10.1109/TCST.2013.2271505. Volumen 22, Issues 3, pp 1103 – 1111, 18 abril 2014. Serrano, M.E., Godoy, S.A., Quintero, L. Scaglia, G. (2016). Interpolation Based Controller for Trajectory Tracking in Mobile Robots. Journal of Intelligent & Robotic Systems, 1-13 DOI 10.1007/s10846-016-0422-4. Scaglia, G., Mut V., and F. Di Sciascio, (2006). “Trajectory Control Of Mobile Robots Based On Numerical Methods,” in XII Congreso Latinoamericano de Control Automático. Scaglia Gustavo, Quintero Olga Lucía, Mut Vicente,di Sciascio Fernando (2009). “Numerical Methods Based Controller Design for Mobile Robots”. Robotica, Volume 27, Issue 02, Mar 2009, pp 269-279 doi: 10.1017/S0263574708004669, Published online by Cambridge University Press 23 Jun 2008 ISSN 0263-5747. Scaglia, Gustavo., Serrano, Emanuel., Rosales, Andrés., Albertos, Pedro. (2015). Linear interpolation based controller design for trajectory tracking under uncertainties: Application to mobile robots, Control Engineering Practice, Volume 45, December 2015, Pages 123-132, ISSN 0967-0661, http://dx.doi.org/10.1016/j.conengprac.2015.09.010. (http://www.sciencedirect.com/science/article/pii/S09670 66115300216) Smith C., and A. Corripio, (1997). Principles and practice of automatic process control, New York: John Wiley & Sons, Inc. Watanabe K, Ito M. A process-model control for linear systems with delay. IEEE Transactions on Automatic Control 1981; 26(6):1261–9. Zhang WD, Sun YX. Modified Smith predictor for controlling integrator/time delay processes. Industrial Engineering Chemistry Research 1996;35:2769–72.
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A Dead Time Compensator Based on Linear Algebra (DTCLA) IFAC PapersOnLine 50-1 (2017) 3075–3080 A Dead Time Compensator Based on Linear Algebra (DTCLA) 1,2 3 A Dead Time Compensator Based on Algebra Camacho , Gustavo , O. Lucia Quintero4 (DTCLA) A Dead Time Oscar Compensator BasedScaglia on Linear Linear Algebra (DTCLA)
3 Oscar Camacho1,2, Gustavo Scaglia , O. Lucia Quintero4 1,2 3 4 1,2 3 4 Oscar Camacho 1,2, Gustavo Scaglia3, O. Lucia Quintero4 Oscar Camacho , Gustavo Scaglia , O. Lucia Quintero 1 Universidad de Los Andes, Mérida, Venezuela, (e-mail: [email protected]) 1 2 Universidad de Nacional, Los Andes,Quito, Mérida, Venezuela, (e-mail: [email protected]) Escuela Politécnica Ecuador, (e-mail: [email protected]) 1 2 3 1 Escuela Politécnica Nacional, Quito, Ecuador, (e-mail: [email protected]) Universidad de Los Andes, Mérida, Venezuela, (e-mail: [email protected]) 1 Universidad Nacional de SanMérida, Juan, CONICET, Universidad de Los Andes, Venezuela,(e-mail: (e-mail:[email protected]) [email protected]) 22 3 Universidad 4 (e-mail: Nacional de San Juan, CONICET, (e-mail: [email protected]) Politécnica Nacional, Quito, Ecuador, [email protected]) 2 Escuela Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Escuela Politécnica Nacional, Quito, Ecuador, (e-mail: [email protected]) 3 4 3 Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Nacional 3 Universidad Universidad Nacional de de San San Juan, Juan, CONICET, CONICET, (e-mail: (e-mail: [email protected]) [email protected]) 4 4 4 Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Universidad EAFIT, Medellín, Colombia, (e-mail: [email protected]) Abstract: This paper proposes a dead time compensator based on a combination of the linear algebra control Abstract: This(proposed paper proposes a dead time compensator based on a combination of the linear algebratocontrol methodology by Scaglia, 2009) and an internal model structure (Smith predictor), solve Abstract: This paper proposes a dead time compensator based on a combination of the linear algebratocontrol methodology (proposed by Scaglia, 2009) and an internal model structure (Smith predictor), solve elevated dead time system problems. Moreover, the process transfer function of the nonlinear system Abstract: This paper proposes a dead time compensator based on a combination of the linear algebra control methodology (proposed by problems. Scaglia, 2009) and anthe internal model structure predictor), tosystem solve elevated dead timeoperation system Moreover, transfer function of transfer the nonlinear linearized at some point is approximated aprocess second order plus dead(Smith time function. The methodology (proposed by Scaglia, 2009) and anbyinternal model structure (Smith predictor), to solve linearized at some operation point is approximated by a second order plus dead time transfer function. The elevated dead time system problems. Moreover, the process transfer function of the nonlinear system controller performance is judged by simulations and it is evaluated using the ISE performance index. We elevated dead time system problems. Moreover, the process transfer function of the nonlinear system controller performance isagainst judged by simulations and ita is evaluated using the results. ISE performance index. The We linearized at some operation point is approximated by second order plus dead time transfer function. compared this approach a typical DTC-PID controller with improved linearized at some operation point is approximated by a second order plus dead time transfer function. The compared performance this approachisagainst a by typical DTC-PID controller with improved results. controller judged simulations and it is evaluated using the ISE performance index. controller performance is judged by simulations andControl) it is evaluated using the ISELtd. performance index. We We Keywords: dead time systems, linear algebra, chemical processes, internal model, and linear systems. © 2017, IFAC (International Federation of DTC-PID Automatic Hosting by Elsevier All rights reserved. compared this approach against a typical controller with improved results. Keywords:this dead time systems, algebra, chemical processes, model, and linear systems. compared approach againstlinear a typical DTC-PID controller withinternal improved results. Keywords: time processes, internal model, and systems. Keywords:1.dead dead time systems, systems, linear linear algebra, algebra, chemical chemical processes, internal model,damped and linear linear systems. INTRODUCTION very oscillatory or highly when the process has a large 1. INTRODUCTION very oscillatory or highly damped when thewith process a large time delay (Astrom et al, 1994). To deal this has additional very oscillatory or highly damped when the process has a large time delay (Astrom et al, 1994). To deal with this additional 1. INTRODUCTION new or structures were when proposed, decoupling the The presence of