Robust MPC of constrained discrete-time nonlinear

systems based on uncertain evolution sets: application to a

CSTR model

D. Limon Marruedo†, J.M. Bravo, T. Alamo and E.F. Camacho 1

† Departamento de Ingenierıa de Sistemas y Automatica, Universidad de Sevilla

Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n 41092. Sevilla

Telephone: +34 954487357, Fax: +34 954487340, email: limon@cartuja.us.es

Abstract

A robust MPC for constrained discrete-time nonlinearsystem with additive uncertainties is presented. TheMPC is based on the so called uncertain evolution sets,which are the sets containing the predicted evolution ofthe uncertain system under any admissible uncertainty.These sets are incorporated in the MPC formulationin order to achieve robust stability. By choosing asterminal constraint a robust positively invariant set, adual-mode controller with guaranteed robust stabilityis obtained. Stability is guaranteed despite suboptimal-ity of the computed solution. This controller is appliedto a highly nonlinear system: a simulation model ofa CSTR for an exothermic irreversible reaction. Theuncertain evolution sets are computed on-line by us-ing interval arithmetic. This procedure obtains quitegood results with a similar computational effort thanthe nominal prediction.

1 Introduction

The Moving Horizon or Model Predictive Control hasbecome a preferred control strategy in academia andin the process industry. The reasons of this success in-clude the constraints handling and the optimal criteriain the computation of the control action. Underlyingcontrol theoretic problems on linear MPC [1] and onnonlinear MPC are well studied. An interesting sur-vey of nonlinear MPC can be found in [8] where anstandard formulation of the MPC is established andsufficient conditions to guarantee asymptotic stabilityare given.

When uncertainties are present, they must be takeninto account in the computation of the control law inorder to get robust stability. Some authors have tack-led this problem. In [9] a dual-mode receding horizoncontroller is proposed and robustness under decaying

1The authors acknowledge MCYT-Spain (contract QUI99-0663-C02-01) for funding this work.

additive uncertainties is achieved by a proper choiceof the terminal region. In [6] a robust MPC strategybased on H∞ cost function is presented and in [11] aclosed-loop min-max technique is shown.

In this paper, a robust MPC for constrained discrete-time nonlinear system with additive uncertainties ispresented. It is based on the uncertain evolution sets.These are the sets which contain the predicted evo-lution of the uncertain system under any admissibleuncertainty. Since the nonlinearity of the model makesthese sets difficult to be accurately obtained, conditionsare established to compute them by using approximateprocedures.

These sets are incorporated in the MPC formulation inorder to consider the uncertainties in the optimal in-put computation. By choosing as terminal constraint arobust positively invariant set, a dual-mode controlleris proposed. If the initial state is such that the op-timization problem is feasible, then robust stability isguaranteed. Hence, the uncertain closed-loop systemreaches the terminal region in a finite number of steps,and the local controller keeps the system in the termi-nal region. This controller does not require optimalityof the solution to guarantee stability. Furthermore, thecommutation to the local controller may be avoided bya modified terminal constraint.

Interval arithmetic is used for the computation of theuncertain evolution set. This procedure is very usefulfor the on-line implementation of the proposed con-troller, since the computational effort is similar to thenominal prediction. Furthermore, quite good resultsare obtained since the method provides local approxi-mations to the uncertain set.

This controller is applied to a highly nonlinear system:a simulation model of a CSTR for an exothermic re-action. The coolant temperature is the input which isconstrained. Simulation results of the proposed MPCare shown.

The paper is organized as follows: In section 2, somepreliminary results are established, uncertain evolu-tion sets are presented and some basic results in in-terval arithmetic are given. In section 3, the RobustMPC strategy is presented, and in the following sec-tion, closed loop stability is proved. The application ofthe proposed controller to a CSTR is shown in section5, and finally some conclusions are given.

