Abstract
In the paper, fractional discrete cone control systems with n-orders are considered. Some relations between invariance and (asymptotic) stability properties of the presented systems are discussed. Operators employed to the considered systems are Caputo-, Riemman-Louville-, and Grünwald-Letnikov type ones. Cone systems with control, which are particular invariant systems with control, together with their stability and asymptotic stability properties are examined.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Positive systems are the systems whose state and input variables are never negative, with a given nonnegative initial state. These systems appear frequently in practical applications as well as in real phenomena, among other in biology, medicine, economics, electrotechnics, control system design, etc. (see [6, 7, 13, 25, 26] and the references therein). The natural generalization of positive systems are cone systems, i.e., systems whose trajectories always remain in the given cone if they are initialized in this cone. Moreover, the special attention can be put on invariant cone systems, since they can be employed to design stabilizing controllers.
In recent years, there has been observed a growing interest in the theory and applications of fractional differential and difference equations. Many authors proved that such types of equations are more adequate for modeling physical and chemical processes than equations with integer order. Fractional differential and difference equations describe many phenomena arising in engineering, physics, or economics. In fact, one can find several applications in viscoelasticity, electrochemistry, electromagnetic, etc. For example, Machado in [19] gave a novel method for the design of fractional order digital controllers.
In both theory and applications, one can meet several definitions of the fractional derivatives, among which the most popular are Caputo-, Riemman-Louville-, and Grünwald-Letnikov operators, so there appears the problem how to deal with differences resulting from the application of these operators. The first steps in this topic were made in [20]. Properties of the fractional sum, Caputo- and Riemman-Louville-type difference operators, were developed in [1–4, 21]. Basic information on fractional calculus concept, ideas, and applications of these operators can be found for example in [15, 18, 24]. In [8], there was adopted a more general fractional h-difference Riemman-Liouville operator, where on the one hand h represents sample step, on the other hand, for h tending to zero, the solutions of the fractional difference equation may be seen as approximations of the solutions of corresponding Riemann-Liouville fractional equations.
The goal of the paper is to examine under which conditions control systems are cone systems. To this aim, in Section 2, we present the needed notation and properties of the h-difference operators of fractional order (with arbitrary h > 0). Operators, which we consider are the three most important among all fractional operators: Caputo-, Riemman-Louville-, and Grünwald-Letnikov type difference operators. In [11], it is shown that these three types of h-difference fractional operators are related to each other. Moreover, the Grünwald-Letnikov-type fractional h-difference operator can be expressed by the Riemann-Liouville-type fractional h-difference operator. So, systems with these operators can be studied simultaneously. Taking into account this fact, in Section 3, there is introduced a discrete-time cone control system with fractional order and properties of its trajectories are discussed. Since a cone is a special case of a polyhedron, in Sections 4 and 5, basing on some properties of polyhedron contractiveness, the problems of stability and asymptotic controllability of class of consider systems are tackled.
2 Some Preliminaries
For α > 0, h > 0 and \(a\in \mathbb {R}\) let
For a function \(x:(h\mathbb {N})_{a}\rightarrow \mathbb {R}\), then the forward h-difference operator is denoted by
while the h-difference sum is given by
where t = a + (k + 1)h and \(k\in \mathbb {N}_{0}\). Additionally, we define \(\left ({~}_{a}{\Delta }^{-1}_{h} x\right )(a) := 0\).
For arbitrary \(t,\alpha \in \mathbb {R}\) the h-factorial function is defined by
where \(\frac {t}{h}\not \in \mathbb {Z}_{-}:=\{-1,-2,-3,\ldots \}\), and we use the convention that division at a pole yields zero. Notice that if we use the general binomial coefficient \(\binom {a}{b}:=\frac {\Gamma (a+1)}{\Gamma (b+1){\Gamma }(a-b+1)}\), then function (1) can be rewritten as
Lemma 1
[10] If 0 < α ≤ 1, then
For α = 1 one gets \((-1)^{s}\binom {1}{s+1}=0\) .