time1.delays in many industrial processes is a problem INTRODUCTION very oscillatory highly damped the process has a large time delay (Astrom et al, 1994). To deal with this additional problem new structures were proposed, decoupling the The presence of time delays in many industrial processes is a disturbance from the et setal, point response (Zhang 1996) well-recognized problem. Time lag, transportation lag, time time delay (Astrom 1994). To deal with and thisSun, additional problem new structures were proposed, decoupling the disturbance from the set point response (Zhang and Sun, 1996) The presence of time delays in many industrial processes is a well-recognized problem. Time lag, transportation lag, time (Astrom etnew al, 1994). In general, approaches have some delaypresence and dead time delays are common in industrial structures were these proposed, decoupling the The of time in manyphenomena industrial processes is a problem (Astrom et al, 1994). In point general, approaches have some from the set response (Zhang and Sun, 1996) well-recognized problem. Time lag,phenomena transportation lag, time delay and Time dead delay time are in industrial problems: sensitive to these modelling since the processes. can common be produced by measurement lag, disturbance disturbancethey fromare the set point response (Zhangerrors and Sun, 1996) well-recognized problem. Time lag, transportation lag, time (Astrom et they al, 1994). In general, these approaches have some problems: are sensitive to modelling errors since the delay andand dead time are phenomena inlag industrial processes. Time delay can common be produced by measurement lag, design requires the use of a process model, which can be analysis computation time, communication or the (Astrom et al, 1994). In general, these approaches have some delay and dead time are common phenomena in industrial problems: they are sensitive to modelling errors since requires the use of aModelling process model, which can the be processes. Time delay can be produced by measurement lag, analysis and computation time, communication lag orAlso, the design difficult to obtain in practice. errors are unavoidable transport time required for a fluid to flow through a pipe. problems: they are sensitive to modelling errors since the processes. Time delay can be produced by measurement lag, design requires the use of a process model, which can be difficult to obtain in practice. Modelling errors are unavoidable analysis and computation time, communication lag or the transport time required for a fluid to flow through a pipe. Also, and theyrequires result inthe a mismatch modelwhich and thecan actual several concerntime, with dynamics and certain delays design use of abetween processthe model, be analysisbioprocesses and computation communication lag or the difficult to obtain practice.designed Modelling errors areand unavoidable and they result incontrollers ainmismatch between the model themodels actual transport time required for a fluid flow through a pipe.adelays Also, several bioprocesses concern withto dynamics and certain plant. Thus, the using particular related to the capability of microorganisms of reaching good difficult to obtain in practice. Modelling errors are unavoidable transport time required for a fluid to flow through a pipe. Also, and they result a mismatch between the model and themodels actual Thus, theincontrollers designed using several bioprocesses concern with dynamics and certain related to theorcapability ofdue microorganisms of reaching adelays good plant. may perform differently when they areparticular implemented on growth rate inhibition to the concentration of product and they resultquite in a mismatch between the model and the actual several bioprocesses concern with dynamics and certain delays plant. may perform quite differently when they are implemented on Thus, the controllers designed using particular models related to the capability of microorganisms of reaching a good growth rate or inhibition due to the concentration of product the actual process (Quintero et al, 2009). and substrate (Quintero et al, 2008) (Amicarelli et al, 2016). plant. Thus, the controllers designed using particular models related to the capability of microorganisms of reaching a good may perform quite differently when they are implemented on the actual process (Quintero et al, 2009). growth rate or inhibition due to the concentration of product and substrate (Quintero et al, 2008) (Amicarelli et al, 2016). Certainly, of some of the the concentration variables for these kinds may perform quite differently when they are implemented on growth ratetheorcontrol inhibition due to of product In addition, the methodology based on linear algebra and actual process (Quintero et al, 2009). and substrate (Quintero al, 2008) etthese al, 2016). Certainly, the control of et some of the (Amicarelli variables kinds of systems must be performed Nonlinearfor Point view. the In addition, the methodology based on linear algebra and the actual process (Quintero ettoal, 2009). and substrate (Quintero et al, from 2008)a (Amicarelli et al,of2016). numerical methods concepts design control algorithms is a Certainly, the control of we some of thea variables for these kinds of systems must be2009 performed from Nonlinear Point of view. In Quintero al, demonstrate that Linear Algebra addition, the methodology based on linear algebra and numerical methods concepts to design control algorithms isofa Certainly, theetcontrol of some of the variables for these kinds In simple and relatively new method that allows the control of systems must be2009 performed from a Nonlinear Point of view. In addition, the methodology based on linear algebra and In Quintero et al, that Linear Algebra based controllers can, we in demonstrate an easily and understandable methods concepts to design control algorithms is simple and relatively new method that allows the control ofa of systems must be performed from a Nonlinear Point of view. numerical highly nonlinear systems. It has been applied the designisof In Quintero et al, 2009 demonstrate that tracking Linear Algebra based controllers can, in antheeasily and understandable numerical methods concepts to design controltoalgorithms a formulation, perform notwe only trajectory control simple and relatively new method that allows the control highly nonlinear systems. It has been applied to the design of In Quintero et al, 2009 we demonstrate that Linear Algebra differentand class of control systems such asallows trajectory tracking of