# Periodic Switched Control of Dual-Rate Sampled-Data Systems

Authors: Dawei Zhang; Wei Bai; Xinchun Jia
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper is concerned with the periodic switched control of a linear dual-rate sampled-data system. The state
variables of the continuous-time plant are sampled by two types of sensors. The ratio of two sampling rates is assumed
to be a rational number. Depending on whether the sampled-data of state variables at two sampling rates is available
simultaneously or separately, a periodic switched controller is constructed. Applying an input delay approach, the
closed-loop system is modeled as a switched system with subsystems having different input delays. Some delay-dependent
criteria for theH∞ performance of the switched system and the existence of the switched controller are
derived by employing a Lyapunov-Krasovskii functional that includes information about two sampling periods. The
dual-rate sampled-data control of a vehicle dynamic system is given to show that the proposed method is effective and
it can achieve a better H∞ control performance than the single-rate design method.


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## Body

## 1. Introduction

Sampled-data control of continuous-time practical systems, especially complex industrial systems, offers several advantages such as flexibility, low cost, and increased reliability [1, 2]. In a sampled-data control system, where a continuous-time plant is controlled with a digital controller, the sampling rate is a critical design parameter. The choice of the sampling rate mainly depends on some factors like bandwidth and response time of closed-loop systems, physical limitations of sensors and actuators, and the effect of the noise. The sampling rate must be chosen as fast as possible to ensure the high precision and the fast response time of the system but as slow as possible to satisfy the hardware limits and to eliminate the effect of the noise on the control input. Taking these factors into account, the effective range of the sampling rate is determined.In many industrial applications, it is impractical to sample all physical signals uniformly at one single-rate, which demands a multirate sampling scheme. For instance, for an industrial vehicle, laser sensors are used to measure the heave position and the heave velocity, and gyrometers are chosen to measure the angular velocity and the heading angle [3]. Due to the sensor restrictions and the control performance requirement, it is often necessary to sample the signals for different types of sensors at different sampling rates. The multirate sampling technique has received much attention since the early 1950s. Compared with the single-rate sampling scheme, the use of the multirate sampling technique is of two main benefits: (i) it may improve the performance such as improving the transient system behavior and enhancing the disturbance rejection property and (ii) it can provide a better tradeoff between the system performance and the implementation cost, which can be achieved by using analog-to-digital converters and digital-to-analog converters at different rates. Motivated by these benefits, much work has been done to deal with system modeling and identification, stability analysis, and controller synthesis of multirate sampled-data systems in the past few decades [3–12]. For example, in [4], a general framework of a multirate sampled-data control system is presented using nest operators and nest algebras, and an H∞ suboptimal controller satisfying causality constraint is designed by the lifting technique, in which the outputs yi(t)(i=1,2,…,p) and the control inputs ui(t)(i=1,2,…,q) are paired with sampling periods hs=mih and holding periods hu=nih, respectively, where mi∈N+(i=1,2,…,p), ni∈N+(i=1,2,…,q), N+ is the set of positive integers, and h is the base sampling period. Based on this framework, some particular cases of the multirate sampled-data systems are considered in [5–15]. More specifically, when the multirate sampled-data system involves a fast sampling rate 1/hs and a slow control input rate 1/hu (i.e., n/hu=1/hs(n∈N+)), a new multirate sampling method for acceleration control is proposed in [9]; on the other hand, when the multirate sampled-data system involves a slow sampling rate 1/hs and a fast control input rate 1/hu (i.e., n/hs=1/hu), some estimation and/or control problems for several practical systems such as polymer reactors [10], visual servo control systems [11, 12], read-write arm of the