PID-MMAC using an approximate H∞ loop-shaping metric

Proceedings of the 3rd IFAC Conference on Proceedings of the Proceedings of the 3rd 3rd IFAC IFAC Conference Conference on on Advances in Proportional-Integral-Derivative Proceedings of on Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on Control Advances in Proportional-Integral-Derivative Control Advances in Proportional-Integral-Derivative Control Available online at www.sciencedirect.com Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Proportional-Integral-Derivative Control Proceedings of the 3rd IFAC Conference on Control Advances in Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018

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Ghent, Belgium, May 9-11, 2018

IFAC PapersOnLine 51-4 (2018) 479–484

PID-MMAC using an approximate H ∞ PID-MMAC ∞ PID-MMAC using using an an approximate approximate H H∞ ∞ loop-shaping loop-shaping metric PID-MMAC using an metric approximate H∞ ∞ loop-shaping metric ∗ metric Rakesh Joshi ∗∗∗loop-shaping Victoria Serrano ∗∗ ∗∗ ∗∗ Konstantinos Tsakalis∗ ∗

Rakesh Joshi Konstantinos Rakesh Joshi ∗∗ Victoria Victoria Serrano Serrano ∗∗ Konstantinos Tsakalis Tsakalis∗∗ Rakesh Rakesh Joshi Joshi Victoria Victoria Serrano Serrano ∗∗ Konstantinos Konstantinos Tsakalis Tsakalis ∗Rakesh Joshi ∗ Victoria Serrano ∗∗ Konstantinos Tsakalis∗ ∗ School of Electrical, Computer and Energy Engineering, Arizona ∗ School of Electrical, Computer and Energy Engineering, ∗ School of Electrical, Computer and Energy Engineering, Arizona Arizona ∗ School of of Electrical, Electrical, Computer and Energy Energy Engineering, Arizona State University, Tempe, AZ 85281, USA School Computer and Engineering, Arizona

State University, Tempe, AZ 85281, USA State University, Tempe, AZ 85281, USA ∗∗∗ State University, Tempe, AZ 85281, USA Electrical Engineering Department, Universidad Tecnologica de of Electrical, Computer and Energy Engineering, Arizona ∗∗ State University, Tempe, AZ 85281, USA ∗∗School Electrical Engineering Department, Universidad Tecnologica ∗∗ Electrical Engineering Department, Universidad Tecnologica de de ∗∗

