Adaptive MPC for Uncertain Discrete-Time LTI MIMO Systems with Incremental Input Constraints

5th International Conference on Advances in Control and 5th International Conference on Advances Control 5th International Conference on Advances in in Control and and Optimization of Dynamical Systems 5th International Conference on in and 5th International Conference on Advances Advances in Control Control and Optimization of Dynamical Systems Optimization of Dynamical Systems Available online at www.sciencedirect.com February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems 5th International Conference on Advances in Control and Optimization of Dynamical Systems February 18-22, 18-22, 2018. 2018. Hyderabad, Hyderabad, India India February February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 329–334

Adaptive MPC for Uncertain Adaptive MPC for Uncertain Adaptive MPC for Uncertain Adaptive MPC for Uncertain Discrete-Time LTI MIMO Systems Adaptive MPC for Uncertain Discrete-Time LTI MIMO Systems Discrete-Time LTI MIMO Systems Discrete-Time LTI MIMO Systems Incremental Input Constraints Discrete-Time LTI MIMO Systems Incremental Input Constraints Incremental Input Constraints Incremental Input Constraints Incremental Constraints ∗ ∗∗ Abhishek DharInput ∗ ∗ Shubhendu Bhasin ∗∗ ∗∗

with with with with with

Abhishek Dhar Abhishek Dhar ∗∗ Shubhendu Shubhendu Bhasin Bhasin ∗∗ Abhishek Abhishek Dhar Dhar ∗ Shubhendu Shubhendu Bhasin Bhasin ∗∗ ∗∗ ∗ Abhishek Dhar Shubhendu Bhasin of Indian Institute of Technology, ∗ ∗ Department of Electrical Electrical Engineering, Engineering, Indian Institute of Technology, ∗ Department Department of Electrical Engineering, Indian Institute of Technology, ∗ Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi: 110016 ([email protected]). Department of Electrical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi: 110016 ([email protected]). Hauz Khas, New Delhi: 110016 ([email protected]). ∗ ∗∗ Hauz Khas, New Delhi: 110016 ([email protected]). Department of Electrical Engineering, Indian Institute of Technology, Department of Electrical Engineering, Indian Institute of Technology ∗∗ Hauz Khas, New Delhi: 110016 ([email protected]). ∗∗ Department of Electrical Engineering, Indian Institute of ∗∗ Department of Electrical Engineering, Indian Institute of Technology ∗∗ Department of Electrical Engineering, Indian Institute Institute of Technology Technology HauzDelhi, Khas, New Delhi: 110016 ([email protected]). Hauz Khas, New Delhi: 110016, India (e-mail: Department of Electrical Engineering, Indian of Technology Delhi, Hauz Khas, New Delhi: 110016, India (e-mail: Delhi, Hauz Khas, New Delhi: 110016, India (e-mail: ∗∗ Delhi, Hauz Khas, New Delhi: 110016, India (e-mail: Department Electrical Engineering, Indian Institute of Technology [email protected]). Delhi, of Hauz Khas, New Delhi: 110016, India (e-mail: [email protected]). [email protected]). [email protected]). Delhi, Hauz Khas, New Delhi: 110016, India (e-mail: [email protected]). [email protected]). Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for controlling a discrete-time linear MIMO system with parametric uncertainties and subjected Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for controlling a discrete-time linear MIMO system with parametric uncertainties and subjected controlling a discrete-time linear MIMO system with parametric uncertainties and subjected controlling a discrete-time linear MIMO system with parametric uncertainties and subjected Abstract: In this paper, an adaptive model predictive control (MPC) strategy is proposed for to actuator constraints. Compared to previous results in literature which either solve the controlling a discrete-time linear MIMO system with parametric uncertainties and subjected to actuator constraints. Compared to previous results in literature which either solve the to constraints. Compared to previous results in literature which solve the to actuator actuatoraMPC constraints. Compared touncertain previous results in literature which either either solve the controlling discrete-time linear MIMO system with parametric uncertainties and subjected constrained problem for stable systems or the unconstrained MPC problem to actuator constraints. Compared to previous results in literature which either solve the constrained MPC problem for stable uncertain systems or the unconstrained MPC problem constrained MPC problem for stable systems or the unconstrained MPC problem constrained MPC problem for uncertain systems or the unconstrained MPC problem to actuator constraints. Compared touncertain previous results in which either the for unstable uncertain systems, this result, presents aa solution for constrained MPC constrained MPC problem for stable stable uncertain systems or literature theapproach unconstrained MPCsolve problem for unstable uncertain systems, this result, presents solution approach for constrained MPC for unstable uncertain systems, this result, presents a solution approach for constrained MPC for unstable uncertain systems, this result, presents a solution approach for constrained MPC constrained MPC problem for stable uncertain systems or the unconstrained MPC problem problems for fully uncertain and unstable systems. An adaptive law, designed to update the for unstable uncertain systems, this result, systems. presents An a solution approach for constrained MPC problems for fully uncertain and unstable adaptive law, designed to update the problems for fully uncertain and unstable systems. An adaptive law, designed to update the problems for fully uncertain and unstable systems. An adaptive law, designed to update the for unstable uncertain systems, this result, presents a solution approach for constrained MPC estimated parameters of the plant, is combined with a constrained MPC for an estimated problems for fully uncertain and unstable systems. An adaptive law, designed to update the estimated parameters of the plant, is combined with a constrained MPC for an estimated estimated parameters of the plant, is combined with a constrained MPC for an estimated estimated parameters of the the and plant, is combined combined with aadaptive constrained MPC for anupdate estimated problems for fully uncertain unstable systems. An law, designed to system. A sufficient condition is imposed on the adaptation gain to account for feasibility of the estimated parameters of plant, is with a constrained MPC for an estimated system. A sufficient condition is imposed on the adaptation gain to account for feasibility of the system. A sufficient condition is imposed on the adaptation gain to account for feasibility of the system. A condition is imposed on the adaptation gain to feasibility of estimated parameters of thein is combined with a constrained MPCfor an estimated MPC optimization problem the presence actuator constraint. Stability analysis system. A sufficient sufficient condition isplant, imposed on of thethe adaptation gain to account account forfor feasibility of the the MPC optimization problem in the presence of the actuator constraint. Stability analysis MPC optimization problem in the presence