Performance Improvement in MPC with Time-Varying Horizon

via Switching

Sachin Prabhu1 and Koshy George2

Abstract—A crucial design parameter in model pre-dictive control is the choice of the prediction horizon.Procedures to estimate this parameter are off-line andcomputationally intensive. Moreover, a single choiceis perhaps not the best option at all time instants.We resolve this issue in this paper by a time-varyinghorizon achieved by switching between multiple modelpredictive controllers. The stability of such an overallcontroller is discussed and the improvement in per-formance demonstrated.

I. Introduction

Model Predictive Control (MPC) or Receding HorizonControl (RHC) is an on-line finite-horizon optimal con-trol technique. At every time instant k an open-loop op-timization problem, which is initialized with the currentstate, is solved to obtain an optimal control sequence.Typically, the first element of this sequence is the inputto the system at instant k. MPC minimizes tracking errorby predicting system behaviour over a finite time periodin the future based on forward simulation of a systemmodel. This makes it easier to dynamically handle inputand state constraints. However, MPC is highly sensitiveto parameter variations and modelling uncertainties.

The initial ideas of MPC are discussed in the semi-nal papers on dynamic matrix control [1], [2]. Internalmodel control was introduced in [3]–[5] to address therobustness issues of the optimal control of unconstrainedsystems, and generalized predictive control using input-output models in [6] and [7] to overcome the lack ofrobustness in self-tuning regulators. The latter was ex-tended to state-space models in [8]. Conditions for closed-loop stability are derived in [9] and [10]. In-depth reviewsof MPC techniques and associated issues such as theexistence of a solution and the region of validity of thissolution, stability, and robustness feature in [11] and [12].

Though addressing constraints is easier with MPCthan with any of the infinite-horizon optimal controltechniques, MPC provides only an approximation to theglobal optimal solution. The length of the predictionhorizon, which is a crucial design parameter in MPC,affects the accuracy of this approximation — and hencethe performance of MPC — and closed-loop stability of

1Sachin Prabhu is with the Department of TelecommunicationEngineering, PES Institute of Technology (now, PES University),100 Feet Ring Road, BSK 3rd Stage, Bangalore 560085, India.

2Koshy George is with the PES Centre for Intelligent Systemsand with the Department of Telecommunication Engineering, PESInstitute of Technology (now, PES University), 100 Feet Ring Road,BSK 3rd Stage, Bangalore 560085, India. kgeorge@pes.edu

the overall system. Methods to choose this parameteroptimally and its implications on performance and sta-bility are discussed in [13]–[15]. Techniques applicable toinfinite dimensional systems are treated in [16] and [17],and the possibility of reducing the prediction windowsize further with minimal compromise in performancediscussed in [18] and [19]. All these techniques attemptto obtain the best estimate for the optimal predictionwindow size; these are based on off-line computations andare applicable to state-regulation problems. However,they are neither simple in practice nor precise. Even forfinite-dimensional LTI systems, it is possible to arrive atdifferent estimates of this parameter using the methodsuggested in [15]. Thus, choosing on-line an optimalprediction horizon is potentially an open problem.

Tracking a non-zero constant reference can be ad-dressed as a regulation problem by redesigning MPC witha suitable change in coordinates. Redesigning of MPCwhenever desired set-point changes is a tedious processand is impractical for rapidly varying set-points or ar-bitrary references. In order to overcome this problem,reference governors were introduced in [20] and [21] tomodulate the reference signals to obtain artificial refer-ence signals. Alternatively, a new optimization variableto approximate the original reference in an admissibleway is introduced in [22] and [23]. MPC designed forregulation is then used by both methods to ensure thatthe output tracks such admissible artificial referencesignals.

