It has been shown that nonlinear discrete maps can display extremely rich behaviour and under certain parameter conditions to show chaotic phenomenon. This work looks at adaptive control feedback systems which can be represented as nonlinear discrete maps and shows how model mismatch can lead to undesired complicated and chaotic behaviour. Moreover that a discrete-time adaptive control system which can display chaotic behaviour can be extended into higher order systems and the results show that under certain parameter conditions, the higher order systems also behave chaotically. A generalised equation form for the eigenvalues is also given

Khan, Khalid

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013                                                                                         ISSN 2229-5518   IJSER © 2013 http://www.ijser.org  Chaotic Behaviour in Some Discrete –Time Adaptive Control Systems Dr Khalid Khan  Abstract— It has been shown that nonlinear discrete maps can display extremely rich behaviour and under certain parameter conditions to show chaotic phenomenon. This work looks at adaptive control feedback systems which can be represented as nonlinear discrete maps and shows how model mismatch can lead to undesired complicated and chaotic behaviour. Moreover that a discrete-time adaptive control system which can display chaotic behaviour can be extended into higher order systems and the results show that under certain parameter conditions, the higher order systems also behave chaotically. A generalised equation form for the eigenvalues is also given.  Index Terms— Adaptive Feedback Control Systems, Bifurcation, Chaos, Nonlinear Mappings, Simulation  ——————————      —————————— 1 INTRODUCTION                                                                     Chaos theory looks at the study of deterministic dynamical systems that are very sensitive to initial conditions. Small dif-ferences in initial conditions can lead to widely diverging out-comes, for such systems making long term predictions gener-ally becomes impossible. Chaotic phenomena have been ob-served in numerous systems in the science and engineering fields [1].  Lorenz [2] made early studies in the changes in the atmosphere which tended to display erratic and unpredictable behaviour. In more recent years, with potential application in engineering fields the study and control of chaotic systems has become important specifically chaos control and synchroniza-tion [3].      Adaptive feedback control starts with a system model with known or unknown parameters. Parameter adaptive control looks at global behaviour and a set of parameters which are manipulated by the observer. The observed behaviour is com-pared with the desired one and corrections made using the system parameters. It is know that adaptive control inherently leads to a nonlinear closed-loop system, even in the situation where we have a linear plant or a linear model of the plant, when a linear output feedback law is used and where the feedback parameters are estimated from input and output da-ta. Adaptive systems can be thought of as being asymptotical-ly linear when the feedback parameters have converged to some steady state. The study considers what sort of dynamical effects on the performance of the adaptive system the nonline-ar features can have      A discrete adaptive system can be represented as a nonline-ar discrete map. It has been shown that a one-dimensional nonlinear map [4], [5], a one-dimensional map with a quadrat-ic nonlinearity known as the logistic map [6], exhibit unpre-dictable and chaotic phenomena. Ydstie [7] has shown how a simple model-reference adaptive system (MRAC) character-ised by a third order nonlinear discrete map with no external forcing can exhibit chaotic behaviour. More recently, due the potential applications in a variety of disciples, chaos control has become an important consideration like the area of secure communications [8].      In this paper a general nth order plant is taken with in-creasing order model assumptions and using analysis and simulation show that complex dynamics and chaotic behav-iour can occur for higher order systems for certain parameter values. A general closed form equation is developed giving the eigenvalue structure for the nth order plant with a first order model assumption. This provides insight into the unsta-ble behaviour of our systems. 2 GENERAL BACKGROUND In model–reference adaptive control, the basic idea is to com-pare the behaviour of the controlled plant with that of refer-ence model representing the desired performance and attempt to reduce the difference between them by changing the con-troller parameters in an appropriate manner. The basic struc-ture is as shown in fig 1.              