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Sheng Zhu, Xuejie Wang and Hong Liu, "Observer-based Iterative and Repetitive Learning Control for a Class of Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 5, no. 5, pp. 990-998, Sept. 2018. doi: 10.1109/JAS.2017.7510463
Citation: Sheng Zhu, Xuejie Wang and Hong Liu, "Observer-based Iterative and Repetitive Learning Control for a Class of Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 5, no. 5, pp. 990-998, Sept. 2018. doi: 10.1109/JAS.2017.7510463

Observer-based Iterative and Repetitive Learning Control for a Class of Nonlinear Systems

doi: 10.1109/JAS.2017.7510463
Funds:

the Third Level of Hangzhou 131 Young Talent Cultivation Plan Funding, and 2018 Soft Science Research Project of Zhejiang Provincial Science and Technology Department Zhejiang Province Construction and participate in the "The Belt and Road" Technology Innovation Community Path Research 2018C35029

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  • In this paper, both output-feedback iterative learning control (ILC) and repetitive learning control (RLC) schemes are proposed for trajectory tracking of nonlinear systems with state-dependent time-varying uncertainties. An iterative learning controller, together with a state observer and a fully-saturated learning mechanism, through Lyapunov-like synthesis, is designed to deal with time-varying parametric uncertainties. The estimations for outputs, instead of system outputs themselves, are applied to form the error equation, which helps to establish convergence of the system outputs to the desired ones. This method is then extended to repetitive learning controller design. The boundedness of all the signals in the closed-loop is guaranteed and asymptotic convergence of both the state estimation error and the tracking error is established in both cases of ILC and RLC. Numerical results are presented to verify the effectiveness of the proposed methods.

     

  • Iterative learning control (ILC), [1] and repetitive control(RC) [2] are two typical learning control strategies developed for systems performing tasks repetitively. ILC aims to achieve complete tracking of the system output to a desired trajectory over a pre-specified interval, through updating the control input cycle by cycle, while RC addresses the problem of the periodic reference tracking and periodic disturbance rejection. The contraction-mapping-based learning control [1] features simplicity, especially reflected in the only use of the output measurements. However, the learning gains are not easy to be determined because of the difficulty in solving norm inequalities, that may lead to obstacles to the applications of the conventional learning control method.

    Early in 1990s, aiming to overcome the mentioned limitation of the contraction-mapping-based method, there have been intense researches in developing Lyapunov-like based designs of iterative learning control [3]-[5] and repetitive control [6], [7]. Recently, such Lyapunov-like approach has received further attention [8], [9], [20], [21]-[25]. In [8], [9], the learning control problems were formulated for broader classes of uncertain systems with local Lipschitz nonlinearities and time-varying parametric and norm bounded uncertainties. Note that in the mentioned works, the full state information is assumed to be available. However, in many applications, the system state may not be available for the controller design, where it is necessary to design output-based learning controllers in the framework of Lyapunov-like based learning control theory.

    For linear systems, Kalman filter [10] and Luenberger observer [11] are two kinds of basic practical observers that adequately address the linear state estimation problem. Observers for nonlinear uncertain systems have recently received a great deal of attention, and there have been many designs such as adaptive observers [12], robust observers [13], sliding mode observers [14], neural observers [15], fuzzy observers [16], etc. In the published literature, works have been done for output-feedback-based learning control. In [17], a transformation to give an output feedback canonical form is taken for nonlinear uncertain systems with well-defined relative degree. But the uncertainties in the transformed dynamics should be state independent. In [18], an adaptive learning algorithm is presented for unknown constants, linking two consecutive iterations by making the initial value of the parameter estimation in the next iteration equal to the final value of the one in the current iteration. The results are extended to output feedback nonlinear systems. But the nonlinear function of the system is assumed to be concerned with the system output only.

