Compensated hybrid PI controllers for sampled-data controlled systems⁎

9th IFAC Symposium on Robust Control Design 9th IFAC Symposium on Robust Control Design Florianopolis, Brazil, September 3-5, 2018Design 9th IFAC IFAC Symposium Symposium on Robust Robust Control Control Design 9th on Available online at www.sciencedirect.com Florianopolis, Brazil, September 3-5, 2018 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September Florianopolis, Brazil, September 3-5, 3-5, 2018 2018Design Florianopolis, Brazil, September 3-5, 2018

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IFAC PapersOnLine 51-25 (2018) 240–245

Compensated Compensated hybrid hybrid PI PI controllers controllers for for ⋆ Compensated hybrid PI controllers ⋆⋆for sampled-data controlled systems Compensated hybrid PI controllers for sampled-data controlled systems sampled-data controlled systems ⋆ sampled-data controlled systems Mariella M. Quadros ∗∗ Ignacio Rubio Scola ∗,∗∗ ∗,∗∗

Mariella M. Quadros ∗∗∗ Ignacio ∗,∗∗ Rubio Scola ∗,∗∗ ∗,∗∗ Mariella M. M. Quadros Quadros Ignacio Rubio Scola Scola ∗,∗∗ Valter J. S. Leite ∗,∗∗ Mariella Ignacio Rubio Valter J.∗ S. Leite ∗,∗∗ ∗,∗∗ Mariella M. Quadros Ignacio Rubio Scola ∗,∗∗ Valter J. S. Leite Valter J. S. Leite ∗,∗∗ ∗ Valter J. S. Leite ∗ Electrical Electrical Engineering Engineering Graduate Graduate Program Program ∗ ∗ Electrical Engineering Graduate CEFET–MG & UFSJ, Belo Horizonte, MG,Program 30510-000 Brazil. Electrical Engineering Graduate Program ∗ CEFET–MG & UFSJ, Belo Horizonte, MG, 30510-000 Electrical Engineering Graduate CEFET–MG(email: & UFSJ, UFSJ, Belo Horizonte, Horizonte, MG,Program 30510-000 Brazil. Brazil. mariella [email protected]). CEFET–MG & Belo MG, 30510-000 Brazil. (email: mariella [email protected]). ∗∗ (email: CEFET–MG & UFSJ, Belo Horizonte, MG, 30510-000 Brazil. mariella [email protected]). of Mechatronics Engineering ∗∗ Department (email: mariella [email protected]). of Mechatronics Engineering ∗∗ (email: mariella [email protected]). ∗∗ Department Department of Mechatronics Engineering CEFET–MG, Divin´ polis, Brazil. ofo Engineering CEFET–MG, Divin´ oMechatronics polis, MG, MG, 25503-822 25503-822 Brazil. ∗∗ Department Department of Mechatronics Engineering CEFET–MG, Divin´ opolis, polis, MG, MG, 25503-822 25503-822 Brazil. (email: [email protected],[email protected]). CEFET–MG, Divin´ o Brazil. (email: [email protected],[email protected]). CEFET–MG, Divin´ o polis, MG, 25503-822 Brazil. (email: [email protected],[email protected]). [email protected],[email protected]). (email: (email: [email protected],[email protected]). Abstract: Abstract: In In this this paper paper we we consider consider sampled-data sampled-data systems systems controlled controlled by by hybrid hybrid PI PI controllers. controllers. Abstract: In this thisthe paper we consider consider sampled-data systems controlled by hybrid hybrid PI controllers. controllers. In these systems, sample time may induce some undesirable overshoot due to the postponed Abstract: In paper we sampled-data systems controlled by PI In these systems, the sample time may induce some undesirable overshoot due to the postponed Abstract: In this paper we consider sampled-data systems controlled by hybrid PI controllers. In these systems, the sample time may induce some undesirable overshoot due to the postponed reset action on the control integral state, which deteriorates the performance of the closed loop In these systems, the sample time may induce some undesirable overshoot due to the postponed reset action on the control integral state, which deteriorates the performance of the closed loop by by In these systems, the sample time may induce some undesirable overshoot due to the postponed reset action on the control integral state, which deteriorates the performance of the closed loop leading to output overshoot as well as to larger settling time values. We propose a methodology reset action on theovershoot control integral the performance of theaclosed loop by by leading to output as wellstate, as towhich largerdeteriorates settling time values. We propose methodology reset action on the control integral state, which deteriorates the performance of the closed loop by leading to output output overshoot as well well as as to to larger settling time values. We propose propose methodology to compute a bounded compensating signal to be added to the control input that minimizes leading to overshoot as larger settling time values. We aa methodology to compute a bounded compensating signal to be added to the control input that minimizes leading to output overshoot as well as to larger settling time values. We propose a methodology to compute compute a bounded bounded compensating signal to be beofadded added to the the control controlsignal inputsynchronizes that minimizes minimizes the overshoot and time. action the the to a compensating signal to to input that the overshoot and the the settling settling time. The The action ofadded the compensating compensating signal the to compute a bounded compensating signal to be to the control inputsynchronizes that minimizes the overshoot and the settling time. The action of the compensating signal synchronizes the reset instants of the hybrid PI controller with the sample-time for a class of systems. The the overshoot and the settling time. The action of the compensating signal synchronizes the reset instants of the hybrid PI controller with the sample-time for a class of systems. The the overshoot and the settling time. The action of the compensating signal synchronizes the reset instants of the hybrid PI controller with the sample-time for a class of systems. The proposed methodology is tested in a laboratory setup system composed of two connected water reset instants of the hybrid PI incontroller withsetup the sample-time for a ofclass of systems.water The proposed methodology is tested a laboratory system composed two connected reset instants ofcapacity the hybrid PIliters with the by sample-time for a of of systems. The proposed methodology is of tested incontroller a laboratory laboratory