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Port-Hamiltonian modeling of the Memristor and the higher order elements

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of the Memristor and the higher order elements Arnau Dòria-Cerezo Dimitri Jeltsema Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Outline Memristor, the
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of the Memristor and the higher order elements Arnau Dòria-Cerezo Dimitri Jeltsema Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Outline Memristor, the fourth element Memristor, the fourth element Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements 3 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems 4 Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Outline Memristor, the fourth element Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future Memristor, the fourth element Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future 3 4 Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 Classical electrical elements Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future In the classical (linear) circuit theory there are three basic elements, Resistor u(t) = Ri(t) Inductor φ(t) = Li(t) Capacitor q(t) = Cu(t) which describes the relationship between two of the four basic variables: Voltage, u(t), Current, i(t), Charge, q(t), Flux, φ(t), where charges and fluxes t t q(t) = i(τ)dτ, φ(t) = u(τ)dτ Arnau Dòria-Cerezo PH modeling of the Memristor... 4/38 Classical electrical elements Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future All these relationships can be summarized in the following figure i Inductor, L φ Resistor, R d dt d dt??? u Capacitor, C q The question is... how are the fluxes, φ, and charges, q, related? Arnau Dòria-Cerezo PH modeling of the Memristor... 5/38 The missing element Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future L.O. Chua, Memristor- the missing circuit element, IEEE Trans. on Circuit Theory, vol. 8(), pp , 97. The missing relationship between φ and q defines memristor, M. or, alternatively, q = ˆq(φ). Why is called memristor? φ = ˆφ(q) A memrsitor behaves as an ordinary resistor at a given instant of time t, but its resistance (conductance) depends on the complete past history of the memristor current (voltage). u = R(x)i ẋ = i It behaves somewhat like a nonlinear resistor with memory. Arnau Dòria-Cerezo PH modeling of the Memristor... 6/38 The missing element Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future Charge-controlled memristor φ = ˆφ(q). Differentiating with respect to time, and using q = i and φ = u, u = M i (q)i where M i (q) := ˆφ (q) is the incremental memristance. q M i has units of Ohm (Ω), the same units than a linear resistor, R. Flux-controlled memristor q = ˆq(φ). Differentiating with respect to time, and using q = i and φ = u, i = M u(φ)u where M u(φ) := ˆq (φ) is the incremental memductance. φ M u has units of Siemens, (S, or Ω ), the same units than a linear conductance, G. Arnau Dòria-Cerezo PH modeling of the Memristor... 7/38 Memristor properties Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future No discharge property From the voltage/current relationship of a memristor (charge-controlled) u = M i (q)i, notice that u whenever i, regardless of q which incorporates the memory effect. This property is called no energy discharge property. Thus, unlike an inductor or a capacitor, a memristor does not store energy. Passivity A charge-controlled memristor is passive, if, and only if, its incremental memristance, M i (q). Dually for, a flux-controlled memristor, with M u(φ). Arnau Dòria-Cerezo PH modeling of the Memristor... 8/38 Memristor properties Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future A linear memristor is a linear resistor Consider a charge-controlled memristor, φ = ˆφ(q), with a linear function ˆφ = Rq, where R is a parameter. Differentiating with respect to time, and using q = i and φ = u, u = Ri we recover the linear resistor constitutive relationship. Arnau Dòria-Cerezo PH modeling of the Memristor... 9/38 The memristor blueprint Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future Connect a two-terminal u-i memristor (charge-controlled) with a current source u-i characteristic (hysteresis pinched loop) 3 i(t) = sin(t) u(t) M i (q) u[v] with the constitutive relationship φ(t) = 3 q3 (t), i[a] φ[wb] q[c] The obtained relationship between u and i is two-valued (except the origin). We are not able to define this element based only on the measurements. These difficulties suggest that the appropriate model is defined by φ and q. Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Example: a tapered dashpot Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future F F q A tapered dashpot is a mechanical damping which is damping coefficient, b, depends on the displacement q, F = b(q)v where q = v. Its description in terms of F and v yields some complicated constitutive relationship!!! The tapered dashpot as a memristor These difficulties are circumvented by modeling the tapered dashpot as a memristor, with the constitutive relationship p = ˆp(q). Differentiating, and with ṗ = F and q = v, F = ˆp q (q)v, where b := ˆp q (q). Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Memristive systems Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future L.O. Chua and S.M. Kang, Memristive devices and systems, Proc. of the IEEE, vol. 64(), pp. 9-3, 976. Memristors are a special case of broader class of dynamical systems called memristive systems. Current-controlled memristive system u = M i (x,i)i, ẋ = f i (x,i) Voltage-controlled memristive system i = M u(x,u)u, ẋ = f u(x,u) where x denotes the internal state of the system. Corresponds to the q and φ of the pure memristor. Arnau Dòria-Cerezo PH modeling of the Memristor... /38 The memristor, the rediscovery Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future Researchers in HP were studying (very)-small devices for computer applications, and proposed a cross bar architecture with a layer with titanium dioxide (TiO ). They obtained fast switch on-off behavior and also the state remained stable, but they had no physical model for how these devices worked. 3 Current (ma) 4 Pt TiO Pt u[v] 4... Voltage (V) i[a] D.B Strukov, G-S. Snider, D.R. Stewart, and R.S. Williams The missing memristor found, Nature, vol. 453, pp. 8-83, 8. Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 The present of the memristor Classical electrical elements Memristor, the missing element Memristive systems Memristor, present and future Potential applications High-density hard disk. Faster than magnetic disks and use much less power. Non-volatile memories (computers without booting time). The behavior of a learning process in an amoeba can be mapped into the response of a LC circuit and a memristor. Y.V. Pershin, S. La Fontaine, and M. Di Ventra Memristive model of amoeba s learning, Phys. Rev. E, vol. 8 96, 9. A memristor can support important synaptic functions. S.H. Jo, T. Chang, I. Ebong, B.B. Bhadvivya, P. Mazumder, and W. Lu Nanoscale memristor device as synapse in neuromorphic systems, Nano Letters, vol.,. Arnau Dòria-Cerezo PH modeling of the Memristor... 4/38 Outline Memristor, the fourth element The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements Memristor, the fourth element The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements 3 4 Arnau Dòria-Cerezo PH modeling of the Memristor... 5/38 Higher-order elements The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements Higher-order elements provide a logically complete formulation for nonlinear circuit theory. L.O. Chua Device modeling via basic nonlinear circuits elements, IEEE Trans. on Circuits and Systems, vol. 7(), pp. 4-44, 98. There are many devices which cannot be modeled using only conventional circuit elements. Nonlinear circuits containing only conventional circuit elements could exhibit impasse points, thereby implying that the model is nonphysical and inadequate for computer simulations. Each element has an independent identity; nonlinear higher order two-terminal elements can be only synthesized using only conventional and/or other higher-order two-terminal elements. A logically consistent foundation of nonlinear circuit synthesis cannot be built using only conventional circuits elements. Arnau Dòria-Cerezo PH modeling of the Memristor... 6/38 Higher-order elements The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements A (two-terminal) higher-order nonlinear circuit element is represented by u(t) i(t) α β where α,β =,±,±, defines u (α) = dα dt α u, i (β) = dβ dtβi, then u (α) = f (i (β)) Conventional circuit elements α =, β = is a resistor (u = f(i)). α =, β = is an inductor (φ = f(i)). α =, β = is a capacitor (u = f(q)). α =, β = is a memristor (φ = f(q)). Arnau Dòria-Cerezo PH modeling of the Memristor... 