Invariant convex approximations of the minimal robust invariant set for linear difference inclusions

Nonlinear Analysis: Hybrid Systems 27 (2018) 289–297

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Invariant convex approximations of the minimal robust invariant set for linear difference inclusions Mohammad S. Ghasemi, Ali A. Afzalian * Department of Electrical Engineering, Abbaspour School of Engineering, Shahid Beheshti University, Tehran, Iran

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Article history: Received 23 June 2016 Accepted 6 September 2017

Keywords: Minimal robust invariant set Linear difference inclusions Switched linear systems Convex approximation

a b s t r a c t In this article, we propose a new computationally efficient algorithm for computing an outer convex robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for polytypic linear difference inclusion (PLDI) systems with additive disturbances. The LDI modelling framework is useful to analyse parametrically uncertain, time-varying linear system or switching linear discrete-time systems. The disturbance which is considered in this paper, is bounded by a polytypic set and acts additively on the state of the system. It is also assumed that the nominal LDI system is absolutely asymptotically stable by a stabilizing linear state feedback. The accuracy of the approximation can be set in advance. The proposed algorithm has far less computational burden in comparison with existing algorithms. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction It is well known that the state of a stable linear system with additive disturbances converges to an invariant set which is known as the minimal robust positively invariant (mRPI) set, see [1,2]. This is also true in the case of a system described by linear difference inclusion (LDI) with additive bounded disturbances [1,3,4]. Linear difference inclusions (LDI) are simple and practical approaches to describe systems with parametric uncertainty; see, for example, [5,6], time-varying linear system and linear systems with switching dynamics [7]. Robust positively invariant set plays an important role in many robust controller design approaches [8–10]. Recently, the computation of mRPI set for LDI systems attracted attention, partly because of the emergence of new robust controller for hybrid systems [11–14], and also the development of fault tolerant control (FTC) schemes through the use of set-theoretic methods [15,16]. Unfortunately, exact mRPI sets can be obtained only for restricted classes of systems and, in general, outer approximations have to be employed instead. Besides, the mRPI set for LDI system is not convex in general, and its convex approximation is required in many control methods. Several authors have developed procedures for computation of RPI outer approximations of the mRPI set, with prespecified accuracy; see, for instance, a procedure proposed in [1,2]. However, those papers address only approximation techniques for autonomous linear discrete-time invariant systems. Since the exact mRPI set for LDI systems is generally non-convex, a simple computational scheme for constructing convex RPI approximation of the mRPI set is required. To the best knowledge of the authors, the only algorithm for computation of the outer RPI ε -approximation of the mRPI set for linear difference inclusions was given in [3,17]. The main drawback of this algorithm is its very increasing computational burden in many practical situations. Although, the approximation of the mRPI set is computed offline in many cases, the algorithm presented in [3,17] may need so many computational resources, that one cannot reach an approximation set with desirable

*

Corresponding author. E-mail addresses: [email protected] (M.S. Ghasemi), [email protected] (A.A. Afzalian).

https://doi.org/10.1016/j.nahs.2017.09.001 1751-570X/© 2017 Elsevier Ltd. All rights reserved.