Electrical Engineering Department, Universidad Tecnologica de Panama, Lassonde, West 6th Avenue, David, Panama State University, Tempe, AZ 85281,Chiriqui, USA Electrical Engineering Department, Universidad Tecnologica de Lassonde, West 6th Avenue, David, Chiriqui, Panama Lassonde, West 6th Avenue, David, Chiriqui, Panama Lassonde, West 6th Avenue, David, Chiriqui, Panama Electrical Engineering Department, Universidad Tecnologica de Panama, Lassonde, West 6th Avenue, David, Chiriqui, Panama Panama, Lassonde, West 6th Avenue, David, Chiriqui, Panama Abstract: This paper proposes a adaptive control (PID-MMAC) algorithm Abstract: This paper proposes aa PID-multi-model PID-multi-model adaptive control (PID-MMAC) algorithm Abstract: This paper proposes PID-multi-model adaptive control (PID-MMAC) algorithm Abstract: This paper proposes a PID-multi-model adaptive control (PID-MMAC) algorithm using an approximate H metric that represents the frequency loop-shaping (FLS) cost ∞ Abstract: This paper proposes a PID-multi-model adaptive control (PID-MMAC) algorithm using an approximate H metric that represents the frequency loop-shaping (FLS) cost ∞ using an approximate H metric that represents the frequency loop-shaping (FLS) cost ∞ using an an approximate approximate H∞ metric that represents the frequency loop-shaping (FLS) cost objective. Existing MMAC algorithms use L or least squares-based cost functionals on aa Abstract: This paper proposes a PID-multi-model adaptive control (PID-MMAC) algorithm 2 using H metric that represents the frequency loop-shaping (FLS) cost ∞ algorithms objective. Existing MMAC use L or least squares-based cost functionals on 2 objective. Existing MMAC algorithms use L or least squares-based cost functionals on aa 2 objective. Existing MMAC algorithms use L or least squares-based cost functionals on suitable error signal to perform controller switching, but their strong dependency on the using an approximate H metric that represents the frequency loop-shaping (FLS) cost 2 or least ∞ objective. Existing MMAC algorithms use L squares-based cost functionals on a 2 suitable error signal to perform controller switching, but their strong dependency on the suitable error signal to perform controller switching, but their strong dependency on the suitable signal to perform controller switching, but their strong dependency on properties of the excitation makes them sensitive noise, disturbances and modeling errors. objective.error Existing MMAC algorithms use L least squares-based cost functionals onthe a 2 orto suitable error signal to perform controller switching, but their strong dependency on the properties of the excitation makes them sensitive to noise, disturbances and modeling errors. properties of the excitation makes them sensitive to noise, disturbances and modeling errors. properties of the excitation makes them sensitive to noise, disturbances and modeling errors. Alternatively, a system-norm-based cost function is advantageous for MMAC as it is less sensitive suitable error signal to perform controller switching, but their strong dependency on the properties of the excitation makes cost themfunction sensitive to noise, disturbances and modeling errors. Alternatively, a system-norm-based is advantageous for MMAC as it is less sensitive Alternatively, a system-norm-based cost is for as it is less sensitive Alternatively, asignals system-norm-based cost function is advantageous advantageous for MMAC MMAC as it iscost lessobjective sensitive to used adaptation. In the in the FLS properties of the excitation themfunction sensitive to noise, disturbances and modeling errors. ∞ norm Alternatively, a system-norm-based cost function advantageous for MMAC as it is less sensitive to the the specific specific signals used for formakes adaptation. In this this ispaper, paper, the H H norm in the FLS cost objective ∞ to the specific signals used for adaptation. In this paper, the H norm in the FLS cost objective ∞ norm to the specific signals used for adaptation. In this paper, the H in the FLS cost objective is approximated by frequency decomposition of the real-time signals using filter-banks. An Alternatively, a system-norm-based cost function is advantageous for MMAC as it is less sensitive ∞ norm to the specific signals used for adaptation. In this paper, the H in the FLS cost objective ∞ is approximated by frequency decomposition of the real-time signals using filter-banks. An is approximated by frequency decomposition of the real-time signals filter-banks. An is approximated by frequency decomposition of the real-time signals using filter-banks. An MMAC algorithm using is presented and its application to controller switching is to the specific signals usedthis for metric adaptation. In this paper, the H∞ norm in using the FLS cost objective is approximated by frequency decomposition of the real-time signals using filter-banks. An MMAC algorithm using this metric is presented and its application to controller switching is MMAC algorithm using this metric is presented and its application to controller switching is MMAC algorithm using this metric metric isaspresented presented and its application to using controller switching is discussed. The buck converter serves the motivating application where the adaptation seeks is approximated by frequency decomposition of the real-time signals filter-banks. An MMAC algorithm using this is and its application to controller switching is discussed. The buck converter serves as the motivating application where the adaptation seeks discussed. The buck converter serves as the motivating application where the adaptation seeks discussed. The buck converter serves as the motivating application where the adaptation seeks to compensate for degradation in its components (inductors and capacitors). A comparative MMAC algorithm using this metric is presented and its application to controller switching is discussed. The buck converter serves as the motivating application where the adaptation seeks to compensate compensate for for degradation degradation in in its its components components (inductors (inductors and and capacitors). capacitors). A A comparative comparative to to compensate for degradation in its components (inductors and capacitors). A comparative study is conducted of the proposed algorithm and an L -based MMAC algorithm under various discussed. The buck converter serves as the motivating application where the adaptation seeks 2 to compensate for degradation in its components (inductors and capacitors). A comparative study is is conducted conducted of of the the proposed proposed algorithm algorithm and and an an L L22 -based -based MMAC MMAC algorithm algorithm under under various various study study is the proposed algorithm and an L MMAC algorithm various excitation conditions. The results that the algorithm is less susceptible to the to compensate for of degradation in show its components and capacitors). Aunder comparative study is conducted conducted of the proposed algorithm and proposed an(inductors L22 -based -based MMAC algorithm under various excitation conditions. The results show that the proposed algorithm is less susceptible to the excitation conditions. The results show that the proposed algorithm is less susceptible to excitation conditions. Theproposed results show that the the proposed algorithm isalgorithm lessMMAC. susceptible to the the properties of the excitation signals as compared to the least squares-based study is conducted of the algorithm and an L -based MMAC under various 2 excitation conditions. The results show that proposed algorithm is less susceptible to the properties of the excitation signals as compared to the least squares-based MMAC. properties of the the excitation signals show as compared compared toproposed the least least squares-based squares-based MMAC. properties of signals excitation The results that theto is lessMMAC. susceptible to the properties conditions. of the excitation excitation signals as as compared to the the leastalgorithm squares-based MMAC. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. properties the excitation signals as compared the least squares-based MMAC. Keywords: of MMAC-Multi model adaptive control,toFLSFLSfrequency loop shaping, shaping, RSC-robust Keywords: MMAC-Multi model adaptive control, frequency loop RSC-robust Keywords: MMAC-Multi Keywords: MMAC-Multi model model adaptive adaptive control, control, FLSFLS- frequency frequency loop loop shaping, shaping, RSC-robust RSC-robust stability condition Keywords: MMAC-Multi model adaptive control, FLSfrequency loop shaping, RSC-robust stability condition stability stability condition condition Keywords: MMAC-Multi model adaptive control, FLS- frequency loop shaping, RSC-robust stability condition stability 1. condition INTRODUCTION In aa brief brief review, review, the the theoretical theoretical framework of the data1. INTRODUCTION In framework of of the the datadata1. INTRODUCTION In aa brief review, the theoretical framework 1. In brief review, the theoretical framework of the datadriven MMAC algorithms seeking robust stability and 1. INTRODUCTION INTRODUCTION In a brief review, the theoretical framework of the datadriven MMAC algorithms seeking robust stability and driven MMAC algorithms seeking robust stability stability and driven MMAC algorithms seeking robust and 1. INTRODUCTION safe switching is discussed in Stefanovic and Safonov In a brief review, the theoretical framework of the datadriven MMAC algorithms seeking robust stability and safe switching switching is is discussed discussed in Stefanovic and Safonov in Stefanovic and Safonov Proportional-Integral-Derivate (PID) (PID) controllers controllers are are aa safe safe switching is discussed in (2016a), Stefanovic and Safonov Safonov (2008), Buchstaller and French French Buchstaller and driven MMAC algorithms seeking robust stability safe switching is discussed in Stefanovic and Proportional-Integral-Derivate Proportional-Integral-Derivate (PID) controllers are a (2008), Buchstaller and (2016a), Buchstaller and (2008), Buchstaller and French (2016a), Buchstaller and Proportional-Integral-Derivate (PID) controllers are a widely used class of controllers in industrial processes, (2008), Buchstaller and French (2016a), Buchstaller and Proportional-Integral-Derivate (PID) controllers are a French (2016b), Sajjanshetty and Safonov (2014). A multisafe switching is discussed in Stefanovic and Safonov (2008), Buchstaller and French and widely used used class class of of controllers controllers in in industrial industrial processes, processes, French widely French (2016b), (2016b), Sajjanshetty and(2016a), Safonov Buchstaller (2014). A A multimultiand Safonov (2014). widely class in industrial processes, electric used circuits andof incontrollers mechanical systems. Low dimenProportional-Integral-Derivate (PID) controllers are a French Sajjanshetty and Safonov (2014). A widely used class of in controllers in systems. industrial processes, variable(2016b), MMACSajjanshetty algorithm using an estimator and aamulticon(2008), Buchstaller and French (2016a), Buchstaller and French (2016b), Sajjanshetty and Safonov (2014). A multielectric circuits and in mechanical systems. Low dimenelectric circuits and mechanical Low dimenvariable MMAC algorithm using an estimator and convariable MMAC algorithm using an estimator estimator and amulticonelectric circuits and mechanical Low dimensionalityused of the the controller, variety in of systems. readily available tun- variable widely class of in controllers industrial processes, MMAC algorithm using an and a conelectric circuits and in mechanical systems. Low dimentroller that can tolerate structural uncertainties is disFrench (2016b), Sajjanshetty and Safonov (2014). A variable MMAC algorithm using an estimator and a consionality of controller, variety of readily available tunsionality of theand controller, of readily available tuntroller that that can can tolerate tolerate structural structural uncertainties uncertainties is disis dissionality of controller, variety of readily tuning methods methods ease of variety implementation by electronic electric circuits andease in mechanical Low dimentroller can tolerate structural uncertainties is dissionality of the theand controller, variety of systems. readily available available tun- troller cussed that inMMAC Tan etalgorithm al. (2016). (2016). Theanapplication application of MMAC MMAC variable using estimator and a controller that can tolerate structural uncertainties is dising of implementation by electronic ing methods and ease of implementation by electronic cussed in Tan et al. The of in Tan al. (2016). application of MMAC ing methods and easemake of variety implementation by electronic circuits using op-amps them of popular asavailable compared to cussed sionality of theop-amps controller, readilyas tuncussed in et al. The application of ing methods and ease of implementation by electronic in pH pH that control is tolerate discussed inThe B¨ ling et al. al. (2007) (2007) and troller canet structural uncertainties is and discussed in Tan Tan et al. (2016). (2016). The application of MMAC MMAC circuits using make them popular compared to circuits using op-amps make them popular as compared to in control is discussed in B¨ ooling ling et in pH control is discussed in B¨ o et al. (2007) and ˚ circuits using op-amps make them popular as compared to other classes of controllers. A str¨ o m and H¨ a gglund (1995), ing methods and ease of implementation by electronic in pH control is discussed in B¨ ooaling et al. (2007) and circuits using op-amps make them popular as compared to refinement of this algorithm for noisy environment is cussed in Tan et al. (2016). The application of MMAC ˚ in pH control is discussed in B¨ ling et al. (2007) and other classes of controllers. A str¨ o m and H¨ a gglund (1995), ˚ other classes of controllers. A str¨ o m and H¨ a gglund (1995), refinement of of this this algorithm algorithm for for aa noisy noisy environment environment is is ˚ other classes A str¨ ooRivera m and H¨ aaal. gglund (1995), ˚str¨ ˚them circuits using op-amps as compared to refinement A om m and of H¨ acontrollers. gglundmake (2006), et (1986) prorefinement of this algorithm for aaling noisy environment is other classes of controllers. A str¨ mpopular and et H¨ gglund (1995), discussed in Bashivan and Fatehi (2012). All these MMAC in pH control is discussed in B¨ o et al. (2007) and ˚ refinement of this algorithm for noisy environment is A str¨ o and H¨ a gglund (2006), Rivera al. (1986) pro˚ discussed in in Bashivan Bashivan and and Fatehi Fatehi (2012). (2012). All All these these MMAC MMAC A str¨ o m and H¨ a gglund (2006), (1986) prodiscussed ˚ ˚ other classes Astr¨ oRivera m and H¨ aal. gglund A str¨ ooreview m and H¨ aacontrollers. gglund (2006), Rivera et al. (1986) provide ofof various methods used in et design and(1995), implein and these ˚ methods use use error signals that represent prediction or refinement this algorithm for (2012). a noisyAll environment is A str¨ m andof H¨ gglundmethods (2006), Rivera et al. (1986) pro- discussed discussed inofBashivan Bashivan and Fatehi Fatehi (2012). All these MMAC MMAC vide review of various methods used in design and implemethods error signals that represent prediction or vide review various used in design and implemethods use error signals that represent prediction or ˚ vide review of various methods used in design and implementation of PIDs. Even though a well tuned fixed PID methods use error signals that represent prediction or A str¨ o m and H¨ a gglund (2006), Rivera et al. (1986) procontrol error or their combination, to compute the metric discussed in Bashivan and Fatehi (2012). All these MMAC vide review of various methods used in design and implemethods use or that represent prediction or mentation of of PIDs. PIDs. Even Even though though aa well well tuned tuned fixed fixed PID PID control control error error orerror theirsignals combination, to compute compute the metric metric mentation their combination, to the mentation of Even though aa well tuned fixed PID controller yields good performance around anand operating or their combination, to compute the vide reviewyields various methods used in design impleused in in error the controller switching and their cost functionals methods use thatand represent prediction or mentation ofofPIDs. PIDs. Even though well tuned fixed PID control control error orerror theirsignals combination, totheir compute the metric metric controller good performance around an operating used the controller switching their cost functionals controller yields good performance around an operating used in the controller switching and cost functionals controller yields performance around an operating point, it it may may failgood to compensate for the nonlinearities in the controller switching and their cost functionals mentation of PIDs. Even though a for well tuned fixed PID used are signal-norms of the error signals. control error or their combination, to compute the metric controller yields good performance around an operating used in the controller switching and their cost functionals point, fail to compensate the nonlinearities are signal-norms signal-norms of of the the error error signals. signals. point, it may to compensate for the nonlinearities point, it fail to for the and time-varying time-varying characteristics perturbing theoperating nominal are are signal-norms of error signals. controller yieldsfail performance around an used in the controller and their cost functionals point, it may may failgood to compensate compensate for the nonlinearities nonlinearities are signal-norms of the theswitching error signals. and characteristics perturbing the nominal and time-varying characteristics perturbing the nominal In this paper, we discuss the development of an an alternative alternative and time-varying characteristics perturbing the nominal plant. To overcome this shortcoming, the adaptation of point, it may fail to compensate for nonlinearities In this paper, we discuss the development of are signal-norms of the error signals. and