2 Preliminary results

2.1 System descriptionConsider an uncertain nonlinear discrete-time systemgiven by

xk+1 = f(xk, uk) + wk (1)

where xk ∈ IRn is the state of the system and uk ∈ IRm

is the control vector at sample time k. The system issubject to constraints on the state xk ∈ X and on thecontrol action uk ∈ U , where X is a closed set and U acompact set, both of them containing the origin. Thevector wk ∈ IRn is the uncertainty which is modeledas additive. The only assumption on wk is that it isbounded in a compact set W that contains the origin.

wk ∈W (2)

Notice that the additive uncertainty can model per-turbed systems and a wide class of model mismatchestaking into account that these ones might depend onthe state of the system, since

xk+1 = f(xk, uk) = f(xk, uk) + ∆f(xk, uk)︸ ︷︷ ︸

wk

where wk ∈ W forall xk ∈ X, uk ∈ U . The only as-sumption on W is that it is bounded. This kind ofmodel uncertainty has been used in previous papersabout robustness in MPC, as in [9] and [7].

The model given by xk+1 = f(xk, uk) denotes the nom-inal model of the system. For the real state of the sys-tem at sample time k, xk and for a given sequenceof control inputs denoted uF (k) = u(k|k), u(k +1|k), · · · , u(k + N − 1|k), the future state of the sys-tem at time k + j predicted by using the nominalmodel is denoted x(k + j|k). Hence, x(k + j + 1|k) =f(x(k + j|k), u(k + j|k)), where x(k|k) = xk.

2.2 Uncertain evolution setsSince there are mismatches between the real systemand the nominal model, the predicted evolution usingthe nominal model differs from the real evolution of thesystem. In order to consider this effect in the controllersynthesis, it is interesting to compute the region around

the nominal prediction that confines the state of thesystem under any admissible uncertainty.

This idea is the basis of the so-called uncertain evo-lution sets. Consider that the state of the system atsample time k is xk and a sequence of control inputsuF (k) is applied to the uncertain system. The evolu-tion of the system depends on the uncertainties, thatare known to belong to the bounded set W . The un-certain evolution set at sample time k+ j is denoted asXj(xk, uF (k)). This set is the region that confines theevolution of the uncertain system under any admissi-ble uncertainty at sample time k + j. Note that thisset depends on xk, on the sequence of inputs from k tok + j − 1, i.e. u(k|k), · · · , u(k + j − 1|k) and on theset of uncertainties W .

Since the exact computation of these sets is difficultfor non-linear systems, approximations to this sets areused. This approximate sets are easily computed usingdifferent approaches: based on the Lipschitz continu-ity of the system or based on interval arithmetic forinstance. Interval arithmetic has been used in this pa-per.

Hereafter some definitions and results related to theuncertain evolution sets are presented.

Consider sets A and B ⊂ IRn , a vector x ∈ IRn and afunction g(x) : IRn → IRn then the following sets aredefined: x+A = x+a, a ∈ A, g(A) = g(a), a ∈ A,A + B = a + b, a ∈ A, b ∈ B, and A ∼ B = c ∈IRn : c+B ⊆ A.

Definition 1 Consider a system given by (1), considerthat the state at sample time k is xk and consider a se-quence of control inputs uF (k) then the uncertain evo-lution set at sample time k + j is

Xj(xk, uF (k)) = f(Xj−1(xk, uF (k)), u(k+j−1|k))+W(3)

where X1(xk, uF (k)) = f(xk, u(k|k)) +W .

Note that Xj(xk, uF (k)) is the set that contains theuncertain evolution of all the states of Xj−1(xk, uF (k)),that is

Xj(xk, uF (k)) =⋃

x∈Xj−1

f(x, u(k + j − 1|k)) +W

Due to the nonlinear nature of the model, these setsare very difficult to be computed and thus, they are notuseful from a practical point of view. In order to re-duce the complexity of the computation, these sets canbe substituted by approximated approaches. These ap-proaches are based on conservative procedures with lesscomputational burden. The aim of these procedures is

to compute an approximation of f(A, u) for a given setA ⊆ IRn and a given control action u. In the sequelψ(A, u) denotes the approximate set of f(A, u). Theprocedure to compute approximate sets must satisfythe following conditions:

Assumption 1 Let A be a set A ⊆ IRn and let u bea control action. Consider a procedure ψ(·, u) : A ⊂IRn → ψ(A, u) ⊂ IRn, then ψ(·, ·) can be used to com-pute uncertain evolution sets if it satisfies the followingconditions

• Inclusion condition: f(A, u) ⊆ ψ(A, u).

