Finite frequency range robust iterative learning control of linear discrete system with multiple time-delays

Accepted Manuscript

Finite frequency range robust iterative learning control of linear discrete system with multiple time-delays Hongfeng Tao, Wojciech Paszke, Huizhong Yang, Krzysztof Gałkowski PII: DOI: Reference:

S0016-0032(19)30096-1 https://doi.org/10.1016/j.jfranklin.2019.01.040 FI 3787

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

29 October 2018 19 January 2019 25 January 2019

Please cite this article as: Hongfeng Tao, Wojciech Paszke, Huizhong Yang, Krzysztof Gałkowski, Finite frequency range robust iterative learning control of linear discrete system with multiple timedelays, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.01.040

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ACCEPTED MANUSCRIPT

Finite frequency range robust iterative learning control of linear discrete system with multiple time-delays✩ Hongfeng Taoa,∗, Wojciech Paszkeb , Huizhong Yanga , Krzysztof Galkowskib Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education, Jiangnan University, Wuxi 214122, P. R. China b University of Zielona G´ ora, ul. Szafrana 2, 65-516 Zielona G´ ora, Poland, Phone: +48 683283219, Fax: +48 683284751

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Abstract

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This paper uses repetitive process stability theory to design robust iterative learning control law for linear discrete systems with multiple time-delays and polytopic uncertainty. Both dynamic and static forms of the control law are considered and used when designing robust iterative learning control schemes. Also, based on the generalized Kalman-Yakubovich-Popov Lemma, the proposed design procedures a required frequency attenuation over a finite frequency range and the monotonic trialto-trial error convergence. Moreover, linear matrix inequality techniques are applied to formulate the convergence conditions and to obtain formulas for the control law designs. Finally, an illustrative numerical simulation example is given and concludes the paper.

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Iterative learning control (ILC) is a technique for improving tracking response in many industrial systems that repeatedly execute the same task over a finite duration. In particular, industrial applications of robot manipulators that are need to repeat a given task with a high accuracy provide strong evidence that ILC can significantly improve the tracking performance of the resulting control system. The survey papers [1, 2] are the starting points for the literature and these together with subsequent publications show a wide range of successful industrial applications of ILC schemes to robotics, batch processes, traffic systems, multi-agent systems, and recently to robotic-assisted upper limb stroke rehabilitation [3]. As opposed to the standard control schemes, which give the same tracking error on each repetition or trial, the ILC algorithm can effectively improve the tracking performance by updating the control signal for a given system that execute the same task over a finite duration repetitively. The update is generated as a function of previous inputs and tracking errors. Consequently, the tracking performance is successively improved along the repetition axis rather than time axis (transient responses). More importantly, since ILC schemes require the reference and plant input and output signals, which

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1. Introduction

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Keywords: Iterative learning control, finite frequency range, multiple time-delays, uncertainty, dynamic and static control law

✩ This work is supported by National Natural Science Foundation of China (61773181, 61203092), 111 Project (B12018), the Fundamental Research Funds for the Central Universities (JUSRP51733B), the Priority Academic Program Development of Jiangsu Higher Education Institutions and National Science Centre in Poland, grant No. 2017/27/B/ST7/01874 ∗ Corresponding author Email addresses: [email protected] (Hongfeng Tao), [email protected] (Wojciech Paszke), [email protected] (Huizhong Yang), [email protected] (Krzysztof Galkowski)

Preprint submitted to Elsevier

February 13, 2019

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Then the ILC error dynamics can be written as a system of linear difference equations in k of the form Ek+1 = GEk . Then conditions for tracking error convergence are based on the matrix G properties. Due to the finite trial length, error in k can occur even if the system has an unstable state matrix. The solution via the lifting design is to first design a stabilizing feedback control law and then apply ILC to the resulting controlled system. Note that for robust control based on the norm bounded or polytopic uncertainty, the entries in the matrix G will contain products of the matrices describing the uncertainty and this makes the analysis significantly more involved. An alternative setting for ILC analysis and design is 2D systems theory, where the directions of information propagation are from trial-to-trial (k direction) and along the trial (p) direction. However, the particular case of 2D systems, the so-called repetitive process [4, 5] is a natural setting for ILC analysis and design since the dynamics evolve in two independent directions but information in the temporal domain is limited to a finite duration [4, 6]. This setting for analysis and design gives a systematic way to simultaneously consider behaviour along the time axis and from trial-to-trial. This means that it is possible to obtain an optimal complementation of the feedback with the feedforward (learning) action. Clearly, during the past decades, there are many successful applications to repetitive framework since it has been proposed for many classes of linear or nonlinear systems. However, a little number of results for systems with delays exist only. In many industrial engineering systems, time-delays are frequently encountered in the transmission of material or information between different parts of a system. Examples of time-delay are transmission lags in remotely controlled plants or ignition delays in diesel engines and even some engineering applications involve multiple time-delays. For example, the consensus problem in multi-agent systems with distinct communication delays stimulates application-orientated research to alleviate the effects of multiple time-delays. Therefore, time-delay is frequently one of the main causes of instability and poor performance of a control system [7, 8], and currently many ILC algorithms have been applied to time-delay systems with the aim to overcome the encountered problems. For example, a robust 2D closed-loop ILC combined with the output feedback scheme has been applied to batch processes with state delay and time-varying uncertainties [9, 10]. Unfortunately, those design methods use static combinations of state or output feedback and pass profile feedforward information to obtain system performances in the entire frequency domain only. However, practical frequency-domain design specifications are usually restricted to semi-finite or finite frequency ranges. In fact, the reference signals are often in a certain frequency range of relevance, so it is sufficient to ensure attenuation over frequency ranges according to the design specification. In [11, 12], the generalized Kalman-Yakubovich-Popov(GKYP) lemma was used to design iterative learning control algorithm for discrete system over finite frequency ranges with experimental verification, and the preliminary results for differential system were presented in [5]. Furthermore, an iterative