time-varying characteristics perturbing the nominal In this paper, we discuss the development of an alternative plant. To To overcome overcome this this shortcoming, shortcoming, the the adaptation adaptation of of In this paper, we discuss the development of an alternative plant. to the usual update laws that is based on the approximaIn this paper, we discuss the development of an alternative plant. To overcome this shortcoming, the adaptation of PID time-varying parameters is characteristics favorite candidate solution. and perturbing the nominal to the the usual usual update update laws laws that that is is based based on on the the approximaapproximaplant. To overcome shortcoming, the adaptation of to PID parameters is aa this favorite candidate solution. to the update laws that is PID parameters is a favorite candidate solution. tion of usual system norms as cost cost functionals, instead of signal signal In this paper, we discuss the development ofthe anapproximaalternative to the usual update laws that is based based on oninstead the approximaPID parameters is aa this favorite candidate the solution. plant. To overcome shortcoming, adaptation of tion tion of system norms as functionals, instead of PID parameters is favorite candidate solution. of system norms as cost functionals, of signal Adaptive control is an extensively researched field with tion of system norms as cost functionals, instead of signal norms. The general adaptation of MMAC schemes uses to the usual update laws that is based on the approximation of system norms as cost functionals, instead of signal Adaptive controlisis isa an an extensively researched field with with norms. PID parameters favorite candidate solution.field norms. The The general general adaptation adaptation of of MMAC MMAC schemes schemes uses uses aaa Adaptive control extensively researched Adaptive control is an extensively researched field with many adaptive control schemes studied in the literature. norms. The general adaptation of MMAC schemes uses aa metric derived from the on-line data for the selection of tion of system norms as cost functionals, instead of signal Adaptive control is an extensively researched field with norms. The general adaptation of MMAC schemes uses many adaptive adaptive control control schemes schemes studied studied in in the the literature. literature. metric metric derived derived from from the the on-line on-line data data for for the the selection selection of many of many adaptive control schemes studied in the literature. Some of them are gain scheduling, multi-model adaptive metric derived from the on-line data for the selection of Adaptive control is an extensively researched field with the controller from a bank of controllers C and can be norms. The general ofdata MMAC schemes usesbe a many adaptive control studied in the literature. i and metric derived fromadaptation on-line for the selection of Some of of them are are gain schemes scheduling, multi-model adaptive the the controller controller from athe bank of controllers controllers C Some them gain scheduling, multi-model adaptive ii and can from bank of C can be Some them are gain scheduling, multi-model adaptive controlof e.g., Stefanovic and Safonov Safonov (2008), Anderson the controller from bank of controllers C can be many adaptive control schemes studied in the literature. written as follows metric derived from aaathe on-line data for the selection of ii and Some of them are gain scheduling, multi-model adaptive the controller from bank of controllers C and can be control e.g., Stefanovic and (2008), Anderson written as follows control e.g., Stefanovic and Safonov (2008), Anderson written as control e.g., and Safonov (2008), Anderson ∗a bank of controllers Ci and can be et al. al. of(2001), (2001), Hespanha et al. al. (2001), Hespanha and the written as follows follows Some themStefanovic are gain scheduling, multi-model adaptive controller from control e.g., Stefanovic and Safonov (2008), Anderson written as follows C (1) ∗ = arg min J(Ci ; x) et Hespanha et (2001), Hespanha and ∗ et al. (2001), Hespanha et al. (2001), Hespanha and C = arg min (1) ∗ C arg min J(Cii ;;; x) x) (1) Ci J(C et al. (2001), Hespanha et al. (2001), Hespanha and Morse (1996), Anderson et al. (2000), model reference control e.g., Stefanovic and Safonov (2008), Anderson ∗ = written as follows et al. (2001), Hespanha et al. (2001), Hespanha and C = arg min (1) C ii ; x) Morse (1996), (1996), Anderson Anderson et et al. al. (2000), (2000), model model reference reference C = arg min J(C x) (1) Cii J(C Morse Cfrom i Morse (1996), Anderson et al. (2000), model reference ∗ adaptive control and indirect adaptive control, e.g., Sastry denotes controller the controller bank, x Where C et al. (2001), Hespanha et al. (2001), Hespanha and C i denotesCcontroller i J(C ; x) Morse (1996), Anderson et al. (2000), model reference = arg min adaptive control and indirect adaptive control, e.g., Sastry from the controller bank, x Where C i ii denotes controller from the controller bank,(1) adaptive control and indirect adaptive control, e.g., Sastry x Where C Cfrom adaptive control and indirect adaptive control, e.g., Sastry denotes signals controller from the controller bank, x Where Cmeasured i J(Cthe and Bodson (2011), Ioannou and Sun (1996). Interest in denotes and , x) is the cost funcMorse (1996), Anderson et al. (2000), model reference ii denotes i adaptive control and indirect adaptive control, e.g., Sastry controller controller bank, x Where C and Bodson (2011), Ioannou and Sun (1996). Interest in denotes measured signals and J(C , x) is the cost funci and Bodson (2011), Ioannou and Sun (1996). Interest in denotes measured signals and J(C , x) is the cost funci and Bodson (2011), Ioannou and Sun (1996). Interest in denotes measured signals and J(C , x) is the cost funcmulti-model adaptive control (MMAC) has gained traction tional associated with a metric used in the controller adaptive control and indirect adaptive control, e.g., Sastry denotes controller from the bank, x Where C i , x)controller i and Bodson (2011), Ioannou and Sun (1996). Interest in denotes measured signals and J(C is the cost funci multi-model adaptive adaptive control control (MMAC) (MMAC) has has gained gained traction traction tional associated with aa metric used in the controller multi-model tional associated with metric in the controller multi-model control has gained traction tional associated with aa and metric used in the controller significantly in recentIoannou past. In In(MMAC) thisSun scheme, theInterest controller switching. Typical cost functionals are the square erand Bodson adaptive (2011), and (1996). in denotes measured signals J(Cused ismean the cost funci , x) multi-model adaptive control (MMAC) has gained traction tional associated with metric used in the controller significantly in recent past. this scheme, the controller switching. Typical cost functionals are the mean square ersignificantly in recentcontrol past. In this scheme, the controller Typical cost functionals are the mean ersignificantly in In(MMAC) this scheme, the acontroller controller switching. Typical cost functionals are the square erused in in the the closed-loop closed-loop system is selected selected from bank of of switching. ror or exponentially weighted mean square error, where the multi-model adaptive has from gained tional associated a metric used the square controller significantly in recent recent past. past. In this scheme, the switching. Typical with cost functionals are theinmean mean square erused system is a traction bank ror or exponentially weighted mean square error, where the used in the closed-loop system is selected from a bank of ror or exponentially weighted mean square error, where the used in the closed-loop system is selected from aacontroller bank of ror or or represents exponentially weighted mean are square error,and where the controllers based on metrics derived from the on-line data. error difference between expected actual significantly in recent past. In this scheme, the switching. Typical cost functionals the mean square erused in the closed-loop system is selected from bank of ror exponentially weighted mean square error, where the controllers based based on on metrics metrics derived derived from from the the on-line on-line data. data. error error represents represents difference difference between between expected expected and and actual actual controllers controllers on derived from on-line data. difference expected actual used in thebased closed-loop system is selected from a bank of error ror or represents exponentially weightedbetween mean square error,and where the controllers based on metrics metrics derived from the the on-line data. error represents difference between expected and actual controllers based on metrics derived from the on-line data. error represents difference 2405-8963 © 2018, IFAC (International Federation of Automatic Control) by Elsevier Ltd. All rightsbetween reserved. expected and actual Copyright 2018 IFAC 479 Hosting Panama, Panama, ∗∗ Panama,

