3 root locus method - scsettituiasiro

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3 root locus method - scsettituiasiro



    Tiberiu Dinu Teodorescu, Liviu Goraş

    Technical University Iasi, Romania

    Faculty of Electronics and Telecommunications


    ABSTRACT The impedance may have any shape and any order.

     In this communication a comparison between two approaches for studying single-coupled CNN’s is presented. First is related to ui so-called dispersion curve and second to the approach

    introduced in [1], which handles the stability problem through BuBu i-1 i+1Y(s) the roots locus method. First method may be used for studying double-coupled CNN’s as well, while the second is suited only for single-coupled CNN’s, but opens new analytical options for studying this kind Figure 1: General CNN cell coupled with first order of systems, due to it’s generality. neighbors.

    Both methods are based on decoupling technique, and valid for Systems obtained by using other shapes for the above-mentioned the central linear part of the non-linear cell. impedance are introduced in [1] and are subjects for further

    studies using roots locus method. 1. INTRODUCTION

    2. DISPERSION CURVE METHOD Since their invention [2], many significant results regarding the CNN behavior have been obtained. In the following we consider two-port cells connected by means

    of two identical first order neighborhood templates. In the One of the methods for patterns studying and filtering central linear part, the CNN is described by the following set of capabilities makes use of the decoupling technique, which is equations valid for dynamics, restricted to the central linear part of the cell characteristics. This method is fundamentally linear and has dut()ifu(fvD)Ou()iM0..1been applied for the study of the linear filtering properties of the uiviu1Di??dt(1) first order cell and second order neighborhood template as well dvt()as for the pattern formation in second order cell first order i ?gugv()uivineighborhood template. ?dt? The aim of this communication is to make a comparison two where O has the shape (for symmetrical templates): 1Dapproaches for stability analysis. One is related to the dispersion curve method connected with the window method [] and the (2) O(u)BuBuAu1Dii1i1iother to the roots locus method and using a certain part of the

    Using the decoupling technique, by means of the change of roots locus method in connection with the so-called template.

    variable: First method was related with certain circuit implementations of M1^ the cell [3] and is known related to Turing patterns where the u(m,i)ui0..M1template was fixed to discrete laplacean. The second is focused )iMm?(3) ?m0mainly on the general implementation of the CNN cell, which is M1modeled by means of a second order impedance coupled with ^ ?the neighbors be means of voltage-controlled current sources as v(m,i)v)iMm ?in the figure below. This approach is a general one because of m0?the fact that the cell may be in general considered an uniport where : with certain impedance. j(?(m)i~(m))(4) (?(m),~(m),i)eM

    the spatial eigenvalues are (see also figure 2): According to the shape of the curves and the domain of the

    eigenvalues various types of dynamics are possible. The (5) difference between this case and the double-coupled CNN’s is K(m)A2Bcos(?(m))1Dthat one cannot obtain a horizontal middle zone except for the situation when D=0, which is the case when the cells are no uand the dispersion curve for a single layered CNN is: more connected. In addition, one cannot obtain extremum points

    for the dispersion curve in the single-coupled CNN case. ? Re((K))1,21DSeveral relevant points on the above dispersion curves are: fgDKuvu1DThe extremities of the zone where the eigenvalues are complex Re{(6) conjugated, 22 2 (gf)DK??2vuu1D (fg}(;K(fg)2fgvu ?D1uvvuleft22??D u(7) The curves in Fig. 1 represent the locus of the real part of the (;K(fg)2fgsolutions above. D1uvvurightD5u

    4The width of the central zone: 3 Re(Lambda) fg42vu(8) KD11D u 0-10-5510K1DThe center of the central zone: -1 -2;;Kfg(9) 1Dmiddleuv -3Du

    and Figure 2: Dispersion curves for a single layer CNN

     gvThe imaginary parts of the temporal eigenvalues are represented Re()(10) K1Dmiddlein Fig. 3: 2 1.5From (7) and (8) one can easily see that the necessary condition for the middle zone to exist (K and K to be real 1Dleft1Dright1 numbers) is that the product fg have negative sign. When vuIm(Lambda)fg=0, there is only one value K for which the temporal vu1D 0.5eigenvalues are complex conjugated.

     The imaginary part curve (in the case it exists) gives the 0-10-5510K1Dfrequency of the temporal oscillations for each mode: -0.5

    -1? Im((K))1,21D

    (12) -1.52(gf)DK??2vuu1D fg}vu?22 ??

     Figure 3: The imaginary part of the eigenvalues

     The dispersion curves exhibit a central region with complex

    roots and two lateral regions with real roots.

    Selecting a region under the dispersion curve (window) is When coupling the cells with a grid one obtain the decoupled possible by appropriately choosing the template parameters equations:

    according to equation (5). (14) ?ˆ(Y(s)K(m))u 1Dm 0.4 The equation above becomes for the case of single coupled Re(l) second order CNN: 0.2 s?K101D22 (15) s?20051015202530modes

     where -0.2 (fg)uv2 (16) -0.4gv 22 ?(fgfg)0uvvuFigure 4: Selecting a window by using template By using the equation (15) one can sketch the roots locus and parameters A=1 and B=-1 from dispersion curve in Fig. . discuss the stability of the CNN depending on the value of K’1D2. (the real part) The value of K’ is the same function of modes as in equation 1D(5). Consequently, the points of the roots locus may be labeled The imaginary part of this part of the dispersion curve is with modes. represented in Fig. 5:

    1.5Some of them will be placed in the left part of the complex plane and others will stay in the right part. The points located in the right part of the complex plane will correspond to unstable 1modes, while the other points on the root locus plot will Im(l)correspond to stable modes. 0.5One roots locus plot is displayed below: 1.5051015202530 modes 1 -0.5 0.5 -1 0-0.4- -0.5-1.5 -1

    Figure 5: Selecting a window by using template -1.5parameters A=1 and B=-1 from dispersion curve in Fig.

    3. (the imaginary part)

    Figure 6: Roots locus when K’ is located inside the 1D3. ROOT LOCUS METHOD interval [-0.5 1.5] (the equivalent of window method for

    roots locus method) The admittance for the description of the cell introduced in Fig.

    1 for the case of Chua cell [2, 4] may be written in the following


    2fg(13) vuThe simulations have been done with the following set of Yssf()uparameters: =5, fu=0.1, gu=0.1, fv=-1, gv=-0.2, Du=0.5. Fig. sg v(2-5) represents the dispersion curve(s) for this set of parameters.

In Fig. (6) the root locus for this set of parameters is displayed,

    according to equations (13-16).

    As one can see, the range for the real parts of the modes last from 0.5 to 0.5 in both representations in Fig. 4 and

    respectively 6. The imaginary part lies in the interval -1.5, 1.5 in both sets of representations (see Figs 5 and 6). According to the window method (see Figs.2 and 3) one have isolated the part of the dispersion curves located between the

    values 1 and 3 of K. All the roots of the characteristic 1D polynomial are complex conjugated inside this interval. Accordingly, when sketching the root locus in Fig. 6, one must

    take into account that the “strength” of the connection of each cell with the neighbors is multiplied with Du in equation (1).

     Consequently, the relationship between K and K’ may be 1D1Dwritten as in the equation displayed below:

    (17) ?KDK1Du1DFigure 8: Time evolution for the CNN made with The equation (17) suggests that D is not necessarily an usecond order cells as a particular case of a general independent parameter. Other parameters aren’t independent as impedance (equations 13 15) well.

     The equivalent values for root locus parameters method are: 2=0.25, =1 and ?=2. 05. CONCLUDING REMARKS

    One present a set of simulations for the system described above. As a conclusion to our presentation, one can say that the

    root locus method is suitable for a specific class of cellular First, a simulation for the system described by equation (1),

    neural networks: single-coupled CNNs. The main presents the evolution in time of the state variables at port u. The state variables (voltages on capacitors) where all zero advantage of this method is the fact that the cell can be except for the state variable in the middle of 1D network, which modeled as an impedance of any order. The drawback is had initially, value 0.1 (Fig. 7) that one cannot analyze double-coupled CNNs by using

    this method.

    The dispersion curve method is powerful only for impedances with shapes as in the equation 15, but it has

     the advantage of being successfully used for double-

    coupled CNNs. Both analysis methods are also design methods for their class of systems.


     [1] L.Goras, T.D. Teodorescu „On the Dynamics of a Class of

    CNN”, to be publisehd in SCS’01 Proceedings, Iasi, 2000, Romania

    [2] L.O.Chua, L. Yang "Cellular Neural Networks: Theory",

    IEEE Transactions on Circuits and Systems, vol. 35, number 10, October 1988,pp 1257-1272

    [3] L. Goras, L.O. Chua “Turing Patterns in CNNs – Part II: Equations and Behaviors”, IEEE Transactions on Circuits and Figure 7: Time evolution for the CNN made with Systems, vol. 42, number 10, October 1995,pp 612-626. second order cells as in equation (1) [4] K.R. Crounse, "Ph. Thesis: Image Processing Techniques for Cellular Neural Network Hardware", University of California, In Fig. 8 the simulation done by using the general simulator Berkeley, Fall, 1997 designed for the general case gave the same result in the linear part:

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