On positively invariant polyhedrons for discrete-time positive linear systems

In this paper, necessary and sufficient conditions for the polyhedron set to be a positively invariant polyhedron of a discrete-time positive linear system subject to external disturbances are established. By solving a set of inequalities, which is also a linear programming, necessary and sufficient conditions for the existence of positive invariant polyhedra for discrete-time positive linear systems are proposed, and the relationship between Lyapunov stability and positively invariant polyhedron is also investigated, numerical examples illustrate our results.


Introduction
Positive invariance in control theory of dynamical systems has received extensive attention over the past few decades [1].Any state trajectory emanating from a set in the state space still remains within the set, such a set is called a positively invariant set.Invariant sets, especially positively invariant sets, play an important role in the theory and application of dynamical systems.Problems related to disturbance rejection can be analyzed and solved with the help of positively invariant sets [2].Similarly, many constrained control problems of dynamical systems can also be represented and solved by positively invariant sets [3].
For discrete-time linear systems, [4] and [5,6] give descriptions of necessary and sufficient algebraic conditions for the positive invariance of convex polyhedra under both unperturbed and bounded perturbations, respectively.In the form of linear relationship, a set of inequalities is derived, and the invariant set of related systems is defined by the method of linear programming [7].There are also many studies on the computational methods of invariant sets [8], and a different linear programming algorithm is proposed in [9] to give sufficient and necessary conditions for any set of polyhedrons to be positively invariant sets for discrete-time linear systems.However, since the algorithm is limited, some algorithms are not suitable for computing all polyhedron positively invariant sets.Daniel Rubin et al. proposed a special supplementary algorithm [10], and they also proposed a new algorithm to compute the polyhedron positively invariant set [11].Disturbance is also a common problem in the research and analysis of dynamical systems.Reference [12] generalizes not only the concept of self-bounded (A, B) invariant subspaces to sets of convex polyhedra for general discrete-time systems, but also their results to systems subject to control constraints and bounded additive disturbances.A solution to the problem of computing a robustly positively invariant outer approximation of the minimal robustly positively invariant set for a discrete-time linear time-invariant system is proposed in [13].An algorithm for computing the maximal robustly positively invariant set is described, and sufficient conditions for finite termination of this algorithm are given [14].[15] presents an algorithm for the computation of full-complexity polytopic robust control invariant sets, which can be extended to linear discrete-time systems subject to additive disturbances and structured norm-bounded or polytopic uncertainties.
For positive systems, their stability has been extensively studied [16].[17] studied the stability and control problems of positive delayed systems.[18] studied the synthesis problem of interval positive linear systems.About the problems investigated in this paper for positive continuous-time linear systems, reference [19] gives excellent research results, related results can also be found in the references therein.The main contribution of this paper is to give necessary and sufficient conditions for the existence of positively invariant polyhedra for discrete-time positive linear systems by solving a set of linear programming.The same method is applied to discrete-time positive linear systems with external inputs to obtain the conditions for the existence of robustly positively invariant polyhedra.In this paper, some properties of regular invariant polyhedra are elucidated, and their relations with Lyapunov stability are investigated.
The rest of the paper is organized as follows.Section 2 presents a preliminary case of discrete-time positive linear systems.Section 3 defines the positively invariant polyhedron and reveals the close connection between the Lyapunov stability and the positively invariant polyhedron.Section 4 establishes the necessary and sufficient condition for the existence of robustly positively invariant polyhedra under two external input conditions.
Throughout the paper, the following notations are used.N , N + set of integers, set of positive integers N 0 {0} ∪ N R, R n set of real numbers, set of n-dimensional real vectors R m×n set of m × n real matrices Rn + , R n + nonnegative and positive orthants of R n 1, I vector [1, 1, . . ., 1] T , identity matrix [1] matrix with all entries assigned to 1 In this paper, capital letters denote real matrices and lower case letters denote column vectors of scalars.
x ≥ 0 denotes that every component of x is nonnegative, A ≥ 0 denotes that every component of A is nonnegative.It is always assumed that all vectors and matrices have compatible dimensions without specification.

