Stability Analysis of High-Order Iterative Learning Control for a Class of Nonlinear Switched Systems

and Applied Analysis 3 Assumption 3. The resetting condition is satisfied for all the iteration; that is, xk (t) = xd (t) , t ∈ [−μ, 0] , (6) where μ = max{h, d} and xd(t) is the desired initial function. Remark 4. From Assumption 1, since ud(t) exists uniquely, the uniform convergence of the control profile u(t) to ud(t) implies that the state and output tracking errors will vanish. Assumption 2 is a basic condition for switched systems which means that system states are continuous, even for structure switches. Remark 5. Assumption 3 means xk(0) = xd(0) for all k, which is the identical initial condition for ILC system. For a switched system, even though the dynamic behavior is changing between different subsystems, the initial condition reset is often satisfied inmany practical systems. For instance, when a product is processed by several different procedures, then the system can be considered as a switched system, and the different procedures represent the different subsystems. If the product is batch processing, then the switched system is operated repetitively. Even though the processing is changing between different procedures, the initial state of the product is identical. Remark 6. The switched rule σ(t) in system (1) is an arbitrary switched rule on the time domain, and it is invariable on the iteration domain. One may argue that the problem considered in this paper is similar to the ILC for time varying nonlinear system. However, there are some crucial differences. The time varying system admits a family of solutions that can be parameterized solely by the initial condition, whereas the switched system admits a family of solutions that is parameterized both by the initial condition and the switching signal. Besides, the switched system has sudden transient at some time instants. The control target is to find a control input sequence uk(t) = ud(t), such that yk(t) converges to yd(t) as k → ∞; that is, as the learning iteration repeats, the system output converges to the desired trajectory. Now, the following high-order ILC updating law for the system (3) is proposedwhich uses theP, I, andD information of tracking errors: uk+1 (t) = N ∑ j=1 Pjuk−j+1 (t) + N ∑ j=1 {Qjek−j+1 (t) + Rj ̇ ek−j+1 (t) + Sj ∫ t 0 ek−j+1 (t) dt} , (7) where ek(t) = yd(t)−yk(t) is the tracking error, integerN ≥ 1 is the order of the ILC law. Pj,Qj, Rj, and Sj are learning gain matrices. The learning operators Qj, Rj, and Sj are chosen to be bounded, and their upper bounds denoted by bQ, bR, and bS, respectively, are defined by bQ = max 1≤j≤N 󵄩󵄩󵄩󵄩 Qj 󵄩󵄩󵄩󵄩 , bR = max 1≤j≤N 󵄩󵄩󵄩󵄩 Rj 󵄩󵄩󵄩󵄩 , bS = max 1≤j≤N 󵄩󵄩󵄩󵄩 Sj 󵄩󵄩󵄩󵄩 . (8) As usual, it is assumed that uk(t) = 0 and ek(t) = 0 for k < 0. The λ-norm will be used in this paper. It is defined by 󵄩󵄩󵄩󵄩f(⋅) 󵄩󵄩󵄩󵄩λ = sup 0≤t≤T {e 󵄩󵄩󵄩󵄩f (t) 󵄩󵄩󵄩󵄩} , λ > 0, (9) for a vector function f : [0, T] → R. For the sake of brevity, the following notations will be used: hk = h (xk (t) , t) , hd = h (xd (t) , t) , δhk = hd − hk, bh = sup ‖h (x (t) , t)‖ , (10) where h represents a function concerned. The partial derivatives of h(x(t), t) are