7/38 The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements The periodic table of all -terminal circuit elements β 3 i(t) q(t) α The (,)-element is a Resistor The (-,)-element is a Inductor The (,-)-element is a Capacitor The (-,-)-element is a Memristor φ(t) u(t) Arnau Dòria-Cerezo PH modeling of the Memristor... 8/38 The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements The periodic table of all -terminal circuit elements β 3 i(t) α M. di Ventra, Y. Pershin, and L.O. Chua Circuits elements with memory: memristors, memcapacitors and meminductors Proc. IEEE, 97(), pp , 9. q(t) σ(t) 3 The (-,-)-element is Meminductor The (-,-)-element is a Memcapacitor 3 3 ρ(t) φ(t) u(t) Arnau Dòria-Cerezo PH modeling of the Memristor... 9/38 Example: the cable reel system The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements qs mass transmission point g R θ θ, ω: angular position and angular speed ( θ = ω) p, τ: angular momentum and torque (ṗ = τ) I(θ): position-depending inertia The motion of a reel is given by p = I(θ)ω. The cable reel system as a meminductor (meminertia) 3 ω θ ρ p β τ α Defining ρ := t p(s)ds, the constitutive relationship for this element is ρ = ˆρ(θ). Differentiating, and with ρ = p and θ = ω, where I(θ) := ˆρ θ (θ). p = ˆρ θ (θ)ω, Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Example: the cable reel system Suppose that the cable reel has the following inertia I(θ) = I o +µr 3 θ, The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements where I o is the inertia of the empty reel, and µ is the mass per unit length. Apply a sinusoidal angular velocity ω = A sin(πft) I[kg m ] 4 8 p[kg m s ] µ=kgm µ=5kgm µ=kgm 5 µ=kgm µ=5kgm µ=kgm 5 ω[rad s ] 5 5 ω[rad s ] 5 We obtain, again, a pinched hysteresis loop, similarly to the memristor. Arnau Dòria-Cerezo PH modeling of the Memristor... /38 Example: the cable reel system The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements The motion of a reel is given by p = I(θ)ω. Remarks Fluxes and currents in electrical machines are related by φ = L(θ)i, but, they are not mem-elements since θ i, i.e., θ and i are not in the same physical domain. In mechanical systems, changes of coordinates are often used, q = Ψ( q), yielding relationships in the form of p = M( q)ṽ but they are not memelements, because the mass remains constant. Arnau Dòria-Cerezo PH modeling of the Memristor... /38 The Universe of the two-terminal elements The periodic table of all two-terminal circuit elements Properties of the two-terminal elements Properties of the two-terminal elements γ=4 γ=3 γ= γ= β Resistance Conductance Negative Res. Inductance Resistance Conductance Negative Res. Inductance α Resistance All elements with an identical voltage-current exponent difference γ = β α have an identical small-signal impedance Z Q (jω) at all operating points Q, and they are indistinguishable from each other. γ= γ= γ= γ= 3 γ= 4 L.O. Chua Nonlinear circuit foundations for nanodevices, part I: the four-element torus Proc. IEEE, 9(), pp , 3. Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 Outline Memristor, the fourth element Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems Memristor, the fourth element 3 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems 4 Arnau Dòria-Cerezo PH modeling of the Memristor... 4/38 Port-Hamiltonian formalism Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems The port-hamiltonian modeling Physical structure (energy and interconnection) Re-usability (libraries) Multi-physics approach Suited to design/control In this universe of elements, PH model could help... to understand the physics of the memristive and higher order systems, to analyze dynamical properties of these elements, to model huge fields of memristive elements by interconnection. Arnau Dòria-Cerezo PH modeling of the Memristor... 5/38 Port-Hamiltonian formalism Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems A major generalization of the classical Hamiltonian formalism is given by the port-hamiltonian equations { ẋ = (J R) H Σ : x (x)+g Pf P e P = G T P H x (x) where x R n, are the state variables, H(x), is the Hamiltonian (represents the stored energy), f P,e P R m, denotes the port system variables, G P R n m, is the input distribution matrix, J = J T R n n, is the interconnection matrix, R = R T R n n, is the dissipative matrix. Arnau Dòria-Cerezo PH modeling of the Memristor... 