hard disk drive [13, 14], and a pilot plant [15] are addressed. It should be mentioned that for the multirate sampling scheme with hu>hs, in [9], the control signal is calculated by output measurement at each sampling rate 1/hs, but only the one produced at the input rate 1/hu is implemented. Although such a multirate scheme is effective in the realization of acceleration control in wide bandwidth, some of the output measurements may not be used to update the control actuation in time; for the multirate sampling scheme with hu<hs in [3, 5–8, 10–15], the control signal is calculated recursively at each input rate 1/hu only using the available data at each sampling rate 1/hs. Moreover, using the lifting technique, the multirate sampled-data system is converted into an equivalent discrete-time single-rate time-invariant system in [3, 5–8, 10–15]. However, the equivalent conversion is not readily applicable to the continuous-time systems with polytopic uncertainties [16–18]. In [16], an input delay approach is proposed to investigate the sampled-data stabilization of linear systems, which can be extended to deal with the sampled-data control for systems with polytopic uncertainties and the networked control systems [19–24]. Most of the existing results developed by using the input delay approach, such as [16–18], have been largely focused on the sampled-data control of single-rate sampled-data systems. However, there are few results available on the dual-rate or multirate sampled-data control of a continuous-time system using the input delay approach except [20, 25], which provides the main motivation of the current study. In [20], exponential stability and the induced L2-gain of networked control systems are investigated, in which the sampled-data via dual-rate samplings are transmitted one after another by introducing a Round-Robin scheduling protocol. In [25], multirate sampled-data systems are modeled as systems with multiple input delays by reordering the updating instants, and some stability and stabilization conditions are established in terms of linear matrix inequalities. Without reordering sensor instants [20] or updating instants [25], this paper attempts to apply the input delay approach for a dual-rate sampled-data control system.In this paper, we apply the input delay approach to investigate the periodic switched control of a linear continuous-time system system with two different sampling rates1/h1 and 1/h2, where the sampling periods h1 and h2 satisfy h1<h2 and l1h1=l2h2, with li(i=1,2) being two positive integers, lihi being the unique basic time period, and {l1,l2} having no common factors greater than unity. Once the sampled-data of the state variable x1(kh1) or x2(kh2) is available, the control input is computed to update the system. Depending on whether the sampled-data of state variables at two sampling rates are available simultaneously or separately, a periodic switched controller with three switching modes is constructed to implement the sampled-data control. Using such a controller and the input delay approach, the resulting closed-loop system is modeled as a switched system with subsystems that have different input delays. A Lyapunov-Krasovskii functional that involves information about two sampling periods is constructed to derive some delay-dependent criteria for the H∞ performance of the switched system and the existence of the periodic switched controller. Comparing with the existing results for multirate sampled-data systems based on lifting technique [3–15], the proposed results can be trivially extended to handle the multirate sampled-data systems or networked control systems with polytopic uncertainties. The effectiveness of the proposed method and its advantage over a single-rate sampled-data control is shown by performing the dual-rate sampled-data control of a vehicle system.Notation. The superscript “T” stands for the transposition of a vector or a matrix. Rn is the n dimensional Euclidean space. N is the set of nonnegative integers and N+ is the set of positive integers. For symmetric matrices P and Q, P≤Q (resp., P<Q) means that P-Q is negative semidefinite matrix (resp., negative definite matrix). λmax(P) is the maximum eigenvalue of a symmetric matrix P. We use an asterisk “∗” to denote a term induced by symmetry and diag{⋯} to denote the block-diagonal matrix. The space of square-integrable vector functions over [t0,∞) is denoted by L2[t0,∞).


