Robust self-triggered DMPC for linear discrete-time systems with

local and global constraints

Zhengcai Lia, Changbing Maa and Huiling Xua

aSchool of Science, Nanjing University of Science and Technology, Nanjing, China

ARTICLE HISTORY

Compiled December 17, 2020

ABSTRACTThis paper proposes a robust self-triggered distributed model predictive control(DMPC) scheme for a family of Discrete-Time linear systems with local (uncou-pled) and global (coupled) constraints. To handle the additive disturbance, tube-based method is proposed for the satisfaction of local state and control constraints.Meanwhile, A special form of constraints tightening is given to guranteen the globalcoupled constraints. The self-triggering mechanism help reduce the computationburden by skip insignificant iteration steps, which determine a certain samplinginstants to solve the DMPC optimization problem in parallel ways. The DMPC op-timization problem is construct as a dual form, and sloved distributedly based onthe Alternative Direction Multiplier Method (ADMM) with some known simplifi-cations. Recursive feasibility and input-to-state stability of the closed-loop systemare shown, the performance of proposed scheme is demonstrated by a simulationexample.

KEYWORDSRobust distributed model predictive control, Self-triggering, Tube-based method,Coupled constraints, Alternative Direction Multiplier Method

1. Introduction

This paper considers the Robust Distributed Model Predictive Control (RDMPC) ofM discrete-time linear dynamical systems with disturbance, which formulate as

xi(t+ 1) = Aixi(t) +Biui(t) + wi(t) (1)

xi(t) ∈ X i,ui(t) ∈ U i, wi(t) ∈ W i, i = 1, . . . ,M (2)

With all of them have to respect a global coupled constraints as following

M∑i=1

(Ψixx

i(t) + Ψiuu

i(t)) ≤ 1p, for all t (3)

where xi ∈ Rni , ui ∈ Rmi and wi ∈ Rni are the state, input and disturbance of of theith system; X i and U i are the corresponding state and input constraints for the ith

CONTACT Zhengcai Li. Email: lizhengcai0224@njust.edu.cn

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system; W i , {wi ∈ Rni | ‖ wi ‖≤ wi} is a compact set containing the origin in itsinterior; the matrices Ψi

x ∈ Rp×ni and Ψiu ∈ Rp×mi denote the coupled constraints of

all M systems.Model predictive control has been proved to be highly successful in comparison

with alternative methods of multivariable control due to its independent adoption bythe process industries (Mayne, 2014). The study of RMPC is an active area of re-search since the robust stability has been studied using the theory of input-to-statestability (ISS)(Jiang & Wang, 2001)(Sontag, 1995). Some examples of the use of ISStheory include(Limon, Alamo, & Raimondo, n.d.)(Pin, Raimondo, Magni, & Parisini,2009) The classical Lyapunov and input-to-state definitions of stability have subtledifferences especially when the system is discontinuous; these related issues includingimplications for MPC are thoroughly explored in(Lazar, Heemels, & Teel, 2013).

The useful concept of tube-based method originated with the seminal papers in1970s and remains an active area of research(Kurzhanski & Filippova, n.d.). It mainlyconstruct nominal MPC system ignoring the additive distrubance to simplify the opti-mization problem in predictive step. It was initially used in MPC (Chisci, Rossiter, &Zappa, 2001) for linear system, and the tube-based procedure has also been extendedto control nonlinear systems (Mayne, Kerrigan, van Wyk, & Falugi, 2011). AperiodicMPC is one another approach to lose the stress of computation burden, there aretwo main types of aperiodic MPC, which are event-triggered MPC(Shuai Liu, 2017)(Brunner, Heemels, & Allg?wer, 2017) and self-triggered MPC(Berglind, Gommans,& Heemels, 2012) (Gommans, Antunes, Donkers, Tabuada, & Heemels, 2014)(Brun-ner, Heemels, & Allg?wer, 2014). The detail introduction about the trigering mecha-nism have been shown in(Heemels, Johansson, & Tabuada, 2012). Recently,(Dai, Yang,Qiang, & Xia, 2019) have proposed a novel self-triggered MPC algorithm for linearsystem subject to both disturbances and state/input constraints, and an self-triggeredMPC algorithm with adaptive predictive horizon for perturbed nonlinear system(Sun,Dai, Liu, Dimarogonas, & Xia, 2019) have been proposed further. These latest liter-ature shows that tube-based method and self-triggered mechanism can be combinedperfectly in discrete-time linear system with bounded disturbance.

There exist many complex systems require control, but the traditional centralizedcontrol are unsuitable because of their complexity. The complexity gives rise to model-ing and data collection issues, raises computational and communication problems, andmake centralized control impractical. A very large literature on distributed control anddistributed model predictive control (DMPC) has emerged for this reason(C. Wang &Ong, 2010). One popular area in DMPC are state/input globle coupled when the con-straints come to (3). (Richards & How, 2007)gives a method of sequential process, itarrange every subsystem updated in a certain sequnce until all system complete theiriterations. Another approach called cooperative MPC method given by(P. A. Trodden& Richards, n.d.)(P. Trodden, 2014). The main idea of that approach is that all sys-tems within a cooperating set are optimized jointly while systems outside set followingtheir old predictive controls.