based controllers can, in an easily and understandable formulation, perform not only the trajectory tracking control simple relatively new method that the control of but alsocontrollers the positioning point.and Butunderstandable the controller highly different class and of control such as2016), trajectory nonlinear systems. It has been to thetracking design of based can, at in certain an easily mobile robots UAV’ssystems (Capito et al,applied chemical plants formulation, not at only the atrajectory control but alsomust the perform positioning certain point. Buttracking the controller highly nonlinear systems. It has been applied to the design of design be performed under zero error model of the different class and of control such trajectory tracking of mobile robots UAV’ssystems (Capito et al,as 2016), chemical plants formulation, perform not only the trajectory tracking control and bioprocesses (Quintero, et al 2009, Scaglia et al, 2015) just but also the positioning at certain point. But the controller design must be performed under a zero error model of the different class of control systems such as trajectory tracking of nonlinear system, in certain cases too muchBut complicated and mobile robots and UAV’s (Capito et al, Scaglia 2016), chemical plants and bioprocesses (Quintero, etIts al 2009, et al,is2015) just but also the positioning at certain point. the controller to name some applications. main advantage that the design must be performed under a zero error model of the nonlinear system, in certain cases too much complicated and mobile robots and UAV’s (Capito et al, 2016), chemical plants highly must nonlinear. Some approaches wereerror alsomodel explored to and to name some applications. Its main advantage is2015) that the bioprocesses (Quintero, al 2009, et al, just design be performed under a zero of the for the trackinget tendScaglia to zero and control nonlinear certain cases too much also complicated highly Some approaches were to conditions and bioprocesses (Quintero, eterror al 2009, Scaglia et al, 2015) just control nonlinear. thissystem, kind ofin Sciascio et al,explored 2004). and conditions for the tracking error tend to zero and control name some applications. Its main advantage is that the nonlinear system, inbioprocesses certain cases(ditoo much complicated and to obtained with low computational cost which makes highly approaches were also control nonlinear. this kind ofSome bioprocesses (di Sciascio et al,explored 2004). to actions to namearesome applications. Its main advantage is that the actions are obtained with low computational cost which makes for the tracking error tend to zeroconditions and control highly nonlinear. Some approaches were also explored to conditions it easy to implement in a microcontroller. These are On the other hand, the achievable performance of typical control this kind of bioprocesses (di Sciascio et al, 2004). conditions for the tracking error tend to zero and control it easyby to implement in a microcontroller. These are are obtained with low computational cost which makes On the this other hand, the achievable performance of typical control kind ofsystems bioprocesses Sciascio al, 2004). found solving a system of linear equations andconditions conditioning feedback control can (di decline if aetprocess has a actions actions are obtained with low computational cost which makes found by solving a system of linear equations and conditioning it easy to implement in a microcontroller. These conditions are On the other hand, the achievable performance of typical feedback control systems can decline if a process has a theeasy system to havein exact solution. For all conditions the previous relatively largehand, time the delay comparedperformance to the dominant time it to implement a microcontroller. These are On the other achievable of typical by solving a system of linear andby conditioning the system tothehave exact solution. For all the previous feedback control systems decline process has relatively large time delay can compared toifthea dominant timea found applications, controllers wereequations designed using the constant (Smith Corripio, 1997). Predictive structures found by solving a system of linear equations and conditioning feedback controlandsystems can decline if a process hasanda the controllers were designed by system tothehave solution. For all(Quintero the using previous relatively large timeCorripio, delay compared to the(Camacho, dominant constant (Smith and 1997). Predictive structurestime and complete nonlinear andexact discrete process model etthe al, sliding mode controllers (Camacho, al, applications, the system to have exact solution. For all the previous relatively large time delay compared2002), to the dominant ettime complete nonlinear and discrete process model (Quintero al, the controllers were designed by usingetthe constant (Smith andused Corripio, 1997). Predictive structures sliding mode controllers (Camacho, 2002), (Camacho, etand al, applications, 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the 2007) have been to solve such problems. Primarily, applications, the controllers were designed by using the constant (Smith and Corripio, 1997). Predictive structures and complete 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the nonlinear and discrete process model (Quintero et al, sliding mode controllers (Camacho, 2002), (Camacho, et al, 2007) have been used to solve such problems. Primarily, controller algorithm applied toetthe internal modelcontrollers control (IMC) and the 2002), Smith predictor (SP) complete mathematical nonlinear and discrete process modelis (Quintero al, sliding mode (Camacho, (Camacho, et are al, obtained obtained mathematical controller algorithm is Therefore, applied to the Scaglia et al, 2009, Serrano et al, 2016). internal model control and the Smith predictor (SP) are 2009; 2007) have been used(IMC) to solve such problems. Primarily, specific process. Hence, for each different process a different the most popular predictive structures used for time delay 2009; Scaglia et al, 2009, Serrano et al, 2016). Therefore, the 2007) have been used to solve such problems. Primarily, obtained process. for each different process a different mathematical controller algorithm isit applied to thea internal control (IMC) the Smith predictor (SP) are specific the mostmodel popular predictive structures used for time delay controller should Hence, be obtained, in other words, is generated compensation (Marlin, 1991,and and Corripio, 1997). obtained mathematical