of the actuator constraint. Stability analysis of the MPC optimization problem in the presence of the actuator constraint. Stability analysis of system. A sufficient condition is imposed on the adaptation gain to account for feasibility the closed loop system with the proposed adaptive MPC strategy has been shown to guarantee MPC optimization problem in the presence of the actuator constraint. Stability analysis of closed loop system with the proposed adaptive MPC strategy has been shown to guarantee the closed loop system with the proposed adaptive MPC strategy has been shown to guarantee the closed loop system the proposed adaptive MPC strategy has shown to guarantee MPC optimization problem in the presence of the actuator constraint. of the ultimate boundedness of the parameter estimation errors and boundedness as well as asymptotic closed loop system with with the proposed adaptive MPC strategy has been been Stability shown toanalysis guarantee the ultimate boundedness of the parameter estimation errors and boundedness as well as asymptotic ultimate boundedness of the parameter estimation errors and as as ultimate boundedness of the the errors parameter estimation errors and boundedness boundedness as well well as asymptotic asymptotic closed loop system with proposed adaptive MPC strategy has been shown to guarantee the convergence of the tracking to zero. ultimate boundedness of the parameter estimation errors and boundedness as well as asymptotic convergence of the tracking errors to zero. convergence of the tracking errors to zero. convergence of the tracking to zero. ultimate boundedness of the errors parameter estimation errors and boundedness as well as asymptotic convergence of the tracking errors to zero. © 2018, IFAC of (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. convergence thePredictive tracking errors toofzero. Keywords: Model Control, Adaptive Control, Input Constraints. Keywords: Model Predictive Control, Adaptive Keywords: Model Model Predictive Control, Control, Adaptive Control, Control, Input Input Constraints. Constraints. Keywords: Keywords: Model Predictive Predictive Control, Adaptive Adaptive Control, Control, Input Input Constraints. Constraints. Keywords: Model Predictive Control, Adaptive Control, Input Constraints. 1. INTRODUCTION control with MPC. Some contributions to this area of re1. INTRODUCTION INTRODUCTION control with with MPC. MPC. Some Some contributions contributions to to this this area area of of rere1. control 1. INTRODUCTION control with MPC. Some contributions to this area of research include adaptive MPC based on comparison model 1. INTRODUCTION control with MPC. Some contributions to this area of research include adaptive MPC based on comparison model search include adaptive MPC based on on comparison model search include adaptive MPC based comparison model 1. INTRODUCTION control with et MPC. Some contributions to this area of re(Fukushima al., 2007), adaptive predictive control of search include adaptive MPC based on comparison model Model predictive control (MPC) has gained much atten(Fukushima et al., 2007), adaptive predictive control of (Fukushima et al., 2007), adaptive predictive control of Model predictive predictive control control (MPC) (MPC) has has gained gained much much attenatten- (Fukushima Model et al., 2007), adaptive predictive control of search include adaptive MPC based on comparison model nonlinear systems (Adetola et al., 2009) and adaptive Model predictive control (MPC) has gained much atten(Fukushima et al., 2007), adaptive predictive control of tion in the recent years due to its simple and effective nonlinear systems (Adetola et al., 2009) and adaptive Model predictive control (MPC) has gained much attennonlinear systems (Adetola et al., 2009) and adaptive tion in the recent years due to its simple and effective tion in the recent years due to its simple and effective nonlinear systems (Adetola et al., 2009) and adaptive (Fukushima et al., 2007), adaptive predictive control of receding horizon predictive control for discrete time linear tion in the recent years due to its simple and effective Model predictive control (MPC) has gained much attennonlinear systems (Adetola et al., 2009) and adaptive approach towards various control problems and its robust receding horizon predictive control for discrete time linear tion in the recentvarious years due to its simpleand andits horizon predictive control for discrete time linear approach towards various control problems and itseffective robust receding approach towards control problems robust receding horizon predictive control for discrete time linear nonlinear systems (Adetola et al., 2009) and adaptive systems (Kim and Sugie, 2008). approach towards various control problems and its robust tion in the recent years due to its simple and effective receding (Kim horizon control for discrete time linear properties towards external disturbances. The main systems (Kim andpredictive Sugie, 2008). 2008). approach towards problems and its advanrobust systems systems and Sugie, properties towards various externalcontrol disturbances. The main advanproperties towards external disturbances. The main advanand Sugie, 2008). receding (Kim horizon control for discrete time linear properties towards external disturbances. The main advanapproach towards various control problems and its robust Although systems (Kim andpredictive Sugie, 2008). tage of MPC is that it exploits within its design methodproperties towards external disturbances. The main advanthere have been significant contributions in tage of MPC is that it exploits within its design methodtage of MPC is that it exploits within its design methodAlthough there have been significant contributions contributions in in systems (Kim andhave Sugie, 2008). Although there been significant tage of MPC is that it exploits within its design methodproperties towards external disturbances. The main advanology the practical problem of exercising control actions Although there have been significant contributions in tage of MPC is that it exploits within its design methodthe field of adaptive MPC, the control of a completely ology the practical problem of exercising control actions Although there have been significant contributions in ology the practical problem of exercising control actions the field of adaptive MPC, the control of a completely the field of ofMIMO adaptive MPC, the control of completely ology practical exercising control actions tage ofthe MPC isenvironment thatproblem it exploits within its design methodin constrained .. of Some excellent overviews of field adaptive MPC, the control of aaa completely ology the practical problem of exercising control actions Although there have beensubjected significant contributions in uncertain systems, to constraints, has in constrained environment Some excellent overviews of the the field ofMIMO adaptive MPC, the control of completely in constrained environment . Some excellent overviews of uncertain MIMO systems, subjected to constraints, has uncertain systems, subjected to constraints, has in constrained environment . Some excellent