unreachable setpoints in mpc

8/19/2019 Unreachable Setpoints in MPC

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Unreachable Setpoints in Model Predictive Control

James B. Rawlings, Dennis Bonné, John B. Jørgensen,Aswin N. Venkat, and Sten Bay Jørgensen

Abstract— In this work, a new model predictive controller is developedthat handles unreachable setpoints better than traditional model predictivecontrol methods. The new controller induces an interesting fast/slow asym-metry in the tracking response of the system. Nominal asymptotic stabilityof the optimal steady state is established for terminal constraint model pre-dictive control (MPC). The region of attraction is the steerable set. Existinganalysis methods for closed-loop properties of MPC are not applicable tothis new formulation, and a new analysis method is developed. It is shownhow to extendthisanalysisto terminal penalty MPC. Twoexamplesare pre-sented that demonstrate the advantages of the proposed setpoint-trackingMPC over the current target-tracking MPC.

Index Terms— Aymptotic stability, constraints, model predictive control.

I. INTRODUCTION

Model predictive control (MPC) has been employed extensively forconstrained, multivariablecontrol in chemicalplants [1]. The mainmo-

Manuscript received February 25, 2008; revised April 23, 2008. Current ver-sion published October 8, 2008. This work was supported in part by the in-

dustrial members of the Texas–Wisconsin Modeling and Control Consortiumand the National Science Foundation under Grant CTS-0456694, by the Com-puter Aided Process and Products Center, Technical University of Denmark,and by the Danish Department of Energy under Grant EFP 273/01-0012. Rec-ommended by Associate Editor M. Fujita.

J. B. Rawlings is with the Department of Chemical and Biological Engi-neering, University of Wisconsin, Madison,WI 53706-1691 USA (e-mail: [email protected]).

D. Bonné is with the CAPEC, Department of Chemical and Biochemical En-gineering, Technical University of Denmark, DK 2800 Lyngby, Denmark.

J.B. Jørgensen is with the DTU Informatics, Technical University of Den-mark, DK 2800 Lyngby, Denmark.

A.N. Venkat is with the Shell Global Solutions (U.S.), Inc., Houston, TX77084 USA.

S.B. Jørgensen is with theCAPEC, Department of ChemicalandBiochemicalEngineering, Technical University of Denmark, DK 2800 Lyngby, Denmark.

Digital Object Identier 10.1109/TAC.2008.928125

tivation for choosing MPC is to handle constrained systems. Essen-tially all current MPC theory is predicated on the assumption that thesetpoint has been transformed to a desired and reachable steady-statetarget in a separate steady-state optimization [2], [3]. This assumptionallows current theory to address only transient active constraints thatoccur when solving the controller’s dynamic optimization to drive thesystem to the desired and reachable steady state. In practice, the set-point often becomes unreachable due to a disturbance and the choiceto transform the problem to one with a desired and reachable steadystate is a signicant decision that affects controller performance. Weshow in this paper that when the setpoint is unreachable, this two-leveloptimization is suboptimal and does not minimize the tracking error.To improve the performance of MPC when the setpoint is unreachable,we propose dening system performance relative to the unreachablesetpoint rather than the reachable steady-state target. As we show inthe examples, a consequence of the unreachable setpoint formulationis a desirable fast/slow asymmetry in the system tracking response thatdepends on the system initial condition, speeding up approaches to theunreachable setpoint, but slowing down departures from the unreach-able setpoint. One technical difculty that arises is that the proposed

MPC cost function is unbounded on the innite horizon. A more sig-nicant problem is that the existing MPC theory for stability and con-vergence no longer applies because this theory is based on establishingthe controller cost function as a Lyapunov function for the closed-loopsystem, but the controller cost function is not strictly decreasing in thiscase.

Optimal control problems with unbounded cost are not new to con-trol theory. The rst studies of this class of problems arose in the eco-nomics literature in the 1920s [4], in which the problem of interest wasto determine optimal savings rates to maximize capital accumulation.Since this problem has no natural nal time, it was considered on theinnite horizon. A urry of activity in the late 1950s and 1960s ledto generalizations regarding future uncertainty, scarce resources, ex-panding populations, multiple products and technologies, and manyother economic considerations. Much of this work focused on estab-lishing existence results for optimal control with innite horizon andunbounded cost, and the famous “turnpike” theorems [5] that charac-terize the optimal trajectory. Refs. [6] and [7] provides a comprehen-sive and readable overview of this research.

