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Economic Model Predictive Control with Zone Tracking
6th IFAC Conference on Nonlinear Model Predictive 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 Available online at www.sciencedirect.com Madison, WI, USA, USA, August August 19-22, 2018 2018 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, 19-22, Madison, WI, USA, August 19-22, 2018 Madison, WI, USA, August 19-22, 2018
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IFAC PapersOnLine 51-20 (2018) 16–21
Economic Economic Economic Economic
Model Predictive Control Model Predictive Control Model Predictive Control Model Predictive Control Zone Tracking Zone Tracking Zone Tracking Zone∗ Tracking∗,1
with with with with
Su ∗,1 Su Liu Liu ∗∗∗ ,, Jinfeng Jinfeng Liu Liu ∗,1 ∗,1 Su Liu ∗ , Jinfeng Liu ∗,1 Su Liu , Jinfeng Liu ∗ ∗ Department of Chemical & Materials Engineering, University of ∗ ∗ Department of Chemical & Materials Engineering, University of of & University Alberta, Edmonton, AB 1H9, ∗ Department Alberta, Edmonton, AB T6G T6GEngineering, 1H9, Canada. Canada. Department of Chemical Chemical & Materials Materials Engineering, University of of Alberta, Edmonton, AB T6G 1H9, Canada. Alberta, Edmonton, AB T6G 1H9, Canada. Abstract: Abstract: In In this this work, work, we we propose propose a a framework framework for for economic economic model model predictive predictive control control (EMPC) (EMPC) Abstract: In this work, we propose a framework for economic model predictive control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework with zone tracking. A zone cost is incorporated the predictive existing EMPC framework Abstract: In this work, wetracking proposestage a framework for economicinto model control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework to essentially form a multi-objective optimization problem. We provide sufficient conditions for to essentially form aAmulti-objective optimization problem. We sufficient conditions for with zone tracking. zone tracking stage cost is incorporated intoprovide the existing EMPC framework to essentially form a multi-objective optimization problem. We provide sufficient conditions for asymptotic stability of the optimal steady state and characterize exact penalty for the zone asymptotic of the optimal optimization steady state problem. and characterize exact penaltyconditions for the zone to essentiallystability form a multi-objective We provide sufficient for asymptotic stability of the optimal steady state and characterize exact penalty for the zone tracking cost which prioritizes zone tracking objective over economic objective. A comprehensive tracking coststability which prioritizes zone tracking economic exact objective. A comprehensive asymptotic of the optimal steady objective state andover characterize penalty for the zone tracking which zone tracking objective over economic A comprehensive study on the between economic is on study on cost the tradeoff tradeoff between zone zone tracking and economic performance is carried carried out on aa simple simple tracking cost which prioritizes prioritizes zone tracking tracking and objective overperformance economic objective. objective. A out comprehensive study on the tradeoff between zone tracking and economic performance is carried out on simple scalar linear system. Simulation results reveal intrinsic difficulties in parameter tuning to scalar linear system. between Simulation reveal in parameter tuning due to study on the tradeoff zoneresults tracking andintrinsic economicdifficulties performance is carried out on aa due simple scalar linear system. Simulation results reveal intrinsic difficulties in parameter tuning due the inconsistency between the zone tracking and economic objectives. A procedure to modify the inconsistency between the zone tracking economic objectives. A procedure to modify scalar linear system. Simulation results revealand intrinsic difficulties in parameter tuning due to to between tracking and objectives. to the target on economic performance and the steady the inconsistency target zone zone based based on the the zone economic performance and reachability reachability ofprocedure the optimal optimal steady inconsistency between zone tracking and economic economic objectives. A Aof procedure to modify modify the zone on performance and of optimal steady state is The target effectively the zone tracking and state is proposed. proposed. The modified modified target zone zone effectively decouples decouples the dynamic dynamic zone tracking and the target target zone based based on the the economic economic performance and reachability reachability of the the optimal steady state is proposed. The modified target zone effectively decouples the dynamic zone tracking economic objectives and simplifies parameter tuning. economic objectives and simplifies parameter tuning. decouples the dynamic zone tracking and state is proposed. The modified target zone effectively and economic objectives and parameter tuning. economic objectives and simplifies simplifies tuning. © 2018, IFAC (International Federationparameter of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Predictive Predictive control; control; Process Process optimization; optimization; Soft Soft constraint; constraint; Zone Zone control control Keywords: Predictive control; Process optimization; Soft constraint; Zone control Keywords: Predictive control; Process optimization; Soft constraint; Zone control 1. As 1. INTRODUCTION INTRODUCTION As its its name name suggests, suggests, soft soft constraint constraint is is often often dismissed dismissed as as a a 1. INTRODUCTION As its name suggests, soft constraint is often dismissed as a trick to avoid feasibility issue with hard constraint, and trick avoid feasibility with hard constraint, and is 1. INTRODUCTION As itsto name suggests, softissue constraint is often dismissed as is a trick to avoid feasibility issue with hard constraint, and is addressed separately from set-point tracking or economic Process control of chemical plants needs to address a addressed separately from set-point tracking or economic to avoid feasibility issue with hard constraint, and is Process control of chemical plants needs to address a trick set-point tracking economic objectives. To of knowledge, only aa few workProcess control of chemical to address a addressed number objectives including safety, environmental reguTo the the best bestfrom of our our knowledge, only or few workaddressed separately separately from set-point tracking or economic number of objectives includingplants safety, needs environmental reguProcess of control of chemical plants needs to address a objectives. objectives. To the best of our knowledge, only a few works explicitly handle zone tracking objectives (Ferramosca number of objectives including safety, environmental regulations, product energy quality, efficiency, profitability, etc. s explicitly handle zone tracking objectives (Ferramosca objectives. To the best of our knowledge, only a few worklations, quality, energy safety, efficiency, profitability,reguetc. s explicitly handle zone tracking objectives (Ferramosca number product of objectives including environmental et al. (2010); Gonz´ a lez et al. (2009); Gonz´ a lez and Odloak lations, product quality, energy efficiency, profitability, etc. According to (Lu (2003)), an integrated control system s explicitly handle zone tracking objectives (Ferramosca According to (Lu (2003)), an efficiency, integratedprofitability, control system lations, product quality, energy etc. et al. (2010); Gonz´ a lez et al. (2009); Gonz´ a lez and Odloak et al. (2009)). According to (Lu (2003)), an integrated control system consists of facets: regulatory control, conal. (2010); (2010); Gonz´ Gonz´aalez lez et et al. al. (2009); (2009); Gonz´ Gonz´aalez lez and and Odloak Odloak consists of three three facets: regulatory control, constraint constraint con- et According to (Lu (2003)), an integrated control system (2009)). (2009)). consists of three facets: regulatory control, constraint control and maneuvering control. Regulatory control refers to trol and of maneuvering Regulatory refers to (2009)). In this work, we propose an EMPC framework with consists three facets:control. regulatory control,control constraint conthis work, we propose an EMPC framework with inintrol and maneuvering control. Regulatory controlminimizes refers to In the setpoint tracking control In this propose framework with inthe conventional setpoint tracking control which which trol conventional and maneuvering control. Regulatory controlminimizes refers to tegrated zone we tracking. A an zoneEMPC tracking stage cost which In this work, work, we propose an EMPC framework with inconventional setpoint tracking control which minimizes tegrated zone tracking. A zone tracking stage cost which the variance of controlled variables to the setpoint. Conzone tracking. A zone tracking stage cost which the conventional variance of controlled variables to thewhich setpoint. Con- tegrated penalizes weighted l norm and squared l norm distance setpoint tracking control minimizes 1 2 penalizes weighted l norm and squared l norm distance tegrated zone tracking. A zone tracking stage cost which 1 2 the variance of controlled variables to the setpoint. Constraint control, or zone control, prevents the system from 1 2 penalizes weighted norm norm straint control, zone control, prevents system Confrom to the zone into EMPC the variance of or controlled variables to thethe setpoint. to the target target zone is isll11incorporated incorporated into the thell22existing existing EMPC penalizes weighted norm and and squared squared norm distance distance straint control, or zoneand control, the back system violating its steers the into the the target zone is incorporated into the existing violating its boundary boundary steersprevents the system system intofrom the to framework, forming a multi-objective optimization probstraint control, or zoneand control, prevents the back system from framework, a multi-objective optimization probto the targetforming zone is incorporated into the existing EMPC EMPC violating its boundary and steers the system back into the zone whenever constraint violation happens. Maneuvering forming a multi-objective optimization zone whenever constraint happens. Maneuvering lem. We provide sufficient conditions for asymptotic staviolating its boundary andviolation steers the system back into the framework, lem. We provide sufficient conditions for asymptoticprobstaframework, forming a multi-objective optimization probzone whenever constraint violation happens. Maneuvering control moves the system from current operating condition lem. We provide sufficient conditions for asymptotic stacontrol moves the system from current operating condition bility zone whenever constraint violation happens. Maneuvering of the optimal steady state and characterize exact bility of the optimal steady state and characterize exact lem. We provide sufficient conditions for asymptotic stacontrol moves the system from current operating condition to a new operating condition, typically due to economic of the optimal steady state and characterize exact to a new operating condition, typically due to condition economic bility penalty for the zone tracking cost which prioritizes the control moves the system from current operating penalty for the zone tracking cost which prioritizes the bility of the optimal steady state and characterize exact to a new operating condition, typically duedesigned to economic considerations. Generally