1

Stochastic Tubes in Model Predictive Control withProbabilistic Constraints

Mark Cannon, Basil Kouvaritakis, Sasa V. Rakovic and Qifeng Cheng

Abstract—Stochastic Model Predictive Control (MPC) strate-gies can provide guarantees of stability and constraint satis-faction, but their online computation can be formidable. Thisdifficulty is avoided in the current paper through the use oftubes of fixed cross section and variable scaling. A modeldescribing the evolution of predicted tube scalings facilitatesthe computation of stochastic tubes; furthermore this procedurecan be performed offline. The resulting MPC scheme has a lowonline computational load even for long prediction horizons,thus allowing for performance improvements. The efficacy ofthe approach is illustrated by numerical examples.Keywords: constrained control; stochastic systems; probabilisticconstraints; model predictive control.

I. INTRODUCTION

A common framework for robust model predictive control(MPC) is based on the assumption that model uncertainty(whether multiplicative or additive) can be described in termsof bounded compact sets (e.g. [13], [22]) without any spe-cific reference to information that may be available on theprobability distribution of the uncertainty. Yet this information(if available) has a key role to play in the treatment of softconstraints of a probabilistic nature. Such constraints can behandled through the use of second order cone inequalities (seefor example [20], [8], both of which deal with the case ofopen loop stable systems with finite impulse responses), orconfidence ellipsoids (e.g. [21], [15]), or through estimatesof the conditional mean of the state [23], or a transformationthat leads to conditions involving standard multivariate normaldistributions [12], or by multivariate integration [1].

However none of these approaches considered the issuesof recursive feasibility or stability of the MPC strategy inclosed loop operation. Recent work [4], [3], [5] consideredconstraints on the average probability of constraint violationover a given horizon. This involved minimizing a performanceindex subject to constraints that restrict the predicted plantstate either to nested ellipsoidal sets ([4]) or to nested layeredtubes with variable polytopic cross sections ([3], [5]). Byconstraining the probability of transition between tubes andthe probability of constraint violation within each tube, thisstrategy ensures satisfaction of soft (also hard) constraints aswell as recursive feasibility with respect to these constraints.

Tubes have been proposed for different MPC problemformulations: [10], [14] consider linear dynamics with additivebounded uncertainty, whereas [19] and [11] address nonlinearsystems with and without additive uncertainty, respectively.The main drawback of [3], [5] is that the constraints ontransitional and violation probabilities were invoked usingconfidence polytopes for model uncertainty, and this results inlarge numbers of variables and linear inequalities in the MPCoptimization, which can limit application to low-dimensionalsystems with short horizons even if the number of tube layers

is small. Moreover, for small numbers of tube layers thehandling of probabilistic constraints becomes conservative.

The current paper uses tubes with fixed cross sections (forconvenience ellipsoids are used, though more general formsare possible), but the scalings and centres of these crosssections are allowed to vary with time. As in the deterministiccase [18], the evolution of the tubes is described by a scalardynamic system, which implies a significant reduction in thenumber of optimization variables. In this context the dynamicsgoverning the tube scalings are stochastic, and the computationof the predicted distributions of tube scalings is facilitatedusing a process of discretization. The distributions enablebounds to be imposed on the probability of violation ofstate constraints; these are computed offline and are invokedonline by tightening the constraints on the predictions of anominal model. The offline computation of distributions forthe stochastic variables allows many discretized levels to beused, thus allowing for the implicit use of many layered tubeswithout any concomitant increase in the online computation.Unlike [21] and [23], which did not guarantee closed loopfeasibility, the current paper ensures feasibility in a recursivemanner. This means that, given feasibility at initial time,the proposed approach ensures that the online constrainedoptimization problem is feasible at the next time instant andtherefore remains feasible at all subsequent times.

