# Delay-Probability-Distribution-Dependentℋ∞ FIR Filtering Design with Envelope Constraints

Authors: Cheng Peng; Zhandong Yu; Hamid Reza Karimi; Weizhi Wang; Nan Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies the problem ofℋ∞ finite-impulse response (FIR) filtering design of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the requirement of pulse-shape is also considered in filter design. Employing the information about the size and probability distribution of delay, a delay-probability-distribution-dependent criterion is proposed for the filtering error system. Based on a Lyapunov-Krasovskii functional, a set of linear matrix inequalities (LMIs) are formulated to solve the problem. At last, a numerical example is used to demonstrate the effectiveness of the filter design approach proposed in the paper.


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## Body

## 1. Introduction

In the studies about filtering problem, one most significant approach frequently applied in the past decades is Kalman filtering, the main idea of which is to minimize the variance of the estimation error assuming considered system dynamics to be exactly known and the external disturbances to be stationary Gaussian noises with known statistical properties [1, 2]. However, in many practical engineering applications, the statistical details about external noise are not available [3–8]. In these cases, many approaches are introduced to improve systems’ robustness, such as ℋ∞, ℋ2, and mixed ℋ∞/ℋ2 filtering [2, 9–13]. In this paper, the ℋ∞ filtering approach is utilized.On the other hand, time-delays are frequently encountered in practical engineering systems, such as manufacturing systems, power systems, and networked control systems [14–17]. Existence of delay makes the analysis and synthesis of systems a much more difficult task; meanwhile it is also the source of instability and poor performance in many cases [13, 18–20]. The main approaches to solve delay problems can be classified into delay-dependent approach and delay-independent approach. It has been shown in [21, 22] that the results obtained using delay-dependent approaches are generally less conservative than the delay-independent approaches ones [23]. Acknowledging this fact, the delay-dependent approach is applied in this paper.In fact, the variation of delay may often stick to some probability distribution in spite of its varying and underivable property [24, 25]. Furthermore, in many real systems such as networked control systems, the time-varying delay may have some abrupt burst, leading to very large delay with a very small probability [26]. In this sense, the discussion about time-delay should not only depend on its size but also on its probability distribution. In this paper, a new filter design approach and new stability criteria for the filtering error system taking the stochastic characteristic of time-varying delay into account is proposed.While anℋ∞ optimal filter can catch the frequency-domain property, the time-domain constraints such as envelope constraints or bounds on signals cannot be handled by this frequency-domain approach [27]. Among various time-domain specifications, envelope constraints, which make requirement on the pulse-shape, have significant applications in many practical engineering systems, such as communication systems, radar, sonar systems, and signal processing systems [28–31]. For instance, in deconvolution filtering and data channel equalization problems, it is extremely important to achieve a desired pulse-shape through designing an appropriate filter [27].Therefore, aiming at incorporating both frequency-domain and time-domain constraints into the problem, we intend to design a filter satisfying theℋ∞ performance and subject to envelope constraints in outputs. Meanwhile, time-varying delays with certain stochastic characteristics in the transmission channel are also taken into account. With the proposed filter design approach, a more general condition of time-varying delay problem can be solved. As in most situations, although detailed and exact information about delay cannot be achieved, the delay’s probability distribution characteristics can be predicted or observed relatively easily. Once the probability information is gotten, the filter design approach can be developed.In this paper, based on a Lyapunov-Krasovskii functional, we first present anℋ∞ optimal solution to the design of a finite-impulse response (FIR) filter using information about the range of time-varying delay and its probability distribution. Then, the envelope constraints are taken into consideration. The resultant filter is called an ℋ∞ optimal Envelope-Constrained FIR (ECFIR) filter. We obtain the solution via solving an LMI optimization problem. At last, a numerical example is presented to illustrate the effectiveness of the proposed filtering design approach.