Preliminaries
In this section, some definitions and lemmas related to invariant sets of discrete-time linear systems are introduced.Consider a discrete-time linear dynamical system described by a difference equations in the following form: where x(k) ∈ R n is system state, k ∈ N 0 N 0 = {0} ∪ N , and A ∈ R n×n is a constant system state matrix.
Definition 1 Any nonempty convex polyhedron in R n can be characterized by a matrix G ∈ R r×n and a vector γ ∈ R r , r ∈ N + , n ∈ N + , which is defined by In particular, in this paper we mainly study the polyhedron described by a matrix G ∈ R r×n and a vector γ ∈ R r And the polyhedron described by a matrix G ∈ R r×n and two vectors γ 1 , Definition 2 A nonempty subset M ∈ R n is said to be a positively invariant set of system S 0 if for each initial state x 0 ∈ M the motion emanating from x 0 remains in M.
From Definitions 1 and 2, one can derive that a nonempty polyhedron R[G, γ ] is positively invariant polyhedron for system S 0 if and only if Likewise, the polyhedron Q[G, γ 1 , γ 2 ] is a positively invariant polyhedron for system S 0 if and only if for any The following lemma proposed in [9] provides a sufficient and necessary algebraic condition for the positive invariance of R [G, γ ] and Lemma 1 [9] The polyhedron P[G, γ ] is a positively invariant polyhedron of system S 0 in (1) if and only if there exists a nonnegative matrix H ∈ R r×r + such that GA -HG = 0, (H -I)γ ≤0.

Positive invariance and its relationship with stability
A linear system becomes a positive linear system when matrix A is nonnegative, that is, where x 0 ≥ 0 is the initial state.A polyhedron with respect to system S 1 is characterized by [20] and a necessary and sufficient condition for the existence of a positively invariant polyhedron R + [G, γ ] and Q + [G, γ 1 , γ 2 ] with respect to system S 1 can be derived from Lemma 1, as stated in the following theorem.

Theorem 1
The nonempty set R + [G, γ ] is a positively invariant polyhedron of system S 1 in (2) if and only if there exists a matrix H ∈ R r×r such that GA -HG ≤ 0, Proof Only the necessary condition is proven.According to the description of R + [G, γ ] in (3), it can be rewritten as then the polyhedral set R + [G, γ ] can be written in the form In particular, One can set H 31 = H 32 = 0 and H 33 = A, without losing generality, which is equivalent to Note that H 13 is nonnegative, it concludes that GA -HG ≤ 0, Remark 1 For any matrix H that satisfies the algebraic inequalities condition in Theorem 1, Theorem 1 guarantees the positive invariance of R + [G, γ ] for system S 1 and does not have any requirements for the matrix G.
Note that the positively invariant polyhedron R + [G, γ ] is symmetric in Theorem 1. Next we consider the more general case where the polyhedral sets Q + [G, γ 1 , γ 2 ] are not symmetric.In the following theorem, we establish conditions for the positive invariance of polyhedral sets is a positively invariant polyhedron of system S 1 in (2) if and only if there exists a matrix H ∈ R r×r such that GA -HG ≤ 0, Proof Only the necessary condition is proven.The polyhedral sets Q + [G, γ 1 , γ 2 ] on the basis of observation and description of 4) can be rewritten in the form which can be rewritten as One can set H 31 = H 32 = 0 and H 33 = A, without losing generality, which is equivalent to Note that H 13 ≥0 and H 23 ≥ 0, from ( 5) and ( 6) it can be obtained that From ( 7) and ( 8), which can be written as then they can be rewritten as So, which are further equivalent to are positive respectively.A necessary and sufficient condition for the existence of a positively invariant polyhedron Q + [G, γ 1 , γ 2 ] when γ 1 = 0 is given in the following corollary.The proof is omitted since it is similar to the proof of Theorem 2.

Corollary 1
The set Q + [G, 0, γ 2 ] is a positively invariant polyhedron of system S 1 in (2) if and only if there exists a matrix H ∈ R r×r such that GA -HG ≤ 0, (H -I)γ ≤ 0.
The positively invariant polyhedron R + [G, γ ] of system S 1 can also be constructed by an invariant polyhedron of similar systems with nonsingular matrix T ∈ R n×n .
By the transformation of state y(k Since Sufficiency.By a similarity transformation of state x(k) = Ty(k), x(k) satisfies the equation Remark 2 Since T -1 AT is not necessarily a nonnegative matrix, system S * 1 may no longer be a positive system.Theorem 3 clarifies the connection between the invariant polyhedron construction of a positive system and a general system.Meanwhile, the above conclusion is also satisfied for Q + [G, γ 1 , γ 2 ] as shown in the following corollary.The proof is omitted.
Remark 3 In the case of γ 1 = 0, the conclusion that Q + [G, 0, γ 2 ] is a positively invariant polyhedron of positive system S 1 if and only if Q[GT, 0, γ 2 ] ∩ P[-T, 0] is a positively invariant polyhedron of system S * 1 is also valid.