Copyright 2018 IFAC 479 Copyright © 2018 IFAC 479 Peer review© of International Federation of Automatic Copyright ©under 2018 responsibility IFAC 479Control. Copyright © 2018 IFAC 479 10.1016/j.ifacol.2018.06.141 Copyright © 2018 IFAC 479

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response (ex. prediction error, control error etc.). If the signals are persistently exciting, there is no noise or disturbances, and the controller bank is appropriately selected, such an adaptive law produces optimal performance. However, perturbations such as sensor noise and disturbances result in a cost functional whose minimum can be significantly different depending on the signals at hand (e.g., see Tsakalis (1996)) and whose consequence is often poor performance or adaptation bursts. Furthermore, the worstcase performance depends on the parametric uncertainty and the problem is not easily detected with a “noisy simulation”. This may provide a false sense of security, which is further exacerbated in the MMAC case where one considers only a small number of controllers. Potential partial remedies for this problem have been recognized in the incorporation of dead-zones in the adaptive law but their design is far from providing a complete practical answer. In an effort to design an adaptation cost functional that provides a better description of our control objective, hence resulting in a weaker dependence on signal properties, we propose the approximation of the norm of an error system. An example of such an error system is the difference between the actual loop and its target, arising in frequency loop-shaping. Controllers based on the optimization of such objectives have quantifiable robustness properties that have constituted the cornerstone of the development of robust control theory since the 80’s. While robust control objectives have often been used in adaptive controllers, our approach is different in that we incorporate them in the estimator (or optimizer) and not just in the derivation of the control law. A general description of one such cost functional can be broadly described as follows: C ∗ = arg min max J(Ci ; x) (2) Ci

