Robust self-triggered control for time-varying and uncertain constrained systems via reachability analysis

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Automatica

journal homepage:www.elsevier.com/locate/automatica

Brief paper

Robust self-triggered control for time-varying and uncertain constrained systems via reachability analysis

^✩

Yulong Gao

^a,∗

, Pian Yu

^a

, Dimos V. Dimarogonas

^a

, Karl H. Johansson

^a

, Lihua Xie

^b

aDivision of Decision and Control Systems, KTH Royal Institute of Technology, SE-10044, Stockholm, Sweden

bSchool of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore

a r t i c l e i n f o

Article history:

Received 5 March 2018

Received in revised form 24 February 2019 Accepted 3 June 2019

Available online 4 July 2019

Keywords:

Constrained systems Robust control Self-triggered control Reachability analysis

a b s t r a c t

This paper develops a robust self-triggered control algorithm for time-varying and uncertain systems with constraints based on reachability analysis. The resulting piecewise constant control inputs achieve communication reduction and guarantee constraint satisfactions. In the particular case when there is no uncertainty, we propose a control design with minimum number of samplings over finite time horizon. Furthermore, when the plant is linear and the constraints are polyhedral, we prove that the previous algorithms can be reformulated as computationally tractable mixed integer linear programs.

The method is compared with the robust self-triggered model predictive control in a numerical example and applied to a robot motion planning problem with temporal constraints.

© 2019 Elsevier Ltd. All rights reserved.

1. Introduction

Control for constrained systems has been extensively studied for over four decades in the literature, see e.g. Gutman and Hagander(1985) andMayne, Rawlings, Rao, and Scokaert(2000).

It is well-known that model predictive control (MPC) (Mayne et al.,2000) is a prime example of such a method. If one restricts the attention to networked control systems (NCS) subject to con- straints, how to jointly design a controller and a communication protocol which can efficiently utilize network resources is a more recent challenge. In order to handle this problem, recent research efforts have been devoted to self-triggered control (Dai, Gao, Xie, Johansson, & Xia,2018; Heemels, Johansson, & Tabuada,2012), in which the next sampling time is determined in advance by the controller according to the received information.

One intuitive idea is to incorporate the self-triggered con- trol into an MPC framework. Many results have been obtained

✩ The work of Yulong Gao, Pian Yu, Dimos V. Dimarogonas, and Karl H.

Johansson was supported in part by the Knut and Alice Wallenberg Foundation, Sweden, the Swedish Foundation for Strategic Research, and the Swedish Research Council. The work of Lihua Xie was supported by the National Natural Science Foundation of China under NSFC, China 61633014 and NSFC 61720106011. The material in this paper was not presented at any conference.

This paper was recommended for publication in revised form by Associate Editor Franco Blanchini under the direction of Editor Ian R. Petersen.

∗ Corresponding author.

E-mail addresses: yulongg@kth.se(Y. Gao),piany@kth.se(P. Yu), dimos@kth.se(D.V. Dimarogonas),kallej@kth.se(K.H. Johansson), elhxie@ntu.edu.sg(L. Xie).

for deterministic constrained systems (Barradas Berglind, Gom- mans, & Heemels, 2012; Henriksson, Quevedo, Sandberg, & Jo- hansson, 2012). A self-triggered MPC scheme is proposed in Barradas Berglind et al. (2012) for constrained linear time-invariant systems and the inter-sampling time is maximized subject to the constraints on the cost function. For systems with uncertainty, some results of robust self-triggered MPC have been reported in Brunner, Heemels, and Allgöwer (2014, 2016). In Brunner et al.(2016), the tube MPC method is utilized to guar- antee constraint satisfactions despite the presence of a bounded additive disturbance and the principle ofBarradas Berglind et al.

(2012) is followed to obtain the next sampling instant. In addi- tion, the effect of the uncertainty is made use of in the design of the self-triggered mechanism.

Despite recent developments, some fundamental issues re- main for self-triggered MPC. The incorporation of the self-triggered scheme into MPC does not immediately preserve the conventional recursive feasibility and closed-loop stability of MPC. In order to guarantee these two properties, the price of more computational effort at the sampling instants is paid to satisfy the constraints on the cost function when maximizing the inter-sampling time.