II. PROBLEM DEFINITION

Consider the linear time-invariant model with statexk ∈ Rnx , control input uk, and disturbance input wk,

xk+1 = Axk +Buuk +Bwwk . (1)

Here wk, for k = 0, 1, . . . , are zero-mean, independent andidentically distributed (i.i.d.) random variables, and it is knownthat wk ∈ E(W,αk) where E(W,αk) is an ellipsoidal set:

E(W,αk) = w : wTWw ≤ αk, W = WT 0. (2)

Here αk ≥ 0, k = 0, 1, . . ., are i.i.d. random variables. Thedistribution function Fα(a) = Pr(αk ≤ a) is assumed to beknown, either in closed form or by numerically integrating thedensity of w, and we make the following assumptions on α.Assumption 1. Fα is continuously differentiable and α∈ [0, α].

The system is subject to soft constraints of the formPr(gTj xk ≤ hj) ≥ pj , j = 1, . . . nc, for given gj ∈ Rnx ,hj ∈ R and given probabilities pj > 0.5. General linearconstraints on states and inputs can be handled using thepaper’s framework, and hard constraints can be included asa special case of soft constraints invoked with probability 1(w.p. 1). To simplify presentation, the paper’s approach isdeveloped for the case of a single soft constraint (nc = 1),

Pr(gTxk ≤ h) ≥ p, p > 0.5, (3)

2

and the case of nc > 1 can be treated simply by taking theintersection of nc constraint sets. The control problem is tominimize, at each time k, the expected quadratic cost

Jk =

∞∑i=0

Ek(xTk+iQxk+i + uTk+iRuk+i

)(4)

subject to Pr(gTxk+i ≤ h) ≥ p for all i > 0, while ensuringclosed loop stability (in a suitable sense) and convergence ofxk to a neighbourhood of the origin as k →∞.

We decompose the state and input of the model (1) into

xk = zk + ek, uk = Kxk + ck, (5)

so that zk and ek evolve according to

zk+1 = Φzk +Buck (6)ek+1 = Φek +Bwwk (7)

where Φ = A + BuK is assumed to be strictly stable andck+i, i = 0, 1, . . . , N − 1 are the free variables in areceding horizon optimization at time k. This allows the effectof disturbances on the i-step-ahead predicted state, xk+i, to beconsidered separately (via ek+i) from the nominal prediction,zk+i, and thus simplifies the handling of constraints.

III. UNCERTAIN PREDICTIONS

The uncertainty in the i-step-ahead prediction ek+i can becharacterized using (2) and (7) in terms of the scalings βi ofellipsoidal sets containing ek+i, denoted ek+i ∈ E(V, βi)

E(V, βi) = e : eTV e ≤ βi, V = V T 0. (8)

This section constructs a dynamic system to define the randomsequence βi, i = 1, 2, . . . and proposes a method of approx-imating numerically the distribution functions of βi, i ≥ 1.

Given ek+i ∈ E(V, βi) and wk+i ∈ E(W,α) we haveek+i+1 ∈ E(V, βi+1) if and only if

maxe∈E(V,βi)w∈E(W,α)

(Φe+Bww)TV (Φe+Bww) ≤ βi+1. (9)

The problem of computing the minimum βi+1 satisfying (9) isNP-complete, but sufficient conditions for (9) are as follows.

Lemma 1. If βi , βi+1 and V satisfy

βi+1 = λβi + αi (10)

V −1 − 1

λΦV −1ΦT −BwW−1BTw 0 (11)

for some λ > 0, then ek+i+1 ∈ E(V, βi+1) whenever ek+i ∈E(V, βi) and wk+i ∈ E(W,αi).

Proof: Using the S-procedure [2], sufficient conditionsfor (9) are obtained as[

ΦT

BTw

]V[Φ Bw

] λ

[I0

]V[I 0

]+ µ

[0I

]W[0 I

]and βi+1 ≥ λβi + µαi for some λ > 0 and µ > 0. Howeverµ can be removed from these inequalities by scaling βi, βi+1

and V , and conditions (10) and (11) follow directly.The distribution of βi for any i > 0 can be determined from

the distributions of α and β0 using (10). In the sequel λ,Vin (10) (11) are taken to be constants independent of αi,βi (aprocedure for optimizing the values of λ and V is described insection IV). We further assume that the distribution function,Fβ0

, of β0 is known and has the following properties.