x

The above structure allows us to describe cost functional in terms of the worst-case behavior of each controller and the selection of the controller is the one that produces minimum worst-case behavior. Motivated by the results of the H∞ robust control theory, we consider the decomposition of the cost function in terms of the frequency components of the signal x: (3) C ∗ = arg min max J(Ci ; xj ) Ci

j

where now xj represents the energy of measured signals at different frequency directions and J(·) is a system norm. The former can be achieved under a linearity assumption by processing the input and output signals through “filter banks” (FB), an example of which is shown in Fig. 1. In this manner, the signal decomposition may only approximate its frequency content, but it is exact at all times and does not suffer the usual windowing effects of Fast Fourier transforms (FFT). On the other hand, a system norm computation would require the normalization of the system output RMS power by the corresponding input power, something that is achievable by requiring excitation that is persistent and of sufficient magnitude. Thus, the proposed approach enables the approximation of an H∞ cost functional using a single data sequence. Its main drawbacks are in the dimensionality of the solution and the excitation requirements. We must point 480

Fig. 1. Bode diagram of a filter bank out, however, that neither one represents insurmountable limitations for many practical cases. With the advances in computing hardware and parallel implementations, the use of filter-banks of band-pass filters is a viable alternative to FFT algorithms. And excitation is often present in practical problems, or can be requested for short time periods, and can at least be verified by a monitoring program before updating the controller parameters. The same monitoring logic can also limit the use of excitation directions to the ones that contain sufficient excitation, e.g., through the use of an adaptation dead-zone. It is our view that the critical part of adaptive control is the ability of the adaptation to take advantage of short or occasional excitation periods in the most efficient way, instead of attempting to operate with the least assumptions but opening the door to undesirable nonlinear behavior like bursting. Over the past two decades, we have employed the FLS approach for PID controller tuning with excellent results in a variety of practical problems. In FLS the specifications for the closed-loop system are expressed in terms of the target loop and the PID parameters are obtained by solving a minimization of a distance metric between the actual loop and the target loop, defined in terms of an H∞ norm. The off-line FLS algorithm for PID tuning is discussed Tsakalis et al. (2002) Grassi and Tsakalis (2000). An on-line version of FLS algorithm for tuning PID parameters with an approximate H∞ cost functional is discussed in Tsakalis and Dash (2013), while an extension and application to performance monitoring is presented in Tsakalis and Dash (2007). We use this FLSbased approximate H∞ metric in the development of our proposed MMAC algorithm. In the rest of this paper we present some details on the formulation of this problem and its computation for a PID control structure (Section 2).We also discuss some of the typical characteristics of the solution, applied to the PID control of a buck converter. Buck converters with a fixed, op-amp-based PID controller are commonly used in the regulation of DC voltage. However, the degradation of its components (e.g., inductor or capacitor) causes regulation performance to deteriorate beyond the capabilities of the fixed PID controller. This specific application is of particular interest for MMAC since common implementations can include banks of resistances that can be used to switch among a finite number of controllers (Section 3). Using this as a case study, we demonstrate the application of the proposed MMAC and compare with the results obtained with existing approaches. The novelty of the contribution is in the usage of the approximate H∞ metric in the formulation

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e = S(CG − L)u; invoking the fact that y = Gu, we obtain e = SCy − T u. Thus, the RSC can be expressed as: ||e||2 RSC = ||S(GC − L)||∞ = sup (7) ||u||=0 ||u||2

Fig. 2. MMAC schematic diagram of the MMAC algorithm and a systematic construction of a test example that shows its low susceptibility to parameter misadjustment and burst phenomena caused by variations in the signal excitation. 2. MULTI-MODEL ADAPTIVE CONTROLLER Letting G be the plant and C the controller, the FLS objective is to minimize the distance between the loop transfer function GC and the target loop L. A frequencyweighted version of this distance is obtained by the application of the Small Gain Theorem, yielding the robust stability condition (RSC) (4) S(GC − L)∞ < 1 Where S = 1/(1+L) is the target sensitivity. The left-hand side of Eq. 4 can be interpreted as a particular, closedloop-relevant, distance between the target and actual loops and S(CG − L)∞ is the FLS optimization objective. A desirable feature of this RSC metric is that it is normalized such that a value less than 1 guarantees closed-loop stability. Furthermore, while it is not explicitly a robust performance metric, small values of the RSC (e.g., 0.1-0.3) indicate a practically close matching between the nominal and the actual loop sensitivities. Next, consider a parameterization of the PID controller as Kd s Ki C(s; θ) = Kp + + (5) s (τ s + 1) where θ = [Kp , Ki , Kd ] are the PID parameters proportional, integral, and derivative gains, respectively, and τ is the time constant of the filter used in the implementation of derivative control and its value is chosen a priori. For this PID parameterization the RSC minimization problem is convex, and, in fact, linear in the parameters θ: θ∗ = arg min ||S(GC(θ) − L)||∞ (6) θ∈M

Here θ ∈ M are convex parameter constraints, e.g., positivity of the PID gains, or a minimum phase condition to avoid right half-plane cancellations. The quality of the approximation of the target by the PID loop can be assessed by looking at the optimal RSC value. If it is less than 1, the stability of the closed-loop system is guaranteed and lower values of optimal RSC implies the actual loop-performance closer to the target. The convergence of the FLS method to reasonable PID tunings, as compared to standard methods in the literature, e.g., ˚ Astr¨ om and H¨ agglund (1995) ˚ Astr¨ om and H¨ agglund (2006) Rivera et al. (1986), has been established in Grassi and Tsakalis (2000). To convert this method for on-line use, we let (u, y) be the input-output pair of the plant and define the error signal 481

Using filter banks of band-pass filters Fi to decompose the signals, and we arrive the following estimate of RSC ||SCFi y − T Fi u||2,δ ||Fi e||2,δ RSC ≈ max = max (8) i i ||Fi u||2,δ ||Fi u||2,δ Here we also use the exponential weighted exponential 1 ∞ norm ||x||2,δ = { −∞ e2δt |x|2 (t)dt} 2 , δ > 0 in the computation of upper bound of RSC approximation. This allows for the recursive and efficient computation of RSC as the ratio of the outputs of two first order filters. This approximation is similar to an FFT decomposition of the signal when the filter bank is composed of sharp bandpass filters. In a strict sense it is only a lower bound of the actual RSC because it is computed for a finite number of signals. However, for reasonably smooth practical problems, a good approximation can be achieved with a modest number of band-pass filters. The significance of this formulation is the PID tuning can be achieved in the sense of system norms, having reduced sensitivity to the actual frequency content of the signals. Furthermore, the use of normalized quantities that are motivated by the Small-Gain Theorem means that the confidence in the adaptation can be assessed (by a supervisory system) in terms that are problem independent. 2.1 RSC based PID-MMAC algorithm In a quick overview, MMAC uses the plant input-output data to estimate a performance or health metric, assessing the suitability of a controller in the loop. This metric is then used to rank the potential controllers and switching to the best candidate. The key difference of the proposed metric is in the approximation of the H∞ gain of the mismatch operator, instead of the more frequently used L2 -norm of the error signal. Our expectation, supported by simulation studies, is that the proposed metric would translate in a reduced sensitivity of the optimal solution to the input signal properties and in a more reliable performance.