A reasonable appoach for system (1) − (3) without disturbance is proposedinBertsekas, Hager, & Mangasarian (1998). The method can achieve overall opti-mality through dual problem involving the Lagrange function, and the dual variableis function as consensus variable in distributed optimal control problem. Until theAlternative Direction Method of Multipliers(ADMM) is reproposed by(Boyd, 2011),and its nice numerical performance in distributed system in many appliations moti-vated some scholars restudy the distributed Optimal control problem(DOCP) usingdistributed ADMM recently(Farina & Scattolini, 2012)(Wang, Ong, & Hong, 2016).

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While the computation expendiency can be very high according to huge iteration stepsin ADMM, a brand new method has been proposed to handle premature terminationADMM algorithm through constraints tightening(Z. Wang & Ong, 2017).

Inspired by the above work of distributed Model Predictive Control under dis-tributed ADMM. A novel robust self-triggered distributed model pridictive control isproposed for a family of discrete-time linear perturbed systems with local and globalconstraints in this paper. The main contributions of this paper organized as follows.

1) It mainly contributes the tube-based method in robust MPC to distributeddiscrete-time linear system. The distribution of system gives rise in terminal set con-structure, constraints satisfaction and stability issues, especially the perturbed dis-tributed system makes global constraints (3) tough to satisfied. A special form ofconstraints tightening is established so that all the above difficulties are solved.

2) Sufficient condition about local and global constraints of bounded disturbanceis given to gurantee recursive feasibility in Optimal control problem(OCP), whichcan link the tube-based method in RMPC and constraints tightening in distributedADMM properly.

3) A parallel DMPC algorithm based on augmented lagrange method(ALM) is givenin this paper, which applies a self-triggering mechanism in RDMPC to lessen computa-tion burdan of optimization control problem as well. It shows the advantage of partialparallel algorithm in iteration times efficiency than traditional central methods.

The paper is organised as follows. In Section 2, the method of tube-based to han-dle bounded dissturbance is given and some useful properties under that approachis introduced. In Section 3, a robust distributed model predictive control scheme isdesigned and a robust self-triggered DMPC algorithm is proposed. Main results in-volving recursive feasibility and stability are developed in Section 4. Section 5 gives anumerical example to illustrating the results obtained and conclusion of the paper isgiven in Section 6.

Notation 1. The set of ni-dimensional Euclidean space, mi-dimensional Euclideanspace, the set of p × ni real matrices and p × mi real matrices are denotedRni ,Rmi ,Rp×ni , and Rp×mi respectively. 1p denote p-dimensional Column vectorwhose element is full of 1. The notations N[0,N−1],ZM indicate the sets {r ∈ N|0 ≤ r ≤N − 1} and {r ∈ Z|r ≤M} respectively. for W ∈ Rn×n, the notation W > 0 indicateW is positive definite, ‖ W ‖ denotes the 2-norm of W and Sn++ denotes the spaceof symmetric n× n positive definite matrices. for W ∈ Sn++ and x ∈ Rn,‖ x ‖2= xTxand ‖ x ‖2W= xTWx. Given X ,Z ⊆ Rn and A ∈ Rn×n, AX = {Ax|x ∈ X},X ⊕ Z ={x + z|x ∈ X , z ∈ Z},X Z = {x + z|x ∈ X , z ∈ Z}. We denote the smallest andlargest eigenvalues of a matrix as λmax(�), λmin(�). V = {1, 2, . . . ,M} and E ⊂ V × Vare the vertex set and edge set of undirect graph G. The i-step-ahead predicted valueof a variable at time tk denote as (tk + i|tk).

For the sake of readability, some proofs are located in Appendix.

2. Preliminaries and properties of tube-based method

This section reviews some results in tube-based RMPC and other related concepts. Fora expository reason, we using tube-based approach to achieve the control goal. We cantake full advantage of the optimal control input sequence calculated at each samplingtime instead of just utilising the first element of the sequence. Inter-sampling time is

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chosen by a self-triggered condition, which actually through a minimun function. Wefirst define the nominal system with a given prediction horizon N ∈ N by neglectingthe disturbance in(1)

zi(k + l + 1|k) = Aizi(k + l|k) +Biui(k + l|k), l ∈ N[0,N−1] (4)

where zi(k + l|k) is the predicted nominal state at step k, and we denotes the errorbetween the actual state and the nominal one as ei(k+ l|k) , xi(k+ l|k)− zi(k+ l|k),with initial conditions zi(k|k) = xi(k|k) and ei(k|k) = 0. It evolve as

ei(k + l|k) =Aiei(k + l − 1|k) + wi(k + l − 1) (5)

=Ail−1wi(k) +Ai

l−2wi(k + 1) + · · ·+ wi(k + l − 1)

If we can assure the boundness of above error ei(k+ l|k) which we’ll prove in section4, and ignoring the based coupled constraints (3), then we can construct the OCP atstate xi(tk) as

P(xi(tk)) , minui(tk)

J i(xi(tk), ui(tk))

=

N−1∑l=0

[‖ zi(tk + l|tk) ‖2Qi + ‖ ui(tk + l|tk) ‖2Ri ]+ ‖ zi(tk +N |tk) ‖2P i (6a)

subject to

zi(tk|tk) = xi(tk) (6b)

zi(tk +N |tk) ∈ T if (6c)

∀l ∈ N[0,N−1] :

zi(tk + l + 1|tk) = Aizi(tk + l|tk) +Biui(k + l|k) (6d)

zi(tk + l|tk) ∈ Zil (6e)

ui(tk + l|tk) ∈ U i (6f)