    The observers used in learning control are reported in [19] and [20]. The former addresses the finite interval learning control problem while the latter addresses the infinite interval learning control problem. The nonlinear functions in [19] are not parametrized and the observer based learning controller is designed in the framework of contraction mapping approach without the requirement of zero relative degree. The observer used in [20] is special and complicated. By virtue of the separation principle, the state estimation observer and the parameter estimation learning law are taken into account respectively. The Lyapunov-like functions are employed and two classes of nonlinearties, the global Lipschitz continuous function of state variables and the the local Lipschitz continuous function of output variables, are all considered.

    As is known, repositioning is required at the beginning of each cycle in ILC. Repositioning errors will accumulate with iteration number increasing, which may lead the system to diverge finally. The variables to be learned are assumed to be repetitive. Repetitive control requires no repositioning, but the variables to be estimated need to be periodic. It is commonly seen that a repetitive signal may not be periodic. RLC has been developed recently [21]-[25], and formally formulated in [24], given as follows:

    F1) every operation ends in the same finite time of duration;

    F2) the desired trajectory is given a priori and is closed;

    F3) the initial condition of the system at the beginning of each cycle is aligned with the final position of the preceding cycle;

    F4) the time-varying variables to be learnt are iteration independent cycle;

    F5) the system dynamics are invariant throughout all the cycles.

    Unlike the ILC and RC, RLC can handle the finite time interval tracking without the repositioning. In the published literature, however, there are few results on observer-based RLC.

    In this paper, through Lyapunov-like synthesis, the ILC problem is addressed for a class of nonlinear systems that only the system output measurements are available. Compared with the existing works, the main contributions of our paper are given as follows. Firstly, the parametric uncertainties discussed in this paper are state-dependent, while the uncertainties treated in the existing results [17]-[19] are assumed to be output-dependent. The state-dependent terms cannot be directly used in the output feedback controller design due to the lack of the state information. Secondly, a robust learning observer is given by simply using Luenberger observer design. Reference [20] pointed out that for learning control systems many conventional observers are difficult to apply. We clarify the possibility of designing observer-based learning controller by using a Luenberger observer. The estimation of output, instead of the system output itself, is applied to form the error equation, which helps to establish convergence of the system output to the desired one. Finally, the method used in output-feedback ILC design is extended to the RLC design. To the best of our knowledge, the output-feedback RLC problem is still open. In this paper, the fully saturated learning laws are developed for estimating time-varying unknowns. The boundedness of the estimations plays an important role in establishing stability and convergence results of the closed-loop system.

    The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section Ⅱ. The main results of this paper are given in Section Ⅲ and Section Ⅳ, providing performance and convergence analysis of the observer based ILC and RLC, respectively. Section Ⅴ presents simulation results and gives the comparison of the ILC and RLC schemes. The final section draws the conclusion of this work.

    Consider a class of uncertain nonlinear systems described by

    ˙x(t)=Ax(t)+B(u(t)+Θ(t)ξ(x(t),t))y(t)=Cx(t) (1)

    where t is the time. x(t)Rn, u(t)Rm and y(t)Rm represent the state vector, the system output and the control input, respectively. ARn×n, BRn×m and CRm×n are known constant matrices. Θ(t)Rm×n1 is the unknown continuous time-varying matrix-valued function and ξ(x(t),t)Rn1 is the known vector-valued function.

    Remark 1: ||Θ(t)|| is bounded over [0,T], hence let θm be the supremum of the ||Θ(t)||, but the value of θm is unknown.

    Assume that the system operates repeatedly over a specified interval [0,T]. Let us denote by k the repetition index, and system (1) can be rewritten as follows:

    ˙xk(t)=Axk(t)+B(uk(t)+Θ(t)ξ(xk(t),t))yk(t)=Cxk(t). (2)

    Given a desired trajectory yd(t)=Cxd(t) over the interval [0,T], our objective is to design a learning control law uk(t), such that the output yk(t) converges to the desired output yd(t), for all t[0,T], as k. To achieve the perfect tracking, the following assumptions are made.

    Assumption 1: For system (2), there exist positive matrices PRn×n and QRn×n satisfying

    PA+ATP=Q (3)
    BTP=C. (4)

    Assumption 2: Rank(CB)=m.