denoted by hxk = ∂h(x, t) ∂x 󵄨󵄨󵄨󵄨󵄨󵄨x=xk(t) , htk = ∂h(x, t) ∂t 󵄨󵄨󵄨󵄨󵄨󵄨x=xk(t) , hxd = ∂h(x, t) ∂x 󵄨󵄨󵄨󵄨󵄨󵄨x=xd(t) , htd = ∂h(x, t) ∂t 󵄨󵄨󵄨󵄨󵄨󵄨x=xd(t) . (11) Let kh be the Lipschitz constant of the function hwith respect to x in [0, T]. Then it is easy to see that 󵄩󵄩󵄩󵄩δhk 󵄩󵄩󵄩󵄩 ≤ kh 󵄩󵄩󵄩󵄩δxk 󵄩󵄩󵄩󵄩 . (12) 3. Main Result Note that the σ(t) is an arbitrary switching rule during the finite time interval [0, T], which is different from the aforementioned studies [14–19]. The switching rule σ(t) can be described as σ = (i0, t0) , (i1, t1) , (i2, t2) , . . . , (in, tn) , (13) where t0 is the initial time instant, by default t0 = 0. tn is the terminal time instant with tn = T. t1, t2, ⋅ ⋅ ⋅ , tn denote the switching instants, and the pair (in, tn) which represents subsystem in is active during the interval tn−1 ≤ t ≤ tn. tn − tn−1 is the dwell time of the subsystem in. Clearly, the control input of overall systems is a piecewise function, and the discontinuity points are those switching instants. Without loss of generality, we can assume that the arbitrary switching rule σ(t) is given as σ (t) = i = {{{{ {{{{ { 1, t ∈ [0, t1] , 2, t ∈ [t1, t2] , .. m, t ∈ [tm−1, T] . (14) Switched sequence (14) implies each subsystem only operated once during the whole interval [0, T]. To proof the main result, we first give the following lemmas. 4 Abstract and Applied Analysis Lemma 7. Suppose that a real positive series {an} ∞ 1 satisfies an ≤ ρ1an−1 + ρ2an−2 + ⋅ ⋅ ⋅ + ρNan−N + ε, (n = N + 1,N + 2, . . .) , (15) where ρi ≥ 0 (i = 1, 2, . . . , N), ε ≥ 0 and ρ = N ∑ i=1 ρi < 1. (16) Then the following holds: lim n→∞ an ≤ ε 1 − ρ . (17) Proof. Let n1 ∈ {n − 1, n − 2, . . . , n − N} be an index number such that n1 = max{an−1, an−2, . . . , an−N}.Then, from (15), we have an ≤ ρ1an−1 + ρ2an−2 + ⋅ ⋅ ⋅ + ρNan−N + ε, ≤ n1 + ε. (18) Similarly, let n2 ∈ {n1 − 1, n1 − 2, . . . , n1 − N} such that n2 = max n1−1 n1−2 . . . , n1−N . (19) Then n1 ≤ 1n1−1 + 2n1−2 + ⋅ ⋅ ⋅ + Nn1−N + ε, ≤ n2 + ε. (20) Therefore an ≤ ρ 2 n2 + ρε + ε. (21) In general, an ≤ ρ m nm + ρ m−1 ε + ρ m−2 ε + ⋅ ⋅ ⋅ + ρε + ε, (22) wherem and nm are positive integers. Ifm is chosen such that nm ≤ N, then (n/N) − 1 ≤ m ≤ n −N, and thereforem → ∞ when n → ∞. LetM = max{a1, a2, . . . , aN}, and then


Introduction
A switched system is a hybrid dynamical system, which consists of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching between them.During the past decades, switched systems have been widely studied, and many interesting results have been reported in the literature, for instance [1][2][3] and the references therein.The motivation to study switched systems is mainly in twofold.First of all, many engineering systems can be represented by switched systems, such as networked control systems (NCS) [4,5], traffic control [6], automotive engine control, and aircraft control [7].Secondly, the idea of controller switching is introduced in order to overcome the shortcomings of the single controller and improve system performance [8,9].Some methods have been used in the study of switched systems such as the multiple Lyapunov functions [10,11], the concept of average dwell time [12], and piecewise quadratic Lyapunov functions [13].