setup system composed ofclass two connected water reservoirs with 200 each, actioned a variable velocity pump. The control proposed methodology is tested in a setup system composed two connected water reservoirs with capacityis of 200 liters each, actioned by a variable velocity pump. The control proposed tested inofa the laboratory setup system composed of two connected water reservoirs methodology with capacitythe of 200 liters each, actioned by variable velocity pump. The control control objective is to regulate level first tank and to disturbances. We compare the reservoirs with capacity of 200 liters each, actioned by aa reject variable velocity pump. The objective is to regulate the level of the first tank and to reject disturbances. We compare the reservoirs with capacity of 200 liters each, actioned by a variable velocity pump. The control objective is to regulate the level of the first tank and to reject disturbances. We compare the performances achieved by two different hybrid PI schemes with and without the proposed objective is to regulate the level of the first tank and to reject disturbances. We compare the performances achieved by two different hybrid PI schemes with and without the proposed objective is to regulate the level of the first tank and to reject disturbances. We compare the performances achieved by two different hybrid PI schemes with and without the proposed additive compensating signal. The experimental tests illustrate the performance, robustness performances achieved signal. by two The different hybrid PI schemes withthe and without therobustness proposed additive compensating experimental tests illustrate performance, performances achieved byachieved two The different hybrid PI schemes withthe and withoutsystems. therobustness proposed additive compensating signal. The experimental tests illustrate the performance, robustness and disturbance rejection by our approach when applied to uncertain additive compensating signal. experimental tests illustrate performance, and disturbance rejection achieved our approach when applied the to uncertain systems. additive compensating signal. The by experimental tests illustrate performance, robustness and rejection achieved by our when applied systems. and disturbance disturbance rejection achieved by our approach approach when applied to to uncertain uncertain systems. © 2018, IFAC (International Federationby of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and disturbance rejection achieved our approach when applied to uncertain systems. Keywords: Keywords: Hybrid Hybrid systems, systems, Stability Stability of of hybrid hybrid systems, systems, Embedded Embedded systems systems Keywords: Keywords: Hybrid Hybrid systems, systems, Stability Stability of of hybrid hybrid systems, systems, Embedded Embedded systems systems Keywords: Hybrid systems, Stability of hybrid systems, Embedded systems 1. INTRODUCTION approaches to handle 1. INTRODUCTION approaches to handle model model uncertainty uncertainty are are proposed proposed by by approaches to handle model uncertainty are proposed by 1. HosseinNia et al. (2014, 2015). See also the works of n os 1. INTRODUCTION INTRODUCTION approaches to al. handle proposed HosseinNia et (2014,model 2015).uncertainty See also theare works of Ba˜ Ba˜ nby os 1. INTRODUCTION approaches to handle model uncertainty are proposed HosseinNia et al. al. (2014,Barreiro 2015). See See also the works of Ba˜ Ba˜ nby os and Barreiro (2012); and Ba˜ n os (2012); Dav´ o A great interest on hybrid systems has been verified in HosseinNia et (2014, 2015). also the works of n os Barreiro (2012); Barreiro and Ba˜ n os (2012); Dav´ o A great interest on hybrid systems has been verified in and HosseinNia et al. (2014, 2015). See also the works of Ba˜ non os Barreiro (2012); Barreiro and Ba˜ n os (2012); Dav´ o and Ba˜ n os (2016); Dav´ o et al. (2018) for contributions A great interest on hybrid systems has been verified in the literature during the last years mainly because of Barreiro (2012); Barreiro and Ba˜ n os (2012); Dav´ o A great interest on hybrid systems has been verified in and Ba˜ n os (2016); Dav´ o et al. (2018) for contributions on the literature during the last years mainly because of and Barreiro (2012); Barreiro and Ba˜ n os (2012); Dav´ o A great interest on hybrid systems has been verified in Ba˜ n os (2016); Dav´ o et al. (2018) for contributions on handling delay and of Zhao et al. (2013) for some general the literature during the last years mainly because of their complex behavior and high performance possibilities, and Ba˜ n os (2016); Dav´ o et al. (2018) for contributions on the literature during the last years mainly because of handling delay and of Zhao et al. (2013) for some general their complex behavior and last high years performance possibilities, and Ba˜ n os (2016); Dav´ o et al. (2018) for contributions on the literature during the mainly because of handling delay and of Zhao et al. (2013) for some general open problems on reset control. their complex behavior and high performance possibilities, specially for control purposes (Goebel et al., 2012). In case handling delay and of Zhao et al. (2013) for some general their complex behavior and high performance possibilities, problems control. specially for control purposes (Goebel et al., 2012). In case open handling delay on andreset of Zhao et al. (2013) for some general their complex behavior and high performance possibilities, problems on reset control. specially for control purposes (Goebel et al., In of hybrid (proportional-integral) controllers, smaller open problems on reset control. specially forPI control purposes (Goebel et al., 2012). 