Other than MPC, only a few work address the self-triggered con- trol for constrained systems. For example, the focus ofLehmann, Kiener, and Johansson(2012),Seuret, Prieur, Tarbouriech, and Za- ccarian(2013) andWu, Reimann, and Liu(2014) is on the design of an event-triggered controller when the systems are subject to actuator saturation. One recent work (Hashimoto, Adachi, &

Dimarogonas,2018) provides a contractive set-based approach to design self-triggered control for linear deterministic constrained systems.

https://doi.org/10.1016/j.automatica.2019.06.015 0005-1098/©2019 Elsevier Ltd. All rights reserved.

Different from the above results, this paper aims at proposing a robust self-triggered control framework for time-varying and uncertain systems with constraints. To the best of our knowledge, this topic has not been explored up to now and cannot be han- dled by the previous mentioned methods, such as self-triggered MPC. The main challenges are: (1) how to guarantee recursive feasibility in a time-varying setup by self-triggered control; (2) how to ensure constraint satisfaction for any disturbance real- ization. In this work, we make full use of reachability analysis to handle these issues. Although reachability has been widely studied (Bertsekas & Rhodes, 1971; Lygeros, Tomlin, & Sastry, 1999;Raković, Kerrigan, Mayne, & Lygeros,2006), the incorpora- tion of reachability into self-triggered control is novel. One recent work (AlKhatib, Girard, & Dang, 2017) uses reachability-based self-triggered control to design the variable sampling period for sampled-data linear systems. However, neither constraints nor uncertainties are considered inAlKhatib et al.(2017).

The use of reachability analysis in this paper provides a ge- ometric interpretation for the self-triggered control from a set theoretical point of view. Available geometry software tools fa- cilitate the implementation of our algorithms. Some practical ap- plications of our algorithm include control of hybrid systems and robot motion planning (seeExample 2). The main contributions are summarized below.

•

We propose a novel robust self-triggered control algorithm (Algorithm1) for time-varying and uncertain systems with constraints. Constraint satisfactions and recursive feasibility are shown to be guaranteed based on reachability anal- ysis. We calculate the maximum inter-sampling time by solving the corresponding optimization problem (P_[¹_k

i,^N](x_ki)) only once at each sampling instant, which avoids the repet- itive computation required in the self-triggered MPC. In the particular case when there is no uncertainty, we de- velop a control method with minimum number of samplings over a finite time horizon. This is achieved by solving the optimization problem (P_[²_0,N](x₀)) only once.

•

When the plant is linear and the constraints are polyhedral, we prove that all the optimization problems (P_[¹_k

i,^N](x_ki) and P_[²_0,N](x₀)) can be reformulated as mixed integer lin- ear programming (MILP) problems, which are, in our cases, computationally tractable (Theorems 4.1and4.2). The nu- merical comparisons (Example 1) show that our algorithm achieves a better communication reduction and faster online computation than the robust self-triggered MPC inBrunner et al.(2014) without much loss in performance.

The remainder of the paper is organized as follows. The prob- lem statement is given in Section2. Section3 presents the ro- bust self-triggered control algorithm. The specialization to linear plants with polyhedral constraints is provided in Section4. Two examples in Section5illustrate the effectiveness of our approach.

Finally, Section6concludes this paper.

Notation. Let N be the set of nonnegative integers. For some q

,

^s

∈

_{N and q}

<

s, let N^[q,s] denote the set

{

r

∈

_N

|

q

≤

r

≤

s

}

, respectively. When

≤

,

≥

,

<

^{, and}

>

, are applied to vectors, they are interpreted element-wise. A column vector of ones with appropriate dimension is denoted by 1. The Minkowski sum of two sets is denoted by A