For a function \(x\in (h\mathbb {N})_{a}\) the fractional h-sum of order α > 0 is defined by
where t = a + (α + n)h. If \(\psi (r)=(r-a+\mu h)_{h}^{(\mu )}\) for any \(r\in (h\mathbb {N})_{a}\), then (see [8])
where \(t\in (h\mathbb {N})_{a+\alpha h}\). Using the general binomial coefficient, one can rewrite Eq. (2) in the form
Note that if ψ ≡ 1, then for μ = 0, a = (1−α)h, and t = n h + a + α h, it holds
Definition 2
[5] Let α ∈ (0, 1] and \(a\in \mathbb {R}\). The Riemann-Liouville–type fractional h-difference operator \(_{a}^{RL}{\Delta }^{\alpha }_{h} \) of order α for a function \(x: (h\mathbb {N})_{a}\rightarrow \mathbb {R}\) is defined by
where \(t\in (h\mathbb {N})_{a+(1-\alpha ) h}\).
Definition 3
[21] Let α ∈ (0, 1] and \(a\in \mathbb {R}\). The Caputo–type fractional h-difference operator \(_{a}^{C}{\Delta }^{\alpha }_{h} \) of order α for a function \(x: (h\mathbb {N})_{a}\rightarrow \mathbb {R}\) is defined by
where \(t\in (h\mathbb {N})_{a+(1-\alpha ) h}\).
Note that the operator \(_{a}^{C}{\Delta }^{\alpha }_{h}\) for any α ∈ (0, 1] changes the domain of the function x, i.e., it maps real valued functions defined on the set \((h\mathbb {N})_{a}\) into real valued functions defined on the set \((h\mathbb {N})_{a+(1-\alpha )h}\). Moreover, for α = 1, we have \(\left ({~}_{a}^{C}{{\Delta }_{h}^{1}} x\right )(t)=\left ({\Delta }_{h} x\right )(t)\). Similarly, it holds also for the Riemann-Liouville-type h-difference operator. Moreover, for α ∈ (0, 1], it holds (see [11]):
for \(t\in (h\mathbb {N})_{a+(1-\alpha )h}\).
Let us recall that the \(\mathcal {Z}\)-transform of a sequence \(\{y(n)\}_{n\in \mathbb {N}_{0}}\) is a complex function Y(z) given by
where \(z\in \mathbb {C}\) is a complex variable for which the series \({\sum }_{k=0}^{\infty }y(k)z^{-k}\) converges absolutely.
Proposition 4
[22] For \(a\in \mathbb {R}\) , α ∈ (0, 1], let us define \(y(n):= \left ({~}_{a}^{RL}{\Delta }_{h}^{\alpha }x \right )(t)\), where t = a + (q−α)h + nh and t 0 = a + (q − α)h. Then
where \(X(z)=\mathcal {Z}[\overline {x}](z)\) and \(\overline {x}(n):=x(a+nh)\).
Proposition 5
[22] For \(a\in \mathbb {R}\) , α∈(0, 1], \(q\in \mathbb {N}_{1}\) let us define \(y(n):= \left ({~}_{a}^{C}{\Delta }_{h}^{\alpha }x \right )(t)\) , where t=a+(q−α)h+nh and t 0 = a+(q−α)h. Then
where \(X(z)=\mathcal {Z}[\overline {x}](z)\) and \(\overline {x}(n):=x(a+nh)\).
The last operator that we take under our consideration is the fractional h-difference Grünwald-Letnikov–type operator.
Definition 6
[11] Let \(\alpha \in \mathbb {R}\). The Grünwald-Letnikov–type h-difference operator \(_{a}^{GL}{\Delta }^{\alpha }_{h}\) of order α for a function \(x:(h\mathbb {N})_{a}\rightarrow \mathbb {R}\) is defined by
where
with
If a = (α−1)h and x(t): = y(t−a) for any \(t\in (h\mathbb {N})_{a}\) then, see [11]
Proposition 7 [22]
For \(t=a+\alpha h+kh\in (h\mathbb {Z})_{a+\alpha h}\) let us denote \(y(k):= \left ({~}_{a}^{GL}{\Delta }^{-\alpha }_{h}x \right )(t)\) and \(\overline {x}(k):=x(a+kh)\) . Then
where \(X(z):=\mathcal {Z}\left [\overline {x}\right ](z)\).
Since the Grünwald-Letnikov–type h-difference operator can be expressed by the Riemann-Liouville–type fractional h-difference operator (see Eq. 3), we restrict our consideration only to the Caputo– and Riemann-Liouville–type fractional h-difference operators.