This class of problems was transferred to and further generalized inthecontrol literature. Forinnite horizon optimal control of continuoustime systems, [8] established the existence of overtaking optimal trajectories. Convergence of these trajectories to an optimal steady stateis also demonstrated. Ref. [9] extended the results of [8] to innitehorizon control of discrete time systems. Reduction of the unboundedcost, innite horizon optimal control problem to an equivalent opti-mization problem with nite costs is established. Ref. [10] provide a

comprehensive overview of these innite horizon results.For feedback control subject to hard input constraints, we often

employ a receding horizon implementation of a suitably chosen nitehorizon optimal control problem. The controller is implemented bysolving the open-loop control problem at each sample time as the statebecomes available and injecting only the rst move of the open-loopoptimal solution. Ref. [11] provides a review of the methods for ana-lyzing the resulting closed-loop behavior of these controllers. In thisformulation, the existence of the optimal control is not at issue becausethe horizon is nite and the cost therefore bounded, but the stabilityand convergence of the resulting receding horizon implementation of the control law is the interesting question. In this paper, we addressthe closed-loop properties of this receding horizon implementation forthe case of unreachable setpoints. In the formulation introduced here,

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Fig. 1. Closed-loop MPC sequence with accumulation point . Two forecasttrajectories from to are shown. Trajectory 1 is the truncation of theoptimal trajectory starting from point and trajectory 2 is the completeoptimal trajectory from point .

For notational simplicity, we de ne

0 . Hence, we ex-press the MPC control problem as

8 (3a)

(3b)

0 (3c)

3

(3d) 0 (3e)

We denote the optimal input sequence by

0

The MPC feedback law is the rst move of this optimal sequence,which we denote as , and we denote the optimal costby 8 . The closed-loop system is given by

with so

1 1 1

Lemma 4: Theoptimal steady state is a xedpoint of theclosed-loopsystem

3

3

3

3

Proof: For

3 in the MPC control problem, (3), both the initialand terminal states are equal to

3 . Strict convexity of the cost functionimplies the entire trajectory remains at

3 so

3

3 . De nition 2 (Steerable Set): The steerable set is the set of states

steerable to

3 in steps

3

0

1 1 1 0

0

Remark 4: For nonsingular , the steerable set is a nonempty,bounded, convex subset of .

The set is nonempty because it contains

3 . The set is bounded be-cause is nite and is bounded. Allclosed-loop sequences arethere-fore bounded because the set is positive invariant under the MPCcontrol law. 1

For singular , the steerable set is not bounded. The followinglemma, however, shows that the closed-loop sequence starting in the(unbounded) steerable set evolves in a bounded set.

Lemma 5: For singular , there exists such that aclosed-loop sequence under terminal constraint MPC starting inthe steerable set remains in a nonempty, bounded, convex subset of

for times . Lemma 6: Every closed-loop sequence has at least one accumula-

tion point.Proof: Remark 4 and Lemma 5 ensure that every closed-loop

sequence is bounded after nite , and the Bolzano-Weierstrassproperty ensures that every in nite sequence in a closed, boundedsubset of has at least one accumulation point [14].

Let be a closed-loop sequence with an accumulation point . Say the closed-loop sequence enters at time index . De-ne the sequence to be a portion of the closed-loop sequence,

, . Because is an accumulation pointof the closed-loop sequence, for every , we can choose and

large enough so that as depicted in Fig. 1. De nethe means over the closed-loop sequence to be

Note also that the mean state and input satisfy

0 (4)

Now consider the MPCoptimal open-loop trajectory from each

for . The optimal open-loop input trajectories are

0

We wish to compare the optimal costs for two succeeding states

and . For , the cost is 8 . For state , consider rst the feasible but possibly suboptimal sequence

0

3

which is created by dropping the rst input , shifting all theother inputs forward in the sequence, and appending

3 for the nalinput. The corresponding trajectory for is labeled Trajectory 1in Fig. 1. This input evaluated at state is admissible for state

because it satis es the model, terminal constraint, and inputconstraints. 2 Its cost from state is directly related to the op-timal cost from by

8 0

8 0

3

3

Therefore, using the optimal sequence in place of in this equation produces the inequality

8 8

0

3

3

0 (5)

This optimal trajectory for is labeled Trajectory 2 in Fig. 1.