The nominal predictive state constraints set is defined as Z il , X iRil, l ∈ N[1,N−1],

with Ril , ⊕l−1j=0A

ijW i. Qi > 0, Ri > 0 and P i > 0 are the weighting matrices of cost

function. Regarding to the terminal set X iε , we first construct a set X ir , {zi ∈ Rni | ‖zi ‖P i≤ ri} for this nominal system(5), such that the set X ir is maximal Robust ControlInvariant Set(RCIS) under local state feedback control law : ui(zi) = Kizi, which alsoneed to satisfied with following basic constraints:

∀zi ∈ X ir ,Kizi ∈ U i (7)

X ir ⊂ X i (8)

Where Ki is chosen as the unconstrained LQ optimal feedback control law for thenominal system(5), while P i is the solution of the Lyapunov equation

(Ai +BiKi)TP i(Ai +BiKi) +Qi +KiTRiKi = P i (9)

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And then we give the terminal set a define X iε , {zi ∈ Rni | ‖ zi ‖P i≤ εi} with√1− λmin(Qi)

λmax(P i)ri ≤ εi ≤ ri. We’ll use this condition to guarantee recursive feasibility

and stability under a self-triggered mechanism.

Lemma 2.1. (Dai et al., 2019) The nominal predictive state difference between tkand tk+1 = tk +M i

k is norm-bounded by

||zi(tk+1 + l|tk+1)− zi(tk+1 + l|tk)||φ =‖ Ailei(tk +M ik|tk) ‖φ

≤

{B(�)||Ai||l(1− ||Ai||M i

k), if||Ai|| 6= 1

B(�)Mk, if||Ai|| = 1(10)

where φ ∈ Sn++ and B(�) : Sn++ → R is defined as

B(�) ,

{√λmax(�) wi

1−||Ai|| , if||Ai|| 6= 1√λmax(�)wi, if||Ai|| = 1

(11)

Lemma 2.2. (Dai et al., 2019) For any sampling instant tk k ∈ N , let u∗(tk) bethe solution of P(xi(tk)). If the first M i

k step are applied to (1) in a strictly open-loop fashion, then the difference between J∗(xi(tk), u

i(tk)) and J∗(xi(tk+1), ui(tk+1))is bounded by

J∗(xi(tk+1), ui(tk+1))− J∗(xi(tk), ui(tk)) ≤ gi(M ik, x

i(tk), ui,∗(tk), wi) (12)

where

gi(M ik, x

i(tk), u∗(tk), wi) , g

i0(M i

k, xi(tk), u

∗(tk), wi)−M i

k−1∑l=0

[||z∗(tk + l|tk)||2Qi + ||u∗(tk + l|tk)||2Ri ]. (13)

In discrete system ,we get a self-triggering machanism which is proposed as following

M ik , arg min

Mk∈N[1,N]

gi(Mk, xi(tk), u

∗(tk), wi) (14a)

subjec to gi(Mk, xi(tk), u

∗(tk), wi) < 0 (14b)

If the optimizaition problem is infeasible, then we set M ik = 1.

3. RDMPC scheme for the coupled constraints system

Our control goal is to design a robust self-triggered DMPC algorithm, which canstablize the whole system at a fast convergence rate and satisfied with constraints (2)and (3), while simultaneously reducing the amount of compute burden.

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3.1. Constriants tightening

Now considering the coupled constraints(3), as we construct a nominal system(5) whichhas a tube error e with actual system(1), we should tightening the constraints tosatisfied with the feasibility. Specifically, the tightening constraints of (3) are

M∑i=1

(Ψixzi(tk + l|tk) + Ψi

uui(tk + l|tk)) ≤ (1− ε(l))1p, ∀l ∈ N[0,N−1] (15a)

M∑i=1

(ΨiNz

i(tk +N |tk)) ≤ (1− ε(N))1p, ∀zi(tk +N |tk) ∈ T if (15b)

where terminal coupled constriants coefficient matrix formulate as ΨiN = Ψi

x + ΨiuK

i,and supposed there are I subsystems ||Ai|| 6= 1, J subsystems ||Aj || = 1 in the discretesystem. We give a simlified denote that

ε(l) =

I∑i=1

‖ Ψix ‖ wi

1− ‖ Ai ‖l

1− ‖ Ai ‖+

J∑j=1

‖ Ψjx ‖ lwj

ε(N) =

I∑i=1

‖ ΨiN ‖ wi

1− ‖ Ai ‖N

1− ‖ Ai ‖+

J∑j=1

‖ ΨjN ‖ Nwj

Obviously, the tolerance 0 < ε(1) < · · · < ε(N) < 1 should be satisfied to ensureGradual decrease in terminal constraints and 0 ∈ T if . Correspondingly, the tightenedRDMPC formulation is:

Pε(xi(tk)) , minui{M∑i=1

J i(xi(tk),ui(tk)) : (6b− 6f)and(15a)} (16)

ui := {ui(0), ui(1), . . . , ui(N − 1)}, the choice of T if is chosen to beT if , X iε , whichsubject to √

1− λmin(Qi)

λmax(P i)ri ≤ εi ≤ ri. (17)

We can represented the tightend RDMPC formulation as

Pε(x(tk)) : minui(k)

M∑i=1

J i(xi(tk),ui(tk)) (18a)

s.t.(6b)− (6f), i ∈ [1,M ]

M∑i=1

f i(xi(tk),ui(tk)) ≤ b(ε) (18b)

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where f i(xi(tk),ui(tk)) is an appropriate function form rewriting from (15a), b(ε) =

[1p, (1− ε(1))1p, . . . , (1− ε(N − 1))1p]T .

