# Output Information Based Fault-Tolerant Iterative Learning Control for Dual-Rate Sampling Process with Disturbances and Output Delay

Authors: Hongfeng Tao; Yan Liu; Huizhong Yang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

For a class of single-input single-output (SISO) dual-rate sampling processes with disturbances and output delay, this paper presents a robust fault-tolerant iterative learning control algorithm based on output information. Firstly, the dual-rate sampling process with output delay is transformed into discrete system in state-space model form with slow sampling rate without time delay by using lifting technology; then output information based fault-tolerant iterative learning control scheme is designed and the control process is turned into an equivalent two-dimensional (2D) repetitive process. Moreover, based on the repetitive process stability theory, the sufficient conditions for the stability of system and the design method of robust controller are given in terms of linear matrix inequalities (LMIs) technique. Finally, the flow control simulations of two flow tanks in series demonstrate the feasibility and effectiveness of the proposed method.


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## Body

## 1. Introduction

In industrial applications, many engineering plants operate in continuous time while the system inputs and outputs are sampled, yielding discrete-time signals. Moreover, due to the hardware limitations, process characteristics, and other reasons, sampling each variable with the same frequency is not necessary and realistic. Therefore, the measurable output and input information is usually sampled in different rates from different types of sensors, manual sampling, or laboratory analyses [1, 2]. These systems are often termed as multirate sampling process and dual-rate sampling process is a special case; sometimes the sampling periods of the slow rate sampled variables are integer multiples of the fast rate sampled ones [3]. For example, the control of the bottom and top composition products of a distillation column by acting on the reflux and vapor flow rates is a typical case, where the input control signals can be rapidly adjusted, while the infrequent and delayed composition measurements are only obtained by gas chromatography [4]. In some vehicle control systems, displacement and velocity are measured by using ultrasonic sensors; the two different groups of sensors are located at different locations of the vehicle and have different sampling periods [5]. During the last several decades, this corresponding control problem has attracted considerable attention, including the model identification [6–8] and control algorithms [9–11]. But the control problem of dual-rate sampling systems still has achieved relatively little research results compared with the single-rate sampling case. Moreover, to the best of our knowledge, there are few papers dealing with iterative learning control (ILC) problems for dual-rate sampling process with time delay and actuator fault.The idea of ILC arose from Uchiyama in 1978 [12]; it represents a powerful approach for high performance tracking control of systems, which execute the same task over a finite duration repeatedly with a given desired trajectory and reset to the starting location once each execution is complete. Each execution of the task is known as a trial, or pass, and its duration is termed the trial length. ILC is currently mainly used to control single-rate sampling process. Compared with standard control scheme, the distinguishing feature of the ILC dynamic sequence of operations is to use the information from previous trials to update the control signal applied on the next one; the major advantage of ILC is the ability to improve system performance from trial to trial and include temporal information from previous trials that would be noncausal in standard systems. Over the past few decades, ILC has drawn significant research attention and increasingly been employed in many industrial processes, such as traffic system [13], networked stochastic system [14], robotic manipulator system [15], multiagent system [16], chemical pharmaceutical crystallization [17], and industrial injection molding batch processes [18].The design of an ILC law starts, as always, with performance specifications where the novel feature for ILC is the reference trajectory or vector, which is assumed to be the same for all trials in most of the ILC literature. In the case of discrete dynamics, lety(t,k), u(t,k) denote the output and input, respectively, on trial k; t denotes the sampling number over the trial duration. Then if the error on trial k is e(t,k)=yr(t)-y(t,k), where yr(t) denotes the reference signal, and the basic ILC design problem is to construct a sequence of input function that forces the error sequence {ek}k≥0 and input sequence {uk}k≥0 to converge to zero and u∞, respectively, or to within an acceptable tolerance, u∞ is termed learned control. It is a common approach to ILC design for discrete dynamics to use the lifting technique [19]. For the SISO systems with a nature extension to the multiple-input multiple-output (MIMO) systems, the input and output on any trial can be represented by supervectors formed by assembling the values at the sample instants into a column vector. Once the ILC law is applied, the propagation of the error can be represented by a linear difference equation and discrete linear systems theory can be utilized for trial-to-trial error convergence analysis and control law design.Given the finite trial length, trial-to-trial error convergence can occur even if the system is unstable since such a system can only produce a bounded output over a finite time duration. Literature reviews [20–22] confirm that an alternative approach to ILC design is to first apply a feedback control law to produce acceptable dynamics along the trial and then apply ILC to accomplish trial-to-trial error convergence of the resulting 2D system. A drawback of the two-step synthesis procedure is that it does not lead to an optimal combination of the feedback and feedforward actions. Based on an abstract model in a Banach space setting, repetitive process is a particular subclass of 2D system that operates over a subset of the upper right quadrant of the 2D plane and is characterized by a series of sweeps, or passes, through a set of dynamics defined over a finite duration known as the pass length. It is a nature setting for ILC analysis and design; the main advantage is that it gives a systematic way to simultaneously consider behaviour along the time axis and from trial to trial [23]. A detailed treatment of the dynamics of these processes, including their origins in the modeling of mining operations, can be found in [24].Time delay is also frequently encountered in the transmission of material or information between different parts of a system, including biology, chemistry, economics, population dynamics, and engineering applications. Time delay is one of the main causes of instability and poor performance in process control systems [25, 26], and currently many ILC algorithms have been applied to time delay single-rate sampling process by treating them as batch processes in a finite time on every trial. 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 [27, 28]. Composite iterative learning feedback controllers combined with state and output information are designed in [29]; then the sufficient conditions for delay dependent stability are obtained. However, these proposed methods are just based on the single-rate sampling process model, so that they cannot be directly applied to dual-rate sampling processes. Furthermore, these methods only use consistent slow sampling data and do not fully utilize all sampling data with different sampling intervals to improve control performance.Moreover, the involved industrial control systems under challenging environment are vulnerable to faults. A fault in a single component may have major efforts on the large system as a whole. Actuator faults will reduce the stability and performance of control systems and may even cause complete breakdown of these systems. Fault-tolerant control is a special action that ensures a fail-safe operation under real-time conditions if components in the control system fail or become faulty [30]. Fault-tolerant ILC design which is sensitive to faults is especially required in application for ILC scheme due to the repeated nature of the control actuator. The extensions to deal with faults for ILC systems receive increasing attentions [31, 32]. For example, a robust fault-tolerant iterative learning control design method is proposed and illustrated by an electric motor system in [31] and also [32] gives iterative learning fault-tolerant control for a class of linear differential time delay uncertain systems with actuator faults in finite frequency domains. The challenge of fault-tolerant control here is how to design a reliable ILC scheme against fault based on the inconsistent dual-rate sampling input and output information with time delay.This paper develops new results for ILC design applied to time delay dual-rate sampling process with the following contributions:(i)
The output information based ILC law design is extended to the fault-tolerant control problem for dual-rate sampling process with time delay and actuator faults.(ii)
Monotonic trial-to-trial error convergence conditions for the controlled ILC dynamics are derived.(iii)
Robust control issue for disturbance attenuation performance is solved.This paper is organized as follows: Section2 describes a dual-rate sampling process with output delay and disturbance in the ILC setting by the state-space model with actuator faults; then it is transformed into a discrete system model form in slow sampling rate without time delay by using the lifting technology. Section 3 formulates the output information based fault-tolerant ILC design problem in the repetitive process setting. Some repetitive process stability theories are given as background in Section 4. Then the sufficient conditions for the stability of the controlled dynamic and the design method of robust controller are analyzed and given in corresponding linear matrix inequalities form in Section 5. Section 6 verifies the effectiveness of the proposed method by the flow control simulations of two flow tanks in series. Finally, some conclusions are given in Section 7.Throughout this paper, the null and identity matrices with the required dimensions are denoted by 0 andI, respectively, and the notation X≺Y (resp., X≻Y) is used to represent the negative definite (resp., positive definite) matrix X-Y. The notation (⋆) denotes the transpose of elements in the symmetric position in a matrix. The symbol diag⁡X1,X2,…,Xn denotes a block-diagonal matrix with diagonal blocks X1,X2,…,Xn and sym⁡(Λ)=Λ+ΛT. The symbol ⌊x⌋ represents the largest integer which is less than or equal to x.