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are directly available, then improvements in tracking performance can be obtained with less plant or system knowledge. Specifically, simple ILC laws have found use, e.g., ILC phase-lead, which has the structure uk+1 (p) = uk (p) + lek (p + λ), 0 ≤ p ≤ α − 1, k ≥ 0, where α < ∞ is the number of samples along the trial length and k is the trial number, ek (p) = yd (p) − yk (p) is the error on trial k, yd (p) is a given reference trajectory for the output, l is a scalar gain and λ > 0 is the phase-lead coefficient. The second term in this control law is non-causal in p but can be used in ILC as it acts on previous trial error data. Anyway, newly generated input is applied to the plant at iteration k + 1 and then is stored in memory for updating the input for the next trial. If dynamics along the trial is discrete, the lifting approach is recognized as an effective method to design ILC schemes. This approach is based on lifting inputs and outputs variables as super-vectors. For example, for single-input single-output system, let the error super-vector on the trial k is defined as  T Ek = ek (0) ek (1) · · · ek (α − 1) .

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• both static and dynamic forms of ILC laws are considered;

• the finite frequency range iterative learning control algorithm is designed for a class of linear discrete systems with multiple time-delays; • the monotonic trial-to-trial error convergence conditions for the resulting dynamical ILC scheme are rigorously derived in LMI form; • the robust design problems against polytopic uncertainties are also solved. This paper is organized as follows: Section 2 describes a class of uncertain linear discrete systems with multiple time-delays in the ILC setting. Section 3 formulates the ILC design problem for both static and dynamic control laws in the repetitive process setting. In terms of the repetitive process theory and generalized KYP lemma, the monotonic trial-to-trial error convergence stability conditions of the controlled dynamic and ILC law design with the corresponding LMIs constraints are given in Section 4 over finite frequency ranges. Section 5 illustrates the effectiveness of the proposed methods through numerical simulations performed on a system with polytopic uncertainty. Finally, some conclusions are given in Section 6. Throughout this paper, the zero matrix and the identity matrix with the required dimensions are denoted by 0 and I, respectively. For symmetric matrices P and Q, the notation P  Q (respectively P ≺ Q) is used to represent the positive definite (respectively, negative definite) matrix P − Q. Also (?) denotes transposed elements in a symmetric matrix and ρ(·) denotes the spectral radius of its matrix argument. The symbol diag{P1 , P2 , · · · , Pn } denotes a block diagonal matrix with diagonal blocks P1 , P2 , · · · , Pn and sym(P ) = P + P T , ⊗ denotes the Kronecker matrix product, the superscript ∗ denotes the complex conjugate transpose of a matrix. 2. Preliminaries and the problem statement Let p ∈ [0, α − 1] be the sample index or time instant and k ≥ 0 be the discrete iteration number. Then let us consider a case when the plant to be controlled is modeled as an uncertain linear discrete system with multiple time-delays represented in ILC setting by the following state-space model over 0 ≤ p ≤ α − 1, xk+1 (p + 1) = A(ξ)xk+1 (p) +

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yk+1 (p) = C(ξ)xk+1 (p), 99 100

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learning fault-tolerant control law was designed for a class of linear differential batch processes with single time-delay and actuator faults using the repetitive process setting in finite frequency domains in [13]. In order to remove or lessen the effect of LMI conservativeness in ILC law design, further noncausal finite-time interval data and parameter dependent Lyapunov functions are used in [14] and [15], respectively. Moreover, dynamical controller is also an effective approach in such cases [16], hence, in this paper, we bring a novel method to design dynamical ILC schemes in frequency domain for discrete systems. According to the aforementioned issues and challenges, this paper proposes a method that integrates repetitive framework and finite frequency domain into the ILC scheme design procedures for systems with multiple time-delays. The contributions of this paper are summarized as follows:

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Adi (ξ)xk+1 (p − di ) + B(ξ)uk+1 (p),

(1)

where xk+1 (p) ∈ Rn , uk+1 (p) ∈ Rm and yk+1 (p) ∈ Rq are the system state, input and output vectors respectively; di is the unknown time-delay constant satisfying 0 < di ≤ d¯i , and d¯i is a known upper 3

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bound. Since it is required that a given trajectory task is repeated, the initial states need to be reset at begin of each trial. No loss of generality arises from assuming xk (p) = x0,k , p ∈ [−di , 0] on each trial. Moreover, it is assumed that the uncertain model matrices in (1) belong to a convex bounded (polytope-type) uncertain domain Dm , where any uncertainty matrix can be written as a convex combination of its vertices   N     X Dm = (A(ξ), Adi (ξ), B(ξ))| (A(ξ), Adi (ξ), B(ξ)) = ξj Aj , Ajdi , B j , ξ ∈ Duc , (2)   j=1

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where N vertices of the polytopes are given by Aj , Ajdi and B j for j = 1, 2, . . . , N and ξ is a real parameter vector lying in the interval [0, 1], and Duc forms the unit simplex defined as   N   X N (3) Duc = ξ ∈ R : ξj ≥ 0, ξj = 1 .   j=1

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Remark 1. The assumed uncertainty description allows the output matrix C(ξ) to be uncertain. However, when dealing with ILC scheme design, the matrix C represents the contribution of the current trial state vector to the current trial, i.e. measured outputs, and hence it is kept fixed C(ξ) = C since the convexity of the resulting conditions are usually lost. Fortunately, several methods can be used to deal with such the case, see e.g. [17].

