Towards Optimal Tuning of Robust Output Feedback MPC

6th 6th IFAC IFAC Conference Conference on on Nonlinear Nonlinear Model Model Predictive Predictive Control Control 6th on Model 6th IFAC IFAC Conference Conference on Nonlinear Nonlinear Model Predictive Predictive Control Control Madison, WI, USA, August 19-22, 2018 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Madison, WI, USA, August 19-22, 2018 Available online at www.sciencedirect.com Madison, WI, USA, August 19-22, 2018 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Madison, WI, USA, August 19-22, 2018

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IFAC PapersOnLine 51-20 (2018) 134–140

Towards Optimal Tuning of Towards Optimal Tuning of Towards Optimal Tuning of Robust Output Feedback MPC Towards Optimal Tuning of Robust Output Feedback MPC Robust Output Feedback MPC Robust Output Feedback MPC Markus K¨ ogel Rolf Findeisen

Markus K¨ o gel Rolf Findeisen Markus o Markus K¨ K¨ ogel gel Rolf Rolf Findeisen Findeisen Markus K¨ o gel Rolf Findeisen Markus K¨ o gel Rolf Findeisen Laboratory for for Systems Systems Theory Theory and Automatic Control Laboratory for Theory and Automatic Control Laboratory and Control Laboratory for Systems Systems Theory Magdeburg, and Automatic Automatic Control Laboratory for Systems Theory and Automatic Control Otto-von-Guericke University Germany. Otto-von-Guericke University Magdeburg, Germany. Otto-von-Guericke University Magdeburg, Germany. Otto-von-Guericke University Magdeburg, Germany. Laboratory for Systems Theory and Automatic Control e-mail: {markus.koegel, rolf.findeisen}@ovgu.de. Otto-von-Guericke University Magdeburg, Germany. e-mail: {markus.koegel, rolf.findeisen}@ovgu.de. e-mail: rolf.findeisen}@ovgu.de. e-mail: {markus.koegel, {markus.koegel, rolf.findeisen}@ovgu.de. Otto-von-Guericke University Magdeburg, Germany. e-mail: {markus.koegel, rolf.findeisen}@ovgu.de. e-mail: {markus.koegel, rolf.findeisen}@ovgu.de. Abstract: Optimal design and tuning model controllers is important as it Abstract: Optimal design and tuning of of model predictive predictive controllers is important as it Abstract: Optimal design and tuning of model predictive controllers is important as it Abstract: Optimal Optimal design and of predictive controllers is important as it and tuning tuning of model model We predictive controllers is of important as it significantly influences the achievable performance. consider the problem tuning a tube Abstract: design significantly influences the achievable performance. We consider the problem of tuning a tube significantly influences the achievable performance. We consider the problem of tuning a tube significantly influences theoutput achievable performance. We consider the problem problem of tuning aa -as tube Abstract: Optimal design and tuning of model predictive controllers is important it significantly influences the achievable performance. We consider the of tuning tube based robust predictive feedback controller, which utilizes a linear feedback the based robust predictive output feedback controller, which utilizes a linear feedback the based robust predictive output feedback controller, which utilizes aa linear feedback --tube the based robustinfluences predictive output feedback controller, which utilizes linear feedback the significantly the achievable performance. We consider the problem of tuning a tube controller to take the uncertainties into account. The design and the tuning of this based robust predictive output feedback controller, which utilizes a linear feedback the tube controller to take the uncertainties into account. The design and the tuning of this tube --- to the uncertainties into account. The design and the tuning of this tube controller controller to atake take the uncertainties into account. account. The design and the feedback tuning of- this this based robust predictive output feedback controller, which utilizes aand linear the linear feedback big influence on the tube and thus the resulting overall performance. tube controller -has to take the uncertainties into The design the tuning of linear feedback has a big influence on the tube and thus the resulting overall performance. linear feedback has a big influence on the tube and thus the resulting overall performance. linear feedback has a big influence on the tube and thus the resulting overall performance. tube controller to take the uncertainties into account. The design and the tuning of this We propose a tuning the Youla leading to a overall convex optimization linear feedback has a exploiting big influence on the parametrization tube and thus the resulting performance. We propose tuning exploiting the Youla parametrization leading to convex optimization We aa exploiting the Youla leading to convex optimization We propose propose a tuning tuning exploiting the Youla parametrization leading to aaaa overall convex optimization linear feedback has aare bigprovided influence on the parametrization tuberobust and thus the resulting performance. We propose a tuning exploiting the Youla parametrization leading to convex optimization problem. Conditions to guarantee constraint satisfaction and robust setproblem. Conditions are provided to guarantee robust constraint satisfaction and robust setproblem. Conditions are provided to guarantee robust satisfaction and robust problem. Conditions are provided to Youla guarantee robust constraint constraint satisfaction andoptimization robust setsetWe propose a tuning exploiting the parametrization leading to a convex point tracking. The results are illustrated by examples. problem. Conditions are provided to guarantee robust constraint satisfaction and robust setpoint tracking. The results are illustrated by examples. point tracking. The results are illustrated by examples. point tracking. The results are illustrated by examples. problem. Conditions are provided to guarantee robust constraint satisfaction and robust setpoint tracking. The results are illustrated by examples. © 2018, IFAC (International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. point tracking. The resultsFederation are illustrated by examples. 1. INTRODUCTION for 1. INTRODUCTION for standard MPC the penalty matrices are tuned based 1. for standard standard MPC MPC the the penalty penalty matrices matrices are are tuned tuned based based 1. INTRODUCTION INTRODUCTION for standard MPC the penalty matrices are tuned on statistical considerations to maximize the performance 1. INTRODUCTION for standard MPC the penalty matrices are tuned based based on statistical considerations to maximize the performance on statistical considerations to maximize the performance on statistical considerations to maximize the performance 1. INTRODUCTION for standard MPC the penalty matrices are tuned based by reducing the required back off from the optimal steady Model predictive control (MPC) determines the input by on statistical considerations to maximize the performance by reducing the required back off from the optimal steady Model predictive control (MPC) determines the input by by reducing the required back off from the optimal steady Model predictive control (MPC) determines the input by by reducing the required back off from the optimal steady Model predictive control (MPC) determines the input by on statistical considerations to maximize the performance state. Cheng et al. (2009, 2013) use Youla solving an optimal control problem at each time instant reducing the required back off from theparametrization optimal steady Model predictive control (MPC) determines the input by by state. Cheng et (2009, 2013) use Youla parametrization solving an optimal control problem at each time instant Cheng et al. (2009, 2013) use Youla parametrization solving an optimal control problem at each time instant state. Chengpredictive et al. al. (2009, 2013)off usefrom Youla parametrization solvingon anthe optimal control problem at each time instant by reducing the required back thecase optimal steady Model predictive control (MPC) determines by state. state. Cheng et al. (2009, 2013) use Youla parametrization to robustify controllers for the of polytopic at each the timeinput instant based current state. In many applications, the full solving an optimal control problem to robustify predictive controllers for the case of polytopic based on the current state. In many applications, the full robustify predictive controllers for the case of polytopic based the current state. In many applications, the full to robustify predictive controllers for the case of polytopic basedison on the currentcontrol state. Indynamics many at applications, the full to state. Cheng et al. (2009, 2013) use Youla parametrization solving an optimal problem each time instant uncertainties in the dynamics. In Henrion et al. (2001) based on the current state. In many applications, the full state not available and the are uncertain. This to robustify predictive controllers for the case of polytopic uncertainties in the dynamics. In Henrion et al. (2001) state is not available and the are uncertain. This uncertainties in the dynamics. In Henrion et al. (2001) state is not available and the dynamics are uncertain. This uncertainties in the dynamics. In Henrion et al. (2001) state ison not available and theIndynamics dynamics are uncertain. This to robustify predictive controllers for the case of polytopic based the current state. many applications, the full Youla parameterization is used to derive locally stabilizing dynamics are uncertain. This motivates the use of robust MPC schemes that take the uncertainties in the dynamics. In Henrion et al. (2001) state is not available and the Youla parameterization is used to derive locally stabilizing motivates the use of robust MPC schemes that take the Youla parameterization is used to derive locally stabilizing motivates the use of robust MPC schemes that take the Youla parameterization is used to derive locally stabilizing motivates the use of robust MPC schemes that take the uncertainties in the dynamics. In Henrion et al. (2001) state is not available and the dynamics are uncertain. This controllers for linear systems with input constraints. uncertainties into account, see e.g. Maciejowski (2002); Youla parameterization is used to derive locally stabilizing motivates the use of robust MPC schemes that take the controllers for linear systems with input constraints. uncertainties