3 Cone Systems
In order to define cone systems, let us discuss first the problem of existing of solutions of a nonlinear control system of fractional order. The reasoning is similar to the one given for finding solutions of the nonlinear autonomous system of the fractional order α i ∈ (0, 1] given in [23], so we present only main steps of it.
Let i = 1,…,n and 0 < α i ≤ 1. Let us consider the following fractional Caputo h-difference system with n orders α 1, …,α n as follows
with initial values
where a i = (α i −1)h ∈ (−h, 0], \(t_{0i}=a_{i}+n_{0} h\in \left (h\mathbb {N}\right )_{a_{i}}\), \(n_{0}\in \mathbb {N}_{0}\), \(t\in \left (h\mathbb {N}\right )_{n_{0} h}\), \(f_{i}:\left (h\mathbb {N}\right )_{0}\times \mathbb {R}^{n}\times \mathcal {U}\to \mathbb {R}\), i = 1,…,n, \(\mathcal {U}\subseteq \mathbb {R}^{m}\). The set \(\mathcal {U}\) is called the control space and satisfies the following property: \(\mathcal {U}\) is such that \(\mathcal {U}\subseteq \overline {\text {int}\,\mathcal {U}}\) and any two points in the same connected component of \(\mathcal {U}\) can be jointed by a smooth curve lying in \(\text {int}\, \mathcal {U}\), except for end points. Let J 0(m) denotes the set of all sequences u = (u 0, u 1, …), where \(u_{n}:=u(nh)\in \mathcal {U}\), \(n\in \mathbb {N}_{0}\). We assume that function f depends on finite number of elements u i .
Note that if the Riemann-Liouville–type fractional h-difference operator \(_{t_{0i}}^{RL}{\Delta }_{h}^{\alpha _{i}}\) is used instead of the Caputo–type h-difference operator \(_{t_{0i}}^{C}{\Delta }_{h}^{\alpha _{i}}\) in Eq. 3, then one gets the fractional Riemann-Liouville h-difference system with n orders of the form
Recall the constant vector \((X^{\mathrm {e}},u^{e}):=\left (x_{1}^{\mathrm {e}},x_{2}^{\mathrm {e}},\ldots ,x_{n}^{\mathrm {e}},u^{e}\right )^{\mathrm {T}}\) is an equilibrium point from time t 0 = n 0 h of fractional difference system (4) if and only if for any \(t\in (h\mathbb {N})_{a}\)
and
in the case of the Riemann-Liouville h-difference systems.
Remark 8
For the Caputo h-difference system \(\left ({~}_{t_{0i}}^{C}{\Delta }_{h}^{\alpha _{i}}x_{i}^{\mathrm {e}}\right )(t)\equiv 0,\) the constant vector \((X^{\mathrm {e}},u^{e})=\left (x_{1}^{\mathrm {e}},x_{2}^{\mathrm {e}},\ldots ,x_{n}^{\mathrm {e}},u^{e}\right )^{\mathrm {T}}\) is an equilibrium point from time t 0 = n 0 h of the Caputo fractional h-difference system (4) if and only if f i (t, X e,u e)=0, i = 1,…,n for all \(t\in \left (h\mathbb {N}\right )_{n_{0}h}\).
For simplicity, we state all definitions and theorems for the case when the equilibrium point is \((0,0)\in \mathbb {R}^{n}\times \mathcal {U}\), i.e. \(x_{i}^{\mathrm {e}}=0\), i = 1, …, n and u e = 0. There is no loss of generality in doing so because any equilibrium point can be shifted to the origin via certain change of variables.
For \(k\in \mathbb {N}_{n_{0}}\), \(n_{0}\in \mathbb {N}_{0}\), let us define
or
for the Caputo or Riemann-Liouville h-difference systems, respectively, and
Then we can write systems (4) or (6) respectively, in the following forms
or
where \(F:\mathbb {N}_{0}\times \mathbb {R}^{n}\times J_{0}(m)\to \mathbb {R}^{n}\). Therefore, systems (7a) and (7b) can be expressed in one compact form
with the initial condition
Then the solution
of IVP given by Eqs. 8 and 9 can be obtain using the same reasoning as in [23] and in [10]. It is a uniquely defined map \(\gamma :\mathbb {R}^{n}\times (h\mathbb {N})_{a}\times J_{0}(m)\rightarrow \mathbb {R}^{n}\) by initial state X 0 and control sequence u ∈ J 0(m) and described by
where \({\Lambda }_{j}=\left [\begin {array}{llllllll} \binom {-\alpha _{1}}{j} & 0 & {\ldots } & 0\\ 0 & \binom {-\alpha _{2}}{j} & {\ldots } & 0 \\ \vdots &\vdots & {\ddots } & {\vdots } \\ 0 & 0 & {\ldots } & \binom {-\alpha _{n}}{j} \end {array}\right ]\in \mathbb {R}^{n\times n} \), and
So, γ(X 0, ⋅,u) is defined by its values γ(X 0, k h, u(k h)) = X(k), \(k\in \mathbb {N}_{0}\), and denotes the state forward trajectory of system (8).