Switching to improve performance is not new in MPC.For instance, switching between robust control and RH-type control is considered in [24]–[26]. The effect ofswitching in a switched multi-objective MPC setup wasanalyzed in [27], where the switching signal is known apriori. These algorithms focus on improving the robust-ness and performance with MPC. A simple method to au-tomatically arrive at a prediction horizon that is optimalat every instant of time by switching between multipleRH-type controllers was proposed in [28] for the regula-tion problem. The potential of such a switching techniqueto improve the transient performance was demonstratedwhereas the stability issue was not addressed there.

In this paper, we extend the idea in [28] to addressthe problem of tracking changing set-points with con-strained LTI systems. This technique is presented inSection II where we vary the prediction horizon online via switching based on a simple decision criterion.This eliminates laborious off-line computational efforts

2014 11th IEEE InternationalConference on Control & Automation (ICCA)June 18-20, 2014. Taichung, Taiwan

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required to choose this parameter. The goal of this paperis to show that this can not only guarantee stability butas well improve the performance. We also demonstratethe prospective ability of this algorithm to address ro-bustness issues in our simulations.

II. Model Predictive Control

Let an LTI system be described as follows:

xk+1 = Axk + Buk

yk = Cxk (1)

where xk ∈ IRn, uk ∈ IRm, and yk ∈ IRp are respectivelythe state, the input and the output at instant k. We saythat the pair (x, u) is admissible if for all time instantsk, xk ∈ χ and uk ∈ Ω, where χ and Ω are closedconvex sets in IRn and IRm respectively, with χ containingthe equilibrium state. A control strategy is said to berecursively feasible if, for any xk ∈ χ, there exists asequence uk ∈ Ω such that xk+1 ∈ χ ∀k. Let Yd be theclass of desired output trajectories such that for everyyd ∈ Yd, there exists an admissible pair (x, u) such thatyk ≡ yd,k. We say a given signal yd is trackable if and onlyif yd ∈ Yd, and u can be generated by some recursivelyfeasible control law.

Consider a constrained optimal control problem wherethe performance index

JNp,Nc=

Np∑

i=1

‖yk+i − yd,k+i‖2Q

+

Nc∑

i=1

‖uk+i−1‖2R (2)

is to be minimized subject to (1), xk+i ∈ χ, 1 ≤ i ≤ Np,and uk+j ∈ Ω, 0 ≤ j ≤ Nc − 1. Here, Np is the timehorizon over which the tracking error is minimized, Nc

is the horizon over which the optimal control sequence iscomputed, and ‖ · ‖Q and ‖ · ‖R are quadratic normswith Q ≥ 0 and R > 0. Clearly, JNp,Nc

is positivedefinite. We emphasize that only trackable signals (yd)are considered; accordingly, the minimization in (2) isover existing recursively feasible control laws. Further,the horizons in (2) are such that Nc ≤ Np in general.

However, for ease of notation, we choose Nc = Np∆= N ,

and consider the following equivalent representation of(2):

JN = ‖Yk+1 − Yd,k+1‖2Q

+ ‖Uk‖2R ,

where YTk+1

∆=

(yk+1 · · · yk+N

), YT

d,k+1∆=(

yd,k+1 · · · yd,k+N

), and UT

k

∆=

(uk · · · uk+N−1

).

The constrained optimal control problem (2) can beformulated as a Quadratic Program (QP):

min JN , (3)

subject to the constraints stated earlier. This approachesthe standard linear quadratic tracking problem as N −→∞. It is well-known that the LQR problem for (1) hasa closed-form solution with the feedback gain a constantmatrix that is used to determine the control input u∗

k. On

the contrary, the QP (3) with N < ∞ results in an opti-mal control sequence U∗

k =(

u∗

k u∗

k+1 · · · u∗

k+N−1

)

at instant k. When the state is measurable at everyinstant, only u∗

k from this sequence is applied at instantk, and other values discarded. If state measurementsare not available at every instant, the computed futurecontrol inputs are used until the instant when the nextmeasurement is made available [29], [30]. Evidently, thisreduces the on-line computational complexity.