overviews of ology the practical problem of exercising control actions MPC can be found in (Garcia et al., 1989) and (Kouvarisystems, to constraints, has in constrained environment . Some overviews of uncertain the field ofMIMO adaptive MPC,subjected the control of a completely been aa challenging problem. Control strategies proposed MPC can be be found found in (Garcia (Garcia et al., al.,excellent 1989) and and (Kouvariuncertain MIMO systems, subjected to constraints, has MPC can in et 1989) (Kouvaribeen challenging problem. Control strategies proposed aa challenging problem. Control strategies proposed MPC can be in et 1989) and (Kouvariin constrained environment . Some overviews of been takis and Cannon, 2015). The design methodology of this been problem. Control strategies proposed MPC can Cannon, be found found in (Garcia (Garcia et al., al.,excellent 1989) and (Kouvariuncertain MIMO toand constraints, has in (Fukushima et al., 2007) and (Kim Sugie, 2008) takis and Cannon, 2015). The design methodology of this this been a challenging challenging problem. Control proposed takis and 2015). The design methodology of in (Fukushima et systems, al., 2007)subjected and (Kimstrategies and Sugie, 2008) in (Fukushima et al., 2007) and (Kim and Sugie, 2008) takis and Cannon, 2015). The design methodology of this MPC can be found in (Garcia et al., 1989) and (Kouvaricontrol strategy is well depicted in (Wang, 2009). MPC has in (Fukushima et al., 2007) and (Kim and Sugie, 2008) takis and Cannon, 2015). The design methodology of this been a challenging problem. Control strategies proposed have guaranteed closed loop stability for partially uncercontrol strategy is well depicted in (Wang, 2009). MPC has in (Fukushima al., 2007) and (Kimfor Sugie, uncer2008) control strategy is well depicted in (Wang, 2009). MPC has have guaranteedetclosed closed loop stability stability forand partially uncerguaranteed loop partially control strategy is depicted in (Wang, 2009). has takis and Cannon, 2015). The design methodology of such this found its applications in various fields of engineering, have guaranteed loop stability for partially control strategy is well well depicted infields (Wang, 2009). MPC MPC has have in (Fukushima etclosed al., linear 2007) and (KimSome Sugie, uncer2008) tain and constrained systems. recent works found its applications in various fields of engineering, engineering, such have guaranteed closed loop systems. stability forand partially uncerfound its applications in various of such tain and constrained linear systems. Some recent works tain and constrained linear Some recent works found its applications in various fields of engineering, such control strategy is well depicted in (Wang, 2009). MPC has as in flight control (Kale and Chipperfield, 2005), control of tain and constrained linear systems. Some recent works found its applications in various fields of engineering, such have guaranteed closed loop stability for partially uncer(Tanaskovic et al., 2014), (Zhu and Xia, 2016) have proas in in flight flight control control (Kale (Kale and and Chipperfield, Chipperfield, 2005), 2005), control control of of (Tanaskovic tain and constrained linear systems. Some recent as et al., al., 2014), 2014), (Zhu and Xia, Xia, 2016) haveworks pro(Tanaskovic et (Zhu and 2016) have proas in flight control (Kale Chipperfield, 2005), control of found its applications in and various fields2005), of engineering, such electrical drives (Linder Kennel, hybrid electric (Tanaskovic et al., 2014), (Zhu and Xia, 2016) have proas in flight control (Kale Chipperfield, 2005), control of tain and constrained linear systems. Some recent works posed effective adaptive MPC strategies to control fully electrical drives (Linder and Kennel, 2005), hybrid electric (Tanaskovic et al., 2014), (Zhu and Xia, 2016) have proelectrical drives (Linder and Kennel, 2005), hybrid electric posed effective adaptive MPC strategies to control fully effective adaptive strategies to control fully electrical drives (Linder and Kennel, 2005), hybrid electric as in flight control (Kale control of posed vehicles (Borhan et al., 2012), etc. posed effective adaptive MPC strategies to control fully electrical drives (Linder and Chipperfield, Kennel, hybrid electric (Tanaskovic et al., 2014),MPC (Zhu and Xia, 2016) have prouncertain systems. However, in (Tanaskovic et al., 2014), vehicles (Borhan et al., 2012), 2012), etc. 2005),2005), posed effective adaptive MPC strategies to control fully vehicles (Borhan et al., etc. uncertain systems. However, in (Tanaskovic et al., 2014), uncertain systems. However, in (Tanaskovic et al., 2014), vehicles et 2012), etc. electrical drives (Linder and Kennel, 2005), hybrid electric uncertain systems. However, in (Tanaskovic et al., 2014), vehicles (Borhan (Borhan et al., al., 2012), etc. posed effective adaptive MPC strategies to control fully FIR approximation method is used to obtain a tractable uncertain systems. However, in (Tanaskovic et al., 2014), However, as the name suggests, MPC is a model dependent FIR approximation approximation method method is is used used to to obtain obtain aa tractable tractable However, as the the name name suggests, MPC is is aa model model dependent dependent FIR vehicles (Borhan et al., 2012), etc. However, as suggests, MPC FIR method is to obtain tractable uncertain systems. However, (Tanaskovic etaaal., 2014), approximated model corresponding to the uncertain plant However, as the name suggests, MPC is aa model dependent FIR approximation approximation method isinused used tothe obtain tractable control strategy. For this reason, systems with parametric approximated model corresponding to the uncertain plant However, as the name suggests, MPC is model dependent approximated model corresponding to uncertain plant control strategy. For this reason, systems with parametric control strategy. For this reason, systems with parametric approximated model corresponding to the uncertain plant FIR approximation method is used to obtain a tractable for which the stability of the plant is a prerequisite. In control strategy. For this reason, systems with parametric However, as the name suggests, MPC is a model dependent approximated model corresponding to the uncertain plant uncertainties in the model cannot be directly tackled by for which the stability of the plant is a prerequisite. In control strategy. For this reason, systems with parametric which the stability of the plant is aa consideration prerequisite. In uncertainties in in the the model model cannot cannot be be directly directly tackled tackled by by for uncertainties for which the stability of the plant is prerequisite. In approximated model corresponding to the uncertain plant (Zhu and Xia, 2016), the system under is uncertainties in the model cannot be directly tackled by control strategy. For this reason, systems with parametric for which the stability of the plant is a prerequisite. In the conventional MPC strategy, owing to the difficulty (Zhu and and Xia, Xia, 2016), 2016), the the system system under under consideration consideration is is uncertainties in the model