• Monotonic condition: Let B a set such that B ⊆A, then ψ(B, u) ⊆ ψ(A, u).

Based on this procedure it is possible to compute asequence of sets which are outer approximations tothe exact sets of the uncertain evolution of the sys-tem. These approximate sets are approximate uncer-tain evolution sets. With slight abuse of notation, theseapproximate sets are merely called uncertain evolutionsets in this paper.

Definition 2 (Uncertain evolution set) Considera system given by (1), consider a procedure ψ(·, ·)that satisfies assumption 1 for the system, considerthe state at sample time k is xk and a sequence ofcontrol inputs uF (k) then the uncertain evolution setat sample time k + j is

Xj(xk, uF (k)) = ψ(Xj−1(xk, uF (k)), u(k+j−1|k))+W(4)

where X1(xk, uF ) = f(xk, u(k|k)) +W .

Due to the inclusion condition, the uncertain evolutionset Xj(xk, uF (k)) contains all the states where the un-certain system may be at sample time k + j from xk

and applying the sequence of inputs uF (k). That is

Xj(xk, uF (k)) ⊆ Xj(xk, uF (k)) (5)

Another interesting property of these sets is stated inthe following lemma.

Lemma 1 Consider that the state of the systemat time k is xk, consider a sequence of controlinputs uF (k), consider the uncertain evolution setXj(xk, uF (k)). Consider xk+1 the state where the sys-tem evolves at k+1 when at k the input u(k|k) is appliedto the system. Consider a sequence of inputs at k + 1,uF (k+1), such that u(k+ j|k+1) = u(k+ j|k) for allj = 1, · · · , N−1, then the predicted uncertain evolutionset at k + 1 satisfies the condition

Xj−1(xk+1, uF (k + 1)) ⊆ Xj(xk, uF (k)) (6)

2.3 Interval arithmeticIn order to implement the computation of the uncertainevolution sets, it is necessary to find procedures thatsatisfies assumption 1. A procedure based on intervalarithmetic is used in this paper.

Interval mathematics is a generalization of real math-ematics in which intervals numbers replace real num-bers, interval arithmetic replaces real arithmetic, andinterval analysis replaces real analysis [10]. An intervalnumber X = [a, b] is the set of real numbers such thatx : a ≤ x ≤ b. If the dimension of the variable isn, then it is extended to interval vectors, where eachcomponent is an interval variable. Note that an inter-val vector X is a set in IRn. The set of real compactintervals [a, b], a, b ∈ IR is denoted by I, and the setsof interval vectors in IRn is denoted by In.

The four basic interval operations [10] are given by

[a, b] + [c, d] = [a+ c, b+ d][a, b]− [c, d] = [a− d, b− c][a, b]× [c, d] = [min(a·c, a·d, b·c, b·d),

max(a·c, a·d, b·c, b·d)][a, b] / [c, d] = [a, b]×

[1d, 1

c

]if 0 /∈ [c, d]

(7)

The ranges of the four elementary interval arithmeticoperations are exactly the ranges of the correspondingreal operations. Extension of the interval arithmetic toinclude 0 in division can be found in [2].

Consider a function g : IRn → IRm and consider aninterval vector X ∈ In, then the set g(X) denotes therange of g(·) over the interval X. Note that it is not aninterval vector in general. Computing the exact rangeof an arbitrary function g(·) over an interval vector Xis a hard problem. However, interval arithmetic can beused to obtain interval bounds of the exact range g(X).