1This positive invariance is a well-known characteristic of terminal constraintMPC, but for those unfamiliar with MPC, it is also established later in the proof of Theorem 1.

2This admissibility implies the positive invariance of the MPC control lawmentioned previously.



2212 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008

It should be noted that in standard MPCwith a reachable steady-statetarget, the origin can be shifted to

3

3 and the term

3

3 iszero. That leads immediately to a cost decrease from to

and the cost function is a Lyapunov function forthe closed-loop system.With the unreachable setpoint, the situation is completely different.Theterm 0

3

3 changes sign with on typ-ical closed-loop trajectories. The cost decrease is lost and 8 is not a

Lyapunov function for the closed-loop system. We next prove asymp-totic stability for model predictive control with unreachable setpointsby other means.

Lemma 7: Every closed-loop sequence converges to the optimalsteady state.

Proof: Let be an accumulation point of the closed-loop se-quence. For every , we can choose and large enough sothat . Denote the difference in the optimal costsbetween points and by

8 0 8 (6)

Because the optimal cost is continuous in the state, goes to zero

with

. Add the inequalities in (5) to obtain

3

3

0

3

3

Next apply Lemma 1 to the variable to obtain

0

0

3

3

0

0

3

3

(7)

Next, we show

converges to

3

3 with increasing .From (4),

converges to the set of steady states, and, if itdoes not converge to

3

3 , then, by optimality,

achieves

values greater than

3

3 for innitely many . Equation (7) thenprovides a contradiction because the right-hand side goes to 0 for asuitable subsequence with increasing and the left-hand side is non-negative. Next consider optimizing subject to the constraint given by(4)

subject to

0

and denote this solution as

3

3

. From its construction weknow

3

3

and therefore

0

0

3

3

0

3

3

0

3

3

Applying Lipschitz continuity, Lemma 3, gives

0

0

0

0

3

3

(8)

We can choose an increasing sequence of for which the right-handside is bounded above. Therefore the sequence con-verges to its mean

or we violate the inequality above. There-fore converges to

3

3 .Theorem 1 (Asymptotic Stability of Terminal Constraint MPC): The

optimal steady state is the asymptotically stable solution of the closed-loop system under terminal constraint MPC. Its region of attraction is

the steerable set.Proof: Any initial state in the steerable set de nes an in nite

closed-loop sequence in through the iteration .1) Lyapunov Stability: Let denote the solution of

at time with initial condition .

Assume, contrary to what is to be proven, that there exists suchthat for every , 0

3

for some nite for some

3 , in which

3

3

, and we take the

intersection in case does not contain a full neighborhood of

3 .We establish a contradiction. Consider an and

3 .Choose nite that depends on so that

3 for all . We know such a nite existsfor every

since thesequence converges to

3 (Lemma 7). This setting is the sameas depicted in Fig. 1 (with

3 ), and we have the same inequalityderived previously in (8)

0

0

0

0

3

3 (9)

in which , 0 . Since 0

3

forsome (denote one of these as ), and 0

3

,we have the following lower bound on the sum of the following twodistances from the mean:

0

0

0

Since , there exists such that

for all ,and we have established a lower bound on the sum

0

0

0

Substitution into (9) gives

0

0

0

3

3

Taking the limit as , the left-hand side converges to

and the right-hand side converges to zero, and we have established acontradiction. Therefore, for every , there exists suchthat for all for all

3 , and we haveestablished Lyapunov stability.

2) Convergence: Convergence of to

3 for in the steerableset is established in Lemma 7. Convergence and Lyapunov stabilityimply asymptotic stability and the theorem is proved.

IV. E XAMPLES

Two examples are presented to illustrate the advantages of the pro-posed setpoint tracking MPC framework (sp-MPC) compared to tradi-




Fig. 2. Closed-loop performance of sp-MPC and targ-MPC (Example 1).

tional target tracking MPC (targ-MPC). The regulator cost function forthe new controller, setpoint-tracking MPC (sp-MPC), is

8 -

0

0

0

0

(10)

in which , and at least one of . This systemcan be put in the standard form de ned for terminal constraint MPC byaugmenting the state as 0 [3]. The regulatorcost function in traditional target-tracking MPC (targ-MPC) is

8 -

0

0

3

0

3

0

(11)

In the examples, the controller performance is assessed using the fol-lowing three closed-loop control performance measures:

8

0

0

0

8

0

0

8 8

8

in which is the process sample time. For each of the indices de nedabove, we de ne the percentage improvement of sp-MPC compared totarg-MPC by

1

8 -

0 8 -

8 -

2

1) Example 1: The rst example is the single input-single outputsystem

0

(12)

sampled with s. The input is constrained as . The de-sired output setpoint is

which corresponds to a steady-state

input value of 0 . The regulator parameters are

, , ,

. A horizon length of

is used. Between times 50 –130, 200 –270 and 360 –430, a state dis-turbance

causes the input to saturate at its lower

limit. The output setpoint is unreachable under the in uence of thisstate disturbance

. The closed-loop performance of sp-MPC andtarg-MPC under the described disturbance scenario are shown in Fig. 2.

TABLE ICOMPARISON OF CONTROLLER PERFORMANCE (EXAMPLE 1)

The closed-loop performance of the two control formulations are com-pared in Table I.

2) Comments: In the targ-MPC framework, the controller tries toreject the state disturbance and minimize the deviation from the newsteady-state target. This requires a large, undesirable control action that

forces the input to move between the upper and lower limits of opera-tion. The sp-MPC framework, on the other hand, attempts to minimizethe deviation from setpoint and subsequently the input just rides thelower limit input constraint. The bene t here is that the sp-MPC con-troller slows down the departure from setpoint, but speeds up the ap-proach to setpoint. The traditional targ-MPC can be tuned to be fast orslow throughrelative choice of tuning parameters and ,butit isfastor slow from all initial conditions, some of which lead to an approachsetpoint, but others of which lead to a departure from setpoint.

The greater cost of control action in targ-MPC is shown by the costindex 8

in Table I. The cost of control action in targ-MPC exceedsthat of sp-MPC by nearly 100%. The control in targ-MPC causes theoutput of the system to move away from the (unreachable) setpoint

faster than the corresponding output of sp-MPC. Since the control ob- jective is to be close to the setpoint, this undesirable behavior is elimi-nated by sp-MPC.

3) Example 2: The second example is the 2 input –2 output system

(13)

The system is sampled at the rate s. The inputs and are constrained between 0 and 0.5. The desired output setpointis

which corresponds to a steady-state input of

0 . The state and measurement evolution of thesystem are corrupted by noise.The state and measurement noise covari-ance are

0

and

0

, respectively. In addition torandom state noise,a zero-mean square-wave disturbance of magnitude6 0 also affects the state evolution of the system. For this



2214 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 9, OCTOBER 2008

Fig. 3. Closed-loop outputs of sp-MPC and targ-MPC in Example 2.

Fig. 4. Closed-loop inputs of sp-MPC and targ-MPC in Example 2.

Fig. 5. Target and disturbance estimates in Example 2.

example, the state and disturbances are estimated using a steady-stateKalman lter. The regulator parameters are

, ,

, and

. The horizon length is .The closed-loop performance of the sp-MPC and the targ-MPC formu-lations are shown in Figs. 3 and 4. Fig. 5 depicts the target and dis-turbance estimates for the noisy system. Table II quanti es the relativeperformance of the sp-MPC and targ-MPC frameworks.

4) Comments: In this example, the input is near its upper limitat steady state. The presence of state and measurement noise causesinput to saturate and consequently the setpoint to become unreach-able. We note from Table II that the total cost of control action used bytarg-MPC and sp-MPC are nearly the same, but the system outputs be-have quite differently. The overall controller cost of targ-MPC exceedsthat of sp-MPC by nearly 70% (Table II).

TABLE IICOMPARISON OF CONTROLLER PERFORMANCE (EXAMPLE 2)

V. TERMINAL PENALTY MPC

It is well known in the MPC literature that terminal constraint MPCwith a short horizon is not competitive with terminal penalty MPC in




terms of the size of the set of admissible initial states, and the undesir-able difference between open-loop prediction and closed-loop behavior[11]. Here we brie y outline how the previous analysis can be appliedto prove asymptotic stability of terminal penalty MPC. First de ne therotated cost

[9]

0

3

3

and compute the in nite horizon rotated cost-to-go under control law 0

3

3 with chosen so that isasymptotically stable. A simple calculation gives

0

3

5 0

3

0 0

3

in which 5 satises the usual Lyapunov equation and is given by

5 5

0

0

0

3

0

3

Note that the rotated cost-to-go satis es

. Next de ne the terminal penalty MPC costfunction as

8

0

0

and controller

8

(14a)

