Closed-Loop Iterative Learning Control for Fractional-Order Linear Singular Time-Delay System : PD α − Type

In this paper a closed-loop PD type iterative learning control (ILC) of fractional order linear singular time-delay system is considered. The sufficient conditions for the convergence in time domain of the proposed PD-alpha type ILC for a class of fractional order singular system are given by the corresponding theorem together with its proof. Also, for the first time, we proposed a proposed ILC PD type for a given class of uncertain, fractional order, singular systems. Finally, the validity of the proposed PD ILC scheme for a class of fractional order singular time-delay system is verified by a numerical example.


Introduction
TERATIVE learning control (ILC) is one of the recent topics in control theories which belongs to the intelligent control methodology [1][2][3][4].Conventional control algorithms do not take advantage of the repetitiveness and ILC is a powerful control concept that iteratively improves the behavior of processes that are repetitive in nature.Iterative learning control was described by Uchiyama in 1978 [5] in Japanese, but only few people noticed it.Arimoto et al. [6] developed the ILC idea and studied the effective control algorithm for robotic system, and now it has become a hot issue in the field of control theory and has attracted broad attention over the past decades.ILC is an approach for improving the transient performance of systems that operate repetitively over a fixed time interval.Emulating human learning, ILC uses knowledge obtained from the previous trial to adjust the control input for the current trial so that a better performance can be achieved.Namely, ILC is a trajectory tracking improvement technique for control systems, which can perform the same task repetitively in a finite time interval to improve the transient response of a system using the previous motion.The key theme of ILC is to compose an upgraded control command for the next operation with its own proportional, integral, and/or derivative tracking errors at the previous operation.The objective is that the sequential ILC inputs stimulate the system to track a desired trajectory as perfect as possible as the operation approaches infinity [7].It has been widely acknowledged that ILCs perform well for general dynamical systems [8,9].Also, ILC requires less a priori knowledge about the controlled system in the controller design phase and also less computational effort than many other kinds of control.The study of ILC is of great significance for dynamic systems with complex modeling, uncertainty and strong nonlinear coupling, see [10,11].
Since theories and learning algorithms on ILC were firstly proposed, ILC has attracted considerable interests [4] due to its simplicity and effectiveness of learning algorithm, and its ability to deal with problems with nonlinear, time-delay, uncertainties and recently singular systems.Namely, delay is very often encountered in different technical systems.It is of great significance to study time-delay systems in theory and practice, because the existence of pure time delay, regardless if it present in the control or/and state, may cause undesirable system transient response, or generally, even an instability.This motivates researches on iterative learning control to focus on systems with time delay [12].
During the past years, singular systems also known as differential-algebraic systems, semistate systems, descriptor systems, or generalized state-space systems have attracted considerable attention because of their significant applications in diverse areas [13][14][15][16].Practically, many physical systems can be better described by singular systems than by regular systems.The conception about singular systems was originally put forward in 1974 [17].Singular systems have more essential differences than the normal systems, due to the fact that singular systems can preserve the structure of physical systems and impulsive elements, and are widely applied in many practical control systems such as electrical network [17], power systems, robotic systems [18] economic systems, chemical processes, and network analysis [19].Naturally, many theoretical results for regular systems have been extended to singular cases.In that way, issues of concern for singular systems are much more complicated than those for regular systems, because we need to consider not only stability, but also regularity and the absence of impulses at the same time for singular systems.
From the control point of view it is also necessary to study

I
The remainder of this paper is arranged as follows: in Section Preliminaries, some preliminaries for λ -norm as well as the fractional Caputo operators are presented.In Section Convergence analysis of PD α type iterative learning control, the first main result is derived where the convergence is guaranteed by mathematical proof rigorously, which includes the extensions of some of the basic result ILC of singular fractional-order systems with order α ∈ (0, 1) to fractionalorder singular system with time-delay.Further, it is presented the second main result in same manner where it is considered now, singular uncertain fractional order time-delay system.In section Simulation results a suitable numerical examples are included to illustrate the performance of the proposed PD α ILC scheme.Finally, the last section summarizes this work.

