economic model predictive control: some design tools and

Economic Model Predictive Control:some design tools and analysis techniques

David Angeli and Matthias A. Muller

Abstract In recent years, Economic Model Predictive Control has emerged as a vari-ant of traditional (tracking) MPC which aims at maximizing economic profitabilityby directly minimizing, over a receding prediction horizon, the costs incurred insystem’s operation. Several design alternatives have been proposed in the literature,as well as suitable tools for the analysis of stability and performance of the differentdesign methods. This note provides the authors’ perpective on some of the most rel-evant results that appeared within this framework as well as, for interested readers,references to the technical papers where such results have first been discussed.

1 Model-based Control and Optimization

Designing a controller for a complex system is a process that entails multiple stepsand decisions. Normally performance requirements, combined with economical andtechnological considerations, inform the selection of sensors and actuators. Evenwhen such selection has been completed, and the scope of available control author-ity as well as information about the system’s state have been identified, defining adetailed control strategy will normally entail a number of trade-offs and conflictingobjectives. Given limited resources, possible disturbances and incomplete informa-tion, this is to be expected. In a realistic scenario, the task of settling such trade-offsmay be daunting, unless appropriate mathematical tools are developed.

David AngeliImperial College London, Dept. of Electrical and Electronic Engineering, Exhibition Road, SW72AZ London, U.K. e-mail: [email protected] di Firenze, Dip. di Ingegneria dell’Informazione, Firenze, Italy.Matthias A. MullerInstitute for Systems Theory and Automatic Control, University of Stuttgart, 70550 Stuttgart, Ger-many. e-mail: [email protected]

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2 David Angeli and Matthias A. Muller

One of the most compelling techniques for approaching this problem is throughmodel-based control. Rather than heuristically tuning knobs of a predefined controlarchitecture, model-based control attempts, as a preliminary step, the formalizationof a mathematical model underlying the evolution of all key variables involved inthe system’s dynamics. This may be done via first principles or, directly, by suitableidentification techniques based on Input-Output data alone, or a combination of thetwo.

A mathematical model allows one to translate the process of selecting and tun-ing a specific controller into an optimization problem. This is done provided per-formance requirements (often conflicting) can be quantified in terms of a singlecost functional whose minimization is supposed to resolve such trade-offs in thebest possible way. Designing cost functionals might itself be non-trivial as not allperformance specifications have a clear “economic” interpretation. However, onecan foresee a trial and error procedure in which cost functionals may be iterativelyadapted and upgraded if the final control design is found to be underperforming insome respect. All other sorts of concerns, such as actuators and sensors limitations,safety and operational constraints, the allowed dynamics of the considered plant,maximal and minimal inflows and outflows, etc., are instead represented as con-straints of the optimization problem. The field of Optimal Control has developed inorder to investigate mathematical tools for pursuing this type of approach. Whilephysical insight might be useful in some respects, the mathematics is such that evenan user with limited experience and understanding of the process to be controlledcan find the optimal solution. Model Predictive Control, [36], on the other hand, hasdeveloped as an evolution of optimal control (introduced by practitioneers) whichis meant to allow a treatment of problems normally out of the range of classicaloptimal control applications, often due to the so called “curse of dimensionality”,[18]. Namely, every realistic problem, involving more than a few variables and con-straints, can hardly be treated by means of the analytical set of tools available inoptimal control. Model Predictive Control, in its simplest form, attempts to solveon-line a finite-horizon optimal control problem for a single given initial configu-ration, and deploys the optimal control action determined in this way according toa rolling-horizon strategy, viz. only implementing the first part of the optimal inputsequence and reformulating, afterwards, a similar optimal control problem over ashifted time window.

When the process to be controlled is meant to spend most of the time in steady-state conditions, with all variables at equilibrium except for possible minor fluctu-ations due to process or sensor disturbances, it is tempting to separate profitabilitymaximization (deciding at which point in state and input space it is best to operatethe plant) from dynamical considerations, involving how to actually steer the plant’sstate towards the desired region and how to possibly stabilize it at some desirableequilibrium in the presence of disturbances that might lead it to drift away.

This hieararchical approach has, historically, developed in the control of chem-ical plants. Such systems, while often nonlinear and subject to hard constraints of

Economic Model Predictive Control: some design tools and analysis techniques 3

different nature, have relatively slow process dynamics, so that one could afford tocompute some solution to an optimal control problem within the inter-sample timeinterval, even with relatively modest computational power.

According to this paradigm, [25], a higher level device, called the Real TimeOptimization (or RTO layer), is responsible for choosing set-points for all sys-tem’s variables in order to maximize profit (in steady state) given the current opera-tional constraints, market conditions (prices of raw materials, energy and products)or other factors which may affect profit. This is a static optimization problem inthat the feasible set of considered input and states corresponds to equilibria whichmeet all operational constraints. It is in general a non-convex and non-linear (andfor both reasons hard) optimization problem. Algorithms for its solutions mightnot have guaranteed convergence within any reasonable amount of time but, on theother hand, this is solved basically off-line, only when the operational constraints or,more likely, the market conditions have changed. Architectures with an even largernumber of layers are often also envisioned, again as a result of tackling phenomenaoccurring at different time-scales.

The RTO forwards the computed set-points to an Advanced Control Layer, wherean often linearized Model Predictive Control algorithm is responsible for solvingon-line an optimal Tracking problem, namely, in the case of constant set-points,steering the systems’ state towards the best possible equilibrium as quickly as pos-sible. Notice that, during transient operations, when the Advanced Control Layeris acting and devising the appropriate control action, profitability concerns are nolonger affecting the selection of the control variable. Indeed, while the predictivecontroller is still an optimization-based controller and operates on the basis of anunderlying cost functional, the latter is devised in order to induce tracking towardsthe desired set-point, and need not bear any resemblance to the original profitabilityfunction maximized by the RTO.