Relation with stability
A well-known result in [4] is that if system S 0 is asymptotically stable, then it possesses positively invariant sets of the form where P ∈ R n×n is a symmetric positive-definite matrix and c is a positive real number.Furthermore, for a symmetric and positive-definite matrix P ∈ R n×n , the corresponding hyperellipsoid is a positively invariant set of system S 0 if and only if there exists a positive semidefinite matrix Q ∈ R n×n such that A T PA -P = -Q.For the positive system S 1 , the following theorem reveals the close connection between the Lyapunov stability and the existence of a positively invariant polyhedron.Theorem 4 Positive system S 1 in (2) possesses at least a positively invariant polyhedron R + [G, γ ] with nonzero vector γ ∈ R n + if and only if system S 1 is Lyapunov stable.
Proof Necessity.Since R + [G, γ ] is a closed convex set, it can be defined by the expression where for any i = 1, 2, . . ., r and V (x) > 0 for all x = 0.
From Theorem 1, there must exist a matrix H ∈ R r×r such that GA -HG ≤ 0 and (|H| - which implies that V (Ax) ≤ V (x), that is, V (x) ≤ 0. Hence system S 1 is Lyapunov stable.Sufficiency.Since system S 1 is Lyapunov stable, there must exist a nonzero vector γ ∈ R n + such that (A -I)γ ≤ 0, which is equivalent to [(A -I)γ ] i ≤ 0. Due to A is a Schur matrix with all the eigenvalues in absolute value smaller than 1, that is, ρ(A) < 1 [20], then A k γ ≤ γ , k ∈ N 0 .Furthermore, taking into account the fact that A is nonnegative and ] is a positively invariant polyhedron of system S 1 .

Theorem 5 Positive system S 1 in (2) possesses at least a positively invariant polyhedron
is a closed convex set, it can be defined by the expression where for any i = 1, 2, . . ., r and V * (x) > 0 for all x = 0.
From Theorem 2, there must exist a matrix H ∈ R r×r such that GA -HG ≤ 0, which implies that V (Ax) ≤ V (x), that is, V (x) ≤ 0. Hence system S 1 is Lyapunov stable.Sufficiency can be obviously evaluated from the sufficient proof of Theorem 4 by assigning -γ 1 = -γ and γ 2 = γ .

Invariant polyhedron with exogenous inputs
In this section, we consider a positively invariant polyhedron for discrete-time positive linear systems with external inputs.Consider a discrete-time linear dynamical system described by the difference equations where , is a closed convex set, and k ∈ N 0 .A nonempty polyhedron R + [G, γ ] is said to be a robustly positively invariant polyhedron of system S 2 with respect to if for each initial state x 0 ∈ R + [G, γ ] the motion emanating from x 0 remains in R + [G, γ ] for all possible ω(k) ∈ .When = {0}, the positive invariance is equivalent to the definition of positive invariance characterized in Sect.3. The necessary and sufficient condition for the existence of positively invariant polyhedra R + [G, γ ] based on (∞, 1)-norm is given below. When 1 is that the sum of the components of ω(k) is not to exceed 1.

Theorem 6
The polyhedron R + [G, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,1 if and only if there exists a matrix H ∈ R r×r such that GA -HG ≤ 0, in which [1] is an r × m-dimensional matrix with all elements being 1.
Proof Necessity.An augmented system can be constructed from system S 2 in (10) as follows: , which is defined as Emanating from any , where x 0 and ω 0 are the initial state and the disturbance vector of system S 2 , there must exist a matrix with After a few algebraic manipulations that is identical to One can get the relationship as follows from the last two inequalities ( 12) and ( 13): Now set H = H 11 , one can get the conditions in the theorem as follows: GA -HG ≤ 0, Sufficiency.Denote a new variable which is followed by a dynamical equation as follows: It follows from Lemma 1 in [9], in which G is assigned to the identity matrix I, that μ(k) ≤ 1 for any Hence, R + [G, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,1 .

Theorem 7
The polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,1 if and only if there exists a matrix H ∈ R r×r such that GA -HG ≤ 0, Proof Necessity.An augmented system can be established from system S 2 in (10) as follows: Emanating from any , where x 0 and ω 0 are the initial state and the disturbance vector of system S 2 , there must exist a matrix with After a few algebraic manipulations, it can be obtained One can obtain the relationship as follows from the last two inequalities ( 15) and ( 16): One can obtain the conditions in the form of theorem as follows: GA -HG ≤ 0, Sufficiency.Denote a new variable which is followed by a dynamic equation It follows from Lemma 1 in [9], in which G is assigned to the identity matrix I, that μ (k) ≤ [1, 0, 1] T for any Hence, the polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,1 . When

Theorem 8
The polyhedron R + [G, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,∞ if and only if there exist two matrices H 1 ∈ R r×r and H Proof Necessity.Consider the following system derived from system S 2 : Similar to the proof of Theorem 6, by virtue of Theorem 1, there must exist a matrix with After a few algebraic manipulations that is identical to Then set H 1 = H 11 , H 2 = H 12 , one can get the conditions in the theorem as follows: Sufficiency.Denote a new variable which is followed by a dynamic equation And it satisfies It follows from Lemma 1 in [9], in which G is assigned to the identity matrix I, that ξ (k) ≤ 1 for any Hence, the polyhedron R + [G, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,∞ .