⊕

_B

= {

a

+

b

| ∀

a

∈

_A

, ∀

^b

∈

_B

}

. The Minkowski difference of two sets is denoted by A

⊖

_B

= {

c

| ∀

b

∈

_B

,

^c

+

b

∈

_A

}

. For a vector x

∈

_Rⁿ, define

∥

x

∥

∞

=

max_i

|

x_i

|

. For a polyhedron P

= {

x

∈

_Rⁿ

|

Px

≤

p

}

, define

∥

_P

∥

∞

=

max_x∈P

{∥

Px

−

p

∥

∞

}

_{. For X}

⊆

_Rⁿand A

∈

_R^n×n, define A⁻¹X

= {

x

∈

_Rⁿ

|

Ax

∈

_X

}

. For x_l

∈

_Rⁿ, l

∈

_{N, define}

∑

j

l=kx_l

=

0 if k

>

j. For Xl

⊆

_Rⁿand A_l

∈

_R^n×n, l

∈

_{N, define}

j

⨁

l=_k

Xl

= {∅ ,

^k

>

^j

,

Xk

⊕

_Xk+₁

⊕ . . .

Xj

,

^k

≤

j

,

j

∏

l=_k

A_l

=

{

I

,

^k

>

^j

,

AjAj−1

. . .

^Ak

,

^k

≤

j

.

Given two sets X and_{X, define}

˜

1_X(x)

=

{

1

,

^x

∈

_X

,

0

,

^x

/∈

X

,

^and1X(_X)

˜ =

{

1

, ˜

X

⊆

_X

,

0

, ˜

X

̸⊆

_X

.

2. Problem statement

Consider a discrete-time dynamic control system

x_k+₁

=

f_k(x_k

,

^uk)

+ w

k

,

⁽¹⁾

where x_k

∈

_R^nxand u_k

∈

_R^nu,

w

k

∈

_R^nx, and f_k

:

_R^nx

×

_R^nu

→

_R^nx. The control input u_k at time k is constrained by a set Uk

⊂

R^nu. The additive disturbance

w

k at time instant k belongs to a compact set Wk

⊂

_R^nx. The initial state x₀ is contained in a given set X0

⊂

_R^nx. In addition, given a finite time horizon N

∈

N, the system (1) is subject to a target tube, denoted by

{

_(Xk

,

^k)

,

^k

∈

_N[₁,N]

}

_{, where Xk}

⊆

_R^nx,

∀

k

∈

_N[₁,N]. It is assumed that the function f_kand the disturbance set Wkare known for all k

∈

_N[₀,N−₁].

Assumption 2.1. The function f_k(x

,

^u),

∀

k

∈

_N[₀,^N−₁], is continu- ous in x and u, respectively.

Assumption 2.2. The sets Uk,

∀

k

∈

_N[₀,N−₁], and Xk,

∀

k

∈

_N[₀,N], are compact.

The objective of this paper is to develop a self-triggered con- trol algorithm for the system(1), thereby yielding a sequence of piecewise constant control inputs. More specifically, we aim to determine a sequence of sampling instants

{

k₀

,

^k1

, . . . ,

^kT

}

with k₀

=

0, k_i+₁

=

k_i

+

M_i, and k_T

=

N such that u_j

=

u_l,

∀

j

,

^l

∈

N^[ki,ki+1−₁]and all the constraints are satisfied at each time instant k

∈

_N[0,^N]. Here, T

+

1 is the total number of samplings within N time instants, which quantifies the communication consumption, and M_idenotes the inter-sampling time between k_iand k_i+₁. 3. Self-triggered control for constrained systems via reacha- bility analysis

In this section, we will provide a reachability-based self- triggered control algorithm for the uncertain constrained system (1). Furthermore, a control method with minimum number of samplings will be designed when the system (1) is reduced to be deterministic, i.e., Wk

= {

0

}

.

3.1. Robust self-triggered control 3.1.1. Computation of reachable sets

Definition 3.1. The target tube

{

_(Xk

,

^k)

,

^k

∈

_N[₁,^N]

}

of the system (1) is reachable from the initial state x₀

∈

_X0 if there exists a sequence of control inputs u_k

∈

_Uk,

∀

k

∈

_N[₀,N−₁], such that the state x_k

∈

_Xk,

∀

k

∈

_N[₁,^N], for all possible disturbance sequences

w

k

∈

_Wk,

∀

k

∈

_N[₀,N−₁].