Assumption 2. Fβ0 is right-continuous with only a finitenumber of discontinuities and β0∈ [0, β0].Note that Assumption 2 allows for fixed, deterministic β0 (e.g.β0 = 0 corresponds to Fβ0

(x) = 1 for all x ≥ 0). Thefollowing result shows that Fβi

is well-defined if λ ∈ (0, 1).

Theorem 2. If 0 < λ < 1, then:(i) βi is bounded for all i: βi ∈ [0, βi], where βi+1 = λβi+

α, and βi ≤ β = maxβ0,1

1−λ α.(ii) Fβi is continuously differentiable for all i ≥ 1

(iii) βi converges in distribution to the random variableβL =

∑∞k=0 λ

kαk as i→∞.

Proof: Assumption 1 and (10) together imply that βi ∈[0, βi] with βi+1 = λβi + α. Hence if λ ∈ (0, 1), then βi ismonotonic and βi → 1

1−λ α as i→∞, which proves (i).The distribution of βi+1 is given for any i ≥ 0 by the

convolution integral (see e.g. [6]):

Fβi+1(x) =

1

λ

∫ β

0

Fα(x− y) fβi(y/λ) dy (12)

where fβidenotes the density of βi. It follows from Assump-

tion 1 and the dominated convergence principle that Fβi+1is

continuously differentiable, as claimed in (ii).To prove (iii), let β′i denote the random variable β′i = γi +

λiβ0 for i = 0, 1, . . . where

γi+1 = γi + λiαi, γ0 = 0. (13)

Then β′i =∑i−1k=0 λ

kαk + λiβ0, and since (10) gives βi =∑i−1k=0 λ

kαi−1−k +λiβ0 where αi is i.i.d., the distributionsof βi and β′i are identical for all i ≥ 0. Furthermore the boundson αi and β0 in Assumptions 1 and 2 imply that for everyε > 0 there exists n such that

Pr(|β′j − β′i| < ε) = 1 ∀ i > n, j > i

and it follows that β′i converges a.e. to a limit as i→∞ [17].From the definition of βL, we therefore have β′i → βL a.e.,and hence βi converges in distribution to βL [17].

The simple form of (10) facilitates the approximate com-putation of the distribution of βi. Consider for example anumerical integration scheme based on a set of points xj , j =0, 1, . . . , ρ in the interval [0, β] with

0 = x0 < x1 < · · · < xρ = β. (14)

Let πi,j be an approximation to Fβi(x) in the interval x ∈

[xj , xj+1) for j = 0, . . . , ρ − 1, and let πi,ρ = 1. Then sincethe convolution in (12) can be written equivalently as

Fβi+1(x) = λ

∫ β

0

Fβi(y) fα(x− λy) dy , (15)

a generic quadrature method enables the vectors πi =[πi,0 · · · πi,ρ

]Tto be computed for i = 1, . . . , N by setting

π0,j = Fβ0(xj) for j = 0, . . . , ρ and using the recursion

πi+1 = Pπi (16)

for i = 0, 1, . . . , N − 1, where the (ρ+ 1)× (ρ+ 1) elementsof the matrix P are determined by the particular numericalintegration scheme employed and the density, fα, of α.

If the data points xj satisfy |xj+1 − xj | ≤ δ for j =0, . . . , ρ − 1, then the approximation error can be madearbitrarily small if δ is sufficiently small, as we show below.

3

We denote Fπi,δ as the piecewise constant function withFπi,δ(x) = πi,j for x ∈ [xj , xj+1), j = 0, . . . , ρ− 1.

Lemma 3. For any finite horizon N we have Fπi,δ → Fβi asδ → 0 for i = 1, . . . , N . Also FπL,δ → FβL

as δ → 0, whereπL is the eigenvector of P associated with eigenvalue 1.