## 2. Modeling of a Dual-Rate Sampled-Data System with a Switched Controller

Consider the linear system described by(1)x˙t=Axt+But+Eωt,zt=Cxt+Dωt,where x(t)∈Rn, u(t)∈Rm, and z(t)∈Rp are the state, the control input, and the controlled output, respectively; ω(t)∈Rq is the external disturbance acting on system (1) and ω(t)∈L2[t0,∞); x(t0)=x0 is the initial state; A, B, C, D, and E are constant matrices of appropriate dimensions. It is assumed that all state variables of system (1) are sampled by two different types of sensors. Let xt=[x1T(t)x2T(t)]T, where x1(t)=[x1,1T(t)x1,2T(t)⋯x1,p1T(t)]T∈Rp1, x2(t)=[x2,1T(t)x2,2T(t)⋯x2,p2T(t)]T∈Rp2, and p1+p2=n. Without loss of generality, we assume that x1(t) and x2(t) are paired with two sampling periods h1 and h2, respectively, and l1h1=l2h2(h1≤h2), where li∈N+(i=1,2) and l1≥l2. Then the sequences of sampled-data of the state variables x1(t) and x2(t) are {x1(kh1):k∈N} and {x2(kh2):k∈N}.In the proposed dual-rate sampling scheme, oncex1(kh1) or x2(kh2) is available, the control signal is computed immediately for input update. By taking full advantage of x1(kh1) and x2(kh2) (k∈N) in a real-time way, we construct the following switched controller, which is shown in Figure 1:(2)ut+=F1,σtx1kh1+F2,σtx2kh2,∀k∈N,where Fσ(t)=[F1σ(t)F2σ(t)] are the control gain matrices to be determined and σ(t):[0,∞)→S={1,2,3} is the switching signal. Notice that there are three cases of available sampled-data of the state variables for control computation of the controller (2): (i) both x1(kh1) and x2(kh2) are available, (ii) only x1(kh1) is available, and (iii) only x2(kh2) is available. Define the switching rules as follows. Set σ(t)=1 when x1(kh1) and x2(kh2) are available simultaneously, σ(t)=2 when only x1(kh1) is available, and σ(t)=3 when only x2(kh2) is available, respectively.Figure 1
The multirate sampled-data system architecture using a digital switched controller.Due to the fact thatl1h1=l2h2, the interval of the control input is [kl1h1,(k+1)l1h1) or equally [kl2h2,(k+1)l2h2)(k∈N). Define mi=max{k∣kh1-ih2<0,0<k<l1,k∈N}(i=1,2,…,l2) and m0=0. Based on the switching rules of controller (2), we propose the following interval partition:(3)kl1h1,k+1l1h1=I1∪I2∪⋯∪Imν-1+ν∪Imν-1+ν+1∪⋯∪Imν+ν∪⋯∪Iml2-1+l2∪⋯∪Iml2+l2,where (4)Imν-1+ν=kl1h1+ν-1h2,kl1+mν-1+1h1,Imν-1+ν+1=kl1+mν-1+1h1,kl1+mν-1+2h1,⋮,Imν+ν=kl1+mνh1,kl1h1+νh2,ν=1,2,…,l2.To clearly show the partition process, a special case for the interval partition of [kl1h1,(k+1)l1h1) with l1=8 and l2=3 is depicted by Figure 2.Figure 2
A special case for the interval partition of[kl1h1,(k+1)l1h1) with l1=8 and l2=3.Considering interval partition (3) and the availability of x1(kh1) and x2(kh2), one can see that switching controller (2) is periodically activated on the time instants {kl1h1+(mi-1+1)h1,kl1h1+(i-1)h2,i=1,…,l2}k=0∞. Then the control input can be described by(5)ut=F11x1kl1h1+F21x2kl1h1,t∈I1,σt=1F12x1kl1+1h1+F22x2kl1h1,t∈I2,σt=2⋮F12x1kl1+m1h1+F22x2kl1h1,t∈Im1+1,σt=2F13x1kl1+m1h1+F23x2kl1h1+h2,t∈Im1+2,σt=3F12x1kl1+m1+1h1+F22x2kl1h1+h2,t∈Im1+3,σt=2⋮F12x1kl1+mνh1+F22x2kl1h1+ν-1h2,t∈Imν+ν,σt=2F13x1kl1+mνh1+F23x2kl1h1+νh2,t∈Imν+ν+1,σt=3F12x1kl1+mν+1h1+F22x2kl1h1+ν-1h2,t∈Imν+ν+2,σt=2⋮F12x1kl1+ml2h1+F22x2kl1h1+l2-1h2,t∈Iml2+l2,σt=2.Define the following input delays on the subintervals in (3):(6)t∈Imν-1+ν:τmν-1+ν1t=t-kl1+mν-1h1,τmν-1+ν2t=t-kl1h1-ν-1h2t∈Imν-1+ν+1:τmν-1+ν+11t=t-kl1+mν-1+1h1,τmν-1+ν+12t=t-kl1h1-ν-1h2⋮t∈Imν+ν:τmν+ν1t=t-kl1+mνh1,τmν+ν2t=t-kl1h1-ν-1h2,where ν=1,2,…,l2. It follows that(7)σt=1:0≤τ1t<h1,t∈I1σt=2:0≤τi1t<h1,0≤τi1t<τi2t<h2,t∈Ii,i=mν-1+ν+1,…,mν+ν,ν=1,2,…,l2σt=3:0≤τmν-1+ν2t<τmν-1+ν1t<h1,t∈Imν-1+ν,ν=2,…,l2.Using (1), (5), and (6), the resulting closed-loop system can be described by(8)Σ11:x˙t=Axt+BF1xt-τ1t+Eωt,t∈I1,σt=1,Σ2ν2:x˙t=Axt+BF2Ip1xt-τi1t+BF2Ip2xt-τi2t+Eωt,t∈⋃mν-1+ν+1mν+νIi,i=mν-1+ν+1,…,mν+ν,ν=1,2,…,l2,σt=2,Σ2ν-13:x˙t=Axt+BF3Ip1xt-τmν-1+ν1t+BF3Ip2xt-τmν-1+ν2t+Eωt,t∈Imν-1+ν,ν=2,…,l2,σt=3,where Ip1=diag{Ip1×p1,0},Ip2=diag{0,Ip2×p2} and Ip1×p1 and Ip2×p2 denote p1×p1 and p2×p2 identity matrices, respectively.The purpose of this paper is to investigate the periodic switched control such that system (8) is exponentially stable with a prescribed H∞ performance, which means that(i)
system (8) with ω(t)=0 is exponentially stable; that is, there exist constants β>0 and λ>0 such that xt2≤βx0c1e-λt for t≥0, where xtc1=sup-h2≤s≤0{x(t+s),x˙(t+s)} [23, 26];(ii)
∫0∞‍e-αszT(s)z(s)ds≤γ2∫0∞‍ωT(s)ω(s)ds can be ensured for all nonzero ω(t)∈L2[0,∞) under the zero initial condition x(t)=ϕ(t)=0(t∈[-h2,0]), where α and γ are positive constants.To end this section, we introduce the following lemma.Lemma 1.