Let us take a nonsingular matrix
with ith, i = 1,…,n, row given as p i = (p i1, …, p i n ). Then the set
is called a linear cone of state generated by the matrix P in \(\mathbb {R}^{n}\) (see [12,14]). If \(\mathcal {X}_{n}:=\{X:\mathbb {N}_{0}\rightarrow \mathbb {R}^{n}\}\), then the set
is called a linear cone of states with the vertex at 0 generated by the matrix P in the space \(\mathcal {X}_{n}\) where \(x_{i}:\mathbb {N}_{a_{i}}\rightarrow \mathbb {R}\).
Definition 9
Let \(P\in \mathbb {R}^{n\times n}\) be a given matrix. The nonlinear fractional difference system (8) together with initial condition (9) is called a \(\mathcal {P}\) cone fractional system if \(X(\cdot )\in \mathcal {P}\) for any X 0 ∈ K P .
Theorem 10
Let K and \(\mathcal {P}\) be given as in Eqs. 12 and 13 together with X 0 ∈K P . If there exists a control u∈J 0 (m) such that for every x ∈ K P it holds
then for every \(k\in \mathbb {N}_{0}\) system (4) or (6) is a \(\mathcal {P}\) cone system.
Proof
The proof uses the analogous reasoning as the one given in [10] for the similar result but without a control. For the proof, we use mathematical induction. For k = 1, the formula holds, since if X 0 ∈ K P , then we get X(1) = F(0,X 0, u(0))+Λ1,1 X 0 ∈ K P . Now, we assume that the hypothesis is true for some k, i.e., F(k, x, u)+Λ k, k ⋅x ∈ K P for every x ∈ K P and u ∈ J 0(m). This means that X(k) ∈ K P . Then, by assumption and Lemma 1, it holds
Hence from mathematical induction, the thesis holds for any natural k. □
Note that for system (8) together with initial condition (9) and with the right-hand side autonomous also the implication “only if” in Theorem 10 is true.
Corollary 11
Let P be a nonsingular n×n matrix. Then system (8) together with initial condition (9) with the right-hand side F(k, X (k), u(k)) = AX (k) + Bu(k), where \(A\in \mathbb {R}^{n\times n}\) and \(B\in \mathbb {R}^{n\times m}\) , is a ( \(\mathcal {P}\,,\mathcal {Q}\)) cone system Footnote 1 if and only if \(P\cdot \left [ A+{\Lambda }_{k,k}\right ]P^{-1}\in \mathbb {R}^{n\times n}_{+}\) and \(PBQ^{-1}\in \mathbb {R}^{n\times m}_{+}\) for nonsingular matrix Q of the form
with ith, i=1,…,m, rows given as q i = (q i1 ,…,q im ).
Let us define a feedback control law by
where κ(x) is a vector function with values in \(\mathbb {R}^{m}\). We say that a function \(\kappa :\mathbb {R}^{n}\rightarrow J_{0}(m)\) is an admissible feedback law for system (8) if for every \(k\in \mathbb {N}_{0}\) there exists a map γ(X 0, ⋅,u) given by Eq. 11 such that u(k) = κ(γ(X 0, a + k h, u(k h))). If κ is admissible feedback law for Eq. 8 (or respectively for Eqs. 7a or 7b)), then the closed loop system is of the form
where X(k) is given by Eq. 10. As an immediate consequence of Theorem 10, we have the following.
Corollary 12
Let K P and \(\mathcal {P}\) be given as in Eqs. 12 and 13 together with X 0 ∈K P . If there exists a feedback law (14) such that for every x ∈ K P , it holds
then for every \(k\in \mathbb {N}_{0}\) system (15) is a \(\mathcal {P}\) cone system.
