Chance Constrained Stochastic Optimal Control for Linear Systems with Time Varying Random Plant Parameters Shawn Priore Ali Bidram and Meeko Oishi

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Chance Constrained Stochastic Optimal Control for Linear Systems with

Time Varying Random Plant Parameters

Shawn Priore, Ali Bidram, and Meeko Oishi

Abstract— We propose an open loop control scheme for linear

systems with time-varying random elements in the plant’s

state matrix. This paper focuses on joint chance constraints

for potentially time-varying target sets. Under assumption of

ﬁnite and known expectation and variance, we use the one-

sided Vysochanskij–Petunin inequality to reformulate joint

chance constraints into a tractable form. We demonstrate our

methodology on a two-bus power system with stochastic load

and wind power generation. We compare our method with

situation approach. We show that the proposed method had

superior solve times and favorable optimally considerations.

I. INTRODUCTION

In much of the linear controls literature, stochasticity is

regarded as a factor external to the system modeling process.

Additive noise is often a placeholder for systemic uncertainty

that is difﬁcult to account for. For example, wind speeds

can affect the output of a wind turbine in a local grid, yet

state-of-the-art models have considerable difﬁculty in making

accurate predictions of their power output [1]. New control

techniques that can incorporate this stochasticity systemically

have the potential to enable more efﬁcient controllers that can

be robust to natural phenomena. In this paper, we develop

an optimal control derivation scheme for discrete time linear

systems with time-varying stochastic elements in the state

matrix subject to joint chance constraints.

Early work in the 1960s and 1970s illuminated the need

for incorporating random elements into the plant with ap-

plications in industrial manufacturing, communications sys-

tems, and econometrics [2], [3], [4]. Several works consid-

ered minimization strategies for linear quadratic regulator

problems. Without the addition of joint chance constraints,

dynamic programming techniques can easily be employed

to ﬁnd optimal controllers [5], [6], [7]. These works have

been extended to account for unknown distributions associ-

ated with the random parameters. Sampling techniques and

feedback mechanisms have been used to overcome these

hurdles [8], [9]. Unfortunately, these regulation problems

are often limited in scope and cannot readily be extended

to solve for chance constraints. Random plants with more

complex structure have been investigated [10] but have

typically been limited to Gaussian disturbances. Since the

This material is based upon work supported by the National Science

Foundation under NSF Grant Numbers CMMI-2105631 and OIA-1757207.

Any opinions, ﬁndings, and conclusions or recommendations expressed in

this material are those of the authors and do not necessarily reﬂect the views

of the National Science Foundation.

Shawn Priore, Ali Bidram, and Meeko Oishi are with the Department

of Electrical and Computer Engineering, University of New Mexico, Albu-

querque, NM; e-mail: shawnpriore@unm.edu (corresponding author),

bidram@unm.edu, oishi@unm.edu.

late 1970s research in this area has been sparse, appearing

only occasionally in econometric literature [11], [12] where

plant uncertainty has been used to model economic trends.

A similar problem, in which the uncertainty in the plant

is modeled either by bounded parameterization or a bounded

column space, has been extensively studied in the robust

model predictive control community [13], [14], [15], [16].

By exploiting the bounded parameter and column spaces,

estimation [17], [18] and stability techniques [19], [20] allow

for closed loop controller synthesis. While several of these

techniques can address uncertainty in the plant, they do

not address uncertainty that is random in nature [21], [22],

such as unknown but deterministic parameters. Further, these

methods can address uncertainty that result from bounded

random variables, such as discrete distributions with ﬁnite

outcomes, and uniform or beta distributions, but cannot ad-

dress random variables on semi-inﬁnite or inﬁnite supports.

We propose to address stochastic optimal control for

systems with uncertain state matrices in a manner that is

amenable to convex optimization techniques. To achieve this,

we use Boole’s inequality [23] and the one-sided Vysochan-

skij–Petunin inequality [24] to transform the chance con-

straint into a biconvex constraint that can be solved with

the alternate convex search method. Our approach offers

a closed form reformulation of the chance constraints that

is biconvex and can readily be solved. Further, this ap-

proach enables optimization under a wide range of distri-

butional assumptions and any solution guarantees chance

constraint satisfaction. However, our method also introduces

conservatism and relies on open loop controller synthesis.

In general, open loop control has known limitations with

respect to stability and convergence. As is common in model

predictive control literature, this approach could be combined

with stabilizing controllers which introduce an extraneous

input [25]. The proposed approach accommodates that well

established framework which implicitly addresses issues of

stabilization and convergence. Hence, many of the known

limitations typically associated with open loop control can

be accommodated. In addition, there are systems, such as

those with limited actuation or sensing, for which feedback

is simply not possible [26], [27]. The main contribution of

this paper is the construction of a tractable optimization

problem that solves for convex joint chance constraints in

the presence of random elements in the state matrix.

The paper is organized as follows. Section II provides

mathematical preliminaries and formulates the optimiza-

tion problem. Section III derives the reformulation of the

chance constraints with Boole’s inequality and the one-sided

arXiv:2210.09468v2 [eess.SY] 22 Mar 2023

Vysochanskij–Petunin inequality. Section IV demonstrates

our approach on two problems involving power generation

and labor allocation, and Section V provides concluding

remarks.