Proof: By Assumptions 1-2 and Theorem 2, the integrandin (15) is piecewise continuous for i = 0 and continuous fori > 0. It can therefore be shown that, as δ → 0

maxx∈[0,β]

|Fπi,δ(x)− Fβi(x)| = O(mδ + iκδν

), i = 1, . . . , N

(17)where m denotes the number of discontinuities in Fβ0

andκ, ν ≥ 1 are constants dictated by the numerical integrationscheme employed. This implies Fπi,δ → Fβi as δ → 0 sincem and N are finite. Combining the error bounds used toderive (17) with πi+1 = Pπi, we have, as δ → 0

Fπi+1,δ(x) = λ

∫ β

0

Fπi,δ(y) fα(x− λy) dy +O(κδν) ,

and since πL satisfies πL = PπL, it follows that, as δ → 0

FπL,δ(x) = λ

∫ β

0

FπL,δ(y) fα(x− λy) dy +O(κδν) . (18)

But by Theorem 2, FβLis the unique solution of

FβL(x) = λ

∫ β

0

FβL(y) fα(x− λy) dy .

Therefore (18) implies FπL,δ → FβLas δ → 0.

Lemma 3 shows that the eigenvector πL provides an ap-proximation to the steady state distribution of (10) despitethe linear growth of the error bound in (17) with N . In thesequel we assume that δ is chosen sufficiently small that theapproximation errors associated with πi for i = 0, . . . , N andπL may be neglected.

Remark 1. The matrix T−1PT , where T is a lower-triangularmatrix of 1’s, is the transition matrix associated with a Markovchain. The (j, k)th element of T−1PT gives the probabilityof xj−1 ≤ βi+1 < xj given xk−1 ≤ βi < xk. The elementsof T−1PT are therefore non-negative and each column sumsto 1, so P necessarily [16] has an eigenvalue equal to 1.

Remark 2. The definitions of P and π0 ensure that each πigenerated by (16) belongs to the set S = π ∈ Rρ+1 : 0 ≤π0 ≤ · · · ≤ πρ+1 = 1.

We next show how the sequence πi can be used toconstruct ellipsoidal sets that contain the predicted state of (7)with a specified level of confidence. For a given sequence xjsatisfying (14), and for any π ∈ S and p ∈ [0, 1], let ind(π, p)and b(π, p) denote the functions that extract respectively theindex and the value of xj corresponding to a confidence level p

ind(π, p) = minj : πj ≥ p, (19a)b(π, p) = xj , j = ind(π, p). (19b)

Theorem 4. The i-step-ahead prediction ek+i satisfies ek+i ∈E(V, b(πi, p)

)with probability p.

Proof: This follows from ek+i ∈ E(V, βi), since, if π0

is determined using the known distribution for β0, then byconstruction βi ≤ b(πi, p) holds with probability p.

Theorem 4 implies that the predicted values of ek+i lieinside tubes with ellipsoidal cross sections (defined by V andb(πi, p)) with probability p; hence we refer to these tubes asstochastic tubes. Although the stochastic tubes apply to ek+i,they can be referred to xk+i through the use of (5), whichtranslates their centres to the nominal state prediction zk+i.

IV. MPC STRATEGY

To avoid an infinite dimensional optimization problem, weadopt the usual dual prediction mode MPC paradigm [13]: attime k, mode 1 consists of the first N steps of the predictionhorizon over which the control moves ck+i in (6-7) are free,and mode 2 comprises the remainder of the horizon, withck+i = 0 for all i ≥ N . Because of the additive uncertaintyappearing in (1), the quadratic cost (4) is unbounded. Hencethe modified cost of [3] is used:

Jk =

∞∑i=0

(Ek(Lk+i)− Lss

), Lss = lim

i→∞Ek(Lk+i) (20)

where Lk+i = xTk+iQxk+i + uTk+iRuk+i. This cost is shownin [3] to be a quadratic function of the vector ck of freevariables ck+i, i = 0, . . . , N − 1 at time k:

Jk =[xTk cTk 1

]Ω[xTk cTk 1

]T(21)

for a suitably defined positive definite matrix Ω. Constraintsare invoked explicitly in mode 1 and implicitly, through theuse of a terminal set, S, in mode 2.