cannotowing be directly by (Zhu the conventional MPC strategy, owing to the thetackled difficulty the conventional MPC strategy, to difficulty (Zhu and Xia, 2016), the system under consideration is for which the stability of the plant is a prerequisite. In unconstrained. the conventional MPC strategy, owing to the difficulty uncertainties in the model cannot be directly tackled by (Zhu and Xia, 2016), the system under consideration is in predicting the future states of the plant. For systems unconstrained. the conventional MPC strategy, owing to the difficulty unconstrained. in predicting the future states of the plant. For systems in predicting the future states of the plant. For systems unconstrained. (Zhu and Xia, 2016), the system under consideration is in predicting the future states of the plant. For systems the conventional MPC strategy, owing to the difficulty unconstrained. with unknown model, system identification has become an in predicting future states of the plant. systems paper presents method by which adaptive MPC with unknownthe model, system identification hasFor become an This with unknown model, system identification has become an This paper presents presents aaa method method by by which which adaptive adaptive MPC MPC unconstrained. paper with unknown model, system identification has become an in predicting future states of the a systems well accepted methodology to obtain viable model for This paper presents aa fully method by which adaptive MPC with unknownthe model, system hasFor become an This can be implemented for uncertain discrete-time linear well accepted methodology toidentification obtain aplant. viable model for This paper presents method by which adaptive MPC well accepted methodology to obtain a viable model for can be be implemented implemented for for fully fully uncertain uncertain discrete-time discrete-time linear linear can well accepted methodology to obtain a viable model for with unknown model, system identification has become an designing controllers. However, it is difficult to do offline be implemented fully uncertain discrete-time well accepted methodology to obtain a viable for can This paper presentsfor aplant method by which adaptive linear MPC MIMO time-invariant subjected to incremental input designing controllers. However, it is is difficult difficult to model do offline offline can be implemented for fully uncertain discrete-time linear designing controllers. However, it to do MIMO time-invariant plant subjected to incremental input time-invariant plant subjected to incremental input designing controllers. However, it is difficult to do offline well accepted methodology to obtain aunstable viable model for system identification for an inherently plant. In MIMO MIMO time-invariant plant subjected to incremental input designing controllers. However, it is difficult to do offline can be implemented for fully uncertain discrete-time linear constraint. The main contribution of this paper is the insystem identification for an inherently unstable plant. In MIMO time-invariant subjected incremental input system identification for an inherently unstable plant. In constraint. The main main plant contribution of to this paper is is the the inconstraint. The contribution of this paper insystem identification for an inherently unstable plant. In designing controllers. However, it is difficult to do offline contrast, adaptive control does online adjustments of the constraint. The main contribution of this paper is the insystem identification for an inherently unstable plant. In MIMO time-invariant plant subjected to incremental input troduction of a sufficient condition on the adaptation gain contrast, adaptive control does online adjustments of the constraint. The main contribution of this paper is the incontrast, adaptive control does online adjustments of the troduction of a sufficient condition on the adaptation gain troduction of a sufficient condition on the adaptation gain contrast, adaptive control does online adjustments of the system for andoes inherently unstable plant. In troduction controller/system parameters on the basis of measured of aa sufficient condition on the adaptation gain contrast,identification adaptive parameters control online of the constraint. The main contribution of this paper is the inin order to satisfy the constraint conditions in the MPC controller/system parameters on the adjustments basis of measured measured troduction of sufficient condition on the adaptation gain controller/system on the basis of in order order to to satisfy satisfy the the constraint constraint conditions conditions in in the the MPC MPC controller/system parameters on the basis of measured contrast, adaptive control does online adjustments of the in input-output data, at the same time ensuring stability. in order to satisfy the constraint conditions in the MPC controller/system parameters on the basis of measured troduction of a sufficient condition on the adaptation gain optimization problem. The adaptive law guarantees the input-output data, at the same time ensuring stability. in order to satisfy the constraint conditions in the MPC input-output data, at thefor same time ensuring stability. optimization problem. The adaptive adaptive law guarantees guarantees the optimization problem. The law the input-output data, at same time ensuring stability. controller/system parameters on control the basis of measured Hence, a possible solution the of uncertain and problem. The adaptive law guarantees the input-output data, at the thefor same time ensuring stability. in order to satisfy the of constraint conditions in theerrors. MPC ultimate boundedness the parameter estimation Hence, a possible possible solution for the control of uncertain uncertain and optimization optimization problem. The adaptive law guarantees the Hence, a solution the control of and ultimate boundedness of the parameter estimation errors. ultimate boundedness ofThe the adaptive parameterlaw estimation errors. Hence, solution for the control of and input-output data, at same time ensuring constrained systems can be the amalgamation of adaptive boundedness errors. Hence, aa possible possible solution the control of uncertain uncertain and ultimate optimization problem. of guarantees the constrained systems canthe befor the amalgamation of stability. adaptive ultimate boundedness of the the parameter parameter estimation estimation errors. constrained systems can be the amalgamation of adaptive constrained systems can be the amalgamation of adaptive Hence, a possible solution for the control of uncertain and constrained systems can be the amalgamation of adaptive ultimate boundedness of the parameter estimation errors. constrained systems be the amalgamation of adaptive 2405-8963 © 2018, IFAC can (International Federation of Automatic Control) Copyright 2018 IFAC 345 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 345 Copyright © 2018 IFAC 345 Peer review© of International Federation of Automatic Copyright ©under 2018 responsibility IFAC 345Control. Copyright © 2018 IFAC 345 10.1016/j.ifacol.2018.05.040 Copyright © 2018 IFAC 345