This approach, which is widely adopted, has some important advantages:

• Computational: the hardest nonconvex programs are only solved off-line or ona much slower time-scale than Advanced Control Layer and do not involve dy-namics, thus drastically reducing their size and complexity;

• Robustness: stability and robustness guarantees are easier to be tackled on a lin-earized model; moreover, nonlinear models, if not derived by first principles areoften affected by uncertainty and may only be reliable near equilibrium or insmall regions of state space so that envisioning a global optimization involvingboth the transient and the regime of operations may be unrealistic or even dan-gerous;

On the other hand, the hierarchical separation might not be ideal in situations where

• market conditions change frequently and on a time-scale which is comparable tothe time-constants of the process dynamics;

• nonlinearity and non-convexity may result in complex optimal regimes of opera-tions, viz. achievable only by keeping the system on a trajectory which is not an


equilibrium state; in this respect, large profitability enhancements can sometimesoccur, for specific models and within certain parameter ranges.

When the quality of the model at hand is deemed appropriate within a regionmuch larger than the equilibrium manifold and if computational power is not a criti-cal issue, it seems therefore appropriate to integrate these two layer into a single onethat is responsible to optimize the dynamics (and not only the “static behaviour”) ofthe system on the ground of an optimal control problem deployed within a rolling-horizon approach and explicitly taking into account a profitability measure in itsdefinition. This is the goal of Economic Model Predictive Control. While such anapproach has a certain intuitive and practical appeal, it is only in recent years thata more systematic analysis of its implications regarding performance, stability androbustness (to name a few) has been attempted. This chapter will provide the readera self-contained introduction to the main results in this relatively new area of re-search. Pointers to the relevant literature will be provided when space constraints donot allow an in depth discussion of the material presented.

2 Formulation of Economic Model Predictive Control

In its basic formulation Economic Model Predictive Control looks at deterministicplants governed by finite-dimensional difference equations of the following type:

x(t +1) = f (x(t),u(t)) (1)

where x(t) ∈X⊂Rn is the state variable, and u(t) ∈U⊂Rm is the control variable.For the sake of simplicity, X and U are assumed to be compact sets. The functionf : X×U→ X is continuous, and ideally known without uncertainty.

In addition we assume that state and control variables be, in the light of opera-tional and safety considerations, constrained to a certain compact set Z ⊂ X×U.We say that a solution x(t) and corresponding control input u(t) are feasible if thefollowing is fulfilled:

(x(t),u(t)) ∈ Z ∀ t ∈ N. (2)

The goal of the control design is to maximize profit or, equivalently, minimizecosts, both during transient and steady-state operation. The latter are quantified bymeans of a scalar valued function ` : X×U→ R, with `(x,u) representing the costincurred for operating the plant at state x, subject to input u, throughout a samplinginterval. In more general scenarios, of course, both f and ` might be time-dependentbut, for the sake of simplicity, it is useful to consider situations in which costs anddynamics do not change significantly over the considered time window.

As customary in traditional MPC and Optimal Control, the stage cost is inte-grated over a (discrete) interval of length N, which is usually referred to as theprediction horizon. The rationale for this choice is to have a sufficiently long time


window to assess (in a way which is not too short-sighted) the value of taking oneparticular course of action over another. Mathematically, the integrated cost is de-fined as below:

J`(x,u) =N−1

∑t=0

`(x(t),u(t))+ψ f (x(N)). (3)

In the following we adopt the convention of denoting by bold fonts finite sequencesof indexed variables: for instance x := [x(0),x(1), . . . ,x(N)], u= [u(0),u(1), . . . ,u(N−1)] where the length of the sequence should be clear from the context. Notice thata final weighting function ψ f (·) might or might not be present, depending on theconsidered formulation of EMPC. Its usefulness is intuitively understood consid-ering that it could mitigate the effect of taking short-sighted actions by providingsome bound to the best achievable cost incurred in operation over an infinite or verylong horizon. More on this later, when the so called terminal ingredients will bediscussed in detail.

The main difference between tracking MPC and Economic MPC, at the defini-tion level, is in the stage cost, `(x,u). Typically, this is taken to be a positive definitequadratic form of state and input, in the former, while it may be an arbitrary con-tinuous function in the latter case. For instance `(x,u) = x′Qx+ u′Ru is a typicalchoice in tracking MPC. If a different equilibrium state-input pair is of interest, say(xs,us), then a suitable expression is normally:

`(x,u) = (x− xs)′Q(x− xs)+(u−us)

′R(u−us). (4)

In other words, the stage-cost ` is designed in order to penalize deviations froman assigned set-point, rather than optimize the plant’s profits. While non-quadraticand more general expressions could be used to such endevour, the usual approachto induce convergence of the closed-loop system’s trajectories towards the desiredset-point is to take `(x,u) positive definite with respect to the point (xs,us). In otherwords:

0 = `(xs,us)< `(x,u) ∀(x,u) 6= (xs,us). (5)

Inequality (5) need not hold for ` in Economic MPC set-ups, even if (xs,us) ischosen to be the best feasible equilibrium, viz.