A. Constraint handling in mode 1

The distribution of β0 is dictated by the information avail-able on the plant state at the beginning of the predictionhorizon. Assuming that xk is known at time k, we set zk = xk,ek = 0 and β0 = 0, so that Fβ0

(x) = 1 for all x ≥ 0. Then,using v·|k to denote the corresponding predictions of variablev at time k, we have from (7),

ek|k = 0, ek+i|k = Φi−1Bwwk + · · ·+Bwwk+i−1, ∀i ≥ 1.

These predictions are independent of k, so the sets that definethe stochastic tube containing ek+i|k can be computed offline,namely ek+i|k ∈ E

(V, b(πi|0, p)

)with probability p, where

πi|0 = P iπ0|0, i = 1, 2, . . . , π0|0 = [1 · · · 1]T .

In this setting the probabilistic constraint of (3) can beenforced as follows.

Lemma 5. The constraint Pr(gTxk+i|k ≤ h) ≥ p for p > 0.5is ensured by the condition

gT zk+i|k ≤ h−[b(πi|0, q)

]1/2√gTV −1g (22)

where q = 2p − 1 and zk+i+1|k = Φzk+i|k + Buck+i|k fori = 0, 1, . . ., with zk|k = xk.

Proof: Assume that the distribution of gT ek+i|k is sym-metric about 0 (note that there is no loss of generality in thisassumption because of the symmetric bound employed in (8)).Then gT zk+i|k+gT ek+i|k ≤ h holds with probability p > 0.5iff gT zk+i|k + |gT ek+i|k| ≤ h with probability q = 2p − 1.The sufficient condition (22) then follows from ek+i|k ∈E(V, b(πi|0, p)

)and maxe∈E(V,b) |gT e| = b1/2

√gTV −1g.

Note that the replacement of |gT e| by its maximum valueb1/2

√gTV −1g describes the process of constraint tightening

4

whereby the constraint gT zk+i|k ≥ h on the nominal pre-dicted state is replaced by the tighter condition given in (22).Although (22), when invoked for i = 1, . . . , N , ensures thatat time k the predicted sequence xk+i|k satisfies (3), itdoes not ensure recursive feasibility, namely that there existsa predicted sequence at time k + 1 satisfying (3). This isbecause constraints of the form Pr(gTxk+i|k ≤ h) ≥ p donot guarantee that gTxk+i|k+1 ≤ h holds with probability p.However the future state is necessarily contained at all timeswithin the robust tube associated with upper bounds on βi:

ek+i|k ∈ E(V, b(πi|0, 1)

), b(πi|0, 1) =

1− λi

1− λα, ∀i ≥ 1

(23)with probability 1. The stochastic tubes that define constraintssuch as (22) at times k + 1, k + 2, . . . are therefore basedon initial distributions for β0 which can be inferred from therobust tube at time k, and this enables the construction ofconstraints to ensure recursive feasibility.

Let πi|j denote the discrete approximation to the distributionof βi that is obtained if βj (j < i) is equal to the upperbound b(πj|0, 1), so that Fβj

(x) = 0 for x ∈ [0, b(πj|0, 1))and Fβj

(x) = 1 for x ≥ b(πj|0, 1) . Then applying (16) forj, . . . , i− 1 gives

πi|j = P i−jπj|j , πj|j = u(ind(πj|0, 1)

)(24)

where u(j) = [0 · · · 0 1 · · · 1]T denotes the jth column ofthe lower-triangular matrix of 1s.

Lemma 6. For any i > j, the value of ek+i|k+j predicted attime k (given xk) satisfies

ek+i|k+j ∈ E(V, b(πi|j , p)

)with probability p. (25)

Proof: From ek+j+1 = Φek+j + Bwwk+j and (23), theprediction at time k of ek+j+1|k+j satisfies

ek+j+1|k+j ∈ ΦE(V, b(πj|0, 1)

)⊕ E

(V, b(π1|0, p)

)w.p. p

where ⊕ denotes Minkowski addition. But, by Lemma 1and (24) we have

ΦE(V, b(πj|0, 1)

)⊕ E

(V, b(π1|0, p)

)⊆ E

(V, b(πj+1|j , p)

),

which establishes (25) for i = j + 1. Similarly

ek+j+2|k+j ∈ ΦE(V, b(πj+1|j , p)