6/38 Port-Hamiltonian formalism Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems Domains and variables used in the port-hamiltonian framework (with the Generalized Bond Graph Framework). Physical domain Flow, f Effort, e State variable, x electric current voltage charge magnetic voltage current flux linkeage potential translation velocity force displacement kinetic translation force velocity momentum potential rotation angular velocity torque angular displacement kinetic rotation torque angular velocity angular momentum potential hydraulic volume flow pressure volume kinetic hydraulic pressure volume flow flow tube momentum chemical molar flow chemical potential number of moles thermal entropy flow temperature entropy Arnau Dòria-Cerezo PH modeling of the Memristor... 7/38 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems Memrisitive port-hamiltonian systems D. Jeltsema and A.J. van der Schaft, Memristive port-hamiltonian systems, Mathematical and Computer Modelling of Dynamical Systems, 6(), pp.75-93,. The memristor constitutive relationship (φ = ˆφ(q), or q = ˆq(φ)), in terms of the port system variables e M,f M, ( ) e ( ) M = M f ( ) M A memristor is a non-energetic system, and it can be represented by an implicit ph system with a feedthrough term of the form where M f := M f ( ) M { ẋf = f M Σ M : e M = H M (x x f )+M f (x f )f M f, and H M, is called the null-hamiltonian, which simplifies in { ẋf = f Σ M : M e M = M f (x f )f M Arnau Dòria-Cerezo PH modeling of the Memristor... 8/38 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems Port-Hamiltonian systems with memristors A (n-port) memristor { ẋf = f M Σ M : e M = H M (x x f )+M f (x f )f M f can be easily interconnected with a standard ph system { ẋ = (J R) H Σ P : x (x)+g Pf P e P = G T H P x (x) via f P = e M and f M = e P, yielding the ph system with memristive dissipation ( ) ( ẋ J R M(xf ) G = P ẋ f G T P ) ( H x (x) H M x f (x f ) ), where the memristive structure matrix M(x f ) := G P M f (x f )G P, and the Hamiltonian H T := H +H M = H. Arnau Dòria-Cerezo PH modeling of the Memristor... 9/38 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems Example: Mechanical system with a tapered dashpot q q d m k m Associate port-hamiltonian equations: ṗ ṗ q k q d = with the Hamiltonian M v(q d ) M v(q d ) M v(q d ) M v(q d ) H(p,p,q k ) = m p + m p + k q k. H p H p H q k H q d, Remark p,p and q k represent the energy stored in the system, since q d represents the memory effect. Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems PH systems with meminductances Mem-inductanctors (and mem-capacitors) can be also represented with the port-hamiltonian formalism. Similarly to the memristor, the ph is extended with a memory variable, but in this case, the Hamiltonian function is not null (this element stores energy). A dissipative term appears (indicates the variation of the inductance/capacitance). Let us to illustrate with the cable reel mechanical example. R θ qs mass transmission point g Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 Example: the cable reel system Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems qs mass transmission point g R θ The motion of a reel which its mass is varying on the position θ is given by The energy stored is Since ṗ = τ and θ = ω, we obtain ( ) ( ṗ D(p,θ) = θ where ω = ( )( H p H θ H x D(p,θ) := m H x p = I(θ)ω. H(p,θ) = I(θ) p. ) ( ) H p H ) θ = I θ (I(θ)) p. ( + ) τ Arnau Dòria-Cerezo PH modeling of the Memristor... 3/38 Example: the cable reel system Port-Hamiltonian formalism Memrisitive port-hamiltonian systems Meminductive(-capacitive) ph systems where and Remarks ( ṗ θ ) = ( D(p,θ) ω = ( )( H p H θ ) ( H p H ) θ H(p,θ) = I(θ) p, H θ D(p,θ) := H p ) = I θ (I(θ)) p. ( + ) τ D(p,θ) can be also written as D(p,θ) = I(θ), and represents the gained or lost inertia. This element stores energy. The energy is also modulated by θ. Two kind of states: energy state p, and memory state θ. Arnau Dòria-Cerezo PH modeling of the Memristor... 33/38 Outline Memristor, the fourth element Memristor, the fourth element 3 4 Arnau Dòria-Cerezo PH modeling of the Memristor... 34/38 The higher-order elements All elements of the α β Universe are physically possible? Analysis of higher order elements: passivity (cyclo-passivity), losslessness... Modeling higher-order elements It is possible to extend the ph formalism to the higher order elements? How we can represent the memory elements using other frameworks (Lagrangian, Brayton-Moser)? Control applications If memristors has the ability of learning, they can be used for adaptive, predictive or gain scheduling control point of view? Using the port-hamiltonian, memory elements could h
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