Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions Shuo Liu1 Jun Zeng2 Koushil Sreenath2and Calin A. Belta1

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Iterative Convex Optimization for Model Predictive Control

with Discrete-Time High-Order Control Barrier Functions

Shuo Liu∗1, Jun Zeng∗2, Koushil Sreenath2and Calin A. Belta1

Abstract— Safety is one of the fundamental challenges in

control theory. Recently, multi-step optimal control problems

for discrete-time dynamical systems were formulated to en-

force stability, while subject to input constraints as well as

safety-critical requirements using discrete-time control barrier

functions within a model predictive control (MPC) framework.

Existing work usually focus on the feasibility or the safety for

the optimization problem, and the majority of the existing work

restrict the discussions to relative-degree one control barrier

functions. Additionally, the real-time computation is challenging

when a large horizon is considered in the MPC problem for

relative-degree one or high-order control barrier functions. In

this paper, we propose a framework that solves the safety-

critical MPC problem in an iterative optimization, which is

applicable for any relative-degree control barrier functions. In

the proposed formulation, the nonlinear system dynamics as

well as the safety constraints modeled as discrete-time high-

order control barrier functions (DHOCBF) are linearized at

each time step. Our formulation is generally valid for any

control barrier function with an arbitrary relative-degree.

The advantages of fast computational performance with safety

guarantee are analyzed and validated with numerical results.

I. INTRODUCTION

A. Motivation

Safety-critical optimal control is a central problem in

robotics. For example, reaching a goal while avoiding ob-

stacles and minimizing energy can be formulated as a

constrained optimal control problem by using continuous-

time control barrier functions (CBFs) [1], [2]. By dividing

the timeline into small intervals, the problem is reduced to

a (possibly large) number of quadratic programs, which can

be solved at real-time speeds. However, this approach can be

too aggressive due to the lack of predicting ahead.

Model predictive control (MPC) with CBFs [3] considers

the safety problem in the discrete-time domain, and provides

a smooth control policy as it involves future state information

along a receding horizon. However, the computational time

is relatively large and increases dramatically with a larger

horizon, since the optimization itself is usually nonlinear

and non-convex. An additional issue of this nonlinear model

predictive formulation is the feasibility of the optimization.

For CBFs with relative-degree one, relaxation techniques

∗Authors contributed equally.

This work was supported in part by the NSF under grants IIS-2024606

and CMMI-1931853.

1S. Liu and C. Belta are with the department of Mechanical Engi-

neering, Boston University, Brookline, MA, 02215, USA {liushuo,

cbelta}@bu.edu.2J. Zeng and K. Sreenath are with the Uni-

versity of California, Berkeley, CA, 94720, USA {zengjunsjtu,

koushils}@berkeley.edu

Implementation code is released on https://github.com/

ShockLeo/Iterative-MPC-DHOCBF.

have been introduced in [4]. In this paper, we address

the above challenges with a proposed convex MPC with

linearized, discrete-time CBFs, under an iterative approach.

In contrast with the real-time iteration (RTI) approach in-

troduced in [5], which solves the problem through iterative

Newton steps, our approach solves the optimization problem

formulated by a convex MPC iteratively for each time step.

We show that the proposed approach can signiﬁcantly reduce

the computational time, compared to the state of the art

introduced in [4], even for CBFs with high relative-degree,

without sacriﬁcing the controller performance. The feasibllity

rate of our proposed method also outperforms that of the

baseline method in [4] for large horizon lengths.

B. Related work

1) Model Predictive Control (MPC): MPC is widely used

in modern control systems, such as controller design in

robotic manipulation and locomotion [6], [7] to obtain a

control strategy as a solution to an optimization problem.

Stability was achieved in [8] by incorporating discrete-time

control Lyapunov functions (DCLFs) into a general MPC-

based optimization problem to realize real-time control on a

robotic system with limited computational resources. More

and more recent work like [9] emphasizes safety in robot

design and deployment since it is an important criterion for

real-world tasks. Some works consider safety criteria through

the introduction of additional repelling functions [1], [10]

while some works regard obstacle avoidance as one concrete

scenario in terms of safety criteria for robots [11]–[13].