Fig. 1 Model – reference adaptive control system structure. ————————————————  Dr Khalid Khan is currently a Lecturer in Fire Engineering at the Univer-sity Of Central Lancashire, Centre for Fire and Hazard Sciences, Preston,  Lancashire, PR1 2HE, United Kingdom,  PH-01772895684. E-mail: kkhan5@uclan.ac.uk            PLANT REFERENCE MODEL ADJUSTABLE CONTROLLER ADAPTATION     LAW Error Signal Reference Signal + - + - International Journal of Scientific & Engineering Research Volume 4, Issue 10, October-2013                                                                                  ISSN 2229-5518  IJSER © 2013 http://www.ijser.org   The general adaptive control system is designed by combin-ing a particular estimation technique with a control law.   ( )    (   ) (   )    (   ) (   )                    (   )                    (   )                     (   )                       (   )                          ( )    (   )  For simplicity   (   )    and (          ( )  ( )      )   It is assumed that the system can be adequately represent-ed as a first-order model,   ( )    ̂ (   ) (   )   (   )  where    ̂  is an estimate of   .  The desired output  ( ) is be equal to some reference value   , we can choose  ( ) using an adaptive control law to achieve closed-loop stability and to asymptotically achieve zero tracking error.  ( )        ̂ ( ) ( ) An algorithm we use to give   ̂  , an estimate of the  actual parameter   .  ̂ ( )    ̂ (   )    (   ) ( )  Where   is the adaptation rate and  ( ) denotes model error.    3  GENERAL NTH ORDER PLANT, FIRST  ORDER MODEL  Plant:           ( )   ∑     (   )      (   ) Model:        ( )    ̂ (   ) (   )   (   ) Control Law:         ( )        ̂ ( ) ( ) Estimation Algorithm:  ̂ ( )    ̂ (   )    (   )   ( )    ̂ (   ) (   )  (   )   This produces a sequence of parameters {  ̂ ( )       with controls  {  ( )      , given      ̂ ( ). Closed-loop system equations become:  ( )   ∑     (   )          ̂ (   ) (   )  ̂ ( )    ̂ (   )      (   )    ( )        In order to facilitate analysis and simulation we can rewrite the above equations as a set of (n+1) first order equations: Let          ( )     (   )    ( )     (   )       ( )       (   )                ( )    (   )  So,   ( )       ̂ (   )  (   )  ∑          (   )           ̂ ( )    ̂ (   )      (   )    ( )         4 EQUILIBRIUM POINTS  Replacing   by     in the left hand side equations gives:   (   )   (   )  ( )       ̂     ∑          From which is it can be seen that if there is no plant to model mismatch ( i.e.  n=1 ) then our parameter estimate   ̂ ( ) at equilibrium is equal to   .   5 EIGENSTRUCTURE FOR THE GENERAL MODEL  The set of (n+1) first order equations represents a nonlinear discrete mapping. The stability of the adaptive system can be determined by looking at the eigenvalue structure of the Jaco-bian matrix at the fixed point. Linearization of the system equations at the equilibrium point gives the Jacobian of first partial derivatives     :    International Journal of Scientific & Engineering Research Volume 4, Issue 10, October-2013                                                                                  ISSN 2229-5518  IJSER © 2013 http://www.ijser.org    [            ( )   ( )         ( )   ( )  ̂ ( ) ]                  [            (   )     (   )         (   )  (   )  ̂ (   ) ]          where, the matrix of first partial derivatives      is given by :      =   [                                                                  ̂   (     ̂ )                   ]        at             ̂     ∑             The characteristic equation  ( ) for     is given  by         ( )      (       )   ( ) = ||                                                                  ̂       (     ̂ )                        ||  A concise algebraic form for   ( ) is:         So,    ( )    is a polynomial equation giving the eigenvalues for a general nth order plant with a first order model assump-tion.   6 ANALYSIS AND SIMULATION RESULTS The simplest case in our general representation model with n=2 has been shown to exhibit chaotic behaviour under certain parameter conditions, Ydstie [3]. We consider a plant which is now of third order assuming it can be modelled by a first or-der system ( i.e. n=3 ). 6.1  THIRD ORDER PLANT WITH FIRST ORDER MODEL Our system equations become:   ( )     (   )   ( )    (   )  ( )     (   )      (   )      (   )       ̂ (   ) (   )  ̂ ( )    ̂ (   )      (   )    ( )         The above equations can be represented as a nonlinear dis-crete mapping   (         ̂ )           where (         ̂ ) is the state vector,   (            ) is a vector of parame-ters.  