)⊕ E

(V, b(π1|0, p)

)w.p. p

so ek+j+2|k+j ∈ E(V, b(πj+2|j , p)

)with probability p,

and (25) follows by induction for all i > j.A simple way to ensure the recurrence of feasibility is to

require the probabilistic constraints (3) to hold for all futurepredictions. Lemma 6 shows that the worst case predictionsof (25) for j = 0, 1, . . . can be handled by selecting at eachprediction time i the largest scaling:

bi(q) = maxb(πi|0, q), . . . , b(πi|i−1, q) (26)

and by replacing b(πi|0, q) with bi(q) in (22):

gT zk+i|k ≤ h−[bi(q)

]1/2√gTV −1g, q = 2p− 1. (27)

The sets E(V, bi(p)), for i = 1, 2, . . ., define the cross sec-tions of a recurrently feasible stochastic tube. The precedingargument is summarized as follows.

Theorem 7. If at k = 0 there exists c0 satisfying (27) for alli > 0, then for all k > 0 there exists ck satisfying (27) for all

i > 0, and hence the probabilistic constraint (3) is feasiblefor all k > 0.

Proof: Follows directly from Lemma 5 and Lemma 6.Remark 3. Note that the computation of recurrently feasiblestochastic tubes does not depend on the information availableat time k, and hence can be performed offline. Consequentlythe online computation does not depend on the dimension of π,so this can be chosen sufficiently large that the approximationerrors discussed in Lemma 3 are negligible.

B. Constraint handling in mode 2

The constraint handling framework of mode 1 can alsobe used to define a terminal constraint, zk+N |k ∈ S, whichensures that (27) is satisfied for all i > N . Since (27)constitutes a set of linear constraints on the deterministicpredicted trajectory zk+i|k, the infinite horizon of mode2 can be accounted for using techniques similar to thosedeployed in [7] for the computation of maximal invariant sets.Before describing the construction of S, we first derive somefundamental properties of the constraints (27).

Lemma 8. The scaling of the tube cross section defined in (26)satisfies bi(q) ≤ bi+1(q) for all i and all q ∈ [0, 1], andconverges as i→∞ to a limit which is bounded by

limi→∞

bi(q) ≤ b∞, b∞ =α

1− λ. (28)

Proof: The monotonic increase of the scaling b(πi|0, 1)of the robust tube with i in (23) and the definition of πj|j interms of the robust tube in (24) implies (by Lemma 1) thatb(P i−jπj|j , q) ≤ b(P i−jπj+1|j+1, q), and hence b(πi|j , q) ≤b(πi+1|j+1, q) for any i > j. This implies bi(q) ≤ bi+1(q) forall i, and since bi(q) ≤ b(πi|0, 1) for all i, bi(q) must convergeas i → ∞ to a limit no greater than the asymptotic value ofthe robust tube scaling, b∞.

Lemma 9. Let S∞ be the maximal set with the propertythat (27) is satisfied for all i > N whenever zk+N |k ∈ S∞,then a sufficient condition for S∞ to be non-empty is:

h ≥ b1/2∞√gTV −1g . (29)

Proof: Satisfaction of constraints (27) over the infinitehorizon of mode 2 requires that

gTΦiz ≤ h−[bN+i(q)

]1/2√gTV −1g, i = 1, 2, . . . (30)

where for convenience the initial state of mode 2 is labelledz. By Lemma 8, bi(q) increases monotonically with i, and Φis strictly stable by assumption, so (29) is obtained by takingthe limit of (30) as i→∞ and using the upper bound in (28).

Lemma 8 implies that bN+i(q) in (30) can be replaced byb∞ for all i ≥ N , for some finite N , in order to define theterminal set. In this setting, the terminal set is given by S =SN , where

SN = z : gTΦiz ≤ h−[bN+i(q)

]1/2√gTV −1g, i = 1, . . . , N

gTΦN+jz ≤ h− b1/2∞√gTV −1g, j = 1, 2, . . . .

