Distributionally Robust Covariance Steering with Optimal Risk Allocation Venkatraman Renganathan Joshua Pilipovsky Panagiotis Tsiotras

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Distributionally Robust Covariance Steering

with Optimal Risk Allocation

Venkatraman Renganathan, Joshua Pilipovsky, Panagiotis Tsiotras

Abstract—This article extends the optimal covariance steering

(CS) problem for discrete time linear stochastic systems modeled

using moment-based ambiguity sets. To hedge against the un-

certainty in the state distributions while performing covariance

steering, distributionally robust risk constraints are employed

during the optimal allocation of the risk. Speciﬁcally, a distri-

butionally robust iterative risk allocation (DR-IRA) formalism

is used to solve the optimal risk allocation problem for the CS

problem using a two-stage approach. The upper-stage of DR-IRA

is a convex problem that optimizes the risk, while the lower-

stage optimizes the controller with the new distributionally robust

risk constraints. The proposed framework results in solutions

that are robust against arbitrary distributions in the considered

ambiguity set. Finally, we demonstrate our proposed approach

using numerical simulations. Addressing the covariance steering

problem through the lens of distributional robustness marks the

novel contribution of this article.

Index Terms—Covariance Steering, Distributional Robustness,

Iterative Risk Allocation, Stochastic Systems, Moment Based

Ambiguity Sets.

I. INTRODUCTION

Intelligent and adaptive systems of the “smart world” that

work under operational constraints seek to solve some instance

of a constrained optimal control problem for optimizing their

performance. Such constrained optimal control problems can

now be increasingly solved efﬁciently using several numerical

optimization techniques. For instance, robot path planning in

uncertain environments [1]–[6] has gained the attention of

researchers worldwide as robots are being increasingly de-

ployed to solve many real-world problems. Apart from realistic

constraints, reliability of operation of these systems is often

thwarted by the ineffective handling of system uncertainties,

which can be either deterministic or stochastic.

Control of stochastic systems can be best formulated as a

problem of controlling the distribution of trajectories over time

subject to constraints. Recently, the ﬁnite horizon covariance

steering (CS) problem, namely, the problem of steering an

initial distribution to a ﬁnal distribution at a speciﬁc ﬁnal

time step subject to linear time varying dynamics has been

explored [7]. Speciﬁcally, the control problem in the CS

problem setting involves steering the mean and the covariance

to the desired terminal values. When the decision-making

process relies blindly on the functional form of the process

that models the stochastic uncertainty, it is known to result

in potentially severe miscalculation of risk. For instance,

Gaussianity assumptions made in the name of tractability in

V. Renganathan is with the Department of Automatic Control LTH,

Lund University, Lund, Sweden. J. Pilipovsky and P. Tsoitras are with the

School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,

GA 30332-0150 USA. E-mail: venkatraman.renganathan@control.lth.se,

jpilipovsky3@gatech.edu, tsiotras@gatech.edu.

several modeling regimes are actually rarely justiﬁable, as

the true distribution that governs the uncertain data might

be non-Gaussian. Such shortcomings can be mitigated with

risk-based stochastic optimization where the risk of wrong

decisions can be appropriately handled to result in risk-averse

decision making. One such tool is the Distributionally Robust

Optimization (DRO) advocated in [8], [9] which enables

modelers to explicitly incorporate ambiguity in probability

distributions into the optimization problem.

Control of stochastic systems often involves optimizing

the system’s objective subject to chance constraints, where

one assumes that the system uncertainties follow a known

distribution and enforces that the system constraints hold with

high probability as a function of the decision variables. The

number of constraint violations, called the total risk budget,

is usually a user-deﬁned a priori speciﬁcation and is a natural

metric to assess risk. Hence, one may consider the problem

of ﬁnding a risk allocation procedure that will allocate the

probability of violating each individual chance constraint at

each time step. Given a number of chance constraints across a

ﬁnite horizon, the total risk budget has to be allocated for all

chance constraints across all time steps [10]. It is a common

practice to consider a uniform risk allocation, i.e., allocate the

same risk for all constraints and across all time steps. However,

risk allocation can be optimized as in [11], [12] to reduce the

conservatism resulting from a uniform risk allocation. If the

probability distribution of the system uncertainties are known

exactly, then non-uniform risk allocation can be performed

effectively. However, risk allocation optimization with arbi-

trary distribution of the system uncertainties has not yet been

explored till now. This article addresses this shortcoming.