controller algorithm is applied to the internal model control (IMC) andSmith the Smith predictor (SP) are specific controller should be obtained, in other words, it is generated process. Hence, for each different process a different the most popular predictive structures used for time delay compensation (Marlin, 1991, Smith and Corripio, 1997). differentprocess. control law in each case. different process a differenta Furthermore, whenpredictive the processstructures presents an integral behaviour specific Hence, for each the most popular used for time delay controller control in each case. shouldlaw be obtained, in other words, it is generated a compensation (Marlin, 1991, be Smith Corripio, 1997). Furthermore, when the process presents an integral behaviour the original structures used and since a constant load different controller should be obtained, in with otherthe words, generated compensation (Marlin,cannot 1991, Smith and Corripio, 1997). Besides, there arelaw two problems use ofit aismodel as fara different control in each case. Furthermore, when the process presents an integral behaviour the original structures cannot be used since a constant load disturbance results erroran (Camacho and De la different Besides, there are two problems with the use of a model as far control law in each case. Furthermore, when in thea steady-state process presents integral behaviour the original structures used a constant load disturbance in a cannot steady-state errorsince (Camacho and De la as industrial processes are concerned. First, the development Cruz, 2004).results To overcome thisbe obstacle different approaches as industrial processes are concerned. First, the development there are two problems with the use of a model as far the original structures cannot be used since a constant load Besides, a complete difficult due complexity Cruz, 2004).proposed To overcome this obstacle approaches disturbance results in a steady-state error different (Camacho andstudies De la of Besides, there model are twoisproblems withmainly the usetoofthe a model as far have been Simulation of a complete model is difficult due mainly to the complexity industrial processes are concerned. First, the development disturbance results in a(Watanabe, steady-state1981). error (Camacho and De la as of industrial the process itself, and the lack of knowledge of some Cruz, 2004).proposed To the overcome this obstacle different approaches have been 1981). Simulation as processes aretoconcerned. First, the development have set(Watanabe, point and load disturbances arestudies either of processmodel itself, isand to thedue lackmainly of knowledge of some athe complete difficult to the complexity Cruz,shown 2004).that To overcome this obstacle different approaches have shown been proposed 1981). Simulationarestudies that the set(Watanabe, point and load disturbances either of a complete model is difficult due mainly to the complexity have been proposed (Watanabe, 1981). Simulation studies of the process itself, and to the lack of knowledge of some have shown that the set point and load disturbances are either of the process itself, and to the lack of knowledge of some have shown thatIFAC the set point and load disturbances are either3130 Copyright © 2017
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Proceedings of the 20th IFAC World Congress 3076 Oscar Camacho et al. / IFAC PapersOnLine 50-1 (2017) 3075–3080 Toulouse, France, July 9-14, 2017
process parameters. Second, most process models relating the controlled and the manipulated variables are of higher-order. Therefore, the traditional numerical controller procedure can produce more complex controllers and also their application is just for the process of analysis. An efficient alternative modelling method for process control is the use of empirical models, which use low-order linear models with dead time (Marlin, 1991; Smith and Corripio, 1997). Most times, first-order-plus dead-time (FOPDT) models are adequate for process control analysis and design. In many cases, mobile robotics as well as chemical processes, can be represented by first-order plus dead time (FOPDT) models (Capito, et al 2016). The internal model control design is based on the idea of forming an ideal and simple controller based on the transfer function of the process model, and it has presented satisfactory results, as in (Camacho et al 2002). When controlling chemical processes, it is important to simplify the process model, because these processes are often represented by several state space equations, and then the controllers must be resistant to modelling errors and uncertainties. Guevara et al, 2016 designed a Linear Algebra (and Numerical Methods) based controller (NMCr) designed on a first order plus dead time (FOPDT) model of the actual process. The overall idea is to develop a general NMCr, which can be used for different self-regulating industrial processes, if they have a similar open loop response, such as an FOPDT model. Hence, this work summarizes the easily and simplicity of numerical methods procedures along to a reduced order model of the process to obtain a simple and versatile controller. The performance of the proposed controller in this article is tested by simulations, but the performance degrades when the 𝑡𝑡 controllability relationship ( 0) increases. 𝜏𝜏
This paper proposes from a Smith Predictor structure the synthesis of a Deadtime compensator that uses the linear algebra methodology. This new approach uses a second order plus dead-time model obtained from the reaction curve of the nonlinear process. This novel proposal has the capability of solving elevated dead time problems. The proposed scheme is tested and compared against the previous approach built from reduced first order plus dead time model of the process (Guevara et al, 2016). This work is organized as follows: section 2 presents the fundamentals of the linear algebra based controllers and IMC control and section 3 explains the Dead Time Compensator Based on Linear Algebra (DTCLA). Then section 4, presents some study cases where the DTCLA scheme is presented and compared with a regular DTC-PID. Finally, conclusions are presented in section 5. 2. BASIC CONCEPTS This section presents briefly some basics of the technique developed by Scaglia in 2009 and then basic ideas of Internal model control (Francis and Wonham, 1976). These concepts are the foundation for the approach that is presented in section 3. It is important to mention that the internal model structure is used as a Smith predictor.