    Assumption 3: The nonlinear function ξ(x(t),t) satisfies global Lipschitz condition, i.e., x1(t),x2(t)Rn, ||ξ(x1(t),t)ξ(x2(t),t)||γ||x1(t)x2(t)||, where γ is the unknown Lipschitz constant.

    Remark 2: Assumption 1 is the common strictly positive real(SPR) condition. It guarantees the asymptotic stability of the linear part of the system which helps us construct the Lyapunov-like function easily. It also indicates that if (A,B) is completely controllable, (A,C)=(A,BTP) is observable because the matrix P is positive, which makes that the observer can be constructed. Assumption 2 is needed to guarantee the existence of the learning gain in the controller design. The Lipschitz condition in Assumption 3 is commonly seen in observer design and ILC design for nonlinear systems.

    For the learning controller design, saturation function sat() which is defined in the following ensures the boundedness of the parameters estimation. For scalar f,

    sat(f)={ˉf1,if f<ˉf1f,if ˉf1fˉf2ˉf2,if f>ˉf2 (5)

    where ˉf1 and ˉf2 are the lower and upper limits, respectively. For m×n1 matrix F={fij}m×n1, the function sat(F) is defined as {sat(fij)}m×n1, where the saturation limits are set to be the same for each entry.

    Lemma 1: For m×n1-dimensional matrixes F1 and F2, if F1=sat(F1), and the saturation limits of sat(F1) and sat(F2) are the same, then

    tr((F1sat(F2))T(F2sat(F2)))0. (6)

    Proof: It follows that for the matrices F1=(f1ij)m×n1 and F2=(f2ij)m×n1

    tr((F1sat(F2))T(F2sat(F2)))    =n1j=1mi=1(f1ijsat(f2ij))(f2ijsat(f2ij)). (7)

    Let ˉf1 and ˉf2 are the lower and upper limits of the saturation function sat() such that ˉf1f1ijˉf2 and ˉf1f2ijˉf2. If f2ij>sat(f2ij), then f1ijsat(f2ij)=f1ijˉf20, if f2ij<sat(f2ij), then f1ijsat(f2ij)=f1ijˉf10, and if f2ij=sat(f2ij), then f2ijsat(f2ij)=0. Therefore, we obtain

    (f1ijsat(f2ij))(f2ijsat(f2ij))0. (8)

    The following property of trace is used as below

    tr(GTg2gT1)=tr(GTg2gT1)T=gT2Gg1 (9)

    where GRm×n1,g1Rn1×1 and g2Rm×1.

    To establish stability and convergence of the repetitive learning control systems in Section Ⅳ, the following lemma is given.

    Lemma 2: The sequence of nonnegative functions fk(t) defined on [0,T] converges to zero uniformly on [0,T], i.e.,

    limkfk(t)=0,t[0,T]

    if

    limkT0fk(τ)dτ=0 (10)

    and fk(t) is equicontinuous on [0,T].

    The proof of lemma 2 can be found in [19].

    Let ˆxk(t) represent the state estimation, and a robust learning observer is constructed in the following form:

    ˙ˆxk(t)=Aˆxk(t)+Buk(t)+BˆΘk(t)ξ(ˆxk(t))+12Bˆμk(t)(yk(t)Cˆxk(t)) (11)

    where ˆΘk(t) is the estimation of the unknown time-varying function Θ(t), and ˆμk(t) is introduced to approximate an unknown constant μ=(2θ2mγ2)/λ1, where λ1 is the minimum eigenvalue of the matrix Q. ˆΘk(t) and ˆμk(t) are updated by the learning laws (22) and (23) respectively.