In particular, if system (8) is a linear one, i.e., F(k, X(k),u(k)) = A X(k) + B u(k), application of linear feedback law
with a matrix \(F\in \mathbb {R}^{m\times n}\) gives the closed loop system of the form
4 Contractive Sets and Stability
Consider a polyhedron
where \(H\in \mathbb {R}^{r\times n}\) and \(w=(\omega _{1},\ldots ,\omega _{r})^{T}\in \mathbb {R}^{r}\) is a positively defined vector.
Definition 13
Polyhedron Ω given by Eq. 18 is λ-contractive set with respect to closed loop system (15) if there is λ ∈ (0, 1] such that
for all ε ∈ (0, 1] and all X(k) ∈ Ω(H, w ε).
Recall that a set S is an invariant set for the system (15) if and only if every trajectory of this system starting within S remains inside it. Then λ-contractiveness with respect to system (15) implies invariance property for this system.
The one step admissible set to Ω(H, w) with respect to system (15) is defined by
The q step admissible set to Ω(H, w) with respect to system (15) is defined by
Proposition 14
Polyhedron Ω given by Eq. 18 is λ-contractive set with respect to closed loop system (15) if and only if there is λ∈(0, 1] such that
for any ε∈(0, 1] and natural q.
Proof
The result is an immediate consequence of Definition 13 and of polyhedron (18). □
Assumption 1
Let us assume that function \(\bar {F}\) given in Eq. 15 is continuous in all variables and (classically) continuously differentiable at the equilibrium point of the given system.
Under Assumption 1, let us define matrix
and consider a linear fractional order discrete-time system
System (20) is called a linear approximation of the nonlinear one given by Eq. 15. Note that for a given initial condition and for an arbitrary sequence of controls u ∈ J 0(m), there exists the unique solution of linear approximation (20). Recall that a constant vector X e = (0,…,0) is an equilibrium point of fractional difference system (20) if and only if
for all \(k\in \mathbb {N}_{0}\). Let us notice that the trivial solution X ≡ 0 is an equilibrium point of system (20). The equilibrium point X e = 0 of Eq. 20 is said to be
-
(a)
stable if, for each 𝜖 > 0, there exists δ = δ(𝜖) > 0 such that ∥X 0∥ < δ implies ∥X(k)∥ < 𝜖, for all \(k\in \mathbb {N}_{0}\);
-
(b)
attractive if there exists δ > 0 such that ∥X 0∥ < δ implies
$$\lim_{k\to \infty} X(k)=0\,;$$ -
(iv)
asymptotically stable if it is stable and attractive.
System (20) is called stable/asymptotically stable if its equilibrium point X e = 0 is stable/asymptotically stable.
In the linear case, the effective characterization of asymptotic stability was given in [22].
Proposition 15
For \(a\in \mathbb {R}\) and α ∈ (0, 1] let \(\overline {x}(s):=x(sh)\) . Then
where \(X(z)=\mathcal {Z}[x](z)\) and β i = α i , i = 1, …, n, for Riemann-Liouville-type operator and β i = 1 for the Caputo type operator.
From Proposition 14, it follows that for any initial condition X 0, the corresponding trajectories of system (20) are contained in Ω(H, w). The asymptotic stability of this system in polyhedron Ω(H, w) is guaranteed if all roots of the equation
where A is the matrix given in Eq. 20, \(\mathcal {H}:=\text {diag} \left \{h^{\alpha _{1}},\ldots ,h^{\alpha _{p}}\right \}\), 0 < α i ≤ 1, \({\Lambda }:=\text {diag} \left \{\left (\frac {z}{z-1}\right )^{\alpha _{1}},\ldots ,\left (\frac {z}{z-1}\right )^{\alpha _{p}}\right \}\) and z is a complex variable, are strictly inside the unit circle, see [22]. Then we have the following.
Proposition 16
Assume that there exists λ∈(0, 1] and a feedback law u(k)=FX(k) such that Ω(H,w) is λ-contractive set with respect to system (17). Then system (17) is locally asymptotically stable in Ω(H, w).