For any constant matricesR∈Rn×n, Si∈Rn×n (i=1,2), R1=RS1TS1R≥0, R2=RS2TS2R≥0, and scalars τi, τi(t) (i=1,2) satisfying 0≤τ1≤τ1(t)<τ2(t)≤τ2 and a vector function x˙:[-τ2,-τ1]→Rn such that the integration concerned is well defined, the following inequality holds:(9)-τ2-τ1∫t-τ2t-τ1x˙TsRx˙sds≤-∑i=17Ei,where (10)E1=e3TtRe3t,E2=e2TtRe2t,E3=e1TtRe1t,E4=e1TtS1Te2t,E5=e2TtS1e1t,E6=e2TtS2Te3t,E7=e3TtS2e2t,e1t=xt-τ1-xt-τ1t,e2t=xt-τ1t-xt-τ2t,e3t=xt-τ2t-xt-τ2.Proof.
Ifτ1(t)≡τ1 (resp., τ2(t)≡τ2), inequality (9) reduces to be the one in Lemma 1 [27]. If 0≤τ1≤τ1(t)<τ2(t)≤τ2, using Jensen inequality to obtain(11)-τ2-τ1∫t-τ2t-τ1x˙TsRx˙sds≤-1+α1α2E1-1+α2α1+α4α3E2-1+α3α4E3,where (12)α1=τ2t-τ1τ2-τ1,α2=τ2-τ2tτ2-τ1,α3=τ2-τ1tτ2-τ1,α4=τ1t-τ1τ2-τ1,for any Ri≥0(i=1,2), the following inequalities hold:(13)α3α4e1t-α4α3e2tTR1α3α4e1t-α4α3e2t≥0,α2α1e2t-α1α2e3tTR2α2α1e2t-α1α2e3t≥0.It follows from (13) that(14)α3α4E3+α4α3E2≥E4+E5,(15)α2α1E2+α1α2E1≥E6+E7.Combining (14) and (15), we obtain inequality (9). This completes the proof.


## 3.H∞ Performance Analysis and Periodic Switched Controller Design

In this section, we first derive a delay-dependent criterion such that system (8) is exponentially stable with prescribed H∞ performance by using the Lyapunov-Krasovskii functional method. Based on the derived performance criterion, we establish the delay-dependent criterion on the existence of a switched controller for system (8).Proposition 2.
Givenγ>0, h2>h1>0, l1>l2>0, 0≤a1≤a2≤a3≤((l2-2)a1+(l2+3)a2)/(l2+1), μ≥1 satisfying ln⁡(μ)<[(a1(l2-2)+a2(l2+3)-a3(l2+1))/l2](h2/2), and the gain matrices Fi(i=1,2,3), system (8) is exponentially stable with a given H∞ performance γ for any switching signal with ln(μ)<[(a1(l2-2)+a2(l2+3)-a3(l2+1))/l2](h2/2), if there exist matrices S1, S2, Si,1, and Si,2(i=2,3) and symmetric matrices Pj>0, Qij>0, and Rij>0(i=1,2;j=1,2,3) such that P2≤μP1, P1≤μP2, P2≤μP3, P3≤μP2, Qi2≤μQi1, Qi1≤μQi2, Qi2≤μQi3, Qi3≤μQi2, Ri2≤μRi1, Ri1≤μRi2, Ri2≤μRi3, Ri3≤μRi2(i=1,2), and (16)R11S1TS1R11≥0,(17)h1R12-h12R22S2TS2h1R12-h12R22≥0,(18)R1iSi,1TSi,1R1i≥0,i=2,3(19)R1iSi,2TSi,2R1i≥0,i=2,3(20)Ξ1<0,t∈I1,(21)Ξi1<0,t∈Ii,i=mν-1+ν+1,…,mν+ν,ν=1,2,…,l2,(22)Ξi2<0,t∈Ii,i=mν-1+ν,ν=2,…,l2,where (23)Ξ1=Ξ111Ξ121Ξ1310P1E+CTD∗Ξ221Ξ23100∗∗Ξ331Ξ3410∗∗∗Ξ4410∗∗∗∗-γ2I+DTD+Ψ1Th12R11+h122R21Ψ1,h12=h2-h1,Ξ111=ATP1+P1A+a1P1+Q11-σ11R11+CTC,Ξ121=P1BF1+σ11R11-S1T,Ξ131=σ11S1T,Ξ221=-σ112R11-S1T-S1,Ξ231=σ11R11-S1T,Ξ331=-σ11R11-σ21R21+σ11Q21-Q11,Ξ341=σ21R21,Ξ441=-σ21R21+Q21,Ψ1=ABF100E,σ11=e-a1h1,σ21=e-a1h2,Ξi1=Ξ11i1Ξ12i1Ξ13i1Ξ14i10Ξ16i1∗Ξ22i1Ξ23i1Ξ24i1Ξ25i10∗∗Ξ33i10Ξ35i10∗∗∗Ξ44i100∗∗∗∗Ξ55i10∗∗∗∗∗Ξ66i1+Ψ2Th12R12+h122R22Ψ2,Ξ11i1=ATP2+P2A+a2P2+Q12+CTC-σ22R22-σ12h1R12-h12R22,Ξ12i1=P2BF12+σ12h1R12-h12R22-S2T+σ22R22-S2,1T,Ξ13i1=P2BF22+σ22S2,1T,Ξ14i1=σ12S2T,Ξ22i1=σ12-2h1R12+2h12R22+S2T+S2+σ22-2R22+S2,1T+S2,1,Ξ16i1=P2E+CTD,Ξ23i1=σ22R22-S2,1T-S2,2T,Ξ24i1=σ12h1R12-h12R22-S2T,Ξ25i1=σ22S2,2T,Ξ33i1=σ22-2R22+S2,2T+S2,2,Ξ35i1=σ22R22-S2,2T,Ξ44i1=σ12-h1R12+h12R22+e-a2h1Q22-Q12,Ξ55i1=-σ22R22-e-a2h2Q22,Ξ66i1=-γ2I+DTD,Ψ2=ABF12BF2200E,σ12=e-a2h1h1,σ22=e-a2h2h12h2,Ξi2=Ξ11i2Ξ12i2Ξ13i200Ξ16i2∗Ξ22i2Ξ23i2Ξ24i200∗∗Ξ33i2Ξ34i200∗∗∗Ξ44i2Ξ45i20∗∗∗∗Ξ55i20∗∗∗∗∗Ξ66i2+Ψ3Th12R13+h122R23Ψ3,Ξ11i2=ATP3+P3A+a3P3+Q13+CTC-σ13R13,Ξ12i2=P3BF13+σ13S3,1T,Ξ13i2=P3BF23+σ13R13-S3,1T,Ξ16i2=E+P¯3CTD,Ξ22i2=-σ132R13-S3,2T-S3,2,Ξ23i2=σ13R13-S3,1-S3,2,Ξ24i2=σ13R13-S3,2T,Ξ33i2=σ13-2R13+S3,1T+S3,1,Ξ34i2=σ13S3,2T,Ξ44i2=-σ13R13-σ23R23+σ13Q23-Q13,Ξ45i2=σ23R23,Ξ55i2=-σ23R23+Q23,Ξ66i2=-γ2I+DTD,Ψ3=ABF13BF2300E,σ13=e-a3h1,σ23=e-a3h2.Moreover, the parameters in the exponential stability and H∞ performance are given by (24)α=a1l2-2+a2l2+3-a3l2+1l2,β=ba1/2,λ=a1l2-2+a2l2+3-a3l2+12l2-ln⁡μh2with (25)a=min⁡λmin⁡P1,λmin⁡P2,λmin⁡P3,b=λmax⁡P1+1-e-a1h2a1bq+h11-e-a1h1a12br+h12e-a1h1-e-a1h2a12br,bq=max⁡λmax⁡Q11,λmax⁡Q21,br=max⁡λmax⁡R11,λmax⁡R21.Proof.
See AppendixA.We now state and establish the delay-dependent criterion for the switched controller design.Proposition 3.
Givenγ>0, h2>h1>0, l1>l2>0, v>0, 0≤a1≤a2≤a3≤((l2-2)a1+(l2+3)a2)/(l2+1), and μ≥1 satisfying ln(μ)<[(a1(l2-2)+a2(l2+3)-a3(l2+1))/l2](h2/2), system (8) is exponentially stable with a given H∞ performance γ for any switching signal with ln(μ)<[(a1(l2-2)+a2(l2+3)-a3(l2+1))/l2](h2/2), if there exist matrices S¯1, S¯2, S¯i,1, S¯i,2(i=2,3), F¯1, F¯12, F¯22, F¯13, and F¯23 and symmetric matrices P¯j>0, Q¯ij>0, and R¯ij>0(i=1,2;j=1,2,3) such that