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The aim of this paper is to provide robust design procedures stated as LMI-based conditions for obtaining convergent ILC schemes for systems represented by (1) and where the uncertainty is modelled as (2) and (3). This means that the control requirement is to force the error sequence {ek }k≥0 converge to zero or a possible minimum in the trial domain. In other words, the objective is to construct a sequence of ILC inputs {uk }k≥0 such that the performance is gradually improving with each successive trial. Therefore, the convergence condition on the input and error can be defined as

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lim kek k = 0, lim kuk − u∞ k = 0, k→∞

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3. ILC design in repetitive process form

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where k·k denotes the norm on the underlying function and u∞ is termed the learned control. The reason for the condition on the control input sequence is to prevent large increases in the control vector from one trial to the next one. According to the aforementioned issues, this paper proposes analysis and design over a repetitive process setting since the dynamics of repetitive processes evolve in two independent directions and information in the temporal domain is limited to a finite duration. As the result this setting gives a systematic way to simultaneously consider behaviour along the time axis and from trial-to-trial. Therefore, the next section provides the adaptation of repetitive processes setting for ILC analysis design of systems with multiple delays.

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3.1. Static ILC design To formulate the ILC design problem in the repetitive process setting, consider a system described (1) and a classical ILC law that constructs the current trial input as that used on the previous trial plus a corrective term, i.e., a control law of the general form uk+1 (p) = uk (p) + rk+1 (p), 4

(5)

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where rk+1 (p) is the correction term computed using previous trial data, and introduce for analysis purposes only the vector ηk+1 (p) = xk+1 (p) − xk (p). (6) Without loss of generality, it is assumed that yd (0) = yk (0) = Cxk (0) and, due to the initial conditions assumed for (1), ηk (0) = 0. When these assumptions are violated then the achievable performance of ILC is limited. In particular, reset errors (the initial state is not reset to the same values, i.e. there are initial state shifts) cause additional transients which propagate across trial and then may obviously cause an increase in the tracking error. Suppose also that the modification term in the ILC law (5) takes the form

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rk+1 (p) = K1 ηk+1 (p + 1) + K2 ek (p + 1),

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where K1 and K2 are the control law matrices to be determined. This control law correction term is the sum of state feedback control based on ηk+1 (p + 1) plus a feedforward term based on the previous trial error ek (p + 1). Then the application of this control law to (1) gives the controlled dynamics state-space model in the form of discrete repetitive process with multiple time-delays as r X

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ηk+1 (p + 1) =Aηk+1 (p) +

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where

Adi ηk+1 (p − di ) + Bek (p), (8)

Cdi ηk+1 (p − di ) + Dek (p),

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C = − CA, Cdi = −CAdi (ξ), D = I − CB(ξ)K2 .

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In particular, the formulas for constructing K1 and K2 in (7) can be obtained in an LMI setting. In ILC design, the use of the LMI techniques to ensure the stability along the trial can be problematic in some cases due to the sufficient but not necessary conditions for stability along the trial, resulting in a poorly conditioned solution or even no solution at all. One possible way of reducing such effects is to include more non-causal (along the trial) data and with a particular emphasis on robustness. An alternative is to use dynamic ILC controllers, i.e., controllers with their own internal dynamics, that is why we chose dynamical ILC law for robust control scheme.

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3.2. Dynamic ILC design Previous research has examined the use of ILC laws that have their own internal dynamics, but not beyond examining their basic structure and some progress on design for time-delay systems. In this paper, the dynamic ILC law has state dynamics described by

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c c ηk+1 (p + 1) = Ac ηk+1 (p) + Bc ek (p),

where c ηk+1 (p + 1) = xck+1 (p) − xck (p)

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(9) (10) xck (p)

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is the trial-to-trial increment of the controller internal state vector ∈ the dimension of this vector can be different from that of the system (1) state vector, i.e., n = 6 nc . Hence, we chose the updating dynamic control of (5) as c rk+1 (p) = Cc ηk+1 (p + 1) + Ec ηk+1 (p + 1) + Dc ek (p + 1),

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where Cc , Ec and Dc are matrices of dynamic ILC law to be determined. Furthermore, let us introduce the augmented state vector as   ηk+1 (p) Xk+1 (p)= (12) c (p) ηk+1 and then application of the ILC law (5) and (11) yields the controlled dynamics as Xk+1 (p + 1) =AXk+1 (p) + ek+1 (p) =CXk+1 (p) +

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Remark 2. Clearly, the dynamic ILC law structure in (9) and (11) includes a general static control law as special case. To see this let us consider the situation when Cc is 0, or Ac and Bc are zero in (9) and (11). Then immediately Ec = K1 and Dc = K2 and the dynamic controller transforms to the static one. However, the dynamic ILC law uses more system internal dynamics to construct updating dynamic controller and aim to improve system performance along the trial. Before presenting main developments of this paper, we give the following theorem which is essential for later developments on application of the dynamic control law for discrete systems with multiple time-delays.

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i=1

where θ ∈ [−π, π]. Then application of repetitive process stability theory [4] leads to the first theorem of this paper, which provides necessary and sufficient conditions for stability along the trial of the controlled dynamics.

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Theorem 1. Suppose that an uncertain linear system of (1) is repeatedly attempting to follow the same reference trajectory yd . Also, let ILC law (7) or (11) be applied. Then the resulting system can be represented by (8) or (13) and this system is robustly stable along the trial ∀ ξ ∈ Duc and di ∈ [0, di ] if and only if i) ρ(D) < 1, r P ii) ρ(A + Adi e−jθdi ) < 1, ∀θ ∈ [−π, π] and di ∈ [0, di ],

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which is again in the special form of the state-space model of a discrete linear repetitive process with multiple delays. Additionally, for any fixed ξ ∈ Duc , we introduce the following frequency response matrix ! !−1 r r   X X jθ jθ −jθdi −jθdi G e = C+ e I −A− Cdi e Adi e B+D, i=1

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Cdi Xk+1 (p − di ) + Dek (p),

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Adi Xk+1 (p − di ) + Bek (p),

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iii) ρ(G(ejθ )) < 1, ∀θ ∈ [−π, π] and di ∈ [0, di ].