5th International Conference on Advances in Control and 330 Abhishek Dhar et al. / IFAC PapersOnLine 51-1 (2018) 329–334 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

It is further theoretically proved that the tracking errors of the closed loop system, with the proposed adaptive MPC, are bounded and asymptotically converging to zero. 2. PROBLEM STATEMENT In this paper, the linear discrete time MIMO system is considered to be of the form xr (k + 1) = Ar xr (k) + Br u(k) (1) yr (k) = Cr xr (k) (2) where xr (k) ∈ Rn is the system state vector, u(k) ∈ Rm is the input vector and yr (k) ∈ Rl is the output vector. Ar ∈ Rn×n and Br ∈ Rn×m are the uncertain and constant system matrix and input gain matrix, respectively. It is assumed that the pair (Ar , Br ) is controllable. Full state feedback is considered to be available and Cr ∈ Rl×n is assumed to be known. The target is to make the system output yr (k) track a given set point r, where T r = [r1 , r2 , ..., rl ] and r ≤ r¯, in the presence of the following incremental input constraint (3) ∆u(k) ≤ ∆Umax For the feasibility of the MPC problem, the desired set point must be reachable with the given input constraints, i.e., there must exist some sequence of ∆u(k), satisfying (3) at all k, such that u(k) reaches some steady control input uss that can keep the output yr (k) equal to r as k → ∞. Let the tracking error associated with the system be denoted by er (k), given as (4) er (k) = yr (k) − r In case of set point tracking, the knowledge of the steady value uss is required for solving the optimization problem in MPC. Since the system (1) is uncertain, obtaining exact knowledge of uss is not possible. Instead, the system given in equations (1) and (2) can be modified and represented in an incremental form (Wang, 2009), given as x(k + 1) = Ax(k) + B∆u(k) (5) y(k) = Cx(k) (6) T

where, x(k)  [∆xr (k)T , er (k)T ] , ∆xr (k)  xr (k) − xr (k −1), ∆u(k)  ur (k)−ur (k −1) and y(k) = er (k); and A=



   Br Ar 0n×l ,B = , Cr Ar Il×l C r Br

which makes the output y(k) of the uncertain system (5)(6) track zero (which corresponds to yr (k) tracking the set point r), in the presence of the incremental input constraint (3). Definition 1. For any two vector a = [a1 , a2 , · · · , an ] and b = [b1 , b2 , · · · , bn ] with a, b ∈ Rn for some finite n ∈ I, the relation a ≤ b is equivalent to ai ≤ bi where i = 1, 2, · · · , n. Assumption 1. It is assumed that Θ is bounded as Θ ≤ Θ, where Θ is a known positive scalar. 3. PRELIMINARIES 3.1 Adaptive Update Law Development An estimated system corresponding to (5) is designed as follows   x (k + 1) = A(k)x(k) + B(k)∆u(k) (8) (n+l)×(n+l) (n+l)×m   and B(k) ∈ R are where A(k) ∈ R the time-varying estimated parameters corresponding to  T  [A(k),   A and B, respectively. Define Θ(k) B(k)] ∈ (n+l)×(n+l+m) . The estimated system state equation (8) R can be written as  T Φ(k) (9) x (k + 1) = Θ(k) Subtracting equation (9) from equation (7) yields  T (k)Φ(k) (10) x (k + 1) = Θ   where x (k)  x(k)− x(k) and Θ(k)  Θ− Θ(k). A gradient descent based update law, given as T  + 1) = Θ(k)  (11) Θ(k + λΦ(k) x(k + 1)

where λ > 0, can be used to minimize the cost function (k)T x (k). Jx (k) = x Theorem 2. For the uncertain system defined by (1)-(2), if there exists an input ur (k) (or ∆u(k)) satisfying the following constraint 2−α T Φ(k) Φ(k) ≤ (12) λ where 0 < α < 2 and λ > 0, then with the adaptive update law (11) the following holds true  • The error Θ(k) is ultimately bounded; • The error x (k) is asymptotically stable.

Proof. For the detailed proof of the above theorem, refer to (Zhu and Xia, 2016).

C = [0l×n Il×l ]

where A ∈ R(n+l)×(n+l) , B ∈ R(n+l)×m and C ∈ Rl×(n+l) . So y(k) of the augmented system is in fact the tracking error er (k) of the original system. Such a formulation aids the concerned MPC problem as for successful set point tracking, ∆u(k) will always go to zero. It can be seen that A and B are dependent on Ar and Br . Consequently, in the incremental model (5), we consider A and B as the uncertain constant system parameters. It can be proved that if pair (Ar , Br ) is controllable, then the pair (A, B) will also be controllable. Define a lumped system parameter matrix Θ as ΘT  [A, B]. The system (5) can be represented using the lumped parameter matrix as (7) x(k + 1) = ΘT Φ(k) T T T n+l+m ∆u(k) ] ∈ R is the regreswhere Φ(k) = [x(k) sor vector. Objective: To design a control input ∆u(k) (or u(k)), 346

3.2 Model Predictive Control for Estimated System In this section, an MPC is developed for the estimated system (8) . Formulation of Predictive Equations for Estimated System. Suppose Np is the length of the prediction horizon and Nc is the length of the control horizon. Let x (k + i|k) denote the predicted state of the estimated system (8) at the (k + i)th instant, using the knowledge of x (k) (here x (k) = x(k)). The predictive equations can be represented in a compact form as σ  m  σ−j B(k)∆u(k   A(k) + j|k) x (k + m|k) = A(k) x(k) + j=0

(13)

5th International Conference on Advances in Control and Abhishek Dhar et al. / IFAC PapersOnLine 51-1 (2018) 329–334 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

Where σ = min(m − 1, Nc − 1) and m = 1, 2, ..., Np . The predictive equation associated with (13) can be written in a compact form as   X(k) = F (k)x(k) + φ(k)∆U (k) (14)

 (k + 2|k)T , ..., x (k + Np |k)T ]T where X(k)  [ x(k + 1|k)T , x T T and ∆U (k)  [∆u(k) , ∆u(k + 1) , ..., ∆u(k + Nc − 1)T ]T and the predictive matrices are given as       B 0 . 0 A  A A    2  B . 0   , φ(k)  B  = F (k) =    .   . . . . Np  A Np −2 B  . A Np −Nc B  Np −1 B A A (15)   represents Θ(k),  ∈ where Θ F (k) ∈ RlNp ×(n+l) and φ(k) RlNp ×mNc .