speaking, aa well confor the zone tracking cost which prioritizes the considerations. Generally speaking, well con- penalty zone tracking objective over the economic objective. The to a new operating condition, typically duedesigned to economic penalty for the zone tracking cost which objective. prioritizes The the considerations. Generally speaking, a well designed conzone tracking objective over the economic trol system necessarily integrates all three control types, tracking objective over the economic objective. The trol system necessarily integrates allathree proposed EMPC framework is tested in a simple numerical considerations. Generally speaking, well control designedtypes, con- zone proposed EMPC framework tested in a simple numerical zone tracking objective overisthe economic objective. The trol system necessarily integrates all three control types, although emphasis on the three facets of control may vary EMPC framework is aa simple although emphasis on the three facets of control maytypes, vary proposed example which we the tradeoff trol system necessarily integrates all three control example in in which we systematically systematically study thenumerical tradeoff EMPC framework is tested tested in in study simple numerical although emphasis the three facets of control may vary proposed from application to application. example in which we systematically study the tradeoff from application toon application. although emphasis on the three facets of control may vary between zone tracking and economic performance based between zone tracking and economic performance based example in which we systematically study the tradeoff from application to application. between zone tracking and economic performance based from application to application. on the l norm and squared l norm penalties. SimulaModel predictive control (MPC) has been the most widely 1 2 on the l11zone norm and squared l22 norm performance penalties. Simulatracking and economic based Model predictive control (MPC) has been the most widely between on the l norm and squared l norm penalties. Simulation results reveal intrinsic difficulties in parameter tuning Model predictive control (MPC) has been the most widely 1 2 applied advanced control technique, due to its ability to tionthe results revealand intrinsic difficulties parameterSimulatuning l1 norm squared l2 norminpenalties. applied advancedcontrol control(MPC) technique, due the to its ability to on Model predictive has been most widely results intrinsic difficulties parameter tuning due the inconsistency the tracking and applied advanced control to its ability to tion handle and incorporate economic considerdue to to the reveal inconsistency between thein zone tracking and results reveal intrinsicbetween difficulties inzone parameter tuning handle constraints and to to technique, incorporatedue economic considerapplied constraints advanced control technique, due to its ability to tion due to the inconsistency between the zone tracking and economic objectives. Finally, to resolve the difficulties in handle constraints and to incorporate economic considerations. The literature is rich with theories for conventional economic objectives. Finally, to resolve the difficulties in due to the inconsistency between the zone tracking and ations. The literature is rich with theories for conventional handle constraints and to incorporate economic consider- economic objectives. Finally, to resolve the difficulties in parameter tuning, we propose a procedure to modify the ations. The literature is rich with theories for conventional setpoint tracking MPC (Mayne et al. (2000)). The past economic objectives. Finally, toa resolve the to difficulties in setpoint tracking MPC (Mayne al. (2000)). The past parameter ations. The literature is rich with et theories for conventional tuning, we propose procedure modify the parameter tuning, we propose a procedure to modify the target zone based on the economic performance and reachsetpoint tracking MPC (Mayne et al. (2000)). The past decade has seen an increasing academic interest in economtarget zonetuning, based on economic performance and reachwethe propose a procedure to modify the decade seen anMPC increasing academic economsetpointhas tracking (Mayne et al. interest (2000)).inThe past parameter target based on economic performance and ability of optimal state. The target decade haspredictive seen an increasing academic(Diehl interest ic model control et al. (2011); ability zone of the the optimal steady state. The modified modified target zone based on the thesteady economic performance and reachreachic model control (EMPC) (EMPC) et in al.econom(2011); target decade haspredictive seen an increasing academic(Diehl interest in economability of the optimal steady state. The modified target zone effectively decouples the dynamic zone tracking and ic model predictive control (EMPC) (Diehl et al. (2011); Heidarinejad et al. (2012); Gr¨ u ne (2013)) which integrates zone effectively decouples the dynamic zone tracking and of the optimal steady state. The modified target Heidarinejad et al. (2012); une (2013)) which integrates ic model predictive controlGr¨ (EMPC) (Diehl et al. (2011); ability zone effectively decouples the dynamic zone tracking economic objectives and simplifies parameter tuning. Heidarinejad et al. (2012); Gr¨ u ne (2013)) which integrates economic objective into regulatory setpoint control. On the economic objectives and simplifies parameter tuning. and effectively decouples the dynamic zone tracking and economic objective into regulatory setpointwhich control. On the zone Heidarinejad et al. (2012); Gr¨ une (2013)) integrates economic objectives and simplifies parameter tuning. economic objective intohas regulatory control. On contrary, zone received less In contrary, zone control control receivedsetpoint less attention. attention. In the the economic objectives and simplifies parameter tuning. economic objective intohas regulatory setpoint control. On contrary, zone control has received less dealt attention. In the MPC framework, zone is with 2. PROBLEM SETUP MPC framework, zone control control is usually usually with by by contrary, zone control has received less dealt attention. In the 2. PROBLEM SETUP MPC framework, zone control is usually dealt with by the 2. so-called soft constraint technique (Scokaert and Rawlings so-called soft constraint technique (Scokaert and Rawlings MPC framework, zone control is usually dealt with by the 2. PROBLEM PROBLEM SETUP SETUP so-called soft constraint technique (Scokaert and Rawlings (1999); Kerrigan and Maciejowski (2000b); de Oliveira and 2.1 Notation (1999); and Maciejowski Oliveira and 2.1 Notation so-calledKerrigan soft constraint technique (2000b); (Scokaertdeand Rawlings (1999); Kerrigan and Maciejowski (2000b); and 2.1 Notation Biegler Zeilinger et Askari al. Biegler (1994); Zeilinger et al. al. (2014); (2014); Askaride etOliveira al. (2017)). (2017)). (1999); (1994); Kerrigan and Maciejowski (2000b); deet Oliveira and Notation Biegler (1994); Zeilinger et al. (2014); Askari et al. (2017)). 2.1 this the lp norm of Biegler (1994); Zeilinger et al. (2014); Askari et al. (2017)). Throughout Throughout this work, work, x x ppp denotes denotes the of the the 1 Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: 1/p llppp norm 1 Throughout this work, x denotes the norm of the 1 Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: (i) p p Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: 1/p .lpThe Throughout this work, x denotes the norm of the 1 = |x | operator vector x such that x 1/p +1-780-492-2881. Email: [email protected]. (i) p Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: p 1 |x(i) . The operator vector x such that x ppp = p +1-780-492-2881. Email: +1-780-492-2881. Email: [email protected]. [email protected]. Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: (i) |p 1/p 1/p . The operator = |x | vector x such that x +1-780-492-2881. Email: [email protected]. (i) p p = |x | . The operator vector by x Elsevier such that +1-780-492-2881. [email protected]. 2405-8963 © 2018,Email: IFAC (International Federation of Automatic Control) Hosting Ltd. x All rights reserved. p
Copyright 2018 IFAC 16 Control. Peer review© under responsibility of International Federation of Automatic Copyright 16 Copyright © © 2018 2018 IFAC IFAC 16 Copyright © 2018 IFAC 16 10.1016/j.ifacol.2018.10.168 Copyright © 2018 IFAC 16
2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018
Su Liu et al. / IFAC PapersOnLine 51-20 (2018) 16–21
| · | denotes the l2 norm or Euclidean norm of a scalar or a vector. The symbol IN M denotes the set of integers {M, M + 1, ..., N }. The symbol projX (O) denotes projection of the set O onto its subspace X.
x0 = x(n)
n=0
∞
e (x(n), u(n)) + K
x ,u
z
(2)
min z z
ui ,xi ,ui
i=0
e (xi , ui ) + c1 (xi − xzi 1 + ui − uzi 1 ) 2
2
+ c2 (xi − xzi 2 + ui − uzi 2 )
(6)
−1 (xzi , uzi ) ∈ Zt , i ∈ IN 0
We note that the zone tracking treatment in the above EMPC formulation of Eq. (6) is closely connected to the so-called soft constraint technique. In the following, we establish sufficient conditions for asymptotic stability of the optimal steady state xs . and convergence into the target zone Zt in finite steps. Definition 1. (Strictly dissipative systems) The system of Eq. (1) is strictly dissipative with respect to the supply rate s : X × U → R if there exists a continuous storage function λ(·) : X → R and a K∞ function αl (·) such that the following holds for all x ∈ X and u ∈ U: λ(f (x, u)) − λ(x) ≤ s(x, u) − αl (|x − xs |) Assumption 1. (Strict dissipativity, Amrit et al. (2011)) The system of Eq. (1) is strictly dissipative with respect to the supply rate s(x, u) = e (x, u) − e (xs , us ) Assumption 2. (weak controllability, (Diehl et al. (2011))) There exists a K∞ function γ(·) such that for all x ∈ XN , there exists a feasible solution to Eq. (4) such that:
(3)
(x, u) ∈ Zt
3. EMPC WITH ZONE TRACKING In this section, a general framework for economic model predictive control with zone tracking is presented. A pointwise terminal constraint is employed for simplicity of exposition.
N −1
Specifically, at a sampling time n, the following finitehorizon optimization problem is solved:
i=0
|ui − us | ≤ γ(|x − xs |)
Theorem 1. If Assumptions 1 and 2 hold, then the optimal steady state xs is asymptotically stable under the EMPC of Eq. (4) with a region of attraction XN .
(4a)
i=0
−1 s.t. xi+1 = f (xi , ui ), i ∈ IN 0
N −1
s.t. (4b) − (4e)
We assume that (xs , us ) uniquely solves the above steadystate optimization problem. Note that the economic cost function e (·) is not necessarily positive definite with respect to (xs , us ).
e (xi , ui ) + z (xi , ui )
(5)
z
s.t. (x , u ) ∈ Zt
x,u
min
2
where c1 and c2 are positive scalars. Equation (5) characterizes a weighted l1 norm and squared l2 norm zone tracking penalty. The incorporation of the l1 norm penalty allows for the so-called exact penalty which will be discussed in the sequel. With the zone tracking penalty defined by Eq. (5), the optimization of Eq. (4) amounts to the following optimization:
Let (xs , us ) denote the economically optimal steady state in the target zone Zt . That is: (xs , us ) = arg min e (x, u)
u0 ,··· ,uN −1
2
c2 (x − xz 2 + u − uz 2 )
subject to the constraint: n≥K +1 (x(n), u(n)) ∈ Zt , The above problem is essentially a multi-objective optimization problem in which the magnitude, duration of zone target violation, as well as the economic performance are tradeoffed. The zone tracking objective is prioritized in that it needs to be satisfied in finite steps (K is finite).