    By defining the estimation error δxk(t)=xk(t)ˆxk(t), it can be easily derived

          δ˙xk(t)=˙xk(t)˙ˆxk(t)=Aδxk(t)+B˜Θkξ(ˆxk)+BΘ(t)(ξ(xk)ξ(ˆxk))12Bˆμk(t)(yk(t)Cˆxk(t)) (12)

    where ˜Θk(t)=Θ(t)ˆΘk(t). Using the following Lyapunov function candidate

    W1k(t)=δxTk(t)Pδxk(t) (13)

    where P is defined in Assumption 1, we obtain

    ˙W1k(t)=2δxTk(t)PAδxk(t)+2δxTk(t)PB˜Θkξ(ˆxk)+2δxTk(t)PBΘ(t)(ξ(xk)ξ(ˆxk))δxTk(t)PBˆμk(t)(yk(t)Cˆxk(t)). (14)

    According to Assumptions 1 and 2, we have

    ˙W1k(t)λ1||δxk||2+2δxTk(t)PB˜Θkξ(ˆxk)+2||ykCˆxk||θmγ||δxk||ˆμk||ykCˆxk||2. (15)

    Using the inequality

    2||ykCˆxk||θmγ||δxk||λ12||δxk||2+2θ2mγ2λ1||ykCˆxk||2 (16)

    it can be verified that

    ˙W1k(t)λ12||δxk||2+2δxTkPB˜Θkξ(ˆxk)+˜μk(t)||ykCˆxk||2 (17)

    where ˜μk(t)=μˆμk(t). In order to counteract the second and third terms to the right of (17), full saturated parameter updating laws are given in the following part.

    Let us define the novel error function ek(t)=ˆyk(t)yd(t), where ˆyk(t)=Cˆxk(t). The objective of the learning controller design is to make yk(t)yd(t), t[0,T], as k. Now that the proposed robust learning observer(11) ensures ˆxk(t)xk(t), t[0,T], as k, and the observer-based iterative learning controller will be designed to make ˆyk(t)yd(t), t[0,T], as k firstly, then the objective can also be achieved. Using (11), the derivative of ek(t) is

    ˙ek(t)=C˙ˆxk(t)˙yd(t)=CAˆxk(t)+CBˆΘk(t)ξ(ˆxk(t))+12CBˆμk(t)(yk(t)Cˆxk(t))˙yd(t)+CBuk(t) (18)

    from which we can easily obtain the control law

    uk=ˆΘk(t)ξ(ˆxk(t))+(CB)1(˙yd(t)CAˆxk(t)L1ek(t))12ˆμk(t)(yk(t)Cˆxk(t)) (19)

    where L1Rm×m is a given positive matrix.

    Using the following Lypunov function candidate

    W2k(t)=12||ek(t)||2 (20)

    and considering the control input (19) and the error dynamic (18), we obtain

    ˙W2k(t)=eTk(t)˙ek(t)=eTk(t)L1˙ek(t)=λ2||ek||2 (21)

    where λ2 is the minimum eigenvalue of the matrix L1.

    It should be noted that the error dynamics (18) is independent of nonlinear uncertainties in system (1), and all variables in (18) are available for controller design. This is the reason why we use ˆyk(t) instead of yk(t) in error definition. The controller (19) and the observer (11) work concurrently, where ˆΘk(t) and ˆμk(t) are updated by the following learning laws

    {ˆΘk(t)=ˆΘk1(t)+2L2(yk(t)Cˆxk(t))ξT(ˆxk)ˆΘk(t)=sat(ˆΘk(t))ˆΘ1(t)={0}m×n1,t[0,T] (22)

    and

    {ˆμk(t)=ˆμk1(t)+l3||ykCˆxk||2ˆμk(t)=sat(ˆμk(t))ˆμ1(t)=0,t[0,T] (23)

    where L2=LT2Rm×m is a given positive matrix and l3>0 is a constant. Let us define Θ(t)={θi,j(t)}m×n1. Since Θ(t) is bounded, we assume ˉθ1<θi,j(t)<ˉθ2, where ˉθ1 and ˉθ2 are the saturation limits of the matrix-valued function sat(ˆΘk(t)). The saturation limits of the scalar function sat(ˆμk(t)) are ˉμ1 and ˉμ2, and we assume ˉμ1<μ<ˉμ2.

    Assumption 4: At the beginning of each cycle, ˆxk(0) =xk(0)=xd(0).

    Remark 3: Assumption 4 is about the initial states resetting condition. This part focuses on the design of observers. An extension of initial states condition is given in Section Ⅳ. See Assumptions 5 and 6.