Preliminaries
The λ -norm, maximum norm, induced norm For later using in proving the convergence of proposed learning control, the following norms are introduced [3] for n -dimensional Euclidean space n R : the supnorm, , ,...
and the λ -norm for a real function: Induced norm of a matrix A is defined as: where ( ) .denotes an arbitrary vector norm.In case ( ) where A ∞ denotes the maximum value of the matrix A. For the previous norms, note that The λ -norm is thus equivalent to the ∞ -norm.For simplicity, in applying the norm ( ) . ∞ the index ∞ will be omitted.Before giving the main results, we first give the following Lemma 1, [3].
where 0, 0 ρ ε ≥ ≥ and 1 ρ < .Then the following holds: One can notice that in case of 0, lim 0 Fractional calculus-Caputo operator Fractional calculus (FC) is a mathematical topic with more than 300 years old history, but its application to physics and engineering has been reported only in the recent years.The fractional integro-differential operators are a generalization of integration and derivation to non-integer order (fractional) operators [23][24][25][26][27][28].The three most frequently used definitions for the general fractional differ-integral are: the Grunwald-Letnikov (GL) definition, the Riemann-Liouville (RL) and the Caputo definitions, [23][24][25][26][27][28].The definition of fractional integral is described by: where ( ) .Γ is the well-known Euler's gamma function, which is defined by ( ) . In this paper, Caputo fractional-order operator is used, where definition of the left Caputo fractional-order derivatives is given [14,15] as follows: ( ) ( ) where In the following sections, D α will denote 0 D α for brevity of notation.

Description of Fractional order linear singular time-delay system
In this paper, a non-integer (fractional) order singular time delay system is considered.Some examples of fractional order singular time delay systems are presented in [33][34][35].To explain the essence of our proposed learning control, consider a following non-autonomous fractional order linear singular time-delay system (FOLSTDS) described by the pseudo-state space equations with associated consistent function of initial state ( ) ( ), 0.
Here, for singular systems mentioned above, matrix E is considered to be singular, i.e.,

[ ] rank E r n
= < , otherwise, the system (11) reduces to a standard (normal) system.In practical system analysis and control system design, some system models may be established in the form of (11,12) while they could not be described by standard forms.Here, we are interested in dynamical behavior of system (11) over time interval , where i denotes the iteration index or the operation number, , ∈ are the state, control input and output of the system, respectively; matrices as well as E is a singular square matrix.Also, D α denotes the α th-order Caputo fractional derivative, 0 1 α < < , , and T τ ≤ denotes pure time-delay.
Definition 1. a)The SFOS system ( 11) is said to be regular if b) The SFOS system ( 11) is said to be impulse free if (11) is regular and ) Lemma 2. The triplet ( ) , , E A α is regular, we call SFOS system (11) regular, and consequently SFOS system is solvable.Here, t is the time in the operationinterval ( ) It is well known that the initial value problem (IVP) ( 11) is equivalent to the following Volterra integral equation with memory ( ) ( ) and its solutions are continuous [53].As discussed in paper [33], applying the method of steps, we obtain the following Lemma 2 which generalizes well-known results of integerorder singular systems without delay [13] to fractional-order singular delay differential systems [54].
To obtain our main result, the following useful lemma is first listed.
Lemma 3. (see [52]) Assume that (E, A) is regular and is the consistent initial function, then system (11) has a unique solution on [ ] 0, +∞ .The following assumptions on the system (11), (12) A3.The system ( 7) is causal, and when 0 t < , is assumed is the initial function of system (11) at i th − iteration, as well as ( ) ( ) , holds for all iterations.