`(xs,us) = min(x,u) ∈ Zx = f (x,u)

`(x,u). (6)

This fact is emphasized in Fig. 1.We are now ready to formally define an Economic Model Predictive Control

scheme. In particular, at each time t, and assuming exact knowledge of current statex(t), the following optimization problem is solved:


−50

5

−10−505100

10

20

30

40

50

60

70

u

l(x,u)

x

(x,u): f(x,u)=x−5

0

5 −5

0

5

−10

−5

0

5

10

15

u

l(x,u)

x

( xs,u

s )

(x,u):x=f(x,u)

Fig. 1 Stage cost `(x,u) for Tracking and Economic MPC

minz,v J`(z,v)subject to(z(k),v(k)) ∈ Z k ∈ 0, . . . ,N−1z(k+1) = f (z(k),v(k)) k ∈ 0, . . . ,N−1z(N) ∈ X fz(0) = x(t),

(7)

where X f is a compact set, whose properties will be later addressed when discussingthe role of terminal ingredients in Economic MPC. Let z?,v? denote any optimal so-lution of problem (7) (this is non unique in general), we define a state feedback bychoosing the current control input u(t) according to u(t) = v?(0).

At the following sampling time the v? vector previously computed is discarded(though normally used as a warm start for the optimizer in a suitably shifted ver-sion of itself), and the problem is solved again from the resulting value of initialstate x(t + 1). The iterative application of this procedure implicitly defines a state-feedback law,

u(t) = k(t,x(t)) := v?(0). (8)

This is, in general, a time-varying, nonlinear and possibly set-valued feedback. Timedependence, in particular, may arise as a result of time-varying terminal penaltyfunctions or constraints, as well, of course, of time-varying costs and constraintswhich, for the sake of simplicity, we are not addressing.

Throughout this chapter, when we talk about the closed-loop system, we there-fore understand the system evolving according to the following difference equation(or more precisely in the case of multiple optimal solutions, difference inclusion):

x(t +1) ∈ f (x(t),k(t,x(t))). (9)

Given the current state x(t), the feasible set Ft is defined as follows:

Ft := x(t) ∈ X : ∃(z,v) fulfilling constraints in (7) . (10)


More sophisticated formulations of EMPC are possible to allow for instance aver-age constraints (on infinite or finite windows) or take into account uncertainty inpredictions. We will mention such variants in a later Section.

3 Properties of Economic MPC

While solutions of the closed-loop system might not be uniquely defined, typicallythey share the same guaranteed stability, feasibility and performance properties.These are normally guaranteed through the adoption of suitable terminal ingredi-ents. These are also commonly adopted in Tracking MPC and are usually terminalconstraints (equality or set-membership) and terminal penalty functions. They havetechnical implications concerning several features of the closed-loop systems dy-namics, such as recursive feasibility, average asymptotic performance and stability.We discuss them separately throughout the following Section.

3.1 Recursive feasibility

Recursive feasibility is the property that for any initial condition x(0)∈F0 solutionsof (9) fulfill for all t ∈ N x(t) ∈Ft . Indeed, due to the presence of hard constraintsin (7), feasible solutions may fail to exist. The property of recursive feasibility thenensures that, provided the system’s state is updated according to its nominal dynam-ics, feasibility is preserved at all times. This fact is normally shown by induction,by directly proving the following implication:

x(t) ∈Ft ⇒ x(t +1) = f (x(t),k(t,x(t))) ∈Ft+1. (11)

There are several ways by which this can be guaranteed.

3.1.1 Terminal equality constraints

One way to ensure recursive feasibility is to constrain the final predicted state z(t) tobe equal to a predesigned feasible solution x?(t). In particular, by endowing problem(7) with the following constraint:

z(N) = x?(t). (12)

This can alternatively be formulated as z(N) ∈ X f where X f = x?(t). Notice thatin such a case the penalty function ψ f does not play any role in defining the optimalinput sequence and may, without loss of generality, be taken as 0. A particularlysimple choice of x?(t) is when equilibrium solutions are considered. In particularone may take x?(t) = xs, as defined in (6). This results in a time-invariant state-


feedback law. Alternatively, another relevant situation arises when x?(t) is periodicwith some period T . In particular, then, x?(t) = x?(t mod T ). The periodic solutioncan potentially be selected optimally, by solving the minimization problem shownbelow:

minx,u ∑T−1t=0 `(x(t),u(t))

subject to

x(1) = f (x(0),u(0))...x(T −1) = f (x(T −2),u(T −2))x(0) = f (x(T −1),u(T −1))

(x(t),u(t)) ∈ Z ∀t ∈ 0, . . . ,T −1.

(13)

3.1.2 Terminal set (or terminal inequality constraint)

Another possibility is to enforce a terminal set-membership constraint

z(N) ∈ X f .

The closed set X f is chosen to be a control invariant set. In particular, then:

∀x ∈ X f , ∃u ∈ U : (x,u) ∈ Z and f (x,u) ∈ X f .

Sometimes it may be useful to have an explicit expression for the control action uthat keeps the state x in X f . We denote such a feedback policy by κ f (x). A specialcase of control invariant set is, for example, the equilibrium manifold. This choicecorresponds to:

X f = x ∈ X : ∃u : (x,u) ∈ Z and x = f (x,u). (14)

This was proposed in [24] in the context of tracking MPC and in [13] in the contextof economic MPC and is normally referred to as Generalized Terminal EqualityConstraint. The main advantage of such a generalized terminal equality constraintis that a possibly (much) larger feasible set is obtained than when using a fixedterminal equality constraint (or a region around some fixed equilibrium).

While, given the economic nature of the problem at hand, selection of X f shouldseek to optimize the economic cost within X f in some sense (for instance on averagewhen operating a viable solution), standard techniques for designing control invari-ant sets can be adopted as in classical MPC. For instance, the paper [1], providessimple linearization-based techniques for designing invariant ellipsoids (around pre-selected equilibrium states) and associated penalty functions. Other design tech-niques are of course possible, such as time-varying terminal sets, see [3]. System-


atic numerical approaches to the design and selection of X f in an economic andnonlinear context are, however, yet to be developed.