N −1
(4d) (4e)
The zone tracking penalty z (x, u) is defined by the following function: z (x, u) = min c1 (x − xz 1 + u − uz 1 )+ z z
n=K+1
s.t. x = f (x, u)
−1 IN 0
−1 Let u∗ (i|n), i ∈ IN denote the optimal solution, the 0 input injected to the system at time n is: u(n) = u∗ (0|n). At the next sampling time n+1, the optimization of Eq. (4) is re-evaluated, generating an implicit feedback control law u(n) = κN (x(n)). We denote the feasibility region of the optimization problem of Eq. (4) by XN . XN is forward invariant under the feedback law u(n) = κN (x(n)) due to the terminal constraint Eq. (4e). In other words, the EMPC design is recursively feasible.
We consider the following nonlinear discrete time system: x(n + 1) = f (x(n), u(n)) (1) where x(n) ∈ X ⊂ Rnx , u(n) ∈ U ⊂ Rnu , n ∈ I≥0 , denote the state and input at time n, respectively. The vector function f (·) : Rnx ×Rnu → Rnx is continuous. The system is subject to coupled state and input constraint: (x(n), u(n)) ∈ Z ⊆ X × U, n ∈ I≥0 where X, U, Z are all compact sets. The primary control objective is to steer and maintain the system in a compact set Zt ⊂ Z. The target zone Zt may correspond to, for example, product specifications or safety considerations. The distance to the target zone is measured by the function z (x, u) : Z → R which is positive definite with respect to the target zone Zt . There is also a secondary economic objective to minimize the operational cost characterized by the function e (x, u) : Z → R. Both z (·) and e (·) are continuous functions. Stated formally, the controller seeks to minimize the following function: z (x(n), u(n)) +
(4c)
(xi , ui ) ∈ Z, i ∈ x N = xs
2.2 System description and control objective
K
17
(4b) 17
2018 IFAC NMPC 18 Madison, WI, USA, August 19-22, 2018
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Theorem 2 implies that if the l1 norm penalty c1 is sufficiently large, then the zone tracking objective is prioritized over the economic objective. Note that the constraints (7c) - (7e) can be combined by canceling the slack variables xzi −1 and uzi into the compact form: (xi , ui ) ∈ Zt , i ∈ IN . 0 Thus the EMPC of Eq. (7) yields the same solution to the following EMPC constrained by the target zone:
Proof We provide a sketch of the proof based on the results in (Diehl et al. (2011)) and (Liu and Liu (2016)). Define the rotated cost: ¯e (x, u) = e (x, u) − e (xs , us ) + λ(x) − λ(f (x, u)) From Assumption 1, the rotated cost satisfies ¯e (x, u) ≥ αl (|x − xs |). Moreover, it can be shown that the solution to Eq. (4) is equivalent to the following problem: min
u0 ,··· ,uN −1
N −1
min
¯e (xi , ui ) + z (xi , ui )
u0 ,··· ,uN −1
i=0
x0 = x(n)
Let L(x, u) = ¯e (x, u) + z (x, u) ≥ αl (|x − xs |), the above optimization is then reduced to a classical MPC with positive-definite stage cost and point-wise terminal constraints. Assumption 2 ensures the existence of a K∞ function β(|x − xs |) which bounds the value function of EMPC from above. It is then straightforward to apply standard MPC theory. The details can be found in Diehl et al. (2011)) and are omitted for brevity.
x N = xs
The constraint (4d) is removed because Zt ⊂ Z. Therefore, sufficiently large l1 norm penalty term c1 (xi − xzi 1 + ui − uzi 1 ) in effect can convert the zone tracking objective into hard constraints (xi , ui ) ∈ Zt . Note that exact penalty cannot be achieved by the quadratic term 2 2 c2 (x − xz 2 + u − uz 2 ) unless c2 can be made infinitely large (de Oliveira and Biegler (1994)).
An important question to ask is: will the system leave the target zone once it is in the zone? To answer this question, we need the following definition of the N -step reachable set (Kerrigan and Maciejowski (2000a)): Definition 2. (N -step reachable set) We use XN (Zt , xs ) to denote the set of states that can be steered to xs in N steps while satisfying the state and input constraints (x, u) ∈ Zt . That is, −1 , ∃ (x(n), u(n)) ∈ Zt , n ∈ IN 0 XN (Zt , xs ) = x(0) | x(n + 1) = f (x(n), u(n)), x(N ) = xs
e (xi , ui )
(xzi , uzi ) ∈ Zt , i ∈
−1 IN 0 −1 i ∈ IN 0
4. EXAMPLE
(7a) (7b)
4.1 l1 norm penalty
(7c)
Fig. 1 shows the closed-loop input trajectories of the EMPC of Eq. (6) with different l1 -norm penalties for c1 while the quadratic penalty c2 = 0. The corresponding economic performances are shown in Fig. 2. It is seen that all closed-loop trajectories reach the target zone in finite steps and asymptotically converge to the optimal value us = 0.9. As c1 increases, the closed-loop trajectory reaches the target zone faster but also stays on the zone boundary for longer periods. Similar phenomina of riding the constraint boundary due to l1 norm penalty was observed in Scokaert and Rawlings (1999). When c1 = 100, the EMPC leads to deadbeat zone tracking.
i=0
s.t. (4b) − (4e)
Note also that to ensure x(n) ∈ Zt for all n ∈ I≥0 and x(0) ∈ XN (Zt , xs ), the condition c1 > λ∞ needs to be satisfied for Lagrange multipliers λ associated with all x(n) ∈ XN (Zt , xs ). Finding the exact lower bound for such c1 is in general difficult. The task is possible for linear systems with quadratic tracking costs and polyhedral constraints (Kerrigan and Maciejowski (2000b)).
In this Section, we use a simple example to illustrate the zone tracking and economic performance of the EMPC of Eq. (6) with different choices of c1 , c2 . Special attentions are paid to the difficulties encountered in parameter tuning. The system investigated is a linear scalar system: x(n + 1) = 1.25x(n) + u(n) with state and input constraints X = [−5, 5], U = [−5, 5] respectively. The target set is Zt = {(x, u) | x ∈ [−5, 5], u ∈ [−1, 1]}. An economic cost e = (u − 0.9)2 is used which corresponds to an optimal steady state (xs , us ) = (−3.6, 0.9). Two different initial states: x(0) = −5 and x(0) = 5 are compared in parallel to indicate the asymmetric closed-loop performance. The control horizon is N = 20 when not specified.
Based on the above definition, the EMPC is capable of maintaining the system in the target zone only if x(0) ∈ XN (Zt , xs ). Let λ denote the vector of the KKT multipliers to the following optimization problem: ui ,xi ,ui
(8)
−1 (xi , ui ) ∈ Zt , i ∈ IN 0
If the optimal steady state lies in the interior of the target zone (i.e., (xs , us ) ∈ Zt ), then asymptotic stability of the optimal steady state implies finite-time convergence into the target zone. However, if the optimal steady state lies on the boundary of the target zone, finite-time convergence to the target zone will be more tricky to achieve. Typically the task can be achieved by employing sufficiently large l1 norm penalty value c1 which is known to result in deadbeat control policy (Rao and Rawlings (2000)).
N −1
e (xi , ui )
i=0
−1 s.t. xi+1 = f (xi , ui ), i ∈ IN 0
s.t. (4b) − (4e)
min z z
N −1
(xi , ui ) = (xzi , uzi ) (7d) Theorem 2. If x(n) ∈ XN (Zt , xs ) and c1 > λ∞ , then the solutions to the optimization of Eq. (6) and Eq. (7) are identical. Proof If c1 > λ∞ , then the objective function of Eq. (6) is an exact penalty function for Eq. (7). The solution to Eq. (6) is identical to Eq. (7), and thus identical to Eq. (8). The proof can be found in Theorem 14.3.1, (Fletcher (2013)). 18
2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018
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Also very noticeable is the asymmetric performance with different initial states. With x(0) = −5 (the upper part in Fig. 1 and lower part in Fig. 2), economic performance is less of concern and the choice c1 = 100 which results in deadbeat zone tracking is preferable. However, with x(0) = −5 (the lower part in Fig. 1 and upper part in Fig. 2), transient economic performance significantly deteriorates because of the boundary riding, which suggests the use of smaller penalty value c1 . The asymmetric behavior, which essentially arises from the inconsistency between zone tracking and economic objectives, poses challenge for parameter tuning.
19
the magnitude of zone tracking violation and increasing the duration of violation. Therefore the quadratic penalty c2 provides a way to tradeoff the duration and magnitude of zone target violation.
1
u(n)
0
-1
-2 2 -3
1
0
10
20
30
40
u(n)
n 0
Fig. 3. Closed-loop input trajectories of EMPC of Eq. (6) with c1 = 0 and c2 = 0.1 (solid), c2 = 1 (dotted), c2 = 10 (dashed), c2 = 100 (dash dotted), respectively. Shaded area depicts the input target zone.
-1 -2
15
-3 0
10
20
30
40
n 10
le(n)
Fig. 1. Closed-loop input trajectories of EMPC of Eq. (6) with c2 = 0 and c1 = 0.1 (solid), c1 = 1 (dotted), c1 = 10 (dashed), c1 = 100 (dash dotted), respectively. Shaded area depicts the input target zone. The upper and lower part correspond to initial state x(0) = −5 and x(0) = 5, respectively.