Let X^∗_N

=

_XN. For k

∈

_N[₀,N−₁], the backward reachable set X^∗_k for the system(1)is recursively computed by:

Pk

= {

z

∈

_R^nx

| ∃

u_k

∈

_Uk

,

^fk(z

,

^uk)

⊕

_Wk

⊆

_X^∗_k+1

} ,

^(2a)

X^∗k

=

_Pk

∩

_Xk

.

^(2b)

Proposition 3.1 (Bertsekas & Rhodes,1971). The target tube

{

_(Xj

,

^j)

,

j

∈

_N[_k+₁,^N]

}

of the system (1) is reachable from x_k if and only if x_k

∈

_Pk. Furthermore, the target tube

{

_(Xk

,

^k)

,

^k

∈

_N[₁,N]

}

is reachable from the initial state x₀

∈

_X0if and only if x₀

∈

_X^∗₀.

According toRaković et al. (2006), Assumptions 2.1and 2.2 make the resulting reachable sets compact.Proposition 3.1indi- cates that if the state x_j

∈

_X^∗_j, the recursive feasibility and the constraint satisfactions can be guaranteed.

Remark 3.1. There exist some methods to compute the reachable sets for a nonlinear system (1), e.g., Chen, Herbert, Vashishtha, Bansal, and Tomlin(2018) andRaković et al.(2006).

In addition, there are results on the inner approximations of the reachable sets X^∗_k, e.g., Althoff and Krogh (2014) and Mitchell (2011). Note that these inner approximations are applicable also for the following algorithms, since they provide constraint satis- faction and recursive feasibility guarantees.

Remark 3.2. Given the initial state x₀, one can choose the minimal horizon N such that

{

_(Xj

,

^j)

,

^j

∈

_N[₀,N]

}

is reachable from x₀.

3.1.2. Algorithm

Define the self-triggered condition for the system(1)as k_i+₁

=

max

{

k

|

k_i

<

^k

≤

N such that u_j

=

u

∈

_Uj

,

j

∈

_N[_ki,^k−₁]

,

and the target tube

{

_(Xj

,

^j)

,

^j

∈

_N[_ki,^N]

}

of(1)is reachable

} .

Recall that k_i+₁

=

k_i

+

M_i. The following lemma provides the formulation to compute M_i.

Proposition 3.2. Given the state x_ki

∈

_X^∗_k

i, k_i

∈

_N[₀,^N−₁], the inter- sampling time M_i is obtained by solving the following optimization problem, denoted byP_[¹_k

i,N](x_ki):

max

u N

∑

j=_ki+₁

r_j (3a)

subject to

X

˜

^[k_i,^ki]

= {

x_ki

} ,

^(3b)

∀

j

∈

_N[_ki,N−₁]

: ˜

_X[_ki,j+₁]

=

f_j(_X

˜

_[

ki,j]

,

^u)

⊕

_Wj

,

^(3c)

∀

j

∈

_N[_ki+₁,N]

:

r_j

=

{

1_X^∗

ki+1(_X

˜

_[

ki,ki+₁])1_U

ki(u)

,

^j

=

k_i

+

1

,

r_j−₁1_X^∗

j(_X

˜

_[k

i,j])1_Uj−1(u)

,

^j

>

^ki

+

1

,

^(3d)

where f_j(X

,

^u)

= {

z

∈

_R^nx

|

z

=

f (x

,

^u)

, ∀

^x

∈

_X

}

. That is, M_i

= ∑

N

j=_ki+₁r_j^∗, where r_j^∗ corresponds to the optimal solution of P_[¹_k

i,N](x_ki).

Proof. The definition of r_j characterizes the successive con- straint satisfactions from time k_i for all possible disturbances

w

l

∈

_Wl

, ∀

^l

∈

_N[_ki,j−₁]. Then, the proof directly follows from Proposition 3.1and the objective function ofP_[¹_k

i,^N](x_ki).