Proof. The proof is an immediate consequence of extending known conditions for stability along the pass [4] to work with uncertain processes represented by (8) or (13). Simply, the condition i) ensures asymptotic stability of the resulting process. The condition ii) states delay-dependent stability conditions for the state dynamics on each trial. Finally, the condition iii) is established by imposing the frequency attenuation of the previous trial error over the complete spectrum for any delay di ∈ [0, di ]. 6

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4. Stability analysis over a finite frequency domain This chapter presents the general convergence results for the considered class of systems. These results are then further adopted to formulate the design procedures in LMI-based fashion. Notably, the notions of asymptotic and monotonic convergence are introduced using z-domain representation. However, application of the Laplace transform requires that input and output signals be defined over an infinite time horizon. This clearly can be done by considering the trial length to be infinite. Then the below analysis is based on infinite trial length (but all practical applications of ILC requires finite trial length) then the z-domain representation is a good approximation of the real situation. In particular, see [4] and the cited references on how detrimental effects due to the finite trial length can be avoided. 4.1. Monotonic trial-to-trial error convergence Applying the Z-transform to (8) or (13) gives Ek+1 (ejθ ) = G(ejθ )Ek (ejθ ).

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It is important to mention that it not easy to apply (16) to conclude on asymptotic convergence since it has to be checked for entire frequency range, i.e. ∀θ ∈ [−π, π]. Anyway, a possible way to overcome this difficulty is to use Lyapunov methods and hence (16) can be replaced by the requirement that there exists a Hermitian matrix P(ejθ )  0 such that
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but unfortunately the dependence of P(ejθ ) on θ is unknown. To overcome this difficulty, at the expense of conservatism, P(ejθ ) is replaced by piecewise constant matrix variables. This means that P is kept constant over the entire or limited frequency range. Furthermore, as known, see, e.g. [18], most of ILC schemes exhibit poor transients during the convergence process even if (16) is satisfied. Specifically, the tracking error may grow over the initial trials before converging as k → ∞. To avoid this problem, a stronger convergence criteria is required and, in particular, monotonic trial-to-trial error convergence holds if and only if σ(G(ejθ )) < 1, ∀θ ∈ [−π, π],

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where σ(·) denotes the maximum singular value of its matrix argument. In common with much of the ILC literature (18) is used from this point onwards. Moreover, since

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Clearly, the tracking error (asymptotically) converges as k → ∞, if and only if   ρ G(ejθ ) < 1, ∀θ ∈ [−π, π].

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σ(G(ejθ )) < 1 ⇔ kG(ejθ )k∞ < 1, ∀θ ∈ [−π, π]

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and therefore if



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Lemma 1. [20] For a given real symmetric matrix Π of compatible dimensions and any delay di satisfying 0 < di ≤ d¯i , a transfer-function matrix G(ejθ ) satisfies the condition of (16) over a specified frequency range Ω0 if there exist P  0, Q  0, Zi  0, and symmetric Xi such that 

A Ad1 I 0  C Cd1 + I 0

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then monotonic trial-to-trial error convergence occurs in `2 . Additionally, (18) holds provided (17) is feasible when P(ejθ ) = I. In applications, it may be required to impose different frequency constraints over finite frequency ranges, e.g., over a low range of frequencies, depending on the application considered. Moreover, at least finite frequency ranges for any disturbances present can often be obtained for physical examples. The stability theory imposes the same condition over the complete frequency spectrum and this may be conservative in some cases. An alternative whereby different specifications can be specified over different frequency ranges is possible as developed in the remainder of this paper using the generalized KYP lemma. Thus a set of specifications would generally consist of different in various

requirements jθ

frequency ranges. One way of dealing with this case is to modify (19) to G(e ) ∞ < 1, ∀θ ∈ Ω0 , where Ω0 is a finite frequency range of interest. In general, it could be required to impose different specifications over a low frequency range specified by |θ| < θl (Ω0 = [−θl , θl ]), a middle frequency range specified by θ1 ≤ θ ≤ θ2 (Ω0 = [θ1 , θ2 ]) and a high frequency range specified by |θ| > θh (Ω0 = [θh , π]) where θl , θ1 , θ2 , θh are given scalars in [−π, π] respectively. Note that the specified frequency limits are application dependent - see [19] for results with experimental verification for discrete systems without time delays. The proposed methodology is based on versions of KYP and elimination lemmas respectively which are reproduced below for convenience.

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Moreover, Ω0 is specified by the following choices for Ψ    0 1   if |θ| < θl (low f requency range)   1 −2 cos(θl )      0 ejθc Ψ := if θ1 ≤ θ ≤ θ2 (middle f requency range) −jθ c e −2 cos(θm )       0 −1   if |θ| > θh (high f requency range) −1 2 cos(θh ) and θc = (θ1 + θ2 )/2, θm = (θ2 − θ1 )/2.

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Lemma 2. [21] Given a symmetric matrix Υ ∈ Rp×p and two matrices Λ, Σ of column dimension p, there exists a matrix W that satisfies  Υ + sym ΛT W Σ ≺ 0 if and only if the following two projection inequalities with respect to W hold T

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It should be noted that the matrix Ξ in (22) is the only matrix whose block entries depend on chosen frequency range, i.e., low, middle or high, and can be partitioned as • for the low frequency range

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• for the middle frequency range 

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where Λ⊥ and Σ⊥ are arbitrary matrices whose columns form a basis of the null spaces of Λ and Σ respectively.