    Theorem 1: For system(1) satisfying Assumptions 1-4, let controller (19) together with full saturated learning laws (22) and (23), where ˆxk is given by observer (11), be applied. Then,

    1) all signals in the closed-loop are bounded on [0,T], and

    2) limk||δxk||2=0, limk||ek||2=0, for t[0,T].

    Proof: Let us consider the following Lyapunov-like function

    Wk(t)=W1k(t)+W2k(t)+12l3t0˜μ2k(τ)dτ+12t0tr[˜ΘTk(τ)L12˜Θk(τ)]dτ (24)

    where W1k(t) and W2k(t) are given by (13) and (20), respectively.

    For k=1, 2, , and t[0,T], the difference of (24) is

    ΔWk(t)=Wk(t)Wk1(t)=W1k(t)+W2k(t)+12t0{tr[˜ΘTk(τ)L12˜Θk(τ)]tr[˜ΘTk1(τ)L12˜Θk1(τ)]}dτ+12l3t0[˜μ2k(τ)˜μ2k1(τ)]dτW1k1(t)W2k1(t). (25)

    Assumption 4 implies W1k(0)=0 and W2k(0)=0. In view of (17) and (21), we obtain

    W1k(t)=W1k(0)+t0˙W1k(τ)dτt0(λ12||δxk||2+2δxTkPB˜Θkξ(ˆxk)+˜μk||ykCˆxk||2)dτ (26)

    and

    W2k(t)=W2k(0)+t0˙W2k(τ)dτ=λ2t0||ek||2dτ. (27)

    Using the equalities

    12[tr(˜ΘTkL12˜Θk)tr(˜ΘTk1L12˜Θk1)]=tr[(ˆΘkˆΘk1)TL12˜Θk]             12tr[(ˆΘkˆΘk1)TL12(ˆΘkˆΘk1)] (28)

    and

    12l3t0˜μ2k˜μ2k1dτ=1l3t0˜μk(ˆμkˆμk1)dτ12l3t0(ˆμkˆμk1)2dτ (29)

    and substituting (26), (27) into (25), it can be verified that

    ΔWk(t)λ12t0||δxk||2dτλ2t0||ek||2dτ+t02δxTkPB˜Θkξ(ˆxk)dτ+t0˜μk||ykCˆxk||2dτt0tr((ˆΘkˆΘk1)TL12˜Θk)dτ1l1t0˜μk×(ˆμkˆμk1)dτW1k1(t)W2k1(t). (30)

    Applying learning laws (22) and (23), inequality (6) and Lemma 1, we obtain

    2δxTkPB˜Θkξ(ˆxk)tr[(ˆΘkˆΘk1)TL12˜Θk]=tr[(ˆΘksat(ˆΘk))TL12(Θsat(ˆΘk))]0 (31)

    and

    ˜μk||ykCˆxk||21l3˜μk(ˆμkˆμk1)=1l3(μksat(ˆμk))(ˆμksat(ˆμk))0. (32)

    Substituting (31) and (32) into (30) gives rise to

    ΔWk(t)λ12t0||δxk||2dτλ2t0||ek||2dτW1k1(t)W2k1(t)λ12t0||δxk||2dτλ2t0||ek||2dτλ3||δxk1||212||ek1(t)||20 (33)

    where λ3 is minimum eigenvalue of matrix P. Obviously, Wk(t) is a monotone non-increasing sequence over [0,T], therefore, in order to prove the boundedness of Wk(t),t[0,T], we need to prove the boundedness of W0(t) for all t[0,T]. From (24), when k=0, we have

    W0(t)=δxT0Pδx0+12||e0||2+12t0tr[˜ΘT0L12˜Θ0]dτ+12l3t0˜μ20dτ. (34)

    Since ˆΘ1(t)={0}m×n1, ˆμ1(t)=0, t[0,T], from (22) and (23), we obtain

    {ˆΘ0(t)=2L2(y0(t)Cˆx0(t))ξT(ˆx0)ˆΘ0(t)=sat(ˆΘ0(t)) (35)
    {ˆμ0(t)=l3||y0Cˆx0||2ˆμ0(t)=sat(ˆμ0(t)). (36)