Convergence analysis of PD α type iterative learning control
In this section, the convergence of ILC scheme given in the previous section is investigated.In order to track the desired output trajectory ( ) d y t , and due to the complicity of the system (11), we take into account the PD α -type ILC algorithm to derive the control input sequences ( ) i u t such that the system output Here, the following fractional order PD α learning algorithm is considered which comprises control law a closedloop PD α law.In that way, closed-loop fractional order PD α learning algorithm takes the form of where , Γ Π are gain matrices of appropriate dimensions, ( ) u t the value of the function at time.The expression ( ) ( ) ( ) denotes the tracking error of the fractional order singular system at i th − iteration.First of all, let us consider the updating law (19) and system (11), (12).A sufficient condition for convergence of a proposed feedback ILC is given by Theorem 1 and proved as follows.
Theorem 1: Consider the fractional order linear singular time-delay system governed by (11), (12) to which an closedloop PD α -type ILC algorithm (19) is applied, and assume that assumptions (A1)-(A4) are satisfied.If matrix Π , exist provided that where is It follows from (18) and ( 21) one observes that tracking error is: From ( 19), ( 22) we obtain: Consequently, we obtain from (7,18): Substituting ( 23) into (24) we have Using suitable gain matrix Π , as well as taking into account previously introduced assumptions, one can determine matrix ( ) Multiplying on the left side expression ( 25) by ( ) where are ) By ( 21) via (26), we get (28) Taking the standard norm, (.) of the above equation, and using the condition of Theorem 1, we find where Also, one can write the solutions of ( 24) in form of the equivalent Volterra integral equations and applying Lemma 2. and assumption A3, as: Applying the norm ( ) . on the equation ( 31), we derive: ( )  .Moreover, taking λ norm, we have ( ) ( ) Furthermore, due to the fact ( ) ( ) which implies that Introducing ( ) Γ + , which implies that (33) simplifies to ( ) ( ) Consequently, it follows that It then follows that there exists λ large enough such that ( ) ( )( ) Summarizing, we see that taking the λ -norm again on the expression (29) leads to: Concerning equations ( 36) and (38) we have ) It then follows that there exists λ large enough such that ( ) ( ) ( ) Therefore, according to Lemma1, [4] it can be concluded that: Moreover, due to the uniqueness and existence theorem for fractional order singular system, [32,33] one can conclude that lim ( ) ( ), This completes the proof.Further, we consider the case of the fractional order ( ) α ∈ uncertain singular system nonautonomous singular linear system which can be written as pseudo state space equation and output equation: (46) with associated consistent function of initial state (13).
Function ( ) where ∑ .Then the following holds: Theorem 2. For the fractional order singular system ( 45), (46) with the PD α -type ILC scheme (19), and the assumptions A1-A5 where the convergence condition is given by (20), then when i → ∞ the bounds of the tracking errors ( ) ( ) , u t u t − converge asymptotically to a residual ball centered at the origin.
Proof: The proof follows from the proof of Theorem 1. Namely, from (45), (46) one can easily find that Multiplying on the left side expression ( 49) by we obtain (50) in the form where we adopted ( , By replacing (50) into (25), we obtain Estimating the norms of ( 52) with ( ) . and using the condition of Theorem 2 one gets Also, one can write the solutions of (45) in the form of the equivalent Volterra integral equations using assumption A5, as: In similar manner, applying the norm ( ) . on the equation (54), if there exists unique solution, [32,33] where are, , , and applying λ norm, we have ) ) Defining respectively ( ) ( ) , then we can get Then, if a sufficiently large λ is used, one can obtain that: ( ) ( ) Taking the λ -norm of the above, equation ( 53) leads to: Finally, taking into account (59) we have So that, there exists a sufficient large λ satisfying ( ) ( ) Therefore, taking into account Lemmas 4,1 [54], it yields: This completes the proof of Theorem 2.
Remark 1.In the case of no uncertainty, i.e., ( )

Simulation results
In this section, we provide an example to illustrate the applicability of the proposed method.Consider the following fractional order linear singular time delay system in state space form described by where ( ) ( ) The desired trajectories are given by ( ) ( ) The gain matrices are 0.9 0.3 0.7 0.2 , 0 0.9 0 0.7 It is easy to show that the pair (E, A) is regularand 0.7 1 . Simulation results in Figures 1-4 show the effectiveness of the developed ILC control scheme for the system (45,46).It can be seen that the system output ( ) y t is capable of approaching the desired trajectory accurately within few iterations.Also, it can be seen (see Figures 1 and  2) that the proposed requirement for the tracking performance is achieved at the seventh iteration.The ILC rule ( 19) is used, (Figures 3 and 4) show the tracking performance of the ILC system outputs on the interval [ ] 0,1 t ∈ .

Conclusions
In this contribution, PD α type of iterative learning feedback control for given class of fractional order linear singular time-delay system is investigated.The sufficient conditions for the convergence in time domain of a proposed ILC were given by the corresponding theorem and proved.Also, for the first time, we proposed a robust PD α type of iterative learning feedback control for given class of uncertain, fractional order, singular systems which guaranteed the convergence conditions in presence of bounded uncertainty of the system.Finally, the theoretical results have been verified through numerical simulations which demonstrate the effectiveness of the proposed robust PD α ILC scheme for a class of fractional order linear singular time-delay system.

Iterativno upravljanje učenjem u zatvorenoj petlji
as well as A, B and C are matrices with appropriate dimensions.It is assumed that det 0 E = as well as SFOS system is regular.Throughout this paper, let [ ] following initial value problem (IVP) for fractional order nonsingular time delay system: accurately as possible when i goes to infinity for all[ ]

Figure 1 .
Figure 1.The sup-norm of tracking error ( ) 1 e t in each iteration