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Ξ=



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i=1



 r r P P jθc Q− ¯i Zi ¯i Zi −P + e d d   i=1 i=1  = r r P ¯ P ¯ ,  −jθc e Q− di Zi P − 2 cos(θm )Q + di Zi i=1

• and for the high frequency range 

(24)

(25)

i=1



 r r P P ¯ ¯ −Q − di Zi  −P + i=1 di Zi  i=1 . = r P   d¯i Zi (?) P + 2 cos(θh )Q +

(26)

i=1

The application of Lemmas 1 and 2 leads to the next theorem which provides new design conditions for uncertain systems represented by (1) to obtain monotonically convergent ILC schemes over limited frequency intervals and any delay di such that 0 < di ≤ d¯i .

9

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for all ξ ∈ Duc where   Ξ11 Ξ12 −W 0 r r   P P ¯−1 Zi + Ξ22 + C T C +sym{AT W } , (?) X + d M Θ11 =  i 1 i   i=1 i=1 −1 T ¯ (?) (?) Cd1 Cd1 −X1 − d1 Z1   0 ··· 0 ··· 0 0 ··· Mi ··· Mr C T D+W T B  , Θ12 =  M2 T C T T T D Cd1 Cd1 d2 · · · Cd1 Cdi · · · Cd1 Cdr  T T C T ··· ··· Cd2 Cd2 Cd2 −X2 − d¯−1 dr 2 Z2 Cd2 Cd3 ..  .. T .  Cd(i−1) Cdi ··· . (?)   .. .. T C −X − d¯−1 Z  . . (?) (?) Cdi i i di i Θ22 =   . .. T  Cd(r−1) Cd r (?) (?) (?)  T  (?) (?) (?) (?) C Cdr −Xr − d¯−1 Zr (?)

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(?)

(?)

TD Cdi .. . T D Cdr T D D−I

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T and Mi = C T Cdi + d¯−1 i Zi + W Adi , i = 1, 2, · · · , r.



     ,    

Proof. Suppose (27) holds for some P  0, Q  0, Zi  0, Xi and W . Therefore DT D−I ≺ 0 and condition i) of Lemma 1 must hold. Next, by Schur’s complement formula, (27) is equivalent to  Ms +sym ΛT W Σ ≺ 0,

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r

dr

T D Cd2 .. .

where

T   T    I 0 I 0 0 0 0 0  0 I   0 I   CT 0   CT 0           0 0   0 0   CT 0   CT 0        d1   d1 Ms =  . . Ξ . .  +  . ..  ..  Π  .. . . . . .   . .   . .   . .  .          .  0 0   0 0   CT 0   CT 0  dr dr DT I DT I 0 0 0 0  0 0 0 ··· 0 r r  P P  (?) Xi + d¯i−1 Zi d¯−1 ··· d¯−1 r Zr 1 Z1  i=1 i=1   .. (?) −d¯−1 0 . + 1 Z1 −X1  (?)  . .  (?) . (?) (?) 0  −1 ¯  (?) (?) (?) (?) −dr Zr −Xr (?) (?) (?) (?) (?)     Λ = −I A Ad1 · · · Adr B , Σ = 0 I 0 · · · 0 0 

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(?)

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(?)

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Theorem 2. Suppose that an uncertain linear system of (1) is repeatedly attempting to follow the same reference trajectory yd and control law is based on (5) and (11). Then the resulting ILC scheme can be described as a discrete linear repetitive process of the form (8) or (13) and this process is robustly stable along the trial and hence monotonic trial-to-trial error convergence occurs for the performance specifications over the finite frequency ranges θ ∈ Ω0 any delay di ∈ [0, di ], if there exist matrices P  0, Q  0, Zi  0, W and symmetric Xi such that the following inequality holds   Θ11 Θ12 ≺0 (27) (?) Θ22

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0



 0   ..  .  ,  0  0 0

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and where the matrices Ξ, Π are the same as in Lemma 1. Also, the matrices Λ⊥ and Σ⊥ , whose columns form a basis for the null spaces of Λ and Σ, are respectively given by     A Ad1 · · · Adr B I 0 ··· 0 0  I  0 0 ··· 0 0  0 ··· 0 0          . . . . . . . . . . . .  0  0 I . . . .  I . .     Λ⊥ =   , Σ⊥ =  . .. ..  0   . . 0 0 0 0 0   0 0   ..   .. .. . .  .. ..  .   . . I 0 . I 0 . . 0 0 ··· 0 I 0 0 ··· 0 I Now, in view of Lemma 2, the feasibility of (27) implies that the inequality   Φ11 Φ12 ⊥T ⊥ Σ Ms Σ = ≺0 (?) Θ22 holds, where    Ξ11 0 0 ··· Φ11 = , Φ12 = T C · ·· C Z 0 C1T C1 − X1 − d¯−1 1 d2 1 d1

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0 ··· T C Cd1 di · · ·

0 0 T T C Cd1 dr Cd1 D



.

Furthermore, by observing that the inequality (20) in Lemma 1 can be rewritten as T

Ψs = Λ⊥ Ms Λ⊥ ≺ 0

277

then the feasibility of (27) implies feasibility of (20) for Ξ given in (24)-(26). This means that the iii) of  Lemma 1 holds. Moreover, (20) requires P  0 and Q  0 to guarantee that  condition r P ρ A+ Adi e−jθdi for all θ ∈ Ω0 and di ∈ [0, di ] and hence the condition ii) of Lemma 1 also

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Remark 3. Depending on the frequency range of interest, i.e. low, middle or high, the matrices Ξ11 , Ξ12 and Ξ22 are given by (24), (25) and (26), respectively. The above result provides a new procedure for ensuring based on stability along the trial. Unfortunately, the inequality (27) is not LMI as it includes bilinear terms in W and the matrices defining a controller. Clearly, no computationally effective method exists for checking (27). However, a particular set of transformations allows us to reformulate the result of Theorem 2 to a convex optimization problem involving set of LMIs for computing the required matrices. The following result provides sufficient constructive conditions for a dynamic controller od the form (11), i.e. the matrices Ac , Bc , Cc , Dc and Ec .