The basic idea behind most recursive feasibility proofs is that the optimal solutionz?t , v?t computed at time t can be used to generate a feasible solution at time t +1. Infact, in the absence of exogenous disturbances the following equality holds:

x(t +1) = f (x(t),k(t,x(t))) = f (z?(0),v?(0)) = z?(1)

In particular, then the sequence [z?(1), . . .z?(N)], [v?(1), . . . ,v?(N− 1)] may serveas the initial section of a solution feasible at time t + 1. The last control move andterminal state can be obtained, in the case of terminal equality constraints simply byconsidering the sequences:

z = [z?(1), . . .z?(N),x?(t +1)], v = [v?(1), . . . ,v?(N−1),u?(t)].

For the case of terminal set-membership constraints:

z= [z?(1), . . .z?(N), f (z?(N),κ f (z?(N)))], v= [v?(1), . . . ,v?(N−1),κ f (z?(N))].

It is worth pointing out that the set of feasible initial conditions, F0, is, in general,dependent on both the terminal set and the length of the prediction horizon. Weemphasize this dependence by denoting it F0(N,X f ). Having a larger feasible setat time 0 is of course of great practical interest. This can be achieved by consideringthe following monotonicity property:

N1 ≤ N2 and X1f ⊆ X2

f ⇒F0(N1,X1f )⊆F0(N2,X2

f ). (15)

3.2 Asymptotic Average Cost

One way to assess the validity of an economic controller is to verify the long-runaverage cost incurred in closed-loop operation of the system. In particular, lettingx(t) be a solution of (9) and u(t) ∈ k(t,x(t)) the associated control input, we maydefine the average asymptotic cost as:

J := limsupT→+∞

∑T−1t=0 `(x(t),u(t))

T. (16)

Many Economic MPC schemes come with apriori bounds on the possible values ofJ as a result of the chosen terminal ingredients.


3.2.1 Terminal feasible trajectory

Given a feasible trajectory x?(t), u?(t), that we adopt as a terminal constraint as inequation (12), we may let:

J? := limT→+∞

∑T−1t=0 `(x?(t),u?(t))

T. (17)

Then, the following inequality holds:

J ≤ J? (18)

as shown in Lemma 1 of [3]. In other words, the asymptotic average cost in closed-loop is never worse than the average cost of the feasible solution adopted as theterminal constraint. Both equality or strict inequality are possible, in fact. The sim-plest form of such bound is found when x?(t) is a constant (i. e. the best feasibleequilibrium), or periodic, [2]. A general method to derive such bounds is based onfeasibility at time t+1 of the shifted sequences z and v. This property, as customaryin traditional MPC, allows one to derive a useful dissipation inequality fulfilled bythe cost-to-go function defined below

V`(x) := minz,v J(z,v)subject to(z(k),v(k)) ∈ Z k ∈ 0, . . . ,N−1z(k+1) = f (z(k),v(k)) k ∈ 0, . . . ,N−1z(N) ∈ X fz(0) = x

(19)

along solutions of the closed-loop system. In particular, exploiting suboptimality ofthe feasible shifted state and input sequences at time t +1, we see that

V`(x(t +1))≤ J`(z, v) = J`(z?,v?)− `(x(t),k(t,x(t)))+ `(x?(t),u?(t))= V`(x(t))− `(x(t),k(t,x(t)))+ `(x?(t),u?(t)) (20)

Adding the previous inequality over a finite time interval implies:

V`(x(T ))−V`(x(0)) =T−1

∑t+0

V`(x(t +1))−V`(x(t))

≤T−1

∑t=0−`(x(t),k(t,x(t)))+ `(x?(t),u?(t)).

Dividing by T and taking liminfs letting T grow to infinity in both sides yields thedesired inequality.


3.2.2 Terminal Penalty Function

When adopting a more general form of terminal constraint, such as the control-invariant set X f , apriori guaranteed bounds on the asymptotic closed loop averagecost may still be possible. To this end, however, the terminal penalty function ψ f (·)needs to fulfill a suitable inequality. This generalizes the idea of a control Lyapunovfunction, usually adopted for tracking MPC, to the context of Economic MPC. Anappropriate condition, in the case of X f being a control invariant neighborhood ofan equilibrium (xs,us) of interest, is the following:

ψ( f (x,κ f (x)))≤ ψ(x)− `(x,κ f (x))+ `(xs,us), (21)

a condition first proposed in [1]. In this case, suboptimality of the feasible shiftedstate and input sequences at time t +1 yields

V`(x(t +1))≤ J`(z, v) = J`(z?,v?)− `(x(t),k(t,x(t)))+ `(z?(N),κ f (z?(N)))

+ψ f ( f (z?(N),κ f (z?(N))))−ψ f (z?(N))

= V`(x(t))− `(x(t),k(t,x(t)))+ `(z?(N),κ f (z?(N)))

+ψ f ( f (z?(N),κ f (z?(N))))−ψ f (z?(N)). (22)

Thanks to (21), the previous inequality can be further simplified into:

V`(x(t +1))≤V`(x(t))− `(x(t),k(t,x(t)))+ `(xs,us). (23)

This is of the same form as (20) when the terminal feasible solution is constant.Hence, the following apriori performance bound can be guaranteed:

J ≤ `(xs,us). (24)

3.2.3 Adaptive terminal weight

When Generalized Terminal Equality constraints are used, the following terminalweighting function is proposed:

ψ(x) := minu `(x,u),subject to(x,u) ∈ Zx = f (x,u)