2.1 Linear Algebra methodology The numerical methods and linear algebra tracking control methodology has been proved over highly nonlinear systems for bioprocesses (Quintero et al, 2009) (Scaglia et al, 2009, 2014), and also in under-actuated mechanical systems, dynamic nonlinear systems controlled with a kinematic model for tracking and positioning control tasks for fast and slow dynamics systems such as mobile robots (Scaglia et al, 2009, 2010, 2015; Serrano et al 2014, Rosales et al 2015). Stability and robustness demonstrations can be found in (Scaglia et al, 2010, 2015) and parameters optimization through Monte Carlo methods (Serrano et al, 2016). The methodology for calculating the control actions can be summarized in the following steps: 1.
To describe the system using state equations and then approximate them using a numerical method.
2.
To set out the control actions calculation as a problem of solving a system of linear equations.
3.
To choose the state variables to be followed and under what conditions the system of linear equations, of the previous point, has exact solution, and then to define the reference trajectory of the remaining state variables.
4.
To solve the system of equations.
2.2 Internal Model Structure as Smith Predictor This controller structure can be seen in Francis and Wonham, 1976. In this structure 𝐺𝐺𝑃𝑃 (𝑠𝑠) represents the process, 𝐺𝐺𝐶𝐶 (𝑠𝑠) is the controller transfer function. 𝐺𝐺𝑚𝑚 (𝑠𝑠) is the process model, em represents the modelling error, 𝑦𝑦𝑦𝑦 is the process output, 𝑦𝑦𝑦𝑦 is − is the output of the non-invertible the model output and 𝑦𝑦𝑚𝑚 part of the model. To implement it, it is necessary to obtain the process model and separate it into two parts: a non-invertible 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ and an invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠)− , as shown in (1). 𝐺𝐺𝑚𝑚 (𝑠𝑠) = 𝐺𝐺𝑚𝑚 (𝑠𝑠)− 𝐺𝐺𝑚𝑚 (𝑠𝑠)+
(1)
To obtain this controller, only the invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠) is used, because the non-invertible part 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ can present problems such as being unstable, not causal or cannot be found. −
3. A DEAD TIME COMPENSATOR BASED ON LINEAR ALGEBRA (DTCLA) In this section is shown the development of the proposed Dead Time Compensator Based on Linear Algebra (DTCLA). This proposal is based on the improvement of the control scheme by (Guevara et al, 2016) in which the linear algebra controller design is based on a first order plus dead time model of the system. For this new approach, it is assumed that the process presents an overdamped dynamics and the parameters of a transfer function are obtained from the reaction curve method identification (Smith and Corripio, 1997). The process model is represented as a second order system plus delay transfer function, which is divided into two components to be added to the control scheme (Astrom et al, 1994).
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In figure 1, it is represented the proposed scheme, it can be seen that the controller based on linear algebra needs as the signal to its calculations the output of the invertible part of the transfer function from the internal model structure in a similar way as in the Smith predictor architecture. Therefore, the Linear Algebra method is applied to the part of the reduced model without time delay.