The eigenstructure for this system is given by the characteris-tic equation,   ( )    with n=3,  (   )(    (    ̂ )          )             By varying different parameters of  , we can observe the system’s dynamical behaviour. Letting parameter    vary over a range of values from 0.2 to 0.8, with   [  ( )  ( ) ( ) ̂ ( )]   [            ]     and   [            ]     [                    ]     Fig. 2 shows  ( ) as    ranges from 0.2 - 0.8.                    0 50 100 150 200 250 3000510Timey(i)Plot of y(i) with a2=0.2, ( a1=0.6, a3=0.01, p=0.005 ) 0 50 100 150 200 250 3000510Timey(i)Plot of y(i) with a2=0.5, ( a1=0.6, a3=0.01, p=0.005 ) 0 50 100 150 200 250 300-50050Timey(i)Plot of y(i) with a2=0.76, ( a1=0.6, a3=0.01, p=0.005 ) 0 50 100 150 200 250 300-50050Timey(i)Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 ) 𝜒(𝑧)  (  𝑧)  (  )𝑛( 𝑧𝑛  ?̂? 𝑧𝑛     ∑  𝑎𝑟𝑧𝑛 𝑟  )𝑛𝑟        𝑝𝑦   (  )𝑛    𝑧𝑛 Typ   qua ion h r  Typ   qua ion h r  Fig. 2 – Plot of 𝑦(𝑖) 0 50 100 150 200 250 300 0 5 10 Time y(i) Plot of y(i) with a2=0.2, ( a1=0.6, a3=0.01, p=0.005 )  0 50 100 150 200 250 300 0 5 10 Time y(i) Plot of y(i) with a2=0.5, ( a1=0.6, a3=0.01, p=0.005 )  0 50 100 150 200 250 300 -50 0 50 Time y(i) Plot of y(i) with a2=0.76, ( a1=0.6, a3=0.01, p=0.005 )  0 50 100 150 200 250 300 -50 0 50 Time y(i) Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 )  International Journal of Scientific & Engineering Research Volume 4, Issue 10, October-2013                                                                                  ISSN 2229-5518  IJSER © 2013 http://www.ijser.org    Fig. 3 below shows   ( ) chaotic for 2000 iterations.                              Fig. 4 below shows the corresponding phase-plane plot of y(i) and a(i).                          It is also seen that as    increases from 0.2 to 0.8 and the sys-tem becomes more unstable and eventually chaotic, that one of the eigenvalues becomes greater than one in modulus.  Parameter    with corresponding eigenvalues:                                                      [                     ]       [                         ]       [                         ]          [                         ]  Similar type of behaviour can be observed if other parameters are varied  [            ]     [                   ]     and let      [ 0.0,  0.032 ] be the bifurcation parameter.  Fig. 5 below shows the results for the parameter      [ 0.001,  0.032 ]. as                               300 350 400 450 500 550 600 650 700 750 800 -50 0 50 Time y(i) Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 )  800 850 900 950 1000 1050 1100 1150 1200 -50 0 50 Time y(i) Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 )  1200 1250 1300 1350 1400 1450 1500 1550 1600 -50 0 50 Time y(i) Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 )  1600 1650 1700 1750 1800 1850 1900 1950 2000 -50 0 50 Time y(i) Plot of y(i) with a2=0.8, ( a1=0.6, a3=0.01, p=0.005 )  Fig. 3 -  Plot of 𝑦(𝑖) -0.5 0 0.5 1 1.5 2 -10 -5 0 5 10 15 20 25 a(i) y(i) Phase-Plane plot of (a,y) with a2=0.8,  Fig. 4 -  Plot of 𝑦(𝑖) vs a(i) 0 20 40 60 80 100 120 140 160 180 200 0 5 10 Time y(i) Plot of y(i) with p=0.001, ( a1=0.1, a2=0.44, a3=0.096 )  0 20 40 60 80 100 120 140 160 180 200 0 5 10 Time y(i) Plot of y(i) with p=0.01, ( a1=0.1, a2=0.44, a3=0.096 )  0 20 40 60 80 100 120 140 160 180 200 0 5 10 Time y(i) Plot of y(i) with p=0.028, ( a1=0.1, a2=0.44, a3=0.096 )  0 20 40 60 80 100 120 140 160 180 200 0 10 20 Time y(i) Plot of y(i) with p=0.031, ( a1=0.1, a2=0.44, a3=0.096 2800 2820 2840 2860 2880 2900 2920 2940 2960 2980 3000 0 10 20 Time y(i) Plot of y(i) with p=0.031, ( a1=0.1, a2=0.44, a3=0.096 Fig. 5 - Plot of 𝑦(𝑖) International Journal of Scientific & Engineering Research Volume 4, Issue 10, October-2013                                                                                  ISSN 2229-5518  IJSER © 2013 http://www.ijser.org   Again from the simulation results in Fig. 5, the system going through various bifurcations, from stable to limit cycle to peri-od doubling through to chaotic behaviour as parameter  p gradually increases.   6.2  THIRD ORDER PLANT WITH SECOND ORDER MODEL  It is now possible to assume a model which is second order and thus a better approximation to our third order plant. It is  expected that system to be more structurally stable as parame-ter values are varied than our first order model assumption. The corresponding set of first order equations become:    ( )     (   )   ( )    (   )  ̂( )   ̂ (   )     ( )     (   )      (   )      (   )                   ̂ (   ) (   )    ̂(   )  (   )     ̂ ( )    ̂ (   )      (   )    ( )                        ̂ ( )    ̂ (   )       (   )    ( )         Since we have a single reference input value  ( )      and we are estimating two parameters   ̂  ,  ̂  then both  ̂  and  ̂  cannot be determined explicitly and so we do not have a unique equilibrium point. We can still simulate our system equations above to see our system behaviour for different pa-rameter values.   