(25)

and ψ f (x) is then replaced by β (t)ψ(x), where β (t) is an adaptive coefficient, up-graded in order to tune the relative weight of the cost associated to the final equi-librium state. Under suitable upgrading rules for β , interesting asymptotic perfor-mance bounds are derived in [27]. In particular, the rationale behind the proposedupdate rules for the terminal weight is such that β is increased whenever the terminalsteady-state of the optimal predicted trajectory is far away from the best reachable


equilibrium (in N steps). To be more precise, define the best reachable steady-statecost (in N steps) from a given initial condition x as

`min(x) := minz,v `(z,v),subject to(z,v) ∈ Zz = f (z,v)

x ∈F0(N,z)

A simple update rule can now be defined as

β (t +1) = β (t)+α(‖ψ(z?(N))− `min(x(t))‖). (26)

This update rule ensures that β→∞ if the terminal predicted steady-state z?(N) doesnot converge to the best reachable steady-state. This property (together with sometechnical conditions) can then be used to show that for the resulting closed-loop sys-tem, the terminal predicted steady-state cost ψ(z?(N)) can be upper bounded by thecost of the best steady-state which is robustly reachable from the ω-limit set of theclosed loop [27]. Since as shown above in (18), the asymptotic average performancecan be upper bounded by J? as defined in (17), this result leads the conclusion thatalso the closed-loop asymptotic average performance can be upper bounded by thecost of the best steady-state which is robustly reachable from the ω-limit set of theclosed loop. Besides the simple update rule (26), also more elaborate update ruleshave been proposed in [27], for which similar closed-loop performance statementscan be derived. For example, one can allow resets of β (to some constant c) in orderto avoid unnecessarily large terminal weights.

The above obtained closed-loop performance bounds for economic MPC with anadaptive terminal weight are rather of conceptual nature and not necessarily verifi-able a priori, since they depend on the resulting ω-limit set of the closed loop. Im-proved bounds have been obtained in [30], for a setting where a generalized terminalregion constraint instead of a generalized terminal equality constraint was used. Thismeans that we use a terminal region around the whole equilibrium manifold insteadof a terminal region around some (fixed) steady-state. In this case, it can be shownthat if this generalized terminal region is designed properly, the terminal predictedsteady-state cost ψ(z?(N)) (and hence also the closed-loop average performance)converges to a local minimum of the stage cost ` on the set of feasible steady-states.In case of linear systems with convex stage cost and constraints, convergence to theglobally optimal steady-state cost can be shown, recovering the above result (24)when using a terminal equality constraint at the optimal equilibrium or a terminalregion around this equilibrium.


3.3 Stability of Economic MPC

An advantage of designing terminal ingredients for tracking MPC is that the stan-dard argument for recursive feasibility also allows a Lyapunov-based analysis of thecontrolled system’s behaviour. In fact, the cost to go function V`(x), is seen to fulfilla dissipation inequality along solutions of the closed-loop system and, under minortechnical assumptions, can be used as a candidate Lyapunov function to carry outproofs of stability and convergence for the desired equilibrium state, [22].

This is no longer the case for Economic Model Predictive Control. For instance,inequality (23) does not guarantee monotonicity with respect to time of V`(x(t)),because it is not necessarily true that `(xs,us) ≤ `(x(t),k(t,x(t))). In fact, the costat the terminal equilibrium is not necessarily a global minimum of the stage-costfunction, even if (xs,us) might be the feasible state-input equilibrium pair of lowestcost.

As a result, even when initialized at xs, the closed-loop system’s solution mightdrift towards a different regime of operation: because of instability phenomena orbecause xs is not an equilibrium in closed-loop. Bounds on asymptotic average cost,such as (24), tell us that when this happens it is in order to yield a regime of oper-ation that is at least as economically rewarding as the best equilibrium. In general,however, stability of the underlying best equilibrium state xs cannot be expected.

While convexity-based arguments were employed at first for the case of linearsystems and convex cost functionals [37], only later Lyapunov-based insights weregained into stability of Economic MPC, [9]. The major breakthrough in [9] was tointroduce a rotated stage-cost function, defined as `(x,u)+ λ T [x− f (x− u)]. Un-der suitable terminal equality constraints, it is seen that the extra-term in the cost-function does not affect the optimal solution of (7). A similar construction was laterproposed in [2]. In particular, for any function λ : X→ R, one may define the asso-ciated rotated cost as:

L(x,u) = `(x,u)+λ (x)−λ ( f (x,u)). (27)

where the previous construction is thus a special case obtained for linear functionsλ (x) = λ T x. Notice that, along any solution:

N−1

∑t=0

L(x(t),u(t)) =

(N−1

∑t=0

`(x(t),u(t))

)+λ (x(0))−λ (x(N)). (28)

Hence, assuming for instance z(N) = x?(t) is the terminal constraint, one may rec-ognize that the optimal solution of (7) is not affected by having either ` or L as thestage-cost. This is of interest, for stability analysis, because choosing λ appropri-ately, the rotated stage cost might admit a global minimum at (xs,us), namely:


L(xs,us) = min(x,u)∈Z

L(x,u), (29)

so that an Economic MPC scheme ends up being equivalent to a tracking MPCscheme with rotated stage cost. In fact, in such a case,

L(xs,us)≤ L(x,u) ∀(x,u) ∈ Z. (30)

Rearranging the different terms and exploiting the equation xs = f (xs,us), the latterinequality reads:

λ ( f (x,u))−λ (x)≤ `(x,u)− `(xs,us), (31)

and is therefore a dissipativity condition for system (1) with respect to the supplyrate s(x,u) = `(x,u)− `(xs,us). For technical reasons, this condition needs to beslightly strengthened.