Proof
Note that inside of polyhedron Ω(H, w), the origin is the only equilibrium point. Then the thesis follows from the compactness and from Proposition 14 and (20). □
Proposition 17
Suppose that Assumption 1 is satisfied. Then there exists a neighborhood \(\mathcal {V}\) of the origin such that the closed loop system (17) is asymptotically stable.
Proof
The result follows from Proposition 16 and from classical reasoning, see for example [9]. □
Proposition 18
Let \(\bar {F}\) fulfill Assumption 1 and suppose there exist λ ∈ (0, 1] and a feedback law u (k) = F X (k) such that Ω(H, ωελ), for every X ∈ Ω (H, ωελ), is a non-empty λ-contractive set with respect to linearized system (20). Then there exists α ∈(0, 1] such that set Ω (H, ωεα), for every X ∈ Ω (H, ωεα), is an invariant set for nonlinear system (15).
Proof
We follow the reasoning from [9]. From Assumption 1 and the fact that A X ∈ Ω(H, ω ε λ), for every X ∈ Ω(H, ω ε λ), it is easy to infer that there exists α ∈ (0, 1] such that \(\bar {F}(k,X(k),\kappa (X(k)))\in {\Omega }(H,\omega \varepsilon \alpha )\). This means that the set Ω(H, ω ε α) is an invariant set for the nonlinear system (15). □
5 Remarks on Asymptotic Controllability of Cone Systems
Let us draw our attention to the problem of controllability of system (8). Classically, controllability of the given system means that it is possible to transfer the considered system from a given initial state to a final state using controls from a certain set, see for example [16,17].
Suppose that a set V is a subset of the state space of system (8).
Definition 19
Let x 0, x f ∈ V. Then
-
i.
X is asymptotically controlled to a final state X f without leaving V if there exists a control u ∈ J 0(m) such that \(\lim _{k\rightarrow \infty }\gamma (X_{0},a+kh,u(kh))=X_{f}\) and γ(X 0, a + k h, u(k h))∈V for all \(k\in \mathbb {N}_{0}\).
-
ii.
If X e is an equilibrium, then system (8) is asymptotically controlled to X e if for each neighborhood V of X e there is some neighborhood W of X e such that each X ∈ W can be asymptotically controlled to X e without leaving V.
Proposition 20
Let K P and \(\mathcal {P}\) be given as in Eqs. 12 and 13 together with X 0 ∈K P . Suppose that there exists some feedback law u(k)=κ(X(k)) so that X e is a local asymptotically stable state for \(\mathcal {P}\) cone system (15). Then system (8) is asymptotically controlled to X e.
Proof
Since asymptotical stability of \(\mathcal {P}\) cone system (14) at X e means that \(\lim _{k\rightarrow \infty }\gamma (X_{0},a+kh,\kappa (k))=X^{e}\) and γ(X 0, a + k h, κ(X(k)) is in a neighborhood of X e for all \(k\in \mathbb {N}_{0}\), hence the thesis follows from Definition 19. □
Let us assume function F given in Eq. 8 fulfills Assumption 1.
Under this assumption, let us define matrices
and consider a linear fractional order discrete-time system
System (20) is called a linear approximation of the nonlinear one given by Eq. 7.
Proposition 21
Suppose there exist λ ∈ (0, 1] and a feedback law u(k) = κ(X(k)) such that Ω(H, ωελ) is λ-contractive set with respect to closed loop system (20) and Ω(H,ωελ) is a compact polyhedron. Then linear system (22) is asymptotically controllable at X 0.
Proof
If there is law u(k) = κ(X(k)) such that Ω(H, ω ε λ) is λ-contractive set with respect to system (20) and Ω(H, ω ε λ) is a compact polyhedron, then by Proposition 17 the closed loop system (15) is asymptotically stable. Hence, by Definition 19, system (22) is asymptotically controllable at X e = X 0. □
6 Conclusions
We examined fractional discrete cone control systems with n-orders. Some relations between invariance and (asymptotic) stability properties of the presented systems where discussed. Since there are several definitions and notations of the fractional derivatives, among which the most popular are Caputo-, Riemman-Louville-, and Grünwald-Letnikov operators, we employ right them as fractional discrete Caputo-, Riemman-Louville-, and Grünwald-Letnikov type operators to the systems. In the paper, there were considered cone systems with control, which are particular invariant systems with control, together with their stability and asymptotic stability properties.