into account, see e.g. Maciejowski (2002); controllers for linear systems with input constraints. uncertainties into account, see e.g. Maciejowski (2002); controllers for linear systems with input constraints. uncertainties into account, see e.g. Maciejowski (2002); Youla parameterization is used to derive locally stabilizing motivates the use of robust MPC schemes that take the controllers for linear systems with input constraints. Goodwin et al. (2014); Mayne (2016, 2014); Rawlings uncertainties into(2014); account, see e.g. Maciejowski (2002); The contribution of this work is a method to tune the Goodwin et Mayne (2016, 2014); Rawlings Goodwin et al. (2014); Mayne (2016, 2014); Rawlings Goodwin et al. al. (2014); Mayne (2016, 2014); Rawlings The contribution of this work is method to tune the the for linear systems with input constraints. uncertainties into account, see e.g. Maciejowski (2002); controllers The contribution of this work is aa method to tune Goodwin et al. (2014); Mayne (2016, 2014); Rawlings et al. (2017); Lucia et al. (2016); K¨ o gel and Findeisen The contribution of this is method to the et al. (2017); Lucia et al. (2016); K¨ o gel and Findeisen et al. Lucia et al. (2016); K¨ oogel and Findeisen tube controller in output feedback MPC schemes using The contribution ofoutput this work work is aa MPC method to tune tuneusing the et al. (2017); (2017); Lucia etschemes, al. (2016); K¨ gel and Findeisen tube controller in feedback schemes Goodwin et al. (2014); Mayne (2016, 2014); Rawlings tube controller in output feedback MPC schemes using (2017b). One of such called tube based MPC, et al. (2017); Lucia et al. (2016); K¨ o gel and Findeisen tube contribution controller in ofoutput output feedback MPC schemes using (2017b). One of such schemes, called tube based MPC, The this work is a method to tune the tube controller in feedback MPC schemes using (2017b). One of such schemes, called tube based MPC, convex optimization. We outline an optimization problem (2017b). One of such schemes, called tube based MPC, convex optimization. We outline an optimization problem et al. (2017); Lucia et al. (2016); K¨ o gel and Findeisen convex optimization. We outline an optimization problem uses aa feedback feedback controller in the prediction and handles the (2017b). One ofcontroller such schemes, called tube based MPC, convex optimization. We outline an optimization problem uses a in the prediction and handles the tube controller in output feedback MPC schemes using uses feedback controller in the prediction and handles the to evaluate the asymptotic worst case performance of convex optimization. We outline an optimization problem uses aa feedback feedback controller in the thefuture prediction andbased handles the to to evaluate the asymptotic worst case performance of (2017b). One ofcontroller such schemes, called tube MPC, evaluate the asymptotic worst case performance of uses in prediction and handles the influence of the yet unknown disturbances in form to evaluate the asymptotic worst case performance of influence of the yet unknown future disturbances in form convex optimization. We outline an optimization problem influence of the yet unknown future disturbances in form the output feedback for a given tube based controller. to evaluate the asymptotic worst case performance of influence oftubes, thecontroller yet unknown future disturbances in form the output feedback for a given tube based controller. uses a feedback in the prediction and handles the the output feedback for a given tube based controller. influence of the yet unknown future disturbances in form of sets: the see e.g. Mayne et al. (2006, 2009); Mayne the output feedback for a given tube based controller. of sets: the tubes, see e.g. Mayne et al. (2006, 2009); Mayne to evaluate the asymptotic worst case performance of the output feedback for a given tube based controller. of sets: the tubes, see e.g. Mayne et al. (2006, 2009); Mayne We illustrate that a tube controller with a prespecified of sets: the tubes, see e.g. Mayne et al. (2006, 2009); Mayne We illustrate that a tube controller with a prespecified influence oftubes, the yet unknown future in form We that aa tube controller with aa prespecified (2016); Brunner et al. (2018); Rawlings et al. (2017); of sets: the see e.g. Mayne etRawlings al.disturbances (2006,et 2009); Mayne We illustrate illustrate that tube controller with prespecified (2016); Brunner et al. (2018); al. (2017); the output feedback for a given tube based controller. We illustrate that a tube controller with a prespecified (2016); Brunner et al. (2018); Rawlings et al. (2017); complexity can be tuned optimally by solving a convex (2016); Brunner et (2015a, al. (2018); (2018); Rawlings et al. 2017a). (2017); can be tuned optimally by solving a convex of the tubes, see e.g. Mayne etRawlings al.2015b, (2006,2016b, 2009); Mayne complexity complexity can be optimally by solving a convex K¨ gel and Findeisen 2016a, 2015b, 2016b, (2016); Brunner et al. et al. (2017); complexity can be tuned optimally by solving a convex K¨ oooosets: gel and Findeisen (2015a, 2016a, 2017a). We illustrate that atuned tube controller with a prespecified K¨ gel and Findeisen (2015a, 2016a, 2015b, 2016b, 2017a). optimization problem using Youla parametrization, see complexity can be tuned optimally by solving a convex K¨ gel and Findeisen (2015a, 2016a, 2015b, 2016b, 2017a). optimization problem using Youla parametrization, see (2016); Brunner et al. (2018); Rawlings et al. (2017); optimization problem using Youla parametrization, see K¨ o gel and Findeisen (2015a, 2016a, 2015b, 2016b, 2017a). Basically, the additional, often linear, feedback controllers optimization problem using Youla parametrization, see Basically, the additional, often linear, feedback controllers complexity can be tuned optimally by solving a convex Basically, the additional, often linear, feedback controllers e.g. Youla et al. (1976); Zhou et al. (1996). optimization problem using Youla parametrization, see Basically, the additional, often linear, feedback controllers e.g. Youla et al. (1976); Zhou et al. (1996). K¨ o gel and Findeisen (2015a, 2016a, 2015b, 2016b, 2017a). e.g. Youla et al. (1976); Zhou et al. (1996). Basically, the additional, often linear, feedback controllers guarantees that the size of the tubes stays bounded and e.g. Youla et al. (1976); Zhou et al. (1996). guarantees that the size of the tubes stays bounded and optimization problem using Youla parametrization, see e.g. Youla et al. (1976); Zhou et al. (1996). guarantees that the size of the tubes stays bounded and guarantees that the size of the tubes stays bounded and Basically, additional, often linear, feedback controllers close to aa the nominal trajectory. The size and shape of these remainder of the paper is structured as follows. In guarantees that thetrajectory. size of the tubes stays bounded and The close to a nominal The size and shape of these The remainder of the paper is structured as follows. In e.g. Youla et al. (1976); Zhou et al. (1996). close to nominal trajectory. The size and shape of these The remainder of the paper is structured as follows. In close to todepend a nominal nominal trajectory. The size and shape shape of has these remainder of the paper is structured as follows. In guarantees that the size of feedback the tubes stays bounded andaa The tubes on the tube feedback controller and Section 2 we introduce the problem setup and illustrate the close a trajectory. The size and of these The remainder of the paper is structured as follows. In tubes depend on the tube controller and has Section 2 we introduce the problem setup and illustrate the tubes depend on the tube feedback controller and has a Section 2 we introduce the problem setup and illustrate the tubes depend on the tube feedback controller and has a Section 2 we introduce the problem setup and illustrate the close to a nominal trajectory. The size and shape of these The remainder of the paper is structured as follows. In tubes depend on the tube feedback controller and has a Section 2 we introduce the problem setup and illustrate the large impact on the achievable performance. Therefore, considered controller design objective. Section 3 presents large impact on the achievable performance. Therefore, considered controller design objective. Section 3 presents large impact on the achievable performance. Therefore, considered controller design objective. Section 3 presents presents large impact on the achievable performance. Therefore, considered controller design objective. Section 3 tubes depend on the tube feedback controller and has a Section 2 we introduce the problem setup and illustrate the large impact on the achievable performance. Therefore, considered controller design objective. Section 3 presents it is desirable to develop methods to efficiently and systhe idea of tube-based output feedback MPC and discusses it to develop methods to efficiently and systhe idea of tube-based output output feedback MPC and discusses it is desirable to develop methods to efficiently and sysidea of tube-based feedback MPC and discusses it is is desirable desirable to the develop methods tothus efficiently and loop sys- the the idea of output feedback MPC and discusses large impactoptimize on achievable performance. Therefore, considered controller design objective. Section 3proposed presents tematically optimize the tube and the closed closed loop properties. In Section 444 we outline the it is desirable to develop methods to efficiently and systhe idea of tube-based tube-based output feedback MPC and discusses tematically the tube and thus the closed loop closed loop properties. In Section we outline the proposed tematically optimize the tube and thus the closed loop closed loop properties. In Section we outline the proposed tematically optimize the tube and thus the closed loop closed loop properties. In Section 4 we outline the proposed it is desirable to develop methods to efficiently and systhe idea of tube-based output feedback MPC and discusses performance by tuning the tube controller. tuning approach using convex tematically optimize the tube and thus the closed loop closed loop properties. In Section 4optimization. we outline theSection proposed5 performance by the tube controller. tuning approach using convex optimization. Section 5 performance by tuning the tube controller. approach