II. PRELIMINARIES AND PROBLEM FORMULATION

We denote the interval that enumerates all natural numbers

from ato b, inclusively, as N[a,b]. Random components will

be denoted with bold case, such as ~

xfor vectors and Afor

matrices, regardless of dimension. We use the notation aij

to denote the (i, j)th element of the matrix A. For a random

variable x, we denote the expectation as E[x], and variance

as Var(x), and standard deviation as Std(x). We use `b

i=a

for when a > b to denote the multiplication of elements over

the index ias it decreases from ato bby −1. For a matrix

A, the operator vec(A)vertically concatenates the columns

of Ainto a column vector. For two matrices Aand B, we

denote the Kronecker product as A⊗B. For matrix entries

A1, . . . , Am, we denote a block diagonal matrix constructed

with these elements as blkdiag(A1, . . . , Am). We denote an

identity matrix of size nas Inand the ith column of an

appropriately sized identity matrix as ~ei.

A. Problem Formulation

We consider a discrete-time linear system given by

x(k+ 1) = A(k)~

x(k) + B~u(k)(1)

with state x(k)∈ X ⊆ Rn, input ~u(k)∈ U ⊆ Rm,

and time index k∈N[0,N]. We presume initial conditions,

~x(0), are known, and the set Uis convex. The state matrix

A(k)contains real valued random variables, aij , each with

probability space (Ω,B(Ω),Paij )with outcomes Ω, Borel

σ-algebra B(Ω), and probability measure Paij [23].

We write the concatenated dynamics as an afﬁne combi-

nation of the initial condition and the concatenated control

sequence,

x(k) =

i=k−1

A(i)~x(0) + Ck−1B~

U(2)

with

Ck="1

i=k

A(i)· · · A(k)In0n×(N−k−1)n#∈Rn×Nn

(3a)

B= (IN⊗B)∈RNn×Nm

(3b)

U=h~u(0)>. . . ~u(N−1)>i>

∈ UN(3c)

Assumption 1. All random components aij (k)are mutually

independent within their matrix. Further, the random matri-

ces A(k)are mutually independent for all time steps.

Assumption 2. Each random element aij (k)has a ﬁnite

expectation and variance.

Both assumptions are easily met in most scenarios. We

would expect the parameters to be independent in many

biological and physical processes, and most distributional as-

sumptions would provide for ﬁnite expectation and variance.

Of notable exception are certain parameterizations of the t,

the Pareto, and the inverse-Gamma distributions.

We presume desired polytopic sets, represented by the

linear inequalities ~

Gik~x(k)≤hik , that the state must stay

within at each time step with a desired likelihood

P∩N

k=1 ∩ck

i=1 ~

Gik~

x(k)≤hik≥1−α(4)

where ckis the number of linear inequalities.

We presume convex, compact, and polytopic sets

x(k)∩ck

i=1 ~

Gik~

x(k)≤hiko⊆ X , and probabilistic

violation threshold α < 1/6.

Assumption 3. The distribution describing each probabilis-

tic constraint P~

Gik~

x(k)≤hikis marginally unimodal.

This is likely to be the most restrictive assumption as

verifying unimodality can be challenging in cases where

the distributional assumptions are not strongly unimodal

[28]. For a thorough review of unimodality in distributions

and strong unimodality, we recommend [29]. The primary

concern for unimodality within this framework is maintain-

ing unimodality through both additive and multiplicative

operations. As the terminal time increases the more likely

a non-unimodal distribution can arise from the complex and

intricate interactions of the random state and the random

plant parameters.

We seek to minimize a convex performance objective J:

XN× UN→R.

minimize

J~

x(1),...,~

x(N),~

U(5a)

subject to ~

U∈ UN,(5b)

Dynamics (1) with ~x(0) (5c)

Probabilistic constraint (4) (5d)

Problem 1. Under Assumptions 1-3, solve the stochastic

optimization problem (5) with open loop control ~

U∈ UN,

and probabilistic violation threshold α.

The main challenge in solving Problem 1 is assuring (5d).

The interaction of multiplying the random state matrices

makes enforcing the constraints challenging. Even if closed

form expressions exist for a single time step there is no

guarantee an expression will exist at the next time step.

III. METHODS

Our approach to solve Problem 1 involves reformulating

the joint chance constraint (4) into a series of constraints

that are afﬁne in the constraint’s expectation and standard

deviation, Eh~

Gik~

x(k)iand Std~

Gik~

x(k), respectively.

This form is amenable to the use of the one-sided Vysochan-

skij–Petunin inequality which guarantees the synthesized

controller satisﬁes the probabilistic constraint. The refor-

mulation results in an easy to solve biconvex optimization

problem.

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ChanceConstrainedStochasticOptimalControlforLinearSystemswithTimeVaryingRandomPlantParametersShawnPriore,AliBidram,andMeekoOishiAbstractWeproposeanopenloopcontrolschemeforlinearsystemswithtime-varyingrandomelementsintheplant'sstatematrix.Thispaperfocusesonjointchanceconstraintsforpotentiallytime-va...

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Chance Constrained Stochastic Optimal Control for Linear Systems with Time Varying Random Plant Parameters Shawn Priore Ali Bidram and Meeko Oishi

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