2012). In case case open However, none of the previous works of hybrid PI (proportional-integral) controllers, smaller open problems on reset control. specially for control purposes (Goebel et al., 2012). In case However, none of the previous works are are concerned concerned with with of hybrid PI (proportional-integral) controllers, smaller settling times and rising times can be achieved while of hybridtimes PI (proportional-integral) controllers, smaller However, none of the previous works are concerned with the effects of discretization on hybrid controllers. Such an settling and rising times can be achieved while However, none of the previous works are concerned with of hybrid PI (proportional-integral) controllers, smaller the effects of discretization on hybrid controllers. Such an settling times and rising times can be achieved while overshoot is avoided (Neˇ s i´ c et al., 2011). Therefore, linear settling times and rising beTherefore, achieved linear while issue However, none ofrelevant the previous works areenergy concerned with the effects of discretization on hybrid controllers. Such an is specially in case of high consumpovershoot is avoided (Neˇsi´c times et al., can 2011). the effects of discretization on hybrid controllers. Such an settling times and rising times can be achieved while issue is specially relevant in case of high energy consumpovershoot is avoided (Neˇ ssi´ cc et al., 2011). Therefore, linear PI controllers broadly employed in industrial processes, overshoot is avoided (Neˇ i´ et al., 2011). Therefore, linear the effects of discretization on hybrid controllers. Such an issue is specially relevant in case of high energy consumption processes, which have great economic and environPI controllers broadly employed in industrial processes, issue is specially relevant in case of high energy consumpovershoot is avoided (Neˇ s i´ c et al., 2011). Therefore, linear tion processes, which have great economic and environPI controllers broadly employed in industrial processes, can be enhancement with hybrid action. PI controllers broadly employed in industrial processes, issue is specially relevant inthe case ofeconomic high energy tion processes, processes, which have great economic andconsumpenvironmental impact. which Moreover, sampling period can lead can be enhancement with hybrid action. have great and environPI controllers broadly employed in industrial processes, tion mental impact. Moreover, the sampling period can lead can be enhancement with hybrid action. can be enhancement with hybrid action. tion processes, which have great economic and environmental impact. Moreover, the sampling period can lead the hybrid controller to lose its properties such as the finite Some recent developments on hybrid PI controllers include mental impact. Moreover, the sampling period can lead can be enhancement with hybrid action. hybrid controller to lose its properties such as the finite Some recent developments on hybrid PI controllers include the mental impact. Moreover, the sampling period can lead the hybrid controller to lose its properties such as the finite time response and overshoot avoidance, because reset Some recent developments on hybrid PI controllers include the L bound estimate on the disturbance rejection in the hybrid controller to lose its properties such as the finite 2 Some recent developments on hybrid PI controllers include time response and overshoot avoidance, because the reset the L2recent bound estimate on the disturbance rejection in the hybrid controller to lose its properties such as the finite Some developments on hybrid PI controllers include time response and overshoot avoidance, because reset or jump instant may be postponed up to one sampling the L bound estimate on the disturbance rejection in case of model mismatch (Neˇ s i´ c et al., 2011); the asymptime response and overshoot avoidance, because the reset 2 the Lof2 model bound mismatch estimate on the disturbance in or jump instant may be postponed up to one sampling pepecase (Neˇ si´c et al., 2011); rejection the asymptime response and overshoot avoidance, because the reset the Lof bound mismatch estimate on the disturbance rejection in riod. or jump instant may be up to one peFor instance, consider aa known first system con2 model case of model mismatch (Neˇ si´ i´cc provided et al., 2011); 2011); the asympasymptotic rejection of disturbances by Panni et al. or jump instant may be postponed postponed up to order one sampling sampling pecase (Neˇ s et al., the riod. For instance, consider known first order system contotic rejection of disturbances provided by Panni et al. or jump instant may be postponed up to one sampling pecase of model mismatch (Neˇ s i´ c et al., 2011); the asympriod. For instance, consider a known first order system controlled by a hybrid PI as proposed in (Neˇ s i´ c et al., 2011). totic rejection of disturbances provided by Panni et al. (2014); Cordioli et al. (2015) with first order reset elements riod. For instance, consider a known first order system contotic rejection of disturbances provided by Panni et al. trolled by a hybrid PI as proposed in (Neˇ s i´ c et al., 2011). (2014); Cordioliofet disturbances al. (2015) withprovided first order reset elements riod. For instance, consider a known first order system contotic rejection by Panni et al. trolled by a hybrid PI as proposed in (Neˇ s i´ c et al., 2011). We have run simulations with the hybrid PI controller (2014); Cordioli et al. (2015) with first order reset elements (FORE) for precisely known process models; the PI controlled by a hybrid PI as proposed in (Neˇ s i´ c et al., 2011). (2014); Cordioli et al. (2015) with first models; order reset elements We have simulations with theinhybrid controller (FORE) for known process the PI trolled by run aincluding hybrid PIthe aszero-order-hold proposed (Neˇ si´c PI etLoan, al., 2011). (2014); et al. (2015) first models; order reset elements We have run simulations with PI controller (Van 1978) (FORE)Cordioli for precisely precisely knownwith process models; the PI concon- discretized trollers with a Clegg integrator (Clegg, 1958) in parallel, We have run simulations with the the hybrid hybrid PI controller (FORE) for precisely known process the PI condiscretized including the zero-order-hold (Van Loan, 1978) trollers with a Clegg integrator (Clegg, 1958) in parallel, We have run simulations with the hybrid PI controller (FORE) for precisely known process models; the PI condiscretized including the zero-order-hold (Van Loan, 1978) with