Recent researches in switched system typically focus on the analysis of dynamic behaviors, such as stability [14], controllability, reachability [15,16], and observability [17] aiming to design controllers with guaranteed stability and performance [18,19].Besides the aforementioned problem, designing a controller to achieve tracking for switched systems is a challenging problem [20][21][22].In [20], the tracking control problem for switched linear systems with timevarying delays is investigated, and the average dwell time approach and piecewise Lyapunov functional methods are utilized to the stability analysis and controller design.In [21], an observer-based tracking control approach is proposed for switched linear time-varying delay systems with unavailable states.The single Lyapunov-Krasovskii functional method is utilized to the stability analysis and controller design.In [22], the tracking control problem for switched nonlinear systems subject to an output constraint is considered.It is worth pointing out that most of these approaches are given based on the accurate model of nonlinear switched systems, and thus their control performances depend on the accuracy of the models.In addition, various unmodelled dynamics and uncertainties always exist in practical systems.The abovementioned model-based control approaches for switched systems may lead to bad performance or cause closed-loop system unstable in practice.Therefore, it is necessary and practical to design tracking control methods for switched systems requiring less knowledge about the system dynamics.
Fortunately, for repetitive systems, iterative learning control (ILC) offers a systematic design that can improve the tracking performance by iterations in a fixed time interval.The key feature of this technique is to use information from the previous operation in order to enable the controlled system to perform better progressively from operation to operation.It seems that the main advantage of the iterative learning control strategy is to require less a priori knowledge about the system dynamics and less computational effort than many other types of control strategies.Hence, iterative learning control for repetitive dynamical systems has received considerable attention, and it has made significant progresses over the past two decades (see [23][24][25][26][27][28][29][30] and references therein).However, to the best of our knowledge, no one has been studied the iterative learning control for switched systems.This observation motivates the present study.
In practice, some switched systems are in general repeated, such as traffic system and batch process with multiprocedure.The traffic system can be viewed as a switched system [6].We may easily find that traffic flow patterns in two consecutive days, or the same weekday of two consecutive weeks, are very close.Ruling out the occasional occurrence of accidents, the routine traffic flow on freeway in the macroscopic level will show inherent repeatability every day.Likely we can find the similarities on a monthly basis, or even a yearly basis [26].For the batch process with multiprocedure, the system can also be considered as a switched system, and the different procedures represent the different subsystems.If the product is batch processing, then the switched system is operated repetitively.In this paper, the problem of iterative learning control for a class of nonlinear switched systems with arbitrary switched rules is considered.Despite much progress, significant research remains to be done in the direction of linear switched systems, especially since the majority of practical switched systems exhibit inherently nonlinear dynamics and delays [31][32][33].Hence, we are focusing on ILC for nonlinear switched systems with time delay.
Most of the existing ILC schemes are based on the firstorder updating laws; that is, only the information of one previous iteration is employed.[34] used a high-order ILC law for tracking control of nonlinear systems, where, to form the control in current iteration, the information of several previous learning iterations, including control functions, tracking errors as well as their derivatives, is used.It is demonstrated that high-order ILC schemes have potential to give a better convergence performance than the first-order ILC schemes.Therefore, it is beneficial to investigate highorder ILC schemes for the tracking control of nonlinear switched systems.
The rest of this paper is organized as follows.In Section 2, the problem formulation is described.In Section 3, a sufficient condition which guarantees the stability of highorder PID-type ILC for the nonlinear switched ILC system is given.In Section 4, two examples are presented to validate the theoretical results.Finally, some conclusions are given in Section 5.
Abstract and Applied Analysis 3 Assumption 3. The resetting condition is satisfied for all the iteration; that is, where  = max{ℎ, } and   () is the desired initial function.
Remark 4. From Assumption 1, since   () exists uniquely, the uniform convergence of the control profile () to   () implies that the state and output tracking errors will vanish.Assumption 2 is a basic condition for switched systems which means that system states are continuous, even for structure switches.
Remark 5. Assumption 3 means   (0) =   (0) for all , which is the identical initial condition for ILC system.For a switched system, even though the dynamic behavior is changing between different subsystems, the initial condition reset is often satisfied in many practical systems.For instance, when a product is processed by several different procedures, then the system can be considered as a switched system, and the different procedures represent the different subsystems.If the product is batch processing, then the switched system is operated repetitively.Even though the processing is changing between different procedures, the initial state of the product is identical.