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accuracy. [13] proposed a method to compute the outer bound convex RPI approximation of mRPI set for piecewise linear (PWL) systems. However, no condition was elaborated in [13] to a priori specify the accuracy of this approximation. In [18,19], characterization and computation of the minimal and maximal Disturbance Dwell-Time invariance sets for constrained switching systems under dwell-time switching are considered. In this paper, we develop a new computationally efficient algorithm for calculation of the convex ϵ-approximation to the minimal RPI set for LDI systems. The accuracy of this approximation can be established in advance. As we will show in a comparison study, the proposed algorithm has far less computational burden in comparison with the algorithm presented in [3,17]. The paper is organized as follows. Firstly, some preliminaries on the minimal robust invariant set of the LDI systems are reviewed. Most of the material in this section is borrowed from [3,17]. Section 3 describes the proposed novel algorithm to compute outer convex RPI approximation of the minimal robust invariant set for LDI system. In Section 4, the efficiency of the proposed algorithm is investigated by comparative examples. Finally, concluding remarks are outlined. 1.1. Notations n n Let N ≜ {0, 1, . . .} , N+ ≜ {1, 2, . . .} , Nq ≜ {0, 1, . . . , q} and N+ q ≜ {1, 2, . . . , q}. Given two sets A ⊂ R and B ⊂ R n and a vector x ∈ R , the Minkowski set addition is defined by A ⊕ B := {x + y|x ∈ A, y ∈ B}, and we write x ⊕ A instead of {x} ⊕ A. A polyhedron is the (convex) intersection of a finite number of open and/or { closed half-spaces } and a polytope is a closed and bounded polyhedron we use co () to denote convex hull. Let Bnζ (µ) = z ∈ Rn | ∥z ∥ζ ≤ µ , where ∥z ∥ζ refer to the ζ -norm of vector z ∈ Rn .

2. Preliminaries We use a convention of notation similar to the one in [17] for describing polytopic linear difference inclusion: x+ ∈ F (x, A, W) F (x,{A, W) ≜ {Ax + w|} A ∈ co (A) , w ∈ W} A ≜ Ai ∈ Rn×n |i ∈ N+ q

(1)

where, x ∈ Rn denotes the current state, x+ denotes the successor state, w ∈ Rn is an unknown disturbance, and q ∈ N+ is a finite integer. The system transition matrix A is uncertain and is known only to the extent that it belongs to the convex hull of a finite set A of known matrices Ai ; furthermore, A is in principle time-varying and different A from the set co (A) can occur at different times, so that: A=

q ∑

λi Ai , λi ∈ R,

i=1

q ∑

λi ≤ 1

i=1

where, λi can vary with time. Assumption 1. W ⊂ Rn is a convex polytope and contains the origin in its interior. Assumption 2. When W = {0} , the linear difference inclusion (1) is Absolutely Asymptotically Stable (AAS), for all A ∈ co (A). Recalling a set of relevant results on stability of linear difference inclusions when W = {0}, [7,20], one common approach to hold Assumption 2 is to find a pair (P , ψ) ∈ Rn×n × (0, 1), such that P = P T > 0 and ATi PAi − P ≤ −ψ P ,

∀i ∈ N+ q .

(2)

However, many LDI systems may not admit a common P but they are still stable. It should be noted that the proposed algorithm can be used in all LDI systems, provided the system stability is guaranteed, either by a common P or by any other methods [7,20,21]. We also need the following definition Definition 1 ([22]). The support function for a set V ⊂ Rn , evaluated at a ∈ Rn , is defined as hV (a) ≜ sup aT v. v∈V

If V is a polytope, then hV (a) is finite. Furthermore, if V is described by a finite set of affine inequality constraints, then hV (a) can be computed by a linear programming solver [1]. Among properties of the support function, see [22], set inclusion condition can be given in terms of support functions. Property 1. Given a set V ⊆ Rn , and a closed, convex set U ⊆ Rn , then V ⊆ U if and only if hV (η) ≤ hU (η) , for all η ∈ Rn . Testing the inclusion V ⊆ U is much easier when U is a polyhedron, U = u|sTi u ≤ ri , i = 1, . . . , N .

{

}

Then V ⊆ U, if and only if hV (si ) ≤ ri , i = 1, . . . , N. Also it can be easily confirmed that hU ⊕V (η) = hU (η) + hV (η).

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2.1. Minimal Robust invariant set for LDI systems Definition 2 (One-step Forward Disturbance Reachable Set). The one-step forward disturbance reachable set D (Ω , A, W) for the difference inclusion (1) is the set of states in which system (1) can evolve at the next time step from x ∈ Ω , for all allowable disturbances w ∈ W. To characterize D (Ω , A, W) for system (1), let D (Ω , A, W) ≜ {Ax + w|x ∈ Ω , A ∈ co (A) , w ∈ W} .