The connection of nominal system and real system is shown in the following lemma3.1, while its proof locate at appedix A.

Lemma 3.1. The coupling constraints (3) in real system can be satisfied when (15)is satisfied in nominal system if the upper bound of disturbance satisfies withw

i ≤ ( 1M‖Ψi

N‖− ri√

λmax(P i)) 1−‖Ai‖

1−‖Ai‖N , if ‖ Ai ‖6= 1

wj ≤ ( 1M‖Ψj

N‖− rj√

λmax(P j)) 1N if ‖ Aj ‖= 1

In the end of this subsection, we give some general assumptions to ensure initialfeasibility of OCP and global constraints from predictive system to real system.(A1): (Ai, Bi) is reachable and xi(tk) is measurable for all i ∈ ZM .(A2): W i, X i and U i are polytope containing the origins in its interior for all i ∈ ZM .(A3): The undirected graph G = (V, E) is full connected.

3.2. Distributed ADMM form of OCP

Let λ ∈ Rp×N be the dual variable associated with constraints (17b), the lagrangefunction of OCP can formulate as:

L(ui, λ) =:

M∑i=1

J i(xi,ui) + λT (

M∑i=1

f i(xi,ui)− b(ε)) (19)

The dual problem is

maxλ≥0

minui∈U

L(ui, λ) := minλ≥0

maxui∈U

−L(ui, λ) = minλ≥0

M∑i=1

gi(λ) (20)

where

gi(λ) := maxui∈U

−J i(xi,ui)− λ(f i(xi,ui)− 1

Mb(ε)) (21)

The dual problem is not distributed as λ is a common variable in gi(λ). According toassumption (A3), the dual problem (19) can rewritten as a consensus problem in thefollowing

minλi≥0

M∑i=1

gi(λi) s.t.λi = λj , (i, j) ∈ E (22)

where λi is the local copy of λ in the ith system , and the conditions λi = λj ensurethe consensus of whole system dual variable. It can be further rewritten by using a

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new set of reference variable in the form of

maxλi≥0

M∑i=1

gi(λi) s.t.

M∑i=1

Eiλi = c. (23)

The augmented lagrangian methods form of (23) is

Lρ(λi, ω) =

M∑i=1

gi(λi) + ωT (

M∑i=1

Eiλi − c) +ρ

2‖

M∑i=1

Eiλi − c ‖22 . (24)

To apply the ALM form to solve problem(22), the distributed ADMM is giving inthe following.

(λ1k+1, . . . , λ

Mk+1) = arg min

λi≥0Lρ(λ1, . . . , λM , ωk), (25)

uik+1 = arg minui∈U i

L(ui, λik+1) (26)

ωk+1 = ωk − ρ(

M∑i=1

Eiλik+1 − c) (27)

For every subproblem with regard to λi, we can introduce neighbor term 12 ‖ λ

i −λik ‖2s and relaxation factor γ to reduce iteration burden. Then the updating step ofλik+1, ωk+1 comes as

λik+1 = arg minλi≥0Lρ(λ1

k, . . . , λi, . . . , λMk , ωk) +

1

2‖ λi − λik ‖2s, (28)

ωk+1 = ωk − ργ(

M∑i=1

Eiλik+1 − c) (29)

We noted that uitk = {uik(0), uik(1), . . . , uik(N − 1)} is the optimal solu-

tion of Pε(x(tk)),and define the applied optimal control sequence as ui,∗(tk) ,{uik(0), uik(1), . . . , uik(Mk)} when we get Mk from algorithm 2. Then we give the closed-loop sysytem under the Robust self-triggered DMPC before next sampling time in thefollowing:

xi(tk + l + 1) = Aixi(tk + l) +Biui,∗(tk + l) + wi(tk + l) (30)

3.3. Self-triggered DMPC algorithm

The overall procedure of the distributed ADMM algorithm at time tk is summarizedin the following algorithm 1.

From (Dai et al., 2019) we know that self-triggered machanism can reduce theburden on compuation largely, we use it in the overall Robust self-triggerd DMPCscheme at time tk. The framework is shown in the algorithm 2.

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Algorithm 1 Consensus ADMM algorithm

Input: Input measured system state xi,i ∈ ZMOutput: ui,∗k , i ∈ ZM

Initiallization: choose ρ > 0,set k = 0, λi0 = 0, ω0 = 0 for all i ∈ ZM , j ∈ Ni.repeatforall i ∈ ZM (parallel)Doobtain λik+1 from (28);

obtain uik+1 from (26);obtain ωk+1 from (29);end for k ← k + 1 until stop criterion is satisfied

Algorithm 2 Robust self-triggered DMPC

1:Every subsystem i measure its own state xi(tk);IF xi(tk) ∈ X iεi , Subsystem i obtains local feedback control law ui(tk) = Kixi(tk) andapplied itself until maximum iteration step.IF ELSE2:Other subsystem i calls algorithm 1 with xi(tk) and get ui,∗;3:Subsystem i calculate optimization problem (14a) to get M i

k;4:Choose Mk = min{M i

k} as next sampling step;5:Subsystem i obtains ui,∗(x(tk)) and applied it to Every subsystem i ;6:Let tk = tk +Mk and go back to step 1.

4. Main properties of the RDMPC scheme

This section presents the main properties of RDMPC schme in algorithm 2, that isrecursive feasibility, satisfaction of both the local state and input constraints and theglobal coupled constraints, and closed-loop stability. The following lemma ensure thatif OCP (18) is feasible at some sampling time, then its feasiblity remains at the nextsampling time.