5
0
15
0
10
20
30
40
n
Fig. 4. Closed-loop economic performance of EMPC of Eq. (6) with c1 = 0 and c2 = 0.1 (solid), c2 = 1 (dotted), c2 = 10 (dashed), c2 = 100 (dash dotted), respectively.
le(n)
10
5
4.3 l1 norm with squared l2 norm As suggested by (Scokaert and Rawlings (1999)), the l1 norm penalty c1 should be chosen to result in exact penalty when used together with the quadratic penalty c2 . Fig. 5 and Fig. 6 show the closed-loop input trajectories and economic performances with c1 = 100 and c2 = 1, 10, 100, 1000 respectively. It is seen that combined use of the l1 norm and squared l2 norm zone tracking penalty further polarizes zone tracking and economic objectives. Symmetric zone tracking with respect to the two initial states is obtained, which implies a higher priority of zone tracking over economic objective. On the other hand, asymmetric economic performance with respect to the two initial states are intensified because of longer duration of target zone riding.
0 0
10
20
30
40
n
Fig. 2. Closed-loop economic performance of EMPC in Eq. (6) with c2 = 0 and c1 = 0.1 (solid), c1 = 1 (dotted), c1 = 10 (dashed), c1 = 100 (dash dotted), respectively. The upper and lower part correspond to initial state x(0) = 5 and x(0) = −5, respectively. 4.2 Squared l2 norm The closed-loop input trajectories and economic performances under different squared l2 norm penalty c2 with the l1 norm penalty c1 = 0 are shown in Fig. 3 and Fig. 4, respectively. Similar asymmetric behaviors with different initial states are observed. We note that when quadratic penalties are employed, no boundary riding takes place. Comparing the cases (c1 = 100, c2 = 0) in Fig. 1 with (c1 = 0, c2 = 100) in Fig. 3, we see that the sum of squares penalty smoothens the control actions by reducing
5. MODIFIED TARGET ZONE The results in Section 4 reveal the intrinsic difficulties of control parameter tuning due to the inconsistency between the zone tracking and economic objectives. Naive implementation of the exact penalty could lead to arbitrarily 19
2018 IFAC NMPC 20 Madison, WI, USA, August 19-22, 2018
Su Liu et al. / IFAC PapersOnLine 51-20 (2018) 16–21
involve computation difficulties. For linear systems with polyhedral target set Zt and polyhedral performance level set {(x, u) : e (x, u) ≤ α}, Algorithm 1 can be performed using the algorithms in (Keerthi and Gilbert (1987)).
2
u(n)
1
Apply Algorithm 1 to the example in Section 4 with M = 10 and α = 1, the modified target zone is obtained as follows: Z t = {(x, u) | Ex + F u ≤ G (9) where E = [1.25, −1.25, 0, 0]T , F = [1, −1, 1, −1]T , G = [−0.7436, 3.9571, 1.0000, 0.1000]. The constraint set Z, target zone Zt and modified target zone Z t are illustrated in Fig 7.
0 -1 -2 0
10
20
30
40
n
Fig. 5. Closed-loop input trajectories of EMPC of Eq. (6) with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. Shaded area depicts the input target zone. 12 10
le(n)
8 6 4 2
Fig. 7. The constraint set Z (box), target zone Zt (shaded rectangle) and modified target zone Z t (parallelogram). The circle indicates the optimal steady state (xs , us )
0 0
10
20
30
40
n
Fig. 6. Closed-loop economic performance of EMPC of Eq. (6) with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively.
Fig. 8 and Fig. 9 show the closed-loop input trajectories and economic performances of the EMPC tracking the modified target zone Z t in Eq. (9), with both l1 norm and squared l2 norm penalties. Compared to Fig. 5 and Fig. 6, we can see that tracking the modified target zone Z t leads to more balanced economic performance in the target zone while accomplishing fast zone tracking for the target zone Zt . Boundary riding takes place at the boundary of the modified target zone Z t which corresponds to the economic performance bound specified by e (x, u) ≤ α = 1. This makes α a good tuning parameter. 40 Let Jez = n=K e (x(n), u(n)) denote the summation of the economic performance in the target zone over 40 steps, where K is the time instant when the system enters the zone and never leaves. Table 1 compares the transient economic performance Jez of the EMPC tracking the target zone Zt and EMPC tracking the modified target zone Z t . It is seen that the EMPC tracking the modified target zone has improved transient performance in the target zone. In fact, the transient performance Jez can be further improved by reducing α.
poor transient economic performance because of the zone boundary riding. Motivated by the difficulties encountered in parameter tuning, we propose tracking a subset of the target zone which can be steered to the optimal steady state in finite steps with guaranteed economic performance. The following conceptual procedure can be taken to find the new target zone: Algorithm 1. (1) Choose some M ∈ I≥1 and α ∈ R such that M ≤ N and α ≥ e (xs , us ). (2) Set Z0 = (xs , us ) (3) for i = 0 : M − 1 f (x, u) ∈ projX (Zi ), Zi+1 = (x, u) (x, u) ≤ α, (x, u) ∈ Z e
t
end (4) The modified target zone is Z t = ZM
The modified target zone Z t , obtained by Algorithm 1, is a zone in which the system can be steered to the optimal steady state in M steps in the target zone Zt while the economic performance of each step is upper bounded by α. The parameters M and α are user-specified to tradeoff fast zone tracking and transient economic performance in the target zone. Note that step (3) of Algorithm 1 requires set projection and set intersection operations which could
Table 1. Transient performance in the Zt tracking Zt x(0) = −5 x(0) = 5 tracking Z t x(0) = −5 x(0) = 5 20
c2 = 1 0.18 58.7 c2 = 1 0.18 17.8
c2 = 10 0.18 58.2 c2 = 10 0.18 20.8
c2 = 100 0.18 60.0 c2 = 100 0.18 25.5
c2 = 1000 0.18 60.1 c2 = 1000 0.18 37.4
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Askari, M., Moghavvemi, M., Almurib, H.A.F., and Haidar, A.M.A. (2017). Stability of soft-constrained finite horizon model predictive control. IEEE Transactions on Industry Applications, 53(6), 5883–5892. doi: 10.1109/TIA.2017.2718978. de Oliveira, N.M.C. and Biegler, L.T. (1994). Constraint handing and stability properties of model-predictive control. AIChE journal, 40(7), 1138–1155. Diehl, M., Amrit, R., and Rawlings, J.B. (2011). A Lyapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56, 703–707. Ferramosca, A., Limon, D., Gonz´alez, A.H., Odloak, D., and Camacho, E. (2010). MPC for tracking zone regions. Journal of Process Control, 20(4), 506–516. Fletcher, R. (2013). Practical methods of optimization. John Wiley & Sons. Gonz´ alez, A.H., Marchetti, J.L., and Odloak, D. (2009). Robust model predictive control with zone control. IET Control Theory & Applications, 3(1), 121–135. Gonz´alez, A.H. and Odloak, D. (2009). A stable MPC with zone control. Journal of Process Control, 19(1), 110–122. Gr¨ une, L. (2013). Economic receding horizon control without terminal constraints. Automatica, 49, 725–734. Heidarinejad, M., Liu, J., and Christofides, P.D. (2012). Economic model predictive control of nonlinear process systems using lyapunov techniques. AIChE Journal, 58(3), 855–870. Keerthi, S. and Gilbert, E. (1987). Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints. IEEE Transactions on Automatic Control, 32(5), 432–435. Kerrigan, E.C. and Maciejowski, J.M. (2000a). Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In Proceedings of the 39th IEEE Conference on Decision and Control, volume 5, 4951–4956. Kerrigan, E.C. and Maciejowski, J.M. (2000b). Soft constraints and exact penalty functions in model predictive control. In Proc. UKACC International Conference (Control. Liu, S. and Liu, J. (2016). Economic model predictive control with extended horizon. Automatica, 73, 180– 192. Lu, J.Z. (2003). Challenging control problems and emerging technologies in enterprise optimization. Control Engineering Practice, 11(8), 847–858. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Constrained model predictive control: stability and optimality. Automatica, 36(6), 789–814. Rao, C.V. and Rawlings, J.B. (2000). Linear programming and model predictive control. Journal of Process Control, 10(2), 283 – 289. doi: https://doi.org/10.1016/S0959-1524(99)00034-7. Scokaert, P. and Rawlings, J.B. (1999). Feasibility issues in linear model predictive control. AIChE Journal, 45(8), 1649–1659. Zeilinger, M.N., Morari, M., and Jones, C.N. (2014). Soft constrained model predictive control with robust stability guarantees. IEEE Transactions on Automatic Control, 59(5), 1190–1202. doi:10.1109/TAC.2014.2304371.
3 2 1
u(n)
0 -1 -2 -3 -4 -5
0
10
20
30
40
n
Fig. 8. Closed-loop input trajectories of EMPC of Eq. (6) with modified zone Z t in Eq. (9), with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. Shaded area depicts the input target zone.