Definition 3 (Inclusion function) A function G :IRn → IRm is called an inclusion function for g(·) ifg(X) ⊆ G(X) for any X of In.

Definition 4 (Inclusion monotonic function)The inclusion function G(·) is inclusion monotonic ifand only if for any X,Y ∈ I such that X ⊆ Y thenG(X) ⊆ G(Y ).

Definition 5 (Natural interval extension [3]) Ifg : IRn → IRm is a function computable as an expres-sion, algorithm or computer program involving the fourelementary operations interspersed with evaluations ofstandard functions, then a natural interval extensionof g(·) is obtained by replacing each occurrence of eachcomponent xi of x by the corresponding interval Xi

of X, by executing all operations according to formu-las (7) and by computing exact ranges of standardfunctions.

Theorem 1 [3] Natural interval extensions are in-clusion monotonic functions, i.e. for any X ∈ In,g(X) ⊆ ψ(X) and for any Y ⊆ X, ψ(X) ⊆ ψ(Y ).

Let ψ(X,u) be a natural interval extension of themodel f(x, u), considering the inputs u as a param-eter . Hence, it can be used as inclusion functions.From theorem 1, we get that for any X, Y ∈ In suchthat X ⊆ Y and for any u, f(X,u) ⊆ ψ(X,u), andψ(X,u) ⊆ ψ(Y, u). Therefore this procedure satisfiesassumption 1 and it can be used to compute uncertainevolution sets. Note that the set of uncertainties Wmust be an interval vector, since the set ψ(X,u) +Wmust be an interval vector in order to compute the fol-lowing set in (4). This is a mild condition since aninterval vector which contains W can be used for thecomputation of the uncertain evolution sets. The onlyconsequence is that the obtained sets might be over-conservative.

3 Robust MPC strategy

Model predictive control is a well established opti-mal control strategy which considers constraints on thestates and on the control actions [8]. The control lawKMPC(xk) is obtained solving a constrained optimiza-tion problem and applying the optimal control actionto the system in a receding horizon manner. Considerthe finite horizon MPC optimization problem stated asfollows

minuF (k)

JN (xk, uF (k))

subject to:

x(k + j|k) ∈ X ∀j = 1, · · · , N

u(k + j|k) ∈ U ∀j = 0, · · · , N − 1

x(k +N |k) ∈ Ω

where

JN (xk, uF (k)) =

i=N−1∑

i=0

L(x(k + i|k), u(k + i|k))

+V (x(k +N |k))

and the vector of decision variables uF (k) =u(k|k), u(k+1|k), · · · , u(k+N −1|k) denotes the fu-ture sequence of control inputs of the system along theprediction horizon N and x(k + i|k) is the predictednominal state of the system applying uF (k). Noticethat the MPC includes a terminal cost V (·) in the cost

function and a terminal constraint given by the regionΩ.

Taking into account that the optimal minimizer u∗F (xk)only depends on xk, and the receding horizon policy,the control law is given by uk = KMPC(xk) = u∗(k|k).In absence of uncertainties, this control law asymptot-ically stabilizes the system under some assumptions onthe terminal cost and the terminal region [8]. More-over, the optimal cost function J∗N (xk) is a Lyapunovfunction of the closed loop system. The domain of at-traction of the controller XN is the set where the opti-mization problem is feasible.

If the system is uncertain, then the stability, and prob-ably, the feasibility of the nominal MPC may be lost.Due to the receding horizon policy, the feedback loopprovides some degree of robustness to the closed loopsystem [4, 12]. In order to achieve robustness, the con-troller must stabilize the system for all possible realiza-tions of the uncertainty along the prediction horizon.Different robust MPC techniques have been proposed:some of them are based on a nominal prediction as in[9], others are based on H∞ [6] or on the worst possibleuncertainty as in min-max MPC [11].

In this paper a robust dual-mode MPC is proposed. Itis based on the computation on the uncertain evolutionsets shown in section 2.2. These sets allow that all pos-sible realizations of the uncertainty to be considered inthe computation of the MPC control law. The systemmust satisfy the following assumption.