331

The above inequality is in accordance with the relation given in (12). Therefore, 2

2

2

x(k) < x(k) + ∆u(k) ≤ Φmax From (23), a bound on x(k) can be given as

(23)

2

(24) x(k) < Φmax Lemma 3. For the optimization problem (19) to be always feasible, subject to (3) and (12), the following condition must hold 2−α (25) λ< 2 ∆Umax + Φmax

Proof. The constraint equation (12) can be written as 2−α 2 2 ∆u(k) + x(k) < λ 2−α 2 2 − x(k) (26) ∆u(k) < λ Inequality (26) is a state dependent constraint on ∆u(k) Choice of Cost Function and the Optimization Problem The cost function J(k) associated with the system (5) is . Using the result from (24) and (26), the smallest bound on ∆u(k) can be found as chosen as T T 2−α 2 J(k) = X(k) QX(k) + ∆U (k) Ru ∆U (k) (16) − Φmax (27) ∆u(k) < λ where Q = C T C, X(k)  [x(k+1|k)T , x(k+2|k)T , ..., x(k+ The objective is to make the above constraint inactive Np |k)T ]T and Ru = diag(ru ) ∈ R(mNc )×(mNc ) is a diagonal for all instances. This can be achieved by imposing the weight matrix with ru ∈ Rm×m being positive definite. following condition  Corresponding to (16), an estimated cost function J(k), 2−α 2 2 − Φmax < ∆u(k) ≤ ∆Umax associated with the estimated system (8), is chosen as λ T T  = X(k)   J(k) QX(k) + ∆U (k) Ru ∆U (k) (17) Therefore, the condition on λ is derived as 2−α  Replacing X(k) from (14) in (17) and solving, the following λ< (28) 2 ∆Umax + Φmax expression is obtained    = ∆U (k)T E(k)∆U (k) + 2∆U (k)T H(k) (18) Remark 1. The above result is true when ∆Umax is finite. J(k)   For unconstrained adaptive MPC problem, results from T   T φ(k)+R    where E(k)  φ(k) u and H(k)  φ (k) F (k)x(k) . (Zhu and Xia, 2016) must be followed. The objective is to minimize (18) with respect to ∆U (k) Remark 2. The condition (28) is a sufficient condition, subject to (3) and (12). which makes the constraint (12) inactive for all instants. Therefore, it is only the actuator constraint (3) that needs 4. MAIN RESULTS (FEASIBILITY AND STABILITY) to be checked and satisfied when solving the optimization problem. 4.1 Feasibility The constrained MPC problem can be formulated as the following  ∆U ∗ (k) = argmin J(k) (19) ∆U (k)

subject to the constraints given in (3) and (12).

Let Φm = max(Φ(i|0)), where Φ(i|0) is obtained by processing the optimization (19) subjected to only a stabilizing terminal constraint as given in (31). According to (Remark 4., (Zhu and Xia, 2016)), for the feasibility of the optimization problem (19), λ must be chosen such that 2−α 2−α >> (20) λ λ0 where, λ0 is given as 2−α 2−α (21) = λ0 = r T Φ max(Φm Φm , Θ ) max r ), r¯ is the bound on the where Φmax = max(ΦTm Φm , Θ reference signal r(k) and Θ is the bound on the uncertain parameter matrix Θ. Φmax can be considered as some 2 upper bound of Φ(k) . Therefore, it can be claimed 2

Φ(k) ≤ Φmax

(22)

347

Following Definition 1, the constraint (3) can be reformulated to be represented in vector form as     ∆Umax 1m Im×m ∆u(k) ≤ √ (29) −Im×m 1m m The last inequality can be written in compact form as M ∆U (k) ≤ γ (30)   Im×m 0 ... 0 ∈ R2m×(mNc ) where M = −Im×m 0 ... 0   1m √max γ = ∆U ∈ R2m and 1m  [1, 1, ..., 1]T ∈ Rm . In m 1m order to guarantee stability of the closed loop estimated system, the following terminal constraint is introduced x (k + Nc |k) = 0l (31) where 0l = [0, 0, ..., 0]T ∈ Rl . In terms of ∆U (k), the above constraint can be written as (32) Mt ∆U (k) = γt Nc −1  Nc −2   Nc  B, A B, ..., B] and γt = −C A where Mt = C[A x(k). Therefore, the MPC optimization can be formulated as (19) subject to (30) and (32). The proposed control strategy is implemented using receding horizon control (33) ∆u(k) = [Im×m , 0, ..., 0]∆U ∗ (k)

5th International Conference on Advances in Control and 332 Abhishek Dhar et al. / IFAC PapersOnLine 51-1 (2018) 329–334 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

Remark 3. The optimization problem mentioned above will be feasible at all instants for proper choice of λ corresponding to a some particular initial condition. If the initial conditions are changed, λ must also be changed following the condition in Lemma 3. Definition 2. The MPC optimization problem (19), subjected to (30) and (32), is said to be feasible at some instant k, if there exists some control input sequence ∆U (k) (dependent on x(k)), which satisfies the constraints (30) and (32) and returns a finite value of the cost function. Assumption 4. If there exists some x(k), for which the MPC optimization problem is feasible for the initial choice  Θ(k), then the optimization problem will be feasible for all  + i) and x(k), where i > k. Θ(k

It is assumed that for some initial choices of x (k) and  Θ(k), the optimization problem (19) is feasible at the k th instant and there exists an optimal ∆U ∗ (k) satisfying (30) and (32), where ∗

∗

T

∗

T

∆U (k) =[∆u (k|k) , ∆u (k + 1|k) , .. th

.., ∆u∗ (k + Np − 1|k)T ]T

(34)

At the (k +1) instant, the initial value of the state would be x (k + 1) = x ∗ (k + 1|k), for which a feasible sequence  exists with the system parameters being Θ(k). Hence, using Assumption 4, it can be claimed that, there exists some feasible control input sequence for the estimated system at (k + 1)th instant with the initial states and  system parameters respectively being x (k+1) and Θ(k+1). Therefore, a feasible solution for the optimization problem (19) at (k + 1)th instant can be given as

∆U (k + 1) = [∆u(k + 1|k + 1)T , .., ∆u(k + Np |k + 1)T ]T

=[∆u∗ (k + 1|k)T , ..., ∆u∗ (k + Np − 1|k)T

, ∆u(k + Np |k + 1)T ]T (35) ∗ where ∆u(k+m|k+1) = ∆u (k+m|k), m = 1(1)Np −1. If Assumption 4 holds, then some ∆u(k+Np |k+1), satisfying (30) and the terminal constraint x (k + Np + 1|k + 1) = 0, will exist. Thus recursive feasibility of the adaptive MPC problem is established.