P¯2≤μP¯1, P¯1≤μP¯2, P¯2≤μP¯3, P¯3≤μP¯2, Q¯i2≤μQ¯i1, Q¯i1≤μQ¯i2, Q¯i2≤μQ¯i3, Q¯i3≤μQ¯i2, R¯i2≤μR¯i1, R¯i1≤μR¯i2, R¯i2≤μR¯i3, R¯i3≤μR¯i2(i=1,2), and(26)R¯11S¯1TS¯1R¯11≥0,(27)h1R¯12-h12R¯22S¯2TS¯2h1R¯12-h12R¯22≥0,(28)R¯1iS¯i,1TS¯i,1R¯1i≥0,i=2,3,(29)R¯1iS¯i,2TS¯i,2R¯1i≥0,i=2,3,(30)Ξ¯1=Π¯111Π¯121∗Π¯221<0,t∈I1,(31)Ξ¯i1=Π¯11i1Π¯12i1∗Π¯22i1<0,t∈Ii,i=mν-1+ν+1,…,mν+ν,ν=1,2,…,l2,(32)Ξ¯i2=Π¯11i2Π¯12i2∗Π¯22i2<0,t∈Ii,i=mν-1+ν,ν=2,…,l2,where (33)Π¯111=Ξ¯111Ξ¯121Ξ¯1310E+P¯1CTD∗Ξ¯221Ξ¯23100∗∗Ξ¯331Ξ¯3410∗∗∗Ξ¯4410∗∗∗∗-γ2I+DTD,Π¯121=h1P¯1ATh12P¯1ATP¯1CTh1F¯1TBTh12F¯1TBT0000000h1ETh12ET0,Π¯221=v2R¯11-2vP¯100∗v2R¯21-2vP¯10∗∗-I,Ξ¯111=P¯1AT+AP¯1+a1P¯1+Q¯11-σ11R¯11,Ξ¯121=BF¯1+σ11R¯11-S¯1T,Ξ¯131=σ11S¯1T,Ξ¯221=-σ112R¯11-S¯1T-S¯1,Ξ¯231=σ11R¯11-S¯1T,Ξ¯331=-σ11R¯11-σ21R¯21+σ11Q¯21-Q¯11,Ξ¯341=σ21R¯21,Ξ¯441=-σ21R¯21+Q¯21,σ11=e-a1h1,σ21=e-a1h2,Π¯11i1=Ξ¯11i1Ξ¯12i1Ξ¯13i1Ξ¯14i10Ξ¯16i1∗Ξ¯22i1Ξ¯23i1Ξ¯24i1Ξ¯25i10∗∗Ξ¯33i10Ξ¯35i10∗∗∗Ξ¯44i100∗∗∗∗Ξ¯55i10∗∗∗∗∗Ξ¯66i1,Π¯12i1=h1P¯2ATh12P¯2ATP¯2CTh1F¯12TBTh12F¯12TBT0h1F¯22TBTh12F¯22TBT0000000h1ETh12ET0,Π¯22i1=v2R¯12-2vP¯200∗v2R¯22-2vP¯20∗∗-I,Ξ¯11i1=P¯2AT+AP¯2+a2P¯2+Q¯12-σ22R¯22-σ12h1R¯12-h12R¯22,h12=h2-h1,Ξ¯12i1=BF¯12+σ12h1R¯12-h12R¯22-S¯2T+σ22R¯22-S¯2,1T,Ξ¯13i1=BF¯22+σ22S¯2,1T,Ξ¯14i1=σ12S¯2T,Ξ¯16i1=E+P¯2CTD,Ξ¯22i1=σ12-2h1R¯12+2h12R¯22+S¯2T+S¯2+σ22-2R¯22+S¯2,1T+S¯2,1,Ξ¯23i1=σ22R¯22-S¯2,1T-S¯2,2T,Ξ¯24i1=σ12h1R¯12-h12R¯22-S¯2T,Ξ¯25i1=σ22S¯2,2T,Ξ¯33i1=σ22-2R¯22+S¯2,2T+S¯2,2,Ξ¯35i1=σ22R¯22-S¯2,2T,Ξ¯44i1=σ12-h1R¯12+h12R¯22+e-a2h1Q¯22-Q¯12,Ξ¯55i1=-σ22R¯22-e-a2h2Q¯22,Ξ¯66i1=-γ2I+DTD,σ12=e-a2h1h1,σ22=e-a2h2h12h2,Π¯11i2=Ξ¯11i2Ξ¯12i2Ξ¯13i200Ξ¯16i2∗Ξ¯22i2Ξ¯23i2Ξ¯24i200∗∗Ξ¯33i2Ξ¯34i200∗∗∗Ξ¯44i2Ξ¯45i20∗∗∗∗Ξ¯55i20∗∗∗∗∗Ξ¯66i2,Π¯12i2=h1P¯3ATh12P¯3ATP¯3CTh1F¯13TBTh12F¯13TBT0h1F¯23TBTh12F¯23TBT0000000h1ETh12ET0,Π¯22i2=v2R¯13-2vP¯300∗v2R¯23-2vP¯30∗∗-I,Ξ¯11i2=P¯3AT+AP¯3+a3P¯3+Q¯13-σ13R¯13,Ξ¯12i2=BF¯13+σ13S¯3,1T,Ξ¯13i2=BF¯23+σ13R¯13-S¯3,1T,Ξ¯16i2=E+P¯3CTD,Ξ¯22i2=-σ132R¯13-S¯3,2T-S¯3,2,Ξ¯23i2=σ13R¯13-S¯3,1-S¯3,2,Ξ¯24i2=σ13R¯13-S¯3,2T,Ξ¯33i2=σ13-2R¯13+S¯3,1T+S¯3,1,Ξ¯34i2=σ13S¯3,2T,Ξ¯44i2=-σ13R¯13-σ23R¯23+σ13Q¯23-Q¯13,Ξ¯45i2=σ23R¯23,Ξ¯55i2=-σ23R¯23+Q¯23,Ξ¯66i2=-γ2I+DTD,σ13=e-a3h1,σ23=e-a3h2.Moreover, the control gains of controller (2) are obtained by F1=F¯1P¯1-1, F2=F¯2P¯2-1, and F3=F¯3P¯3-1, where F¯2=F¯12+F¯22 and F¯3=F¯13+F¯23. The parameters in the exponential stability and H∞ performance are given by (34)α=a1l2-2+a2l2+3-a3l2+1l2,β=b¯a¯1/2,λ=a1l2-2+a2l2+3-a3l2+12l2-ln⁡μh2with (35)a¯=min⁡λmin⁡P¯1-1,λmin⁡P¯2-1,λmin⁡P¯3-1,b¯=λmax⁡P¯1-1+1-e-a1h2a1b¯q+h11-e-a1h1a12b¯r+h12e-a1h1-e-a1h2a12b¯r,b¯q=max⁡λmax⁡P¯1-1Q¯11P¯1-1,λmax⁡P¯1-1Q¯21P¯1-1,b¯r=max⁡λmax⁡P¯1-1R¯11P¯1-1,λmax⁡P¯1-1R¯21P¯1-1.Proof.