(3)

Definition 3. A set Ω is a robust positively invariant (RPI) set of the difference inclusion (1), if D (Ω , A, W) ⊆ Ω . Consider the set sequence {Dk } , k ∈ N defined by the following recursion: Dk+1 ≜ D (Dk , A, W) , k ∈ N+ with D0 = {0} .

(4)

A fundamental computational problem with respect to the set sequence {Dk } is the fact that the sets {Dk } are not necessarily convex [3,5]. Definition 4. A set D∞ is the minimal robust positively invariant (mRPI) set for the difference inclusion (1), if D∞ is an RPI set and D∞ is contained in every closed RPI set for the difference inclusion (1). Theorem 1 ([3]). Under Assumption 2, the set sequence Dk , k ∈ N has the following properties: I. Dk ⊆ Dk+1 for all k ∈ N. II. There exists a compact set D∞ such that H (D∞ , Dk ) → 0 as k → ∞, where H (D∞ , Dk ) is Hausdorff distance and D∞ is the minimal RPI (mRPI) set of system (1). Theorem 1 shows the existence and uniqueness of the minimal closed RPI set D∞ for linear difference inclusion (1). Therefore, it can be easily shown that under Assumption 2 for any x0 , the evaluation of xk of the system (1) is robustly steered to D∞ [13]. Therefore, this set is the limit set of all trajectories of system (1). Unfortunately, the set D∞ does not generally admit an explicit representation, and it is difficult to be computed exactly. On the other hand, since sets Dk are not convex, the set D∞ is not convex for the LDI systems in general. Furthermore, it is more appropriate for constrained control problems, when the constraints are convex, to compute its convex outer bound of D∞ . Therefore, we consider the set sequence: Fk+1 ≜ co (D (Fk , A, W)) ,

k ∈ N+ w ith F0 = {0} .

An alternative form for the set sequence {Fk } is given by:

⎞ ⋃ ⎟ ⎜ {Ai Fk }⎠ ⊕ W, k ∈ N+ , F0 = {0} . = co ⎝ ⎛

Fk+1

(5)

i∈N+ q

Proposition 1. Suppose Assumption 1 holds, and consider the set sequences {Dk } and {Fk } defined by (4) and (5), respectively, then the set sequence Fk has the following properties: Fk = co (Dk ) , ∀k ∈ N+

and F∞ = co (D∞ ) is a convex RPI set. Proof. See Proposition 3 in [3]. By Proposition 1, since set F∞ is the convex hull of the set D∞ , then it is the tightest convex approximation of the D∞ . Now, the main problem is to obtain an explicit description for F∞ by means of practical computation. Unfortunately, it is difficult to obtain an explicit description of the set F∞ , and it is only possible to obtain a family of outer RPI approximation of the set F∞ . To the best knowledge of the authors, the only algorithm for computation of the outer, convex, RPI ϵ-approximation of F∞ was given in [17]. The main drawback of this algorithm is its increasing computational burden, when q and k grow. Therefore, it is not viable in many practical situations. In the next section, we develop a novel algorithm for calculation of the outer convexϵ-approximation of the F∞ . This algorithm has far less computational burden in comparison with the algorithm presented in [17]. Firstly, we propose a proposition which will be used in the next section. Proposition 2. Under Assumption 2, the set sequence Fk , k ∈ N has the following properties: I. Fk ⊆ Fk+1 for all k ∈ N. II. There 1 exist an η ∈ (0, 1) and a norm-ball Bnζ (θ), such that F∞ ⊆ 1−η Bnζ (θ). Proof. I: By Theorem 1, Dk ⊆ Dk+1 for all k ∈ N. Therefore, co (Dk ) ⊆ co (Dk+1 ) . Then, ⋃ by Proposition 1, Fk ⊆ Fk+1 . II: Assumption 2 implies that there exist an η ∈ (0, 1) and a ball Bnζ (θ) ⊇ W, such that i∈N+ Aki Bnζ (θ)⊆ηBnζ (θ). We assume q