3077
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 = 𝜒𝜒̇ 2 + 𝐾𝐾𝐴𝐴 𝜒𝜒2 + 𝐾𝐾𝐵𝐵 𝜒𝜒1
(10)
Applying Euler approximation to discretise the derivative of the state variables, the following equations are obtained:
𝜒𝜒2 𝑛𝑛 =
𝜒𝜒1
𝑛𝑛+1 −𝜒𝜒1 𝑛𝑛
𝑇𝑇
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 𝑛𝑛 =
𝜒𝜒2
(11)
𝑛𝑛+1 −𝜒𝜒2 𝑛𝑛
𝑇𝑇
+ 𝐾𝐾𝐴𝐴 𝜒𝜒2 𝑛𝑛 + 𝐾𝐾𝐵𝐵 𝜒𝜒1 𝑛𝑛
(12)
Where, 𝑇𝑇 is the sampling time. The sampling time for the discrete controller will be in terms of the time constants of 𝜏𝜏𝑚𝑚1 and 𝜏𝜏𝑚𝑚2 . min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 )/15 < 𝑇𝑇 < min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 )/4
Fig.1 Proposed scheme for the DTCLA. First of all, let us show the model of the process, the system model to approximate the process is described by (2): 𝐺𝐺𝑚𝑚 (𝑠𝑠) = 𝑘𝑘𝑚𝑚
𝑒𝑒 −𝑡𝑡0 𝑠𝑠
(2)
(𝜏𝜏𝑚𝑚1 𝑠𝑠+1)(𝜏𝜏𝑚𝑚2 𝑠𝑠+1)
Where: − (s) = (𝜏𝜏 𝐺𝐺𝑚𝑚
𝑘𝑘𝑚𝑚
𝑚𝑚1 𝑠𝑠+1)(𝜏𝜏𝑚𝑚2 𝑠𝑠+1)
; 𝐺𝐺𝑚𝑚 (𝑠𝑠)+ = 𝑒𝑒 −𝑡𝑡0𝑠𝑠
(3)
Where: 𝑘𝑘𝑚𝑚 : Static gain. 𝜏𝜏𝑚𝑚1 : Time constant 1. 𝜏𝜏𝑚𝑚2 : Time constant 2. 𝑡𝑡0 : Dead time. − Rearranging some terms in Eq. (2), the invertible part 𝐺𝐺𝑚𝑚 (s), can be represented as: − (s) = 𝐺𝐺𝑚𝑚
− (𝑠𝑠) 𝑦𝑦𝑚𝑚
𝑢𝑢(𝑠𝑠)
=
𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑠𝑠 2 +𝐾𝐾𝐴𝐴 𝑠𝑠+𝐾𝐾𝐵𝐵
(4)
Where: 𝐾𝐾𝐴𝐴 =
𝐾𝐾𝐵𝐵 =
𝜏𝜏𝑚𝑚1 +𝜏𝜏𝑚𝑚2 𝜏𝜏𝑚𝑚1 𝜏𝜏𝑚𝑚2
(5)
𝜏𝜏𝑚𝑚1 𝜏𝜏𝑚𝑚2
1
(6)
𝜒𝜒̈ + 𝐾𝐾𝐴𝐴 𝜒𝜒̇ + 𝐾𝐾𝐵𝐵 𝜒𝜒 = 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢
(7)
Representing this second order equation in state variables form, the following expression is obtained: 0 𝜒𝜒̇ [ 1 ] = [−𝐾𝐾 𝐵𝐵 𝜒𝜒̇2
1 𝜒𝜒1 0 −𝐾𝐾𝐴𝐴 ] [ 𝜒𝜒2 ] + [ 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 ] [𝑢𝑢]
(8)
If a faster response is required, the value of 𝐾𝐾𝑥𝑥 should be close to 0 and if a slower response is required, the value should be close to 1. To introduce this parameter into the algorithm is required change the nomenclature of (11) to have a variation of 𝐾𝐾𝑥𝑥 related with the difference between the reference and output value (error). Having the next resulting expression with reference terms:
𝜒𝜒2 𝑛𝑛 =
(9)
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 −𝜒𝜒1 𝑛𝑛 −𝐾𝐾𝑥𝑥 (𝜒𝜒1 𝑇𝑇
𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 −𝜒𝜒 𝑛𝑛 )
(14)
Where:
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 : is the reference. 𝜒𝜒1 𝑛𝑛 : is the measured output.
To reduce the resulting expressions, the difference between the reference 𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛 and the actual output 𝜒𝜒 𝑛𝑛 will be represented by 𝑒𝑒 𝑛𝑛 . Assuming that the current value of variable 𝜒𝜒2 𝑛𝑛 is the same required for a next discrete time 𝜒𝜒2 𝑛𝑛+1 ( see remark 1). Then it can replace (14) in (12) to obtain: 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑢𝑢 𝑛𝑛 =
𝜒𝜒1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 − 𝜒𝜒1 𝑛𝑛 − 𝐾𝐾𝑥𝑥 𝑒𝑒 𝑛𝑛 𝑇𝑇 2
𝐾𝐾𝐵𝐵 𝜒𝜒1 𝑛𝑛
−
𝜒𝜒2 𝑛𝑛 𝑇𝑇
+ 𝐾𝐾𝐴𝐴 𝜒𝜒2 𝑛𝑛 +
(15)
Finally, from previous equation, the control law 𝑢𝑢 𝑛𝑛 , is obtained: 1 [𝜒𝜒 − 𝜒𝜒1 𝑛𝑛 + 𝐾𝐾𝑥𝑥 𝑒𝑒 𝑛𝑛 − 𝑇𝑇 𝜒𝜒2 𝑛𝑛 ] + 𝑢𝑢 𝑛𝑛 = 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵 𝑇𝑇 2 1 𝑟𝑟𝑟𝑟𝑟𝑟 𝑛𝑛+1 𝜒𝜒1 𝑛𝑛 𝑘𝑘𝑚𝑚
Where:
Where 𝜒𝜒1 and 𝜒𝜒2 are the state variables of the system, being the derivate of the controlled variable and u is the controller output. From the state variables representation, the following equations are obtained:
𝜒𝜒̇ 1 = 𝜒𝜒2
Where min(𝜏𝜏𝑚𝑚1 , 𝜏𝜏𝑚𝑚2 ) means the minimum between the time constants. In order to improve the algorithm and makes decrease slowly the variation of error, a constant 𝐾𝐾𝑥𝑥 is added and it takes values between 0 and 1 depending on how fast the control action is needed. This becomes the first tuning parameter to calibrate the response of the controller.