Definition 1. (Strict dissipativity) System (1) is strictly dissipative with respect tothe supply rate s(x,u) := `(x,u)− `(xs,us), if there exists a continuous functionλ : X→ R and a positive definite function ρ , such that:

λ ( f (x,u))−λ (x)≤−ρ(|x− xs|)+ `(x,u)− `(xs,us) (32)

Notice that condition (32) implies that xs is (strictly) the best feasible equilibriumand that the system is, in fact, suboptimally operated outside this equilibrium (in anasymptotic average sense), [4].

As anticipated, dissipativity plays a crucial role in establishing stability of Eco-nomic MPC. We sketch the basic argument below, for the case of a terminal equalityconstraint, z(N) = xs. In fact, under strict dissipativity (32), we see that:

L(xs,us)+ρ(|x− xs|)≤ L(x,u), ∀(x,u) ∈ Z. (33)

Moreover, by virtue of (28), the optimal solution of

minz,v JL(z,v)subject to(z(k),v(k)) ∈ Z k ∈ 0, . . . ,N−1z(k+1) = f (z(k),v(k)) k ∈ 0, . . . ,N−1z(N) = xsz(0) = x(t),

(34)

where a rotated stage cost L(x,u) is adopted, is the same as the one for (7). LettingVL(x) define the minimum of problem (34) and exploiting (33), we see that,

VL(x(t +1))≤VL(x(t))−L(x(t),k(t,x(t)))+L(xs,us)≤−ρ(|x(t)− xs|). (35)

Since VL(x)−VL(xs) is seen to be a positive definite function, condition (35) implies,by standard Lyapunov analysis, asymptotic stability of the equilibrium xs within the


feasibility region of problem (7).

When adopting a formulation with a terminal penalty function (rather than aterminal equality constraint), one may realize that:

N−1

∑t=0

L(x(t),u(t))+Ψf (x(N))=

(N−1

∑t=0

`(x(t),u(t))

)+λ (x(0))+Ψf (x(N))−λ (x(N))

=

(N−1

∑t=0

`(x(t),u(t))

)+ψ f (x(N))+λ (x(0)).

provided the rotated terminal cost is defined as:

Ψf (x) = ψ f (x)+λ (x). (36)

The previous derivation shows that solution of (7) is not affected when L and Ψfreplace ` and ψ f respectively in the definition of the cost-functional. Thus, stabilityanalysis can then be carried out under strict dissipativity, along the same lines as inthe case of terminal equality constraints (see [1]).

Remarkably, dissipativity conditions are seen to be not only sufficient, but alsonecessary for optimal steady-state operation, in a suitable sense. Namely, as dis-cussed above, under the dissipativity condition (31), the system is optimally oper-ated at steady-state, and under the strict dissipativity condition (32) suboptimally op-erated outside the optimal equilibrium. This means that each other feasible state andinput sequence pair yields a worse (strictly worse) asymptotic average performancethan the optimal steady-state cost `(xs,us). One can show that the converse state-ment is also true under a certain controllability condition [29]. While the proof ofsufficiency of dissipativity for optimal steady-state operation follows rather straight-forwardly from the dissipation inequality and the definition of the latter property, theconverse result is a bit more involved and can be shown by a contradiction argument.Namely, assuming that the system is not dissipative and using the controllabilitycondition, one can construct a specific (periodic) state and input sequence whichresults in a lower average cost than the best equilibrium cost, contradicting optimalsteady-state operation [29]. This converse result together with the above stabilityanalysis allows the following interpretation. If steady-state operation is optimal, thesystem is dissipative with respect to the supply rate s(x,u) := `(x,u)− `(xs,us),which in turn (in its strict form) can be used to conclude that the closed loop con-verges to the optimal steady-state (xs,us). This means that the closed loop ”does theright thing”, i.e., ”finds” the optimal operating behavior.

It is worth mentioning that dissipativity conditions can not only be used to estab-lish optimality of steady-state solutions over other more complex regimes of opera-tion, but also exploited in order to find the best asymptotic average performance ofthe system. This may be defined as:


`? := infx(.),u(·) liminfT→+∞

∑T−1t=0 `(x(t),u(t))

Tsubject to(x(t),u(t)) ∈ Zx(t +1) = f (x(t),u(t))

(37)

and happens to be related to dissipativity by the following equality:

`? := sup` ∈ R : ∃λ (·) continuous : λ ( f (x,u))≤ λ (x)+ `(x,u)− `, ∀(x,u) ∈ Z.

This equality was first derived for continuous time systems in [16] and later adaptedto discrete-time systems in [21].

Finally, we remark that not only optimal steady-state operation can be charac-terized via a suitable dissipativity condition, but also the more general case whereperiodic operation is optimal [40, 28].

3.4 EMPC without terminal ingredients

In economic MPC schemes without terminal ingredients, the terminal constraintz(N) ∈ X f in (7) is omitted and the terminal cost in (3) is chosen to be zero. Usingno terminal constraints may lead to algorithmic advantages (since the possibly re-strictive terminal constraint is absent), and also the optimal steady-state, the optimalperiodic orbit or some feasible terminal trajectory x?(·), u?(·) (depending on how theterminal region constraint is formulated, cf. Section 3.2) need not be known for im-plementing the economic MPC scheme. On the other hand, guaranteeing recursivefeasibility is not as straightforward as in the case of suitably defined terminal in-gredients, neither is the establishment of (a priori) closed-loop performance boundsthat typically requires knowledge of the optimal operating behavior.