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M.S. Ghasemi, A.A. Afzalian / Nonlinear Analysis: Hybrid Systems 27 (2018) 289–297 1−ηk+1

that Fk+1 ⊆

1−η

Bnζ (θ). Thus, we have

⎞ ⎛ ⎞ { } k+1 ⋃ ⋃ 1−η ⎟ ⎜ ⎟ ⎜ {Ai Fk+1 }⎠ ⊕ W ⊆ co ⎝ Ai Bnζ (θ) ⎠ ⊕ W = co (D (Fk+1 , A, W)) = co ⎝ 1 − η + + ⎛

Fk+2

i∈Nq

i∈Nq

1 − ηk+1

⊆

1−η

ηBnζ (θ) + Bnζ (θ) =

We conclude that F∞ ⊆

1 Bn 1−η ζ

1 − ηk+2 1−η

Bnζ (θ) .

(θ) . ■

3. Main contribution: outer convex approximation of the F∞ Motivated by the fact that it is very difficult to obtain a simple computational scheme for constructing the set F∞ , we propose a description of convex sets that are outer-bounds for the set F∞ , in this section. Following the idea presented in [13], the outer approximation of F∞ has the form σ Fk for some scalar σ ≥ 1 and some index k in such a way that, for a given error bound ϵ > 0 the following relation holds F∞ ⊆ σ Fk ⊆ F∞ + Bζ (ϵ)

and σ Fk is convex RPI set of system (1). By( Definition 3, it)is obvious that σ Fk is RPI, if and only if D (σ Fk , A, W) ⊆ σ Fk . Using (5), it can be stated that σ Fk is RPI if co

⋃

i∈N+ q

{Ai σ Fk } ⊕ W ⊆ σ Fk .

{

( )T

Assumption 3. Fk is a polytope that contains the origin. Thus, it can be written as Fk ≜ x ∈ Rn | fjk

}

x ≤ 1, ∀j ∈ I , where

fjk ∈ Rn and I is a finite index set. This assumption simply holds for the set sequence {Fk } according to Assumption 1. Lemma 1. Let

⎛

⎞ ⋃ ⎜ ⎟ {Ai Fk }⎠ , Ck ≜ co ⎝ i∈N+ q

( k)

{

and K ≜ k|hCk fj occurs when

≤ 1 , ∀j ∈ I × N + q }. Then, for all k ∈ K, the tightest inclusion σk Fk , provided that σk Fk is convex RPI for (1),

( )

σk = max j∈ I

hW f j k

( ).

(6)

1 − hC k f j k

Proof. For a given index k, the tightest inclusion σk Fk , provided that σk Fk is RPI occurs when

⎧ ⎫ ⎛ ⎞ ⏐ ⎪ ⎪ ⎨ ⏐ ⎬ ⏐ ⎜⋃ ⎟ + {Ai σ Fk }⎠ ⊕ W ⊆ σ Fk , ∀i ∈ Nq . σk = min σ ⏐co ⎝ ⎪ ⎪ ⎩ ⏐ ⎭ i∈N+ q

Using Property 1 and by Assumption 3, co

(⋃

i∈N+ q

) ( ) ( ) {Ai σ Fk } ⊕ W ⊆ σ Fk can be expressed as hCk fjk + σ1 hW fjk ≤ 1, for

all j ∈ I. By definition of the set K, hCk fjk ≤ 1, for all k ∈ K. Thus it follows that for all

( )

( )

σ ≥ max j∈I

hW fjk

( ),

1 − hC k f j k

the set σ Fk is a convex RPI set of system (1). ■ According to Proposition 1, F∞ is the tightest convex approximation of the mRPI set for (1). Therefore, F∞ ⊆ σk Fk , for all k ∈ K. Since Fk → F∞ as k → ∞, and F∞ is a convex RPI set, then it can be clearly concluded that σk → 1 as k → ∞. Also, the set σk Fk is an outer convex RPI approximation of F∞ . However, the accuracy of this approximation cannot

be priori identified. Besides, it is obvious that for a given set K, a lower value of k generally results in lower complexity for description of Fk . Therefore, it is worthwhile to provide an algorithm to compute the minimum value of k ∈ K such that the sets σk Fk is ϵ -approximation of F∞ i.e. σk Fk ⊆ F∞ + Bζ (ϵ). In Theorem 2, we propose a condition that allows one to a priori specify the accuracy of this approximation.