Contributions: Since authors in [13] solved the CS problem

for the Gaussian case, this article extends it with arbitrary

distributions using the theory of distributional robustness (DR).

To the best of our knowledge, this article is the ﬁrst one to ex-

tend the CS problem using distributionally robust optimization

techniques for both polytopic and convex conic state constraint

sets. Our main contributions in this article are as follows:

1) We extend the covariance steering problem tailored

between Gaussian distributions to arbitrary distributions

modeled using moment based ambiguity sets.

2) We enforce distributionally robust risk constraints for

both polytopic and convex cone state constraint satisfac-

tion, while solving the covariance steering problem, and

obtain the optimal risk allocation through a distribution-

ally robust iterative risk allocation (DR-IRA) algorithm.

3) We demonstrate our approach using simulation examples

and show the effectiveness of the proposed generaliza-

tion for covariance steering problems between arbitrary

distributions in moment-based ambiguity sets.

arXiv:2210.00050v2 [math.OC] 7 Nov 2022

Following a short summary of notation and preliminaries,

the rest of the paper is organized as follows: The main problem

statement of distributionally robust covariance steering prob-

lem with iterative risk allocation is presented in Section II.

Then, the proposed Distributionally Robust Iterative Risk

Allocation (DR-IRA) algorithm is discussed in Section III.

Subsequently, the proposed approach is demonstrated using

simulation results in Section IV. Finally, the paper is con-

cluded in Section Valong with directions for future research.

NOTATION AND PRELIMINARIES

The set of real numbers and natural numbers are denoted

by Rand N, respectively. The subset of natural numbers

between and including aand bwith a < b is denoted by

[a, b]. An identity matrix in dimension nis denoted by In.

For a non-zero vector x∈Rnand a matrix P∈Sn

++ or

P0, we denote kxk2

P=x>P x, where Sn

++ is the set

of all positive deﬁnite matrices. We say PQor PQ

if P−Q0or P−Q0respectively. We denote by

B(Rd)and P(Rd)the Borel σ−algebra on Rdand the space

of probability measures on (Rd,B(Rd)) respectively. A prob-

ability distribution with mean µand covariance Σis denoted

by P(µ, Σ) and, speciﬁcally Nd(µ, Σ), if the distribution is

normal in Rd. The cumulative distribution function (cdf) of

the normally distributed random variable is denoted by Φ.

A uniform distribution over a compact set Ais denoted by

U(A). Given a constant q∈R≥1, the set of probability

measures in P(Rd)with ﬁnite q−th moment is denoted by

Pq(Rd) := µ∈ P(Rd)|RRdkxkqdµ < ∞.

II. PROBLEM FORMULATION

Consider a linear, stochastic, discrete and time-varying

system as follows

xk+1 =Akxk+Bkuk+Dkwk, k = [0, N −1],(1)

where xk∈Rnand uk∈Rmis the system state and input

at time k, respectively and Ndenotes the total time horizon.

Further, Ak∈Rn×nis the dynamics matrix, Bk∈Rn×mis

the input matrix and Dk∈Rn×ris the disturbance matrix.

The process noise wk∈Rris a zero-mean random vector

that is independent and identically distributed across time.