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holds. The proof is complete.

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Theorem 3. Suppose that an uncertain linear system of (1) is repeatedly attempting to follow the same reference trajectory yd and control law is based on (11). Then the resulting ILC scheme can be described as a discrete linear repetitive process of the form (13) and this process is robustly stable ˆ  0, Zˆi  0, S1 , S2 , L1 , L2 , L3 , L4 , L5 and symmetric along the trial if there exist matrices Pˆ  0, Q ˆ i such that the following LMIs hold X   Υ1j Υ2j Υ3j Υj :=  (?) Υ4j Υ5j  ≺ 0, (28) (?) (?) Υ6j 11

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for j = 1, 2, . . . , N where

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 ˆ  ˆ 12 −S T Ξ 0 Ξ11 ˆ + Zˆ + Ξ ˆ 22 + sym{Aˆj } d¯−1 Zˆ1 + Aˆj  , Υ1j =  (?) X 1 d1 ˆ 1 − d¯−1 Zˆ1 (?) (?) −X 1     0 0 0 ··· 0 ··· 0 ˆ j Cˆ jT  , ¯−1 ˆ ¯−1 ˆ ˆ ˆj ˆj ˆj  , Υ3j =  B Υ2j =  d¯−1 2 Z2 + Ad2 · · · di Zi + Adi S · · · dr Zr + Adr jT 0 ··· 0 ··· 0 0 Cˆd1 ˆ 2 − d¯−1 Zˆ2 , · · · ,− X ˆ i − d¯−1 Zˆi , · · · , −X ˆ r − d¯−1 ˆ Υ4j = diag{− X r Zr }, S = diag{S1 , S2 }, 2 i  T    j  ˆ jT 0 ··· 0 ··· 0 B L −I D 4 j ˆ = Υ5j = ˆ j , Υ6j = , B , j j L5 (?) −I Cd2 · · · Cˆdi · · · Cˆdr  j   j  jL jL A S +B B A S 0 j 2 1 1 j ˆ j = I −CB j L4 , di Aˆ = , Aˆdi = ,D 0 L3 0 0 h i  j  = −CAjdi S1 0 Cˆ j = −CAj S1 −CB j L1 −CB j L2 , Cˆdi

ˆ 11 , Ξ ˆ 12 and Ξ ˆ 22 have the same form as Ξ11 , Ξ12 and Ξ22 (see the equations (24)-(26) and and Ξ ˆ dependency on the particular frequency range of interest) but with P , Q, and Zi replaced by Pˆ , Q, and Zˆi respectively ∀i = 1, 2, . . . , r. Moreover, the ILC law matrices of (11) given by Ec = L1 S1−1 , Cc = L2 S2−1 , Ac = L3 S2−1 , Dc = L4 , Bc = L5

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ensure that the resulting ILC scheme is monotonically convergent from trial to trial and satisfy the finite frequency range specifications θ ∈ Ω0 for any delay di such that 0 < di ≤ d¯i . Proof. Suppose that the LMIs (28) are feasible for all j = 1, . . . , N . Then multiplying Υj by ξj , with ξ ∈ Duc , and summing for j = 1, . . . , N gives

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(29)

Υ(ξ) := ξ1 Υ1 + ξ2 Υ2 + . . . + ξN ΥN .

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Consequently, the feasibility of LMIs in (28) ensures that Υ(ξ) ≺ 0 and the matrix variable S in nonsingular and hence invertible. Now, setting L1 = Ec S1 , L2 = Cc S2 , L3 = Ac S2 , L4 = Dc , L5 = Bc , let us show that Υ(ξ) ≺ 0 guarantees the feasibility of the matrix inequality in Theorem 2 applied to the system controlled under law of (11). To proceed, introduce the following change of variables (∀i = 1, 2, . . . , r)

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ˆ T , Zi = W Zˆi W T , Xi = W X ˆiW T . S = W −1 , P = W Pˆ W T , Q = W QW

Multiplying Υ(ξ) by diag{W T , W T , · · · , W T , I, I} obtain  Ω11 Ω12  (?) Ω22 (?) (?)

12

on the left and by its transpose on the right, we  Ω13 Ω23  ≺ 0, (30) Ω33

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where



Ξ11

 Ω11 =   (?) (?) 

Ξ12 − W

0



r  P T T ¯−1  d¯−1 i Zi +Ξ22 +sym{A W } d1 Z1 + W Ad1  , i=1 i=1 (?) −X1 − d¯−1 1 Z1  0 ··· 0 ··· 0 T T T ¯−1 ¯−1 , Ω12 =  d¯−1 2 Z2 +W Ad2 · · · di Zi + W Adi · · · dr Zr +W Adr 0 ··· 0 ··· 0 −1 −1 −1 ¯ ¯ ¯ Ω22 = diag{ − X2 − d2 Z2 , · · · , − Xi − di Zi , · · · , −Xr − dr Zr },  T  T   0 BT W 0 0 ··· 0 ··· 0 −I DT Ω13 = , Ω23 = , Ω33 = . 0 C Cd1 Cd2 · · · Cdi · · · Cdr (?) −I