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293

Theorem 2. If 0 ∈ int (W), then for all ϵ > 0 and k ∈ N+ , there exists a σ > 1, such that

(σ − 1) Fk ⊆ Bζ (ϵ) , holds. Furthermore, if σk ≤ σ , where σk is calculated from (6), then σk Fk is a convex RPI, outer ϵ -approximation of F∞ . Proof. From Proposition 2, F∞ ⊆ then it follows that

1 Bn 1−η ζ

(σ − 1) Fk ⊆ (σ − 1) F∞ ⊆ The last term holds for σ ≤ 1 +

(θ ) and Fk ⊆ F∞ , for all k ∈ N+ . Since, Fk and F∞ are convex and contain the origin,

σ −1 Bζ (θ) ⊆ Bζ (ϵ) . 1−η ϵ(1−η) . θ

Let σk ≤ σ in which the value of σk is calculated from (6) for a specific k, then

σk Fk ⊆ σ Fk = (σ + 1 − 1) Fk ⊆ Fk ⊕ Bζ (ϵ) ⊆ F∞ ⊕ Bζ (ϵ) . ■ According to Theorem 2, for a given ϵ , it is sufficient to increase k until (σk − 1) Fk ⊆ Bζ (ϵ). In this case, the set σk Fk ⊆ F∞ ⊕ Bζ (ϵ) is an outer convex RPI approximation of F∞ for linear difference inclusion (1). Now, we propose an algorithm for computation of the minimum value of k, such that the set G (σk − 1, k) ≜ σk Fk be the ϵ -approximation of F∞ . Algorithm 1 initially sets k to 1 and increases it at each step. The values of σk are calculated in each iteration using (6). According to Theorem 2, the algorithm stops when (σk − 1) Fk ⊆ Bζ (ϵ), in which the a priori specified accuracy ϵ>0 has been obtained. The ϵ-approximation G (σk − 1, k) of F∞ can then be computed by a simple scaling, i.e. G (σk − 1, k) = σk Fk .

The comparison study on the proposed algorithm and the algorithm presented in [3,17] shows that: (1) The algorithm in [3,17] involves the solution of a number of the following linear programming problems in each iteration:

α o (s) = max

maxis ∈Is maxω∈W wjT Ais ω gi

j∈Nl

M (s) = max

j∈{1,...,n}

{ s−1 ∑ k=0

max max eTj Aik ik ∈Ik ω∈W

ω,

s−1 ∑ k=0

} max max eTj Aik ik ∈Ik ω∈W

ω

where,

{ } • W ≜ ω ∈ Rn |wjT ω ≤ 1, ∀j ∈ Nl , wj ∈ Rn and l is the number of the inequalities of the polyhedral set W. • ej is the jth standard basis vector in Rn . • ik is defined as a set of switching sequences from the first iteration to iteration k, and defineAik ≜ Aik . . . Ai1 Ai0 , for each ik ∈ Ik . • the set Ik is defined as a set of all possible switching sequences ik from the first sample time to sample time s.

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M.S. Ghasemi, A.A. Afzalian / Nonlinear Analysis: Hybrid Systems 27 (2018) 289–297 Table 1 Comparative computation time of the proposed algorithm and algorithm presented in [3] for Example 1. Approximation set D ( 3.78 × 10−7 , 28) G 1.36 × 10−6 , 26

(

)

ϵ

Computation time (s)

10−4 10−4

10 000 1.7

This means that in the sth iteration:

• One should solve lqs number of LP problems for calculation of α o (s). • One should solve, at least, 2nqs number of LP problems in order to calculate M (s). After working out the number of iterations s needed for calculation of the appropriate approximation of the minimal invariant set, one should calculate the set D (s, α) = (1 − α)−1 Fs , in which Fs is computed from the set sequence (5). Therefore, one should solve (l + 2n) qs number of LP problems only in the sth iteration. We see that the computational burden of algorithm in [3,17] increases exponentially in the number of iterations. ( ) with the(increase ) Our algorithm involves the calculation of hW fjk and hCk fjk for all j ∈ I in each iteration, where I is the number of inequalities of the polyhedral sets Fk . This means that we should solve just 2I number of LP problems in each iteration. Since Fk → F∞ as k → ∞, the number of inequalities of the polyhedral set Fk , no matter how large it is, remains relatively constant after some iteration. (2) As it is mentioned in 1, one should take the convex hull of the propagated disturbance set Ck and Minkowski sum Fk in each iteration in both algorithms. The computational effort for Minkowski sum and convex-hull operation are known to increase with the dimension of the system. In this respect, both algorithms have the same computational burden. Reduction of the computational burden of the sets Ck and Fk (Minkowski sum and convex-hull operation) is not the focus of this research, and can be studied in future work. However, since Fk → F∞ as k → ∞, the number of vertices of the polyhedral set Fk will remain relatively constant after some iteration. Hence, the computational burden of the calculation of the set Fk , no matter how much it is, remains relatively constant after some iterations. Summing up, we see that the main computational burden of the proposed algorithm is the calculation of the sets Fk , while the algorithm presented in [3,17], regardless of the calculation of the sets Fk , leads to a combinatorial explosion of LP problems. 4. Illustrative example and comparison study In this section, the efficiencies of the proposed procedure are illustrated using two examples of uncertain, linear discretetime systems, which are modelled as: x+ ={Fx + Bu + w, (F , B) ∈ co (C)}, w ∈ W C ≜ (Fi , Bi ) ∈ Rn×n × Rn×m |i ∈ N+ q .

(7)

The stabilizing state feedback controller takes the form u = Ki x,. The closed loop system has the form (1) with Ai = Fi + Bi Ki . In order to demonstrate the efficiency of the proposed algorithm, we compute the ϵ-approximation D (s, α) of F∞ using the algorithm in [3,17], and also the ϵ-approximation G (σk − 1, k) of F∞ using Algorithm 1. We compare the computation time of the two algorithms Example 1. In the first example, we consider the following 2-dimensional switched linear system with two modes of operations, that was reported in [3,17]:

[

1 F1 = 0

]

[

1 1 , F2 = 1.1 1

]

1 , 1.9

and B = B1 = B2 = [11]T . The additive disturbance set is W ≜ {w ∈ Rn | ∥w∥∞ ≤ 10}. Assumption 3 is satisfied by (2) which holds for ψ = 0.22, K1 = K2 = [−1.4936 − 1.5073], and 0.07133 P = 0.046751

[

0.046751 . 0.094124

]

The set G 1.36 × 10−6 , 26 which is computed using the procedure ( described in )Algorithm 1 is depicted in Fig. 1. This set is computed for ϵ = 10−4 . It also shows the approximation set D (3.78 × 10−7 , 28 )which is computed by the algorithm reported in [3], for ϵ = 10−4 . This approximation set fits exactly on 1.36 × 10−6 , 26 . Table 1 shows the elapsed time for computation of the two approximation sets. Although, the system presented in this example is very simple, the computation time of the algorithm presented in [3] takes approximately 2.8 h (10 000 s) while calculation of this set using our proposed algorithm takes only 1.7 s. The number of inequalities of the polyhedral set F26 is 12. Therefore, our algorithm involves 24

(

)

M.S. Ghasemi, A.A. Afzalian / Nonlinear Analysis: Hybrid Systems 27 (2018) 289–297

295

Fig. 1. Convex approximation sets of mRPI set in Example 1.