sampling periods T ∈ {0.01, 0.5, 1, 1.5, 2}. Details trollers with a Clegg integrator (Clegg, 1958) in parallel, called PI-CI controller, which is used to compute a bias discretized including 1978) trollers with acontroller, Clegg integrator (Clegg, 1958) in parallel, with sampling periodsthe Tsszero-order-hold ∈ {0.01, 0.5, 1,(Van 1.5,Loan, 2}. Details called PI-CI which is used to compute a bias discretized including the zero-order-hold (Van Loan, 1978) trollers with a Clegg integrator (Clegg, 1958) in parallel, with sampling periods T ∈ {0.01, 0.5, 1, 1.5, 2}. Details of the simulations are discussed later. As one can easily called PI-CI controller, which is used to compute a bias value for the jump (reset) (Vidal et al., 2008; Vidal and s with sampling periods T ∈ {0.01, 0.5, 1, 1.5, 2}. Details called for PI-CI controller, which is used to 2008; compute a bias s of the simulations are discussed later. As one can easily value the jump (reset) (Vidal et al., Vidal and with sampling periods T ∈ {0.01, 0.5, 1, 1.5, 2}. Details called PI-CI controller, which is used to compute a bias s of the thethe simulations are discussed later. As one can canperiods easily note, overshoot increases with larger sampling value for the Paesa jump (reset) (reset) (Vidaland et al., al., 2008; Vidal Vidal and of Ba˜ n 2009; et an scheme simulations discussed one easily value the jump (Vidal et 2008; and note, the overshoot are increases withlater. largerAs sampling Ba˜ nos, os,for 2009; et al., al., 2011); 2011); and an adaptive adaptive scheme of the simulations are discussed later. As onelost. canperiods easily value for thebyPaesa jump (reset) (Vidal et(2017), al., 2008; Vidal and note, the overshoot increases with larger sampling periods and also the finite time convergence is clearly Ba˜ n os, 2009; Paesa et al., 2011); and an adaptive scheme introduced Rubio Scola et al. that yields a note, the overshoot increases with larger sampling periods Ba˜ n os, 2009; Paesa et al., 2011); and an adaptive scheme and also the finite time convergence is clearly lost. introduced byPaesa Rubio Scola et al. (2017), that yields a note, the overshoot increases with larger sampling periods Ba˜ n os, 2009; et al., 2011); and an adaptive scheme and also the finite time convergence is clearly lost. introduced by Rubio Scola et al. (2017), that yields a robust hybrid PI controller that rejects piecewise constant and also the finite time convergence is clearly lost. introduced byPIRubio Scolathat et rejects al. (2017), that constant yields a Motivated by such a practical issue, we propose the design robust hybrid controller piecewise also the finite time convergence is clearly lost. introduced byand Scola et on al. regulation (2017), that yields a and Motivated by such aa practical issue, we propose the design robust hybrid hybrid PIRubio controller that rejects piecewise constant disturbances avoids ripple mode even robust PI controller that rejects piecewise constant Motivated by such practical issue, we propose the design aa compensating control signal that mitigates bad disturbances and avoids ripple on regulation mode even of Motivated by such a practical issue, we propose thethe design robust hybrid PI controller that rejects piecewise constant of compensating control signal that mitigates the bad disturbances and avoidsand ripple onThis regulation mode even even with uncertain models delay. last approach endisturbances and avoids ripple on regulation mode Motivated by such a practical issue, we propose the design of a compensating control signal that mitigates the bad effects of time-discretization of hybrid controllers. Our apwith uncertain models and delay. This last approach enof a compensating control signal that mitigates the bad disturbances and avoids ripple on regulation mode even effects of time-discretization of hybrid controllers. Our apwith uncertain models and delay. This last approach encompasses the one in (Neˇ s i´ c et al., 2011) (by setting α = 1) with uncertain models and delay. This (by last setting approach enof a compensating control signal thatcontrollers. mitigates the bad effects of time-discretization of hybrid Our approach is tested in a level control system with 2 reservoirs compasses the one in (Neˇ s i´ c et al., 2011) α = 1) effects of time-discretization of hybrid controllers. Our apwith uncertain models and delay. This lastαsetting approach is tested in a level control system with 2 reservoirs compasses the one one in (Neˇ (Neˇ si´ i´cc et et al., 2011) (by setting αOther =en1) proach and the conventional linear PI (by setting = 0). compasses the in s al., 2011) (by α = 1) effects of time-discretization of hybrid controllers. Our approach is tested in a level control system with 2 reservoirs of 200 liters capacity each and with two types of hybrid and the conventional linear PI (by setting α = 0). Other proach is tested in a level control system with 2 reservoirs compasses the one in (Neˇ s i´ c et al., 2011) (by setting α = 1) of 200 liters capacity each and with two types of hybrid and the conventional linear PI PI (by setting α αAgency = 0). 0). CAPES Other proach ⋆ This and the conventional linear setting = Other is tested in acontroller level control system with 2 reservoirs of 200 liters capacity each and with two types of hybrid work has been supported by (by the Brazilian PI controllers. The runs under Python language of 200 liters capacity each and with two types of hybrid ⋆ and the conventional linear PI setting αAgency = 0). CAPES Other PI work has been supported by (by the Brazilian controllers. The controller runs under language ⋆ This of liters capacity each and with twoPython types of hybrid and PNPD ⋆ This work has PI controllers. The runs under Python language This work1474377. has been been supported supported by by the the Brazilian Brazilian Agency Agency CAPES CAPES PI 200 controllers. The controller controller runs under Python language and PNPD 1474377. ⋆ This work1474377. has been supported by the Brazilian Agency CAPES PI controllers. The controller runs under Python language and PNPD 1474377. and PNPD