Remark 6.The switched rule () in system ( 1) is an arbitrary switched rule on the time domain, and it is invariable on the iteration domain.One may argue that the problem considered in this paper is similar to the ILC for time varying nonlinear system.However, there are some crucial differences.The time varying system admits a family of solutions that can be parameterized solely by the initial condition, whereas the switched system admits a family of solutions that is parameterized both by the initial condition and the switching signal.Besides, the switched system has sudden transient at some time instants.
The control target is to find a control input sequence   () =   (), such that   () converges to   () as  → ∞; that is, as the learning iteration repeats, the system output converges to the desired trajectory.Now, the following high-order ILC updating law for the system (3) is proposed which uses the , , and  information of tracking errors: where As usual, it is assumed that   () = 0 and   () = 0 for  < 0.
The -norm will be used in this paper.It is defined by for a vector function  : [0, ] →   .
For the sake of brevity, the following notations will be used: where ℎ represents a function concerned.The partial derivatives of ℎ((), ) are denoted by

Main Result
Note that the () is an arbitrary switching rule during the finite time interval [0, ], which is different from the aforementioned studies [14][15][16][17][18][19].The switching rule () can be described as where  0 is the initial time instant, by default  0 = 0.   is the terminal time instant with   = . 1 ,  2 , ⋅ ⋅ ⋅ ,   denote the switching instants, and the pair (  ,   ) which represents subsystem   is active during the interval  −1 ≤  ≤   .  −  −1 is the dwell time of the subsystem   .Clearly, the control input of overall systems is a piecewise function, and the discontinuity points are those switching instants.Without loss of generality, we can assume that the arbitrary switching rule () is given as Switched sequence (14) implies each subsystem only operated once during the whole interval [0, ].
To proof the main result, we first give the following lemmas.
Remark 10.Theorem 9 is given for the nonlinear switched system with the switched sequence (14), which assumes each subsystem only operated once during the whole interval [0, ], and the active sequence for subsystems is {1, 2, . . ., }.
From the proof process of Theorem 9, we know the result can also be extended to the arbitrary switch sequence with more times active for each subsystem.
Remark 11.High-order ILC updating law utilizes the past experiences comprehensively and has more flexibilities in choosing learning operators and parameters.Hence, the better ILC performance can be expected.If the system dynamics is totally unknown, like the selection of learning parameters in traditional ILC laws, the order  selection is also based on a trial-and-error method, which should be also in an iterative learning way.In practice,  should normally be chosen to be less than or equal to 3.
Remark 12.In many practical systems, stochastic disturbances are unavoidable [36][37][38].In this case, the nonlinear switched system (1) can be described by where   () ∈   and V  () ∈   are state disturbances and output disturbances, respectively.If the disturbances are bounded in the sense of ‖  ()‖ ≤   and ‖V  ()‖ ≤  V for all  ≥ 0, where   , and  V are positive constants, from the proof of Theorem 9, it is easy obtained that, when the PID ILC law ( 7) is used for system (12), if the conditions in Theorem 9 are guaranteed, then the system output converges to the desired output with a bounded error.The error bound depends on the   and  V .The similar results can also be founded in ILC for general nonlinear systems with disturbances [39,40].

Simulation Illustrations
In this simulation test, let us consider the following SISO nonlinear switched system with two subsystems: The nonlinear switched system is operated during the two subsystems with arbitrary switched law in time domain, and the desired output is given as   () = sin(30) +  2 , and  ∈ [0, 1].For the initial state, it is assumed that   (0) =   (0) = 0 for all .The ILC law (7) is applied by adopting the zero initial control input  0 () = 0 for all  ∈ [0, 1].In simulation, we produces a random sequence (),  ∈ [0, 1] with the value is 1 and 2, as shown in Figure 1.If () = 1, the system (64) is  1 ; otherwise, if () = 2, the system (64) is  2 .