15
le(n)
10
5
0 0
10
20
30
21
40
n
Fig. 9. Closed-loop economic performance of EMPC of Eq. (6) with modified zone Z t in Eq. (9), with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. 6. CONCLUSION In this work, an MPC framework with zone tracking and economic objectives was presented. Tradeoff between the fast zone tracking and transient economic performance based on the l1 norm and squared l2 norm penalties was systematically studied. A simple numerical example was used to illustrate the intrinsic difficulties in parameter tuning due to the inconsistency between the economic and zone tracking objectives. A procedure to modify the target zone was proposed which showed promising results. By modifying the target zone, we effectively decoupled the zone tracking and dynamic economic objectives so that parameter tuning can be made simpler and less counterintuitive. Future work will focus on more generalized procedure to modify the target zone as well as the computation of the modified target zone. REFERENCES Amrit, R., Rawlings, J.B., and Angeli, D. (2011). Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control, 35, 178– 186. 21
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IFAC PapersOnLine 51-20 (2018) 16–21
Economic Economic Economic Economic
Model Predictive Control Model Predictive Control Model Predictive Control Model Predictive Control Zone Tracking Zone Tracking Zone Tracking Zone∗ Tracking∗,1
with with with with
Su ∗,1 Su Liu Liu ∗∗∗ ,, Jinfeng Jinfeng Liu Liu ∗,1 ∗,1 Su Liu ∗ , Jinfeng Liu ∗,1 Su Liu , Jinfeng Liu ∗ ∗ Department of Chemical & Materials Engineering, University of ∗ ∗ Department of Chemical & Materials Engineering, University of of & University Alberta, Edmonton, AB 1H9, ∗ Department Alberta, Edmonton, AB T6G T6GEngineering, 1H9, Canada. Canada. Department of Chemical Chemical & Materials Materials Engineering, University of of Alberta, Edmonton, AB T6G 1H9, Canada. Alberta, Edmonton, AB T6G 1H9, Canada. Abstract: Abstract: In In this this work, work, we we propose propose a a framework framework for for economic economic model model predictive predictive control control (EMPC) (EMPC) Abstract: In this work, we propose a framework for economic model predictive control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework with zone tracking. A zone cost is incorporated the predictive existing EMPC framework Abstract: In this work, wetracking proposestage a framework for economicinto model control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework to essentially form a multi-objective optimization problem. We provide sufficient conditions for to essentially form aAmulti-objective optimization problem. We sufficient conditions for with zone tracking. zone tracking stage cost is incorporated intoprovide the existing EMPC framework to essentially form a multi-objective optimization problem. We provide sufficient conditions for asymptotic stability of the optimal steady state and characterize exact penalty for the zone asymptotic of the optimal optimization steady state problem. and characterize exact penaltyconditions for the zone to essentiallystability form a multi-objective We provide sufficient for asymptotic stability of the optimal steady state and characterize exact penalty for the zone tracking cost which prioritizes zone tracking objective over economic objective. A comprehensive tracking coststability which prioritizes zone tracking economic exact objective. A comprehensive asymptotic of the optimal steady objective state andover characterize penalty for the zone tracking which zone tracking objective over economic A comprehensive study on the between economic is on study on cost the tradeoff tradeoff between zone zone tracking and economic performance is carried carried out on aa simple simple tracking cost which prioritizes prioritizes zone tracking tracking and objective overperformance economic objective. objective. A out comprehensive study on the tradeoff between zone tracking and economic performance is carried out on simple scalar linear system. Simulation results reveal intrinsic difficulties in parameter tuning to scalar linear system. between Simulation reveal in parameter tuning due to study on the tradeoff zoneresults tracking andintrinsic economicdifficulties performance is carried out on aa due simple scalar linear system. Simulation results reveal intrinsic difficulties in parameter tuning due the inconsistency between the zone tracking and economic objectives. A procedure to modify the inconsistency between the zone tracking economic objectives. A procedure to modify scalar linear system. Simulation results revealand intrinsic difficulties in parameter tuning due to to between tracking and objectives. to the target on economic performance and the steady the inconsistency target zone zone based based on the the zone economic performance and reachability reachability ofprocedure the optimal optimal steady inconsistency between zone tracking and economic economic objectives. A Aof procedure to modify modify the zone on performance and of optimal steady state is The target effectively the zone tracking and state is proposed. proposed. The modified modified target zone zone effectively decouples decouples the dynamic dynamic zone tracking and the target target zone based based on the the economic economic performance and reachability reachability of the the optimal steady state is proposed. The modified target zone effectively decouples the dynamic zone tracking economic objectives and simplifies parameter tuning. economic objectives and simplifies parameter tuning. decouples the dynamic zone tracking and state is proposed. The modified target zone effectively and economic objectives and parameter tuning. economic objectives and simplifies simplifies tuning. © 2018, IFAC (International Federationparameter of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Predictive Predictive control; control; Process Process optimization; optimization; Soft Soft constraint; constraint; Zone Zone control control Keywords: Predictive control; Process optimization; Soft constraint; Zone control Keywords: Predictive control; Process optimization; Soft constraint; Zone control 1. As 1. INTRODUCTION INTRODUCTION As its its name name suggests, suggests, soft soft constraint constraint is is often often dismissed dismissed as as a a 1. INTRODUCTION As its name suggests, soft constraint is often dismissed as a trick to avoid feasibility issue with hard constraint, and trick avoid feasibility with hard constraint, and is 1. INTRODUCTION As itsto name suggests, softissue constraint is often dismissed as is a trick to avoid feasibility issue with hard constraint, and is addressed separately from set-point tracking or economic Process control of chemical plants needs to address a addressed separately from set-point tracking or economic to avoid feasibility issue with hard constraint, and is Process control of chemical plants needs to address a trick set-point tracking economic objectives. To of knowledge, only aa few workProcess control of chemical to address a addressed number objectives including safety, environmental reguTo the the best bestfrom of our our knowledge, only or few workaddressed separately separately from set-point tracking or economic number of objectives includingplants safety, needs environmental reguProcess of control of chemical plants needs to address a objectives. objectives. To the best of our knowledge, only a few works explicitly handle zone tracking objectives (Ferramosca number of objectives including safety, environmental regulations, product energy quality, efficiency, profitability, etc. s explicitly handle zone tracking objectives (Ferramosca objectives. To the best of our knowledge, only a few worklations, quality, energy safety, efficiency, profitability,reguetc. s explicitly handle zone tracking objectives (Ferramosca number product of objectives including environmental et al. (2010); Gonz´ a lez et al. (2009); Gonz´ a lez and Odloak lations, product quality, energy efficiency, profitability, etc. According to (Lu (2003)), an integrated control system s explicitly handle zone tracking objectives (Ferramosca According to (Lu (2003)), an efficiency, integratedprofitability, control system lations, product quality, energy etc. et al. (2010); Gonz´ a lez et al. (2009); Gonz´ a lez and Odloak et al. (2009)). According to (Lu (2003)), an integrated control system consists of facets: regulatory control, conal. (2010); (2010); Gonz´ Gonz´aalez lez et et al. al. (2009); (2009); Gonz´ Gonz´aalez lez and and Odloak Odloak consists of three three facets: regulatory control, constraint constraint con- et According to (Lu (2003)), an integrated control system (2009)). (2009)). consists of three facets: regulatory control, constraint control and maneuvering control. Regulatory control refers to trol and of maneuvering Regulatory refers to (2009)). In this work, we propose an EMPC framework with consists three facets:control. regulatory control,control constraint conthis work, we propose an EMPC framework with inintrol and maneuvering control. Regulatory controlminimizes refers to In the setpoint tracking control In this propose framework with inthe conventional setpoint tracking control which which trol conventional and maneuvering control. Regulatory controlminimizes refers to tegrated zone we tracking. A an zoneEMPC tracking stage cost which In this work, work, we propose an EMPC framework with inconventional setpoint tracking control which minimizes tegrated zone tracking. A zone tracking stage cost which the variance of controlled variables to the setpoint. Conzone tracking. A zone tracking stage cost which the conventional variance of controlled variables to thewhich setpoint. Con- tegrated penalizes weighted l norm and squared l norm distance setpoint tracking control minimizes 1 2 penalizes weighted l norm and squared l norm distance tegrated zone tracking. A zone tracking stage cost which 1 2 the variance of controlled variables to the setpoint. Constraint control, or zone control, prevents the system from 1 2 penalizes weighted norm norm straint control, zone control, prevents system Confrom to the zone into EMPC the variance of or controlled variables to thethe setpoint. to the target target zone is isll11incorporated incorporated into the thell22existing existing EMPC penalizes weighted norm and and squared squared norm distance distance straint control, or zoneand control, the back system violating its steers the into the the target zone is incorporated into the existing violating its boundary boundary steersprevents the system system intofrom the to framework, forming a multi-objective optimization probstraint control, or zoneand control, prevents the back system from framework, a multi-objective optimization probto the targetforming zone is incorporated into the existing EMPC EMPC violating its boundary and steers the system back into the zone whenever constraint violation happens. Maneuvering forming a multi-objective optimization zone whenever constraint happens. Maneuvering lem. We provide sufficient conditions for asymptotic staviolating its boundary andviolation steers the system back into the framework, lem. We provide sufficient conditions for asymptoticprobstaframework, forming a multi-objective optimization probzone whenever constraint violation happens. Maneuvering control moves the system from current operating condition lem. We provide sufficient conditions for asymptotic stacontrol moves the system from current operating condition bility zone whenever constraint violation happens. Maneuvering of the optimal steady state and characterize exact bility of the optimal steady state and characterize exact lem. We provide sufficient conditions for asymptotic stacontrol moves the system from current operating condition to a new operating condition, typically due to economic of the optimal steady state and characterize exact to a new operating