and PNPD©1474377. 2405-8963 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 2018 351 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 351 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 351 Copyright © 2018 IFAC 351 10.1016/j.ifacol.2018.11.112 Copyright © 2018 IFAC 351

IFAC ROCOND 2018 Mariella M. Quadros et al. / IFAC PapersOnLine 51-25 (2018) 240–245 Florianopolis, Brazil, September 3-5, 2018

241

Rubio Scola et al. (2017) and omitting the time and the uncertainty dependency, we can the hybrid closed loop system as (considering r˙ = 0):

2

Fig. 1. Effect of sampling period on hybrid controller. on a notebook communicating with a programmable logic computer (PLC) through Ethernet connection. The PLC receives the level measures by means of a current loop and send control signals to an inverter to control the velocity of a hydraulic pump. The performances of the tested controllers are evaluate under classical indexes like IVE (integral of variability error), IAE (integral of absolute error) and IVU (integral of variability control signal) and such indexes may be improved up to 35% when our approach is used. The remain of this paper is organized as follows: in the next section the framework as well as the hybrid PI controllers used in this paper are presented. This section also includes a presentation of the technique used to design the PI controllers. In section III the main contribution is presented: the additive compensating control signal. In section IV, the proposed approach is applied in set of experiments in a laboratory setup. The paper ends with the conclusions (section V) and the references. Notation: The independent variable (hybrid time) is (t, i) defined by [0, ∞] × N0 which can be obtained by (finite or infinite) union of [ti , ti+1 ] × {i}, with i indicating jump instants. The periods of the dynamics flow of a system are given by [ti , ti+1 ]. x(t, i) denotes the state vector. Also, we use the simplified notation x ≡ x(t, i). After a jump, the state is represented by x+ ≡ x(ti+1 , i + 1). The set of matrices with dimensions n × m and real entries is represented by Rn×m . λmin (P ) (λmax (P )) denote the smallest (largest) eigenvalue of P = P T > 0. 2. PRELIMINARIES AND PROBLEM STATEMENT We assume that the controlled system has relative degree one ans is modeled as � � x(t) ˙ = Ap (ζ)x(t) + Bp (ζ) u(t) + d(t) , (1) y(t) = Cp (ζ)x(t), (2) n where x(t) ∈ R is the state vector, u(t) ∈ R is the control signal, y(t) ∈ R is the output vector, d(t) ∈ R is a finite energy perturbation, i.e., d(t) ∈ L2 , the uncertain �N parameter ζ ∈ Ω, Ω = {ζ ∈ RN : i=1 ζi = 1, ζi ≥ 0}, and matrices N � [Ap , Bp , Cp ](ζ) = ζi [Ap,i , Bp,i , Cp,i ], ζ ∈ Ω (3) i=1

where matrices Ap,i ∈ Rn×n , Bp,i ∈ Rn×1 , and Cp,i ∈ R1×n , i = 1, . . . , N , are known. The control signal is given by u(t) = kP e(t) + kI xI (t), with kP and kI being the proportional and integral gains, respectively, e(t) = r(t) − y(t) is the regulation error, r(t) is the reference signal assumed to be piecewise constant, and xI (t) is the integrator state. Moreover, xI (t) can be reseted once some conditions are achieved. By following the ideas of

352

x+ = x x˙ =(Ap − Bp kP Cp )x + Bp kI xI + Bp kP r + Bp d e+ = e e˙ = − Cp Ap x − Cp Bp kP e x+ I = xI − αξ (4) − Cp Bp kI xI − Cp Bp d ξ+ = 0 x˙ I =e, ξ˙ = e τ˙ = 1 τ+ = 0 �� � � � �� � η ≥ 0 or τ ≤ ρ η ≤ 0 and τ ≥ ρ

where τ ≥ 0 is an auxiliary state used fot time regularization, ρ > 0 is a given constant, η = 2eξ +ǫe2 and α ∈ [0, 1] is the tunable adaptation rate. An interesting issue on this hybrid PI adaptive controller is that it can recover either a) the classical linear PI controller (without reset action) by setting α = 0; or b) the hybrid PI controller with reset action proposed in (Zaccarian et al., 2007) (without adaptation) by setting α = 1. As discussed by Rubio Scola et al. (2017), the use of α ∈]0, 1[ allows to improve the rejection and tracking properties by melting the nice properties of the classical linear PI with those from the hybrid one. Moreover, description (4) can be cast under Goebel, Sanfelice, and Teel’s framework by defining assuming an aug�T � mented state vector xa = xT eT xTI ξ T and matrices     Ap − Bp kP Cp Bp kP 0 Bp kI 0 −Cp Bp kP −Bp kI 0  −Cp Ap  0  B = A= 0 1 0 0 r  0  0 0 1 0 0 C = [Cp 0 0 0] (5)       0 0 00 Bp Id 0 0 0 0 ǫ 0 1 −Cp Bp  0 1 0 0  Bd =  M = A = 0 0 0 0 0  r  0 0 1 −α 0 1 00 0 0 0 0 0 which allows to write x˙ = Axa + Bd x+ a = Ar xa + τ˙ = 1 τ =0 (6) � �� � �� � � xa ∈C or τ ≤ρ

xa ∈D and τ ≥ρ

where the closed sets C ⊂ Rn and D ⊂ Rn , respectively the flow and the jump sets, verify C ∪ D = Rn such that C := {xa ∈ Rn : xTa M xa ≥ 0},

(7) D := {xa ∈ Rn : xTa M xa ≤ 0}. The following theorem adapted from (Rubio Scola et al., 2017, Th. 1) guarantees the robust closed-loop stability of this hybrid controller and, thus it proof is omitted. Theorem 1. Consider the closed-loop system described by (6) with matrices given by (5) with (3). If there exists symmetric positive definite matrices Pi ∈ Rn×n , i = 1, . . . , N , matrices F1 , F2 , G1 , G2 belonging to Rn×n , constants λF , λJ , such that ΘJ,i � 0 and ΘF,i ≺ 0, i = 1, . . . , N, (8) where � � ⋆ F A +ATr F2T −Pi −λJ Mi (9) ΘJ,i ≡ 2 r Pi − (G2 + GT2 ) G2 Ar − F2T � � ⋆ λF Mi + F1 Ai + ATi F1T (10) ΘF,i ≡ −(G1 + GT1 ) Pi + G1 Ai − F1T