## 2. Problem Formulation and Preliminaries

Consider a filtering system shown in Figure1, where Σl represents a linear dynamic system with state-space realization given by
(1)Σl:{xl(k+1)=Alxl(k)+Blw(k)s(k)=Clxl(k),
where xl(k)∈ℜnl is the model state vector, w(k)∈ℜnw is the input signal, s(k)∈ℜns is the source signal generated by the model, and Al, Bl, Cl are known constant matrices with appropriate dimensions. Then the output s(k) is transmitted through a channel with time-varying delay modeled by
(2)Σc:{xc(k+1)=Acxc(k)+Ads(k-d(k))+Bcv(k)y(k)=Ccxc(k)+Cds(k-d(k))+Dcv(k),
where xc(k)∈ℜnc is the channel state vector, d(k)∈[0,d2] is the time-varying delay with an upper bound of d2, y(k) is the output of the channel, and v(k) is the disturbance input; Ac, Ad, Bc, Cc, Cd, Dc are all known constant system matrices with appropriate dimensions. As is shown in (2), the source signal s(k) suffers from influence of time-varying delay d(k) and disturbance from the environment represented by v(k). The output of transmission channel is y(k), which is also the input signal of the filter. We are going to use the corrupted signal y(k) to reconstruct original source signal.Figure 1
Filtering system.Assumption 1.
d
(
k
) changes randomly and for a constant d1∈[0,d2], and the probability of d(k)∈[0,d1) and d(k)∈[d1,d2] can be known. The following sets and functions are defined:
(3)Ω1={k:d(k)∈[0,d1)},Ω2={k:d(k)∈[d1,d2]},d1(k)={d(k),fork∈Ω10fork∉Ω1,d2(k)={d(k),fork∈Ω2d1,fork∉Ω2.
Obviously, it can be seen from the definition that k∈Ω1 is equal to the occurrence of event d(k)∈[0,d1) and k∈Ω2 means that the event d(k)∈[d1,d2] occurs. Therefore, a stochastic variable β(k) can be defined as
(4)β(k)={1,k∈Ω10,k∈Ω2.Assumption 2.
β
(
k
) is a Bernoulli distributed sequence with
(5)Prob{β(k)=1}=𝔼{β(k)}=β0,Prob{β(k)=0}=1-𝔼{β(k)}=1-β0,
where 0≤β0≤1 is a constant.Remark 3.
From Assumption2, it is easy to see that 𝔼{β(k)-β0}=0 and 𝔼{(β(k)-β0)2}=β0(1-β0). As Prob{d(k)∈[0,d1)}=Prob{β(k)=1}=β0 and Prob{d(k)∈[d1,d2]}=Prob{β(k)=0}=1-β0,  β0 and 1-β0 also denote the probability of d(k) taking values in [0,d1) and [d1,d2], respectively.
According to Assumptions1 and 2, the system model described by (2) can be rewritten as
(6)xc(k+1)=Acxc(k)+β(k)Ads(k-d1(k))+(1-β(k))Ads(k-d2(k))+Bcv(k),y(k)=Ccxc(k)+β(k)Cds(k-d1(k))+(1-β(k))Cds(k-d2(k))+Dcv(k).
At the receiving end, we are interested in designing a linear filter with state-realization as follows:
(7)Σf:{xf(k+1)=Afxf(k)+Bfy(k)s^(k)=Cfxf(k)+Dfy(k),
where xf(k)∈ℜnf is the filter state vector, s^(k), is the estimated signal of source signal s(k) and Af, Bf, Cf, Df have the following form:
(8)Af=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0]nf×nf,Bf=[00⋮01]nf×1,Cf=[f(nf)f(nf-1)…f(1)]Df=f(0).
The transfer function of the filter is given by
(9)Φf(z)=Cf(zI-Af)-1Bf+Df=f(0)+f(1)z-1+f(2)z-2+⋯+f(nf)z-nf,
where f(0), f(1),…, and f(nf) are parameters to be determined. Define the filtering error as e(k)=s(k)-s^(k). Then, via augmenting the models Σl and Σc, the filtering error system is given as follows:
(10)Σe:{xe(k+1)=Aexe(k)+β(k)Aedxe(k-d1(k))+(1-β(k))Aedxe(k-d2(k))+Bewe(k)e(k)=Cexe+β(k)Cedxe(k-d1(k))+(1-β(k))Cedxe(k-d2(k))+Dewe(k),
where
(11)xeT(k)=[xlT(k)xcT(k)xfT(k)]T,we(k)=[wT(k)vT(k)]T,Ae=[Al000Ac00BfCcAf],Be=[Bl00Bc0BfDc],Ce=[Cl-DfCc-Cf],De=[0-DfDc],Aed=[000AdCl00BfCdCl00],Ced=[-DfCdCl00].
Before giving the main results, we need following definitions at first.Definition 4.
For a given functionV(x(k)), its stochastic difference operator is defined as
(12)ΔV(x(k))=𝔼{V(x(k+1))∣x(k)}-V(x(k)).Definition 5 (see [32]).
The filtering error system in (10) is said to be stochastically stable if for any initial condition xe(0) and zero exogenous noise we(k), there exists a positive definite W independent of xe(0), such that the following condition is satisfied:
(13)𝔼{∑k=0∞‍|xe(k)|2∣xe(0)}<xeT(0)Wxe(0).Definition 6.
System (10) is said to be stochastically stable with an ℋ∞ norm bound γ, if the following conditions hold.(1)
The filtering error system withwe(k)=0 is stochastically stable.(2)
For all nonzerowe(k)∈l2[0,∞) and under zero initial conditions, the following inequality holds:
(14)∥e(k)∥2≤γ∥we(k)∥2.
Now, with the definitions above, we present the objective of this paper.
Given the filtering system shown in Figure1, we are interested in designing a filter in the form of (7)-(8) such that (a)
the filtering error system (10) is asymptotically stable in the stochastic sense;(b)
the filtering error system (10) possesses a minimized ℋ∞ performance level γ;(c)
a time-domain envelope constraint is imposed on the output signals^(k) as follows:
(15)l(k)≤s^(k)≤u(k),where l(k) and u(k) are the lower and upper bounds of the time-domain mask, respectively.