using convex 5 performanceoptimize by tuning tuningthe thetube tubeand controller. tuningloop approach using convex optimization. Section 5 tematically thus the closed loop tuning closed properties. In Section 4optimization. we outlinebefore theSection proposed performance by tuning the tube controller. tuning approach using convex optimization. Section 5 illustrates the results using a toy example, Section illustrates the results using a toy example, before Section illustrates the results using a toy example, before Section For output feedback tube based approaches often comillustrates the results using a toy example, before Section For output feedback tube based approaches often comperformance by tuning the tube controller. tuning approach using convex optimization. Section 5 For output feedback tube based approaches often com6 summarizes the results. Appendix B reviews results with illustrates the results using a toy example, before Section For output output feedback tube based based approaches oftena comcomsummarizes the results. Appendix B reviews results with 66illustrates summarizes the results. Appendix B reviews results with bine observers with robust MPC laws employing preFor feedback tube approaches often 66respect summarizes the results. Appendix B reviews results with bine observers with robust MPC laws employing a prethe results using a toy example, before Section bine observers with robust MPC laws employing a preto the Youla parametrization. summarizes the results. Appendix B reviews results with bine output observers with robust MPC lawscontroller employing prerespect to the Youla Youla parametrization. For feedback tubetube based approaches often respect to the parametrization. bine observers with robust MPC laws employing aa compredictive part and linear tube based based on respect to Youla parametrization. dictive part and aaaa linear based controller based on summarizes results. Appendix B reviews results with dictive part and linear tube based controller based on to the the the Youla parametrization. dictive part andwith linear tube based controller based on 6respect bine observers robust MPC laws employing a prethe state estimates, see e.g. Mayne et al. (2006, 2009); The notation is mainly standard. For sets A, B and a dictive part and a linear tube based controller based on the state estimates, see e.g. Mayne et al. (2006, 2009); The notation is mainly standard. For sets A, B and a respect to the Youla parametrization. the state estimates, see e.g. Mayne et al. (2006, 2009); The notation is mainly standard. For sets A, B and a the state state estimates, see e.g. e.g. Mayne et al.ogel (2006, 2009); The notation is mainly standard. For sets A, B and a dictive part and a linear tube based controller based on Mayne (2016); Rawlings et al. (2017); K¨ and Findmatrix M , A ⊕ B, A  B, M A denote the Minkowski the estimates, see Mayne et al. (2006, 2009); The notation is mainly standard. For sets A, B and a Mayne (2016); Rawlings et al. (2017); K¨ o gel and Findmatrix M , A ⊕ B, A  B, M A denote the Minkowski Mayne (2016); Rawlings et al. (2017); K¨ o gel and Findmatrix M , A ⊕ B, A  B, M A denote the Minkowski Mayne (2016); Rawlings et al. (2017); K¨ o gel and Findmatrix M , A ⊕ B, A  B, M A denote the Minkowski the state estimates, see e.g. Mayne et al. (2006, 2009); The notation is mainly standard. For sets A, B and a eisen (2016b, 2017a). Instead of an observer linear moving sum, the Minkowski difference, and the set multiplication, Mayne (2016); Rawlings et al. (2017); K¨ o gel and Findmatrix M , A ⊕ B, A  B, M A denote the Minkowski eisen (2016b, 2017a). Instead of an observer linear moving sum, the Minkowski difference, and the set multiplication, eisen 2017a). Instead of observer linear moving sum, difference, and the set multiplication, eisen (2016b, (2016b, 2017a). Instead of an an observer linear moving sum, the theMMinkowski Minkowski difference, and theMiani set the multiplication, Mayne (2016); Rawlings et al. (2017); K¨ o gel and Findmatrix , A ⊕ B, A  B, M A denote Minkowski eisen (2016b, 2017a). Instead of an observer linear moving sum, the Minkowski difference, and the set multiplication, horizon estimators, see e.g. Sui et al. (2008); K¨ o gel and respectively, see e.g. Blanchini and (2015). v[i] horizon estimators, see e.g. et al. (2008); K¨ ooogel and respectively, see e.g. Blanchini and Miani (2015). v[i] horizon estimators, see e.g. Sui et al. (2008); K¨ gel and respectively, see e.g. Blanchini and Miani (2015). v[i] horizon estimators, see e.g. Sui Sui etofobserver al. (2008); K¨ gel and denotes respectively, see e.g. Blanchini and Miani (2015). v[i] eisen (2016b, 2017a). Instead of an linear moving sum, thethe Minkowski and the set multiplication, Findeisen (2015a) or combination linear estimators ith of aaa vector v. For aaa positive definite horizon estimators, see e.g. Sui et al. (2008); K¨ ogel and respectively, seeentry e.g.difference, Blanchini and Miani (2015). v[i] Findeisen (2015a) or combination of linear estimators denotes the ith entry of vector v. For positive definite Findeisen (2015a) or combination of linear estimators and denotes the ith entry of vector v. For positive definite 2 T Findeisen (2015a) or combination of linear estimators denotes the ith entry of a vector v. For a positive definite horizon estimators, see e.g. Sui et al. (2008); K¨ o gel and respectively, see e.g. Blanchini and Miani (2015). v[i] 2 entry Tof set-membership estimators can be used, see e.g. Brunner matrix M : x = x M x. The  -gain of linear systems Findeisen (2015a) or combination of linear estimators and denotes the ith a vector v. For a positive definite T 2 ∞ set-membership estimators can be used, see e.g. Brunner M matrix M x = x M x. The -gain of linear systems 2 T ∞ -gain set-membership estimators can be used, see e.g. Brunner matrix M ::: x = x M x. The v. of linear systems M ∞ 2 entry Tof set-membership estimators can be used, see e.g. Brunner M matrix M x = x M x. The -gain of linear systems 2 T Findeisen (2015a) or combination of linear estimators and denotes the ith a vector For a positive definite ∞ M et al. (2018); K¨ gel and Findeisen (2015b, 2016a). In the is defined set-membership can be (2015b, used, see2016a). e.g. Brunner matrix M :as: xM ∞ -gain of linear systems et K¨ ooooestimators gel and In defined as: M = x M x. The ∞ et al. (2018); K¨ gel and Findeisen (2015b, In the is defined et al. al. (2018); (2018); K¨ gel(2005); and Findeisen Findeisen (2015b, 2016a). In the the is is defined as: set-membership can beand used, see2016a). e.g. Brunner matrix M :as: x2M = xT M x. The ∞ -gain of linear systems et al. (2018); K¨ oestimators gel and Findeisen (2015b, 2016a). In the is defined as: works Peng et al. Omell and Chmielewski (2014) works Peng et al. (2005); Omell Chmielewski (2014) Definition 1. ( -gain of a linear system) ∞ works Peng al. Omell Chmielewski Definition 1. ( -gain of a linear system) ∞ works Peng et etK¨ al. (2005); Omell and and Chmielewski (2014) 1. ( -gain of a linear system) et al. (2018); ogel(2005); and Findeisen (2015b, 2016a). (2014) In the Definition is defined as: ∞ Definition 1. ( -gain of a linear system) works Peng et al. (2005); Omell and Chmielewski (2014) ∞ For an asymptotic stable, linear system given by Definition 1. ( -gain of a linear system) ∞ The authors acknowledge partial support by the EU-program ∞ For an asymptotic stable, linear system given by For an asymptotic stable, linear system given The authors acknowledge partial support by the EU-program works Peng et al. (2005); Omell and Chmielewski (2014) The authors acknowledge partial support by the EU-program For an asymptotic stable, linear system given by by Definition 1. ( -gain of a linear system) The authors acknowledge partial support by the EU-program ∞ For an asymptotic stable, linear system given by ERDF (European Regional Development Fund) within the research The authors acknowledge partial support by the EU-program ERDF (European (European Regional Regional Development Development Fund) Fund) within within the the research research =Φζ + Θρ , ψ =Γζ ,,, (1) ζ k+1 k k k k ERDF =Φζ + Θρ , ψ =Γζ (1) ζ k+1 k k k k ERDF (European Regional Development Fund)bywithin the research For an asymptotic stable, linear system given =Φζ + Θρ , ψ =Γζ (1) ζ k+1 k k k k center of dynamic systems (CDS), the International Max Planck The authors acknowledge partial support the EU-program =Φζkk + + Θρ Θρkk ,, ψkk =Γζ =Γζkk by , (1) ζk+1 ERDF (European Regional Development Fund) within Max the research center of dynamic systems (CDS), the International Planck =Φζ ψ , (1) ζ center of dynamic systems (CDS), the International Max Planck k+1 k+1 k k k k center of dynamic systems (CDS), the International Max Planck -gain γ from ρ to ψ implies that if ζ = 0 and an  Research School (IMPRS-ProEng) and the German Research FounERDF (European Regional Development Fund) within the research ∞ ∞ 0 -gain γ from ρ to ψ implies that if ζ = 0 and an  center of dynamic systems (CDS), the International Max Planck Research School (IMPRS-ProEng) and the German Research Foun∞ -gain ∞=Φζ 0, = + Θρ , ψ =Γζ (1) ζ γ from ρ to ψ implies that if ζ 0 and an  k+1 k k k k Research School (IMPRS-ProEng) and the German Research Foun∞ ∞ 0 max max -gain γ from ρ to ψ implies that if ζ = 0 and an  Research School (IMPRS-ProEng) and the German Research Founmax max -gain γ∞ from ρ0, tothen ψ implies that ifρ ζ000 ..= 0 and an dation the frame of research program SPP 1914. ρ ∞ ≤ ρ k ≥ ψ ∞ ≤ γ center of School dynamic systems (CDS), thethe International Max Planck Research (IMPRS-ProEng) and German Research Foun∞ ∞for max max k k ∞ ∞ ∞ dation in in the frame of the the focal focal research program SPP 1914. ρ ≤ ρ for k ≥ 0, then ψ ≤ γ ρ max max k ∞ k ∞ ∞ ρ ≤ ρ for k ≥ 0, then ψ ≤ γ ρ . dation in the frame of the focal research program SPP 1914. k ∞ k ∞ ∞ max max dation in the frame of the focal research program SPP 1914. ρ  ≤ ρ for k ≥ 0, then ψ  ≤ γ ρ . max max -gain γ from ρ to ψ implies that if ζ = 0 and an  k ∞ ∞ ∞ρ 0 . Research (IMPRS-ProEng) and the German Research dation in School the frame of the focal research program SPP 1914. Founρkkk ∞ ∞ ∞ ≤ ρmax∞for k ≥ 0, then ψk k ∞ ≤ γ∞ ∞ dation in the frameIFAC of the focal research program of SPP 1914. Control) Hosting ρk ∞ by ≤ Elsevier ρ for k All ≥ 0, then ψk ∞ ≤ γ∞ ρmax . 2405-8963 © 2018, (International Federation Automatic Ltd. rights reserved.

