Stochastic perspective and parameter estimation for RC and RLC electrical circuits

Transcription

1 Int. J. Nonlinear Anal. Appl. 6 (5 No., 53-6 ISSN: 8-68 (electronic Stochastic perspective and parameter estimation for C and LC electrical circuits P. Nabati a,,. Farnoosh b a Department of Mathematics, Urmia University of Technology, Urmia, Iran. b Department of Mathematics, Iran University of Science and Technology, Tehran, Iran. (Communicated by M. Eshaghi Gordji Abstract The main focus of this paper is to examine the effects of noise perturbations on the parameters of C and LC electrical circuits. For this purpose the input voltage is assumed to be corrupted by the noise and the current is observed at discrete time points. The deterministic models will be transferde to stochastic differential equations and then these models will be solved analytically. Scince there is no always a closed form for stochastic differential equations, then these models will be solved numerically based on Euler- maruyama scheme. The parameter estimation for these stochastic models are investigated when the resistance is missing data that it is a concern in electrical engeineering. This paper can be used in various types of electrical engineering real time projects. The projects include electrical circuits, electrical machines theory and drives. Some numerical simulations via Matlab programming are carried out inorder to show the efficiency and accuracy of the present work. Keywords: Stochastic Differential Equation, White noise, Simulation, Electrical Circuits. MSC: Primary 6H; Secondary 6H35.. Introduction Fluctuations in statistical mechanics are usually modeled by adding a stochastic term to the deterministic differential equation. By doing this one obtains what is called a Stochastic Differential Equations (SDEs, and the term stochastic called noise []. Thus, a SDE is a differential equation in which one or more of the terms is a stochastic process, hence resulting in a solution which is itself a Corresponding author addresses: nabati@iust.ac.ir (P. Nabati, rfarnoosh@iust.ac.ir (. Farnoosh eceived: June 4 evised: Desember 4

2 54 Nabati, Farnoosh stochastic process. SDE models play a relevant role in many application areas including environmental modeling, engineering and biological modeling. They topically describe the time dynamics of the evolution of a state vector, based on the (approximate physics of the real system, together with a driving noise process. The noise processes can be thought of in several ways. It often represents processes not included in the model, but present in the real system. The accurate parameter estimation of both the drift and diffusion terms of the aformentioned models is a concern. Two type of noises can exist in an electrical circuit, internal noise and external noise. Internal noise is assumed to be Gaussian noise. In the internal noise, value of the random field in a given point at a given time does not depend on its value in other points or at other times. Internal noise be called white noise. The effects of this type of noise in electrical circuits has been studied by many researchers. For example, Kampowsky et al, described classification and numerical simulation of electrical circuits with white noise [6]. Furthermore, Penski, was presented a new numerical method for SDEs with white noise and its application in circuit simulation [3]. ecently, awat, showed an application of the Ito stochastic calculus to the problem of modeling a series C circuit with white noise and colored noise, including numerical solution [5]. Farnoosh et al, presents a stochastic perspective for L electrical circuits with different noise terms [3]. External noise is assumed to be color noise which correlation of the random field between differential points and different times could be nonzero. Many researchers have been studied about colored noise and that applications [5],[5]. The aim of this paper is to analysis the effects of white noise perturbations on the parameters of electrical networks and then investigate the problem of parameter estimation for these models using Least Square Estimator (LSE. The strong consistency of the LSE is studied in [7]. The convergence in probability is discussed in [] and the asymptotic distribution was studied in [4]. The rest of this paper is organized as follows. In section, the deterministic and stochastic models for C and LC circuits are presented. The SDEs of these circuits with white noise are established in subsections. and. respectivly. The numerical solutions for these models are performs in section 3. Section 4 describes parameter estimation based on least square estimator and apply it to finding the resistance in C and LC circuits when it is an unavailable data.. Problem formulation Any electrical circuit consists of resistor(, capacitor(c and inductor(l. These circuit elements can be combined to form an electrical circuits in four distinct ways: the C, L, LC and LC circuits. The effects of noise terms in the L electrical circuit presented in [3], then now we investigate this problem for C and LC networks... Modeling C circuit with stochastic source When a capacitor is placed with a battery and a resistor, the capacitor charges up to the voltage of the battery. This kind of circuit is called a C circuit because the only two components besides a power supply are a resistor and a capacitor. If the charge on the capacitor is Q and the current flowing in the circuit is I, the voltage across and C are I and Q C respectively. By the Kirchhoff s law that says the voltage between any two points has to be independent of the path used to travel between the two points,