    Taking the derivative of W0(t) and using (35), (36) and Lemma 1, we have

    ˙W0λ12||δx0||2λ2||e0||212tr(ˆΘT0L12ˆΘ0)+tr((ˆΘ0sat(ˆΘ0))TL12(Θsat(ˆΘ0)))+12tr(ΘTL12Θ)+1l3(μsat(ˆμ0))(ˆμ0sat(ˆμ0))+12l3s212l3ˆμ2012tr(ΘTL12Θ)+12l3μ2. (37)

    Since ˉθ1<θi,j(t)<ˉθ2, ˉμ1<μ<ˉμ2, ˙W0 has an upper bound. By the boundedness of W0(0) and continuity of W0(t) on [0,T], W0(t) is bounded. From (33), Wk(t) is uniformly bounded on [0,T]. Therefore, ||δxk|| and ||ek|| are all uniformly bounded on [0,T]. It is seen that the full saturated learning laws (22) and (23) ensure the boundedness of ˆΘk(t) and ˆμk(t). We can conclude that from (19) uk(t) is uniformly bounded on [0,T], and from (18), ˙ek(t) is uniformly bounded on [0,T], and from (12), δ˙xk(t) is uniformly bounded on [0,T].

    From (33), we obtain

    Wk(t)=W0(t)+ki=1ΔWi(t)W0(t)λ12ki=1t0||δxi||2dτλ2ki=1t0||ei||2dτλ3k1i=1||δxi||212k1i=1||ei||2. (38)

    Since Wk(t) is a monotone non-increasing series with an upper bound, its limit exists such that

    limkWk(t)W0(t)λ12limkki=1t0||δxi||2dτλ2limkki=1t0||ei||2dτλ3limkk1i=1||δxi||212limkk1i=1||ei||2. (39)

    By the positiveness of Wk(t) and the finiteness of W0(t), we have limk||δxk||2=0, limk||ek||2=0, for t[0,T].

    In this section, we extend the observer-based ILC design into RLC design for uncertain nonlinear systems. The following properties are assumed according to the repetitive learning control formulation.

    Assumption 5: The desired trajectory is given to satisfy

    yd(0)=yd(T) (40)

    and yd(t) is bounded for t[0,T].

    Assumption 6: At the beginning of each cycle,

    xk(0)=xk1(T) (41)
    ˆxk(0)=ˆxk1(T) (42)

    where ˆxk(t),t[0,T] is given by observer (11).

    Remark 4: Assumptions 5 and 6 satisfy F2) and F3). The initial state estimation condition (42) is required for the observer (11). We do not need Assumption 3 which is a strict condition in practical system. No extra limits of the unknown time-varying function Θ(t) are needed in RLC, that implies Θ(t) is also repetitive over [0,T] instead of being periodic in repetitive control.

    Theorem 2: Considering system(1) with controller (19) and full saturated learning laws (22) and (23), where the states are given by the observer (11) over a specified time interval [0,T], if Assumptions 1-3 and Assumptions 5 and 6 are satisfied,

    1) all signals in the closed-loop are bounded on [0,T], and

    2) limk||δxk||2=0, limk||ek||2=0, for t[0,T].

    Proof: Assumptions 5 and 6 imply

    W1k(0)+W2k(0)=(xk(0)ˆxk(0))TP(xk(0)ˆxk(0))+12||ˆyk(0)yd(0)||2=(xk1(T)ˆxk1(T))TP(xk1(T)ˆxk1(T))+12||Cˆxk1(T)yd(T)||2=W1k1(T)+W2k1(T) (43)

    where W1k(t) and W2k(t) are given by (13) and (20), respectively. We choose the same Lyapunov-like function Wk(t) as (24), and use the same control law uk(t) as (19), where ˆxk(t) is obtained by the observer (11). It follows that for k=1,2,,