Assumption 2 There is a region Ω ⊆ X with an as-sociated control law u = h(x) such that Ω is a ro-bust positively invariant set for the uncertain systemi.e. ∀x ∈ Ω, f(x, h(x)) + w ∈ Ω, ∀w ∈ W andu = h(x) ∈ U for all x ∈ Ω.

The optimization problem to be solved at time k is:

Robust dual-mode MPC optimization problem(P d

k (xk))minuF (k)

JN−k(xk, uF (k))

subject to:

Xj(xk, uF (k)) ⊆ X ∀j = 1, · · · , N − k (8)

u(k + j|k) ∈ U ∀j = 0, · · · , N − k − 1(9)

XN−k(xk, uF (k)) ⊆ Ω (10)

Note that the prediction horizon is reduced at eachsample time. Therefore, this optimization problem isonly defined for k = 0 to k = N − 1. In next sectionit is proved that xN ∈ Ω, and, hence, the local controllaw u = h(x) makes the system to remain in Ω.

The robust dual-mode control law is given by

KdMPC(xk) =

u∗(k|k) if xk /∈ Ωh(xk) if xk ∈ Ω

For k ≥ 1 it is possible to find a feasible solution(uF (k)) of the optimization problem in k, based on theoptimal solution in k−1 (u∗F (k−1)). It is the sequenceof N − k inputs, uF (k), such that

u(k+j−1|k) = u∗(k+j−1|k−1) for j = 1, ..., N−k

The approach proposed in here is different to the onepresented in [9]: the uncertain evolution sets are in-corporated in the optimization problem and then, theeffect of uncertainty is considered in the solution of theoptimization problem. It is not necessary to use a moreconservative terminal region and the effect of the un-certainty is considered for all the prediction horizon.Furthermore, the constraints on the states are consid-ered in a simpler way. The procedure used for the com-putation of the uncertain evolution sets are local, andhence, less conservative than global approaches as, forinstance, the ones based on a global Lipschitz constantused in [9].

It is worthy of remark that the proposed controller, asthe one proposed in [9], is over-conservative since bothof them are based in open-loop predictions of the sys-tem. Closed-loop formulation are obtained by posingthe MPC problem in terms of a sequence of controllaws. These approaches are not conservative but theirimplementation is quite difficult.

4 Stability analysis

Since the uncertainties are merely bounded and theymay not be decaying, the origin is not an steady stateof the uncertain system. Therefore, it is necessary thenext definition of stability:

Definition 6 A system is asymptotically ultimatelybounded if the system evolves asymptotically to abounded set, i.e. there exist positive constants b andc such that for every α ∈ (0, c), there is a k∗ such thatfor all ‖x0‖ ≤ α then ‖xk‖ ≤ b, ∀k > k∗.

Hence, the aim of an stabilizing controller is to steer thestate to a neighborhood of the origin and keep the stateevolution in it. This set is a robust positively invariantset for the closed loop system and its size depends onthe bound on the uncertainties. Therefore, the systemis ultimately bounded to this set. The controller pro-posed in this paper steers the uncertain system to theterminal region, which is a robust invariant set. Hence,the closed loop system is ultimately bounded.

Theorem 2 (Robust stability) Consider a systemgiven by (1) with additive uncertainties subject to (2)and with constraints on the states xk ∈ X and on theinputs uk ∈ U . Consider a robust invariant set for thesystem Ω with an associated local controller u = h(x)such that both satisfy assumption 2. Consider that it isavailable a procedure to compute the uncertain evolu-tion sets, then the system controlled by u = Kd

MPC(xk)is ultimately bounded for all x0 such that the optimiza-tion problem P d

0 (x0) is feasible.

Stability is guaranteed by the feasibility of the com-puted control action at each sample time. Hence, op-timality is not required and a suboptimal solution ofthe optimization problem suffices to guarantee stabil-ity. Furthermore, the cost function only affects theperformance but not the stability of the closed-loopsystem.