IFAC ROCOND 2018 242 Mariella M. Quadros et al. / IFAC PapersOnLine 51-25 (2018) 240–245 Florianopolis, Brazil, September 3-5, 2018

and if x ∈ D ⇒ Ar xa ∈ C, then there exists a small enough ρ > 0 such that the considered closed-loop system is exponentially stable, which is ensured by a Lyapunov function, given by N  V (x) = xTa P (ζ)xa ; P (ζ) = ζi Pi . (11) i=1

2.1 Discretization Problem The problem addressed in this work arises when the controller dynamics is discretized to be implement in digital platform with sampling time constraints such as the PLC industrial devices or stand alone devices where energy economy is crucial. In this case, the jump moment may be postponed by a sampling period what may lead to very poor performance. In Fig. 1 we present the time response of a system modeled with Ap = −0.0588, Bp = 1, Cp = 0.0698, for a unitary step at t = 0s, driven by the hybrid controller proposed in Neˇsi´c et al. (2011) (α = 1 in (4)) with kP = 1.1, kI = 1.8, and sampling periods Ts ∈ {0.01, 0.5, 1, 1.5, 2}. The controller discretization has included the holder zero-order-hold (Van Loan, 1978). Note that, for Ts = 0.01s the time behavior is almost continuous and no overshoot is verified. However, as the sampling period is increased, we clearly verify that the finite time convergence is lost and the overshoot can reaches values close to 60%. The respective control signals can be seen in Fig. 2. We can note that, the bigger the sampling period, the more delayed the reset instant is. By inspection in Fig. 1, we can note that the reset is expected around t = 4s, when the output crosses the reference signal. Also, in the worst case (bigger overshoot) the integral state jump instant is postponed to t = 6.0s (see Fig. 2). This discussion clearly illustrate how the sampling period can degenerate the closed loop response of hybrid controllers. Therefore, we propose the following problem to be investigate: Problem 1. Propose a strategy to mitigate the effects of the sampling period in digital implementations of hybrid PI controllers, improving the closed-loop robustness. 3. COMPENSATING STRATEGY The discrete-time implementation of the continuous part of the controller given by (4) is done with a sample period Ts as follows: uk = kP ek + kI xIk , xIk+1 = xIk + ek Ts , (12) ξk+1 = ξk + ek Ts .

Input

4 2 0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

(1) to adjust the value of uk avoiding overshoots, and (2) to vanish exponentially after a few samples. It is worth to say that wk should start to be computed whenever the output prediction one step ahead, yˆk+1 , crosses the reference signal rk before the next sampling; otherwise, wk = 0. We propose to compute yˆk+1 by ¯p (uk + wk ), yˆk+1 = C¯p A¯p xk + C¯p B (13) ¯ ¯ ¯ with Ap , Bp , Cp are the discrete-time versions of the nominal or mean values of A(ζ), B(ζ), C(ζ), respectively. Thus, this simple prediction, based on nominal model, guides the decision on the compensating strategy. Observe that other estimations may be used, including a filter for the uncertain system with some performance index as, for instance, H∞ guaranteed cost. We use the following compensating signal to be added to the control signal: ¯k e−β max(0,−k1 +k−κ) (14) wk = w where β and κ are design parameters discussed later, w ¯k is computed as rk − yˆk+1 rk − C¯p A¯p xk = − uk , (15) w ¯k = ¯ ¯ ¯p Cp Bp C¯p B and k1 ≤ k is called the activation sample, i.e., the sample in which the predicted output crosses the reference signal. Such a sample can be computed in real time through Algorithm 1. Data: sampling k, measure yk , estimate yˆk+1 , setpoint rk , noise amplitude ̺. Result: Activation sample of wk1 . if (yk < rk − ̺ and yˆk+1 > rk ) or (yk > rk + ̺ and yˆk+1 < rk ) then k1 = k; end

Because wk is a signal with finite energy, it does not interfere with the closed loop stability. Also, it is implicit in our approach that we assume that the reference does not change at sample k1 , i.e. rk1 = rk1 +1 , and the estimated error in the next sampling is ek1 +1 = rk − yˆk1 +1 = 0 ⇒ yˆk1 +1 = rk1 . Thus, the compensating signal given by (14) can be applied in a quite general family of process models with relative degree equals to 1.

−2 0.0

We propose to compute a prediction of the output one step ahead and verify if the activation of the jump condition occurs before the next sampling. If this is the case, a signal wk is compute as an additive perturbation on uk such that the conditions to activate the jump (reset) are fulfilled only in the next sampling, but not before. Moreover, because hybrid PI controllers (4) with α = 1 can reject perturbations with a L2 gain bound — see (Zaccarian et al., 2007; Neˇsi´c et al., 2011) — we compute a compensating signal wk to match two objectives:

Algorithm 1: Activation time algorithm

Ts=2 Ts=1.5 Ts=1.0 Ts=0.5 Ts=0.01

6

Note that, the postponed reset problem occurs whenever the reset condition is achieved between two consecutive sampling instants. Therefore, our approach to provide a solution to Problem 1 consists in adding a compensating signal wk to the control signal uk to change the moment when the reset condition is active.