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Markus Kögel et al. / IFAC PapersOnLine 51-20 (2018) 134–140

2. PROBLEM FORMULATION

135

3.1 Controller structure

In this section we introduce the considered class of systems and formulate the considered problem. 2.1 System class We consider linear, discrete time systems given by xk+1 =Axk + Buk + Bd dk (2a) (2b) yk =Cxk + Ddk , where x ∈ Rn denotes the system’s state, u ∈ Rp the applied control input and y ∈ Rq are the available measurements. d ∈ Rr is an unknown disturbance affecting the dynamics and the measurements, which is bounded by dk ∈ D, D = {d s.t. d∞ ≤ 1}. (3) We assume that (A, B) is controllable and (A, C) is observable. The state xk and control input uk need to satisfy polytopic constraints given by: gk =Exk + F uk ∈ G, G ={gk s.t. gk ≤ g max }, (4) where 0 < g max . Note that G contains a neighborhood of its origin and the definition is rather general. An estimate x ˆ0 of the initial state x0 is available, as well as a convex set E0 bounding the difference between the real and estimated initial state: e0 = x0 − x ˆ 0 ∈ E0 . 2.2 Control objective We aim for a dynamic output feedback controller of the form ζk+1 =λk (ζk , yk ), uk =κk (ζk ), (5) for the constrained, uncertain system (2), which guarantees: Definition 2. (Robust constraint satisfaction) The controller (5) guarantees robust constraint satisfacˆ 0 ⊃ {0}, if for any initial tion for system (2) and a set X ˆ ˆ 0 ⊕ E0 and any estimate x ˆ0 ∈ X0 , any initial state x0 ∈ X disturbance sequence {dk }, dk ∈ D the constraints (4) are satisfied for any k ≥ 0. Moreover, the output feedback controller (5) should minimize an upper bound on the worst case performance given by: Definition 3. (Worst case performance bound) For the closed loop system given by (2), (5) J˜ is an ultimate, upper bound on the performance objective J(xk , uk ) = cx xk + cu uk , (6) ˆ and ˆ 0 , if for any initial estimate x ˆ0 ∈ X for sets X admissible disturbance sequence {dk }, dk ∈ D: (7) J(xk , uk ) ≤ J˜ + k , where k ≥ 0 and k → 0 as k → ∞.