5 Application to a CSTR model

As illustrative example of the robust MPC controllerproposed the technique is applied to a highly nonlin-ear system: a continuous stirred tank reactor (CSTR)simulation model. The continuous time model of anCSTR for an exothermic, irreversible reaction A → Bwith constant liquid volume is given by [5]:

dCA

dt=

q

V·(CAf − CA) − k0· exp

(

−

E

R·T

)

·CA

d T

dt=

q

V·(Tf − T ) −

∆H·k0

ρ·Cp

· exp(

−

E

R·T

)

·CA +

+U ·A

V ·ρ·Cp

·(Tc − T )

where CA is the concentration of A in the reactor, Tis the reactor temperature and Tc is the temperatureof the coolant stream. The parameters of the modelare: ρ = 1000 g/l, Cp = 0.239 J/g K, ∆H = −5× 104

J/mol, E/R = 8750 K, k0 = 7.2× 1010 min−1, U ·A =5× 104J/min K. The nominal operating conditions aregiven by: q = 100 l/min, Tf = 350 K, V = 100 l,CAf = 1.0 mol/l. The steady state is Co

A = 0.5 mol/l,T o = 350 K, T o

c = 300 K. The temperature of thecoolant is constrained to 280K ≤ Tc ≤ 370. The stateof the system is defined as x = [CA − Co

A, T − T o]T ,and the input as u = Tc − T

oc .

The model is discretized with a sampling period Ts =0.03 min. The additive uncertainty on the discrete-time model of the system are bounded by w1 ∈[−3·10−3, 3·10−3

]and w2 ∈ [−0.15, 0.15], where w =

[w1, w2]T . The different scales of w1 and w2 are due to

the different scales of the corresponding states variables(concentration and temperature). The robust posi-tively invariant set is Ω = x ∈ IR2 : xT ·P ·x ≤ 0.6226

for the system controlled by uk = K·xk, where

P =

[54.3945 0.59280.5928 0.0351

]

K = [−91.3362− 4.7002]

Figure 1: Uncertain evolution sets at x0.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15−20

−15

−10

−5

0

5

x1

x 2

Figure 2: Closed loop state portrait.

In Fig.1 the solution of P d0 (x0) for a couple of feasi-

ble initial states with a prediction horizon of N = 5 isshown. The uncertain evolution sets are also depicted.It can be seen that for all possible realization of theuncertainties, the system evolution is bounded by theuncertain evolution sets. These sets have been obtainedusing interval arithmetic. Note that the optimal solu-tion found is such that X5 ⊆ Ω.

In Fig.2 closed loop trajectories of the uncertain systemare shown. For all the initial states, the optimizationproblem is feasible and hence, the system is steeredto Ω despite the uncertainties. It is worthy of remarkthat the system to be controlled is highly nonlinear,and quite difficult to control in some regions due tothe bounds on the input. Furthermore, the effect ofthe uncertainties is critical. This effect and the open-loop nature of the MPC controller makes the controllerover-conservative. The main contribution is that it ispossible to guarantee robust stability if the optimiza-tion problem is feasible at the initial state.

6 Conclusions

In this paper, a robust dual-mode MPC controller forconstrained discrete-time nonlinear systems with addi-tive uncertainties is presented. It is based on the ad-dition of the uncertain evolution sets to the MPC op-timization problem and considering a robust positivelyinvariant set as terminal region. The main contributionof the proposed MPC is that there is not an explicituncertainty bound to guarantee robust stability. It isguaranteed under feasibility of the optimization prob-lem. Furthermore, suboptimal solution of the problemalso guarantees stability.

The paper has also shown that interval arithmetic isan appropriate technique for the on-line computationof the uncertain evolution sets, because of the smallcomputational burden required. Using this technique,the proposed dual-mode controller has been applied toa simulation model of a CSTR. The highly nonlinearmodel of the system and the bounded inputs make thesystem difficult to be controlled.

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