20.0

Time [s]

Fig. 2. Control signals associate with Fig. 1. 353

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To illustrate our approach, consider the worst case of the simulations presented in Fig. 1, i.e., using Ts = 2s. We have simulated this case again, at this time adding wk given by (14)-(15) with β = 0.15 and κ = 5. The new simulation is presented in Fig. 3 where it is shown in the top the reference signal (black dashed line), the output with the compensating signal (continuous line) and the previous behavior of the output (dashed line) with wk = 0. Note that the overshoot (previously about 60%) is completely mitigated. In the bottom part, it is presented the control signals: the computed one, uk (blue line), the proposed compensating signal wk (black line) and the applied signal, uk + wk (magenta line). Observe that the hatched areas in Fig. 3 can be associated with the extra energy provided by the controller due to the postponed reset. The reader can note that in the simulation with wk = 0 (dashed line) the reset should be activated at the instant tj , but such an instant does not match with the available sampling times ti and ti+1 . Therefore, the compensating signal wk synchronizes the moment of the reset with an available sampling time recovering the finite convergence. 4. EXPERIMENTAL TESTS 4.1 Experimental Setup We have tested our approach to control the level on a experimental setup with two reservoirs each of them with a capacity of 200 liters, height of 80cm and diameters of 62cm. The reader can find in Figure 4 a representation of such a system, where the left tank contains a nonlinear solid inside. The nonlinearity is a solid of revolution of a Gaussian curve that has been made with expanded polyurethane, yielding a circular cross section function with diameters varying from 23.3cm to 59.8cm. The controlled level is the one in the tank on the left side, where the inlet flow is a consequence of the control signal sent to an inverter used to command a variable velocity pump. This tank has two connections through valves V1 and V2 that allows two operating modes: (1) mode 1: valve V1 open and valve V2 closed;

Fig. 3. Compensated signals associate with Fig. 1, Ts = 2s. 354

Perturbation

62cm

Input

62cm

23.3cm 80cm

Parameters κ and β are used to adjust the control signal and vanishing of the compensation signal, respectively. The value of κ means the number of samples after instant k1 that the compensation computed by (15) is entirely added to the control signal. After such a number of samples, the signal wk starts to vanish exponentially depending on β. Our experience indicates that it is reasonable to choose these parameters by considering the dominant time constant, τ , of the ideal hybrid closed loop system: choose (6τ )−1 ≤ β ≤ (4τ )−1 and 6τ /Ts ≤ κ ≤ 10τ /Ts , with Ts being the sampling period.

243

Nonlinear volume

Q2 Q1

V1

V2

Fig. 4. Diagram of the water reservoirs. (2) mode 2: valve V1 closed and valve V2 opened. Note that in the operating mode 2, the tank on the right acts as a perturbation w.r.t. the first operating mode. The local linearization around h1 = 45.9cm results in a first order system for the mode 1:     −0.0588 0.0698 Ap1 Bp1 = (16) G1 = Cp1 Dp1 1 0 and for mode 2, around 49.88cm:     G2 =

Ap2 Bp2 Cp2 Dp2

=

−0.1118 0.1118 0.0710 0 0.0083 −0.0130 1 0 0

(17)

For details on these models, please see Rubio Scola et al. (2017). Polytopic uncertain models for each operating mode were obtained by taking 10 equilibrium points around the nominal operating point: for mode 1 it was computed 10 linear models for heights among y eq = 41.15cm and 1 y eq1 = 57.78cm; and similarly for mode 2 for heights between yeq = 42.97cm and y eq2 = 55.22cm. From these 2 models we take the maximum and minimum values of each entry of the matrices of the systems. Thus, in mode 1 we have 2 parameters yielding a 4 vertex polytope; and in mode 2, 5 uncertain parameters, 4 in matrices Api and 1 in Bpi , yield a 32 vertices polytope made up with all the combinations of extreme values of the uncertain parameters. The used data for mode 1 was [Ap1 ]11 ∈ [−0.06413, − 0.02216], [Bp1 ]11 ∈ [0.0263, 0.07619]; and for mode 2: [Ap2 ]11 ∈ [−0.12300, − 0.05421], [Ap2 ]12 ∈ [0.05421, 0.12300], [Ap2 ]21 ∈ [0.00770, 0.00906], [Ap2 ]22 ∈ [−0.01374, − 0.01238], [Bp2 ]11 ∈ [0.03696, 0.07629]. 4.2 Controllers Our approach has been evaluated in the level control system by using the hybrid PI controller (4) with α = 1, hybrid PI controller (PIH) proposed in Neˇsi´c et al. (2011), and with α = 0.4, the adaptive hybrid PI controller introduced in Rubio Scola et al. (2017) (PIHA) under two conditions: with and without the compensating signal proposed in Section 3. Therefore, for better relate the time-response signals with the employed controllers, all experiments presented in this section follow a color code: signals from hybrid controllers PIH and PIHA with the compensating signal are noted as PIHW and PIHAW , respectively. In all cases, the respective nominal models were used to compute yˆk+1 given in (13). The controllers were designed for both modes using specifications (for the linear PI) of overshoot of 60% and settling time of 90s for the case of nominal models. Note that, these specifications ensures that the hybrid part of the controllers acts on the

IFAC ROCOND 2018 244 Mariella M. Quadros et al. / IFAC PapersOnLine 51-25 (2018) 240–245 Florianopolis, Brazil, September 3-5, 2018

Operating mode 1: Tests have been performed for constant piecewise reference tracking around the specified operating point (h1 = 45.9cm). First, a positive step of magnitude 6cm is applied and after the steady state a negative step of 9cm — thus leading the output to 42.9cm — as shown in Figure 5 (top). The control signals are show in

Level [cm]

PIH

PIHA

PIHW

PIHAW

Ref.

wk

50 45 40

uk from PIHA & PIHAW

100

50

PIH & PIHW

uk from

0 100

50

0 440

460

480

500

520

Time [s] Fig. 5. Control mode 1: Level (top), control signals PIHA and PIHAW (middle), PIHA and PIHAW (bottom), wk (dotted lines). this same figure: on the middle plot for PIHA and PIHAW , and on the bottom for PIH and PIHW ; the compensating signal wk is presented on the middle and on the bottom plots in black line. Note the superior behavior of the controllers PIHW and PIHAW w.r.t. their respective versions without the compensating signal. The PIHA controller shows an overshoot of approximately 54.6% and settling time of 41s, while the PIHAW presents no overshoot and only 6s of settling time, that is, about 14.6% of the settling without our approach. A similar behavior has been verified with the PIHW controller, with no overshoot and almost the same settling time of PIHAW . On the other hand, PIHcontroller shows an overshoot of 74.3% and a settling time of 25s which is more than 4 times greater than the same controller with the proposed compensation. Operating mode 2: We have performed experiments with constant piecewise reference tracking around the equilibrium point for operating mode 2 (h1 = 49.88cm). In this case, besides the controlled system be a second order one, 355