The geometric interpretation of the optimization problem P_[¹_k

i,N](x_ki) is to seek a fixed control input u such that starting from time k_i, the time length, during which the state constraints and the control input constraints are satisfied for all possible disturbances, is maximized.

We denote by u^∗ the optimal solution to the optimization problemP_[¹_k

i,N](x_ki). The following Algorithm 1presents the ro- bust self-triggered control for the uncertain constrained system (1).

3.2. Control with minimum number of samplings

This subsection will provide a control method with minimum number of samplings, denoted by T^∗

+

1, over a given finite

Algorithm 1 Robust self-triggered control Offline:

Determine a sequence of backward reachable sets

{

_(X^∗_j

,

^j)

,

^j

∈

N^[0,N]

}

by(2).

Online:

1: Initialize i

=

0. If x₀

∈

_X^∗₀, continue. Else, stop.

2: Sample the state x_ki, solveP_[¹_k

i,N](x_ki) to obtain u^∗and M_i.

3: Set k_i+₁

=

k_i

+

M_i. Implement u_j

=

u^∗,

∀

j

∈

_N[_ki,ki+1−₁] to system(1)for some realizations

w

j

∈

_Wj, j

∈

_N[_ki,ki+1−₁].

4: Set i

=

i

+

1.

5: If k_i

<

N, go to step 2. Else, stop.

horizon for the system(1). Without loss of generality, we assume that N

≥

2.

Proposition 3.3. The minimum number of samplings T^∗

+

1 is obtained by solving the following optimization problem, denoted by P_[²_0,N](x₀),

min

u0,∆j N−₂

∑

j=₀

(1

−

1{₀}(∆j)) (4a)

subject to

∀

j

∈

_N[₀,N−₁]

:

x_j+₁

=

f_j(x_j

,

^uj)

,

^(4b)

∀

j

∈

_N[₀,N−₁]

:

u_j

=

{

u₀

,

^j

=

0

u_j−1

+

_∆j−1

,

^j

≥

1

,

^(4c)

∀

j

∈

_N[₁,N]

:

x_j

∈

_X^∗_j

,

^(4d)

∀

j

∈

_N[₀,N−₁]

:

u_j

∈

_Uj

.

^(4e) That is, T^∗

= ∑

N−₂

j=₀(1

−

1{₀}(∆^∗_j^{)), where}∆^∗_j corresponds to the optimal solution ofP_[²_0,N](x₀).

Proof. In (4c), ∆j−1 denotes the difference between uj and u_j−₁. The objective function ofP_[²_0,N](x₀) aims at maximizing the number of zero (i.e.,∆j

=

0) over the time interval N^[0,N−₁]. Thus, the optimal solution generates a sequence of piecewise constant control inputs with minimum number of switching times.

Note that Algorithm1is applicable for deterministic systems.

In this case, the difference is that Algorithm 1 cannot guar- antee the achievement of minimum number of samplings for deterministic systems.

Remark 3.3. For uncertain constrained systems, it is difficult to design the control with minimum number of samplings since at each sampling instant k_i, the exact executions of the future disturbance

w

j,

∀

j

∈

_N[k_i,^N−1], are unknown.

4. Self-triggered control for linear systems with polyhedral constraints

The development of geometry software allows us to compute the sets X^∗_kexactly and efficiently if the system is linear and the constraint sets are polyhedral (Herceg, Kvasnica, Jones, & Morari, 2013). This section will specialize the systems(1)to be linear and the constraint sets to be polyhedral. We can reformulate the opti- mization problemsP_[¹_k

i,N](x_ki) andP_[²_0,N](x₀) to be computationally tractable MILP problems.

If the model f_kis linear, the system(1)becomes

x_k+₁

=

A_kx_k

+

B_ku_k

+ w

k

.

⁽⁵⁾

Here A_kand B_kare deterministic real matrices with appropriate dimensions at each time k

∈

_N[₀,^N−₁]. The control input sets Uk, k

∈

_N[₀,N−₁], are compact polyhedra. Each set Xkof the target tube

{

_(Xk

,

^k)

,

^k

∈

_N[₁,N]

}

is a compact polyhedron. The disturbance sets Wk, k

∈

_N[₀,N−₁], are compact polyhedra.