A. Prediction Horizon and Stability

The effect of the design parameters Q, R, and N on theclosed-loop stability, performance, and computationalefforts are discussed in [6], [7]. Of these, N is the mostimportant as its effect on computational complexity isexponential in nature. Thus far no method has beenproposed to precisely determine its optimal value N ∗, thesmallest N for which the closed loop system is stable.Off-line techniques that provide estimates N∗ of N∗

have appeared in the past [13]–[17]; clearly, N∗ ≥ N∗.However, these methods are computationally intensiveand the resulting estimates conservative.

The authors in [15] propose a technique to determine a

bound N∗ on N∗ that guarantees closed-loop stability forcontrollable finite-dimensional linear systems. The keyadvantage of this technique is that additional terminalconstraints are not necessary to enforce stability. Indeed,for any N > N∗ the following has been shown:

Theorem 1 ([15]): If I is any compact set in the feasi-

ble region, there exists a natural number N∗ such that forany other N ≥ N∗, the predictive control is stabilizing forany initial condition in I . Moreover, J cN∗

is a Lyapunovfunction.

The proof in [15] requires the prior knowledge of theinfinite horizon cost J∞. This is not available in generaland hence the scheme is not implementable. Therefore,the following computational scheme is proposed in [15]:For different values of N > 1, compute the quantitiesαN = supx

JN+1(x)JN (x) and σN = infx

xT PxJN (x) , for some

positive definite matrix P > 0, and for all x ∈ W .Here, W =

x ∈ IRn : xT Px ≤ µ

is a sub-level set in the

feasible region of the state-space for a prescribed constantvalue µ ≤ J∞. Clearly, for N > N∗, αN ≤ 1 + ε forsome ε > 0, and αN −→ 1 as N −→ ∞. The estimatedprediction window N∗ is the minimum value of N > 1such that α cN∗−1

(1−σ cN∗) < 1. It is to be noted that N∗

not only results in a stabilizing MPC, but also ensuresthat the QP in (3) is recursively feasible. The possibility

of determining an N that is smaller than this N∗ isdiscussed in [18] and [31].

Note that the quantities αN and σN can be computedby solving a couple of non-convex optimization problems.However, there is no guarantee on the global optimality[15]. Alternatively, it is possible to compute these quan-tities at every point in W . A more practical approachis to discretize W at some desired resolution, and to

169

compute αN and σN at these points for each choice of N .Evidently, the computational complexity grows with N ,the resolution, and the size of W . Hence, the scheme isnot readily scalable to higher order systems.

Each trackable fixed set-point corresponds to a steady-state admissible pair (x, u). Evidently, MPC with a fixed

horizon N∗ requires a reference governor after introduc-ing a change of variables: x − x. However, the MPCformulation in (3) is designed to handle explicitly thetracking problem without such a change in variables.A single value for N∗ is perhaps not sufficient in thiscase. This is primarily due to the fact that the techniqueproposed in [15] targets only state-regulation and maynot guarantee good tracking performance. Moreover,such a choice of N∗ may fall short in addressing robust-ness issues. Therefore, exorbitant off-line computationalefforts to compute N∗ is not justified for systems withmodelling uncertainties, or for those which are expectedto track arbitrary references. Varying the predictionhorizon dynamically avoids the need to recompute N∗

for every change in the reference.

B. MPC Based on Multiple-Controllers and Switching

Our goal in this paper is to track changing set-points.As pointed out in Section (II-A), we present a techniquethat automatically and dynamically chooses an appro-priate value of the prediction horizon N via switching.The overall scheme is shown in Fig. 1. It consists of a setof N RH-type controllers Cw, 1 ≤ w ≤ N , operating inparallel, denoted K in Fig. 1. Each Cw solves the QP (3)with the prediction window w in (2). (We note that thisstructure was first proposed in [28] in the context ofstate-regulation.) Clearly, there exists some N ∗ that isstabilizing, and using the procedure outlined earlier it ispossible to arrive at a conservative N∗. By Theorem 1,any N > N∗ is stabilizing. Thus, in our approach,we merely choose a sufficiently large prediction horizon.This naturally reduces the design overhead of explicitlycomputing N∗ a priori. We have the following result:

Theorem 2: For (1), let SN∆= 1, 2, . . . , N, where N

is sufficiently large. Suppose at each time instant k

N ′

k = arg minw∈SN

J (w), (4)

where J (w) ∆=

∥∥∥y(w)k+1 − yd,k+1

∥∥∥2

Q+

∥∥u(w)∥∥2

R, y(w) is

the output of (1) with u(w) as the input, and u(w) isthe solution of QP (3) with N = w. Then, for anytrackable output, the receding-horizon with predictionwindow N ′

k is both recursively feasible and stabilizing.Moreover, the resulting control action results in a lowercost compared to minimizing (2) with fixed predictionhorizon.Proof: Evidently, N sufficiently large implies that N ∗ ≤N∗ ≤ N . By Theorem 1, a receding horizon policy withany N ≥ N∗ is stabilizing. Moreover, JN is a Lyapunovfunction; therefore, JN (k + 1) < JN (k) ∀ k. Further, by

u - Plant -r y

t

t

t

rrr

t

rrr

K

yd

r

r

r

rrr

D

6

Fig. 1. MPC with time-varying horizon.

construction, Ji(k) ≤ JN (k) ∀ i ≤ N and ∀ k. For any k,N ′

k ∈ SN , and N ′ ≤ N . Accordingly,

JN ′

k(k + 1) ≤ JN (k + 1) < JN (k),

clearly preserving the monotonicity. Therefore, JN con-tinues to be the Lyapunov function and stability ofthe closed loop is guaranteed with the proposed MPCwith time-varying prediction horizon. Moreover, when-ever N ′

k+1 = w < N , the RH controller with predictionhorizon w places the pair (xk+1, uk+1) closer to thesteady-state value (x, u) than any other RH controller inK, proving that tracking performance is improved withswitching.

III. Simulation Results

In this section, we demonstrate through simulation theadvantages of using the proposed MPC with time-varyinghorizon and switching for tracking a changing set-point.We consider an unstable second-order system (Exam-ple I) and the Shell benchmark system (Example II) asthe two example systems.

A. An Unstable System

The example problem considered in [15] is as follows:

xk+1 =

(1 1.1

−1.1 1

)xk +

(01

)uk

yk =(

1 2)xk (5)

Using the technique described earlier in Section II, anestimate of the optimal prediction horizon obtained in[15] for this system is N∗ = 4 for the state-regulation

problem. The computation of N∗ requires the choiceof the sub-level set W in the feasible region. Following[15], the feasible region is χ =

x ∈ IR2 : |Cx| ≤ 2

.

Clearly W∆=

x ∈ IR2 : |x(1)| ≤ 0.5, |x(2)| ≤ 0.5

is in

the interior of χ. Discretizing W with a resolution of

170

0.1 results in 100 distinct points where αN and σN arecomputed for different values of N > 1. Improving theresolution to 0.01 results in 10,000 points where suchcomputations are to be carried out. Evidently, there isan exponential growth in the off-line computational effortif better estimates of N∗ are required. On the contrary,as in Theorem 2, we choose N sufficiently high for thetracking problem. The desired and actual output signalswith N∗ = 4 are shown in Fig. 2(a) where the initialcondition is assumed to be xT