See AppendixB.Remark 4.
Suppose that the system matricesΩ:=[AB] are not exactly known and they reside in the uncertain polytope [16, 17, 26]. Consider (36)Ω∈∑i=1MμiΩi,0≤μi≤1,∑i=1Mμi=1,where the M vertices of the polytope are described by Ωi≔[AiBi]. For the H∞ performance analysis and the periodic switched controller design of the dual-rate system with polytopic uncertainties, one can solve the LMIs in Propositions 2 and 3 for all the M vertices Ωi by applying the same decision matrices, respectively.For comparison purpose, a criterion for the existence of a sampled-data controller for system (1) via single-rate sampling is developed and given by the following corollary.Corollary 5.
Givenh>0,a>0, and v>0, system (1) with a single-rate sampled-data controller is exponentially stable with a given H∞ performance γ, where the exponential decay rate α=a, if there exist a matrix S¯ and symmetric matrices P¯>0,Q¯>0, and R¯>0 such that(37)Ξ¯=Ξ¯11Ξ¯12σS¯TΞ¯14hP¯ATP¯CT∗Ξ¯22Ξ¯230hF¯TBT0∗∗Ξ¯33000∗∗∗Ξ¯44hET0∗∗∗∗Ξ¯550∗∗∗∗∗-I<0,where (38)Ξ¯11=P¯AT+AP¯+aP¯+Q¯-σR¯,Ξ¯12=BF¯+σR¯-S¯T,Ξ¯14=E+P¯CTD,Ξ¯22=σ-2R¯+S¯T+S¯,Ξ¯23=σR¯-S¯T,Ξ¯33=-σR¯+Q¯,Ξ¯44=-γ2I+DTD,Ξ¯55=v2R¯-2vP¯,σ=e-ah.Moreover, the control gain is obtained as F=F¯P¯-1.Remark 6.
For the casel1=Nl2 (N>1 and N∈N), periodic switched controller (2) reduces to a switched controller with switching modes σ(t)=1 and σ(t)=2. Correspondingly, Proposition 3 with P¯3=0, Q¯i3=0, S¯3,i=0, and R¯i3=0 (i=1,2) reduces to a sampled-data H∞ control design result for system (1) via two sampling rates 1/h1 and 1/Nh1. Particularly, when N=2, the design result can be reduced to the one in [28].Remark 7.
In this paper, we obtain some results on periodic switched control of dual-rate sampled-data systems. It should be pointed out that these results can be extended to multirate sampled-data systems although the process is very tedious. The corresponding results are omitted.Remark 8.