We assume that the system state and the process noise is

independent of each other at all time steps, meaning that

E[xkwj] = 0, for 0≤k≤j≤N. The distribution Pwof

wkis unknown but is assumed to belong to a moment-based

ambiguity set of distributions, Pwgiven by

Pw=Pw|E[wk] = 0,E[wkw>

k]=Σw.(2)

Note that there are inﬁnitely many distributions present in the

considered set Pw. For instance, both multivariate Gaussian

and multivariate Laplacian distributions with zero mean and

covariance Σwbelong to Pwwith the latter having heavier

tail than the former. We assume that the system is controllable

under zero process noise meaning that given any x0, xf∈Rn,

there exists a sequence of control inputs {uk}N−1

k=0 that steers

the system state from x0to xf. Since the considered system

is stochastic, the initial condition x0is subject to a similar

uncertainty model as the noise, with the distribution belonging

to a moment-based ambiguity set, Px0∈ Px0, where

Px0=nPx0|E[x0] = µ0,E[(x0−µ0)(x0−µ0)>]=Σ0o.(3)

Provided that the control law ukis selected as an afﬁne

function of state xkat any time k, similar moment-based

ambiguity sets Pxkcan be written for the distribution of

states at any time k∈[1, N]using the propagated mean and

covariance at time k. Note that Pxkwould not be empty

at all time steps k∈[1, N]as a Gaussian distribution

would be a guaranteed member in that set. This is because,

Gaussianity is preserved under linear transformations deﬁned

by the dynamics. However, the initial state x0need not be

Gaussian, and thereby it would be inappropriate to assume

that the ﬁnal state xNwould also be Gaussian. Hence, the

terminal state xNis subject to a similar uncertainty model as

the x0, with its distribution PxNbelonging to a moment-based

ambiguity set PxN, given by

PxN=nPxN|E[xN] = µf,E[(xN−µf)(xN−µf)>]=Σfo.(4)

The objective is to steer the trajectories of (1) from x0∼

Px0∈ Px0to xN∼PxN∈ PxNin Ntime steps. This

covariance steering objective is usually achieved by minimiz-

ing a cost function under speciﬁed convex state and input

constraints. We deﬁne the concatenated variables required for

the problem formulation as follows

X=x>

0x>

1. . . x>

N>,

U=u>

0u>

1. . . u>

N−1>,

W=w>

0w>

1. . . w>

N−1>,

Ek=0n,kn In0n,(N−k)n,

X=µ>

0µ>

1. . . µ>

N>.

Using the concatenated variables, (1) can be written as

X=Ax0+BU+DW,(5)

where the matrices A,Band Dare of appropriate dimensions

containing the time varying system matrices Ak, Bk, and Dk

respectively. See [13] for more details on this transformation. It

is evident that PXis not known exactly but a similar ambiguity

set like Px0can be constructed for PX1. That is,

PX:= PX|E[X] = ¯

X,E[(X−¯

X)(X−¯

X)>] = ΣX.

(6)

The cost function to optimize is given as follows

J(U) := EX>¯

QX+U>¯

RU,(7)

where ¯

Q=diag(Q0, Q1, . . . , QN−1)is the state penalty ma-

trix and ¯

R=diag(R0, R1, . . . , RN−1)is the control penalty

matrix with each Qk0, Rk0for all k∈[0, N −1].

Over the whole time horizon, we want the states to respect

certain state constraints. Since the state is stochastic, constraint

violation can be imposed through a chance constraint for-

mulation. Speciﬁcally, a distributionally robust risk constraint

1Note that PXis the joint distribution formed with Pxk, k = 0,...,N

being its marginal distributions.

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DistributionallyRobustCovarianceSteeringwithOptimalRiskAllocationVenkatramanRenganathan,JoshuaPilipovsky,PanagiotisTsiotrasAbstractThisarticleextendstheoptimalcovariancesteering(CS)problemfordiscretetimelinearstochasticsystemsmodeledusingmoment-basedambiguitysets.Tohedgeagainsttheun-certaintyinthes...

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Distributionally Robust Covariance Steering with Optimal Risk Allocation Venkatraman Renganathan Joshua Pilipovsky Panagiotis Tsiotras

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