3. TUBE BASED OUTPUT FEEDBACK MPC We consider a tube based output feedback MPC scheme, which uses the measurements yk and takes into account that the system as well as the measurements are affected by disturbances. Moreover, the scheme should optimize the performance (Definition 3). 156

The idea of tube based MPC is to use two parts for the feedback: a part based on the so-called nominal behavior, which uses an MPC control law and a second part handling the influence of disturbances using a linear feedback. The first part uses the nominal dynamics: xk+1 =Axk + Buk , y k =Cxk , g k =Exk + F uk , (8) i.e. it neglects the effect of disturbances and uses a nominal state xk instead of the real, but unknown state xk . This simplification enables a straightforward and efficient prediction and optimization of the future behavior using an MPC based control law as defined in Section 4.3. The second part of the feedback handles the mismatch between the nominal model and the real system: ∆xk+1 =A∆xk + B∆uk + Bd dk (9a) (9b) ∆yk =C∆xk + Ddk , where ∆uk = uk − uk , ∆xk = xk − xk and ∆yk = yk − y k . To “stabilize” and attenuate the mismatch ∆xk a linear output feedback is used: ξk+1 =Ac ξk + Bc ∆yk , ∆uk =Cc ξk , (10) m where ξ ∈ R is the internal state of the output feedback controller and Ac , Bc and Cc define its dynamics. The overall feedback uk consist of the nominal input uk together with a correction ∆uk generated by (10): uk = ∆uk + uk . (11) A suitable choice of Ac , Bc , Cc (suitable design of (10)) stabilizes the closed loop (9), (10) given by     A BCc Bd ωk+1 = ωk + d , (12) Bc C Ac Bc D k       

Aω

∆xTk

Bω

T ξkT .

where ωk = This guarantees that ∆xk remains bounded. Remark 4. (Existing works) Existing works do not focus on the optimization of the controller (10) and use less general controller structures than (10): Mayne et al. (2006, 2009); K¨ ogel and Findeisen (2017a) use for (10) a combination of an observer with a linear feedback based on the state estimate, which results in Ac = A + BK + LC, Bc = −L and Cc = K where L/K (observer/controller gain) are such that A + BK and A+LC is asymptotic stable. In Sui et al. (2008) and K¨ ogel and Findeisen (2015a) instead of the observer a linear moving horizon state estimator is used. 3.2 Tubes and constraint satisfaction

Constraint satisfaction demands that gk ∈ G for any k ≥ 0. Using the combined feedback (11) one obtains gk =E(xk + ∆xk ) + F (uk + ∆uk ) (13a) (13b) = Exk + F uk + (E F Cc ) ωk .       =g k

=∆gk

Note that the nominal values xk and uk are known. In contrast, ωk for the current time instant is only partly known (∆xk depends on the real state) and the future

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evolution of ωk is affected by the yet unknown, uncertainties. Therefore, the idea is to derive bounds on ωk in form of sets Ωk , the so-called tube, and use this tube to tighten the constraints for the predictive control law part of the output feedback. For the bounds the following two results holds : Proposition 5. (Set-based bounds on ωk ) Let Aω be Schur stable and let ω0 ∈ Ω0 , where Ω0 is a polytope. Then ωk ∈ Ωk for any admissible disturbance sequence {dj }, dj ∈ D, j = 0, . . . , k − 1 where (14) Ωk+1 = Aω Ωk ⊕ Bω D. Moreover, there exist sets Ω∞ such that (15) Ω∞ ⊇ Aω Ω∞ ⊕ Bω D, and ωk converges to Ω∞ in the Hausdorff metric. This proposition follows from standard results, compare Kolmanovsky and Gilbert (1998); Rawlings et al. (2017). There is a minimum set Ωmin ∞ satisfying (15) with equality and Ωmin ∞ ⊆ Ω∞ for all Ω∞ such that (15). Proposition 6. (Constraint tightening) If Aω is Schur stable, ω0 ∈ Ω∞ and g j ∈ G where

Clearly, this optimization problem is challenging as the stabilization constraint is highly nonlinear, the dimension and the structure of Ac , Bc and Cc are not fixed and the set Ω∞ is involved. Moreover, depending on the employed tube controller the constraint restricting the location of the nominal steady state xss changes, compare Fig. 1. To tackle this challenge, we describe in the next section a method to solve (18) based on convex optimization and the Youla parametrization (see Appendix B for a review). Remark 7. (Solution of (18) using existing approaches) One possibility to solve (18) is to use for (10) a combination of an observer and a linear feedback, i.e. parametrize Ac , Bc , Cc using a controller gain K and an observer gain L such that A + BK and A + LC are Schur stable (compare Remark 4). This allows for stabilizing gains K, L to compute Jss as follows: First the corresponding set Ω∞ is determined and then Jss is evaluated (18). This enables to evaluate the performance for given K and L. However optimizing K and L is challenging, as one needs to determine Ω∞ , which itself is computationally expensive, see e.g. Rakovic et al. (2005).

G = G  (E F Cc ) Ω∞ , (16) for j = 0, . . . , k, then for any admissible disturbance sequence {dj }, dj ∈ D, gj ∈ G, j = 0, . . . , k.

The tightened constraints g k ∈ G is enforced by the proposed MPC. These constraints depend also on the tube Ω∞ and thus on the design of the tube controller (10). 3.3 Evaluation of the performance criteria We aim to improve the asymptotic, worst case performance, compare Definition 3 by optimizing the tube controller (3). The basic idea is that we use for the MPC part of the feedback, i.e. the part based on the nominal system (8), a set-point tracking framework as outlined in the next Section. This implies that one needs to tune the tube controller (10) such that the set-points reachable by the nominal system (8) optimize the performance. For the nominal system (8) the reachable steady states feasible with respect to the tightened constraint (16) are xss = Axss + Buss , (17a) F C (17b) Exss + F uss ∈ G  (E c ) Ω∞ . The reachable set points (17) depend on Cc and on the tube Ω∞ , which itself is influenced by the choice/tuning of the output feedback controller (10), i.e., Ac , Bc and Cc . Thus optimizing the worst case performance (Definition 3) corresponds to optimizing the nominal steady state xss /uss as well as the linear output feedback controller (10) at the same time, compare Fig. 1. This can be formulated as the optimization problem:  T T   cx xss + (I 0) ω Jss = min max T uss + (0 Cc ) ω xss ,uss ,Ac ,Bc ,Cc ω∈Ω∞ cu s.t. (15), (17) and Aω is Schur stable. (18) 157

x1,⋆ ss

(I 0)Ω2∞ ⊕ {x2,⋆ ss } x[2]

Proof (Sketch): Since Aω is Schur stable and ω0 ∈ Ω∞ using Proposition 5 one has ωk ∈ Ω∞ , for any k ∈ N. Combined with (4), (13) this yields the above result. 