    Wk(t)=W1k(t)+W2k(t)+12t0{tr[˜ΘTk(τ)L12˜Θk(τ)tr[˜ΘTk1(τ)L12˜Θk1(τ)]}dτ+12l1t0[˜μ2k(τ)˜μ2k1(τ)]dτW1k1(t)W2k1(t)+Wk1(t). (44)

    In view of (17) and (21), substituting (28) and (29) into (44), we obtain

    Wk(t)λ12t0||δxk||2dτλ2t0||ek||2dτ+W1k(0)+t02δxTkPB˜Θkξ(ˆxk)dτ+t0˜μk||ykCˆxk||2dτt0tr((ˆΘk(t)ˆΘk1(t))TL12˜Θk)dτ+W2k(0)1l1t0˜μk(τ)(ˆμk(τ)ˆμk1(τ))dτW1k1(t)W2k1(t)+Wk1(t). (45)

    Applying inequalities (31) and (32), we have

    Wk(t)λ12t0||δxk||2dτλ2t0||ek||2dτ+W1k(0)+W2k(0)W1k1(t)W2k1(t)+Wk1(t). (46)

    In addition, by the definition of Wk(t), we obtain

    Wk1(t)W1k1(t)W2k1(t)                         =12t0tr(˜ΘTk1L12˜Θk1)dτ+12l3t0˜μ2k1dτ. (47)

    Therefore, in view of (43), we have

    Wk(t)λ12t0||δxk||2dτλ2t0||ek||2dτ+12l3t0˜μ2k1dτ+W1k1(T)+W2k1(T)+12t0tr(˜ΘTk1L12˜Θk1)dτ. (48)

    It follows that

    Wk(t)12t0tr(˜ΘTk1L12˜Θk1)dτ+12l3t0˜μ2k1dτ+W1k1(T)+W2k1(T)12T0tr(˜ΘTk1L12˜Θk1)dτ+12l3T0˜μ2k1dτ+W1k1(T)+W2k1(T). (49)

    It is obvious that the right-hand side of the last inequality is actually the Wk1(T), which implies

    Wk(t)Wk1(T) (50)

    for all t[0,T]. By setting t=T, we obtain

    Wk(T)Wk1(T). (51)

    From above, it is clearly seen that Wk(T) is monotonically decreasing. Taking the derivative of W0(t), we can obtain the same result as (37) such that W0(t) is bounded on [0,T]. Therefore, Wk(T) is uniformly bounded. Using (50), Wk(t) is uniformly bounded on [0,T]. Therefore, ||δxk|| and ||ek|| are all uniformly bounded on [0,T]. It is seen that the full saturated learning laws (22) and (23) ensure the boundedness of ˆΘk(t) and ˆμk(t). We can claim that from (19), uk(t) is uniformly bounded on [0,T], and from (18), ˙ek(t) is uniformly bounded on [0,T], and from (12), δ˙xk(t) is uniformly bounded on [0,T].

    Setting t=T in (48) results in,

    Wk(T)λ12T0||δxk||2dτλ2T0||ek||2dτ+12T0tr(˜ΘTk1L12˜Θk1)dτ+12l3T0˜μ2k1(τ)dτ+W1k1(T)+W2k1(T)λ12T0||δxk||2dτλ2T0||ek||2dτ+Wk1(T). (52)

    Therefore,

    Wk(T)Wk1(T)λ12T0||δxk||2dτλ2T0||ek||2dτ. (53)

    Since Wk(T)0 is monotonically decreasing and bounded, its limit exists such that

    limkT0||δxk||2dτ=0 (54)
    limkT0||ek||2dτ=0. (55)

    Now, based on the above analysis and using Lemma 1, we can summarize the stability and convergence results as Theorem 2.

    In this section, two illustrative examples are presented to show the design procedure and the performance of the proposed controller for the cases of ILC and RLC, respectively.

    Example 1: Consider the following system

    [˙x1k˙x2k]=[1     3  2  2][x1kx2k]+[01](uk(t)+η(t,x1k,x2k)) (56)
    yk(t)=[0  1][x1kx2k] (57)

    where t[0,1], and η(t,x1k,x2k)=[t  sin(t)] [sin(x1k)x2k].