Theorem 5. If the optimization (19), subject to (30) and (32), is feasible at the initial time, then with the receding horizon control (33), the tracking  error of the closed loop system defined by (1) and (2) or (5) and (6) , with the adaptive update law (11), is bounded and asymptotically converging to zero. Proof. Without loss of generality, it is assumed that Np = Nc . The cost function described in (17) can be reformulated as Np  x (k + m|k)T Q x(k + m|k)+ Jy (k) = 

∆u(k + m|k)T ru ∆u(k + m|k)

m=0

x ∗ (k + m|k) +

Np −1



∆u∗ (k + m|k)T ru ∆u∗ (k + m|k)

m=0

(37) V (k) can be upper and lower bounded as (Grune and Pannek, 2011)   2 2 V (k) ≥ α1 er (k) + ∆u∗ (k)    (38) 2 2 V (k) ≤ α2 er (k) + ∆u∗ (k) 

where α1 (·), α2 (·) ∈ κ∞ are invertible functions. Accordingly, the Lyapunov function candidate, at (k + 1)th instant, is then formulated as V (k + 1) = x(k + 1)T Qx(k + 1) + J∗ (k + 1) y

Np +1

=



m=1 Np



x ∗0 (k + m|k + 1)T Q x∗0 (k + m|k + 1)+

∆u∗ (k + m|k + 1)T ru ∆u∗ (k + m|k + 1)

(39)

m=1

A feasible control sequence at (k + 1)th instant can be chosen according to (35) with the assumption that some feasible ∆u(k + Np |k + 1) exists, satisfying x (k + 1 + Np |k + 1) = 0 (40) Thus ∆U (k + 1) satisfies the constraint given by (32). The constraint given by (30) can be satisfied by following Lemma 3. Hence, feasibility of the optimization problem (19) is guaranteed at (k + 1)th instant. Hence at the (k + 1)th instant, some positive definite function V (k + 1), is obtained as Np  V (k + 1) = x (k + m|k + 1)T Q x(k + m|k + 1) m=1

4.2 Stability of the Closed-Loop System

m=1 Np −1

the k th instant and there exists a ∆U ∗ (k), satisfying (30) (by following Lemma 3) and (32), so that the cost function (17) reaches its optimal value Jy∗ (k). Considering x ∗ (k|k) = x(k) a Lyapunov function candidate is now chosen as Np  x ∗ (k + m|k)T Q V (k) = x(k)T Qx(k) + Jy∗ (k) =

(36)

m=0 T

where Q = C C and ru ∈ Rm×m is positive definite. It is assumed that the optimization (19) is possible at 348

+

Np −1



∆u∗ (k + m|k)T ru ∆u∗ (k + m|k)

m=1

(41) + ∆u(k + Np |k + 1)T ru ∆u(k + Np |k + 1) It is to be noted that, (41) is obtained by replacing the optimal input sequence ∆U ∗ (k + 1) by the feasible sequnece ∆U (k + 1) in (39). Therefore, V (k + 1) is an upper bound on V (k + 1) and hence the following hold true V (k + 1) − V (k) ≤ V (k + 1) − V (k) (42) Due to the update of the system parameters from    + 1), B(k  + 1)] from k th instant to [A(k), B(k)] to [A(k th (k + 1) instant, the following holds true  x (k + 1|k + 1) = x ∗ (k + 1|k) + δ0 (∆Θ(k + 1))   x (k + 2|k + 1) = x ∗ (k + 2|k) + δ1 (∆Θ(k + 1)) .   x (k + Np |k + 1) = x ∗ (k + Np |k) + δNp −1 (∆Θ(k + 1)) (43)

5th International Conference on Advances in Control and Abhishek Dhar et al. / IFAC PapersOnLine 51-1 (2018) 329–334 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

 + 1) − Θ(k)  where ∆Θ(k + 1)  Θ(k and

333

2

x∗ (k+1|k)+∆ΘAB(k+1, i) (44) δi (k+1) = ∆ΘA(k+1, i)

 + 1)c − A(k)  c and ∆Θ (k + where ∆ΘA(k + 1, c)  A(k  AB∗ c   j−1   j−1 B(k)  ∆u (k+ 1, c)  j=1 A(k+1) B(k+1)− A(k) j|k). δi (k+1) is the explicit expression for δi (∆Θ(k+1)). It is seen that ∆Θ(k +1) → 0 asymptotically, because of (11)  and Theorem 2, which results in ∆ΘA(k+1, c), ∆ΘAB(k+  1, c) → 0. Consequently, it is claimed that lim δi (k + 1) = 0

(45)

2

2

V (k + 1) − V (k) ≤ −λ1 x(k) − ru ∆u(k) +

ξ1 (k) 4λ2

Np−1

+ ξ2 (k) +



m=0

2

2

Q δm  + ru ∆u(k + Np |k + 1)

(48)

Let Np−1 2  ξ1 (k) 2 + ξ2 (k) + Q δm  Ω= 4λ2 m=0 2

Using the result from (13), (37), (41) and (43)

+ ru ∆u(k + Np |k + 1) Therefore, the inequality (48) is reformulated as

V (k + 1) − V (k) = −x(k)T Qx(k) − ∆u∗ (k)T ru ∆u∗ (k) Np  σ T   ∗  m x(k) +   σ−j B(k)∆u A(k) (k + j|k) A(k) +2

V (k + 1) − V (k) ≤ −λ1 x(k) − ru ∆u(k) + Ω (49) Since ξ1 (k), ξ2 (k) and δm are bounded and ∆u(k + Np |k + 1) is finite, Ω is also bounded. Moreover, as δm → 0 asymptotically, it implies lim ξ1 (k), ξ2 (k) = 0

k→∞

m=1

2

j=0

Np−1

Qδm−1 (k + 1) +



k→∞

δm (k + 1)T Qδm (k + 1)

m=0

+ ∆u(k + Np |k + 1)T ru ∆u(k + Np |k + 1)

(46)

Define Np   T     m Qδm−1 (k + 1) A(k) ξ1 k, Θ(k), δi (k + 1)  2 

 δi (k + 1)  ξ2 k, ∆U ∗ (k), Θ(k), 2

Np  σ  

m=1

j=0

∗  σ−j B(k)∆u  (k + j|k) A(k)