For the MMAC problem, we consider the bank of plants [Gj ], j = 1, 2 . . . n, and a corresponding bank of controllers [Cj ], designed with an off-line FLS algorithm and with a target loop-shape L. The RSC estimate of each controller Cj is computed using the plant input-output data according to Eq. 8 and is denoted by RSCj . (Here, it helps to evaluate the controllers using the same criterion as for their design.) The objective of a multi-model adaptive switching control is to select the controller that minimizes the estimated RSC, which can thus be written as the following optimization problem: ||SCj Fi y − T Fi u)||2,δ Cj∗ = arg min max (9) i Cj ||Fi u||2,δ Although the above switching logic can be used with any type controllers, the PID structure (Eq. 5) enables us to draw on the extensive available insight to design the filter banks and determine the set of reasonable controllers. For the PID-MMAC, the parameter adaptation becomes

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θj ∗ = arg min max θj

i

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||(SFi W T y)θj − T Fi u)||2,δ ||Fi u||2,δ

(10)

s ] and θj = [Kpj Kij Kdj ]T are where W = [1; 1s ; τ s+1 the parameters of the controller Cj (Eq. 5). To avoid excessive switching, a common approach is to introduce a switching threshold or a hysteresis logic, trading off transient performance for a less oscillatory behavior.

Thus, the PID-MMAC algorithm, based on the approximation of the H∞ RSC, becomes: • Design PID controllers [Cj ] for the set of plants Gj for the selected the target loop L(s) . • Initialize to the nominal controller. • For each time step k, computethe RSC values for all controllers [Cj ] RSCj (k) = max

||SFi W T yθj − T Fi u||2,δ ||Fi u||2,δ

(11) i . • If RSCselected > (1 + h)minRSCj (k), then switch to j

the controller C(θj ∗ ), where RSCselected is the RSC value of the currently selected controller, j ∗ = arg minRSCj (k) and h is a hysteresis parameter (h > 0). j

The evaluation of the RSC metric (as all similar metrics) involves the inherent trade-off between the long and short term memory. The former has noise immunity while the latter offers fast adaptation to rapidly changing conditions. In our case, a comparison between the estimated exponentially weighted norm and the actual 2-norm of the RSC indicates that a value δ = BW/10, where BW is the bandwidth of target loop transfer function is a reasonable choice. The RSC computation also needs to handle the possibility of insufficient excitation. The disturbance threshold term described in Tsakalis and Dash (2007) can be used in the RSC computation to avoid bias in the RSC values because of lack of excitation. To avoid bias in the parameter estimate because of disturbances or noise, the plant inputoutput pair used in the RSC estimation is filtered using a band pass filter that only allows signals in the frequencies of interest, namely around the loop crossover frequency. 3. PID-MMAC DESIGN FOR A BUCK CONVERTER A schematic diagram of a buck converter is shown in Fig. 3. It consists of a MOSFET, a LC circuit with a power inductor (H) and capacitor (Q) and the goal of the converter is to regulate the voltage across the resistive load to a set-point by controlling gate of MOSFET. The MOSFET gate is controlled by varying the duty cycle of a pulse-width-modulated (PWM) signal and an op-ampbased PID controller is typically employed to generate the PWM signal based on the output voltage measurement. Degradation in the inductor and capacitor components can result in the degradation of the performance of the buck converter, which the fixed PID may be unable to compensate. To solve this problem, multiple PID controllers can be made available, e.g., through a resistor bank, with a MMAC algorithm used in the selection the best controller. 482

Fig. 3. Schematic diagram of a buck converter In this section, we show the results from application of the proposed MMAC in the buck converter and we also demonstrate pros and cons of MMAC using the proposed approximate H∞ cost functional over the standard, signalbased, L2 cost functionals. Details of the buck converter components and justification for their selection are discussed in Serrano (2016). For this study we restrict ourselves to transfer functions of the buck converter derived using first principles with degradation in its components. It can be also seen from Serrano (2016) that degradation in the capacitor and inductance only affects the resonance frequency (fo = √ 1 ) of the buck converter’s transfer HQ

function, so for this study we only consider combined degradation in capacitance and inductance. The transfer functions of the buck converter with the percent degradation in its components from the nominal are listed below. The frequency units are rad/µsec 0.0098(s+14.29) (s2 +0.1419s+0.2778) = (s20.01535(s+14.29) +0.1774s+0.4341) = (s20.0273(s+14.29) +0.2365s+0.7717) 0.04(s+14.29) = (s2 +0.2838s+1.111) = (s20.0614(s+14.29) +0.3548s+1.736))

• 0 percent: G1 =

• 20 percent: G2 • 40 percent: G3 • 50 percent: G4 • 60 percent: G5

The target loop shape is selected to achieve a closedloop bandwidth 0.7 rad/µsec and is used to tune PID controllers. Its transfer function is: 0.65019(s2 + 0.467s + 0.2531) (12) L(s) = s(s2 + 0.1419s + 0.2778) which includes resonance characteristics from the plant, since a PID by itself has no degrees of freedom to alter plant zeros. (Also see Grassi and Tsakalis (2000), Tsakalis et al. (2002) for comments on target selection.) Finally, the MMAC controller bank is formed using PID controllers designed for the above-listed plants, with a derivative time constant τ = 0.01 and for the target loop-shape L(s) (Eq. 12). We also include some spurious controllers in the controller bank and PID parameters of all controllers in Table 1 to demonstrate the difference of parameter misadjustment in system-norm and signal-norm adaptation. 4. MMAC RESULTS To test the proposed MMAC algorithm, the plant with no degradation G1 is switched to a plant with degradation(G2G5) at 400 micro seconds. Excitation for the adaptation algorithms is defined as a square wave (−5V to +5V ) in the reference signal and a sine wave (magnitude 0.2V ) with different frequencies is injected at the plant input (ui in Fig 2) as a disturbance. White noise is also added at

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483

This cost is consistent with the FLS objective used for the initial tuning of all controllers in the controller bank. Then, the controller switching is performed based on minimization of the same cost estimated from input-output data. A buck converter with degradation of its components is used as a motivating example to demonstrate the application proposed algorithm and it is tested under different cases of excitation and disturbances. Our simulation results show that the proposed H∞ -MMAC algorithm is far less susceptible to the type of excitation and the disturbance, compared to a more common L2 -MMAC algorithm whose cost objective is based on the error signal (for the same FLS objective).