Lemma 4.1 (Recursive feasibility of nominal predictive system). Assume thatu(tk) = [u1(tk),u

2(tk), . . . ,uM (tk)] where ui(tk) = [ui(tk|tk), ui(tk + 1|tk), . . . , ui(tk +

N − 1|tk)]T is a feasible solution at time tk for OCP Pε(x(tk)), Then the OCPPε(x(tk+1)) is feasible at time tk+1.If the bound on disturbance satisfies with(1)the local condition w

i ≤ (ri−εi)(1−||Ai||)√λmax(P i)(1−||Ai||)N

, if||Ai|| 6= 1

wj ≤ (rj−εj)√λmax(P j)N

, if||Aj || = 1

(2)the global conditionwi ≤ ( 1

M‖ΨiN‖− ri√

λmax(P i)) 1−‖Ai‖

1−‖Ai‖N , if ‖ Ai ‖6= 1

wj ≤ ( 1M‖Ψj

N‖− rj√

λmax(P j)) 1N if ‖ Aj ‖= 1

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The proof of lemma 4.1 locate at appedix B. Through the recursive feasibility lemmaof nominal system, we can conclude the theorem of recursive feasibility and constraintssatisfaction in real system.

Theorem 4.2 (Recursive feasibility and constraints satisfaction). If the Pε(x(t0)) isfeasible and the conditions in lemma 4.1 is satisfied, then for the distributed system(1) under algorithm 2, it holds that(1)Pε(x(tk)) is feasible for all tk and k ∈ N;

(2)xi(t) ∈ X i, ui(t) ∈ U i,∑M

i=1(Ψixx

i(t) + Ψiuu

i(t)) ≤ 1p for every realization wi(t) ∈W i.

Proof. By lemma 4.1, the assumption that feasible of initial optimization problemPε(x(t0)) implies all Pε(x(tk)) is feasible for all tk and k ∈ N, which proves (1).

To show the satisfaction of local constriants, the time horizon is divided into twoparts, k ∈ N[0,ti] and k ∈ N[ti,∞]. Where ti denotes the time when the state of system

i first enter X iε . for every sampling time tk before the time ti, from (30) and (15), forl ∈ N≤Mk

, we have the close-loop system formulat as

xi(tk + l + 1) = Aixi(tk + l) +Biui,∗(tk + l) + wi(tk + l)

= Ai(zi,∗(tk + l|tk) + ei(tk + l|tk)) +Biui,∗(tk + l|tk) + wi(tk + l)

= zi,∗(tk + l + 1|tk) + ei(tk + l + 1|tk)∈ Zil+1 ⊕Ril+1 ⊆ X

Further, according to constraints (6f), ui(tk + l + 1) = ui,?(tk + l + 1) ∈ U i.Now considering the sampling time after ti, the state of system i have already enter

X iε . The local control ui(ti + l) = Kixi(ti + l) ∈ U i, l ∈ N is applied to the real systemas dual control strategy. So the close-loop system i after ti is

xi(ti + l + 1) = Aixi(ti + l) +BiKixi(ti + l) + wi(ti + l)

= (Ai +BiKi)xi(ti + l) + wi(ti + l)

from the definition of RCIS X iε , xi(ti) ∈ X iε implies (Ai + BiKi)xi(ti) ∈ X iε , l ∈ N,according to the relation of disturbance and RCIS, λmax(P i)wi ≤ ri−εi, it holds from(8) that xi(ti + 1) ∈ X ir ∈ X . Consider xi(ti + l) ∈ X ir and (B3) we have

‖ (Ai +BiKi)xi(ti + l) ‖P i≤ εi

‖ xi(ti + 1 + 1) ‖P i≤‖ (Ai +BiKi)xi(ti + l) ‖P i + ‖ wi(ti) ‖P i

≤ εi + ri − εi = ri

Then xi(ti + l) ∈ X ir ∈ X i, ui(ti + l) = Kixi(ti + l) ∈ U i, l ∈ N.We denote tout = min{ti}, tin = max{ti}, for every sampling time tk before the

time tout, according to constraints (15a) and lemma 3.1, we have

M∑i=1

(Ψixx

i(tk) + Ψiuu

i(tk)) ≤ 1p

For every sampling time tk+l between tout and tin, assume L subsystem have already

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get in terminal set. As we take ui(tk+l) = ui(tk+l|tk) and ui(tk+l|tk) = Kixi(tk+l|tk)when xi(tk + l|tk) ∈ X ir , according to constraints (15a) and lemma 3.1, then we have

M−L∑i=1

(Ψixx

i(tk + l) + Ψiuu

i(tk + l)) +

L∑i=1

(Ψixx

i(tk + l) + ΨiuK

ixi(tk + l))

=

M−L∑i=1

(Ψixx

i(tk + l|tk) + Ψiuu

i(tk + l|tk)) +

L∑i=1

(Ψixx

i(tk + l|tk) + ΨiuK

ixi(tk + l|tk))

=

M−L∑i=1

(Ψixx

i(tk + l|tk) + Ψiuu

i(tk + l|tk)) +

L∑i=1

(Ψixx

i(tk + l|tk) + Ψiuu

i(tk + l|tk))

=

M∑i=1

(Ψixx

i(tk + l|tk) + Ψiuu

i(tk + l|tk)) ≤ 1p

For every sampling time tk after the time tin, according to last conclusion in lemma3.1, we have

M∑i=1

ΨiNx

i(tk) ≤ (1− ε(N))1p ≤ 1p

Hence, we have already proved (2) for different situation of all sampling time.