condition, typically due to condition economic bility penalty for the zone tracking cost which prioritizes the control moves the system from current operating penalty for the zone tracking cost which prioritizes the bility of the optimal steady state and characterize exact to a new operating condition, typically duedesigned to economic considerations. Generally speaking, aa well confor the zone tracking cost which prioritizes the considerations. Generally speaking, well con- penalty zone tracking objective over the economic objective. The to a new operating condition, typically duedesigned to economic penalty for the zone tracking cost which objective. prioritizes The the considerations. Generally speaking, a well designed conzone tracking objective over the economic trol system necessarily integrates all three control types, tracking objective over the economic objective. The trol system necessarily integrates allathree proposed EMPC framework is tested in a simple numerical considerations. Generally speaking, well control designedtypes, con- zone proposed EMPC framework tested in a simple numerical zone tracking objective overisthe economic objective. The trol system necessarily integrates all three control types, although emphasis on the three facets of control may vary EMPC framework is aa simple although emphasis on the three facets of control maytypes, vary proposed example which we the tradeoff trol system necessarily integrates all three control example in in which we systematically systematically study thenumerical tradeoff EMPC framework is tested tested in in study simple numerical although emphasis the three facets of control may vary proposed from application to application. example in which we systematically study the tradeoff from application toon application. although emphasis on the three facets of control may vary between zone tracking and economic performance based between zone tracking and economic performance based example in which we systematically study the tradeoff from application to application. between zone tracking and economic performance based from application to application. on the l norm and squared l norm penalties. SimulaModel predictive control (MPC) has been the most widely 1 2 on the l11zone norm and squared l22 norm performance penalties. Simulatracking and economic based Model predictive control (MPC) has been the most widely between on the l norm and squared l norm penalties. Simulation results reveal intrinsic difficulties in parameter tuning Model predictive control (MPC) has been the most widely 1 2 applied advanced control technique, due to its ability to tionthe results revealand intrinsic difficulties parameterSimulatuning l1 norm squared l2 norminpenalties. applied advancedcontrol control(MPC) technique, due the to its ability to on Model predictive has been most widely results intrinsic difficulties parameter tuning due the inconsistency the tracking and applied advanced control to its ability to tion handle and incorporate economic considerdue to to the reveal inconsistency between thein zone tracking and results reveal intrinsicbetween difficulties inzone parameter tuning handle constraints and to to technique, incorporatedue economic considerapplied constraints advanced control technique, due to its ability to tion due to the inconsistency between the zone tracking and economic objectives. Finally, to resolve the difficulties in handle constraints and to incorporate economic considerations. The literature is rich with theories for conventional economic objectives. Finally, to resolve the difficulties in due to the inconsistency between the zone tracking and ations. The literature is rich with theories for conventional handle constraints and to incorporate economic consider- economic objectives. Finally, to resolve the difficulties in parameter tuning, we propose a procedure to modify the ations. The literature is rich with theories for conventional setpoint tracking MPC (Mayne et al. (2000)). The past economic objectives. Finally, toa resolve the to difficulties in setpoint tracking MPC (Mayne al. (2000)). The past parameter ations. The literature is rich with et theories for conventional tuning, we propose procedure modify the parameter tuning, we propose a procedure to modify the target zone based on the economic performance and reachsetpoint tracking MPC (Mayne et al. (2000)). The past decade has seen an increasing academic interest in economtarget zonetuning, based on economic performance and reachwethe propose a procedure to modify the decade seen anMPC increasing academic economsetpointhas tracking (Mayne et al. interest (2000)).inThe past parameter target based on economic performance and ability of optimal state. The target decade haspredictive seen an increasing academic(Diehl interest ic model control et al. (2011); ability zone of the the optimal steady state. The modified modified target zone based on the thesteady economic performance and reachreachic model control (EMPC) (EMPC) et in al.econom(2011); target decade haspredictive seen an increasing academic(Diehl interest in economability of the optimal steady state. The modified target zone effectively decouples the dynamic zone tracking and ic model predictive control (EMPC) (Diehl et al. (2011); Heidarinejad et al. (2012); Gr¨ u ne (2013)) which integrates zone effectively decouples the dynamic zone tracking and of the optimal steady state. The modified target Heidarinejad et al. (2012); une (2013)) which integrates ic model predictive controlGr¨ (EMPC) (Diehl et al. (2011); ability zone effectively decouples the dynamic zone tracking economic objectives and simplifies parameter tuning. Heidarinejad et al. (2012); Gr¨ u ne (2013)) which integrates economic objective into regulatory setpoint control. On the economic objectives and simplifies parameter tuning. and effectively decouples the dynamic zone tracking and economic objective into regulatory setpointwhich control. On the zone Heidarinejad et al. (2012); Gr¨ une (2013)) integrates economic objectives and simplifies parameter tuning. economic objective intohas regulatory control. On contrary, zone received less In contrary, zone control control receivedsetpoint less attention. attention. In the the economic objectives and simplifies parameter tuning. economic objective intohas regulatory setpoint control. On contrary, zone control has received less dealt attention. In the MPC framework, zone is with 2. PROBLEM SETUP MPC framework, zone control control is usually usually with by by contrary, zone control has received less dealt attention. In the 2. PROBLEM SETUP MPC framework, zone control is usually dealt with by the 2. so-called soft constraint technique (Scokaert and Rawlings so-called soft constraint technique (Scokaert and Rawlings MPC framework, zone control is usually dealt with by the 2. PROBLEM PROBLEM SETUP SETUP so-called soft constraint technique (Scokaert and Rawlings (1999); Kerrigan and Maciejowski (2000b); de Oliveira and 2.1 Notation (1999); and Maciejowski Oliveira and 2.1 Notation so-calledKerrigan soft constraint technique (2000b); (Scokaertdeand Rawlings (1999); Kerrigan and Maciejowski (2000b); and 2.1 Notation Biegler Zeilinger et Askari al. Biegler (1994); Zeilinger et al. al. (2014); (2014); Askaride etOliveira al. (2017)). (2017)). (1999); (1994); Kerrigan and Maciejowski (2000b); deet Oliveira and Notation Biegler (1994); Zeilinger et al. (2014); Askari et al. (2017)). 2.1 this the lp norm of Biegler (1994); Zeilinger et al. (2014); Askari et al. (2017)). Throughout Throughout this work, work, x x ppp denotes denotes the of the the 1 Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: 1/p llppp norm 1 Throughout this work, x denotes the norm of the 1 Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: (i) p p Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: 1/p .lpThe Throughout this work, x denotes the norm of the 1 = |x | operator vector x such that x 1/p +1-780-492-2881. Email: [email protected]. (i) p Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: p 1 |x(i) . The operator vector x such that x ppp = p +1-780-492-2881. Email: +1-780-492-2881. Email: [email protected]. [email protected]. Corresponding author: Jinfeng Liu. Tel.: +1-780-492-1317. Fax: (i) |p 1/p 1/p . The operator = |x | vector x such that x +1-780-492-2881. Email: [email protected]. (i) p p = |x | . The operator vector by x Elsevier such that +1-780-492-2881. [email protected]. 2405-8963 © 2018,Email: IFAC (International Federation of Automatic Control) Hosting Ltd. x All rights reserved. p
Copyright 2018 IFAC 16 Control. Peer review© under responsibility of International Federation of Automatic Copyright 16 Copyright © © 2018 2018 IFAC IFAC 16 Copyright © 2018 IFAC 16 10.1016/j.ifacol.2018.10.168 Copyright © 2018 IFAC 16
2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018
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| · | denotes the l2 norm or Euclidean norm of a scalar or a vector. The symbol IN M denotes the set of integers {M, M + 1, ..., N }. The symbol projX (O) denotes projection of the set O onto its subspace X.
x0 = x(n)
n=0
∞
e (x(n), u(n)) + K
x ,u
z
(2)
min z z
ui ,xi ,ui
i=0
e (xi , ui ) + c1 (xi − xzi 1 + ui − uzi 1 ) 2
2
+ c2 (xi − xzi 2 + ui − uzi 2 )
(6)
−1 (xzi , uzi ) ∈ Zt , i ∈ IN 0
We note that the zone tracking treatment in the above EMPC formulation of Eq. (6) is closely connected to the so-called soft constraint technique. In the following, we establish sufficient conditions for asymptotic stability of the optimal steady state xs . and convergence into the target zone Zt in finite steps. Definition 1. (Strictly dissipative systems) The system of Eq. (1) is strictly dissipative with respect to the supply rate s : X × U → R if there exists a continuous storage function λ(·) : X → R and a K∞ function αl (·) such that the following holds for all x ∈ X and u ∈ U: λ(f (x, u)) − λ(x) ≤ s(x, u) − αl (|x − xs |) Assumption 1. (Strict dissipativity, Amrit et al. (2011)) The system of Eq. (1) is strictly dissipative with respect to the supply rate s(x, u) = e (x, u) − e (xs , us ) Assumption 2. (weak controllability, (Diehl et al. (2011))) There exists a K∞ function γ(·) such that for all x ∈ XN , there exists a feasible solution to Eq. (4) such that:
(3)
(x, u) ∈ Zt
3. EMPC WITH ZONE TRACKING In this section, a general framework for economic model predictive control with zone tracking is presented. A pointwise terminal constraint is employed for simplicity of exposition.
N −1
Specifically, at a sampling time n, the following finitehorizon optimization problem is solved:
i=0
|ui − us | ≤ γ(|x − xs |)
Theorem 1. If Assumptions 1 and 2 hold, then the optimal steady state xs is asymptotically stable under the EMPC of Eq. (4) with a region of attraction XN .
(4a)
i=0
−1 s.t. xi+1 = f (xi , ui ), i ∈ IN 0
N −1
s.t. (4b) − (4e)
We assume that (xs , us ) uniquely solves the above steadystate optimization problem. Note that the economic cost function e (·) is not necessarily positive definite with respect to (xs , us ).
e (xi , ui ) + z (xi , ui )
(5)
z
s.t. (x , u ) ∈ Zt
x,u
min
2
where c1 and c2 are positive scalars. Equation (5) characterizes a weighted l1 norm and squared l2 norm zone tracking penalty. The incorporation of the l1 norm penalty allows for the so-called exact penalty which will be discussed in the sequel. With the zone tracking penalty defined by Eq. (5), the optimization of Eq. (4) amounts to the following optimization:
Let (xs , us ) denote the economically optimal steady state in the target zone Zt . That is: (xs , us ) = arg min e (x, u)
u0 ,··· ,uN −1
2
c2 (x − xz 2 + u − uz 2 )
subject to the constraint: n≥K +1 (x(n), u(n)) ∈ Zt , The above problem is essentially a multi-objective optimization problem in which the magnitude, duration of zone target violation, as well as the economic performance are tradeoffed. The zone tracking objective is prioritized in that it needs to be satisfied in finite steps (K is finite).