(31)The constraints in (31) are necessarily conservative, but theimplied constraint tightening can be made insignificant with

5

sufficiently large N . This follows from the assumption that Φis strictly stable, which implies that SN → S∞ as N →∞.

Although (31) involves an infinite number of inequalities,SN has the form of a maximal admissible set. The procedureof [7] can therefore be used to determine an equivalent repre-sentation of SN in terms of a finite number of inequalities, theexistence of which is ensured if S∞ is bounded. This involvessolving a sequence of linear programs to find the smallestinteger n∗ such that gTΦn

∗+1z ≤ h− b1/2∞√gTV −1g for all

z satisfying gTΦjz ≤ h − b1/2∞√gTV −1g for j = 1, . . . , n∗,

then SN in (31) is given by

SN = z : gTΦiz ≤ h−[bN+i(q)

]1/2√gTV −1g, i = 1, . . . , N

gTΦN+jz ≤ h− b1/2∞√gTV −1g, j ≤ 1, . . . , n∗ .

(32)

Remark 4. The scalings of the tube cross sections in (31) arecomputed on the basis of (10); this is done for conveniencegiven the simple form of (10). However tighter tubes wouldbe obtained if (9) were used in place of (10) to generate therobust bounds b(πi|0, 1) of (23) and the probabilistic boundsbi(q) of (26). The implied computation is practicable eventhough (9) involves an implicit optimization because of thediscretization of the range for β in (14).

C. Stochastic MPC algorithm

The offline computation of the tube scalings requires knowl-edge of V and λ. A possible choice is given by the pairV, λ that minimizes the steady state effect of the uncer-tainty on the constraints (27). This effect is given by thesteady state value of [bi(q)]

1/2√gTV −1g, which by (28) is

[α/(1−λ)]1/2√gTV −1g, and this suggests defining V, λ via:

(V −1, λ) = arg minV −1,λ

1

(1− λ)gTV −1g subject to (11).

(33)For a fixed value of λ > 0, (33) is a semidefinite program(SDP) in the variable V −1, and since λ is univariate andrestricted to the interval (0, 1), (33) can be solved by solv-ing successive SDPs for V −1 with alternate iterations of aunivariate optimization (such as bisection) for λ.

Given the definition of the cost (21), the constraints (27),and the terminal set (31), the MPC algorithm is as follows.

Algorithm 1 (Stochastic MPC). Offline: Compute V , λ via(33). Determine P in (16), and hence, for i = 1, . . . , N + N ,compute πi,j , j = 0, . . . , i − 1 in (24) and the scalings bi(q)in (26). Compute the smallest n∗ such that SN in (31) can beexpressed as (32).Online: At each time-step k = 0, 1, . . .

1. solve the quadratic programming (QP) optimization

c∗k = arg minck

Jk subject to (27) for i = 1, . . . , N − 1

and zk+N |k ∈ SN(34)

2. implement uk = Kxk + c∗k.

Theorem 10. Given feasibility at k = 0, Alg. 1 is feasible forall k ≥ 1. The closed loop system satisfies the probabilistic

constraint (3) and the quadratic stability criterion

limn→∞

1

n

n∑k=0

E0(Lk) ≤ Lss. (35)

Proof: Recurrence of feasibility follows from Theorem 7,which implies that the tail: ck+1|k = ck+1, . . . , ck+N−1, 0of the minimizing sequence: c∗k = ck, . . . , ck+N−1 of (34)at time k is feasible for (34) at time k + 1. The feasibility ofthe tail also implies that the optimal cost satisfies the bound:J∗k − Ek(J∗k+1) ≥ Lk − Lss, which implies (35).

V. ILLUSTRATIVE EXAMPLES

Example 1. Consider the 2nd order system with

A =

[1.8 −0.5−1.5 1.2

], Bu =

[21

], Bw = I, W = I

Pr(gTxk ≤ h) ≥ p, g = [−1.75 1]T , h = 1.5, p = 0.8.

The disturbance wk has a truncated normal distribution, withzero mean, covariance E(wkw

Tk ) = W−1/122 and bounds

−2.68 ≤ (wk)1,2 ≤ 2.68. The distribution of α is thena modified χ-squared distribution with α = 0.1. The costweights are Q = I , R = 5, and K in (5) is chosenas the LQG optimal feedback for the unconstrained case:K = [1.518 − 0.645].