## 3. Main Results

In this section, based on the Lyapunov-Krasovskii stability theorem, a delay-probability-distribution-dependent approach is proposed to solve theℋ∞ FIR filter design problem subject to envelope constraints described in (15). First, a stability criterion for the filtering error system described in (10) is proposed. Then the envelope constraints are taken into consideration. An ℋ∞ optimal ECFIR filter design approach is given at last.Theorem 7.
Given the system in Figure1, for some given constants 0≤d1≤d2, β0, and γ, the filtering error system (10) is stochastically stable with ℋ∞ performance γ if there exist matrices P>0, Q1>0, Q2>0, R1>0, R2>0 of appropriate dimensions such that the following optimization problem has solutions,
(16)minP>0,Q1>0,Q2>0,R1>0,R2>0,fγ,
subject to the following LMI constraint:
(17)Ξ=[Ξ11Ξ12Ξ13*Ξ22Ξ23**Ξ33]<0,
where
(18)Ξ11=[-PPAeβ0PAed*Q-P-1d1R11d1R1**-Q1-1d1R1-1d2-d1R2],Ξ12=[(1-β0)PBePBe000d1(AeT-I)R11d2-d1R20d1β0AedTR1],Ξ13=[00d2-d1(AeT-I)R2CeTβ0d2-d1AedTR2β0CedT],Ξ22=|-Q2-1d2-d1R20d1(1-β0)AedTR1*-γ2Id1BeTR1**-R1|,Ξ23=[d2-d1(1-β0)AedTR2(1-β0)CedTd2-d1BeTR2DeT00],Ξ33=[-R20*-I],Q=(1+d1)Q1+(d2-d1+1)Q2,
and Ae, Aed, Be, Ce, Ced, and De are defined in (11).Proof.
First, define a Lyapunov-Krasovskii functional as follows:(19)V(k)≜V1(k)+V2(k)+V3(k)+V4(k),
where
(20)V1(k)≜xeT(k)Pxe(k),V2(k)≜∑i=k-d1(k)k-1‍xeT(i)Q1xe(i)+∑i=k-d2(k)k-1‍xeT(i)Q2xe(i),V3(k)≜∑i=-d1+2-1‍∑j=k+i-1k-1‍xeT(j)Q1xe(j)+∑i=-d2+2-d1+1‍∑j=k+i-1k-1‍xeT(j)Q2xe(j),V4(k)≜∑i=k-d1k-1‍∑j=ik-1‍δT(j)R1δ(j)+∑i=k-d2k-d1-1‍∑j=ik-1‍δT(j)R2δ(j),δ(j)≜xe(j+1)-xe(j),
and P=PT>0, Q1=Q1T>0, Q2=Q2T>0, R1=R1T>0, and R2=R2T>0 are Lyapunov matrices to be determined.
Then using the stochastic difference operator defined in (12), we obtain
(21)ΔV1(k)=[xeT(k)AeT+β0xeT(k-d1(k))AedT+(1-β0)xeT(k-d2(k))AedT+weT(k)BeT]×P[Aexe(k)+β0Aedxe(k-d1(k))+(1-β0)Aedxe(k-d2(k))+Bewe(k)]-xeT(k)Pxe(k),ΔV2(k)=xeT(k)(Q1+Q2)xe(k)-xeT(k-d1(k))×Q1xe(k-d1(k))-xeT(k-d2(k))Q2xe(k-d2(k))+∑i=k+1-d1(k+1)k-1‍xeT(i)Q1xe(i)-∑i=k-d1(k)+1k-1‍xeT(i)Q1xe(i)+∑i=k+1-d2(k+1)k-1‍xeT(i)Q2xe(i)-∑i=k-d2(k)+1k-1‍xeT(i)Q2xe(i)≤xeT(k)(Q1+Q2)xe(k)-xeT(k-d1(k))×Q1xe(k-d1(k))-xeT(k-d2(k))Q2xe(k-d2(k))+∑i=k-d1+1k‍xeT(i)Q1xe(i)+∑k-d2+1k-d1‍xeT(i)Q2xe(i),ΔV3(k)=d1xeT(k)Q1xe(k)+(d2-d1)xeT(k)Q2xe(k)-∑i=k-d1+1k‍xeT(i)Q1xe(i)-∑i=k-d2+1k-d1‍xeT(i)Q2xe(i),ΔV4(k)=d1δT(k)R1δ(k)+(d2-d1)δT(k)R2δ(k)-∑i=k-d1k-1‍δT(i)R1δ(i)-∑i=k-d2k-d1-1‍δT(i)R2δ(i)≤d1δT(k)R1δ(k)+(d2-d1)δT(k)R2δ(k)-∑i=k-d1(k)k-1‍δT(i)R1δ(i)-∑i=k-d2(k)k-d1-1‍δT(i)R2δ(i).
Using the Jensen inequality [33], the following expressions are obtained:
(22)-∑i=k-d1(k)k-1‍δT(i)R1δ(i)≤-1d1(k)(∑i=k-d1(k)k-1‍δT(i))R1(∑i=k-d1(k)k-1‍δ(i))≤-1d1(∑i=k-d1(k)k-1‍δT(i))R1(∑i=k-d1(k)k-1‍δ(i)),-∑i=k-d2(k)k-d1-1‍δT(i)R2δ(i)≤-1d2(k)-d1(∑i=k-d2(k)k-d1-1‍δT(i))R2(∑i=k-d2(k)k-d1-1‍δ(i))≤-1d2-d1(∑i=k-d2(k)k-d1-1‍δT(i))R2(∑i=k-d2(k)k-d1-1‍δ(i)).
Thus, we have
(23)ΔV4(k)≤δT(k)[d1R1+(d2-d1)R2]δ(k)-1d1[xeT(k)-xeT(k-d1(k))]×R1[xe(k)-xe(k-d1(k))]-1d2-d1[xeT(k-d1(k))-xeT(k-d2(k))]×R1[xe(k-d1(k))-xe(k-d2(k))].
Thus, we obtain
(24)ΔV(k)=ΔV1(k)+ΔV2(k)+ΔV3(k)+ΔV4(k)≤ηT(k)Υη(k),
where
(25)ηT(k)=[xeT(k)xeT(k-d1(k))xeT(k-d2(k))weT(k)],Υ=[Υ11Υ12Υ13Υ14*Υ22Υ23Υ24**Υ33Υ34***Υ44],Υ11=AeTPAe+(AeT-I)R(Ae-I)+Q-P-1d1R1,Υ12=β0AeTPAed+β0(AeT-I)RAed+1d1R1,Υ13=(1-β0)AeTPAed+(1-β0)(AeT-I)RAed,Υ14=AeTPBe+(AeT-I)RBeT,Υ22=β02AedT(P+R)Aed-Q1-1d1R1-1d2-d1R2,Υ23=β0(1-β0)AedT(P+R)Aed+1d2-d1R2,Υ24=β0AedT(P+R)Be,Υ33=(1-β0)2AedT(P+R)Aed-Q2-1d2-d1R2,Υ34=(1-β0)AedT(P+R)Be,Υ44=BeT(P+R)Be,R=d1R1+(d2-d1)R2.
By Schur complement, it can be concluded from (17) that Υ<0. By similar lines as in [32], the stochastic stability can be guaranteed if condition (17) holds.
Then, define the performance index as follows:(26)J=∑k=0∞‍[eT(k)e(k)-γ2weT(k)we(k)].
Considering the fact that V(k)≥0, under the zero initial condition, we have
(27)J≤∑k=0∞‍[eT(k)e(k)-γ2weT(k)we(k)]+V(∞)-V(0)=∑k=0∞‍[eT(k)e(k)-γ2weT(k)we(k)+ΔV(k)].
Thus, J<0 is equal to
(28)ηT(k)(Θ+Υ)η(k)<0,
where(29)Θ=[CeTCeβ0CeTCed(1-β0)CeTCedCeTDe*β02CedTCedβ0(1-β0)CedTCedβ0CedTDe**(1-β0)2CedTCed(1-β0)CedTDe***DeTDe-γ2I].Through applying Schur complement, it is shown that (Θ+Υ)<0 can be guaranteed by condition (17). That is to say, once (17) is satisfied, the ℋ∞ performance can be guaranteed to be less than γ. Thus, the proof is completed.
At this point, the second desired property of the system will be considered, which is the envelope constraints demand. First, some notations are introduced [34]:
(30)y=[y(0)y(1)⋮y(m)],l=[l(0)l(1)⋮l(n)],u=[u(0)u(1)⋮u(n)],f=[f(0)f(1)⋮f(nf)],Y=[y(0)0⋯0y(1)y(0)⋯0⋮y(1)⋯⋮y(m)⋮y(0)0y(m)⋮y(1)⋮⋮⋮00⋯y(m)],
where Y is an n×(nf+1) matrix, n=m+nf+1,
(31){y(0)y(1)⋯y(m)00⋯}
is a given signal, and
(32){l(0)l(1)⋯l(m)},{u(0)u(1)⋯u(m)}
Are, respectively, the upper and lower bounds. Therefore, the constraint of (15) is equal to
(33)diag(l)≤diag(Yf)≤diag(u),
where diag(•) denotes a conversion from a vertical vector to a diagonal matrix.
Based on Theorem7 and (33), we can establish another theorem to determine the filter that satisfies the envelope constraint meanwhile possessing optimal ℋ∞ performance.Theorem 8.
Anℋ∞ optimal filter of the form (7)-(8) satisfying envelope constraint in (15) can be obtained by solving the following LMI optimization problem:
(34)minP>0,Q1>0,Q2>0,R1>0,R2>0,fγ,
subject to
(35)Ξ=[Ξ11Ξ12Ξ13*Ξ22Ξ23**Ξ33]<0,diag(l)≤diag(Yf),diag(Yf)≤diag(u),
where Ξ is defined in (17).