Those safety criteria are usually formulated as constraints in

optimization problems. This paper can be seen in the context

of MPC with safety constraints.

2) Continuous-Time CBFs: It has recently been shown

that to stabilize an afﬁne control system while also satisfying

safety constraints and control limitations, CBFs can be

uniﬁed with control Lyapunov functions (CLFs) to form

a sequence of single-step optimization programs [1], [2],

[14], [15]. If the cost is quadratic, the optimizations are

quadratic programs (QP), and the solutions can be deployed

in real time [1], [16]. Adaptive, robust and stochastic ver-

sions of safety-critical control with CBFs were introduced

in [17]–[21]. For safety constraints expressed using functions

with high relative degree with respect to the dynamics of

the system, exponential CBFs [22] and high-order CBFs

(HOCBFs) [23]–[25] were proposed.

3) Discrete-Time CBFs: Discrete-time CBFs (DCBFs)

were introduced in [26] as a means to enable safety-critical

control for discrete-time systems. They were used in a

arXiv:2210.04361v3 [math.OC] 13 Jul 2023

nonlinear MPC (NMPC) framework to create NMPC-DCBF

[3], wherein the DCBF constraint was enforced through

a predictive horizon. This method was also utilised in a

multi-layer control framework in [27], where DCBFs with

longer horizons were considered in the MPC problem serving

as a mid-level controller to guarantee safety. Generalized

discrete-time CBFs (GCBFs) and discrete-time high-order

CBFs (DHOCBFs) were proposed in [28] and [29] respec-

tively, where the DCBF constraint only acted on the ﬁrst

time-step, i.e., a single-step constraint. MPC with DCBF has

been used in various ﬁelds, such as autonomous driving [30]

and legged robotics [31]. For the work above, the CBF

constraints are either limited to be activated at the ﬁrst time-

step [26], [28], [29] to improve the optimization feasibility at

the cost of sacriﬁcing the safety performance, or for multiple

or all steps [27], [30], [31] with additional performance

optimization from other modules, such as multi-layer con-

trol [27], [30] or planning [31], which needs to speciﬁed for

different platforms. A decay-rate relaxing technique [32] was

introduced for NMPC with DCBF [4] for all time-steps to

enhance the safety and feasibility at the same time, but the

computation itself is overall still nonlinear and non-convex

which could be computationally slow for large horizons and

nonlinear dynamical systems, and the discussion in [4] is

limited to relative-degree one. In this paper, we generalize

relaxing technique for DHOCBF and largely optimize the

computational time compared to all existing work.

C. Contributions

We propose a novel approach to the NMPC with discrete-

time CBFs that is signiﬁcantly faster than existing ap-

proaches. In particular, the contributions are as follows:

•We present a model predictive control strategy for

safety-critical tasks, where the safety-critical constraints

can be enforced by DHOCBFs. The decay rate in each

constraint can be relaxed to enhance the feasibility in

optimization and to ensure forward invariance of the

intersection of a series of safety sets.

•We propose an optimal control framework for guaran-

teeing safety, where the DHOCBF constraints as well

as the system dynamics are linearized at each iteration,

and considered as constraints in a convex optimization

solved iteratively.

•We show through numerical examples that the proposed

framework is signiﬁcantly faster than existing methods,

without sacriﬁcing safety and feasibility.

II. PRELIMINARIES

In this section, we introduce some deﬁnitions and results

on CBF and MPC.

A. Discrete-Time High-Order Control Barrier Function

(DHOCBF)

In this work, safety is deﬁned as forward invariance of a

set C, i.e., a system is said to be safe if it stays in Cfor all

time, given that it is initialized in C. We consider the set C

as the superlevel set of a discrete-time function h:Rn→R:

C:={xt∈Rn:h(xt)≥0}.(1)

We consider a discrete-time control system in the form

xt+1 =f(xt,ut),(2)

where xt∈ X ⊂ Rnrepresents the state of system (2) at

time step t∈N,ut∈ U ⊂ Rqis the control input, and

function fis locally Lipschitz.