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Corollary 1. Suppose that an uncertain linear system of (1) is repeatedly attempting to follow the same reference trajectory yd and the applied control law is based on a static controller as in (7). Then the resulting ILC scheme can be described as a discrete linear repetitive process of the form (8) and this process is stable along the trial and hence monotonic trial-to-trial error convergence occurs for the performance specifications over the finite frequency ranges θ ∈ Ω0 , and any delay di satisfying ˆ  0, Zˆi  0, S, L1 , L2 and symmetric X ˆ i such that the 0 < di ≤ d¯i if there exist matrices Pˆ  0, Q following LMIs hold   ¯ 1j Υ ¯ 2j Υ ¯ 3j Υ ¯ 4j Υ ¯ 5j  ≺ 0 ¯ j =  (?) Υ Υ (31) ¯ 6j (?) (?) Υ

for j = 1, 2, . . . , N where   ˆ 11 ˆ 12 − S T   Ξ Ξ 0 0 0 r r   P P ˆ i + Zˆi + Ξ ˆ 22 +sym{Aˆj } d¯−1 Zˆ1 + Aˆj  , Υ ˆ j Cˆ jT  , ¯ 3j =  B ¯ 1j =  (?) X Υ 1 d1   i=1 i=1 jT 0 Cˆd1 ˆ 1 − d¯−1 Zˆ1 (?) (?) −X 1     0 ··· 0 ··· 0 ˆ jT −I D j −1 −1 −1 ¯ ¯ , Υ2j =  d¯2 Zˆ2 + Aˆd2 · · · d¯i Zˆi + Aˆdi · · · d¯r Zˆr + Aˆdr  , Υ6j = (?) −I 0 ··· 0 ··· 0  T 0 ··· 0 ··· 0 ¯ 5j = ¯ 4 = diag{ − X ˆ 2 − d¯−1 Zˆ2 , · · · , − X ˆ i − d¯−1 Zˆi , · · · , −X ˆ r − d¯−1 Zˆr }, Υ , Υ j j j r 2 i Cˆd2 S · · · Cˆdi S · · · Cˆdr S ˆ j = B j L2 , Aˆj = Aj S, Cˆ j = −CAj S −CB j L2 , Cˆ j = −CAj S, D ˆ j = I −CB j L2 . Aˆj = Aj S +BL1 , B

AC

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A static controller of (5) can be designed by introducing some relatively simple changes in the above theorem. This leads to the following corollary.

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Finally, it remains to apply Schur’s complement formula to show that (30) is equivalent to a version of LMI in (28). The the proof is complete.

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Xi +

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r P

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di

di

di

di

Moreover, the corresponding matrices of the static ILC law in (7) are given by K1 = L1 S −1 , K2 = L2 .

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Proof. This proof details can be omitted since they follow the same lines of the proof of Theorem 3. One main difference is to use the following change of variables (∀i = 1, 2, . . . , r) ˆ 11 = S T Ξ11 S, Ξ ˆ 12 = S T Ξ12 S, Ξ ˆ 22 = S T Ξ22 S, X ˆ i = S T Xi S, Zˆi = S T Zi S. Ξ 13

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ˆ Pˆ , Zˆi and X ˆ i of Theorems 3 and 1 are It is important to stress that that the matrix variables Q, separated from the uncertain process matrices and therefore it allows us to introduce the affine parameter dependent matrix variables of the following form Pˆ (ξ) =

N X

ˆ ξi Pˆ j , Q(ξ) =

j=1

330 331 332 333 334 335 336 337 338 339 340 341 342 343

j=1

N X

ˆ i (ξ) = ξi Zˆij , X

j=1

N X

ˆ j. ξi X i

(33)

j=1

The above matrix variables can take N different values corresponding to the vertices of the polytope Dm . As the result, less conservatism can be attained. 4.2. Practical implementation issues

In applications terms a critical problem is to achieve the desired shape of σ(G(ejθ )) in specified frequency range. A guideline for the partitioning of the entire frequency range is to cover the reference signal bandwidth with the first frequency range. This frequency range is of primary interest because these frequencies have the most effect on the speed of the trial-to-trial error convergence. Additionally, by minimizing the attenuation level in the low frequency range a higher speed of monotonic trialto-trial error convergence is obtained. The second frequency range frequently should cover these frequencies where the plant model accurately fits real response of the plant and hence the control is effective. In the last frequency range, low values of the frequency response is required since this mitigates modeling errors and non-repetitive disturbances at higher frequencies. The demarcation between low-frequency, mid-frequency and high-frequency range is based on requirements to account for the spectra of exogenous signals, to penalize regulated variables and to specify the amount of plant model uncertainty. To impose different performance specifications in different frequency ranges, divide the entire frequency range, i.e., from θ = 0 to θ = π, into H intervals (not necessarily containing the same number of frequencies) such that

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ˆ j , Zˆi (ξ) = ξi Q

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[0, π] =

H [

[θh−1 , θh ],

(34)

h=1

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5. Case study

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where θ0 = 0 and θH = π. Then the LMI conditions in Theorem 3 or 1, respectively, can be applied over chosen frequency intervals. The last issue is to solve the problem when control specification cannot be satisfied over some frequency ranges. This means that the learning has to be performed over these frequencies where the LMIs of Theorem 3 or 1 are feasible. The remaining frequencies should be cut-off by a low-pass filter (which can be implemented as the zero-phase filter, e.g., by using the filtfilt routine in Matlab) with cut-off frequency equal the highest frequency for which the result of Theorem 3 or 1 is valid.