## 2. System Description

Consider a class of SISO linear continuous processesPc in Figure 1 with output delay and disturbance; system dynamics are described in the ILC setting by the following state-space model:(1)x˙t,k=Acxt,k+Bcut,k+Dcwt,k,yt,k=Ccxt-τ,k,0<t<α,where the symbol k≥0 denotes the trial number, α is the fixed and finite trial length, x(t,k)∈Rn, u(t,k)∈R1, y(t,k)∈R1, and w(t,k)∈Rm are the system state, input, output, and disturbance vectors, respectively, τ is the time delay constant, and Ac, Bc, Cc, and Dc are system matrices of appropriate dimensions. Without loss of generality, assume x(t,k)=x0,k, t∈-τ,0, on each trial.Figure 1
The general dual-rate sampling system.To include actuator faults, letuF(t,k) represent the failed actuator with the following fault model [32]:(2)uFt,k=Γut,k,where the actuator failure parameter Γ is unknown, but it satisfies the following condition:(3)0≤Γ_≤Γ≤Γ¯.The parameters Γ_(Γ_≤1), Γ¯ (Γ¯≥1) in this fault model, are assumed to be known. When Γ=1, it corresponds to the normal case uF(t,k)=u(t,k); when Γ=0, it covers the outage case; 0<Γ_≤Γ<1 and 1<Γ≤Γ¯ correspond to partial failures, for example, partial degradation of an actuator or the abnormal case when the faulty actuator output is larger than the normal controller output. Introduce(4)q=Γ¯+Γ_2,q0=Γ¯-Γ_Γ¯+Γ_,and define Γ0=Γ-q/q; then by using (4), the unknown failure parameter Γ can be written as(5)Γ=I+Γ0q,where(6)Γ0≤q0≤1.Therefore, the continuous process Pc of (1) with actuator fault can be described by(7)x˙t,k=Acxt,k+BcΓut,k+Dcwt,k,yt,k=Ccxt-τ,k,0<t<α.For process (7), the discrete-time signal u(nT1,k) with sampling period T1=ph is maintained as a continuous-time input signal u(t,k) using a zero-order hold HT1 in Figure 1; then(8)ut,k=unT1,k,nT1<t<n+1T1;the continuous-time output signal y(t,k) is sampled by a sampler ST2 with sampling period T2=qh to get discrete-time output signal y(nT2,k), where h>0 is the basic sampling period and p and q are the two positive coprime integers. For such a dual-rate sampling process, the measurable input-output data is {u(nT1,k),y(nT2,k),n=0,1,2,…}; {u(nT1+ih,k),y(nT2+jh,k), i=1,2,…,p-1,j=1,2,…,q-1} are unknown [33]. Moreover, in order to obtain the system model of dual-rate sampling process, the continuous process Pc is first discretized with sampling period h to obtain the discrete system model Ph:(9)xnh+h,k=Ahxnh,k+BhΓunh,k+Dhwnh,k,ynh,k=Chxnh-dh,k,where Ah=eAch, Bh=∫0heActdtBc, Ch=Cc, and Dh=∫0heActdtDc, d=⌊τ/h⌋.For processPh, let T≔pqh be the frame period to denote the cycle sampling period of the system, then using symbol npq instead of n in (9), we have(10)xnT,k=xnpqh,k=Ahxnpq-1h,k+BhΓunpq-1h,k+Dhwnpq-1h,k=AhAhxnpq-2h,k+BhΓunpq-2h,k+Dhwnpq-2h,k+Ahxnpq-1h,k+BhΓunpq-1h,k+Dhwnpq-1h,k=⋯=AhdxnT-dh,k+∑j=pq-d+1pqAhpq-jBhΓunT+j-1h-T,k+∑j=pq-d+1pqAhpq-jDhwnT+j-1h-T,k.Similarly, we can directly obtain(11)xnT+T,k=AhpqxnT,k+∑j=0q-1∑i=1pAhpq-jp-iBhΓunT+jT1+i-1h,k+∑j=0q-1∑i=1pAhpq-jp-iDhwnT+jT1+i-1h,k,(12)xnT+T-dh,k=Ahpq-dxnT,k+∑j=1pq-dAhpq-d-jBhΓunT+j-1h,k+∑j=1pq-dAhpq-d-jDhwnT+j-1h,k;according to (8), (11) can be simplified as(13)xnT+T,k=AhpqxnT,k+∑j=0q-1∑i=1pAhpq-jp-iBhΓunT+jT1,k+∑j=0q-1∑i=1pAhpq-jp-iDhwnT+jT1+i-1h,k.Substituting (10) into (12) and defining m≔pd-d, where we assume pq>d, we have(14)xnT+T-dh,k=AhpqxnT-dh,k+∑j=m+1pqAhpq+m-jBhΓunT+j-1h-T,k+∑j=1mAhm-jBhΓunT+j-1h,k+∑j=1mAhm-jDhwnT+j-1h,k+∑j=m+1pqAhpq+m-jDhwnT+j-1h-T,k.Remark 1.
The assumption conditionpq>d can usually be satisfied for some dual-rate sampling processes; this means the time delay exists in the sampling frame period. Furthermore, if pq≫d, then more information of the time delay system can be obtained in a full frame period when the controller is designed for the sampling process in Figure 1.
In terms of lifting technology, defining the state expanding vector with2n dimensions, (15)XnT,k=xnT-dh,kxnT,k,and the input and disturbance expanding vectors with (pd+d) dimensions,(16)UznT,k=unT-dh,kunT-dh+h,k⋮unT+pq-1h,k,WznT,k=wnT-dh,kwnT-dh+h,k⋮wnT+pq-1h,k,we can directly convert the dual-rate sampling system model (9) with output delay into a lifting linear discrete system model without delay in a single slow sampling rate T=pqh,(17)XnT+T,k=AzXnT,k+BzΓzUznT,k+DzWznT,k,ynT,k=CzXnT,k,where(18)Az=A0n×n⋆A,Bz=B0n×d⋆B,Cz=Ch01×n,Dz=D0n×d⋆D,A=Ahpq,Γz=diag⁡Γ,Γ,…,Γ,B=Ahpq-1BhAhpq-2Bh⋯AhBhBh,D=Ahpq-1DhAhpq-2Dh⋯AhDhDh.
We can obviously find that although (17) is a slow sampling rate system with T=pqh, every real control input variable of expanding vector Uz(nT,k) in (16) is also updated with a relatively fast sampling period h. Particularly, if we force the input variables to be updated just in slow sampling period T, that is, u(nT-dh,k)=u(nT-(d-1)h,k)=⋯=u(nT-h,k), u(nT,k)=u(nT+h,k)=⋯=u(nT+T-h,k), then (17) can be described as(19)UsnT,k=unT-T,kunT,k;thus we can get the discrete state-space model of the dual-rate sampling process (1) with one sampling period T; it is just a special form of (17):(20)XnT+T,k=AsXnT,k+BsΓsUsnT,k+DsWznT,k,ynT,k=CsXnT,k,where(21)As=Az,Cs=Cz,Ds=Dz,Γs=diag⁡Γ,Γ,Bs=Ahpq-1Bh+Ahpq-d-1Bh+⋯+Ahpq-dBhAhpq-d-1Bh+⋯+Bh0Ahpq-1Bh+Ahpq-d-1Bh+⋯+Bh.


## 3. ILC Law Design

For the discrete sampling system described by (17), design an ILC law that constructs the current trial input that is equal to the control input on the previous trial plus a corrective term, that is, a control law of the form(22)UznT,k+1=UznT,k+rnT,k+1,where r(nT,k) is the correction term computed using previous trial data. Given the reference signal yr(nT), the error on trial k is(23)enT,k=yrnT-ynT,k.To formulate the ILC design problem in the repetitive process setting, for analysis purpose only, we define the vectors(24)ηnT+T,k+1=XnT,k+1-XnT,k,ωnT+T,k+1=WznT,k+1-WznT,k,(25)μnT+T,k+1=ynT,k+1-ynT,k=CzηnT+T,k+1.Without loss of generality, it is assumed that yr(0)=y(0,k)=CzX(0,k) and, due to the initial conditions assumed for (17), μ(0,k)=0.Since we can just obtain measurable input and output signal for dual-rate sampling process, suppose that the modification term in the output information based ILC law (22) takes the form(26)rnT,k+1=K1μnT+T,k+1+K2enT+T,k,where K1 and K2 are the control law matrices to be determined. Moreover, by (22)–(26) the controlled ILC dynamics can be written in the form of discrete repetitive process as(27)ηnT+T,k+1=AηnT,k+1+B0enT,k+B11ωnT,k+1,enT,k+1=CηnT,k+1+D0enT,k+B12ωnT,k+1,where(28)A=Az+BzΓzK1Cz,B0=BzΓzK2,B11=Dz,C=-CzAz+CzBzΓzK1Cz,D0=I-CzBzΓzK2,B12=-CzDz.The state-space model (27) represents a linear discrete repetitive process with pass output e(nT,k) and state vectors η(nT,k), respectively, once the initial conditions are specified, that is, the pass state initial vector η(0,k), k≥1, and the initial pass profile e(nT,0).Remark 2.
It is important to stress that, with the ILC law (22) applied, the resulting controlled dynamics (27) are modeled by a linear repetitive process state-space model; the previous trial (pass) error e(nT,k) affects the current trial (pass) error e(nT,k+1). Repetitive process is a particular class of 2D systems where a series of sweeps are made through a set of dynamics defined over a finite duration. Once each sweep is completed, the dynamics are reset and the next sweep starts. On any sweep, the output on the previous sweep explicitly contributes to the current sweep output and hence the link to ILC dynamics [24].Next the representation of discrete repetitive processes and the associated stability theory are given as background.


## 4. Repetitive Process Theory

The state-space model of a linear discrete repetitive process over the finite pass lengthα is(29)xt+1,k+1=Axt,k+1+But,k+1+B0yt,k+B11wt,k+1,yt,k+1=Cxt,k+1+Dut,k+1+D0yt,k+B12wt,k+1,where, on pass k, x(t,k) is the state vector, y(t,k) is the output vector, u(t,k) is the control input vector, and w(t,k+1) is the disturbance input vector. If(30)ωt,k+1l2=∑t=0N1∑k=0N2ωt,k+12<∞for any integers N1,N2>0, then w(t,k+1) is in l2 space, denoted by w(t,k+1)∈l2. To complete the process description, it is necessary to specify the boundary conditions, that is, the pass state initial vector sequence x(0,k),k≥1, and the initial pass profile y(t,0).Define the shift operatorsz1, z2 along the pass (t) and pass-to-pass (k) directions acting, for example, on x(t+1,k) and y(t,k+1), respectively, as(31)xt,k≔z1xt+1,k,yt,k≔z2xt,k+1;then the 2D characteristic polynomial for processes described by (29) is defined as(32)Iz1,z2=det⁡I-z1A-z1B0-z2CI-z2D0;the stability along the pass holds if and only if(33)Iz1,z2=det⁡I-z1A1-z2A2≠0,ifz1≤1,z2≤1,where(34)A1=AB000,A2=00CD0.This reversely promotes the development of LMI based stability condition along the pass which are sufficient but not necessary.Lemma 3 (see [24]).
A linear discrete repetitive process described by (29) is stable along the pass if there exists a block-diagonal matrix P≻0 such that the following LMI holds:(35)ΦTPΦ-P≺0,where(36)Φ=AB0CD0is the so-called augmented plant matrix.Definition 4.
A linear discrete repetitive process described by (29) is said to have H∞ robust disturbance attenuation γ if it is stable along the pass and(37)sup0≠w∈l2⁡yt,k+1l2wt,k+1l2<γ.
Now consider the 2D transfer function matrix coupling the disturbance and current pass profile vectors which is given by(38)Gywz1,z2=0II-z1A-z1B0-z2CI-z2D0-1B11B12and then the 2D Parseval theorem, which states that (37) is equivalent to the requirement that(39)Gywz1,z2=supw1,w2∈0,2π⁡σ¯Gywejw1,ejw2<γ,where σ¯(·) denotes the maximum singular value [34].


## 5. Stability Analysis

The following lemmas are used in the proofs of the main results.Lemma 5 (Schur complement formula [35]).
Given a symmetric matrixS=S11S12S21S22, S11 and S22 are square matrices, and then the following inequalities are equivalent:(i)
S≺0.(ii)
S11≺0 and S22-S12TS11-1S12≺0.(iii)
S22≺0 and S11-S12TS22-1S12≺0.Lemma 6 (see [36]).
Given matricesX, Y, Ω=ΩT, and δ(t) of compatible dimensions, then(40)Ω+XδtY+YTδTtXT≺0,for all δ(t) satisfying δT(t)δ(t)⪯I if and only if there exists some ε>0 such that(41)Ω+εXXT+ε-1YTY≺0.Theorem 7.
Assume repetitive disturbancew(t,k) exists in dual-rate sampling process Pc, that is, ω(nT,k+1)=0 in discrete repetitive process (27); for any ε>0, output information based ILC scheme described as (27) under an actuator fault of the form (3)–(6) is stable along the trial, if there exist matrices Q=diag⁡{Q1,Q2}≻0, N, M, and L such that(42)CzQ1=MCz,(43)-Q⋆⋆R1-Q⋆R2R3-εI≺0hold, where qz=diag⁡q,q,…,q, q0z=diag⁡q0,q0,…,q0, and(44)R1=AzQ1+BzqzNCzBzqL-CzAzQ1-CzBzqzNCzQ2-CzBzqzL,R2=00qzNCzqzL,R3=εq0BzT-εq0zBzCzT00;then the corresponding matrices in output information based control law (26) are given by(45)K1=NM-1,K2=LQ2-1.Proof.
According to Lemma3, we have ΦTPΦ-P≺0, where P=diag⁡P1,P2≻0, and(46)Φ=Az+BzΓzK1CzBzΓzK2-CzAz-CzBzΓzK1CzI-CzBzΓzK2.Applying Lemma 5 (Schur’s complement formula) to ΦTPΦ-P≺0 gives(47)-P1⋆⋆⋆0-P2⋆⋆Az+BzΓzK1CzBzΓzK2-P1-1⋆-CzAz-CzBzΓzK1CzI-CzBzΓzK20-P2-1≺0.Introduce Q1=P1-1 and Q2=P2-1, and pre- and postmultiply (47) by matrix diag⁡{Q1,Q2,I,I} to yield(48)-Q1⋆⋆⋆0-Q2⋆⋆Az+BzΓzK1CzQ1BzΓzK2Q2-Q1⋆-CzAz-CzBzΓzK1CzQ1I-CzBzΓzK2Q20-Q2≺0.Due to Γz=diag⁡{(I+Γ0)q,(I+Γ0)q,…,(I+Γ0)q}=(I+Γ0z)qz and Γ0z=diag⁡{Γ0,Γ0,…,Γ0}, we can rewrite (48) as(49)H+XΓ0zY+XΓ0zYT≺0,where(50)X=00BzT-CzBzTT,Y=qzK1CzQ1qzK2Q200,H=-Q1⋆⋆⋆0-Q2⋆⋆Az+BzqzK1CzQ1BzqzK2Q2-Q1⋆-CzAz+CzBzqzK1CzQ1Q2-CzBzqzK2Q20-Q2;then by Lemma 6 and (6), inequality (49) is feasible if and only if(51)H+ε1/2Xq0zε-1/2YTε1/2q0zXTε-1/2Y≺0holds, where q0z=diag⁡{q0,q0,…,q0}; then application of Schur’s complement formula to (51) gives(52)H⋆⋆ε1/2q0zXT-I⋆ε-1/2Y0-I≺0.Pre- and postmultiplying the above inequality by diag⁡{I,I,I,I,ε1/2I,ε1/2I} and applying condition (42) immediately give the LMI of (43). This completes proof of the theorem.Suppose nonrepetitive disturbancew(t,k) exists in dual-rate sampling process Pc, that is, ω(nT,k+1)≠0 in repetitive process (27); then the H∞ robust disturbance attenuation performance (37) should be modified as(53)sup0≠w∈l2⁡enT,k+1l2ωnT,k+1l2<γ;consequently, we can have the following result for robust controller design.Theorem 8.
Assumeω(nT,k+1)≠0; for any ε>0, output information based ILC scheme described as a discrete linear repetitive process of the form (27) under an actuator fault of the form (3)–(6) is stable along the trial and has H∞ disturbance attenuation γ>0; hence if there exist matrices Q=diag⁡{Q1,Q2}≻0, N, M, and L such that(54)-Q⋆⋆⋆⋆0-γI⋆⋆⋆R1F-Q⋆⋆R400-γI⋆R20R30-εI≺0holds, where R1, R2, R3 are given in (44), R4=0Q2, F=DzT-DzTCzTT, and the robust control law matrices K1 and K2 are given by (45).Proof.
Suppose there existsH∞ robust disturbance attenuation γ if the associated Hamiltonian function(55)JnT,k=ΔVnT,k+γ-1enT,k+12-γωnT,k+12<0holds, where ΔV(nT,k) is the associated increment of Lyapunov function for repetitive process (27):(56)VnT,k=ηTnT,k+1P1ηnT,k+1+eTnT,kP2enT,k,where P1≻0, P2≻0, and(57)ΔVnT,k=ηTnT+T,k+1P1ηnT+T,k+1+eTnT,k+1P2enT,k+1-ηTnT,k+1P1ηnT,k+1-eTnT,kP2enT,k.It is clear that (55) requires ΔV(nT,k)<0 and hence the result of Lemma 3 holds. Using (27), it is easily shown that(58)JnT,k=ηnT,k+1enT,kωnT,k+1TΨηnT,k+1enT,kωnT,k+1<0,where(59)Ψ=ATP1A+CTP2C-P1ATP1B0+CTP2D0ATP1B11+CTP2B12⋆B0TP1B0+D0TP2D0-P2-γ-1IB0TP1B11+D0TP2B12⋆⋆B11TP1B11+B12TP2B12-γI=ΦTFTPΦF-P-γ-1GTG00γI≺0,and G=0I and P=diag⁡{P1,P2}. Applying the Schur complement formula to (59) and defining Q=P-1=diag⁡{Q1,Q2} give(60)-P⋆⋆⋆0-γI⋆⋆ΦF-Q⋆G00-γI≺0.Moreover, pre- and postmultiplying (60) by diag⁡{Q,I,I,I} and its transpose, respectively, we have(61)-Q⋆⋆⋆0-γI⋆⋆ΦQF-Q⋆GQ00-γI≺0;therefore, (61) also can be modified as(62)H′+X′Γ0zY′+X′Γ0zY′T≺0,where(63)X′=000BzT-CzBzT0T,Y′=qzK1CzQ1qzK2Q20000,H′=-Q1⋆⋆⋆⋆⋆0-Q2⋆⋆⋆⋆00-γI⋆⋆⋆Az+BzqzK1CzQ1BzqzK2Q2Dz-Q1⋆⋆-CzAz-CzBzqzK1CzQ1Q2-CzBzqzK2Q2-CzDz0-Q2⋆0Q2000-γI.Similar to the proof of Theorem 7, another application of Lemmas 6 and 5 immediately gives (54).
The following analysis is to prove theH∞ robust disturbance attenuation γ. For any positive integers N1, N2, when all the initial conditions η(0,k) and e(nT,0) in (57) are assumed to be zero, we have(64)∑n=0N1∑k=0N2ΔVnT,k=∑k=0N2ηN1+1T,k+1TP1ηN1+1T,k+1+∑n=0N1enT,N2+1TP2enT,N2+1≥0.Using (55) and (64) yields(65)∑n=0N1∑k=0N2γ-1enT,k+1l22-γωnT,k+1l22≤∑n=0N1∑k=0N2ΔVnT,k+γ-1enT,k+1l22-γωnT,k+1l22=∑n=0N1∑k=0N2JnT,k<0;then (53) is satisfied; this means the ILC control process (27) has robust disturbance attenuation γ and the proof is complete.It should be emphasized that condition (42) is not an inequality form; it cannot be directly solved by LMI tools. Therefore, one can first determine Q1, Q2, N, and L by solving the LMI (43) or (54) and then get(66)M=CzQ1Cz+,where Cz+ is the right inverse of matrix Cz. However, in some cases, it is impossible to obtain the matrix M by using the above formulation. Therefore, condition (42) in Theorems 7 and 8 can be converted into the following inequality [37]:(67)MCz-CzQ1TMCz-CzQ1≺ζI,where ζ is a sufficiently small positive scalar. Observe that if the value of the scalar ζ is sufficiently small, condition (42) is approximately satisfied. Thus one obtains the LMI condition using the Schur complement formula: (68)Minimizeζsubject toζICzQ1-MCz⋆ζI≻0.


## 6. Case Study

To illustrate the validity of the proposed design method, a system of two noninteracting flow tanks in series including pipelines and valves is considered for process control [38]. The material balance equations that govern the system in the disturbance-free and fault-free case shown in Figure 2 are(69)Ac1dh1dt=Fi-c1h1,Ac2dh2dt=c1h1-c2h2,where Ac1 and Ac2 are the cross-sectional areas of tanks 1 and 2, unmeasurable states h1 and h2 are the liquid levels for tanks 1 and 2, c1 and c2 are constants which depend on the valves, and input Fi is the measured inlet flow rate. Meantime, amplifier element is used when the measured output signal outlet flow rate Fo2 is very small and nominally equal to 5c2h2. Thus (69) can be written in state-space form(70)x˙t,k=Acxt,k+Bcut,kyt,k=Ccxt,k,where(71)Ac=-c1Ac10c1Ac2-c2Ac2,Bc=1Ac10,Cc=05c2,ut,k=Fi,yt,k=Fo2,xTt,k=h1h2.Figure 2
Simple structure diagram of two flow tanks in series.Due to the differences of distance between the control points of tanks and flow rates to the sensors at the bottom of each tank, output time delay commonly appears in the system, which gives rise to instability and poor performance. Consider the case where the unknown external bounded disturbance stream (Stream 6 in Figure2) is introduced into tank 2 for t>0. The state-space form (71) becomes(72)x˙t,k=Acxt,k+Bcut,k+Dcwt,k,yt,k=Ccxt-τ,where w(t,k) represents the unknown external bounded disturbance stream and τ>0 is the fixed delay. In the following simulations, the model parameters used are Ac1=Ac2=0.8, c1=0.3, c2=0.2, τ=1, Dc=00.1T, the input updating time T1=h and the output sampling period T2=5h, and then frame period T=5h. Applying the Euler concretization method with a sampling period h=1s to (72) results in the following discrete system model of the form (9), where(73)Ah=0.687300.27450.7788,Bh=1.04240.1910,Ch=01,Dh=00.0834T,d=1.Partial degradation and wear from repeated control operation can lead to faults arising during the trials and it is assumed that an actuator fault occurs in the operation valve. As a numerical example, it is assumed that0.6≤Γ_≤Γ≤Γ¯=1; hence q=0.8 and q0=0.25. Suppose no actuator faults occur before k=15 trials have elapsed. Applying the lifting technology, this dual-rate sampling tank model can be converted into a lifting linear discrete model of the form (17), where(74)Az=0.15340000.39940.286500000.15340000.39940.2865,Bz=0.23260.33840.49240.71641.042400.52290.55210.53540.43490.191000.23260.33840.49240.7161.042400.52290.55210.53540.43490.191,Cz=0100,Dz=0000000.01860.02710.03940.05730.0834000000000.01860.02710.03940.05730.0834.If we also force the input variable Fi to be just updated in slow sampling period T, then u(nT-h,k)=u(nT-T,k) and u(nT,k)=u(nT+h,k)=u(nT+2h,k)=u(nT+3h,k)=u(nT+4h,k); hence the dual-rate sampling tank model can be of the form (19), where As=Az, Cs=Cz, Ds=Dz, Bs=Ah4BhAh3Bh+Ah2Bh+AhBh+Bh0Ah4Bh+Ah3Bh+Ah2Bh+AhBh+Bh.The initial state vector on each trial is assumed to be zero,∀k≥0, and the reference trajectory is(75)yrnT=0.001nT,0≤n<1000.5,100≤n<2000.0003nT+0.2,200≤n<3000.65,300≤n≤400.Introduce the root mean square (RMS) to evaluate tracking performance from trial to trial:(76)Hk=1400∑n=1400e2nT,k,where the smaller the value of H(k), the better the tracking performance along the kth trial. Next two possible scenarios are considered.Scenario 1 (repetitive disturbance with a constant fault).
It is assumed thatw(nh,k)=0.1sin⁡(0.2n) in this scenario; hence ω(nT,k+1)=0. Consider the case when the operating valve is always partially blocked in tank system (9). This constant partial fault causes the actuator to drop to 80% of its normal value; hence Γ=0.8. For single-rate sampling system model with matrices {As,Bs,Cs}, using LMI toolbox to solve inequality (68) in Theorem 7 gives the corresponding matrices in (45) as(77)K1=-0.4120-0.2312T,K2=0.20010.1058Tand achieves minimum ζ=0.6148. Furthermore, we can also obtain the dual-rate sampling controller matrices from matrices {Az,Bz,Cz}:(78)K1=-0.1457-0.1167-0.1017-0.06060.02410.0256T,K2=0.28210.20540.17160.0861-0.0846-0.0476T.ζ=0.0126. Substituting (77) and (78) into (26) gets single-rate sampling and dual-rate sampling fault-tolerant ILC law (22), respectively. Based on (16), we can find that single-rate sampling controller is just updated in sampling time nT, but dual-rate sampling controller is updated in every sampling time nT, nT+h, nT+2h, nT+3h, and nT+4h; the total sample number of dual-rate sampling controllers is five times that of the single-rate sampling controllers. The simulation results of input, output, and tracking error are shown in Figures 3–9.
Before the constant fault occurs, the tracking error rapidly converges into a steady state from trail to trial. As soon as the fault occurs, the output responses deviate from the reference trajectory and the tracking performance deteriorates. However, the tracking performance can achieve an excellent level again after some trials and even restore it to the original level under the fault-tolerant controller. This result can be reflected clearly in Figure3, where the values of RMS for different trials are presented. Obviously, compared with the single-rate sampling controller, the dual-rate sampling controller has the advantage of faster convergence which ensures the reliable stability of the tank system. The effectiveness of the proposed scheme is thus illustrated under both normal and fault scenario.Figure 3
The RMS performance for Scenario1.Figure 4
The single-rate input 3D curve for Scenario1.Figure 5
The single-rate output curve for Scenario1.Figure 6
The dual-rate input 3D curve for Scenario1.Figure 7
The dual-rate output curve for Scenario1.Figure 8
The single-rate tracking error 3D curve for Scenario1.Figure 9
The dual-rate tracking error 3D curve for Scenario1.Scenario 2 (nonrepetitive disturbance with a time-varying fault).
In this scenario, the operation valve is partly blocked by a time-varying fault in tank system (9), and the time-varying fault matrix Γ=0.8+0.2sin⁡(0.2n) is used here. In this case, nonrepetitive disturbance exists, and w(nh,k)=0.1sin⁡(0.01k) and ω(nT,k+1)≠0. Solving the LMI of (68) in Theorem 8 also gives the single-rate and dual-rate ILC law matrices:(79)K1=-0.6204-0.6236T,K2=0.30030.2233T.ζ=0.6662, and(80)K1=-0.4622-0.3833-0.3392-0.21280.05070.0811T,K2=0.51410.38440.30720.1231-0.2366-0.0405T.ζ=0.5205, and the H∞ disturbance attenuation is γ=1.3244 and γ=1.1040, respectively. The simulation results are shown in Figures 10–16. From these figures, the robustness and convergence of the designed control system along both sampling number and trial directions can be guaranteed even with a certain degree of nonrepetitive external disturbance.Figure 10
The RMS performance for Scenario2.Figure 11
The single-rate input 3D curve for Scenario2.Figure 12
The single-rate output curve for Scenario2.Figure 13
The dual-rate input 3D curve for Scenario2.Figure 14
The dual-rate output curve for Scenario2.Figure 15
The single-rate tracking error 3D curve for Scenario2.Figure 16
The dual-rate tracking error 3D curve for Scenario2.


## 7. Conclusion and Future Works

This paper deals with the iterative learning fault-tolerant tracking control problem for a class of linear dual-rate sampling processes with actuator fault and output time delay. Firstly, the dual-rate sampling system is described in slow sampling rate model by lifting technology. Then we designed the output information based robust fault-tolerant ILC law based on the repetitive process theory. Sufficient conditions for the ILC law with stability and disturbance attenuation performance are given in terms of the corresponding linear matrix inequalities. Finally, the control simulation of the two noninteracting flow tanks in series verifies the effectiveness of this design method.This proposed approach is not applicable for nonuniform sampling processes now; the uncertainty and time variation of the refresh time interval in nonuniform sampling processes make the system modeling very complicated by lifting technology. Moreover, the extension to multiple-input multiple-output systems may also cause computation burden issue when using expanding vector to design ILC law. But there are still many areas to which further research could profitably be directed. Possible further research directions include the optimization ofH∞ robust performance γ. Another area is to use the proposed control algorithm to deal with dual-rate sampling processes with external bounded disturbances in both the state and output sides. Extending the theory to simultaneously dispose state delay and input delay is also a possible topic for future work.



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*Source: 7072032-2018-03-19.xml*