7 6 x2,⋆ ss

xss = Axss + Buss 5

(I 0)Ω1∞ ⊕ {x1,⋆ ss }

1 Jss = −12.68

2 Jss = −8.69

4 5

6

7

x[1]

8

9

10

Fig. 1. Illustration of the paper’s objective for a SISO system (Appendix A) and gains K 1 , L1 (blue), K 2 , i L2 (magenta) showing the costs Jss and the boundary of optimally placed tubes Ωi∞ ⊕ {xi, ss } (solid blue/magenta); optimal nominal steady states xi, ss , (Diamond), the extent of Ωi∞ into the direction of constraints (dashed arrows). Red: state constraints. Black: set of steady states (17a). 4. DESIGN OF OPTIMAL TUBE-CONTROLLER USING YOULA PARAMETRIZATION In the following we outline an efficient evaluation of (18) and the convex, optimal tuning of the tube controllers. 4.1 Evaluating (18) via ∞ -gains For a given compact set Ω∞ satisfying (15) and a matrix Cc one can reformulate (18) as (19) Jss (Ω∞ , Cc ) =J1 (Ω∞ , Cc ) + J2 (Ω∞ , Cc ),  T T   cx xss J1 (Ω∞ , Cc ) = min subject to (17), T uss xss ,uss cu  T T   cx I 0 ω. J2 (Ω∞ , Cc ) = max T 0 Cc ω∈Ω∞ cu

As illustrated in Fig. 1 the precise shape of Ω∞ is not needed to evaluate (19), important is only how much Ω∞ extends into the s + 1 directions given by the constraints

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(4) (arrows in the figure) and the performance objective (cx cu ). The key observation is that one can compute these quantities depending on Ω∞ and Cc by investigating the ∞ -gain (see Definition 1) of the dynamical system (12) choosing these directions as “performance output” ζk :   E F Cc (20) ωk . ζk = cx cu Thus to find a controller (10) optimizing (18) requires to tune the s + 1 ∞ -gains from dk to each of the components and the nominal steady state (17a) simultaneously. Theorem 8. (Cost evaluation via ∞ -gains) Let Ac , Bc , Cc in (10) be given such that Aω (12) is Schur stable. The ∞ -gain γ∞ [i] from dk to ζk [i] of the closed loop system (12), (20) satisfy for i = 1, . . . , s G ={g k ≤ g max }, g max [i] =g max [i] − γ∞ [i] and for γ∞ [s + 1] it holds that J2 (Ωmin ∞ , Cc ) ≤ γ∞ [s + 1].

(21a) (21b)

Proof (Sketch): For ω0 = 0, ωk given by (12) satisfies |ζk [i]| = |((E F Cc ) ωk )[i]| ≤ γ∞ [i] due to the definition of the ∞ -gain and the fact that dk is bounded by the ∞norm ball: dk ∞ ≤ 1, ∀k ≥ 0. Similarly, | (cx cu Cc ) ωk | ≤ γ∞ [s + 1]. Note that ω0 = 0 instead of ω0 ∈ Ωmin ∞ is not min restrictive as ω∞ = D ⊕ Aω D ⊕ A2ω D ⊕ . . . 

In a nutshell this results allows to determine the tightened constraints G (16) and the second part of the cost function in (19) based on the ∞ -gain, i.e. it is not necessary to determine a (the minimum) robust positive invariant set Ω∞ (Ωmin ∞ ). Thus, if the ∞ -gains are available, then one can evaluate (19) straightforwardly for a given controller (10) by solving a linear program to evaluate J1 . 4.2 Youla parametrization for tube controller It is well known that one can design linear controllers minimizing the ∞ -gain (L1 -norm) of a closed loop system using the Youla parametrization (Khammash, 2000). As we show in the following one can also tune Ac , Bc and Cc to minimize (19) based on the Youla parametrization (see Appendix B for details), although here we are interested in the ∞ -gain of every single performance output (in contrast to only the maximum one (Khammash, 2000)). To do so we observe: Proposition 9. (Structure of the Youla parametrization) Let K, L be such that A+BK and A+LC are Schur stable, then for any proper, asymptotic stable Q(z) the Youla parametrization of (12), (20) with performance output ζk [i] is given by P (z)i = H(z)i − U (z)i Q(z)V (z). The verification follows directly from the observation that the system V (z) and the resulting control law K(z) (i.e. (10)) are for a given Q(z) independent of E[i] and F [i], i.e. the performance output, see Lemma 16. To practically tune via Q(z) the ∞ -gains and “shape” the set Ωmin ∞ one can use a design approach similar to the one proposed in Khammash (2000): Proposition 10. (Design via a single linear program) Let K and L be such that A+BK and A+LC are nilpotent. Let Q(z)[i, j] be given as a FIR filter of order ni,j Q as in 158

137

i,j Prop. 19 with filter coefficients φi,j m , m = 1, . . . , nQ , then (18) is a linear program.

Proof (Sketch): If A + BK and A + LC are nilpotent and Q(z) is a FIR filter, then all systems involved in the Youla parametrization have a finite impulse response. So one can compute the L1 -norm of the impulse response exactly and thus the ∞ -gain (Proposition 18). Moreover, parameterizing Q(z) in such a way guarantees that Q(z) is proper, asymptotic stable and the impulse responses H(z)i − U (z)i Q(z)V (z) are affine in the optimization variables φ. Thus one can express the ∞ -gains using inequalities affine in φ in Theorem 8). Consequently, as the constraints for the evaluation of J1 are affine in the gains, the overall problem is a linear program.  Remark 11. (Limitations) The described design approach can lead to a challenge: in theory the choice of K and L as deadbeat gains is possible without any approximation as (A, B) is controllable and (A, C) is observable. However, if (2) is not a FIR system, then in practice the required computation can be numerically ill conditioned, see e.g. Ogata (1995). Note that the need to fix ni,j Q results in an approximation error, however by choosing ni,j Q large enough one can try to make the approximation error arbitrary small, compare Prop. 19. In summary we have a design technique, which allows to efficiently determine (suboptimal) solutions to (18). 4.3 Model predictive control law To determine the predictive control law part of the combined feedback (11) we propose to uses a set-point tracking scheme based on (8) and (Simon et al., 2014). Therefore, one needs to find the input sequence u and the state sequence x defined over a horizon N     uk|k xk|k     .. (22a) uk =  xk =  ...  , . uk+N −1|k

and a target state

xtg k , xtg k

utg k

xk+N |k

tg = Axtg k + Buk , which are subject to the nominal dynamics (8) xi+1|k =Axi|k + Bui|k , xk|k = xk and satisfy the tightened constraint (21a)

xtg k

Exi|k + F ui|k ∈ G

(22b) (22c) (22d)

and the state sequence xk are coupled The target state via the terminal set and an optimization variable α ∈ [0, 1]

(xk+N |k − xtg k ) ∈ αT. Moreover, the target state needs to satisfy tg Extg k + F uk ∈ (1 − α)G.

(22e) (22f)

xtg k

The input sequence u and the target state need to be chosen to minimize the convex quadratic cost function tg 2  2 (22g) J =xtg k − xss M + xk+N |k − xk P +

k+N −1 i=k

tg 2 2 xi|k − xtg k Q + ui|k − uk R ,

where Q, R, M and P are positive definite.

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To show recursive feasibilit and convergence the following assumptions are made Assumption 12. (Conditions on T and P ) The terminal set T contains a neighborhood of its origin and satisfies for a terminal control gain Kf (A + BKf )T ⊆ T, (E + F Kf )T ⊆ G, and the terminal penalty P satisfies P = (A + BKf )T P (A + BKf ) + Q + KfT RKf .