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In this section, an example of ILC design is given to illustrate the proposed results. In particular, the previous sections’ result is applied to the discrete system with multiple time-delays and polytopic uncertainties in state-space model where the time-delay range is 0 < d1 ≤ d¯1 = 1, 0 < d2 ≤ d¯2 = 2, i.e. d1 = 1, d2 = 2. According to these assumptions, one can rewrite the system (1) as xk+1 (p + 1) = A(ξ)xk+1 (p) +

2 X i=1

yk+1 (p) = Cxk+1 (p),

Ai (ξ)xk+1 (p − di ) + B(ξ)uk+1 (p),

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where it is supposed that       0.7+0.1(−ξ1 + ξ2 ) 0.1ξ2 0.1 A(ξ) = , B(ξ) = ,C = 0 1 , 0 0.8 + 0.2(ξ1 − ξ2 ) 0.2 + 0.1(ξ1 − ξ2 )     −0.1 0.2 0.2 + 0.1(ξ1 − ξ2 ) −0.1 A1 (ξ) = , A2 (ξ) = . −0.15 0.2(−ξ1 + ξ2 ) + 0.3 −0.15 0.15ξ1 The parameters ξ1 and ξ2 can vary from [0, 1] and ξ1 + ξ2 = 1 is satisfied. This gives a polytopic uncertainty with two vertices for robust design. Specifically, the vertices are

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Vertex 1:

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A = 362

Vertex 2:

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A = 364 365 366 367





0.6 0 0 1



0.8 0.1 0 0.6

1

,B =





2

,B =



0.1 0.3



0.1 0.1

, A11



=

, A21



=

−0.1 0.2 −0.15 0.1



−0.1 0.2 −0.15 0.5



, A12



=

, A22



=

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0.3 −0.1 −0.15 0.15



0.1 −0.1 −0.15 0

.



.

The operating time during a trial is p ∈ [0, 400], we assume the initial state for each trial is 0, ∀k ≥ 0. The reference trajectory yref (p) = −0.5 sin(0.04p) + 1.3 sin(0.025p) + 3 sin(0.085p) is shown in Figure 1(a) and associated frequency spectrum in Figure 1(b). Inspecting the amplitudes in the frequency spectrum, it is shown that significant harmonics in the range from 0 to 10Hz only. 450

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yd(p)

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|yd(f)|

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100 50

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0

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6

8

10 12 Frequency(Hz)

14

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20

(b)

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(a)

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Figure 1: (a)The output reference trajectory and (b)the corresponding frequency spectrum.

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To evaluate the tracking performance from trial to trial, we introduce the following performance index by the root mean square(RMS) formula v u 400 u 1 X RMS(ek (p)) = t e2k (p). (35) 401 p=0

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The control signal for implementation is (12) and the analysis is general and applies for any controller state dimension. However, for simplicity of implementation the lowest possible state dimension should 15

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be used and in this particular case this is equal to one. Then applying Theorem 3, i.e., solving the LMI (28) for low frequency interval, the updating control gain matrices of (11) are derived as follows   Cc = 0.8161, Ec = 1.3453 −4.1453 , Dc = 1.8182, Ac = 0.5766, Bc = −0.3092.

The simulation results for this design are shown in Figure 2, 3. This demonstrates that the dynamical ILC can deliver high performance with monotonic trial-to-trial error convergence.

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20 trial number (k)

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40

(a)

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trial number (k)

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time point (p)

(b)

Figure 2: (a)The RMS performance and (b)the tracking error with dynamical ILC.

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y2(p) y4(p)

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y6(p)

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20

200

10 250

300

350

400

trial number (k)

(a)

100 0

0

time point (p)

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Figure 3: (a)The output and (b)the input with dynamical ILC.

To examine the possible advantages of dynamical ILC, its performance is compared with the static control law designed (7), where   K1 = 0.3864 −1.6600 , K2 = 1.5921.

The comparison between these laws is also given in Figure 4, 5 in terms of the error, output and input for each trial. These plots compare convergence of the trial-to-trial error and for the dynamical 16

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ILC law the reference trajectory is achieved with high tolerance after the 10th trial but similar accuracy with the static control law requires approximately 30 trials, the control input signals for the dynamical ILC law and the maximum value required over the trials is almost the same as its static counterpart. 2.5

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Figure 4: (a)The RMS performance and (b)the tracking error with static ILC. 383 5

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y40(p)

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(b)

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Figure 5: (a)The output and (b)the input with static ILC.

6. Conclusion

This paper has developed new results on ILC design for discrete systems with multiple time-delays in the repetitive process setting. Particularly, it has been shown that it is possible to apply the control law with dynamics and allows to combine frequency domain specifications together with robustness against model polytopic uncertainty. Sufficient conditions for the existence of the feasible robust controllers are presented in terms of LMIs and these controllers are capable to enforce monotonic trial-to-trial error convergence. Finally, the numerical example is given to illustrate the usefulness of the proposed methods. 17

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Future work will address the potential of an inclusion of non-causal previous data in ILC design. Also, the main result of the current control scheme still uses state feedback and if all states are not available for measurement either an observer is required or the theory is extended to a control law that uses only output information. Furthermore, the H∞ /H2 robust performance subject to external bounded non-repetitive disturbances in both the state and output vectors also should be considered over limited frequency ranges.

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References

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[1] Ahn, H.S., Chen, Y.Q., and Moore, K.L. (2007), “Iterative learning control: brief survey and categorization,” IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, vol. 37, no 6, pp. 1109–1121. [2] Bristow, D.A., Tharayil, M., and Alleyne, A. (2006), “A survey of Iterative Learning Control,” IEEE Control Systems Magazine, vol. 26, no 3, pp. 96–114. [3] Freeman, C. T.(2015), “Upper limb electrical stimulation using input-output linearization and iterative learning control.” IEEE Transactions on Control Systems Technology, vol. 23, no 4, pp. 1546–1554.

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