As shown in [20, 29], analysis and synthesis of research on networked control systems with multiple samplings are promising and significant issues. Compared with [20], this paper is distinguished based on the following evidences: (i) The dual-rate sampling schemes in this paper and [20] are different, resulting in two essentially different switched system models. In [20], by introducing a Round-Robin scheduling protocol, the sampled-data via two different sampling rates is available one after another. In this paper, however, the sampled-data is available in three cases (both x1(kh1) and x2(kh2), only x1(kh1), and/or only x2(kh2)). In other words, under the sampling relationship l1h1=l2h2 in this paper, the case that x1(kh1) and x2(kh2) are available simultaneously must occur, but this surely does not occur in [20]. (ii) The problems addressed in this paper and [20] are different. Stability and L2-gain analysis carried out for dual-rate networked control systems is considered in [20], while this paper focuses on controller design for a dual-rate sampled-data system without considering the effect of network-induced delays. It should be mentioned that the constant constraint on network-induced delays in [20] does not significantly change the nature of the proposed results. (iii) The methodologies of performance analysis in this paper and [20] are different. In [20], a common discontinuous Lyapunov-Krasovskii functional is constructed to derive some delay-dependent stability and L2-gain criteria. However, in this paper, a switched continuous Lyapunov-Krasovskii functional is proposed to establish delay-dependent criteria for the H∞ performance and switched controller design by fully utilizing the relation of multiple input delays and the information about two sampling periods.


## 4. The Dual-Rate Sampled-Data Control of a Vehicle Dynamic System

In this section, we consider the dual-rate sampled-data control of a vehicle dynamic system described by the following state-space representation [30]:(39)x˙t=Axt+But+Eωt,zt=Cxt+Dωt,where (40)A=01022.30-4.2796-19.4355001.4391-4.274300010,B=025.065517.75480,C=0100,D=0.1,E=00.10.10T,x(t)=[yϕrψ]T, y is the inertial lateral displacement of the vehicle mass center, ϕ is lateral velocity in the vehicle body axis system, r is the angular velocity, and ψ is the vehicle heading angle.The state variablesy and ϕ are measured by using laser sensors at a sampling rate 1/h1, and the state variables r and ψ are measured by using gyrometers at a sampling rate 1/h2, where hi(i=1,2) are two sampling periods.We now show the effectiveness of the proposed dual-rate sampling design result. Seta1=0.75, a2=0.85, a3=0.9, v=0.25, l1=3, l2=2, μ=1.01, h1=0.16 s, and h2=0.24 s. Using Proposition 3, we can obtain the minimum H∞ performance γmind=0.1539, an exponential decay rate α=0.775, and corresponding control gain matrices F1=[-0.0141-0.0487-0.0532-0.7625], F2=[-0.0142-0.0492-0.0539-0.7672], and F3=[-0.0142-0.0490-0.0546-0.7686], which means that switched controller (2) with the above Fi(i=1,2,3) can stabilize system (39) and achieve an H∞ performance γmind=0.1539 and α=0.775 for system (39) with an external disturbance. Using Corollary 5, the minimum H∞ performance γmins=0.2317, the exponential decay rate a=α=0.775, and the control gain is F=[-0.0092-0.0252-0.0386-0.6028]. Clearly, Proposition 3 can be used to search for a better H∞ control performance than Corollary 5. For comparison purpose, we choose different delay bounds of h2 for both dual-rate sampling case and single-rate sampling case. The solvability of Proposition 3 and Corollary 5 is shown in Table 1, from which one can see that the dual-rate sampling design result can provide a larger sampling period than the single-rate design result.Table 1
Solvability comparison between Proposition3 with 3h1=2h2 and Corollary 5 with h=h2.
h
2
0.24 s
0.27 s
0.30 s
0.36 s
Proposition3 (dual-rate)
Feasible
Feasible
Feasible
Feasible
Corollary5 (single-rate)
Feasible
Feasible
Infeasible
InfeasibleThen we compare the proposed dual-rate sampling design result and the single-rate design result in simulation. Choose the initial statex0=[-2010]T and the external disturbance ω(t)=2e-0.3tsin⁡(2.7t+0.05). Using the switched controller with Fi(i=1,2,3), we can depict the state responses of system (39) by Figure 3 and the corresponding control input by Figure 4, respectively. Under the zero initial condition, L2-norm zd(t)2 using the switched controller and zs(t)2 using the single-rate sampled-data controller are shown in Figure 5. From Figures 3–5, one can conclude that the dual-rate sampled-data can stabilize the system (39) and achieve a better H∞ performance than the single-rate sampled-data controller.Figure 3
The state responses of system (39) using the switched controller.Figure 4
The control input of system (39) using the switched controller.Figure 5
The comparison ofe-αt/2z(t)2≤γω(t)2 under the zero initial condition by two different methods.


## 5. Conclusion

This paper has dealt with theH∞ sampled-data control for a multirate system with two sampling rates by using an input delay approach. A periodic switched controller has been proposed to implement the sampled-data control such that the sampled-data of state variables with different sampling rates can be used real-timely. The resulting closed-loop system has been modeled as a switched system, where the subsystems have different input delays. Some delay-dependent criteria for the H∞ performance of the switched system and the existence of the switched controller have been established by using the Lyapunov-Krasovskii functional method. By comparing with single-rate design methods, the effectiveness of the proposed design method has been illustrated by an example.



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*Source: 515232-2015-07-15.xml*