Deﬁnition 1 (Relative degree [33]).The output yt=h(xt)

of system (2) is said to have relative degree mif

yt+i=h(¯

fi−1(f(xt,ut))), i ∈ {1,2, . . . , m},

s.t. ∂yt+m

∂ut

̸= 0q,∂yt+i

∂ut

= 0q, i ∈ {1,2, . . . , m −1},(3)

i.e., mis the number of steps (delay) in the output ytin

order for the control input utto appear.

In the above deﬁnition, we use ¯

f(xt)to denote the

uncontrolled state dynamics f(xt,0). The subscript iof

function ¯

f(·)denotes the i-times recursive compositions of

f(·), i.e., ¯

fi(xt) = ¯

f(¯

f(. . . , ¯

| {z }

(¯

f0(xt))))

i-times

with ¯

f0(xt) = xt.

We assume that h(xt)has relative degree mwith respect

to system (2) based on Def. 1. Starting with ψ0(xt):=h(xt),

we deﬁne a sequence of discrete-time functions ψi:Rn→

R,i= 1, . . . , m as:

ψi(xt):=△ψi−1(xt,ut) + αi(ψi−1(xt)),(4)

where △ψi−1(xt,ut):=ψi−1(xt+1)−ψi−1(xt), and

αi(·)denotes the ith class κfunction which satisﬁes

αi(ψi−1(xt)) ≤ψi−1(xt)for i= 1, . . . , m. A sequence

of sets Ciis deﬁned based on (4) as

Ci:={xt∈Rn:ψi(xt)≥0}, i ={0, . . . , m −1}.(5)

Deﬁnition 2 (DHOCBF [29]).Let ψi(xt), i ∈ {1, . . . , m}

be deﬁned by (4) and Ci, i ∈ {0, . . . , m −1}be deﬁned by

(5). A function h:Rn→Ris a Discrete-Time High-Order

Control Barrier Function (DHOCBF) with relative degree m

for system (2) if there exist ψm(xt)and Cisuch that

ψm(xt)≥0,∀xt∈ C0∩ · · · ∩ Cm−1.(6)

Theorem 1 (Safety Guarantee [29]).Given a DHOCBF

h(xt)from Def. 2 with corresponding sets C0,...,Cm−1

deﬁned by (5), if x0∈ C0∩ · · · ∩ Cm−1,then any Lipschitz

controller utthat satisﬁes the constraint in (6),∀t≥0

renders C0∩ · · · ∩ Cm−1forward invariant for system (2),

i.e., xt∈ C0∩ · · · ∩ Cm−1,∀t≥0.

Remark 1. The function ψi(xt)in (4) is called a ith order

discrete-time control barrier function (DCBF) in this paper.

Since satisfying the ith order DCBF constraint (ψi(xt)≥

0) is a sufﬁcient condition for rendering C0∩ · · · ∩ Ci−1

forward invariant for system (2) as shown above, it is not

necessary to formulate DCBF constraints up to mth order as

(6) if the control input utcould be involved in some optimal

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IterativeConvexOptimizationforModelPredictiveControlwithDiscrete-TimeHigh-OrderControlBarrierFunctionsShuoLiu∗1,JunZeng∗2,KoushilSreenath2andCalinA.Belta1Abstract—Safetyisoneofthefundamentalchallengesincontroltheory.Recently,multi-stepoptimalcontrolproblemsfordiscrete-timedynamicalsystemswereformula...

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Iterative Convex Optimization for Model Predictive Control with Discrete-Time High-Order Control Barrier Functions Shuo Liu1 Jun Zeng2 Koushil Sreenath2and Calin A. Belta1

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