N −1
(4d) (4e)
The zone tracking penalty z (x, u) is defined by the following function: z (x, u) = min c1 (x − xz 1 + u − uz 1 )+ z z
n=K+1
s.t. x = f (x, u)
−1 IN 0
−1 Let u∗ (i|n), i ∈ IN denote the optimal solution, the 0 input injected to the system at time n is: u(n) = u∗ (0|n). At the next sampling time n+1, the optimization of Eq. (4) is re-evaluated, generating an implicit feedback control law u(n) = κN (x(n)). We denote the feasibility region of the optimization problem of Eq. (4) by XN . XN is forward invariant under the feedback law u(n) = κN (x(n)) due to the terminal constraint Eq. (4e). In other words, the EMPC design is recursively feasible.
We consider the following nonlinear discrete time system: x(n + 1) = f (x(n), u(n)) (1) where x(n) ∈ X ⊂ Rnx , u(n) ∈ U ⊂ Rnu , n ∈ I≥0 , denote the state and input at time n, respectively. The vector function f (·) : Rnx ×Rnu → Rnx is continuous. The system is subject to coupled state and input constraint: (x(n), u(n)) ∈ Z ⊆ X × U, n ∈ I≥0 where X, U, Z are all compact sets. The primary control objective is to steer and maintain the system in a compact set Zt ⊂ Z. The target zone Zt may correspond to, for example, product specifications or safety considerations. The distance to the target zone is measured by the function z (x, u) : Z → R which is positive definite with respect to the target zone Zt . There is also a secondary economic objective to minimize the operational cost characterized by the function e (x, u) : Z → R. Both z (·) and e (·) are continuous functions. Stated formally, the controller seeks to minimize the following function: z (x(n), u(n)) +
(4c)
(xi , ui ) ∈ Z, i ∈ x N = xs
2.2 System description and control objective
K
17
(4b) 17
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Theorem 2 implies that if the l1 norm penalty c1 is sufficiently large, then the zone tracking objective is prioritized over the economic objective. Note that the constraints (7c) - (7e) can be combined by canceling the slack variables xzi −1 and uzi into the compact form: (xi , ui ) ∈ Zt , i ∈ IN . 0 Thus the EMPC of Eq. (7) yields the same solution to the following EMPC constrained by the target zone:
Proof We provide a sketch of the proof based on the results in (Diehl et al. (2011)) and (Liu and Liu (2016)). Define the rotated cost: ¯e (x, u) = e (x, u) − e (xs , us ) + λ(x) − λ(f (x, u)) From Assumption 1, the rotated cost satisfies ¯e (x, u) ≥ αl (|x − xs |). Moreover, it can be shown that the solution to Eq. (4) is equivalent to the following problem: min
u0 ,··· ,uN −1
N −1
min
¯e (xi , ui ) + z (xi , ui )
u0 ,··· ,uN −1
i=0
x0 = x(n)
Let L(x, u) = ¯e (x, u) + z (x, u) ≥ αl (|x − xs |), the above optimization is then reduced to a classical MPC with positive-definite stage cost and point-wise terminal constraints. Assumption 2 ensures the existence of a K∞ function β(|x − xs |) which bounds the value function of EMPC from above. It is then straightforward to apply standard MPC theory. The details can be found in Diehl et al. (2011)) and are omitted for brevity.
x N = xs
The constraint (4d) is removed because Zt ⊂ Z. Therefore, sufficiently large l1 norm penalty term c1 (xi − xzi 1 + ui − uzi 1 ) in effect can convert the zone tracking objective into hard constraints (xi , ui ) ∈ Zt . Note that exact penalty cannot be achieved by the quadratic term 2 2 c2 (x − xz 2 + u − uz 2 ) unless c2 can be made infinitely large (de Oliveira and Biegler (1994)).
An important question to ask is: will the system leave the target zone once it is in the zone? To answer this question, we need the following definition of the N -step reachable set (Kerrigan and Maciejowski (2000a)): Definition 2. (N -step reachable set) We use XN (Zt , xs ) to denote the set of states that can be steered to xs in N steps while satisfying the state and input constraints (x, u) ∈ Zt . That is, −1 , ∃ (x(n), u(n)) ∈ Zt , n ∈ IN 0 XN (Zt , xs ) = x(0) | x(n + 1) = f (x(n), u(n)), x(N ) = xs
e (xi , ui )
(xzi , uzi ) ∈ Zt , i ∈
−1 IN 0 −1 i ∈ IN 0
4. EXAMPLE
(7a) (7b)
4.1 l1 norm penalty
(7c)
Fig. 1 shows the closed-loop input trajectories of the EMPC of Eq. (6) with different l1 -norm penalties for c1 while the quadratic penalty c2 = 0. The corresponding economic performances are shown in Fig. 2. It is seen that all closed-loop trajectories reach the target zone in finite steps and asymptotically converge to the optimal value us = 0.9. As c1 increases, the closed-loop trajectory reaches the target zone faster but also stays on the zone boundary for longer periods. Similar phenomina of riding the constraint boundary due to l1 norm penalty was observed in Scokaert and Rawlings (1999). When c1 = 100, the EMPC leads to deadbeat zone tracking.
i=0
s.t. (4b) − (4e)
Note also that to ensure x(n) ∈ Zt for all n ∈ I≥0 and x(0) ∈ XN (Zt , xs ), the condition c1 > λ∞ needs to be satisfied for Lagrange multipliers λ associated with all x(n) ∈ XN (Zt , xs ). Finding the exact lower bound for such c1 is in general difficult. The task is possible for linear systems with quadratic tracking costs and polyhedral constraints (Kerrigan and Maciejowski (2000b)).
In this Section, we use a simple example to illustrate the zone tracking and economic performance of the EMPC of Eq. (6) with different choices of c1 , c2 . Special attentions are paid to the difficulties encountered in parameter tuning. The system investigated is a linear scalar system: x(n + 1) = 1.25x(n) + u(n) with state and input constraints X = [−5, 5], U = [−5, 5] respectively. The target set is Zt = {(x, u) | x ∈ [−5, 5], u ∈ [−1, 1]}. An economic cost e = (u − 0.9)2 is used which corresponds to an optimal steady state (xs , us ) = (−3.6, 0.9). Two different initial states: x(0) = −5 and x(0) = 5 are compared in parallel to indicate the asymmetric closed-loop performance. The control horizon is N = 20 when not specified.
Based on the above definition, the EMPC is capable of maintaining the system in the target zone only if x(0) ∈ XN (Zt , xs ). Let λ denote the vector of the KKT multipliers to the following optimization problem: ui ,xi ,ui
(8)
−1 (xi , ui ) ∈ Zt , i ∈ IN 0
If the optimal steady state lies in the interior of the target zone (i.e., (xs , us ) ∈ Zt ), then asymptotic stability of the optimal steady state implies finite-time convergence into the target zone. However, if the optimal steady state lies on the boundary of the target zone, finite-time convergence to the target zone will be more tricky to achieve. Typically the task can be achieved by employing sufficiently large l1 norm penalty value c1 which is known to result in deadbeat control policy (Rao and Rawlings (2000)).
N −1
e (xi , ui )
i=0
−1 s.t. xi+1 = f (xi , ui ), i ∈ IN 0
s.t. (4b) − (4e)
min z z
N −1
(xi , ui ) = (xzi , uzi ) (7d) Theorem 2. If x(n) ∈ XN (Zt , xs ) and c1 > λ∞ , then the solutions to the optimization of Eq. (6) and Eq. (7) are identical. Proof If c1 > λ∞ , then the objective function of Eq. (6) is an exact penalty function for Eq. (7). The solution to Eq. (6) is identical to Eq. (7), and thus identical to Eq. (8). The proof can be found in Theorem 14.3.1, (Fletcher (2013)). 18
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Also very noticeable is the asymmetric performance with different initial states. With x(0) = −5 (the upper part in Fig. 1 and lower part in Fig. 2), economic performance is less of concern and the choice c1 = 100 which results in deadbeat zone tracking is preferable. However, with x(0) = −5 (the lower part in Fig. 1 and upper part in Fig. 2), transient economic performance significantly deteriorates because of the boundary riding, which suggests the use of smaller penalty value c1 . The asymmetric behavior, which essentially arises from the inconsistency between zone tracking and economic objectives, poses challenge for parameter tuning.
19
the magnitude of zone tracking violation and increasing the duration of violation. Therefore the quadratic penalty c2 provides a way to tradeoff the duration and magnitude of zone target violation.
1
u(n)
0
-1
-2 2 -3
1
0
10
20
30
40
u(n)
n 0
Fig. 3. Closed-loop input trajectories of EMPC of Eq. (6) with c1 = 0 and c2 = 0.1 (solid), c2 = 1 (dotted), c2 = 10 (dashed), c2 = 100 (dash dotted), respectively. Shaded area depicts the input target zone.
-1 -2
15
-3 0
10
20
30
40
n 10
le(n)
Fig. 1. Closed-loop input trajectories of EMPC of Eq. (6) with c2 = 0 and c1 = 0.1 (solid), c1 = 1 (dotted), c1 = 10 (dashed), c1 = 100 (dash dotted), respectively. Shaded area depicts the input target zone. The upper and lower part correspond to initial state x(0) = −5 and x(0) = 5, respectively.