In the following part, we concentrate our attentions to the stability of whole dis-tributed system. Fisrt we give a normal definition of stability in discrete-time system.

Definition 4.3. (Sontag, 1995) System x(k+1) = f(x(k), u(k)) is (globally) input-to-state stable(ISS) if there exist a KL-function β : R≥0 ×R≥0 → R≥0 and a K-functionγ such that, for each input u ∈ lm∞ and each ξ ∈ Rn, it holds that

x(k, ξ, u) ≤ β(k, |ξ|) + γ(u) (31)

for each t ∈ Zn.

When the dynamics comes to a closed-loop perturbed system, we introduce a com-mon lemma to simplify the prove of ISS.

Lemma 4.4. (Jiang & Wang, 2001) A system of the form x(k + 1) = f(x(k), w(k))is input-to-state stability(ISS) if and only if there exists a continuous ISS-lyapunovfuction V (x(k)) such that for K∞ functions ρ1(�), ρ2(�), and ρ3(�), and a K functionα(�), it satisfies with

ρ1(‖ x(k) ‖) ≤ V (x(k)) ≤ ρ2(‖ x(k) ‖)V (x(k + 1))− V (x(k)) ≤ α(w)− ρ3(‖ x(k) ‖)

Then we give the input-to-state stability theorem of RDMPC scheme as follows.

Theorem 4.5. Given fesibility of Pε(x(t0)) and satisfaction of lemma 4, the closed-loop system in (30) is ISS.

11

Proof. The ISS-Lyapunov function can be chosen as V (x(tk)) , J∗(x(tk),u∗(tk)), ap-

parently J∗(x(tk),u∗(tk)) =

∑Mi=1 J

∗(xi(tk), ui(tk)) is a K∞ functions where x(tk)) =

[x1(tk), x2(tk), . . . , x

M (tk)], u∗(tk)) = [u1,∗(tk), u2,∗(tk), . . . , u

M,∗(tk)] is boundness inlocal constriants (2), so the first ISS condition is trivially proved. Then from lemma2,the difference between V (x(tk)) and V (x(tk + 1)) is bounded by

V (x(tk + 1))− V (x(tk)) =

M∑i=1

{J∗(xi(tk+1), ui(tk+1))− J∗(xi(tk), ui(tk))}

≤M∑i=1

{gi(M ik, x

i(tk), ui,∗(tk), wi)}

≤M∑i=1

{gi0(M ik, x

i(tk), u∗(tk), wi)− [||xi(tk)||2Qi + ||ui,∗(tk)||2Ri ]}

≤M∑i=1

{β(wi)− ||xi(tk)||2Qi}

≤ α(wi)−M∑i=1

||xi(tk)||2Qi

where the bound of distrubance in real system defined as wi = [w1, w2, . . . , wM ],

while α(wi) ,∑M

i=1 β(wi) is a K function defined as the maximum of∑Mi=1 g

i(M ik, x

i(tk), ui,∗(tk), wi) and

∑Mi=1 ||xi(tk)||2Qi is absolutely K∞ functions on

x(tk).

5. Numerical simulation

The numerical example is a 4-agent system with same system matrix A1,2,3,4 =[1.1, 0.12; 0.35, 0.0075], B1,2,3,4 = [1.5; 0.5], the local constraints in all subsystemsare X i = {xi||xi,1| ≤ 20, |xi,1| ≤ 5} and U i = {ui||ui| ≤ 2}, while global con-straints is ‖

∑4i=1 Ψi

xxi + 0.01u1 + 0.02u2 + 0.03u3 + 0.04u4 ‖≤ 101p. Where Ψi

x =[0.08, 0.02], p = 1. The set of bounded disturbance are satisfied with condition of lemma3, we set that asW i = [−0.3, 0.3]× [−0.3, 0.3]. Set weight matrix Qi = [1, 0; 0, 1], Ri =0.1, i = 1, 2, 3, 4, we obtain from LQR feedback control law and Riccati equationthat P 1,2,3,4 = [1.0516, 0.0057; 0.0057, 1.0015],K1,2,3,4 = [−0.7033,−0.0710], and ini-tiall state x1(0) = [−19;−4], x2(0) = [−18;−3], x3(0) = [−10; 4], x4(0) = [−18; 3]. Theprediction horizon is chosen to be N = 5 and simulation length is Trun = 30 steps.

Fig.1 indicates the summarize of the cost function in all subsystem, from thatwe can know the objective function in (6a) is convergence to a stable value withinalgorithm tolerance.

Fig.2 shows the sate trajectories of all subsystem under algorithm 2 with differentsequence of disturbances, the system state convergence to the neighborhood of the ori-gin, the state trajectories reflect the excellent performance of algorithm we proposedfrom the stability side.

Fig.3 dispalys the Control inputs trajectories under algorithm 2, the samplinginstants is indicated by scarlet circles while each iteration steps denotes by bluecrossmark. Iteration time of algorithm without and with self-triggering machanism

12

Figure 1. Cost function of overall system under algorithm 2

Figure 2. System trajectories under algorithm 2

Figure 3. Control inputs trajectories under algorithm 2

13

Figure 4. Global constraints of overall system

is T a1run = 1.8190s and T a2

run = 1.1097s, the difference between computation time showsoff the latter algorithm can lossen computation burden apparently.

