On Chaotic Behavior of a Class of Linear Systems with Memristive Feedback Control

Transcription

1 Proceedings of the th WSEAS International Conference on SYSTEMS On Chaotic Behavior of a Class of Linear Systems with Memristive Feedback Control JOSEF HRUSAK, MILAN STORK, DANIEL MAYER Department of Applied Electronics and Telecommunications Theory of Electrical Engineering University of West Bohemia P.O. Box 4, 064 Plzen Czech Republic hrusak@kae.zcu.cz, stork@kae.zcu.cz, mayer@kte.zcu.cz Abstract: - The contribution is concerned on structural properties and general features of the new ideal circuit element, a memristor. By definition, a memristor relates the charge q and the magnetic flux φ in a circuit, and complements a resistor R, a capacitor C, and an inductor L as an ingredient of ideal electrical circuits. The definition of the memristor is based solely on fundamental circuit variables, similar to the resistor, capacitor, and inductor. Unlike conventional electrical circuit elements, memristors do not belong to the class of linear time-invariant systems, because a linear time-invariant memristor is simply a conventional resistor. In the paper a special class of nonlinear feedback systems consisting of finite dimensional linear dynamical subsystems controlled by a memristance-based feedback controller with static nonlinearity is investigated. Key-Words: Memristance, feedback, nonlinearity, state space energy, chaotic behavior, dissipation structure. Introduction The concept of memristance was introduced and named by Leon Chua in his seminal paper [] in the year 97. The existence of the memristor as the fourth ideal electrical circuit element was predicted in 97 based on logical symmetry arguments, but it was clearly experimentally demonstrated thirty seven years later. It was not before April 0, 008, that a team at Hewlett Packard Labs reported about discovery of a switching memristor in the famous Nature article [], []. Remind that conventional circuit element - capacitor was discovered 745, the resistor 87, and inductor 8. Everybody knows that nonlinearity is universal property of any real-world system. Similarly the memristance is now known to be an intrinsic property of any real electronic circuit. Its existence could even have been deduced already by James Clerk Maxwell or by Gustav Kirchhoff, if either had consider nonlinear circuits in the 800 s. However, the scales at which the electric and electronic devices have been build for most of the past two centuries have prevented experimental observation of the memristance effect. Now, the situation changes dramatically. As we build smaller and smaller electronic devices, memristance is becoming significantly more noticeable and in nano-size it becomes even dominant. One immediate application offers an enabling low-cost technology for non-volatile memories. As a result, e.g. future computers would turn on instantly without the usual "booting time", currently required in all personal computers. Chua also speculates that they may be useful in the construction of artificial neural networks [4], [5]. Chua strongly believed that a fourth device existed to provide conceptual symmetry with the resistor, inductor, and capacitor. This symmetry follows from the description of basic passive circuit elements as defined by a relation between two of the four fundamental circuit variables, namely voltage, current, charge and flux. In principle, the memristor is a device linking charge and flux as time integrals of current and voltage. It is obvious that in connection with the concept of memristance the restriction on electrical interpretation of state space variables is not necessary. In this paper, fundamental properties of memristance based feedback systems are investigated. Memristor fundamentals As the electrical engineer Leon Chua pointed out, the fourth passive element - the memristor - should in fact be added to the list of basic circuit's elements. It is just the inability to duplicate the properties of the memistor with a combination of the other three passive circuit's elements, what makes the existence question of the memristor fundamental. The memristor is mathematically described by two nonlinear constitutive relations v = M ( q) i, or i = W ( ϕ ) v () between the terminal voltage v and terminal current i. The nonlinear functions M(q) and W(φ), are called the memristance and memductance, and are defined by [6]: dϕ( q) M( q) =, dq dq( ϕ) W ( ϕ) = () dϕ ISSN: ISBN:

2 Proceedings of the th WSEAS International Conference on SYSTEMS Notice that the functions M(q) and W(φ) determine the slope of static nonlinearities represented by nonlinear functions φ=φ(q) and q= q(φ), respectively. Recall that resistance R, is the rate of change of voltage with current; capacitance C, is the rate of change of charge with voltage; and similarly inductance L, is that of flux with current. For better understanding is the situation illustrated in the Fig.. i Resistor dv = Rdi Inductor dϕ = Ldi v dϕ = vdt ϕ Capacitor dq = Cdv dq = idt Memristor dϕ = Mdq Fig.. Logically complete set of passive circuit elements Thus, the memristor acts as a nonlinear resistor the dissipation power of which depends on the history of the circuit state variables, e.g. the voltage across it. On the other hand, it implies that if the memristance increases rapidly, current and power consumption will quickly stop. This is the essence of the memory effect. Dissipative system structures Let the system is defined by a given representation 0 () t = f[ x()] t + B u(), t x( t0 ) = x, () y() t = C x() t Let s notice that the vector field f of the state space velocity vector can, without any restriction of generality, be expressed in the semilinear form: f x () t = A ( x ) x () t (4) Recall that according to Liouville s theorem of vector analysis, dissipative systems have the important property that any volume of the state space strictly decreases under the action of the system flow. For linear as well as for a nonlinear systems with the state velocity vector given by a vector field f, the property of dissipativity is defined by: n f ( ) i div f( x ) = x < 0 (5) i= xi Thus a system defined by a matrix is dissipative if the matrix A(x) as negative trace. q It follows from the state space energy conservation principle that a special form of a structurally dissipative state equivalent system representation can be derived. It has been called generalized dissipation normal form. Its internal structure is determined by the following matrix: A α α α α α α α 0 0 = 0 0 αn αn αn, n α n αn αnn, (6) The structure of this representation is shown in Fig.. It is completely characterized by minimal number of independent (internal) state energy storage elements, represented by minimal number of state variables. ut () yt () ut () un() t α x β γ β β n x -α -α α α 4 -α -α -α -α nn x -α 4 n xn γ n γ yt () β γ Fig.. Structure of the dissipation normal form ut () yt () yt n() In some situations not only the state minimality, but also a property of parametric minimality is required. ISSN: ISBN:

3 Proceedings of the th WSEAS International Conference on SYSTEMS In the important special case of parametrically minimal system representation the internal structure reduces to: α α α 0 α α A = 0 0 αn 0 (7) α n 0 α n α n 0 Because both the derived system structures satisfy an abstract form of state space energy conservation principle, we have called them physically correct. It is worthwhile to notice that any of internal or external power-informational interactions, as depicted in the Fig.. may be nonlinear with respect to inputs, as well as to state variables. 4 Dissipativity and conservativity System representations having zero divergence preserve volume along state trajectories and are referred to as conservative. If a representation is neither dissipative, nor conservative, instability appears. Time evolution of the abstract state space energy E[x(t)] illustrating corresponding typical situations for the 6th order linear system is displayed in the Fig.. 5. i, i {,,,n}:α i 0,α 0 is equivalent to parametric and state minimality 6. i, i {,,,n}:α i 0,β 0 is equivalent to structural controlability 7. i, i {,,,n}:α i 0,γ 0 is equivalent to structural observability 8. i, i {,,,n}:α i 0,α > 0 is equivalent to structural asymptotic stability 5 Chaoticity due to memristive feedback Recall that the memristor represents a new nonlinear passive element of electrical circuits theory which may relate some state variable to flux without storing a magnetic field. As a motivating example, let s consider a 4th order nonlinear circuit with the internal structure corresponding to the matrix (7) with the constant parameter α (0)= interpreted as a resistance: = αx+ αx α + 0x 0 = αx+ αx α α 0 = = = αx + α4x4 α, x (0) = 0.5 (8) 4 = α4x α Consequently the corresponding nonlinear function α (.) should consistently be considered as a memristance. Typical course of the state space energy E(x) defined by E( x ) = x + x + x + x = ρ [ x,0] (9) and demonstrating chaotic nature of system behavior caused by the memristance is shown in Fig. 4. Fig. 4. Time evolution of the chaotic state space energy Fig.. Time evolution of state space energy E[x(t)] a) conservative case α = 0, e) instability α < 0, b) dissipativity α >0, α =0, c) dissipativity α >0, α 5 =0,d) asymptotic stability α >0, α k 0, k =,,,n Following elementary consequences of the state space energy conservation principle for parametrically minimal dissipation structure are in order:. α > 0 is necessary and sufficient for dissipativity. α < 0 is sufficient for structural instability. α = 0 is necessary and sufficient for conservativity 4. α > 0 is necessary for asymptotic stability Notice that in the first phase the state space energy displayed in the Fig. 4. increases. The course of the curve corresponds exactly to the time evolution in the case e) instability α < 0 shown in the Fig.. After the first transient phase the course changes in the way, which is characteristic for dissipativity. This process is repeating irregularly. It is easy to deduce that irregular alternating of both two typical regimes dissipativity and instability belongs to the most important attributes of chaotic behavior. Even more, it seems to be defining property of the deterministic chaos as a situation in which any motion of a given system is non-periodical, locally unstable and globally bounded. ISSN: ISBN:

4 Proceedings of the th WSEAS International Conference on SYSTEMS In order to give a simple explanation, let s assume that the system output is defined by the relation y(t)=x and and a system input is defined by the nonlinear feedback control u(t)=- Φ(x, x ) given by the memristance term Φ( x, x) = 0xx (0) Obviously for small values of control signal the system behaves as almost linear, the state space energy increases monotonically and local instability follows. On the other hand for large values of control signal the system behavior becomes strongly nonlinear, and as a result the state space energy increase will irregularly stop and, as a consequence of the memristive feedback, the global boundedness results. In the next section an analysis of memristive feedback from the energy dissipation rate point of view follows. 6 Effect of memristive feedback control At this point it seems to be evident that the concept of memristance can be interpreted as a special case of a significantly more general property, occurring in a broad class of nonlinear dynamical systems, including e.g. chaos generating systems. Thus a deep understanding of the memristor s dynamic nature, as well as the non-linear energy dissipation effects is necessary. From this point of view a concept of generalized memristor can be introduced as an abstract power dissipation subsystem of a nonlinear feedback system, the dissipation rate of which depends nonlinearly on the time history of some system state variables. Recall that the memristor represents a new passive element, which may relate some state variable to flux without storing a magnetic field. It means that nonzero current implies instantanously varying charge. On the other hand, it implies that if the memristance increases rapidly, current and power consumption will quickly stop. This is the essence of the memory effect. It follows that memristance can simply be seen as a property of charge-dependent resistance. If no current is applied, the memristance is constant, and consequently memristor reduces to a static circuit element ordinary linear resistor. In order to illustrate the effect, let s consider a simple nd order linear conservative system with input v(t) given by: y( t) + a y( t) = v( t) () Structure of the state representation is assumed: = + α x + u( t) () = α x The state space energy E(x) can be defined by Ex (, x) = E( x) + E( x) = x + x = ρ [ x,0] () We assume the output is given by the relation y(t)=x and a nonlinear control u(t)=-φ(y) has to be specified in such a way that the zero equilibrium state will be locally asymptotically stable in a region D X R. The state space energy of the representation () is observable if and only if a >0, and for power balance relation we get det () = Pt () = x. Φ( x) 0 (4) dt R( s) It means that the Taylor expansion of the mapping Φ(x ) should contain odd terms only, i.e. we define Φ( x) = ε α x βx (5) The structure of the matrix A(x) with feedback is given ε α βx, a A ( x, x) = (6) a, 0 and the system with memristive feedback Φ reads yt () + M( y) yt () + ayt () = 0 (7) Obviously the term M ( y) = ε α βy ( t) (8) plays the role of generalized memristance induced by the static nonlinearity Φ( x) = ε α x βx (9) because for yt () = x the defining relationship holds dφ( y) M ( y) = = ε α βy ( t) dy (0) Notice that in linear case (β=0) the memristance reduces to the resistance R=εα, and in the conservative case (ε=0) it reduces to zero, and thus for the dissipation power we get P(t)=0. 7 Structure of memristive feedback As an example of generalized memristive feedback let s consider an even memristance function M(y) induced by derivative of an odd static nonlinearity illustrated by a typicalϕ - q characteristic shown in the Fig. 5., [7], [8]. q 0 Fig. 5. Typical example of odd static nonlinearity ϕ ISSN: ISBN:

5 Proceedings of the th WSEAS International Conference on SYSTEMS It is obvious that, as a straightforward modification, instead of the generalized memristance M(y) it's inverse W(φ) called generalized memductance can also be used. Fig. 6. can be considered as an example. Fig. 6. Example of an even memductance function As an example of the nonlinear dissipation structure a typical internal structure of the nonlinear feedback system representation is displayed in the Fig. 7. The system consists of a linear subsystem (dissipative or conservative) as presented in sections. and 4., and of a memristive feedback control subsystem as discussed in the part 6., and motivated by chaoticity effects as briefly demonstrated in the part 5., [9]. Memristive feedback control The whole 4 th order nonlinear feedback system is described by corresponding four state equations with the last state equation in integral form, representing explicitly the memristive feedback interaction = W( w) x+ x C C = x+ x L L = x + Gx C C t wt () = x( τ)d τ + wt ( ) t 0 0 () Notice that a standard physical interpretation of system parameters from electrical circuit's theory is used. For G=0 and for given choice of state variables the unique physical circuit structure is displayed in the Fig. 8. w W( w) Fig. 8. The oscillator with flux-controlled memductance /C x /C /C x x / L / L Linear conservative subsystem Fig. 7. Dissipation structure with memristive feedback The linear subsystem of the topological structure shown in the Fig. 7. can be described by the standard rd order state representation () with the nonlinear state velocity vector field given in the semilinear form f x () t = A ( x ) x () t () Because the additional linear dissipation term Gx () C can in principle be included in the memristive feedback in the Fig. 7. it is also not explicitly displayed there. Let s now make a further step and define the gradient vector of a state space energy function E(x) grad xe = [ Cx, Lx, Cx] (4) Using the chosen set of system parameters we get by integration E( x, x, x) = E( x) + E( x) + E( x) (5) where conventional terms for energy components follow E( x) = Cx E( x) = Lx (6) E( x) = Cx Obviously, now the state space energy conservation principle can be expressed as a standard dual product of the state space energy gradient vector grad x E and the state space velocity vector, [0]. ISSN: ISBN:

6 Proceedings of the th WSEAS International Conference on SYSTEMS By substitution from (4) and from state equations () we get it in following form of a power balance relation de = gradx E, = R( ϕ ) dt R( ϕ ) = Cx W( wx ) + x C C + Lx( x + x) L G + Cx x + x = C C W( w) x Gx (7) = + In such a way not only the physical meaning of chosen state variables but physical interpretation of individual state equations can be specified as well. For the given choice it follows e.g. from the eqn. (6) that the first and the third state variables have to be voltages on the corresponding capacitors, while the second one is the current of the inductor. Further we can see from (7) that the resistor with the conductance G would have to be physically connected parallel to the capacitor C, and the memristor with the memductance W(w) has to be connected parallel to the capacitor C. From (7) as well as from the Fig. 7. it follows that nonquadratic term in the dissipation power depends exclusively on x, the voltage v. Therefore it seems natural to say that in this case the voltage-controlled dissipation structure results. It is well known that the following linear nonsingular state space equivalence transformation q = Cv ϕ = Li (8) q = Cv representing a physically motivated linear scaling, (or any of its linear or nonlinear generalizations), can be used as well. Hence, from the first row of (8) we can conclude that the scaled state variable x represents the charge q. In conventional terminology it would mean that in this case we have got a nonlinear dissipative circuit with the so called charge-controlled memristor. Thus we can conclude that physical interpretation of any dissipation structure elements follows from the chosen set of the state variables and from theirs relationship to fundamental variables - charges and fluxes. Some sceptics argue that the memristor is not a fourth fundamental circuit element but an example of bad science. The crux of their arguments rests on two fundamental misunderstandings: at first, they overlook the expanded design space arising from structures based on strongly nonlinear circuit elements. The second and more profound misunderstanding concerns Prof. Chua s mathematical definition of a memristor. At first, most people did not understand the real nature of the general concept of memristance. In paradigm of the conventional electrical circuits theory the concept of memristor actually has been defined strictly by means of electromagnetic interactions expressed by a relationship between electric charge and magnetic flux. However the real content of the Chua s concept of memristance is significantly more general and obviously can bypass electromagnetic interactions altogether. We have demonstrated that proposed significantly more general paradigm of nonlinear dissipation structures based on the state space energy and on the concept of memristive feedback can be useful. It is worthwhile to notice that Chua s memristance definition has two parts: The first one generalizes the Ohm s law to nonlinear situation, and defines how the memristor s voltage depends on the current and on a further changing state variable. The second one describes how the changing state variable depends on the amount of charge flowing through the device. The next modification of the example defined by the eq. () dq = W ( ϕ ) q + ϕ dt C L dϕ = q + q dt C C (9) dq G = ϕ q dt L C dϕ = q dt C can be useful for better understanding. The matrix A(x) of the circuit representing a set of state equations (9) of a rd order nonlinear electrical circuit with the last row as an additional feedback interaction is compatible with the structure shown in the Fig. 7., and can be expressed in the form W ( ϕ) 0 0 C L 0 0 C C (0) A(.) = G 0 0 L L C Dissipative feedback structures depending on a variety of memductance functions W(y) as well as ones with negative inductances -L and capacitances -C can be treated by the same way, too. ISSN: ISBN:

7 Proceedings of the th WSEAS International Conference on SYSTEMS In this connection an example of an active electric circuit in the Fig.9., 0. can provide useful motivation. R R + i L i + + C C () v v Wϕ Fig. 9. Negative inductor L and negative capacitor C realization by using operational amplifier. ϕ W ( ϕ ) Fig.. D projection of chaotic system: φ, v, v. /C x q / L /C ϕ / L q /C Fig. 0. Structure of feedback system with flux-controlled memristor. 8 Simulation results In this section, the simulation results of the circuit which is described by eq. (9) and system structure according Fig. 0. are shown. The simulation parameters were: C = 0 nf, L=-7.95 mh, C =-.6 nf and initial conditions: φ=0, v =0.0, i=0, v =0. The results are shown in Fig.. Fig.. D projection of chaotic system: i, φ. The simulations shows that the system described by eq. (9) and structure according Fig. 0., has a chaotic attractor as shown in Fig.. Calculating the Lyapunov exponents from sampled time series give one positive exponent λ =0.. When the nonlinear function is changed (multiplied by 0 on x-axes and y-axes) the pseudo chaotic system with two distinct unstable periodic attractors is simulation result. The D and D projections are shown in Fig. 4, 5 and 6. Fig.. The time diagram of φ(t). Fig. 4. D projection of pseudo-chaotic system with modified nonlinear function W(φ). ISSN: ISBN:

8 Proceedings of the th WSEAS International Conference on SYSTEMS could replace transistors in future computers, taking up a much smaller area. They can also be fashioned into non-volatile solid-state memory, which would allow greater data density than hard drives with access times potentially similar to DRAM. In this paper we demonstrated that proposed significantly more general paradigm of nonlinear dissipation structures based on the state space energy and on the concept of memristive feedback can be useful in electronic circuits design. Fig. 5. D projection of pseudo-chaotic system with modified nonlinear function W(φ). Variables: v, i. Fig. 6. D projection of pseudo-chaotic system with modified nonlinear function W(φ). Variables: v, v. 9 Conclusion Memristor-based memory devices have the potential to lower power consumption and provide greater reliability in the face of power interruptions to a data center. Another potential application of memristor technology could be the development of computer systems that remember and associate series of events in a manner similar to the way a human brain recognizes patterns. This could substantially improve today s facial recognition technology, enable security and privacy features that recognize a complex set of biometric features of an authorized person to access personal information, or enable an appliance to learn from experience. For some memristors, applied current or voltage will cause a great change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent in order to achieve a desired change in resistance.the solid-state memristors can be combined into devices called crossbar latches, which ACKNOWLEDGMENT This work has been supported from Department of Applied Electronics and Telecommunications, University of West Bohemia, Plzen, Czech Republic and GACR project No. 0/07/047. References: [] Chua, L. O., Memristor-The missing circuit element. IEEE Trans. Circ. Theory.CT-8, 97, pp [] Strukov, D. B., Snider, G. S., Stewart, D. R. & Williams, R. S., The missing memristor found. Nature 45, 008,pp [] Yang, J. J., Pickett, M. D., Li, X., Ohlberg, D. A. A., Stewart, D. R. & Williams, R. S., Memristive switching mechanism for metal/oxide/metal nanodevices. Nature Nano-technology, 008, pp [4] Pershin, Yu. V. & Di Ventra, M., Spin memristive systems: Spin memory effects in semiconductor spintronics. Phys. Rev. B 78, 008. [5] Erokhin, V. & Fontana, M. P., Electrochemically controlled polymeric device: amemristor (and more) found two years ago. arxiv: [6] Itoh, Makoto, Chua, L.O, Memristor Oscillators, Intl. Journ. of Bifurcations and Chaos, Vol. 8, No., 008, pp [7] Chua, L. O., Kocarev, L., Eckert, K. & Itoh, M., Experimental chaos synchronization in Chua's circuit, Int. J. Bifurcation and Chaos (), 99, pp [8] Chua, L. O., Itoh, M., Kocarev, L. & Eckert, K., Chaos synchronization in Chua's circuit, J. Circuits Syst. Comput. (), 99, pp [9] Hrusak J., Mayer D., Stork M., On System Structure Reconstruction Problem And Tellegen- Like Relations, Proc. of 8 th World Multiconf.,SCI, Vol. VIII, Florida, USA, 004, pp [0] Hrusak J., Mayer D., J. Lahoda, Stork M., On Synthesis of Controlled Chaos Based On Lyapunov-Tellegen s Principle, Proc. of Intl. Conf. SICPRO 08, V.A.Trapeznikov Inst. of Control Sciences, Moscow, Russia, 008, pp ISSN: ISBN: