# New Designs of Reduced-Order Observer-Based Controllers for Singularly Perturbed Linear Systems

Authors: Heonjong Yoo; Zoran Gajic
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The slow and fast reduced-order observers and reduced-order observer-based controllers are designed by using the two-stage feedback design technique for slow and fast subsystems. The new designs produce an arbitrary order of accuracy, while the previously known designs produce the accuracy ofO(ϵ) only where ϵ is a small singular perturbation parameter. Several cases of reduced-order observer designs are considered depending on the measured state space variables: only all slow variables are measured, only all fast variables are measured, and some combinations of the slow and fast variables are measured. Since the two-stage methods have been used to overcome the numerical ill-conditioning problem for Cases (III)–(V), they have similar procedures. The numerical ill-conditioning problem is avoided so that independent feedback controllers can be applied to each subsystem. The design allows complete time-scale separation for both the reduced-order observer and controller through the complete and exact decomposition into slow and fast time scales. This method reduces both offline and online computations.


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## Body

## 1. Introduction

The presentation in this paper is based on the doctoral dissertation [1]. The fundamental technique used is the two-stage feedback controller design [2, 3]. The full-order observers for singularly perturbed linear systems were considered in [4–10]. The reduced-order observer for singularly perturbed discrete-time systems has been studied only in two papers [11, 12], both of them producing accuracy of O(ϵ) (an O(ϵ) is defined by O(ϵ)≤kϵ, where k is a bounded constant), where ϵ is a small singular perturbation parameter. There are no corresponding results reported in the literature for the reduced-order continuous time observers. Since O(ϵ) accuracy might not be acceptable, our motivation is to design reduced-order observers for this class of systems with the accuracy of O(ϵi)≤cϵi, i=1,2,3,…, where ϵ is a bounded constant, which is along the lines of high accuracy techniques for singularly perturbed systems [5]. Note that, for ϵ<1, ϵi→0 rapidly.Consider a singularly perturbed linear system that contains slow and fast modes [6]: (1)Ex˙t=Axtyt=CxtIn100ϵIn2x˙1tx˙2t=A11A12A21A22x1tx2t,yt=C1C2x1tx2t,where ϵ is a small positive singular perturbation parameter that indicates separation of state variables x(t)∈Rn into slow x1(t)∈Rn1 and fast x2(t)∈Rn2 and n1+n2=n, and y(t)∈Rl are the system measurements. The problem matrices are constant and of appropriate dimensions. It was assumed that the matrix C has full rank equal to l. The singularly perturbed system is studied under the following standard assumption [4].Assumption 1.
A
22 is nonsingular.In the following we consider five cases for the reduced-order observer-based controller design for singularly perturbed linear systems depending on combinations of measured states [1].


## 2. Case I: Controller Design When Only All Slow Variables Are Measured

Consider the linear time invariant singularly perturbed control system [1], in which only slow variables are directly measured:(2)x˙1t=A11x1t+A12x2t+B1utx˙2t=1ϵA21x1t+1ϵA22x2t+1ϵB2utyt=x1t,where u(t)∈Rm is the control input. The reduced-order observer for (2) is given by [13](3)ϵz^˙2t=Azz^2t+Bzut+Kzyt,where(4)Az=A22-K11A12,Bz=B2-K11B1Kz=A21-K11A11+1ϵA22K11-1ϵK11A12K11.The state estimation of the fast variables is obtained from(5)x^2t=z^2t+1ϵK11ytso that(6)x^t=x1tx^2t=ytx^2t.The matrix K11 is chosen to stabilize the reduced-order observer (3); that is,(7)λA22-K11A12=λA22T-A12TK11T=λrobsdesired.To design this reduced-order observer the following assumption is needed [1].Assumption 2.
The pair(A22T,A12T) is controllable, which is equivalent to the pair (A22,A12) being observable, which is equivalent to the requirement that the original system is observable.In the following, the Chang transformation matrix [4] will be needed:(8)Tc=InϵH-LIm-ϵLH,Tc-1=In-ϵHL-ϵHLIm,where matrices L and H satisfy the algebraic equations(9)0=ϵLA11-A12L+A21-A22L0=ϵA11-A12LH+A12-HA22+ϵLA12.The solutions for L and H can be obtained using either the fixed-point iterations or Newton method or eigenvector method [5].Using the separation principle, the observer-based controller design via the two-stage design [2] produces [1](10)ut=-Fx^t=-F1F2x1tx^2t=-Fs+Ff2PFf2Tc-1ytx^2t.The feedback gain Fs is chosen thus to place slow eigenvalues at the desired locations; that is,(11)λAs-BsFs=λsdesired.The matrix P is obtained from the Sylvester algebraic equation(12)ϵPAs-BsFs-BfFs-AfP=0⟹P0=Af-1BfFs,where (13)As=A11-A12L,Bs=B1In-ϵHL-ϵHB2Af=A22+ϵLA12,Bf=ϵLB1+B2.The feedback gain Ff2 is chosen thus to place the fast eigenvalues at the desired locations; that is,(14)λAf-Bf+ϵPBsFf2=λAf-BfnewFf2=λfdesired.To obtain the reduced-order feedback gains Fs and Ff2, the following controllability assumption is needed [14].Assumption 3.
The pairs(As,Bs) and (Af,Bf+ϵPBs) are controllable.Based on information from (3), (7), and (10), we present in Figure 1 the block diagram for the reduced-order observer-based controller when only all slow state variables are perfectly measured. In (11) and (14) we have chosen the feedback gains for the eigenvalue assignment problem. However, any feedback gains F1 and F2 can be used to control the system and provide corresponding design requirements.Figure 1
Case I: reduced-order observer-based controller design for a singularly perturbed linear system when only all slow state variables are directly measured.


## 3. Case II: Controller Design When Only All Fast Variables Are Measured

Consider the linear time invariant singularly perturbed control system [1], when only all fast state variable are directly measured:(15)x˙1t=A11x1t+A12x2t+B1utx˙2t=1ϵA21x1t+1ϵA22x2t+1ϵB2utyt=x2t.The reduced-order observer for system (15) is given by [7](16)z^˙1t=Azz^1t+Bzut+Kzyt,where(17)Az=A11-1ϵK12A21,Bz=B2-1ϵK12B1Kz=A12+A11K12-1ϵK12A22-1ϵK12A21K12.The slow state estimation is obtained from(18)x^1t=z^1t+K12ytso that(19)x^t=x^1tx2t=x^1tyt.The matrix K12 is chosen to stabilize the reduced-order observer (16)-(17); that is,(20)λA11-1ϵK12A21=λA11T-1ϵA21TK12T=λrobsdesired.The reduced-order observer (16) can be designed under the following assumption [14].Assumption 4.
The pair(A11T,1/ϵA21T) is controllable, which is equivalent to the pair (A11,1/ϵA21) being observable. It was shown in Appendix that this is equivalent to (A11,A21) being observable.Additional matrices needed in this design can be obtained from (9) and (12)-(13). Using the separation principle, the observer-based controller can be designed via the two-stage design as(21)ut=-Fxt=-F1F2x^1tx2t=-Fs+Ff2PFf2Tc-1x^1tyt,where Fs and Ff2 are obtained from (11) and (14). The feedback gains Fs and Ff2 can be obtained under Assumption 3.In Figure2, the block diagram for the reduced-order observer-based controller when only all fast variables are perfectly measured is presented.Figure 2
Case II: slow and fast observer-based controller design for a singularly perturbed linear system when only fast state variables are measured directly.


## 4. Cases (III)–(V): Controller Design When Some Combinations of Slow and Fast Variables Are Measured

In Case (III), the measurable statesx11(t) are parts of the slow state vector x1(t) in the singularly perturbed linear system defined in (1), as(22)x˙1t=A11x1t+A12x2tϵx˙2t=A21x1t+A22x2tyt=x11t,where(23)x1t=x11tx12t=ytx12t.Use the following partitioning:(24)A11=a11a12a21a22,A12=a13a23A21=a31a32,A22=a33,where dimensions of matrices a11, a22 and a33 are, respectively, a11∈Rl×l, a22∈R(n1-l)×(n1-l), and a33∈Rn2×n2, with corresponding dimension of aij, i≠j, matrices. System (22) with (23)-(24) can be represented as(25)x˙11t=A11rx11t+A12rx2rtx˙2rt=A21rx11t+A22rx2rtyt=x11t,where(26)x2rt=x12tx2tA11r=a11,A12r=a12a13A21r=a211ϵa31,A22r=a22a231ϵa321ϵa33.The reduced-order observer for this case is derived in [1], and it is given by(27)z^˙2rt=Azrz^2rt+Kzryt,where(28)Azr=A22r-K2A12r=a22a231ϵa321ϵa33-K21K22a12a13,Kzr=A21r-K2A11r+A22rK2-K2A12rK2,z^2rt=z^12tz^2t.For the eigenvalue assignment in Azr, we encounter the singularly perturbed structure, so that the two-stage method is applied for a two time-scale problem.The reduced-order sequential observer configuration obtained using the two-stage method [1] is given by(29)z^˙stϵz^˙fnewt=Asr-KsrCsr0-ϵKf2rCsrAfr-Kf2rCfnewrz^stz^fnewt+Ksr∗Kf2r∗yt,where Ksr∗ and Kf2r∗ are obtained from T4r-1Kzr, with T4r defined by(30)z^12tz^2t=T1rTcrT2r=T4rz^stz^fnewt,where (31)T1r=In1-l001ϵIn2,T2r=In1-lPorT0In2,TcrT=In1-l-ϵLrTHrTIn2-ϵHrTLrT,where LrT and HrT are matrices that satisfy(32)0=ϵa22T-LrTa23TLrT+a32T-LrTa33T0=ϵHrTa22T-LrTa23T+a23T-a33T+ϵa23TLrTHrTwith aij matrices defined in (26). The reduced-order observer (29) has asequential structure. It can be block diagonalized and used in aparallel structure as follows:(33)z^˙st=Asr-KsrCsrz^st+Ksr∗ytϵz^˙fnew2t=Afr-Kf2rCfnewrz^fnew2t+Kf3ryt,where(34)Kf3r=ϵPo2rKsr∗+Kf2r∗,z^stz^fnew2t=In1-l0Po2rIn2z^stz^fnewt=T3rz^stz^fnewt.The original coordinates z^12(t) and z^2(t) and the coordinates z^s(t) and z^fnew2(t) are related via(35)z^12tz^2t=T4rT3r-1z^stz^fnew2t=Trz^stz^fnew2t=In1-l-PorTPo2r-HrPo2rPorT+Hr-Lr+LrPorTPo2r-1ϵIn2-ϵLrHrPo2r-LrPorT+1ϵIn2-ϵLrHrz^stz^fnew2t,where Po2r satisfies the algebraic Sylvester equation represented by(36)ϵPo2rAsr-KsrCsr-ϵKf2rCsr-Afr-Kf2rCfnewrPo2r=0⟹Po2r0=Oϵ.The steps used in the sequential and parallel observer design structures are summarized in Figure 3.Figure 3
Case III: design outlines for the sequential and parallel reduced-order observer.We use the parallel observers structure (33) and consider the reduced-order observer-based controller design for singularly perturbed linear systems. Observer (33) is now driven by the system measurements andcontrol inputs; that is,(37)z^˙st=Asr-KsrCsrz^st+Bsr2ut+Ksr∗ytϵz^˙fnew2t=Afq-Kf2Cfnewz^fnew2t+Bf2rut+Kf3yt,where Bsr2 and Bf2r can be obtained from Tr-1B as(38)Bsr2=In1-l-ϵHrLrBz1-ϵPorTLrBz1-HrBz2-PorTBz2,Bf2r=ϵPo2rIn1-l-ϵHrLrBz1-ϵ2Po2rPorTLrBz1+ϵ2LrBz1-ϵPo2rHrBz2-ϵPo2rPorTBz2+ϵBz2.The control input in the z^s-z^fnew2 coordinates is given by(39)ut=-Frx^t=-F12F2x^12tx^2t=-F12F2K21K22yt+F12F2Trz^stz^fnew2t=-FrK2yt-Fsr2z^st-Ffr2z^fnew2t,F11F12∣F2=F1∣F2=Fs+Ff2P∣Ff2Tc-1sothatF11∣F12F2=F11∣Frwith(40)Fsr2=F12In1-l-PorTPo2r-HrPo2r-F2LIn-PorTPo2r-1ϵF2In2-ϵLrHrPo2rFfr2=F12PorT+Hr-F2LrPorT+1ϵF2In2-ϵLrHr.The corresponding block diagram for the observer driven controller is presented in Figure 4. This block diagram clearly indicates full parallelism of the slow controller driven by the slow observer and the fast controller driven by the fast observer.Figure 4
Case III: slow and fast observer-based controller designs for singularly perturbed linear systems with the system feedback gains obtained in (39) when parts of slow variables are measured.The remaining matrices defined in (29) are given by(41)AsrT=a22T-LTa23T,AfrT=a33T+ϵa23TLTCsrT=a12T-LTa13T,CfrT=ϵHTa12T+In2-ϵHTLTa13TCfnewr=Cfr+ϵCsrPorT.Ksr, Kf2r, and Por can be obtained from the following formulas:(42)λAsrT-CsrTKsrT=λAsr-KsrCsr=λsdesiredλAfr-Kf2rCfnewr=λfdesiredϵPorAsrT-CsrTKsrT-CfrTKsrT-AfrTPor=0⟹Por=Oϵ.In Case (IV), the measurable states x21(t) are parts of the fast state vector x2(t) in the singularly perturbed linear system defined in (1):(43)x˙1t=A11x1t+A12x2tϵx˙2t=A21x1t+A22x2tyt=Ilx22t,where(44)x2t=x21tx22t=x21tyt,using the following partitioning:(45)A11=a11,A12=a12a13A21=a21a31,A22=a22a23a32a33,where dimensions of matrices a11, a22, and a33 are, respectively, a11∈Rn1×n1, a22∈R(n2-l)×(n2-l), and a33∈Rl×l, with the corresponding dimensions of aij,i≠j, matrices. System (43) with (44)-(45) can be represented as(46)x˙1rt=A11rx1rt+A12rx22tx˙22t=A21rx1rt+A22rx22tyt=Ilx22t,where(47)x1rt=x1tx21tA11r=a11a121ϵa211ϵa22,A12r=a131ϵa23A21r=1ϵa311ϵa32,A22r=1ϵa33.The corresponding reduced-order observer structure is given by [1](48)z^˙1rt=Az,2rz^1rt+Bz,2rut+Kz,2ryt,where(49)z^1rt=z^1tz^21tThe observer gain K3 and matrices Bz,2r and Kz,2r [7] are obtained from(50)Az,2r=A11r-K3A21r=a11a121ϵa211ϵa22-K31K321ϵa311ϵa32,Bz,2r=B2-K3B1=Bz1Bz2,Kz,2r=A12r-K3A22r+A11rK3-K3A21rK3.For the eigenvalue assignment in Az,2r, we encounter singularly perturbed structure so that the two-stage design is applied to the slow and fast subsystems.The two-stage method provides the parallel observer [1]. Using the parallel observer we can form the reduced-order observer-based controller given as(51)z^˙s,2tϵz^˙fnew2,2t=Asr,2-Ksr,2Csr,200Afr,2-Kf2r,2Cfnewr,2z^s,2tz^fnew2,2t+Bs2r,2Bf2r,2ut+Ksr,2∗Kf3r,2yt,where(52)Asr,2T=a11T-Lr,2Ta12T,Afr,2T=a22T+ϵa12TLr,2TBs2r,2=In1-ϵHr,2Lr,2Bz1-ϵPor,2TLr,2Bz1-Hr,2Bz2-Por,2TBz2,Bf2r,2=ϵPo2r,2In1-ϵHr,2Lr,2Bz1-ϵ2Po2r,2Por,2TLr,2Bz1+ϵ2Lr,2Bz1-ϵPo2r,2Hr,2Bz2-ϵPo2r,2Por,2TBz2+ϵBz2Csr,2T=1ϵa31T-1ϵLr,2Ta32T,cfr,2T=Hr,2Ta31T+1ϵIn2-l-ϵHr,2TLr,2Ta32T,Cfnewr,2=Cfr,2+ϵCsr,2Por,2Tand Lr,2T and Hr,2T are the solution given as(53)0=ϵa22T-Lr,2Ta23TLr,2T+a32T-Lr,2Ta33T0=ϵHr,2Ta22T-Lr,2Ta23T+a23T-a33T+ϵa23TLr,2THr,2T.The coordinates of the original observer (48) and the parallel observer (51) are related via a transformation as(54)z^1tz^21t=Tr,2z^s,2tz^fnew,2t=T4r,2T3r,2-1z^s,2tz^fnew,2t=In1-Por,2TPo2r,2-Hr,2Po2r,2Por,2T+Hr,2-Lr,2+Lr,2Por,2TPo2r,2-1ϵIn2-l-ϵLr,2Hr,2Po2r,2-Lr,2Por,2T+1ϵIn2-l-ϵLr,2Hr,2z^s,2tz^fnew2,2t,where(55)T4r,2=In1Por,2T+Hr,2-Lr,2-Lr,2Por,2T+1ϵIn2-l-ϵLr,2Hr,2,T3r,2-1=In10-Po2r,2In2-l.Ksr,2∗ and 1/ϵKf2r,2∗ are obtained from T4r,2-1Kz,2rr. The slow feedback gain Ksr,2 can be obtained using the eigenvalue assignment; that is,(56)λAsr,2T-Csr,2TKsr,2T=λAsr,2-Ksr,2Csr,2=λsdesiredrequiring that the following assumption is satisfied [14].Assumption 5.
The pair(Asr,2,Csr,2) is observable.The fast feedback gainKf2r,2 is obtained from the eigenvalue assignment problem; that is,(57)λAfr,2-Kf2r,2Cfnewr,2=λfdesired.The following observability assumption is needed [14].Assumption 6.
The pair(Afr,2,Cfnewr,2) is observable.The control input inz^s,2-z^fnew2,2 coordinates is given by(58)ut=-Frx^t=-F1F21x^1tx^21t=-Fr,2K3yt-Fsr2,2Ffr2,2z^s,2tz^fnew2,2tF1∣F21F22=F1∣F2=Fs+Ff2P∣Ff2Tc-1sothatF1F21∣F22=Fr∣F22with(59)Fsr2,2=F1In1-Por,2TPo2r,2-Hr,2Po2r,2-F21LIn2-l-Por,2TPo2r,2-1ϵF2In2-l-ϵLr,2Hr,2Po2r,2Ffr2,2=F1Por,2T+Hr,2-F21Lr,2Por,2T+1ϵF21In2-l-ϵLr,2Hr,2.The remaining matrices Por,2 and Po2r,2 satisfy the algebraic Sylvester equation represented by(60)ϵPor,2Asr,2T-Csr,2TKsr,2T-Cfr,2TKsr,2T-Afr,2TPor,2=0⟹Por,2=OϵϵPo2r,2Asr,2-Ksr,2Csr,2-ϵKf2r,2Csr,2-Afr,2-Kf2r,2Cfnewr,2Po2r,2=0⟹Po2r,20=Oϵ.The corresponding block diagram for the observer driven controller is presented in Figure 5.Figure 5
Case IV: slow and fast observer-based controller design for a singularly perturbed linear system measurable variables being part of the fast variables only.In Case (V), the measurable statesx11(t) and x21(t) are parts of the slow state vector x1(t) and the fast state x2(t) in the singularly perturbed linear system defined as(61)x˙1t=A11x1t+A12x2tϵx˙2t=A21x1t+A22x2tyt=I00000I0x11tx12tx21tx22t=x11tx21t=y1ty2t,where(62)x1t=x11tx12t,x1t=x11tx12t=y1tx12t,x2t=x21tx22t=x21ty2t,using the following partitioning:(63)A11=a11∗a12∗a21∗a22∗,A12=a13∗a14∗a23∗a24∗A21=a31∗a32∗a41∗a42∗,A22=a33∗a34∗a43∗a44∗,where dimensions of matrices a11∗, a22∗, and a33∗ are, respectively, a11∗∈Rl1×l1, a22∗∈R(n1-l1)×(n1-l1), and a33∗∈Rl2×l2, with the corresponding dimensions of aij, i≠j, matrices. System (61) with (62)-(63) can be represented as(64)x˙mt=A1rxmt+A2rxutx˙ut=A3rxmt+A4rxutyt=I00000I0x11tx12tx21tx22t=x11tx21t=y1ty2t,where(65)xmt=x11tx21t,xut=x12tx22tA1r=a11∗a13∗1ϵa31∗1ϵa33∗,A2r=a12∗a14∗1ϵa32∗1ϵa34∗A3r=a21∗a23∗1ϵa41∗1ϵa43∗,A4r=a22∗a24∗1ϵa42∗1ϵa44∗.xm(t) are the measurable states and xu(t) are the unmeasurable states. A1r and A3r are elements in (62) relevant to the measurable states and A2r and A4r are elements in (62) relevant to the unmeasurable states. The reduced-order observer is given by(66)z^˙1rt=Az,3rz^1rt+Bz,3rut+Kz,3ryt.The observer gains are obtained from(67)Az,3r=A4r-K4A2r=a22∗a24∗1ϵa42∗1ϵa44∗-K41K42a12∗a14∗1ϵa32∗1ϵa34∗,Bz,3r=B2-K4B1=Bz1Bz2,Kz,3r=A3r-K4A1r+A4rK4-K4A2rK4.For the eigenvalue assignment in Az,3r, we encounter again the singularly perturbed structure.The two-stage method provides the parallel observer; we can form the reduced-order observer-based controller given as [1](68)z^˙s,3tϵz^˙fnew2,3t=Asr,3-Ksr,3Csr,300Afr,3-Kf2r,3Cfnewr,3z^s,3tz^fnew2,3t+Bs2r,3Bf2r,3ut+Ksr,3∗Kf3r,3yt,where(69)Asr,3T=a22∗T-Lr,3Ta24∗T,Afr,3T=a44∗T+ϵa24∗TLr,3TBsr2,3=In1-l1-ϵHr,3Lr,3Bz1-ϵPor,3TLr,3Bz1-Hr,3Bz2-Por,3TBz2,Bf2r,3=ϵPo2r,3In1-l1-ϵHr,3Lr,3Bz1-ϵ2Po2r,3Por,3TLr,3Bz1+ϵ2Lr,3Bz1-ϵPo2r,3Hr,3Bz2-ϵPo2r,3Por,3TBz2+ϵBz2Po2r,2Por,2TLr,2Bz1+ϵ2Lr,2Bz1-ϵPo2r,2Hr,2Bz2-ϵPo2r,2Por,2TBz2+ϵBz2Csr,31T=a12∗T-Lr,3Ta14∗T,Csr,32T=1ϵa32∗T-Lr,3T1ϵa34∗TCfr,31T=ϵHr,3Ta12∗T+In2-l2-ϵHr,3TLr,3Ta14∗T,Cfr,32T=ϵHr,3T1ϵa32∗T+In2-l2-ϵHr,3TLr,3T1ϵa34∗TCsr,3T=Csr,31TCsr,32T,Cfr,3T=Cfr,31TCfr,32TCfnewr,3=Cfr,3+ϵCsr,3Por,3Tand Lr,3T and Hr,3T are the solution given as(70)0=ϵa22∗T-Lr,3Ta24∗TLr,3T+a42∗T-Lr,3Ta44∗T0=ϵHr,3Ta22∗T-Lr,3Ta24∗T+a24∗T-a44∗T+ϵa24∗TLr,3THr,3T.The matrices Por,3 and Po2r,3 satisfy the algebraic Sylvester equation represented by(71)ϵPor,3Asr,3T-Csr,3TKsr,3T-Cfr,3TKsr,3T-Afr,3TPor,3=0⟹Por,3=OϵϵPo2r,3Asr,3-Ksr,3Csr,3-ϵKf2r,3Csr,3-Afr,3-Kf2r,3Cfnewr,3Po2r,3=0⟹Po2r,30=Oϵ.The transformation is needed to relate both coordinates in which original observer (48) and parallel observer (68) are located; that is,(72)z^12tz^22t=Tr,3z^s,3tz^fnew2,3t=T4r,3T3r,3-1z^s,3tz^fnew2,3t=In1-l1-Por,3TPo2r,3-Hr,3Po2r,3Por,3T+Hr,3-Lr,3+Lr,3Por,3TPo2r,3-1ϵIn2-l2-ϵLr,3Hr,3Po2r,3-Lr,3Por,3T+1ϵIn2-l2-ϵLr,3Hr,3z^s,3tz^fnew2,3t,where(73)T4r,3=In1-l1Por,3T+Hr,3-Lr,3-Lr,3Por,3T+1ϵIn2-l2-ϵLr,3Hr,3,T3r,3-1=In1-l10-Po2r,3In2-l2.Matrices Ksr,3∗ and 1/ϵKf2r,3∗ are obtained from T4r,3-1Kz,3rr. The slow feedback gain Ksr,3 can be obtained from the eigenvalue assignment problem; that is,(74)λAsr,3T-Csr,3TKsr,3T=λAsr,3-Ksr,3Csr,3=λsdesiredassuming that the following assumption is satisfied [14].Assumption 7.
The pair(Asr,3,Csr,3) is observable.The fast feedback gainKf2r,3 can be obtained from the eigenvalue assignment problem as(75)λAfr,3-Kf2r,3Cfnewr,3=λfdesired.The following observability assumption is needed [14].Assumption 8.
The pair(Afr,3,Cfnewr,3) is observable.The control input in thez^s,3-z^fnew2,3 coordinates is given by(76)ut=-Fr,3x^t=-F12F21x^12tx^21t=-Fr,3K4yt-Fsr2,3Ffr2,3z^s,3tz^fnew2,3t,F11F12∣F21F22=F1∣F2=Fs+Ff2P∣Ff2Tc-1sothatF11∣F12F21∣F22=F11∣Fr,3∣F22with(77)Fsr2,3=F12In1-l1-Por,3TPo2r,3-Hr,3Po2r,3-F21LIn2-l2-Por,3TPo2r,3-1ϵF21In2-l2-ϵLr,3Hr,3Po2r,3Ffr2,3=F12Por,3T+Hr,3-F21Lr,3Por,3T+1ϵF21In2-l2-ϵLr,3Hr,3.The corresponding block diagram for the observer driven controller is presented in Figure 6.Figure 6
Case V: slow and fast observer-based controller design for a singularly perturbed linear system when some parts of the slow variables and some parts of the fast variables are measured.


## 5. Numerical Example for Case I

Consider a 4th-order system with the system matricesA,B, and C defined in [1]. The controllability matrix has full rank and therefore the pair (A,B) is controllable. The results obtained using MATLAB are given below. We locate the feedback system slow eigenvalues at λcsdesired=(-2,-3) and the feedback system fast eigenvalues at λcfdesired=(-7,-8) and the reduced-order observer eigenvalues at λrobsdesired=(-50,-60), given in the previous numerical example. Following the design procedure, the observer matrices Az, Kz, K11, F1, and F2 are given as(78)Az=-108.99990.2620-11034.3511-1.0000,K11=0314.5942031983.6265,Kz=0-259111.10370-35033265.2948, and the feedback gains are obtained as(79)F1=92.930637.8637,F2=2.43560.5349.


## 6. Numerical Example for Case II

Consider a 4th-order system with the system matricesA,B, and C defined in Section 5. We locate the feedback system slow eigenvalues at λcsdesired=(-2,-3) and the feedback system fast eigenvalues at λcfdesired=(-7,-8) and the reduced-order observer eigenvalues at λrobsdesired=(-50,-70), given in the previous numerical example. Following the design procedure, the observer matrices Az, Kz, K12, F1, and F2 are given as(80)Az=-49.99990.0000-0.0000-70.0000,K12=11.073410.3042-6.46606.4112,Kz=-537.2325-556.1992400.7840-440.7330, and the feedback gains are(81)F1=92.930637.8637,F2=2.43560.5349.


## 7. Numerical Example for Case III

Consider a 4th-order magnetic tape system from Section5. We locate the feedback system slow eigenvalues at λcsdesired=(-2,-3) and the feedback system fast eigenvalues at λcfdesired=(-7,-8) and the slow observer eigenvalues at λosdesired=-50 and the fast observer eigenvalues at λofdesired=(-200,-300), given in the previous numerical example. Following the design procedure, the completely decoupled slow and fast observers in the zs-zfnew2 coordinates, driven by the system measurements and control inputs, are(82)z^˙st=-50.0000z^st+-2.323033338614758ut+-6196.4741yt.z^˙fnew2t=-1693.6295113.2762-18376.02291193.6295z^fnew2t+-0.24990.4666ut+-2859901.4120-33059040.2923ytut=-1761.6414z^st-26.98245.1780z^fnew2tThe slow and fast controller gains Fsr2 and Ffr2 are obtained as(83)Fsr2=1761.6414,Ffr2=26.98245.1780.


## 8. Numerical Example for Case IV

Consider a 4th-order system with the system matricesA,B, and C defined in Section 5. We locate the feedback system slow eigenvalues at λcsdesired=(-2,-3) and the feedback system fast eigenvalues at λcfdesired=(-7,-8) and the slow observer eigenvalues at λosdesired=(-50,-60) and the fast observer eigenvalues at λofdesired=-300, given in the previous numerical example. Following the design procedure from Section 4, the completely decoupled slow and fast observer in the zs-zfnew2 coordinates, driven by the system measurements and control inputs, are(84)z^˙s,2t=-0.0000-4127.73670.7267-110.0000z^s,2t+4.7409-3.9951ut+113729.62512281.0457ytz^˙fnew2,2t=-300.0000z^fnew2,2t+-779.5149ut+437703.1601ytut=--6530.324291.6868z^st-332.8512-10.9008z^fnew2t.The slow and fast controller gains Fsr2,2 and Ffr2,2 are obtained as(85)Fsr2,2=-6530.324291.6868,Ffr2,2=332.8512-10.9008.


## 9. Numerical Example for Case V

Consider a 4th-order system with the system matricesA,B, and C defined in Section 5. We locate the feedback system slow eigenvalues at λcsdesired=(-2,-3) and the feedback system fast eigenvalues at λcfdesired=(-7,-8) and the slow observer eigenvalues at λosdesired=-50 and the fast observer eigenvalues at λofdesired=(-200,-300), given in the previous numerical example. Following the design procedure, the completely decoupled slow and fast observers in the zs,3-zfnew2,3 coordinates, driven by the system measurements and control inputs, are(86)z^˙s,3t=-50.0000z^st+-2.323033338614758ut+-6196.4741ytz^˙fnew2,3t=-1693.6295113.2762-18376.02291193.6295z^fnew2t+-0.24990.4666ut+-2859901.4120-33059040.2923ytut=--6530.324291.6868z^st-332.8512-10.9008z^fnew2t.The slow and fast controller gains Fsr2,3 and Ffr2,3 are obtained as(87)Fsr2,3=-6530.324291.6868,Ffr2,3=332.8512-10.9008.


## 10. Conclusion

We have designed with high accuracy the reduced-order observer-based controllers for singularly perturbed linear systems. The numerical ill-conditioning of the original problem is removed. We have demonstrated that the full-order singularly perturbed system can be successfully controlled with the state feedback coming from the reduced-order observer-based controllers fully designed on the subsystem levels. The two-stage method is successfully implemented for both the observer and controller designs.



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*Source: 2859548-2017-12-13.xml*