Theorem 9
The polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 in (10) Simplifying the proof above, we obtain GB ω -H 12 I ≤ 0, (18) Then setting H 1 = H 11 , H 2 = H 12 , we have the conditions in the theorem as follows: Sufficiency.Denote a new variable which is followed by a dynamic equation And it satisfies , in which G is assigned to the identity matrix I, that ξ (k) ≤ [1, 0, 1] T for any Hence, the polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 in (10) with respect to ∞,∞ .
Remark 4 The conclusions and method in Theorem 1 to Theorem 9 can also be extended to Markovian positive systems [21].

Numerical examples
Example 1 Consider a two-dimensional positive system S 1 and a polyhedron R + [G, γ ] with From Theorem 1, it can be verified that R + [G, γ ] is a positively invariant polyhedron of system S 1 since there exists a matrix   According to Theorem 1, this polyhedron is not a positively invariant set of system S 1 since no feasible matrix H can be found.Figure 2 shows the trajectory of system state starting from [0.1 0.1] T .The trajectory of system state is not in the given polyhedron after two iterations.
Example 3 Consider a positive system S 2 and a polyhedron R + [G, 1] with According to Theorem 6, the polyhedron R + [G, 1] is a positively invariant set of system S 2 with respect to ω ∈ ∞,1 with a feasible matrix H = -0.040.40 -0.01 0.34 and initial conditions [0.1 0.1] T to determine whether the given polyhedron is a positively invariant set of system S 2 .The trajectory of system state starting from [0.1 0.1] T exhibits circular motion similar to an ellipse.And the ellipse stays in the given polyhedron in Fig. 3.
Similarly, this example also satisfies GA -HG ≤ 0, So the polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 with respect to ∞,1 .
To illustrate the necessity part of Theorem 6, a counter-example is given in Example 4.   This conclusion can be showed in Fig. 5, which depicts the system trajectory with respect to ω(k) = 0.5 + 0.5 sin(k) 0.5 + 0.5 cos(k) T and the initial state [0.1 0.1] T .Figure 5 is similar to Fig. 3, the system trajectory with respect to ω(k) = 0.5 + 0.5 sin(k) 0.5 + 0.5 cos(k) T and the initial state [0.1 0.1] T exhibits circular motion similar to an ellipse and stays in the given polyhedron.For convenience, it is concluded that the polyhedron Q + [G, 0, 1] is a positively invariant polyhedron of system S 2 with respect to ∞,∞ by Example 5.

Conclusion
Necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a discrete-time positive linear system are presented in this paper.The relationship between Lyapunov stability and positively invariant polyhedra for discrete-time positive linear systems is also studied.Under two types of external perturbations whose (∞, 1)-norm or (∞, ∞)-norm are bounded by a constant, the necessary and sufficient algebraic conditions for the positive invariant polyhedra are both investigated, which can be solved by a linear programming.The results obtained in this paper enrich and complete the results of positively invariant sets for positive linear systems with disturbances.

Figure 1
Figure 1 System trajectories

Figure 1
Figure 1 indicates the trajectory of system state starting from [1.4 1.4] T .The trajectory of system state starting from [1.4 1.4] T approaches the origin gradually.But it will never coincide with the origin.The system state trajectories are completely kept in this invariant polyhedron R + [G, γ ].Set γ 1 = 1 5 and γ 2 = γ , G and A remain unchanged, the result can also illustrate Theorem 2.

Figure 2
Figure 2 System trajectories

Example 4
Consider a positive system S 2 and a polyhedron R + [G, 1] with

Figure 5
Figure 5 System trajectories a positively invariant polyhedron of system S 1 in (2) if and only if there exists a nonnegative matrix H ∈ R ⎤ ⎥ ⎦ with H 11 , H 12 , H 21 , H 22 ∈ R r×r + ,H 13 , H 23 ∈ R r×n + ,H 31 , with respect to ∞,∞ if and only if there exist two matrices H 1 ∈ R r×r and H2 ∈ R r×m H 11 ∈ R r×r , H 12 ∈ R r×1 , H 21 ∈ R 1×r , H 22 ∈ R 1 , such that with