The evolution of global constraints function f =∑M

i=1 fi has shown in Fig.4, in

which the blue dashed line represents the maximum value of the global constraint.Together with Fig.2 and Fig.3, they illustrated the satisfaction of the local constraintson state and control input for every subsystem, as well as the global constraints inoverall system.

6. conclusions

A self-triggered DMPC scheme has been proposed for disturbed linear systems withlocal and global constraints. The proposed scheme uses tube method to capture thebounded disturbance in subsystem, while it also gives a form of global constraintstightening based on upper bound of disturbance to apply parallel distributed ADMMmethods. For closed-loop system under algorithm 2, the properties about the recur-sive feasibility of OCP and ISS stability of overall system are given, some numericalexamples has been shown to demonstrated the good performance of proposed scheme.

Acknowledgement(s)

An unnumbered section, e.g. \section*{Acknowledgements}, may be used for thanks,etc. if required and included in the non-anonymous version before any Notes or Ref-erences.

7. References

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8. Appendices

Appendix A. the proof of lemma 3.1

Proof. From the proof of lemma 1(Dai et al., 2019), we know the error e betweennominal and real system satisfies with:{

‖ ei(tk + l|tk) ‖≤ 1−||Ai||l1−||Ai|| w

i, ||Ai|| 6= 1

‖ ej(tk + l|tk) ‖≤ lwj , ||Aj || = 1

Then we can get

‖M∑i=1

Ψixei(tk + l|tk) ‖≤

M∑i=1

‖ Ψix ‖‖ ei(tk + l|tk) ‖

≤I∑i=1

‖ Ψix ‖ wi

1− ‖ Ai ‖l

1− ‖ Ai ‖+

J∑j=1

‖ Ψjx ‖ lwj = ε(l)

M∑i=1

Ψixei(tk + l|tk) ≤ ε(l)1p (A1)

16

According to the condition about upper bound of disturbance, we havewi ≤ ( 1

M‖ΨiN‖− ri√

λmax(P i)) 1−‖Ai‖

1−‖Ai‖N , if ‖ Ai ‖6= 1

wi ≤ ( 1M‖Ψj

N‖− rj√

λmax(P j)) 1N if ‖ Aj ‖= 1‖ Ψi

N ‖ (wi 1−‖Ai‖N1−‖Ai‖ + ri√

λmax(P i)) ≤ 1

M , if ‖ Ai ‖6= 1

‖ ΨjN ‖ (Nwj + rj√

λmax(P j)) ≤ 1

M , if ‖ Aj ‖= 1

I∑i=1

(‖ ΨiN ‖ wi

1− ‖ Ai ‖N

1− ‖ Ai ‖+ ‖ Ψi

N ‖ri√

λmax(P i))+

J∑j=1

(‖ ΨjN ‖ Nwj+ ‖ Ψj

N ‖rj√

λmax(P j)) ≤ 1

M∑i=1

‖ ΨiN ‖

ri√λmax(P i)

+ (

I∑i=1

‖ ΨiN ‖ wi

1− ‖ Ai ‖N

1− ‖ Ai ‖+

J∑j=1

‖ ΨjN ‖ Nwj) ≤ 1

M∑i=1

‖ ΨiN ‖

ri√λmax(P i)

+ ε(N) ≤ 1 (A2)

‖M∑i=1

ΨiNe

i(tk +N |tk) ‖≤M∑i=1

‖ ΨiN ‖‖ ei(tk +N |tk) ‖= ε(N)

M∑i=1

ΨiNe

i(tk +N |tk) ≤ ε(N)1p (A3)

When l = 0, apparently we have

M∑i=1

(Ψixx

i(tk|tk) + Ψiuu

i(tk|tk)) =

M∑i=1

(Ψixzi(tk|tk) + Ψi

uui(tk|tk)) ≤ 1p

For l ∈ Z[1,N−1],

M∑i=1

(Ψixx

i(tk + l|tk) + Ψiuu

i(tk + l|tk))

=

M∑i=1

(Ψix(zi(tk + l|tk) + ei(tk + l|tk)) + Ψi

uui(tk + l|tk))

≤ (1− ε(l))1p +

M∑i=1

Ψixei(tk + l|tk) ≤ 1p

17

For l = N , from (A2) and (A3), we have

‖M∑i=1

ΨiNz

i(tk +N |tk) ‖≤M∑i=1

‖ ΨiN ‖‖ zi(tk +N |tk ‖

≤M∑i=1

‖ ΨiN ‖

εi

λmax(P i)≤

M∑i=1

‖ ΨiN ‖

ri

λmax(P i)≤ 1− ε(N)

M∑i=1

ΨiNx

i(tk +N |tk) =

M∑i=1

ΨiN (zi(tk +N |tk) + ei(tk +N |tk))

≤ (1− ε(N))1p +

M∑i=1

ΨiNe

i(tk +N |tk) ≤ 1p

Appendix B. the proof of lemma 4.1

Proof. Consider a candidate solution u(tk+1) at time tk+1 given by{ui(tk+1 + l|tk+1) = ui(tk+1 + l|tk), l ∈ N[0,N−M i

k−1]

ui(tk+1 + l|tk+1) = Kizi(tk+1 + l|tk+1), l ∈ N[N−M ik,N−1]

where zi(tk+1) is the prediction state of nominal system with u(tk+1) and x(tk+1).Next we will prove u(tk+1) is a feasible solution to Pε(x(tk+1)), i.e. it satisfies with(6b-6f) and (15). Constraints (6b) and (6d) are obviouly satisfied in Pε(x(tk+1)).For l ∈ N[0,N−M i

k−1], by the definition of u(tk+1), (6f) can trivally obtained. Fromlemma 1 we get,

zi(tk+1 + l|tk+1) = zi(tk+1 + l|tk) +Ailei(tk +M i

k|tk)

zi(tk+1 + l|tk+1) = zi(tk +M ik + l|tk) +Ai

lei(tk +M i

k|tk)

where we know from (4) and (5) that zi(tk+M ik+l|tk) ∈ ZiM i

k+l, ei(tk+M i

k|tk) ∈ RiM ik.