Table 1. PID gains for the buck converter controller bank. Controller Kp Ki Kd Comments C1 2.007 1.185 4.559 Controller designed for plant G1 C2 1.062 1.232 2.835 Controller designed for plant G2 C3 0.487 1.296 1.463 Controller designed for plant G3 C4 0.344 1.317 0.969 Controller designed for plant G4 C5 0.227 1.348 0.651 Controller designed for plant G5 C6 1.326 1.142 2.733 Spurious controller C7 1.102 1.135 1.346 Spurious controller C8 1.018 1.132 0.853 Spurious controller C9 0.511 1.117 1.067 Spurious controller C10 0.933 1.132 0.481 Spurious controller

This result demonstrates the importance of using systembased norms and cost functionals as adaptation metrics,

plant output to mimic sensor noise. The frequencies of the excitation signals were selected to bring out the differences in the controllers as the plant parameters deteriorate. Details of other settings used in the simulations are listed below. • The proposed MMAC with H∞ cost functional: ||(SFi W T y)θj − T Fi u)||2,δ θj∗ = arg min max (13) i θj ||Fi u||2,δ • A common MMAC with L2 cost functional on the error signal θj∗ = arg min ||(SW T y)θj − T u)||2,δ (14) θj

• • • •

Closed-loop bandwidth: BW = 0.7rad/µsec Derivate time constant τ = 0.01 µsecs Forgetting factor (δ)=5 × 10−4 Filter Bank [Fi ]: 20 filters logarithmically placed between 0.1 × BW and 10 × BW • Hysteresis parameter: h = 0 • Frequency of sine wave injected at plant input: · S1: 0.015rad/µsec · S2: 0.15rad/µsec · S3: 1.5rad/µsec

The results of our simulation study are shown in the sequence of figures Fig. 4, 5, 6, for the transition from the plant G1 to the plants G2, G3, G5, respectively. These plots show the controller index selected by the proposed H∞ MMAC in a red trace and the index selected by the common L2 MMAC in a blue trace. Ideally, after an initial transient, we expect both algorithms to converge to the corresponding target controller. It can be clearly seen from these figures that the presence of disturbances cause the L2 algorithm to fail, while the proposed H∞ MMAC algorithm converges to correct controller irrespective of the excitation. Some sample output plots are shown in Fig. 7 showing that the proposed MMAC performance is closer to the target. It goes without saying that these are the interesting cases that bring out the difference between the two controllers. Cases with low levels of perturbations or different frequency contents do not produce mis adjustment and are not included here. 5. CONCLUSION This paper presents an H∞ -based MMAC algorithm using an approximation of a system norm as a cost functional. 483

(a)

(b)

Fig. 4. MMAC results for plant switched from G1 to G2 (a) For input injection S1 (b) or input injection S3; For input injection S2 results are similar to of S1 and S3. For all excitation signals S1-S3 the L2 -based MMAC is switching between the correct controller C2 and a spurious controller C6 whereas the H∞ -MMAC converges to the correct controller C2

(a)

(b)

Fig. 5. MMAC results for plant switched from G1 to G3 (a) For input injection S1, (b) for input injection S3; For input injection S2 results are similar to of S1 and S3. For all excitation signals S1-S3, the L2 -MMAC converges to spurious controller C9, whereas the H∞ MMAC converges to correct controller C3

(a)

(b)

Fig. 6. MMAC results for plant switched from G1 to G5: (a) For input injection S1, (b) for input injection S3; For input injection S2 results are similar to of S1. For excitation signals S1 and S2 the L2 -MMAC is switching between spurious controller C9 and C10, while for the excitation signal S3 it converges to the correct controller C5. The H∞ -MMAC converges to correct controller C5 for all excitation signals.

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even if they are only approximate. On the other hand, signal-based functionals produce results that are excitation dependent and can be very sensitive to the excitation conditions, especially in perturbed cases. For such cases, looking at ideal-case simulations, or injecting random noise offers little indication of the worst-case behavior. The study of robustness in adaptation should involve signals that are designed specifically to produce misadjustment. Finally, it is also interesting to observe that in these terms of robust performance, multi-model adaptation does not seem to offer an advantage over continuous adaptation. Under perturbed or insufficiently excited conditions, they can both be driven to spurious controllers, if such controllers are contained in the admissible set. ACKNOWLEDGEMENTS The authors acknowledge the financial support from the SERDP environmental restoration project 2239. REFERENCES Anderson, B., Brinsmead, T., Liberzon, D., and Stephen Morse, A. (2001). Multiple model adaptive control with safe switching. International journal of adaptive control and signal processing, 15(5), 445–470. Anderson, B.D., Brinsmead, T.S., De Bruyne, F., Hespanha, J., Liberzon, D., and Morse, A.S. (2000). Multiple model adaptive control. part 1: Finite controller coverings. International Journal of Robust and Nonlinear Control, 10(11-12), 909–929. ˚ Astr¨ om, K. and H¨ agglund, T. (1995). PID Controllers. Setting the standard for automation. International Society for Measurement and Control. ˚ Astr¨ om, K. and H¨ agglund, T. (2006). Advanced PID Control. ISA-The Instrumentation, Systems, and Automation Society. Bashivan, P. and Fatehi, A. (2012). Improved switching for multiple model adaptive controller in noisy environment. Journal of Process Control, 22(2), 390 – 396.

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Fig. 7. Output plots of the buck converter: (a) For input injection S3 for plant switching G1 to G3, (b) for input injection S2 for plant switching G1 to G4, (c) for input injection S1 for plant switching G1 to G5, (d) for input injection S2 for plant switching G1 to G5. In all four cases the proposed H∞ -MMAC produces tracking performance closer to the target, as compares to the L2 -based MMAC. 484

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