3 Stochastic perspective and parameter estimation...6 (5 No., I(t + Q(t = V (t C Assuming that, C and V are known, this is still one differential equation in two unknowns, I and Q. However the two unknowns are related by I(t = dq(t so that dt Q (t + Q(t = V (t (. C If V (t is a piecewise continuous function, the solution of the first order linear differential equation ( is: Q(t = Q( exp( t C + exp( (t s V (sds (. C Where Q( is the initial charge stored in the capacitor. Now let us allow some randomness in the potential source. Then voltage may not be deterministic but of the form: V (t = V (t + noise = V (t + αξ(t (.3 Where ξ(t is a white noise process of mean zero and variance one, and α is nonnegative constant, known as the intensity of noise. With substitute (3 in the (, we have dq(t = CV (t Q(t dt + α db(t, (.4 C where, C and α are constant and ξ(t = db(t dt. To solve this equation we define the function, h(t, Q(t = e t C Q(t, using the Ito formula, the derivative of this function is as follows: so dh(t, Q(t = d(e t C Q(t = e t C Q(tdt + e t C dq(t C = e t C Q(tdt + e t C [ V (t Q(t dt dt + α db(t] C = e t C [V (tdt + αdb(t], From this we get the solution Q(t = e t C Q( + dh(t, Q(tdt = Q(te t C = Q( + V (se s t C ds + α V (se s C ds + α V (se s C ds + α e s C db(s, e s C db(s, e s t C db(s. (.5

4 56 Nabati, Farnoosh The solution Q(t is a random process and for it s expectation we have for every t > and E(Q(t = e t C E(Q( + V (se s t C ds (.6 dk(t = ( (t k(t + m(tv dt C + α. Where k( = E[Q (], m(t = E[Q(t] and k(t = E[Q(t ]. Then, solution Q(t is a Gaussian process, based on the properties of the normal distribution. We can compute in any t, that P ( Q(t m(t <.96σ(t =.95 As we are able to compute m(t and σ(t, we can predict with a probability 95 presented the interval (m(t ξ, m(t + ξ, where the trajectories of the stochastic solution take place... Modeling LC circuit with stochastic source An (LC circuit, is an electrical circuit composed of resistor, capacitor and inductor driven by a voltage or current source. The charge Q(t at time t at a fixed point in an electrical circuit according Kirchhoff s law satisfies the differential equation [8] V (t + I(t + L di dt + C where I(t = dq(t dt then I(tdt =, (.7 L Q(t + Q(t + C Q(t = V (t, Q( = Q, Q( = I, (.8 where L is inductance, is resistance, C is capacitance and V (t is the potential source at time t. Now we may have a situation where some of the coefficients, say V (t, are not deterministic but of the form V (t = V (t + noise. Observation indicated that the noise can be described as a multiple of the so called white noise process denoted by ξ(t, we get the following equation, L Q(t + Q(t + C Q(t = V (t + αξ t, Q( = Q, Q( = I, (.9 where α is the intensity of noise. In order ( to solve analytically ( equation (9, we introduce the vector X (t Q(t X = = and obtain X (t Q(t { dx (t = X (tdt LdX (t = ( X (t C X (t + V (tdt + αdb t, (. or in matrix form dx(t = AX(tdt + H(tdt + KdB(t, (.

5 Stochastic perspective and parameter estimation...6 (5 No., where dx = ( dx (t dx (t ( A = CL L ( H(t = V (t L K = ( α L and B t is one dimensional Brownian motion. ewrite equation ( as exp( AtdX(t = exp( AtAX(tdt + exp( At[H(tdt + KdB t ], (. where for a general n n matrix A we define exp(a = n= n! An. Applying a dimensional ( version of the Ito formula for the function g : [, x (t given by g(t, x, x = exp( At x (t we obtain, d(exp( AtX(t = ( Aexp( AtX(tdt + exp( AtdX(t (.3 substituted in ( this gives, or exp( AtX(t X( = exp( AsH(sds + X(t = exp(at[x( + exp( AtKB t + by integration by parts ( []. Corollary: let A = a b where, then, exp( AsKdB s (.4 exp( As[H(s + AKB s ]ds] (.5 exp(at = λt ξ{(ξ cos(ξt + λ sin(ξti + A sin(ξt}, λ = b, ξ = a b 4. The solution X(t is a random process and for it s expectation we have for every t > E(X(t = exp(ate(x + E( exp(a(t sh(sds + Since E( f(sdb s = then it can be easily shown that 3. Numerical Simulation E(X(t = exp(ate(x + e A(s t H(sds exp(a(t skdb s There is no always a closed form solutions for SDEs, hence researchers have looked for solving them numerically. The simplest numerical scheme, the stochastic Euler scheme, is based on numerical methods for ordinary differential equations [9]. For a given positive integer n, let t = T/n and consider the partitions, Π n = {, t, t,..., (n t, T }, of the interval [,T].