The main insight which is employed in economic MPC schemes without termi-nal constraints is the so-called turnpike property [11, 41]. This property means thatopen-loop optimal trajectories z? resulting from application of the optimal solu-tion v? to Problem (7) spend most of the time in a neighborhood N of the optimaloperating behavior (e.g., a neighborhood of the optimal steady-state xs if steady-state operation is optimal, etc.). Importantly, the number of time instants where theoptimal trajectory is outside of this neighborhood depends on the size of N , but isindependent of the prediction horizon N. In [17], it was shown that the same strictdissipativity condition as employed in Section 3.3, i.e., strict dissipativity with re-spect to the supply rate s(x,u) = `(x,u)− `(xs,us), together with suitable controlla-bility conditions is sufficient for the turnpike property at the optimal steady-state xs(compare also [14] for a continuous-time version of this result). In fact, also con-verse statements showing necessity of strict dissipativity for the turnpike behaviorhave recently been obtained [20]. The turnpike property at xs can now be used toconclude practical asymptotic stability of the optimal steady-state for the resultingclosed-loop system, using again VL as a (practical) Lyapunov function, i.e., the op-


timal value function of the MPC problem using the rotated stage cost L, see [19](compare also [15] for a continuous-time version of this result). In particular, it wasshown that the size of the neighborhood of the optimal steady-state xs into which theclosed loop converges depends on the prediction horizon N and decreases to zeroas N→ ∞. Interestingly, a candidate input sequence v for the next time instant t +1is not necessarily constructed by appending some control value at the end (as is thecase when using terminal constraints, see Section 3.1), but at some point where theoptimal predicted trajectory is close to the optimal steady-state xs as guaranteed bythe turnpike property. A generalization of these results to the case where the opti-mal operating behavior is some general periodic orbit (instead of a steady-state) hasbeen presented in [31].

4 EMPC with constraints on average

While classical Model Predictive Control always affords solutions that converge, atleast nominally, towards the desired reference signal, Economic Model PredictiveControl is not always stabilizing and may result in closed-loop behaviours that donot converge towards the best equilibrium or towards the underlying feasible solu-tion adopted as a terminal ingredient. Under such circumstances it makes sense towant to guarantee, for the closed-loop behaviour, specific additional constraints tobe fulfilled on average, rather than pointwise in time. A typical example could be theoutflow of a plant, which rather than constraining to be always greater or equal thana given flow rate, might be required to fulfill a similar inequality only in an averagesense. Relaxing constraints in this way, if deemed suitable from an operational pointof view, can in fact improve profitability margin of a plant simply because we arecarrying out an optimization over a larger set of feasible solutions. Most EMPC con-trol schemes can be endowed with constraints on average quantities. In particular,we may define the asymptotic average of a signal v as the following set:

Av[v] =

v : ∃Tn+∞

n=1 : limn→+∞

Tn =+∞ and v = limn→+∞

∑Tn−1k=0 v(k)

Tn

. (38)

For most signals v of interest, Av[v] is actually a singleton, corresponding to theasymptotic average of the signal. However, in general, a signal might have morethan one asymptotic average when it keeps spending longer and longer time periodsclose to several values. Economic MPC with constraints on average allows to definean auxiliary output variable:

y(t) = h(x(t),u(t)) (39)

where h : X×U→Rp, and require that for the controlled system and given a convexset Y the following holds:

Av[y] ∈ Y.


To this end, the following definitions and augmented set-up were proposed in [2]:

Yt+1 = Yt ⊕Y⊕−h(x(t),u(t)), (40)

where⊕ denotes set-sum. In particular Y0 can be initialized as an arbitrary compactconvex set. The iteration leads to

Yt = tY⊕Y0⊕

−

t−1

∑k=0

h(x(k),u(k))

.

Therefore the set Yt grows in size with linear speed Y/sampling time, in all alloweddirections, while being shifted around by a quantity equal to the “integral” of h.Problem (7) can then be augmented as follows

minz,v J`(z,v)subject to(z(k),v(k)) ∈ Z k ∈ 0, . . . ,N−1z(k+1) = f (z(k),v(k)) k ∈ 0, . . . ,N−1z(N) ∈ X fz(0) = x(t),∑

N−1k=0 h(z(k),v(k)) ∈ Yt .

(41)

Recursive feasibility, guaranteed satisfaction of all constraints in closed-loop andperformance bounds can be derived for this EMPC scheme along similar steps asin the previous cases. When the set Y can be defined as a polyhedron, suitablerelaxed notions of dissipativity on averagely constraints solutions are also proposedin [2]. These are used in [32] to derive asymptotic convergence results for EconomicMPC with constraints on average. Another possibility suggested in [32] is to usesuitably designed constraints on average in order to induce asymptotic convergencetowards equilibria that would otherwise be unstable. Average constraints on finitetime windows are also a possibility, and are proposed in [33].

5 Robust Economic Model Predictive Control

In real-world applications, the presence of disturbances and model uncertainties istypically unavoidable. Hence it is of paramount interest to obtain closed-loop per-formance and stability guarantees despite such disturbances. To this end, some ofthe techniques developed in the context of robust stabilizing (tracking) MPC can behelpful, in particular for addressing issues of robust feasibility, [23, 34, 35]. How-ever, it turns out that just transferring e.g. tube-based MPC approaches to an eco-nomic MPC context without suitable adaptations can lead to a suboptimal closed-loop performance. This observation can be illustrated by the following simple mo-tivating example.


0 2.5

0

xC

ost

ˆ

Fig. 2 Stage cost ˆ in the motivating example of Section 5.