(23a) (23b)

Assumption 13. (Initial time) The initial state x0 , initial nominal state x0 and initial controller state ξ0 (10) are such that ω0 ∈ Ω∞ . Proposition 14. (Recursive feasibility and constraint satisfaction) Let Assumptions 12 and 13 hold and G be given by (21a). If the optimization (22) is feasible at k = 0, then for any k ≥ 0 {dk }, dk ∈ D gk ∈ G holds and the optimization problem (22) is feasible. Proof (Sketch): We only sketch the basic idea due to space limitations. Recursive feasibility of the nominal system (8) and with the tightened constraints (21a) follows directly from (Simon et al., 2014). Constraint satisfaction follows from Proposition 6.  Proposition 15. (Convergence) Let Assumptions 12, 13 hold and let G be as in (21a). If the optimization problem (22) is feasible at k = 0, then for any admissible disturbance realization {dk }, dk ∈ D the state xk converges to the set (I 0) Ω∞ ⊕ {xss }, where xss is the optimal nominal steady state (19). Proof (Sketch): First note that recursive feasibility and constraint satisfaction follow from Proposition 14. Due to the MPC setup (22) and the assumptions made convergence of the nominal state xk to the optimal nominal steady state xss follows from (Simon et al., 2014). From ωk ∈ Ω∞ for k ≥ 0 follows that the real state xk convergences to (I 0) Ω∞ ⊕ {xss } for all disturbance sequences {dk , dk ∈ D}.  5. SIMULATION

Bd =



 0.02 0 0 , 0 0.4 0

Dd = (0 0 0.1)

The state is constrained to X = [−0.4, 0.4] × [−15, 15] and the inputs to U = [−10, 10]. Optimal tube controller design: The objective (compare Definition 3) is that the constraints are satisfied and that the temperature is maximized, i.e. cx = (0 −1) and cu = 0. The proposed approach yields with Q(z) as a FIR filter with orders nQ = 5, nQ = 10 and nQ = 30 the performances Jss = −6.15, Jss = −11.64 and Jss = −13.16, respectively. In contrast using a tracking approach based on (K¨ogel and Findeisen, 2017a) or (Mayne et al., 2006, 2009) yields Jss = −11.8 or Jss = −8.2, c.f. (K¨ ogel and Findeisen, 2017a). So the proposed approach yields better results, if a tube controller with a high order is used. Simulation: The proposed approach and its closed loop performance is illustrated in Fig 2. For the cost function (23a) we utilize Q = I, R = 0.01 and N = 20. The gain of the terminal controller Kf is chosen as the discrete time, infinite horizon LQ regulator equation and M = I takes the difference between the target state and the optimal steady state into account. Figure 2 shows the closed loop simulation results of the proposed approach for different realizations of the uncertainty dk . We observe that the state constraints as well as the input constraints are not violated at all. Moreover, the proposed output feedback control scheme enables to move the set-point close to the boundary of the constraints. Additionally for some realizations of the admissible noise the reactor temperature TR approaches the boundary. 0.15

14

0.1

12

Reactor temp. TR

In summary, the predictive control law is obtained by solving the optimization problem (22), which depends on the current nominal state xk . The first part of the optimal nominal input uk is used as feedback, see (11).

Concentration CE

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0.05 0 −0.05 −0.1 −0.15 −0.2 0

Plant setup: The reactor has two states: the concentration of E given by CE and the reactor temperature TR . The cooling jacket temperature TC is the considered input. The reactor temperature is measured. We consider the linear model around a steady state and use a sampling rate of 0.25 min, which leads to the following discrete-time model:    0.745 −0.002 −5.6 · 10−4 A= , B= , C = (0 1) 5.61 0.780 0.464 The process dynamics and measurement are affected by the disturbance dk as follows: 159

8 6 4 2 0 −2

5

10

0

2

Time [min]

4

6

8

10

12

Time [min]

10

Cooler temp. TC

To illustrate the proposed approach we consider as in K¨ ogel and Findeisen (2017a) a linearized model of a continuous stirred reactor with a constant volume with an irreversible exothermic reaction E → G.

10

8 6 4 2 0 −2 0

2

4

6

8

10

12

Time [min]

Fig. 2. Simulation of the continuous stirred reactor controlled by the proposed scheme. Black: nominal case di = 0. Red: sample paths for random noise (uniform distribution). Green: static noise vertices of D.

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6. SUMMARY AND FUTURE DIRECTIONS This work showed that it is possible to design and optimize the tube controller in the considered tube based output feedback MPC scheme using convex optimization and Youla parametrization. In detail, we outlined that it is possible to quantify the extent of the tube into the direction of the constraints and the cost function using ∞ gains, which enables to shape the tube in order to improve the asymptotic closed loop performance. The resulting output feedback MPC is recursive feasible and convergence of the system state to the set of worst case optimal states is guaranteed. Future works aim to improve the initialization of the robust output feedback model predictive control. Moreover, we want to address the limitations in Remark 11 for example by extending the results such that the controller and observer gain do not need to be deadbeat gains. REFERENCES Blanchini, F. and Miani, S. (2015). Set-Theoretic Methods in Control. Birkhuser, Basel, 2nd edition. Brunner, F.D., M¨ uller, M.A., and Allg¨ ower, F. (2018). Enhancing output-feedback MPC with set-valued moving horizon estimation. IEEE Trans. on Automatic Control. Cheng, Q., Cannon, M., and Kouvaritakis, B. (2013). The design of dynamics in the prediction structure of robust MPC. International Journal of Control, 86(11), 2096– 2103. Cheng, Q., Kouvaritakis, B., Cannon, M., and Rossiter, J.A. (2009). Youla parameter approach to robust constrained linear model predictive control. In Proc. IEEE Conf. Decision and Control, 2771–2776. Goodwin, G.C., Kong, H., Mirzaeva, G., and Seron, M.M. (2014). Robust model predictive control: reflections and opportunities. Journal of Control and Decision, 1(2), 115–148. Henrion, D., Tarbouriech, S., and Kuˇcera, V. (2001). Control of linear systems subject to input constraints: a polynomial approach. Automatica, 37(4), 597 – 604. Khammash, M. (2000). A new approach to the solution of the l1 control problem: the scaled-Q method. IEEE Trans. Automatic Control, 45(2), 180–187. K¨ ogel, M. and Findeisen, R. (2017a). Robust output feedback predictive control for uncertain linear systems with reduced conservatism. In Proc. IFAC World Congress, 11172–11177. K¨ ogel, M. and Findeisen, R. (2015a). Robust output feedback model predictive control using reduced order models. In Proc. IFAC Int. Symp. Advanced Control of Chemical Processes, 1008–1014. K¨ ogel, M. and Findeisen, R. (2015b). Robust output feedback predictive control with self-triggered measurements. In Proc. IEEE Conf. Decision and Control, 5487–5493. K¨ ogel, M. and Findeisen, R. (2016a). Output feedback MPC with send-on-delta measurements for uncertain systems. In Proc. IFAC Works. Distributed Estimation & Control of Networked Systems, 145–150. K¨ ogel, M. and Findeisen, R. (2016b). Sampled-data, output feedback predictive control of uncertain, nonlinear systems. In Proc. IFAC Symp. Nonlinear Control Systems, 47 – 52. 160