    Choosing P=[1001], we have

    Q=(PA+ATP)=[2  5  54] (58)

    and

    BTP=[0  1][1001]=C. (59)

    therefore, Assumption 1 is satisfied.

    Observer (11), control law (19) and full saturated learning laws (22) and (23) are applied. The desired trajectory is given by yd(t)=12t2(1.1t), t[0,1]. We choose L1=0.1, L2=diag{10}, and l3=7. Simulation results can be seen in Figs. 1-4. In Fig. 1, it can be easily seen that the system output yk(t) and the output estimation yk(t)=Cˆxk(t) converge to the desired trajectory yd(t) over [0,1]. The quantities on vertical axis in Fig. 2 and Fig. 3 represent Jk =log10(maxt[0,1]||δxk||) and Jk=log10(maxt[0,1]|ek|), respectively. The learned control quantity uk(t) is shown in Fig. 4.

    Figure  1.  Desired trajectory, output estimation and system output in the case of ILC (k= 50).
    Figure  2.  State estimation errors in the case of ILC.
    Figure  3.  Output tracking errors in the case of ILC.
    Figure  4.  Control input in the case of ILC (k= 50).

    Example 2: Consider the circuit [20] described by

    [˙x1k˙x2k]=[R1M2M1M2M23  R2M3M1M2M23  R1M3M1M2M23R2M1M1M2M23][x1kx2k]+[M2M3M1M2M23M1M3M1M2M23](uk(t)+η(t,x1k,x2k)) (60)
    \begin{eqnarray} y_k(t)= [\!\begin{array}{l} 0\ \ 2 \end{array}]\!\left[\!\begin{array}{l} x_{1k}\\x_{2k} \end{array}\!\right] \end{eqnarray} (61)

    where R_1=1\Omega and R_2=1\Omega are resistors, and M_1=0.36H, M_2=0.5H are inductors, and the mutual inductors M_3 =0.5H. x_{1k}=i_1 and x_{2k}=i_2 are the loop currents. \eta(t) =x_{2k}\sin^3t+0.8\sin^2t\sin{x_{1k}} represents the input perturbation. The desired trajectory is given by y_d(t)=12t^2(1-t), t\in [0, 1]. We set L_1=0.5, L_2={\text{diag}}\{0.8\}, and l_3=1. Simulation results can be seen in Figs. 5-8. Fig. 5 shows the ultimate tracking of the system output and the output estimation and the desired trajectory. In order to show the difference between the ILC and the RLC, we depict the state x_{2k}(t) of the top 10 iterations in Fig. 6. It can be easily seen that the initial value of the kth cycle is equal to the final value of the (k-1)th cycle which satisfies the Assumption 6.

    Figure  5.  Desired trajectory, output estimation and system output in the case of RLC (k= 40).
    Figure  6.  The changes of state x_{2k}(t) in 10 iterations in the case of RLC.
    Figure  7.  Output tracking errors in the case of RLC.
    Figure  8.  Control input in the case of RLC (k= 40).

    The vertical quantities in Fig. 7 represent J_k\, =\, \log_{10} (\mathit{\rm{max}}_{t\in [0, 1]}|e_k|) which implies perfect tracking can be achieved after 50 iterations. The learned control quantity u_k(t) is shown in Fig. 8.

    In this paper, an observer-based iterative learning controller has been presented for a class of nonlinear systems. The uncertainties treated is parameterized into two parts. One is the unknown time-varying matrix-valued parameters and the other is the Lipschitz continuous function, which is also unknown due to unmeasurable system states. The learning controller designed for trajectory tracking composes of parameter estimation and state estimation which is given by a robust learning observer. The parameter estimations are constructed by full saturated learning algorithms, by which the boundedness of the parameter estimations are guaranteed. Further, the extension to repetitive learning control is provided. The observer-based RLC avoids the initial repositioning and does not require the strict periodicity constraint in repetitive control. The global stability of the learning system and asymptotic convergence of the tracking error are established through theoretical derivations for both ILC and RLC schemes, respectively.

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