0 =(

1 1), and the

weighting matrices in (2) are chosen to be Q = 1 andR = 1. In this figure, the desired output is indicated asa solid line, and the actual output as a dashed line. Theactual and desired outputs are compared in Fig. 2(b)when the proposed time-varying horizon MPC is used.As discussed earlier, choosing a horizon larger than N∗

may not yield appreciably different results. Therefore, forthe time-varying horizon MPC we consider four differentRH-type controllers with horizons 1, 2, 3 and 4. Atevery instant, the admissible controller that yields thelowest one-step ahead cost is chosen. A considerableimprovement in the performance can easily be observedwhen the two outputs are compared. With our proposedstrategy, the output is able to follow better the changes inthe set-point. The benefits of using the proposed strategyis further evident by comparing the energy in the trackingerror as tabulated in Table I(A). There is a reductionof about 29.82% in this energy. This is achieved at ahigher control energy; the two input signals are shownin Fig. 2(c), and the choice of the prediction windowin the proposed multiple-controller based MPC shownin Fig. 2(d). The aforementioned improvement in theperformance became possible when the prediction win-dow size is allowed to be time-varying. Interestingly, theprediction window most often used is N = 2; this showsthat the estimate provided by the technique discussed in[15] is rather conservative.

We repeat the experiment with N∗ = 2 and comparethe tracking performances of conventional MPC and theproposed MPC. The results are respectively shown inFig. 3(a) and (b). The available prediction window sizesare now restricted to 1 and 2. The conventional MPCappears to perform better with N∗ = 2 than with N∗ =4. This further indicates that the estimate of the optimalprediction window obtained using the method proposedin [15] is conservative. Moreover, the performance withan MPC with a time-varying horizon is better than thatof conventional MPC. From Table I(B), a reduction of13.56% in the tracking error energy is observed. Thishas been achieved at lesser control cost, quite unlike thesituation with N∗ = 4. Furthermore, it appears fromTable I(A) and (B) that there is very little differencein the performance between the time-varying MPC withN = 4 and N = 2. This indicates that the proposedstrategy attempts to find the optimal prediction windowat every instant, and choosing a conservative N does not

0 20 40 60 80 100−1

0123

y k

(a)

0 20 40 60 80 100−2

−1

0

e k

(b)0 20 40 60 80 100

−10123

y k

(c)

0 20 40 60 80 100−2

−1

0

e k

(d)

0 20 40 60 80 100−1

0

1

2

u k

Sample (k)

(e)

0 20 40 60 80 1001

2

3

4

N

(f)

Sample (k)

Fig. 2. Example I with bN∗ = 4: (a) Output response and(b) error with conventional MPC. (c) Output response and (d) errorwith time-varying horizon MPC. (e) Input signals. (f) Choice ofprediction window.

affect the performance. However, it must be noted thatthe required computational effort grows with N . FromFig. 3(c) it appears that when the set-point is non-zeroN = 2 is preferred.

We now compare the robustness properties. We assumea 20% parametric uncertainty. The results of this simu-lation experiment with N∗ = 2 are shown in Fig. 3(d)–(f), and tabulated in Table I(C). We again observe asimilar trend: The performance of an MPC with time-varying horizon is better than conventional MPC, witha reduction of 14.09% in the tracking error energy. Acomparison between Table I(B) and (C) shows that thecontrol effort required for this experiment is lesser in thelatter case despite the uncertainty. This is due to the factthat the parametric change in the system matrix for thisexample ensures that the poles move closer to the origin.

B. The Shell Heavy Oil Fractionator Benchmark System

This benchmark problem [32] has large dead-timeswith constraints placed on the magnitudes and instan-taneous changes of the controlled and manipulated vari-ables. The distillation column has three product drawsand three side heat circulating loops. The three loopsremove heat to achieve a desired product separation.Since the energy acquired by the heat exchangers isused to reheat other parts of the overall process, theheat-duty requirements are time-varying. The controlledvariables are Top End Point (TEP), Side End Point(SEP) and the Bottom Reflux Temperature (BRT). Themanipulated variables are Top Draw (TD), Side Draw(SD) and Bottom Reflux Duty (BRD). The unmeasured

171

0 20 40 60 80 100−1

0123

y k(a)

0 20 40 60 80 100−1

0123

y k

(b)

0 20 40 60 80 1001

2(c)

N

Sample (k)

0 20 40 60 80 100−1

0123

y k

(d)

0 20 40 60 80 100−1

0123

y k

(e)

0 20 40 60 80 1001

2(f)

N

Sample (k)

Fig. 3. Response of Example I with bN∗ = 2. (a) ConventionalMPC. (b) Time-varying MPC. (c) Choice of prediction window.Example I with 20% uncertainty: (d) Conventional MPC. (e) Time-varying MPC. (f) Choice of prediction window.