LP problems in iteration 26, while the algorithm presented in [3,17] leads to solving (4 + 2) × 228 ≈ 1.6 × 109 LP problems only in iteration s = 28 in order to compute the approximation set with the same accuracy. In Example 2, we consider a more sophisticated system to show further computational advantages of the proposed method. Example 2. This example consists of 3-dimensional switched linear systems under arbitrary switching with four modes of operations. This system is a special case of the LDI system (7). Let

−0.2523 = 0.5290 −0.4415 [ 0.0647 = −0.3131 −0.3085 [ −0.6402 = −0.6693 −0.2812 [ −0.3501 = −0.4808 −0.1217

[

F1

F2

F3

F4

0.4856 −0.2616 −0.2713

0.6467 0.3128 , B1 = −0.6967

0.1729 −0.6691 −0.0613

] [ ] −0.6542 0.6543 −0.6516 , B2 = 0.5266 −0.0099 −0.0558 ] [ ] −0.5629 0.7580 0.1748 , B3 = −0.8050 −0.3526 −0.4059 ] [ ] 0.6695 0.6961 0.3865 , B4 = −0.7619 . −0.0013 −0.2590

−0.5409 −0.6874 −0.4898 0.2590 0.1905 −0.2631

]

0.5656 0.5460 0.9389

[

]

The additive disturbance set is W ≜ {w ∈ Rn | ∥w∥∞ ≤ 10}. Assumption 2 is satisfied by (2) which holds for ψ = 0.1 and k1 = 0.4699

[

k3 = −0.7742

0.1750

0.1591 , k2 = 0.4039

]

[

0.4239

] [ −0.1436 −0.1603 , k4 = −0.0800 [ ] 0.5156 −0.1112 −0.0057 0.5815 0.1579 P = −0.1112 −0.0057 0.1579 0.5825 [

1.1529

]

−0.0405

] −0.2867

Fig. 2 shows G (0.1458, 13) and D (0.1357, 15)which are computed using the proposed algorithm and also the one given in [3] for ϵ = 0.12, respectively. Since the error of the approximation is considered a rather big number, these two sets are slightly different. As we show in Table 2, calculation of the set D (0.1357, 15) takes approximately 21.5 h, while calculation of the set G (0.025, 10)takes only 20 s. Due to the high calculation burden, we cannot compute( ϵ -approximation of F∞ for ) ϵ = 10−5 using algorithm presented in [3]. However, computation of the approximation set G 1 × 10−5 , 76 for ϵ = 10−5 using the proposed algorithm takes only 60 s. The result is shown in Fig. 3. The number of inequalities of the polyhedral set F76 is 94 and the number of vertices of this set is 70. Indeed, in this example, the number of the inequalities (vertices) of the polyhedral sets Fk remained relatively constant for k > 7. Therefore, the proposed algorithm involves just about 192 LP problems in each iteration. While, one should solve (6 + 3) × 415 ≈ 6.4 × 109 LP problems in iteration s = 15 only, in order to compute D (0.1357, 15).

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Fig. 2. The sets D(0.1357, 15) and G(0.1458, 13) in Example 2.

Fig. 3. Convex ϵ -approximation sets of mRPI set in Example 2 for ϵ = 10−5 . Table 2 Comparative computation time of the proposed algorithm and algorithm presented in [3] for Example 2. Approximation set

ϵ

Computation time (s)

D (0.1357, 15) G ((0.1458, 13) ) G 1 × 10−5 , 76

0.12 0.12 10−5

77 400 20 60

YALMIP Toolbox [23] and Multiparametric Toolbox 3. 0 (MPT) [24] are used for the set calculations. These calculations are carried out on a computer with Core i7, 2.3 GHz CPU and 8 GByte RAM. 5. Conclusions We proposed a novel procedure for computation of the outer convex robust invariant (RPI) approximation of the minimal RPI set for linear difference inclusion systems. Conditions are also given that allow one to a priori specify the accuracy of this approximation. The algorithm improves on existing algorithms, since it has far less computational burden. The advantage of the proposed algorithm is illustrated in two examples. The results obtained in this paper can be exploited in robust control of switched linear system subject to constraints and additive but bounded disturbances. Another application is set theoretic method in fault tolerant control design.

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