To check the tracking performance, we consider the following second-order ILC law where  1 = 0.3,  2 = 0.5,  1 = 0.9, and  2 = 1.2.Note that the nonlinear functions sin(), cos() are both uniformly globally Lipschitz in  and uniformly bounded for all  ∈ [0,1] and  ∈ .It is obvious that   and   are also bounded.Checking the condition in Theorem 9, we have then,  1 +  2 = 0.48 < 1.From Theorem 9, we know that the nonlinear switched ILC system is asymptotically stable.Figures 2, 3, and 4 give simulation results for system output trajectories at 5th, 10th, and 30th iterations.In these figures, we plot the system output in solid line and the desired output trajectory in dashed line.It can be observed that the better tracking performance can be obtained after 30th iteration.Intuitively, this result can be understood in the following way: although the switched system has an arbitrary   switched rule on the time domain, the arbitrary switched rule is invariable on the iteration domain.If we update the control signals on the repetitive iteration domain, the perfect tracking performance can be obtained eventually.
In this section, we consider two illustrative examples.
Example 1 (SISO nonlinear switched system).To further check the effectiveness of high-order ILC law, we also did simulation tests using first-order ILC given as  +1 () =   () + 0.6  () + 0.5 ė  (), and the control input at the first iteration is also given as  0 () = 0 for all  ∈ [0, 1].The tracking errors for second-order ILC and first order ILC are given in Figure 5. Clearly, the high-order ILC scheme can be faster convergence than the first-order one.
Example 2 (MIMO nonlinear switched system with state delay).In this example, let us consider the following MIMO nonlinear switched system with time delay: where ℎ =  = 0.1.The switched system is also operated during the two subsystems with arbitrary switched law in time domain, and the desired trajectories are given as For the initial state, it is assumed that  1, (0) =  2, (0) = 0 for all .The ILC law is applied by adopting the zero initial control input  1,0 () = 0,  2,0 () = 0, for all  ∈ [0, 1].In simulation, we produces a random sequence (), and  ∈ [0, 1] with the value is 1 and 2, as shown in Figure 6, If () = 1, the system (2) is  1 ; otherwise, if () = 2, the system (2) is  2 .We consider the following second order ILC law: where  1 = 0.5,  2 = 0.4,  1 = 0.8, and  2 = 1.1.Note that the nonlinear system only contains nonlinear argument sin(), cos(), and they are both uniformly globally Lipschitz in  and uniformly bounded for all  ∈ [0, 1].To guarantee the ILC convergence,  should be less than one.It is easy to obtain that the previous choices of  1 ,  2 clearly satisfy the convergence condition.The simulation results for system output trajectories at 5th, 10th, and 30th iterations are shown in Figures 7, 8, and 9.In these figures, we plot the system output in solid line and the desired output trajectory in dashed line.It can be observed that the switched ILC system is asymptotically stable, and the better tracking performance can be obtained after 30th iteration for the whole time interval.We also did simulation tests using first order ILC   given as  +1 () =   () + 0.6  () + 0.5 ė  (), and the control input at the first iteration is also given as  1,0 () = 0,  2,0 () = 0, for all  ∈ [0, 1].The simulation results of tracking errors for second order ILC and first order ILC are shown in Figure 10.It is presented to illustrate the effectiveness of high-order ILC law over the conventional first-order one.

Conclusions
The stability of high-order ILC law for nonlinear switched systems with state delays is analyzed.It is shown that the asymptotic convergence of the tracking error can be guaranteed under some conditions, and the perfect tracking performance can be obtained during the entire time interval.This result can be understood in the following way: although the switched system has an arbitrary switched rule on the time domain, the arbitrary switched rule is invariable on the iteration domain.If we update the control signals on the repetitive iteration domain, the perfect tracking performance can be obtained eventually.
In the previous works [20][21][22], the asymptotic convergence of tracking control is only guaranteed in the time  domain; that is, the system output tracking errors converge to 0 when  → ∞.When the ILC is considered for switched systems, the perfect tracking performance can be obtained during the entire time interval.ILC can use the previous operation information of the switched systems, and then the tracking can be performed better progressively from operation to operation.It also requires less a priori knowledge about the system dynamics and less computational effort than many other kinds of control approaches.In future work, we will consider the stability of nonlinear switched systems with arbitrary switched rule on the iteration domain.

Figure 5 :
Figure 5: The max tracking error of Example 1 for different ILC laws.