Consider the system x(t+1) = x(t)+u(t) with stage cost `(x,u) = ˆ(x) as shownin Figure 2. Clearly, the system is trivially optimally operated at the optimal steady-state x= u= 0, since the cost function ˆ is positive definite (i.e., a tracking cost func-tion). Now suppose that some additive disturbances are present, satisfying w(t)∈Wfor all t ∈ N and some compact set W. If a standard tube-based MPC scheme suchas the one in [26] is employed, one can show that the closed-loop system convergesto a robust positively invariant (RPI) set Ω1 centered at origin, which is exemplar-ily depicted in red in Figure 2. The size of this RPI set scales with the size of thedisturbance set W. Now if W is large enough, one can see that a better average per-formance might be obtained if the economic MPC scheme is such that the closedloop converges to an RPI set Ω2 centered, e.g., at x = 2.5, which is exemplarilydepicted in green in Figure 2. Namely, while ˆ(2.5) > ˆ(0) = 0, the average of ˆover Ω2 is smaller than that of ˆ over Ω1.

Motivated by this fact, several different robust economic MPC schemes havebeen proposed in the literature where some knowledge about the present distur-bances is incorporated into the cost function employed within the repeatedly solvedoptimization problem. Depending on the available knowledge of the disturbances,different closed-loop performance guarantees can then be derived. To this end, sup-pose that the real (disturbed) system is given by x(t + 1) = f (x(t),u(t),w(t)) withw(t) ∈W for all t ∈ N and some compact set W, and denote the correspondingnominal system by ξ (t + 1) = f (ξ (t),ν(t),0). Furthermore, we assume that somerobust control invariant set Ω has been determined such that if the error between thereal and nominal state at time t is contained in Ω , the same is true at time t+1 inde-pendent of the disturbance w(t) ∈W and the (nominal) input ν(t). This is typicallyachieved by using some prestabilization or additional (error) feedback, i.e., usingu = φ(x,ξ ,ν) (in the linear case, e.g., φ(x,ξ ,ν) = K(x− ξ )+ν can be chosen, inwhich case the computation of Ω reduces to determining an RPI set). As in stan-dard (stabilizing) tube-based MPC, one can show that the real closed-loop systemis contained in the set Ω around the nominal system state. In order to account forthis fact within the repeatedly solved optimization problem, the following two costfunctions have been proposed:


`max(ξ ,ν) := maxω∈Ω

`(ξ +ω,φ(ξ +ω,ξ ,ν)

)(42)

ìnt(ξ ,ν) :=∫

Ω

`(ξ +ω,φ(ξ +ω,ξ ,ν)

)dω (43)

Here, `max is such that the worst case cost within the set Ω around the nominal stateand input is considered, while in ìnt the average over all values in Ω is taken. Usingthese cost functions within a suitably defined tube-based MPC scheme based on theone in [26], the following closed-loop average performance bounds can be derivedin a similar fashion as shown in Section 3.2 for the nominal case. For ìnt as definedin (43), it was shown in [5] that

limsupT→∞

1T

T−1

∑t=0

ìnt(ξ (t),ν(t))≤ ìnt(zs,vs),

where ξ (·) and ν(·) are the resulting nominal state and input sequences and (zs,vs)is the optimal steady-state minimizing ìnt. Since the real closed-loop state is con-tained in the set Ω around the nominal closed-loop state, this can be interpretedas an average performance result for the real closed-loop system, averaged over allpossible disturbances. For `max as defined in (42), one can directly obtain an averageperformance bound for the real closed-loop system (independent of the realizationof the disturbance), as shown in [6]:

limsupT→∞

1T

T−1

∑t=0

`(x(t),u(t))≤ `max(zs,vs).

Here, (zs,vs) is the optimal steady-state minimizing `max.If additional information about the distribution of the disturbance is available,

improved performance bounds can be obtained. This was shown in [7] for linear sys-tems subject to additive disturbances. Here, one can directly minimize the expectedvalue cost of predicted states and inputs, i.e., in (7) the following (prediction-timedependent) cost function is used:

ìntk (z(k),v(k)) := E`(x(t + k),u(t + k))|x(t) .

Using a suitable tube-based MPC scheme based on [8], the following closed-loopaverage performance bound has been obtained in [7]:

limsupT→∞

1T

T−1

∑t=0

E`(x(t),Kx(t)+u(t))|x(0) ≤ ìnt∞ (zs,vs). (44)

Here, ìnt∞ is defined by taking the expected value of the cost ` over the RPI set Ω ac-

cording to the steady-state error distribution, and (zs,vs) is the optimal steady-stateminimizing ìnt

∞ . The bound (44) then says that the expected closed-loop average costgiven the initial condition x(0) is upper bounded by the best expected steady-statecost.


6 Conclusions

The chapter motivates and introduces Economic Model Predictive Control as amethod to merge the Real Time Optimization layer and the Control Layer withina single opimization layer which is responsible of controlling the plant and opti-mizing its economic performance. The main relevant analytical tools and results areillustrated in a self-contained way and pointers to relevant in-depth literature on thetopic are provided. In particular, we emphasized the role played by terminal ingre-dients in determining issues related to recursive feasibility, asymptotic perfomanceand stability of nominal and robust Economic Model Predictive Control. The theorythat emerged in the last few years has reached already a considerable maturity andis still developing with a number of issues of theoretical and practical relevance cur-rently under investigation by the scientific control community. While we have madean effort to propose a coherent perspective on Economic Model Predictive Con-trol we are aware that several recent developments did not find space in the presentchapter. We would like to mention some of them:

• Formulation of tracking MPC schemes matching locally a given EconomicModel Predictive Control, [39];

• Lyapunov-based Economic Model Predictive Control, [12];• EMPC with generalized optimal regimes of operation, [10];• Stochastic Economic Model Predictive Control, [38].

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economic model predictive control: some design tools and

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