Letting parameter    vary over a range of values from 0.1 to 0.63, with               [       ( )  ( ) ( ) ̂( ) ̂ ( ) ̂ ( )]       [                       ]          and   [            ]     [                   ]      Fig. 6 shows  ( ) as    ranges from 0.1 – 0.63.                                        7 DISCUSSIONS AND CONCLUSIONS It is seen that model reference adaptive systems which can be represented as nonlinear discrete maps undergo various bifurcations before the system becomes unstable and eventual-ly unbounded. Before the system becomes unbounded it traverses a region of increasingly complex dynamics charac-terised by random unpredictable behaviour which is termed as chaotic. A similar type of behaviour was observed for dif-ferent bifurcation parameter values. A discrete time adaptive control system which had shown to display chaotic behaviour was extended into a higher order system. It was shown that the higher order system was also likely to display chaotic be-haviour under certain parameter conditions. Observations of chaotic behaviour for higher order plants with first order model assumptions were expected because these  higher order systems can be reduced back into second and third order sys-tems by choosing the appropriate parameter values. A gener-alised analytical form for the characteristic equation was de-termined and this allowed the parameter values to be deter-mined for which the system just becomes unstable. However, 0 20 40 60 80 100 120 140 160 180 200 0 10 20 Time y(i) Plot of y(i) with a3=0.1, ( a1=0.6, a2=0.8, p=0.005 )  0 20 40 60 80 100 120 140 160 180 200 -20 0 20 Time y(i) Plot of y(i) with a3=0.6, ( a1=0.6, a2=0.8, p=0.005 )  0 20 40 60 80 100 120 140 160 180 200 -20 0 20 Time y(i) Plot of y(i) with a3=0.62, ( a1=0.6, a2=0.8, p=0.005 )  0 20 40 60 80 100 120 140 160 180 200 -20 0 20 Time y(i) Plot of y(i) with a3=0.63, ( a1=0.6, a2=0.8, p=0.005 )  300 310 320 330 340 350 360 370 380 390 400 0 5 10 Time y(i) Plot of y(i) with a3=0.63, ( a1=0.6, a2=0.8, p=0.005 )  Fig. 6 - Plot of 𝑦(𝑖)  International Journal of Scientific & Engineering Research Volume 4, Issue 10, October-2013                                                                                  ISSN 2229-5518  IJSER © 2013 http://www.ijser.org  to find the values of the bifurcation parameter for which cha-otic behaviour occurs simulation was used. When a second order model assumption to the third order plant was used, the system was found to be stable for a larger range of parameter values. It was seen in section 6.2 as the parameter a3 was increased beyond a critical point the system no longer converged to the desired value but began to diverge slowly and gradually became unbounded. Chaotic behaviour was not observed as in the previous section, and a possible suggestion for this could be that the system does not have unique equilibrium points as was the case for the earlier sys-tems we studied with a first order model assumption. The chaotic phenomena was generally associated with the equilib-rium point continually bifurcating from a stable fixed point to a limit cycle and through continuous period doubling and eventually leading to complex chaotic behaviour.  From a control aspect, it is important to know the dynamical behaviour of a system for different parameter values. It is rea-sonable to assume that local stability can be maintained if the period doubling phenomena can be avoided. Knowledge of the overall system behaviour for different parameter values is crucial in attaining any desired type of behaviour or maybe avoids an undesirable type of behaviour.   REFERENCES [1]    Guckenheimer, J and P. Holmes, Nonlinear Oscillations, Dynamical           Systems and Bifurcations of Vector Fields,  Springer Verlag, New York,            1983 [2]    E. N. Lorenz, ―Deterministic non-periods flows‖, J. Atmos. Sci., vol.20,           pp.130-141, 1963. [3]   J. Lü, J. Lu, S. Chen, Chaotic time series analysis and its application, Wu        han Univ.  Press, Wuhan (2002) [4]   Devany, R.L., Introduction to chaotic dynamical systems , Benjamin         Cummings Publ. Co., Boston, MA, 1986 [5]   Collet, P and J.P. Eckman, Iterated Maps of  the Interval as Dynamical         Systems, Birkhauser, Boston, 1980 [6]   May, R.M., ―Simple mathematical models with very complicated         dynamic‖’, Nature, Vol.201, 1976,  pp. 459. [7]   Ydstie, B.C, ―Bifurcations and complex dynamics in adaptive control          systems‖, Proc. 25th Conf. on Decision and Control, Greece, 1986,  pp.          2232. [8]   C. Hua, B. Yang, G. Ouyang, and X. Guan, ―A new chaotic secure          communication scheme,” Physics Letters A, vol. 342, no. 4, pp. 305–         308, 2005.  

Chaotic Behaviour in Some Discrete –Time Adaptive Control Systems

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