## 4. An Illustrative Example

In this section, an example is given to support the filter design method proposed in the paper. Consider a filtering system as shown in Figure1. The parameters for Σl are given by (36)Al=[-2.3060-2.9625-2.2590-1.0922-0.3009-0.0325100000010000001000000100000010],Bl=[100000],Cl=[00000.00620.2170].The parameters for the delay channelΣc are given by
(37)Ac=[010-0.1],Ad=[00.1],Bc=[0.10.1],Cc=[01],Ccd=0.2,Dc=1,d1=2,d2=3,β0=0.7.
Using Theorem 8, the ℋ∞ optimal filter is obtained via using the LMI toolbox of MATLAB with nf chosen to be 5. The resultant optimal γ is 8.5057 and filter gains are given as follows:
(38)Cf=[0.0437-0.3344-0.40230.2045-0.5089],Df=4.3664.The expected envelope constraints ands(k) (the output of Σl) corresponding to a particular case where input signal w(k) is chosen to be unit impulse signal are shown in Figure 2. The transmitted signal y(k) through Σc which is generated with no noise added is also given in the figure. The filter output s^(k) and filtering error e(k) are given in Figures 3 and 4, respectively.Figure 2
Source signals(k), filter input signal y(k), and envelope bounds.Figure 3
Output of the filter without disturbance.Figure 4
Filtering errore(k) without disturbance.Furthermore, to illustrate the performance of the designed filter, we add the disturbance signalv(k) chosen as white noise with mean of zero and variance of 1×10-3 into the system. The resultant filter output and filtering error are shown in Figures 5 and 6, respectively. It is shown that the designed filter is effective.Figure 5
Output of the filter with disturbance.Figure 6
Filtering errore(k) with disturbance.


## 5. Conclusions

In this paper, we have solved the filtering design problem of time-delay system. The time-delay considered here is time-varying meanwhile with a certain stochastic characteristic, and the probability of delay distribution is assumed to be known. Furthermore, the envelope constraints are also considered in the process of filtering design. The delay-distribution-dependent criterion is formed for the filtering error system, employing the information about not only the size of delay but also its probability distribution. A set of linear matrix inequalities (LMIs) are formulated to solve the problem. Through solving the LMI optimization problem, theℋ∞ performance is minimized and pulse-shape demand imposed by envelope constraints is satisfied. Finally, an illustrative example is presented to demonstrate the effectiveness of the filtering design approach. For future research directions, extending the filter design approach proposed in this paper to networked control systems and distributed systems is an interesting issue. Besides, more general filter design approaches considering delays in different forms with different characteristics also deserve further investigation.



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