T

Consequently, from (42) and (49) it follows   2 2 V (k + 1) − V (k) ≤ − x(k) + ∆u∗ (k) + Ω (51) T

Qδm−1 (k + 1)

   δi (k + 1) is repFor ease of representation ξ1 k, Θ(k),    δi (k + 1) is resented as ξ1 (k) and ξ2 k, ∆U ∗ (k), Θ(k),  represented as ξ2 (k). According to Theorem 2, Θ(k) is ultimately bounded and due to the constraints (30) and (32) imposed on the input, ∆U ∗ (k) will also be always bounded. Hence, ξ1 (k) and ξ2 (k) will also be bounded at all instants. Using these assumptions, V (k + 1, x(k + 1) − V (k, x(k)) can be upper bounded as follows 2

2

V (k + 1) − V (k) ≤ −λmin (Q)  x(k) − ru ∆u∗ (k) Np−1

+ ξ1 (k) x(k) + ξ2 (k) + 2

+ ru ∆u(k + Np |k + 1)



m=0

Also, the control input ∆u(k + Np |k + 1) is present due to the presence of model mismatch in between two consecutive instants. Since ∆Θ(k + 1) → 0 as k → ∞ (from Theorem 2), ∆u(k + Np |k + 1) → 0 as k → ∞. Therefore, it can be claimed that lim Ω = 0 (50) k→∞

m=1



2

2

Q δm 

(47)

where λmin (·) is the minimum eigen value of the argument matrix. Since Q is a positive definite matrix, Θmin (Q) > 0. Suppose λmin (Q) = λ1 + λ2 , where λ1 , λ2 > 0. Using this result in (47), the following is obtained

where = min(λ1 , ru ). Since x(k)  [∆xr (k)T , er (k)T ] , it implies er (k) < x(k). Therefore,   2 2 V (k + 1) − V (k) ≤ − er (k) + ∆u∗ (k) + Ω (52)

Using the result from (38), equation (52) can be reformulated as   V (k + 1) − V (k) ≤ − α1−1 V (k) + Ω (53)

Using Definition 3.2 of (Jiang and Wang, 2001), it is claimed that V (k) is an ISS-Lyapunov function. Since Ω is bounded and asymptotically converging to zero, it is further claimed that V (k) is bounded and asymptotically converging to zero.Therefore, it can be claimed that the state x(k) and consequently tracking error er (k) of the actual system (1)-(2) is bounded and asymptotically converging to zero. 5. SIMULATION RESULTS

The following simulation example is used from (Zhu and Xia, 2016), to illustrate the results as claimed in the theory. 2 2 2 The plant under consideration is MIMO with two inputs, V (k + 1) − V (k) ≤ −λ1 x(k) − λ2 x(k) − ru ∆u(k) given as [ur1 , ur2 ]T and two outputs given as [yr1 , yr2 ]T . Np−1  The uncertain system matrix, input matrix and the known 2 + ξ1 (k) x(k) + ξ2 (k) + Q δm  output matrix are given as     m=0   0.8 0.4 1.1 0.7 0 2 100 + ru ∆u(k + Np |k + 1) Ar = 0.6 1.5 −0.1 ; Br = 0.2 0 ; Cr = 001 0.1 −1.2 1.8 −0.6 1.4 Completing the squares using the second and the fourth terms on the right hand side of the last inequality, the The initial values of the estimated parameter matrices are following is obtained considered as 349

5th International Conference on Advances in Control and 334 Abhishek Dhar et al. / IFAC PapersOnLine 51-1 (2018) 329–334 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

6. CONCLUSION

0.8 y (k) 1

y(k)

0.6

r

1

y (k)

r

2

2

An adaptive MPC strategy is proposed for controlling a fully uncertain discrete-time linear system and subjected to actuator rate constraints. The update rate of the gradient descent adaptive law is upper bounded to satisfy the actuator rate constraint and to make the constraint imposed by the adaptive law inactive for all instants. This work is a contribution over existing results in literature, where an adaptive MPC problem for completely uncertain linear system has not been considered for the constrained case. With the proposed adaptive MPC, it is proved that the parameter estimation errors are ultimately bounded and the tracking errors of the closed loop system are bounded and asymptotically converging to zero. The derived results are substantiated by the provided simulation results.

0.4 0.2 0

0.5

1

1.5 2 Time (sec)

2.5

3

Fig. 1. Actual and desired trajectories. 0.15 0.1

∆u1(k) ∆u2(k)

∆u(k)

0.05

∆umax ∆umin

0

REFERENCES

-0.05 -0.1 -0.15

0.5

1

1.5 2 Time (sec)

2.5

3

2.5

3

Fig. 2. The incremental control inputs

˜ ˜ T Θ) tr(Θ

0.48

0.47

0.46

0.45

0.5

1

1.5 2 Time (sec)

Fig. 3. The norm and variation of parameter estimation errors.     1 0.5 0.75 0.5 0.1 r (0) = 0.2 0.8 0.5 ; B r (0) = 1 0.1 A 0 0 2 −0.5 1.75 The initial conditions of the system states are chosen as x(0) = [0.15, 0.1, −0.2, 0, 0]T and that of the estimated states are chosen as x (0) = [0, 0, 0, 0, 0]T . The prediction horizon and the control horizon are chosen as Np = Nc = 6. The sampling interval is chosen as h = 0.02. The actuator constraint imposed on the system is given as ∆u(k) ≤ 0.164 Therefore, ∆ui (k), where i = 1, 2, is bounded as (54) −0.12 ≤ ∆ui (k) ≤ 0.12, i = 1, 2 The variable α is chosen as α = 1.8 and correspondingly λ is chosen such that Lemma 3 is satisfied. In this problem λ = 0.1 and ru = 0.1. The reference time varying signals are given as r = [0.5 0.3]T . The tracking performance of the closed loop system is shown in Fig. 1. It is displayed in Fig. 2 that the rate of change of control inputs are contained within the bounds imposed on them. The parameter estimation error, as seen in Fig. 3, is also bounded and its variation also converges to zero. 350

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