Reference Governor for Input-Constrained MPC to Enforce State Constraints at Lower

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Reference Governor for

Input-Constrained MPC to Enforce

State Constraints at Lower

Computational Cost

Miguel Castroviejo-Fernandez1, Jordan Leung1and Ilya Kolmanovsky1∗

October 21, 2022

Abstract

In this paper, a control scheme is developed based on an input con-

strained Model Predictive Controller (MPC) and the idea of modifying the

reference command to enforce constraints, usual of Reference Governors

(RG). The proposed scheme, referred to as the RGMPC, requires opti-

mization for MPC with input constraints for which fast algorithms exist,

and can handle (possibly nonlinear) state and input constraints. Condi-

tions are given that ensure recursive feasibility of the RGMPC scheme and

ﬁnite-time convergence of the modiﬁed command to the the desired ref-

erence command. Simulation results for a spacecraft rendezvous maneu-

ver with linear and nonlinear constraints demonstrate that the RGMPC

scheme has lower average computational time as compared to state and

input constrained MPC with similar performance.

1 introduction

Model Predictive Control (MPC) is informed by optimization of a state and

input dependent cost function. At each time step, the input sequence that

minimizes this cost subject to constraints on the inputs and/or the states [1] is

computed and the input is set to the ﬁrst element of the sequence. While MPC

has emerged as an eﬀective control strategy for constrained systems and is used

in many applications, one of its primary drawbacks is the high computational

cost associated with solving the optimization problem at each time step. This

computational cost can be signiﬁcantly lowered in the case of short horizon

Linear Quadratic MPC (LQ MPC) with only input constraints by exploiting

the underlying structure of the cost to speed up gradient computations as in the

∗1University of Michigan, Ann Arbor, MI 48109 USA mcastrov, jmleung,

ilya@umich.edu. This research is supported by Air Force Oﬃce of Scientiﬁc Research Grant

number FA9550-20-1-0385.

arXiv:2210.10995v1 [eess.SY] 20 Oct 2022

Fast MPC algorithm of [2] or by employing accelerated primal projected gradient

methods [3]. In addition, it is easier to enforce anytime feasibility properties

[4] for input constrained MPC (e.g., by saturating the computed input in the

case of boxed constraints), analyze the impact of inexact implementation [5, 6],

certify an inexact solution [7] and exploit the regularity properties as compared

to the state constrained case. For example, [8] performs the analysis of an

inexact implementation of state and input constrained MPC. Finally, to handle

nonlinear constraints the use of more computationally expensive nonlinear MPC

is required.

To capitalize on advantages of short-horizon input constrained MPC (uMPC)

with polytopic input constraints yet be able to handle state constraints and (pos-

sibly nonlinear) input constraints, in this paper we consider the augmentation

of uMPC with a reference governor (RG). RGs [9] are add-on schemes that en-

sure, at each time step, selection of the reference command so that subsequent

trajectories remain feasible with respect to constraints. However, the direct ap-

plication of existing RGs to uMPC-based closed-loop systems is diﬃcult. For

instance, if RG is based on online prediction [10, 11], a uMPC optimization

problem will need to be solved at each time step over the reference governor

prediction horizon; this will likely exceed the computational cost of a state and

input constrained MPC (cMPC).

In this paper we propose a new scheme which enables a computationally

eﬃcient application of RGs to complement uMPC in controlling linear systems

with (possibly nonlinear) state constraints and nonlinear input constraints. This

scheme, that we refer to as RGMPC, only requires that a single uMPC opti-

mization problem be solved per time step.

For the proposed RGMPC scheme we show, under suitable assumptions,

the recursive feasibility as well as ﬁnite-time convergence of the modiﬁed ref-

erence command to the desired constant reference command, i.e. properties

expected of conventional RGs. Simulation results for a spacecraft rendezvous

(RdV) problem demonstrate low computational requirements and good closed-

loop performance being achieved with the proposed approach.

The paper is organized as follows. In Section 2 the class of systems being

addressed is discussed and the two main ingredients: uMPC and the Incremen-

tal Reference Governor (IRG) of [11], needed for subsequent developments are

reviewed. Section 3 introduces the proposed RGMPC scheme and presents the-

oretical results. Finally, numerical simulations of the proposed scheme applied

to a spacecraft RdV maneuver are reported in Section 4.

Notations: Sn

++,Sn

+denote the set of symmetric n×npositive deﬁnite

and positive semi-deﬁnite matrices respectively. Imdenotes the m×midentity

matrix. Given x∈Rnand W∈Sn

+, the W-norm of xis ||x||W=√x>W x.

Given P∈Sn

++, y ∈Rn,BP(y, r) = {x∈Rn| ||y−x||P≤r}and λ+(P) is

the maximum eigenvalue of P. Given a∈Rn, b ∈Rm,(a, b) = [a>, b>]>. The

sequence made of the αj∈Rn, j =a, . . . , b elements is denoted by {αj}b

j=a.The

set Nis the set of positive integers and N0the set of non negative ones.