PIH

PIHW

PIHA

PIHAW

Ref.

wk

Level [cm]

60 55 50 45

uk from PIHA & PIHAW

40 100 50 0

-50 100

PIH & PIHW

To proceed with the experiments we have used the minimum sampling period allowed by the available PLC, Ts = 2 seconds, and the nominal models (16)-(17) to compute the additive compensating signal (14)-(15); the estimate of the output one step ahead; and the initial reset value for PIH and PIHA controllers. The time response of the experiments are presented in Figures 5-7. The time responses of the closed-loop controlled system are presented at the top and the control signals at the bottom, with the same color code, and the additive compensating signal wk is shown in black. In all experiments, we have used ̺ = 2.5 corresponding to the amplitude of the measurements noise.

the uncertainties are more important. As presented in Fig-

uk from

closed-loop. The obtained gains were (kp1 = 1.1, kI1 = 1.8) for mode 1 and (kp2 = 0.08, kI2 = 1.6) for mode 2.

50 0

-50 800

850

900

950

Time [s] Fig. 6. Control mode 2: Level (top), control signals PIHA and PIHAW (middle), PIHA and PIHAW (bottom), wk (dotted lines). ure 6, PIHAW and PIHAhave almost the same overshoot (54.1% and 59%, respectively) but the settling time of the compensated one is ≈ 55s and of the latter ≈ 135s. Therefore although the compensating scheme has a small impact on the overshoot (a reduction of 5% percent points) it yields a relevant reduction on the settling time. On the other hand, the PIHW and PIH controllers present almost the same overshoot than PIHAW and PIHAones, but they have an oscillatory behavior with an offset due to the modeling errors. Note that such an error does not arrive when PIHAW and PIHAcontrollers are used. Because these modeling uncertainties, the resets are made to with incorrect values as it is clear on the bottom plot in Figure 6. Therefore, we can say that in this case our compensating strategy has a side effect decreasing ripple and improving the time response of the controlled variable, h1 . This can be seen on both the control signals at bottom and the controlled variable at the top of the Fig. 6. The reader can note that the compensating signal wk is activated several times, which can be interpreted as a tentative to adjust the control signal. Another set of experiments has been performed to evaluate our compensating strategy in the presence of disturbances. Then, with the system in equilibrium at h1 = 49.88cm, a constant perturbation of 18% has been added to the control signal from t = 400s to t = 800s. This has been repeated for all 4 controllers considered before. The regulated output is shown for each case in Figure 7 (top). The control signals of PIHA and PIHAW are shown in the middle plot, while the ones of PIH and PIHW are shown in the bottom plot. Due to the adaptive properties of the PIHA controller, the compensating signal wk is less required by PIHAW than by both PIH and PIHW controllers. It is worth to notice that the control effort of PIH controllers is very high, because this controller performs numerous erroneous reset values (due to the model uncertainties) during the perturbation period. To evaluate the performance of each controller, we have computed classical performance indexes IAE, IVE and

IFAC ROCOND 2018 Mariella M. Quadros et al. / IFAC PapersOnLine 51-25 (2018) 240–245 Florianopolis, Brazil, September 3-5, 2018

Level [cm]

PIH

PIHW

PIHA

PIHAW

Ref.

performed where our approach clearly leads to better performance indexes.

wk

55

REFERENCES

50 45

uk from PIHA & PIHAW

40 100 50 0

PIH & PIHW

-50 100

uk from

245

50 0

-50 400

500

600

700

800

900

1000

Time [s] Fig. 7. Operation mode 2: response to piecewise constant perturbation. Level (top), control signals PIHA and PIHAW (middle), PIHA and PIHAW (bottom), wk (dotted lines). IVU for the tests shown in figures 5-7. All indexes have been normalized by the respective value achieved by PIHAW and the relative performance is shown in Table 1. Positive values in Table 1 means a worst performance (a Table 1. Performance indexes percentages with respect to PIHAW Controller IVU IVE IAE Piecewise constant reference test (Mode 1) PIHA 115.28% 29.78% 90.67% PIHW 7.25% −0.88% 16.53% PIH 140.68% 11.40%% 42.83% Piecewise constant reference test (Mode 2) PIHA 143.03% 33.21% 82.50% PIHW 59.96% 9.25% 38.41% PIH 91.34% 8.16% 38.84% Piecewise constant perturbation test (Mode 2) PIHA 88.59% 68.80% 55.10% PIHW 71.66% 48.80% 74.82% PIH 208.79% 123.53% 173.70%

high integral value) w.r.t. PIHAW controller. The reader can observe that, except for the IVE with PIHW , the controller PIHAW leads to the best controller choice. The exception occurs in operating mode 1, where the nominal model is very close to the real dynamics and, thus, the controller PIHW is a better choice. Moreover, it is clear from Table 1 that our compensating strategy improves the performance indexes in all cases. 5. CONCLUSIONS We have handled the practical problem of digital implementation of hybrid PI controllers, where large sample periods may degenerate the performance expected from hybrid PI controllers. We provide a solution to such an issue that consists in a compensating signal added to the control signal, mitigating the sampling period effects. Our approach has been illustrated by simulation examples. Moreover, we have made real-time experiments in a level control system where a number of comparisons have been 356

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