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K¨ogel, M. and Findeisen, R. (2017b). Low latency output feedback model predictive control for constrained linear systems. In Proc. IEEE Conf. Decision and Control, 1925–1932. Kolmanovsky, I. and Gilbert, E.G. (1998). Theory and computation of disturbance invariant sets for discretetime linear systems. Mathematical problems in engineering, 4(4), 317–367. Kuˇcera, V. (1975). Stability of discrete linear feedback systems. IFAC Proceedings Volumes, 8(1), 573–578. Larin, V., Naumenko, K., and Suntsev, V. (1971). Spectral methods for synthesis of linear systems with feedback. Kiev, Ukraine: Naukova Dumka. Lucia, S., K¨ogel, M., Zometa, P., Quevedo, D., and Findeisen, R. (2016). Predictive control, embedded cyberphysical systems and systems of systems - A perspective. Annual Reviews in Control, 41, 193 – 207. Maciejowski, J. (2002). Predictive Control: With Constraints. Prentice Hall. Mayne, D.Q. (2014). Model predictive control: Recent developments and future promise. Automatica, 50(12), 2967 – 2986. Mayne, D.Q. (2016). Robust and stochastic model predictive control: Are we going in the right direction? Annual Reviews in Control, 41, 184 – 192. Mayne, D.Q., Rakovi´c, S., Findeisen, R., and Allg¨ ower, F. (2006). Robust output feedback model predictive control of constrained linear systems. Automatica, 42(7), 1217–1222. Mayne, D.Q., Rakovi´c, S., Findeisen, R., and Allg¨ ower, F. (2009). Robust output feedback model predictive control of constrained linear systems: Time varying case. Automatica, 45(9), 2082 – 2087. Ogata, K. (1995). Discrete-time Control Systems. Prentice-Hall, Englewood Cliffs, NJ, 2nd edition. Omell, B.P. and Chmielewski, D.J. (2014). On the tuning of predictive controllers: impact of disturbances, constraints, and feedback structure. AIChE Journal, 60(10), 3473–3489. Peng, J.K., Manthanwar, A.M., and Chmielewski, D.J. (2005). On the tuning of predictive controllers: The minimum back-off operating point selection problem. Industrial & Engineering Chemistry Research, 44(20), 7814–7822. Rakovic, S.V., Kerrigan, E.C., Kouramas, K.I., and Mayne, D.Q. (2005). Invariant approximations of the minimal robust positively invariant set. IEEE Transactions on Automatic Control, 50(3), 406–410. Rawlings, J.B., Mayne, D.Q., and Diehl, M.M. (2017). Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing, Madison, WI, 2nd edition. Simon, D., L¨ofberg, J., and Glad, T. (2014). Reference tracking MPC using dynamic terminal set transformation. IEEE Trans. Automatic Control, 59(10), 2790– 2795. Sui, D., Feng, L., and Hovd, M. (2008). Robust output feedback model predictive control for linear systems via moving horizon estimation. In Proc. American Control Conf., 453–458. Youla, D., Jabr, H., and Bongiorno, J. (1976). Modern Wiener-Hopf design of optimal controllers–Part II: The multivariable case. IEEE Trans. Automatic Control, 21(3), 319–338.

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Zhou, K., Doyle, J.C., Glover, K., et al. (1996). Robust and optimal control. Prentice Hall, New Jersey. Appendix A. MINI EXAMPLE Figure 1 is based on     0.75 0.1 1 A= ,B= , C = (1 0) , 0.2 0.5 1   0.5 0 0 , D = (0 0 0.05) , Bd = 0 0.5 0

(A.1) (A.2)

and |x[1]| ≤ 10, |x[2]| ≤ 7.8, |u| ≤ 10 as constraints and cx = (−1 −1) , cu = 0.

(A.3)

The tube controller (3) are set up using Remark 4 and T L1 = L2 = (0.55 0.4) , K 1 = (0.748 0.1) and K 2 = (0.026 0.012). Note that with the proposed approach with nQ = 2, nQ = 5 and nQ = 15 the achieved performance is Jss = −13.27, Jss = −13.41 and Jss = −13.65, respectively. Appendix B. YOULA PARAMETRIZATION This work uses the Youla parametrization (also known as Youla-Kucera, Youla-Kucera-Larin parametrization or Q parametrization) as a method, which allows to parametrize all internally stabilizing, time-invariant output feedback controllers with a finite-dimension, see e.g. (Youla et al., 1976; Zhou et al., 1996; Kuˇcera, 1975; Larin et al., 1971). (B.1) (B.2) (B.3)

with disturbance input dk , control input uk , performance output gk and measurements yk by   A Bd B P (z) = E 0 F  (B.4) C D 0

see e.g. (Zhou et al., 1996). For a system P (z)   A Bd B P (z) = E 0 F  C D 0

(B.5)

and gains K, L the Youla parametrization is given by: Lemma 16. (Youla parametrization (Zhou et al., 1996)) Let for the system P (z) a controller gain K and an observer gain L be given such that A + BK, A + LC are Schur stable, then all linear, time-invariant, finite dimensional output feedback controllers K(z) resulting in an internally stable closed loop G are given by G(z) =H(z) − U (z)Q(z)V (z), K(z) =M (z)  Q(z),

(B.6a) (B.6b)

where  denotes the Redheffer star product (Zhou et al., 1996), Q(z) is any proper, asymptotic stable system and the systems H(z), U (z), V (z) and M (z) are given by 161

(B.7a) (B.7b) (B.7c) (B.7d)

Note that the closed loop system G(z) is linear in Q(z), see (B.6a), which enables an efficient tuning. The next results shows that by choosing K and L as deadbeat gains one can enforce a finite impulse response (FIR) for some systems appearing in the Youla parametrization: Proposition 17. (Enforcing a finite impulse response) If A + BK and A + LC have all eigenvalues at 0 (are nilpotent) and Q(z) has a finite impulse response, then also H(z) and U (z)Q(z)V (z) have a finite impulse response. So, if (A, B) is controllable and (A, C) is observable, then in principle one can determine H(z), U (z) and V (z) as FIR systems, which enables an efficient way to determine Q(z) to achieve a specific closed loop performance. The relationship between the ∞ -gain of the system (1) and the 1 -norm of its impulse response is: Proposition 18. (∞ -gain = 1 -norm of impulse response) For (1) with Ψ ∈ R1×β the ∞ -gain γ∞ from ρ to ψ is: γ∞ =

In the following we denote a dynamical system/plant P xk+1 =Axk + Buk + Bd dk gk =Exk + F uk yk =Cxk + Ddk

 Bd A + BK −BK 0 A + LC Bd + LD , H(z) =  E + F K −F K 0   A + BK −B U (z) = , E + F K −F   A + LC Bd + LD , V (z) = C D   A + BK + LC −L B K 0 I . M (z) =  −C I 0 

β  ∞ 

i=1 k=0

|ΓΦk Θ[i]|,

(B.8)

where Θ[i] ∈ Rp×1 is given by Θ = (Θ[1] . . . Θ[β]). Note that (B.8) involves an infinite sum. However, if (1) is a FIR system then Φm+k = 0 for some m and k ∈ N, Proposition 19. (FIR approximation, see c.f. Ogata (1995)) For an asymptotic stable system SISO system (1) with state ζk , input ρk and output ψk and for a FIR filter of order m in the form     0 1 0 ... 1 0 0 1 . . .  0 (B.9a) θk+1 =   θk +   ρk , .. .. .. . . .. . . . . . ψˆk = (φ1 φ2 . . . φm ) θk + φ0 ρk , (B.9b) with filter state θk = (ρk−1 ρk−1 . . .) there exist filter coefficients φi such that if ζ0 = 0, θ0 = 0, ρ0 = 1, ρk = 0, k ≥ 1, then |ψˆk −ψk | = 0 for k ≤ m. Moreover, if m → ∞, m then for these coefficients φi k=0 |ψˆk − ψk | → 0.