2 Preliminaries

2.1 Class of systems

We consider a class of systems represented by the following linear discrete-time

models,

xk+1 =Axk+Buk,(1a)

yk=Cxk,(1b)

where A∈Rn×n,B∈Rn×m,C∈Rp×nand k∈N0. The system is subject to

hard constraints on both states and inputs:

zk=(xk, uk)∈ Z,∀k≥0,(2a)

Z={(x, u)|x∈ X, u ∈ U} ⊆ Rn+m,(2b)

where X ⊂ Rn,U ⊂ Rmare compact, convex sets with the origin in their

interiors. Furthermore, we make the following assumption:

Assumption 1 The pair (A, B)is stabilizable.

2.2 Characterization of the steady states and inputs

We consider the reference command (set-point) tracking problem of bringing

the output, state and input of the system to a speciﬁc set-point r∈Rpand to

the associated steady states and inputs xss, uss, respectively. Using the usual

deﬁnition of a steady state and (1), the set-points must satisfy the following:

A−In−B0n+m×p

C0p×m−Ip



xss

uss

r

=M



xss

uss

r

= 0.(3)

Assumption 1 ensures that (3) has a solution [12]. In the following, we

deﬁne zss(r) = (xss(r), uss(r)), where (zss(r), r) are solutions to (3). Given the

existence of constraints, the following equation describes an inner approximation

of the set of admissible reference commands.

R=r∈Rp|∃ z∈˜

Z, M z

r= 0.

where ˜

Z ⊂ Int Zis a compact and convex set. This, under Assumption 1,

implies that Ris compact and convex.

2.3 Input constrained MPC

As explained in the introduction, uMPC oﬀers several advantages as compared

to state and input constrained MPC (cMPC). In the following we consider short-

horizon uMPC with a quadratic cost function,

J(ξ, µ, v) =

NMPC −1

i=0 ||ξi−xss(v)||2

Q+||µi−uss(v)||2

+||ξNMPC −xss(v)||2

where ξ={ξi}NMPC

i=0 , µ ={µi}NMPC −1

i=0 ,Q∈Rn×n, R ∈Rm×m, P ∈Rn×nand

NMPC ∈N. The MPC law is deﬁned using the solution to the following Optimal

Control Problem (OCP) P r(x, v, NMPC):

min

ξ,µ

J(ξ, µ, v) (4a)

s.t. ξ0=x(4b)

ξi+1 =Aξi+Bµi, i = 0, . . . , NMPC −1,(4c)

µi∈ U, i = 0, . . . , NMPC −1.(4d)

We assume that

Assumption 2 Q∈Sn

++, R ∈Sm

++, P ∈Sn

++ and P=Q+A>P A −

(A>P B)(R+B>P B)−1(B>P A), i.e. Pis the solution to the Discrete Algebraic

Riccati Equation (DARE).

Finally, let

{u∗

j(x, v, NMPC)}NMPC −1

j=0 (5)

denote the solution to P r(x, v, NMPC). Then, at time instant kthe MPC com-

puted input is given by uk=u∗

0(xk, vk, NMPC). Assumption 1 and Q∈Sn

ensure the existence of a stabilizing solution to the DARE in Assumption 2,

and since 0 ∈Int Uthe MPC control law is locally stabilizing at strictly con-

straint admissible equilibria [13]. Note that MPC described in this section does

not handle state constraints which will be handled by the IRG.

2.4 Incremental Reference Governor (IRG)

For the time being, suppose that a control law for system (1),

u=g(x, r),(6)

which depends on the state xand reference command r, is available. We deﬁne

j(x, r) = g(xg

j(x, r), r), j ∈N0and xg

j(x, r) = Ajx+Pj−1

i=0 Aj−1−iBug

ifor j≥

1 and xg

0(x, r) = x. The corresponding state-input vector is zg

j(x, r) = (xg

j, ug

j).

Now, considering (1) in closed-loop with controller (6), the aim of the IRG

is to adjust the reference command that the system follows in such a way as

to ensure that constraints are enforced. The IRG accomplishes this by testing

whether an increment of the current reference command leads to constraint

admissible trajectories.

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ReferenceGovernorforInput-ConstrainedMPCtoEnforceStateConstraintsatLowerComputationalCostMiguelCastroviejo-Fernandez1,JordanLeung1andIlyaKolmanovsky1*October21,2022AbstractInthispaper,acontrolschemeisdevelopedbasedonaninputcon-strainedModelPredictiveController(MPC)andtheideaofmodifyingthereferenceco...

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Reference Governor for Input-Constrained MPC to Enforce State Constraints at Lower

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