5
0
15
0
10
20
30
40
n
Fig. 4. Closed-loop economic performance of EMPC of Eq. (6) with c1 = 0 and c2 = 0.1 (solid), c2 = 1 (dotted), c2 = 10 (dashed), c2 = 100 (dash dotted), respectively.
le(n)
10
5
4.3 l1 norm with squared l2 norm As suggested by (Scokaert and Rawlings (1999)), the l1 norm penalty c1 should be chosen to result in exact penalty when used together with the quadratic penalty c2 . Fig. 5 and Fig. 6 show the closed-loop input trajectories and economic performances with c1 = 100 and c2 = 1, 10, 100, 1000 respectively. It is seen that combined use of the l1 norm and squared l2 norm zone tracking penalty further polarizes zone tracking and economic objectives. Symmetric zone tracking with respect to the two initial states is obtained, which implies a higher priority of zone tracking over economic objective. On the other hand, asymmetric economic performance with respect to the two initial states are intensified because of longer duration of target zone riding.
0 0
10
20
30
40
n
Fig. 2. Closed-loop economic performance of EMPC in Eq. (6) with c2 = 0 and c1 = 0.1 (solid), c1 = 1 (dotted), c1 = 10 (dashed), c1 = 100 (dash dotted), respectively. The upper and lower part correspond to initial state x(0) = 5 and x(0) = −5, respectively. 4.2 Squared l2 norm The closed-loop input trajectories and economic performances under different squared l2 norm penalty c2 with the l1 norm penalty c1 = 0 are shown in Fig. 3 and Fig. 4, respectively. Similar asymmetric behaviors with different initial states are observed. We note that when quadratic penalties are employed, no boundary riding takes place. Comparing the cases (c1 = 100, c2 = 0) in Fig. 1 with (c1 = 0, c2 = 100) in Fig. 3, we see that the sum of squares penalty smoothens the control actions by reducing
5. MODIFIED TARGET ZONE The results in Section 4 reveal the intrinsic difficulties of control parameter tuning due to the inconsistency between the zone tracking and economic objectives. Naive implementation of the exact penalty could lead to arbitrarily 19
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involve computation difficulties. For linear systems with polyhedral target set Zt and polyhedral performance level set {(x, u) : e (x, u) ≤ α}, Algorithm 1 can be performed using the algorithms in (Keerthi and Gilbert (1987)).
2
u(n)
1
Apply Algorithm 1 to the example in Section 4 with M = 10 and α = 1, the modified target zone is obtained as follows: Z t = {(x, u) | Ex + F u ≤ G (9) where E = [1.25, −1.25, 0, 0]T , F = [1, −1, 1, −1]T , G = [−0.7436, 3.9571, 1.0000, 0.1000]. The constraint set Z, target zone Zt and modified target zone Z t are illustrated in Fig 7.
0 -1 -2 0
10
20
30
40
n
Fig. 5. Closed-loop input trajectories of EMPC of Eq. (6) with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. Shaded area depicts the input target zone. 12 10
le(n)
8 6 4 2
Fig. 7. The constraint set Z (box), target zone Zt (shaded rectangle) and modified target zone Z t (parallelogram). The circle indicates the optimal steady state (xs , us )
0 0
10
20
30
40
n
Fig. 6. Closed-loop economic performance of EMPC of Eq. (6) with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively.
Fig. 8 and Fig. 9 show the closed-loop input trajectories and economic performances of the EMPC tracking the modified target zone Z t in Eq. (9), with both l1 norm and squared l2 norm penalties. Compared to Fig. 5 and Fig. 6, we can see that tracking the modified target zone Z t leads to more balanced economic performance in the target zone while accomplishing fast zone tracking for the target zone Zt . Boundary riding takes place at the boundary of the modified target zone Z t which corresponds to the economic performance bound specified by e (x, u) ≤ α = 1. This makes α a good tuning parameter. 40 Let Jez = n=K e (x(n), u(n)) denote the summation of the economic performance in the target zone over 40 steps, where K is the time instant when the system enters the zone and never leaves. Table 1 compares the transient economic performance Jez of the EMPC tracking the target zone Zt and EMPC tracking the modified target zone Z t . It is seen that the EMPC tracking the modified target zone has improved transient performance in the target zone. In fact, the transient performance Jez can be further improved by reducing α.
poor transient economic performance because of the zone boundary riding. Motivated by the difficulties encountered in parameter tuning, we propose tracking a subset of the target zone which can be steered to the optimal steady state in finite steps with guaranteed economic performance. The following conceptual procedure can be taken to find the new target zone: Algorithm 1. (1) Choose some M ∈ I≥1 and α ∈ R such that M ≤ N and α ≥ e (xs , us ). (2) Set Z0 = (xs , us ) (3) for i = 0 : M − 1 f (x, u) ∈ projX (Zi ), Zi+1 = (x, u) (x, u) ≤ α, (x, u) ∈ Z e
t
end (4) The modified target zone is Z t = ZM
The modified target zone Z t , obtained by Algorithm 1, is a zone in which the system can be steered to the optimal steady state in M steps in the target zone Zt while the economic performance of each step is upper bounded by α. The parameters M and α are user-specified to tradeoff fast zone tracking and transient economic performance in the target zone. Note that step (3) of Algorithm 1 requires set projection and set intersection operations which could
Table 1. Transient performance in the Zt tracking Zt x(0) = −5 x(0) = 5 tracking Z t x(0) = −5 x(0) = 5 20
c2 = 1 0.18 58.7 c2 = 1 0.18 17.8
c2 = 10 0.18 58.2 c2 = 10 0.18 20.8
c2 = 100 0.18 60.0 c2 = 100 0.18 25.5
c2 = 1000 0.18 60.1 c2 = 1000 0.18 37.4
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Askari, M., Moghavvemi, M., Almurib, H.A.F., and Haidar, A.M.A. (2017). Stability of soft-constrained finite horizon model predictive control. IEEE Transactions on Industry Applications, 53(6), 5883–5892. doi: 10.1109/TIA.2017.2718978. de Oliveira, N.M.C. and Biegler, L.T. (1994). Constraint handing and stability properties of model-predictive control. AIChE journal, 40(7), 1138–1155. Diehl, M., Amrit, R., and Rawlings, J.B. (2011). A Lyapunov function for economic optimizing model predictive control. IEEE Transactions on Automatic Control, 56, 703–707. Ferramosca, A., Limon, D., Gonz´alez, A.H., Odloak, D., and Camacho, E. (2010). MPC for tracking zone regions. Journal of Process Control, 20(4), 506–516. Fletcher, R. (2013). Practical methods of optimization. John Wiley & Sons. Gonz´ alez, A.H., Marchetti, J.L., and Odloak, D. (2009). Robust model predictive control with zone control. IET Control Theory & Applications, 3(1), 121–135. Gonz´alez, A.H. and Odloak, D. (2009). A stable MPC with zone control. Journal of Process Control, 19(1), 110–122. Gr¨ une, L. (2013). Economic receding horizon control without terminal constraints. Automatica, 49, 725–734. Heidarinejad, M., Liu, J., and Christofides, P.D. (2012). Economic model predictive control of nonlinear process systems using lyapunov techniques. AIChE Journal, 58(3), 855–870. Keerthi, S. and Gilbert, E. (1987). Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints. IEEE Transactions on Automatic Control, 32(5), 432–435. Kerrigan, E.C. and Maciejowski, J.M. (2000a). Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. In Proceedings of the 39th IEEE Conference on Decision and Control, volume 5, 4951–4956. Kerrigan, E.C. and Maciejowski, J.M. (2000b). Soft constraints and exact penalty functions in model predictive control. In Proc. UKACC International Conference (Control. Liu, S. and Liu, J. (2016). Economic model predictive control with extended horizon. Automatica, 73, 180– 192. Lu, J.Z. (2003). Challenging control problems and emerging technologies in enterprise optimization. Control Engineering Practice, 11(8), 847–858. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Constrained model predictive control: stability and optimality. Automatica, 36(6), 789–814. Rao, C.V. and Rawlings, J.B. (2000). Linear programming and model predictive control. Journal of Process Control, 10(2), 283 – 289. doi: https://doi.org/10.1016/S0959-1524(99)00034-7. Scokaert, P. and Rawlings, J.B. (1999). Feasibility issues in linear model predictive control. AIChE Journal, 45(8), 1649–1659. Zeilinger, M.N., Morari, M., and Jones, C.N. (2014). Soft constrained model predictive control with robust stability guarantees. IEEE Transactions on Automatic Control, 59(5), 1190–1202. doi:10.1109/TAC.2014.2304371.
3 2 1
u(n)
0 -1 -2 -3 -4 -5
0
10
20
30
40
n
Fig. 8. Closed-loop input trajectories of EMPC of Eq. (6) with modified zone Z t in Eq. (9), with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. Shaded area depicts the input target zone.
15
le(n)
10
5
0 0
10
20
30
21
40
n
Fig. 9. Closed-loop economic performance of EMPC of Eq. (6) with modified zone Z t in Eq. (9), with c1 = 100 and c2 = 1 (solid), c2 = 10 (dotted), c2 = 100 (dashed), c2 = 1000 (dash dotted), respectively. 6. CONCLUSION In this work, an MPC framework with zone tracking and economic objectives was presented. Tradeoff between the fast zone tracking and transient economic performance based on the l1 norm and squared l2 norm penalties was systematically studied. A simple numerical example was used to illustrate the intrinsic difficulties in parameter tuning due to the inconsistency between the economic and zone tracking objectives. A procedure to modify the target zone was proposed which showed promising results. By modifying the target zone, we effectively decoupled the zone tracking and dynamic economic objectives so that parameter tuning can be made simpler and less counterintuitive. Future work will focus on more generalized procedure to modify the target zone as well as the computation of the modified target zone. REFERENCES Amrit, R., Rawlings, J.B., and Angeli, D. (2011). Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control, 35, 178– 186. 21