The offline optimization of V, λ in (33) gives

λ = 0.41, V =

[0.782 −0.348−0.348 0.168

]The robust tube has asymptotic scaling b∞ = 0.170, and theinterval [0, 0.173] is divided into 1000 equal subintervals todefine xj and hence compute P and bi(q). Horizons N =6, N = 7, give n∗ = 0 and the scalings of the recurrentlyfeasible stochastic tube are

bi(q) = 0.31, 4.41, 6.09, 6.78, 7.06, 7.18, . . . 7.26 × 10−2

Algorithm 1 meets constraints with probability greater than0.8, while allowing some constraint violations (17% at k = 1).As expected, LQG control achieves lower average cost:

Jcl = limn→∞

1

n

n−1∑k=0

[E(xTkQxk + uTkRuk)− Lss

],

the average values over 200 realizations of the disturbancesequence being Jcl = 306 for LQG, and Jcl = 330 for Algo-rithm 1, but this is at the expense of violating the probabilisticconstraint (LQG had 100% violations at k = 1). Conversely,robust MPC (with p = 1) ensures zero violations, but achievesthis at the expense of higher average cost: Jcl = 360.

The online optimization is a QP with 6 optimization vari-ables and 13 constraints, requiring an average CPU time of3.5 ms (Matlab/2.4 GHz processor). By contrast, the approachof [20] requires the solution a second order cone program(SOCP), which for the same number of free variables andconstraints requires an average of 105 ms CPU time.

Example 2. This example considers the DC-DC convertersystem described in [9]. For the operating point defined byx0

1 = 0.389 A, x02 = −16 V, u0 = 0.516, sampling interval

T = 0.65 ms and problem parameters Vin = 15 V, R = 85 Ω,L = 4.2 mH, C = 2200 µF, the linearized state space model

6

−1 0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

x1 − x10

x 2 − x

20

unconstrainedLGQ optimalcontrol

Algorithm 1p = 0.8

Fig. 1. Closed loop responses of Algorithm 1 and LQG control for 103

disturbance sequence realizations, showing ellipsoidal tube cross-sectionspredicted at k = 0 for p = 1 and p = 0.8.

matrices are given by:

A =

[1 0.0075

−0.143 0.996

], Bu =

[4.7980.115

], Bw = I, W = I.

The LQG-optimal feedback gain for Q = diag1, 10, R = 1is K = [0.211 0.0024]. Fluctuations in the inductor current,which arise due to random variations in the source inputvoltage, are subject to the probabilistic constraint defined byPr[1 0](xk − x0) ≤ 2 ≥ 0.8. For additive uncertainty withcovariance W−1/252, and α = 0.02, the solution of (33) is

λ = 0.980, V =

[1.458 7.307.30 114.8

].

The robust tube has asymptotic scaling b∞ = 1.02, andhorizons N = 8, N = 9 were employed. For initial conditionx0−x0 = (2.5, 2.8) and 103 disturbance sequence realizations,Algorithm 1 produced an average cost of Jcl = 221, ascompared with Jcl = 230 for robust MPC (with p = 1) andJcl = 166 for LQG in closed loop operation. Robust MPCgave no constraint violations (Fig. 2), which is clearly over-cautious given the limit of 20% violations at each sample,whereas Algorithm 1 (Fig. 1) allowed constraint violationswith a frequency of 14.4% over k = 1, . . . , 6, and LQGignored the constraint (100% violation rate for k ≤ 3).

VI. CONCLUSIONS

The proposed stochastic MPC strategy handles probabilisticconstraints with online computational load similar to that ofnominal MPC. This is achieved by fixing the cross sectionalshape of a tube containing predicted trajectories, but allowingthe centres and scalings of the cross sections to vary. Theresulting MPC law ensures recursive feasibility and conver-gence.

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−1 0 1 2 3 4 5 6−0.5

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1.5

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2.5

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3.5

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x1 − x10

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