− (𝑠𝑠) 𝑦𝑦𝑚𝑚
= 𝜒𝜒(𝑠𝑠). To facilitate the next calculations, let us call Now for controller design (4) can be represented in differential equation form as follows:
(13)
𝜒𝜒2
𝑛𝑛
=
𝜒𝜒1 𝑛𝑛 −𝜒𝜒1 𝑛𝑛−1 𝑇𝑇
+
𝐾𝐾𝐴𝐴 𝜒𝜒 2 𝑛𝑛 𝑘𝑘𝑚𝑚 𝐾𝐾𝐵𝐵
(16)
(17)
One of the main advantages of the proposed scheme is that it presents a fixed architecture facilitating its implementation, therefore the proposed controller that can be easily used for nonlinear and different kinds of processes that fit the premise
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of second order model plus dead time model; which is a good approximation to the main percentage of chemical processes. Remark 1: 𝑥𝑥2(𝑛𝑛+1) can be estimated using the Taylor’s formula: 𝑥𝑥2(𝑛𝑛+1) = 𝑥𝑥2(𝑛𝑛) +
𝑑𝑑𝑑𝑑2 𝑑𝑑𝑑𝑑
𝑇𝑇 +
𝑑𝑑 2 𝑥𝑥2 𝑇𝑇 2 𝑑𝑑𝑡𝑡 2
2
+ ⋯ + 𝐶𝐶
(18)
Where, 𝐶𝐶 is the complementary term (Hildebrand, 1987). So, if the sampling time is small, 𝑥𝑥2(𝑛𝑛+1) can be approximated in one of following ways: 𝑥𝑥2(𝑛𝑛+1) ≈ 𝑥𝑥2(𝑛𝑛)
(19)
𝑥𝑥2(𝑛𝑛+1) = 𝑥𝑥2(𝑛𝑛) + 𝑥𝑥2(𝑛𝑛+1) ≈ 𝑥𝑥2(𝑛𝑛) +
𝑑𝑑𝑑𝑑2 𝑑𝑑𝑑𝑑
𝑇𝑇 ≈ 2𝑥𝑥2(𝑛𝑛) − 𝑥𝑥2 (𝑛𝑛−1)
𝑥𝑥2(𝑛𝑛) −𝑥𝑥2(𝑛𝑛−1) 𝑇𝑇
𝑇𝑇 +
(20)
𝑥𝑥2(𝑛𝑛) −2𝑥𝑥2(𝑛𝑛−1) +𝑥𝑥2(𝑛𝑛−2) 𝑇𝑇 2 𝑇𝑇
The previous model can be decomposed in two parts 1 𝐺𝐺𝑚𝑚3 − (𝑠𝑠) = (0.9173𝑠𝑠 + 1)(0.7120𝑠𝑠 + 1)
𝐺𝐺𝑚𝑚3 + (𝑠𝑠) = 𝑒𝑒 −9.95𝑠𝑠 The controller design was based on 𝐺𝐺𝑚𝑚3 − (𝑠𝑠) as process model and the procedure was the same as indicated in (17). In Fig. 2 can be seen the response of the system when the set point goes from 1 to 1.5 in time equal to 175 seconds. It is also necessary to establish certain boundaries for stability and robustness against disturbances. For this reason, the system was perturbed with magnitude 0.3 in time 100 seconds until the end of the simulation, from Fig. 2 can be seen that the controller can lead the system to the reference very fast.