Now, the computation of the sets X^∗_kin(2)is given as follows.

Note that the following equations involve only set operations and the corresponding sets can be well-defined even if the matrix A_k is not invertible. Hence, we do not impose any assumption on A_k. Lemma 4.1 (Bertsekas & Rhodes, 1971). For the uncertain linear system(5)with polyhedral constraints, the set X^∗_kin(2)evolves as

Qk

=

_X^∗_k

⊖

_Wk

,

^(6a)

Pk

=

A⁻_k¹(Qk+₁

⊕

(

−

B_kUk))

,

^(6b)

X^∗k

=

_Pk

∩

_Xk

,

X^∗N

=

_XN

.

^(6c)

Remark 4.1. Since the sets Xk, k

∈

_N[₀,N], are compact, the sets X^∗k, k

∈

_N[0,^N], are also compact even when the matrices A_kare not invertible.

The polyhedral sets Ukand X^∗_kin(6)are written as Uk

= {

z

∈

_R^nu

|

E_kz

≤

e_k

} ,

X^∗k

= {

z

∈

_R^nx

|

F_kz

≤

f_k

} ,

where E_k and F_k (or e_k and f_k) are matrices (or vectors) with appropriate dimensions.

4.1. Robust self-triggered control

Before providing the main result, we need some preliminary lemmas.

Lemma 4.2. The set_X

˜

_[k

i,j]in(3c)can be written as

X

˜

^[ki,j]

=

(G[_ki,j]x_k

+

H[_ki,j]u)

⊕

_Z[_ki,j]

,

^j

∈

_N[_ki,N]

,

⁽⁷⁾ where G[_ki,j]

= ∏

j−₁

l=_kiA_l, H[_ki,j]

= ∑

j−₁ m=_ki

∏

j−₁

l=_m+₁A_lB_m, Z^[ki,j]

=

⨁

j−₁ m=_ki

∏

j−₁

l=_m+₁A_lWm. Furthermore, the set Z^[ki,j] is a closed poly- hedron.

Proof. When j

=

k_i,(7)implies that_X

˜

_[

ki,ki]

= {

x_ki

}

. According to the definition of_X

˜

_[k

i,^j], j

∈

_N[k_i+1,^N], in(3c), by induction, it follows

X

˜

^[ki,^j+₁]

=

A_jX

˜

_[

ki,j]

⊕

B_ju

⊕

_Wj

=

A_j(G[_ki,^j]x_ki

+

H[_ki,^j]u

⊕

_Z[_ki,^j])

⊕

B_ju

⊕

_Wj

=

(A_j

j−₁

∏

l=_ki

A_lx_ki

+

(A_j

j−₁

∑

m=_ki j−₁

∏

l=_m+₁

A_lB_m

+

B_j)u)

⊕

(A_j

j−1

⨁

m=_ki j−1

∏

l=_m+₁

A_lWm

⊕

_Wj)

=

(

j

∏

l=ki

A_lx_ki

+

j

∑

m=ki j

∏

l=m+1

A_lB_m)

⊕

(

j

⨁

m=ki j

∏

l=m+1

A_lWm)

=

(G[_ki,^j+₁]x_ki

+

H[_ki,^j+₁]u)

⊕

_Z[_ki,^j+₁]

.

Since the sets Wmare compact polyhedra, we have that the sets Z^[ki,j]are closed polyhedra.

Lemma 4.3 (Blanchini & Miani,2007). Given two polyhedra P

= {

z

∈

_Rⁿ

|

Pz

≤

p

}

_{and Q}

= {

z

∈

_Rⁿ

|

Qz

≤

q

}

_{, P}

⊆

Q holds if and only if there exists a non-negative matrix S such that SP

=

Q and Sp

≤

q.

Assume now that Z^[ki,j]

= {

z

∈

_R^nx

|

V[_ki,j]z

≤ v

^[ki,j]

}

, j

∈

_N[_ki+₁,N], where the matrix V[_ki,j] and vector

v

^[ki,j] can be computed offline according to Z^[ki,j]

= ⨁

j−1

m=_ki

∏

j−1

l=_m+₁A_lWm. By Lemma 4.3, we derive the following result.