Notes
System (8) is called (\(\mathcal {P}\,,\mathcal {Q}\) ) cone system if \(X(\cdot )\in \mathcal {P}\) for every X 0 ∈K P and every u i ∈ Q
References
Abdeljawad T. 2011. On Riemann and Caputo fractional differences. Comput Math Appl.
Abdeljawad T, Baleanu D. Fractional differences and integration by parts. J Comput Anal Appl 2011;13(3):574–582.
Atici FM, Eloe PW. A transform method in discrete fractional calculus. Int J Diff Equa 2007;2:165–176.
Atici FM, Eloe PW. Initial value problems in discrete fractional calculus. Proc Amer Math Soc 2009;137(3):981–989.
Bastos NRO, Ferreira RAC, Torres DFM. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin Dyn Syst 2011;29(2):417–437.
Benvenuti L, Farina L. A tutorial on the positive realization problem. IEEE Trans Autom Control 2004;49(5):651–664.
De Jong H. Modelling and simulation of genetic regulatory systems: a literature review. J Comput Biol 2002;9(1):67–103.
Ferreira RAC, Torres DFM. Fractional h-difference equations arising from the calculus of variations. Appl Anal Discrete Math 2011;5(1):110–121.
Fiacchini M, Alamo T, Camacho EF. 2007. On the computation of local invariant sets for nonlinear systems. In: Proceedings of the 46th IEEE conference on decision and control. p. 3989–3994.
Girejko E, Mozyrska D. Cone solutions of multi-order fractional difference systems. Control Cybern 2013;42(2):419–429.
Girejko E, Mozyrska D, Wyrwas M. Comparison of h-difference fractional operators. In: Mitkowski W, Kacprzyk J, Baranowski J, editors. Advances in the theory and applications of non-integer order systems. Springer; 2013. vol. 257, p. 191–197.
Kaczorek T. Fractional positive continuous-time linear systems and their reachability. Int J Appl Comput Sci 2008;18(2):223–228.
Kaczorek T. Fractional positive linear systems. Kybernetes 2009;38(7–8):1059–1078.
Kaczorek T. Reachability of cone fractional continuous-time linear systems. Int J Appl Math Comput Sci 2009;19(1):89–93.
Kaczorek T. 2011. Selected problems of fractional systems theory. Springer.
Kalman RE. On general theory of control systems. In: Proc. of the first IFAC Word Congres. Moscow; 1960.
Kalman RE, Falb P. Topics in mathematical systems theory. New York: McGraw-Hill; 1969.
Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: North-Holland Mathematics Studies, Elsevier Science B. V.; 2006.
Machado JAT. Analysis and design of fractional-order digital control systems. Syst Anal Model Sim 1997;27(2–3):107–122.
Miller KS, Ross B. Fractional difference calculus. In: Proceedings of the international symposium on univalent functions, fractional calculus and their applications. Kuoriyama; Nihon University; 1988. p. 139–152.
Mozyrska D, Girejko E. Advances in harmonic analysis and operator theory: the Stefan Samko Anniversary Volume, volume 229, chapter Overview of the fractional h-difference operators. Springer; 2013. p. 253–267.
Mozyrska D, Wyrwas M. The Z-transform method and delta-type fractional difference operators. Discret Dyn Nat Soc. 2015. doi:10.1155/2015/852734
Pawluszewicz E, Wyrwas M, Girejko E. Stability of nonlinear h- difference systems with n fractional orders. Kybernetika 2015;51(1):112–136.
Podlubny I. Fractional differential and equations. Mathematics in sciences and engineering. San Diego: Academic Press; 1999.
Tarbouriech S, Burgat C. Positively invariant sets for constrained continuous-time systems with cone properties. Proc 30th IEEE Conf Decis Control 1991:1748–1754.
Shorten R, Wirth F, Leith D. A positive systems model of tcp-like congestion control: asymptotic results. EEE-ACM Trans Netw 2006;14(3):616–629.
Acknowledgments
The project was supported by the founds of the National Science Centre granted on the bases of the decision number DEC-2011/03/B/ST7/03476. The work was supported by Bialystok University of Technology grant G/WM/3/12.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Girejko, E., Pawłuszewicz, E. Remarks on Fractional Discrete Cone Control Systems with n-Orders and Their Stability. J Dyn Control Syst 23, 269–281 (2017). https://doi.org/10.1007/s10883-016-9315-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-016-9315-x