(14b)

0 (14c)

0 (14d)

We compute the change in cost along the closed-loop trajectory. For , the cost is 8

. For state , consider

the candidate sequence 0

in which is the state value at stage using the optimal inputsequence. Because we usethe control law in place ofthe constant

3 as we did in terminal constraint MPC, and

3 is on the boundary of , we must restrict the gain matrix and initial state such that the

control law is feasible [3] and the system is positive invariant. Com-puting the change in cost for this candidate sequence gives

8

8

0

0

The last three terms cancel and noting the optimization at stage ,we have the inequality

8

8

0

8

0

3

3

Notice this inequality is the same as (5) if we replace the cost function8 in (5) by the rotated cost function 8

. The remaining steps after (5) inthe previous development can then be followed to establish asymptoticstability of

3

3 for terminal penalty MPC.If the system is unstable and feasible does not exist to make

stable given the active constraints at

3 , constraints to zero theunstable modes at stage are added to (14) [15]. The set of admissible

initial states is chosen to ensure positive invariance and feasibility of control law . These plus compactness of ensure system evolutionin a compact set. The user has some exibility in choosing , such asthe simple choice . A procedure to choose feasible that moreclosely approximates the in nite horizon solution is given in [3].

VI. C ONCLUSION

In this work, a new MPC controller was presented that handles un-reachable setpoints better than standard MPC methods. The new con-troller was based on a cost function measuring distance from the un-reachable setpoint, and, for simplicity of exposition, a nite horizonwith a terminal constraint. Nominal asymptotic stability of the optimalsteady state was established for this terminal constraint MPC. Two ex-amples were presented that demonstrate the advantagesof the proposedsetpoint-tracking MPC over the existing target-tracking MPC. Cost im-provements between 50% and 70% were obtained for the two exam-ples. It was next shown how to extend the asymptotic stability result tothe case of terminal penalty MPC by introducing rotated cost.

This new unreachable setpoint approach should also prove useful inapplications where optimization of a system ’s economic performanceis a more desirable goal than simple setpoint tracking.

ACKNOWLEDGMENT

The authors would like to thank D. Q. Mayne and S. J. Wright forhelpful discussion of the ideas contained in this paper.

REFERENCES

[1] S. J. Qin and T. A. Badgwell, “A survey of industrial model predictivecontrol technology, ” Control Eng. Prac , vol. 11, no. 7, pp. 733 –764,2003.

[2] K. R. Muske and J. B. Rawlings, “Model predictive control with linearmodels, ” AIChE J.. , vol. 39, no. 2, pp. 262 –287, 1993.

[3] C. V. Rao and J. B. Rawlings, “Steady states and constraints in modelpredictive control, ” AIChE J. , vol. 45, no. 6, pp. 1266 –1278, 1999.

[4] F. P. Ramsey, “A mathematical theory of saving, ” Econ. J , vol. 38, no.152, pp. 543 –559, Dec. 1928.

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[6] L. W. McKenzie, “Accumulation programs of maximum utility andthe von Neumann facet, ” in Value, Capital, and Growth , J. N. Wolfe,Ed. Chicago, IL: Edinburgh Univ. Press/Aldine, 1968, pp. 353 –383.

[7] L. W. McKenzie, “Turnpike theory, ” Econometrica , vol. 44, no. 5, pp.841 –865, Sep. 1976.[8] W. A. Brock and A. Haurie, “On existence of overtaking optimal tra-

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[9] A. Leizarowitz, “Innite horizon autonomoussystems with unboundedcost,” Appl. Math. Opt , vol. 13, pp. 19 –43, 1985.

[10] D. A. Carlson, A. B. Hauie, and A. Leizarowitz , In nite Horizon Op-timal Control , 2nd ed. Berlin, Germany: Springer –Verlag, 1991.

[11] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert,“Constrained model predictive control: Stability and optimality, ” Automatica , vol. 36, no. 6, pp. 789 –814, 2000.

[12] O. Mangasarian , Nonlinear Programming . Philadelphia, PA: SIAM,1994.

[13] W. W. Hager, “Lipschitz continuity for constrained processes, ” SIAM J. Cont. Opt , vol. 17, no. 3, pp. 321 –338, 1979.

[14] W. Rudin , Principles of Mathematical Analysis , 3rd ed. New York:

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unreachable setpoints in mpc

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