# H

Authors: Lan Zhou; Jinhua She; Shaowu Zhou; Qiwei Chen
Journal: International Journal of Engineering Mathematics
 (2014)

---

## Abstract

This paper presents a method of designing a state-observer based modified repetitive-control system that provides a givenH∞ level of disturbance attenuation for a class of strictly proper linear plants. Since the time delay in a repetitive controller can be treated as a kind of disturbance, we convert the system design problem into a standard state-feedback H∞ control problem for a linear time-invariant system. The Lyapunov functional and the singular-value decomposition of the output matrix are used to derive a linear-matrix-inequality (LMI) based design algorithm for the parameters of the feedback controller and the state-observer. A numerical example demonstrates the validity of the method.


---

## Body

## 1. Introduction

In control engineering practice, many systems exhibit repetitive behavior, such as a robot manipulator, a hard disk drive, and many other servo systems. Repetitive control [1], or RC for short, has proven to be a useful control strategy for a system with a periodic reference input and/or disturbance signal [2–4]. The distinguishing feature of RC is that it contains a pure-delay positive-feedback loop, which is the internal model of a periodic signal. For a given periodic reference input, a repetitive controller gradually reduces the tracking error through repeated learning actions [5], which involves adding the control input of the previous period to that of the present period to regulate the present control input. This theoretically guarantees gradual improvement and finally eliminates any tracking error and provides very precise control, which is a chief characteristic of the human learning process.From the standpoint of system theory, an RC system (RCS) is a neutral-type delay system. Asymptotic tracking and stabilization of the control system are possible only when the relative degree of the compensated plant is zero [5]. To use RC on a strictly proper plant, that is, the case that most control engineering applications deal with, the repetitive controller has to be modified by the insertion of a low-pass filter into the time-delay feedback line. The resulting system is called a modified RCS (MRCS). Since a modified repetitive controller is just an approximate model of a periodic signal, there exists a steady-state tracking error; that is, in an MRCS, the low-pass filter relaxes the stabilization condition but degrades the tracking precision [6].RC is similar to iterative learning control (ILC), which is another well-known method that makes use of previous control trials. However, as pointed out by [7–9] and others, there are significant differences between them. First of all, the initial state of a period is different. In an RCS, the state at the beginning of a period is the same as the final state in the previous period. However, in an ILC system (ILCS), the reference trajectory is defined over a finite time interval, and the state of an ILCS is usually reset after a trial. In the literatures of ILC, the initial or boundary conditions, that is, the initial state on each trial and the initial trial profile, are commonly taken to be zero. The difference in initial-condition resetting leads to the different analysis techniques and results.One problem with an RCS is that the improved disturbance rejection at the periodic frequency and its harmonics is achieved at the expense of degraded system sensitivity at intermediate frequencies. In other words, an RCS cannot reject, and may even amplify, an aperiodic disturbance. Various solutions have been presented in the literatures addressing this problem. Time-varying and adaptive RC was proposed in [10]. Kim et al. presented a design method of a two-parameter robust RCS that used discrete-time m-synthesis and H∞ control to reject both periodic and aperiodic disturbances [11]. However, the high order of the controller makes it difficult to implement. She et al. devised an H∞ control method to design the parameters of the stabilization controller for a robust MRCS with time-varying uncertainties [12]. But it only considers the case that there is no disturbance input to the controlled plant. In addition, the full states of a plant are needed to design the state-feedback controller. In many practical applications, the complete state of a plant is not available because of running cost and difficulty of installation.Zhou et al. presented a method based on two-dimensional system theory of designing a robust observer based MRCS [13]. But it only considered the robust stability of the system. To enable RC method to handle a larger class of systems, this paper presents a method of designing an MRCS with a prescribed bound on disturbance attenuation for a class of strictly proper linear systems. First, we present a configuration of an observer based MRCS and formulate the design problem as an equivalent H∞ state-feedback control problem. Then, combining the Lyapunov stability theory and the singular-value decomposition (SVD) of the output matrix, we derive a linear-matrix-inequality (LMI) based sufficient stability condition. The condition can be used directly to design the parameters of the feedback controller and the state-observer. Compared to the one in [13] and other RC methods, the advantage of this method is that it takes into consideration the transient performance of the MRCS. Finally, a numerical example demonstrates the validity of the method.Throughout this paper,L2[0,+∞) is the linear space of square integrable functions from [0,+∞) to Cp; ∥G(s)∥∞:=sup⁡0≤w<∞⁡∥G(jω)∥ is the H∞ norm of a transfer function G(s); and * denotes the transpose of a block entry in a matrix.


## 2. Problem Description

Consider the MRCS in Figure1. The compensated single-input, single-output (SISO) plant is
(1)x˙p(t)=Apxp(t)+Bpu(t)+Bww(t),y(t)=Cpxp(t),
where xp(t)∈ℝn is the state of the plant; u(t),y(t)∈ℝ are the control input and output, respectively; and w(t)∈L2[0,+∞) is the disturbance input. Setting Bω≠0 adds the disturbance to the system, and setting Bω=0 removes it. Ap, Bp, and Cp are real constant matrices. Assume that (Ap,Bp) is controllable and (Cp,Ap) is observable, which are standard for a servo system.Figure 1
Configuration of the state-observer based MRCS.In Figure1, r(t) is a periodic reference signal with a period of T, and
(2)e(t)=r(t)-y(t)
is the tracking error. The modified repetitive controller is
(3)CMR(s)=11-q(s)e-sT.q(s) is a first-order low-pass filter that ensures the stability of the control system. Without loss of generality, we assume that q(s) is a first-order filter; that is
(4)q(s)=ωcs+ωc,
where ωc is the cutoff angular frequency of the filter. The inverse Laplace-transform of (3) yields
(5)v˙(t)=-ωcv(t)+ωcv(t-T)+ωce(t)+e˙(t),
where the output, v(t), of CMR(s) is chosen to be the state of the modified repetitive controller.The following state-observer is used to reproduce the state of the plant(6)x~˙p(t)=Apx~p(t)+Bpu(t)+L[y(t)-y~(t)]y~(t)=Cpx~p(t),
where L is the observer gain.The error between the states of the actual plant and the observer is(7)xe(t)=x~p(t)-xp(t).
Thus, from (1) and (6), we have
(8)x˙e(t)=(Ap-LCp)xe(t)-Bww(t).A linear control law based on the states of the observer and the repetitive controller is(9)u(t)=Kev(t)+Kpx~p(t),Ke∈ℝ,Kp∈ℝ1×n.
where Ke is the feedback gain of the repetitive controller and Kp is the reconstructed state-feedback gain.Since the stability of the system does not depend on an exogenous signal, we setr(t)=0. Choose xp(t), xe(t), and v(t) to be the state variables of the MRCS in Figure 1. Then, from the dynamic equations (1), (5), (8), and (9) yields the following state equation:
(10)x˙(t)=Ax(t)+A1x(t-T)+Bu(t)+B~w(t),
where
(11)xT(t)=[xpT(t)xeT(t)vT(t)],A=[Ap000Ap-LCp0-ωcCp-CpAp0-ωc],A1=[00000000ωc],B=[Bp0-CpBp],B~=[Bw-Bw-CpBw].To enhance the system convergence, applying the method in [12], we treat the delay item v(t-T) as a disturbance. We construct a generalized plant, P(s), and redraw Figure 1 as Figure 2. Considering that the H∞ norm of the time delay is one, we formulate the problem of the controller design as an H∞ state-feedback one (Figure 3). The state-space representation of the generalized plant, P(s), is
(12)x˙(t)=Ax(t)+B1wl(t)+B2u(t),zl(t)=C1x(t),yl(t)=C2x(t),
where
(13)wl(t)=[wT(t)vT(t-T)]T,yl(t)=[xpT(t)0vT(t)]T,B1=[Bw0-Bw0-CpBwωc],B2=[Bp0-CpBp],C1=[000001],C2=[100000001].Figure 2
Argument system.Figure 3
Formulation of design problem forH∞ state-feedback control.Substituting the control input (9) into (12) yields a representation of the closed-loop system in Figure 1:
(14)x˙(t)=(A+B2K)x(t)+B1wl(t),z(t)=C1x(t),
where
(15)K=[KpKpKe].This paper considers the followingH∞ disturbance attenuation problem.Design suitable control gains,Ke and Kp, and the state-observer gain, L, such that(1)
the MRCS in Figure1 is internally stable; and(2)
theH∞ norm from wl to z(t) is less than γ; that is,
(16)∥Gwlz(s)∥∞=∥C1[sI-(A+B2K)]-1B1∥∞<γ,where γ is a positive number.


## 3. Design of theH∞ Controller

In this section we construct an LMI-based design algorithm for theH∞ state-feedback controller by employing a Lyapunov functional and the SVD of the output matrix.Definition 1 (see [14]).
The matrixΠ has full row rank (rank⁡(Π)=p). The SVD of Π is
(17)Π=U[S0]VT,
where S∈ℝp×p is a diagonal matrix with positive, diagonal elements in decreasing order; 0∈ℝp×(n-p) is a zero matrix; and U∈ℝp×p and V∈ℝn×n are unitary matrices.
The following lemma presents an equivalent condition for matrix equation(18)ΠX=X-Π.Lemma 2 (see [15]).
For a givenΠ∈ℝp×n with rank⁡(Π)=p, if X∈ℝn×n is a symmetric matrix, then there exists a matrix, X-∈ℝp×p, such that ΠX=X-Π holds if and only if
(19)X=Vdiag⁡{X11,X22}VT,
where X11∈ℝp×p and X22∈ℝ(n-p)×(n-p).The following two lemmas are also employed in the derivation of the existence condition for theH∞ controller of the system (14).Lemma 3 (Schur complement [16]).
For any real matrixΣ=ΣT, the following assertions are equivalent: (1)
Σ=[S11S12*S22]<0;(2)
S11<0 and S22-S12TS11-1S12<0;(3)
S22<0 and S11-S12S22-1S12T<0.Lemma 4 (see [17]).
For the system(20)x˙(t)=Ax(t)+Bw(t),z(t)=Cx(t)+Dw(t),
the following assertions are equivalent. (1)
A is stable; and the H∞ norm of the transfer function Gwz(s), from w(t) to z(t) satisfies ∥Gwz(s)∥∞<γ.(2)
There exists a symmetric matrixP>0 such that
(21)[PA+ATPPBCT*-IDT**-γ2I]<0,holds.So, we have the following theorem.Theorem 5.
For a given positive scalarγ, if there exist symmetric, positive-definite matrices X1, X11, X22, and X3, and arbitrary matrices W1, W2, W3, and W4 such that the LMI
(22)[Θ11BpW2Θ13Bw000*Θ22-W2TBpTCpTBw000**Θ33-CpBwωc0X3***-I000****-I00*****-γ2I0******-γ2I]<0,
holds, where the SVD of the output matrix Cp is
(23)Cp=U[S0]VT,X2=Vdiag⁡{X11,X22}VT,(24)Θ11=ApX1+X1ApT+BpW1+W1TBpT,Θ13=BpW3-W1TBpTCpT-ωcX1CpT-X1ApTCpT,Θ22=ApX2+X2ApT-W4Cp-CpTW4T,Θ33=-CpBpW3-W3TBpTCpT-2ωcX3,
then the closed-loop repetitive-control system (14) is asymptotically stable and provides a prescribed H∞ disturbance attenuation level, γ. Furthermore, the parameters in control law (15) are
(25)Kp=W1X1-1,Ke=W3X3-1,
and the state-observer gain in (6) is
(26)L=W4USX11-1S-1UT.Proof.
Applying Lemma2 to (23), there exists
(27)X-2=USX11S-1UT,
such that
(28)CpX2=X-2Cp,X-2-1=USX11-1S-1UT.
Thus, the observer gain L in (26) is equal to
(29)L=W4X-2-1.
Let(30)P1=X1-1,P2=X2-1,P3=X3-1W1=KpX1,W2=KpX2,W3=KeX3,W4=LX-2.
Pre- and postmultiplying the matrix on the left-side of LMI (22) by
(31)P=diag⁡{P1,P2,P3,I,I,I,I},
yields the following matrix inequality that is equivalent to LMI (22) (32)[Λ11P1BpKpΛ13P1Bw000*Λ22-KpTBpTCpTP3P2Bw000**Λ33-P3CpBwP3ωc01***-I000****-I00*****-γ2I0******-γ2I]<0,where
(33)Λ11=P1Ap+ApTP1+P1BpKp+KpTBpTP1,Λ13=P1BpKe-KpTBpTCpTP3-ωcCpTP3-ApTCpTP3,Λ22=P2Ap+ApTP2-P2LCp-CpTLTP2,Λ33=-P3CpBpKe-KeTBpTCpTP3-2ωcP3.
Two steps bring us to the conclusion.
First, we consider the stability of the closed-loop MRCS (14) without a disturbance (Bw=0).
Substituting the control input (9) into (10) yields the closed-loop system
(34)x˙(t)=(A+BK)x(t)+A1x(t-T),
where A and A1 are defined in (10).
Choose a Lyapunov functional candidate to be(35)V(t)=xT(t)Px(t),
where
(36)P=diag⁡{P1,P2,P3}.
Along the time trajectory of (34)
(37)dV(t)dt=2xT(t)Px˙(t)=ξT(t)Λ~ξ(t),
where
(38)ξ(t)=[xT(t)xT(t-T)]T,Λ~=[Λ11P1BpKpΛ13000*Λ22-KpTBpTCpTP3000**Λ3300P3ωc***000****00*****0],
and Λ11,Λ13,Λ22,Λ33 are defined in (33).
From (37), Λ~<0 implies that, for any ξ(t)≠0, dV(t)/dt<0. Also, Λ~<0 is equivalent to the matrix inequality
(39)[Λ11P1BpKpΛ130*Λ22-KpTBpTCpTP30**Λ33P3ωc***0]<0.
Furthermore, sinceBw=0, applying Sylvester Criterion [18] and Schur-complement Lemma 3 to (32) yields Λ~<0. So, if LMI (22) holds, then MRCS (34) is asymptotically stable; that is, the closed-loop system (14) is asymptotically stable when Bw=0.
Next, we consider the caseBw≠0. Note that closed-loop MRCS (14) is asymptotically stable.
Applying Lemma4 to (14), it follows that if the matrix inequality
(40)Ω=[P(A+B2K)+(A+B2K)TPPB1C1T*-I0**-γ2I]<0
holds, then the system (14) satisfies the disturbance attenuation performance ∥Gwlz(s)∥∞<γ.
Obviously, from the coefficient matrices and controller parameters in (14), the matrix inequality (40) is equal to (32). Also, LMI (32) is equivalent to (22). So, if LMI (22) holds, then Ω<0.Remark 6.
Theorem5 provides an LMI-based sufficient condition for the closed-loop repetitive-control system (14) with a prescribed H∞ disturbance-attenuation level γ. The condition can be used directly to design the parameters of the controller and the state-observer in Figure 1.In addition, from Theorem5, we obtain a sufficient condition for an H∞ disturbance-attenuation level for the system (1) under the state-feedback-based control law:
(41)u(k)=Kev(t)+Kpxp(k).The representation of the corresponding closed-loop system in Figure1 is
(42)x-˙(t)=(A-+B-2K-)x(t)+B-1wl(t),z-(t)=C-1x-(t),
where
(43)x-(t)=[xpT(t)vT(t)]T,A-=[Ap0-ωcCp-CpAp-ωc],B-1=[Bw0-CpBpωc],B-2=[Bp-CpBp],C-1=[0001],K-=[KpKe].Corollary 7.
For given positive scalarγ, if there exist symmetric, positive-definite matrices X1 and X2 and arbitrary matrices W1 and W2 such that the LMI
(44)[Θ11Θ12Bw000*Θ22-CpBwωc0X3**-I000***-I00****-γ2I0*****-γ2I]<0,
holds, where
(45)Θ11=ApX1+X1ApT+BpW1+W1TBpT,Θ12=BpW2-W1TBpTCpT-ωcX1CpT-X1ApTCpT,Θ22=-CpBpW2-W2TBpTCpT-2ωcX2,
then the closed-loop MRCS (42) is asymptotically stable and the H∞ disturbance-attenuation performance ∥Gwlz(s)∥<γ is guaranteed. Furthermore, the feedback gains in control law (41) are
(46)Kp=W1X1-1,Ke=W2X2-1.


## 4. Numerical Example

In this section, we apply our method to the position control of a chuck-workpiece system with a three-jak chuck (Figure4, [19]).Figure 4
Chuck-workpiece system.Assume that the parameters of plant (1) are
(47)Ap=[01-1-1],Bp=Bw=[0.50],Cp=[10].We consider the problem of tracking the reference input(48)r(t)=sin(2πt),
and rejecting the aperiodic disturbance
(49)w={0,t<5sin(2πt)+5sin(πt)+5sin(3πt)+5(tan(t-10)-tan(t-9)),5≤t≤150,t>15,
below the level γ=0.2.Thus, the repetition period isT=1.Carrying out the design procedure in Theorem5, yields the corresponding control gains
(50)Ke=4.0882×104,Kp=[-63.5735-1.9221],
and the state-observer gain
(51)L=[0.5003-0.3630]T.Simulation results in Figure5 show that the system is asymptotically stable. After it enters the steady state in the third period for the reference input, the disturbance is added to the system. And the largest steady-state peak-to-peak relative tracking error is as small as 0.0893%.Figure 5
Simulation results for observer basedH∞ repetitive-control system.To better assess the disturbance attenuation performance of the observer based MRCS, we design anH∞ output-feedback controller [20] (Figure 6) for the plant (47), where the PI controller in Figure 6 is
(52)Kpi=650+1400s.Figure 6
Block diagram ofH∞ MRCS.Let(53)wl(t)=[vT(t-T)wT(t)]T,z(t)=e(t).
The H∞ disturbance rejection problem was formulated as follows.Find anH∞ output controller K∞(s) such that the MRCS in Figure 6 is internally stable and provides the disturbance attenuation level γ; that is, ∥Gwlz(s)∥∞<γ.The problem of designingK∞(s) is shown in Figure 7, and the resulting controller is
(54)K∞(s)=541667.3035(s+1)(s+2.154)(s+8.157×104)(s+2.154)(s+1).Figure 7
Block diagram for designingK∞(s) in Figure 6.Choose the index(55)Je=sup⁡t≥ts⁡|e(t)|
to evaluate the steady-state tracking performance, where ts is the setting time of the control system.Figure8 shows the tracking error of the system in Figure 6 for the reference input (48) and the aperiodic disturbance (49). We find that Je=0.0098 for the observer based H∞ in Figure 1, but Jeoutput=0.022 for the H∞ MRCS in Figure 6. Moreover, the former has better transient performance.Figure 8
Tracking error ofH∞ MRCS in Figure 6 for reference input (48) and input disturbance (49).


## 5. Conclusion

In a conventional repetitive-control system, the tracking performance may be degraded by aperiodic disturbance even if it contains an internal model of a periodic signal, which theoretically guarantees asymptotic tracking for the periodic reference input. This paper describes a method of designing an observer based repetitive-control system with a prescribedH∞ disturbance attenuation level for a class of strictly proper linear plants. An equivalent system is established to convert the design problem to a static state-feedback H∞ control problem. The Lyapunov stability theory and the SVD of the output matrix are used to derived an LMI-based design algorithm for the parameters of the feedback controller and the state-observer. The validity of the method was demonstrated using the position control of a noncircular cutting workpiece system. The simulation results show that the designed MRCS is asymptotically stable and exhibits satisfactory disturbance-attenuation performance.



---

*Source: 838905-2014-05-13.xml*