Lemma 4.4. For the sets _X

˜

_[k

i,j] and X^∗_j,_X

˜

_[k

i,j]

⊆

_X^∗_j holds if and only if there exists a non-negative matrix S[_ki,j]such that

S[_ki,j]V[_ki,j]

=

F_j

,

⁽⁸⁾

S[_ki,j](

v

^[ki,j]

+

V[_ki,j](G[_ki,j]x_ki

+

H[_ki,j]u))

≤

f_j

.

⁽⁹⁾ Proof. This directly follows fromLemmas 4.2–4.3.

Since u is the decision variable, the constraints(9)are nonlin- ear. To remedy this, we can calculate the nonnegative matrices S[_ki,j] offline to satisfy (8) by an LP (Fleming, Kouvaritakis, &

Cannon,2013):

(S[_ki,^j])_l

=

argmin

a^T

{

1^Ta

|

a^TV[_ki,^j]

=

(F_j)_l

,

^a

≥

0

} ,

⁽¹⁰⁾ where a is a vector with appropriate dimension and (S)_l denotes the lth row of the matrix S.

Remark 4.2. The LP in(10)admits a nonnegative matrix solution with minimum infinity norm, which could lead to a larger fea- sible region for the optimization problemP_[¹_k

i,^N](x_ki) than other nonnegative solutions to(8).

The next theorem shows that the robust self-triggered control for the system(5)with polyhedral constraints can be designed by solving a computationally tractable MILP.

Theorem 4.1. For the uncertain linear system(5)with polyhedral constraints, the problemP_[¹_k

i,N](x_ki) can be reformulated as an MILP, denoted byP_[³_k

i,N](x_ki), max

u,δj N

∑

j=_ki+₁

(1

− δ

j) (11a)

subject to

∀

j

∈

_N[_ki+₁,^N]

:

S[_ki,j](_G

˜

_[

ki,j]x_ki

+ ˜

H[_ki,j]u)

≤ ˜

f[_ki,j]

+ δ

jΓ¹

,

^(11b)

∀

j

∈

_N[_ki,N−₁]

:

E_ju

≤

e_j

+ δ

jΓ¹

,

^(11c)

∀

j

∈

_N[_ki+₁,N−₁]

: δ

j

≤ δ

j+₁

,

^(11d)

∀

j

∈

_N[_ki+₁,^N]

: δ

j

∈ {

0

,

¹

} ,

^(11e) where_G

˜

_[

ki,j]

=

V[_ki,j]G[_ki,j],_H

˜

_[

ki,j]

=

V[_ki,j]H[_ki,j],

˜

_f[

ki,j]

=

f_j

−

S[_ki,j]

v

^[ki,j], andΓ is a positive constant satisfying

Γ

>

^max

{

max

j∈N[_ki,^N−1]

∥

_Uj

∥

∞

,

max

j∈N[_ki+1,N]

max

u∈Uj

∥

S[_ki,^j](_G

˜

_[

ki,^j]x_ki

+ ˜

H[_ki,^j]u)

− ˜

f[_ki,^j]

∥

∞

} .

⁽¹²⁾ Proof. Recall the definition of r_j in (3d). Let us introduce a sequence of 0–1 variables

δ

j, j

∈

_N[_ki+₁,N]. By setting r_j

=

1

− δ

j, we have

⎧

⎨

⎩

∀

l

∈

_N[ki+1,^j]

:

S[_ki,^l](_G

˜

_[

ki,^l]x_ki

+ ˜

H[_ki,^l]u)

≤ ˜

f[_ki,^l]

+ δ

lΓ¹

,

∀

l

∈

_N[_ki,j−₁]

:

E_lu

≤

e_l

+ δ

lΓ¹

,

whereΓ is a positive number satisfying(12). Furthermore,

∀

j

∈

N^[ki+₁,N−₁], r_j

≥

r_j+₁can be rewritten as

δ

j

≤ δ

j+₁. Then, we get the problemP_[²_k

i,N](x_ki).