2
4. SIMULATION RESULTS AND DISCUSSION
(21)
System response without Error
1.6 1.4
Magnitud
1.2
The process model will be described by the following transfer function: 1 𝑒𝑒 −9.7𝑠𝑠 𝐺𝐺𝑝𝑝 (𝑠𝑠) = (𝑠𝑠 + 1)(0.5𝑠𝑠 + 1)(0.25𝑠𝑠 + 1)(0.125𝑠𝑠 + 1)
1
Reference System Response
0.8 0.6 0.4
4.1 Guevara et al, approach
0.2
From reaction curve procedure a FOPDT model is obtained. 𝑒𝑒 −10.38𝑠𝑠 𝐺𝐺𝑚𝑚1 (𝑠𝑠) = (1.305𝑠𝑠 + 1) 𝑡𝑡 10.38 The controllability relationship is given by 0 = = 𝜏𝜏𝑚𝑚
1.305
7.95. The time delay can be replaced by a pole located in 1/10.38. Thus, the former transfer function can be modified as indicated in (Camacho et al, 2003), 1 𝐺𝐺𝑚𝑚2 − (𝑠𝑠) = (1.305𝑠𝑠 + 1)(10.38𝑠𝑠 + 1) Following the procedure as was described in (Guevara et al, 2016), the best results obtained can be seen in the following figure: Original Controller
2 1.8 1.6 1.4
0 0
50
100
150
Time (s)
200
250
300
Fig 2. System response without error. It also can be seen that the system reacts to the events in a time instant equal to the delay, following the setpoint changes and rejecting the constant disturbances with zero error. To test the robustness of the system 1000 tests were calculated through Monte Carlo simulations as in (Serrano et al, 2016), varying the parameters kA, kB y km with a uniform distribution. Results can be seen in Fig. 3. From Fig. 3 can be seen that system follows the reference signal even with a constant disturbance in presence of modeling errors. The proposed control structure allows following different continuous trajectories for each change in the set point.
Magnitude
1.2 1
MC Set Point Changes
1.8
0.8
1.6
System Response Reference
0.6 0.4
1.4
0.2
50
100
150
Time (s)
200
250
1.2
300
Magnitud
0 0
Fig.1 Original Controller (Guevara et al, 2016)
1 0.8 0.6
4.2 Proposed approach
0.4
For this part, the system is described, as was mentioned in the previous section, by a second order model plus time delay and using in the scheme shown in figure 1. The second order plus time delay representation of the process is: 𝑒𝑒 −9.95𝑠𝑠 𝐺𝐺𝑚𝑚3 (𝑠𝑠) = (0.9173𝑠𝑠 + 1)(0.7120𝑠𝑠 + 1)
0.2 0 0
50
100
150
Time (s)
200
250
300
Fig.3 System responses against set point changes and disturbances considering parameter errors.
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1.4
Constant Set point - Advanced reference
1.6 1.4 1.2
Magnitude
Fig. 4 shows the system response when the set point is a continuous signal variant in time and also it is present a load disturbance, the system is capable of following the reference signal. If the reference is known each time instant, it is possible to advance the reference. The system response with an advanced reference can be seen in Fig. 5 and Fig. 6, with variable and constant set points respectively, also the system follows the reference signal in a precise way without the delay shown in Fig. 2 and 4. The closed loop system reacts to the disturbance because of neither the magnitude nor the time instant are known.
3079
1
Advanced Reference
0.8
System response
0.6 0.4 0.2
Variable reference - without advance
0 0
50
100
150
Time (s)
200
250
300
1.2
Fig.6 System response advancing the reference signal and constant disturbance.
Magnitud
1 0.8
1.8 0.6
Reference System response
1.4
0.2
50
100
150
200
250
Magnitude
1.2 300
Time (s)
Fig.4 System response against a variable reference and constant disturbance.
1 0.8 0.6 0.4
Variable Reference - Advanced Reference
1.4
0.2 0 0
1.2 1
Magnitude
Linear Algebra Controller Variable reference PID controller
1.6
0.4
0 0
Closed loop with Linear Algebra and PID Controllers
100
200
300
400 500 Time (s)
600
700
800
900
Fig.7 Comparison of the DTC with a PID controller and a LA controller following a variable reference.
0.8 0.6
System response Reference
Fig. 7 presents the results of the DTC with a PID controller, tuned with Dahlin formula, and the DTC with a LA controller following a variable reference. It can be seen that the DTC-LA follows properly the desired trajectory. In order to compare the performance of the proposed architecture, we used the ISE criterion: ISE_DTC-PID = 52.9709 and ISE_DTC-LA controller = 26.016. This result shows that the proposed scheme overcome the DTC-PID over 100%.
which leads to an easy design with better and faster control actions. The scheme showed accurate results and presented a good performance against disturbances of big magnitude. Also, the robustness of the proposed controller was achieved through Monte Carlo simulations to find the accurate boundary for the parameters. This controller is a good option for elevated dead time systems. Therefore, it can be used for processes with a controllability relationship greater than two. The proposed approach can track variable time references, therefore it can be applied in robotics, teleoperation systems, bioprocesses, chemical processes and so on. The proposed approach was compared against a typical deadtime compensator for the same kind of systems and presented better results. The deadtime compensator based on linear algebra presents an easy algorithm which does it very suitable for industrial implementation.
5. CONCLUSIONS
ACKNOWLEDGMENTS
In this work was proposed a dead time compensator that combines the Smith Predictor control architecture and the Linear algebra controller methodology. For designing purposes, a second order plus dead time model was used. To get the controller only the invertible part of the model was used
Authors want to acknowledge the Research Office from Universidad EAFIT for their support to join the conference. OC thanks to PROMETEO Project of SENESCYT, Republic of Ecuador, for its support for the realization of this work. This work was partially funded by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET),
0.4 0.2 0 0
50
100
150
200
250
300
Time (s)
Fig.5 System response advancing the reference signal and constant disturbance.
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