It follows that

zi(tk+1 + l|tk+1) ∈ ZiM ik+l ⊕A

ilRiM ik

= X i RiM ik+l ⊕A

ilRiM ik

= X i Ril , Z il (B1)

18

which proves the satisfaction of (6e).

‖M∑i=1

ΨixA

ilei(tk +M ik|tk) ‖≤

M∑i=1

‖ Ψix ‖‖ Ai ‖l‖ ei(tk +M i

k|tk) ‖

≤I∑i=1

‖ Ψix ‖ wi

‖ Ai ‖l − ‖ Ai ‖l+M ik

1− ‖ Ai ‖+

J∑j=1

‖ Ψjx ‖M i

kwj

≤ ε(M ik + l)− ε(l)

M∑i=1

(Ψixzi(tk+1 + l|tk+1) + Ψi

uui(tk+1 + l|tk+1))

=

M∑i=1

(Ψixzi(tk+1 + l|tk) + Ψi

uui(tk+1 + l|tk)) +

M∑i=1

ΨixA

ilei(tk +M ik|tk)

≤ (1− ε(M ik + l))1p +

M∑i=1

ΨixA

ilei(tk +M ik|tk)

So we have

M∑i=1

(Ψixzi(tk+1 + l|tk+1) + Ψi

uui(tk+1 + l|tk+1))

≤ (1− ε(M ik + l))1p + (ε(M i

k + l)− ε(l))1p ≤ ε(l)1p

it prove the global constraints (15a).

For l ∈ N[N−M ik,N−1],

zi(tk +N |tk+1) = zi(tk +N |tk) +AiN−M i

kei(tk +M ik|tk)

Then we hold from triangle inequality that

‖ zi(tk +N |tk+1) ‖P i ≤‖ zi(tk +N |tk) ‖P i + ‖ AiN−Mikei(tk +M i

k|tk) ‖P i

≤‖ zi(tk +N |tk) ‖P i + ‖ ei(tk +N |tk) ‖P i{≤‖ zi(tk +N |tk) ‖P i +wi

√λmax(P i)1−‖Ai‖N

1−‖Ai‖ , ||Ai|| 6= 1

≤‖ zi(tk +N |tk) ‖P i +wi√λmax(P i)N, ||Ai|| = 1

Considering the local condition wi ≤ (ri−εi)(1−||Ai||)√λmax(P i)(1−||Ai||)N

(wi ≤(ri−εi)√λmax(P i)N

, when||Ai|| = 1) and zi(tk +N |tk) ∈ X iε , we have zi(tk +N |tk+1) ∈ X ir .

And the set X ir is maximal Robust Control Invariant Set(RCIS) under local statefeedback control law : ui(zi) = Kizi, so we have

zi(tk+1 + l|tk+1) = (Ai +BiKi)zi(tk+1 + l − 1|tk+1)

∈ X ir ⊆ ZiN ⊆ Zil , (B2)

19

which proves the satisfaction of (6e), from (7), we know Kizi ∈ U i,which satisfies (6f)from lemma 3, when zi(tk+1 + l|tk+1) ∈ X ir , we have

M∑i=1

(Ψixzi(tk+1 + l|tk+1) + Ψi

uui(tk+1 + l|tk+1))

=

M∑i=1

(Ψixzi(tk+1 + l|tk+1) + Ψi

uKizi(tk+1 + l|tk+1))

=

M∑i=1

ΨiN z

i(tk+1 + l|tk+1)

≤ (1− ε(N))1p ≤ (1− ε(l)1p

Next we prove the satisfaction of (6c). With a proper choose of P i which obtained by(9), we have

‖ zi(tk+1 +N |tk+1) ‖2P i − ‖ zi(tk+1 +N − 1|tk+1) ‖2P i

= − ‖ zi(tk+1 +N − 1|tk+1) ‖2Qi+KiTRiKi

≤ − ‖ zi(tk+1 +N − 1|tk+1) ‖2Qi

≤ −λmin(Qi) ‖ zi(tk+1 +N − 1|tk+1) ‖2

≤ −λmin(Qi)

λmax(P i)‖ zi(tk+1 +N − 1|tk+1) ‖2P i

from above inequality and (17), we have

‖ zi(tk+1 +N |tk+1) ‖2P i ≤ (1− λmin(Qi)

λmax(P i)) ‖ zi(tk+1 +N − 1|tk+1) ‖2P i

≤ (1− λmin(Qi)

λmax(P i))ri

2 ≤ εi2 (B3)

So we have zi(tk+1 +N |tk+1) ∈ X iε which satisfies with (6c), from lemma 3 we get thefollowing inequality

M∑i=1

ΨiN z

i(tk+1 +N |tk+1) ≤ (1− ε(N))1p

Hence we complete the proof of (6b)− (6f) and (15) in Pε(x(tk+1)).

20