TABLE I

Example I: Comparison of performance.

A: N = 4

Energy Conventional Proposed

‖y‖2

28.3142 5.8352

‖u‖2

237.7930 58.0875

B: N = 2

Energy Conventional Proposed

‖y‖2

26.8391 5.9112

‖u‖2

266.5190 58.0714

C: N = 2 with 20% modeling uncertainty

Energy Conventional Proposed

‖y‖2

29.5577 8.2110

‖u‖2

225.2114 23.6713

disturbances are Inter Reflux Duty (IRD) and UpperReflux Duty (URD). Each transfer function is modelled

as a first order dead-time: Ke−θs

τs+1 , where the units of θ

and τ are in minutes. The values of the parameters K, θ

and τ and the uncertainties are taken from [32].

The continuous-time model in [32] is discretized witha sampling interval of 1 minute and uses a fifth-orderPade approximation for the delays. The dimension ofthe discrete-time state-space model is 100. ComputingN∗ as discussed in Section (II-A) for this system ispractically impossible. In our experience, N = 5 isfound to be sufficient to ensure good performance forthe regulation problem. Although originally proposed asa regulation problem, we consider here a changing set-point for SEP so chosen that it does not violate any

constraints specified in [32] subject to the requirementsthat there are no deviations from the constraints imposedon other variables. Moreover, of the 5 prototype test casesrecommended in [32], we consider here only Cases 1 and3: Case 1 considers addition of the two specified constantdisturbances. No modeling uncertainty is introduced inCase 1. Case 3 is closer to the practical scenario wheremodelling uncertainties and disturbances are present.

SEP tracking for Case 1 with conventional MPC isshown in Fig. 4(a) and that with time-varying MPC inFig. 4(b). (Here, the desired set-point is indicated as asolid line, and the actual output as a dashed line.) Asobserved earlier in Example I, we see an improvement inthe tracking performance with the proposed MPC withtime-varying horizon when compared to conventionalMPC. There are no violations of any other imposed con-straints. The tracking-error energy reduces from 6.1144to 5.6686. The desired and actual outputs for Case 3 withconventional MPC are shown in Fig. 4(c) and with time-varying MPC are shown in Fig. 4(d). The tracking-errorenergy reduces from 9.4389 to 8.5641. Interestingly, thehorizon N = 5 is chosen only for the first few minuteswhen there are larger transients; subsequently, it choosesN = 1. It can easily be observed that in the presenceof uncertainty and disturbances, the response with atime-varying MPC settles down faster and with lessertracking error. Indeed, the tracking-error energies in theperiod from 10 minutes to 90 minutes with the proposedMPC are 0.14 and 0.20 as opposed to 0.19 and 0.97 withconventional MPC for Cases 1 and 3, respectively. Evi-dently, the time-varying MPC makes the overall systemmore robust to disturbances and modelling uncertainties.Further, as mentioned earlier, computing N∗ for this100th order system is nearly impossible. However, bychoosing a sufficiently high upper bound for the hori-zon, and allowing the possibility of optimally choosingthe window every instant permits one to use MPC inpractical applications.

IV. Conclusions

A method to optimally choose at every instant theprediction horizon in a receding-horizon type control isproposed in this paper. This is achieved by switchingbetween multiple such controllers. The overall systemis shown to be stable, and the simulation experimentsshow an improvement in performance when the outputis expected to track changing set-points. Further, theproposed strategy works in the presence of disturbancesand uncertainty.

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0 20 40 60 80

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