6 58 Nabati, Farnoosh To simulate Q(t, in SDE (4 numerical techniques have to be used. The Euler scheme for system equation (4 is as follows [], Q i+ = Q i + CV i Q i C t + α B i, (3. Where B i = B i+ B i N(, t i and Q i = Q(t i. Similarly we can use the EM method for second order SDE (. The simulated Q(t is as folows: { X (i + X (i = X (i t i X (i + X (i = ( L X (i CL X (i + L V ( t i t i + α L B i. (3. Where B i+ B i N(, t i, X ( = Q(, X ( = I(. Eample : Assume, C and V(t be constants. The results of the stochastic solution of the C circuit with stochastic potential source and with α = and α = are showm in Figures and respectivly. Eample : Let us consider the LC electrical circuit, when L,,C and V(t are constants, I =. Using the Euler scheme we compute and graph the mean and three sample pass of stochastic solutions for I(t and Q(t with α = in fig. 3, and α = in fig Parameter Estimation The parameters estimation of second order SDEs have received less attention, thus estimating the parameters both the drift and diffusion terms of these linear and nonlinear models is an interesting topic in itself. For estimating the parameters of C and LC circuits we use the Least Square Estimation (LSE method. The LSE of for SDE (6 is to minimize the following contrast function Φ n ( = i= [(Q i+ Q i C.V i Q i C t], (4. i= Then the LSE ˆ n is defined as: ˆ n = (Q i+ Q i (CV i Q i C t (Q i+ Q i. (4. i= It is important to identify the parameters from the observed data when some of them are unavailable. For estimation the parameter in LC circuit by using the equation (7 we can define the contrast function as follows: Ψ n ( = [L(X (i + X (i + (X (i + C X (i V t n t n ], (4.3 i= the LSE ˆ n is defined as ˆ n = argmin Ψ n (, which can be explicitly represented as, ˆ n = X (i t i [L(X (i + X (i + C X (i t i + V ( t i ] (X (i t i (4.4 i= i= In simulation study we use these estimators for simulated data set (=, T= and different values for n and α. The results are shown in Tables and.

7 Stochastic perspective and parameter estimation...6 (5 No., Q(t t Figure : Numerical simulation for C circuit with stochastic source and α =..5 Q(t t Figure : Numerical simulation for C circuit with stochastic source and α =. Table : The numerical value of estimator ˆ in C circuit where the data in parenthesis is relative error Least Square Estimator n α = α =. α = (5.77e-5.3 (.7e-7. (3.45e (4.6e-8. (.54e-9. (.e-. (.9e (3.95e-9. (.E- Table : The numerical value of estimator ˆ in LC circuit where the data in parenthesis is relative error Least Square Estimator n α = α =. α =..7 (4.83e-5.9 (3.43e (.45e (4.75e (.9e-6. (9.e (3.75e (5.56e (9.e-7

8 6 Nabati, Farnoosh 5 8 Current(I(t 6 Charge(Q(t Time (t Time (t Figure 3: The mean and three sample pass of stochastic solution for LC circuit with α = Current(I(t 8 6 Charge(Q(t Time (t Time (t Figure 4: The mean and three sample pass of stochastic solution for LC circuit with α =. eferences [] L. Arnold, Stochastic Differential Equations, Theory and Applications, Johan Wiley Sons, (974. [] A. L. Breton, On continues and discrete sampling for parameter estimation in diffusion type processes, Math program, (976. [3]. Farnoosh, P. Nabati,. ezaeyan and M. Ebrahimi, A stochastic perspective of L electrical circuit using different noise terms, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol 3, (, 8-8. [4] D. Ham and A. Hajimiri, Complete noise analysis for CMOS switching mixtures via SDEs, IEEE Custom Integrated circuits conference, (, [5] Y. Kamarianakis and N. Frangos, Deterministic and Stochastic Differential Equations modeling for electrical networks, Hellenic and European esearch in Computational Mathematics Conference, Athens University of Economics Businness, (. [6] W. Kampowsky, P. entrop and W. Schmidt, Classification and numerical simulation of electric circuits, Surveys Math. Indust, (99, [7]. A. Kasonga, The consistency of a nonlinear least square estimator for diffusion processes, Stochastic processes Appl, (988. [8] C. Kim and E. K. Lee, Numerical Method for Solving Differential Equations with Dichotomous noise, Physical eview, E73, 6 (6. [9] P. E. Kloeden and E. Platen, Numerical solution of Stochastic Differential Equations, Springer, Berlin, (995. [] E. Kolarova, Modeling L electrical circuits by SDEs, International conference on computer as a tool, (5, [] G. N. Milstein, Numerical integration of SDE, Kluwer Academic Publishers, Dordrecht, (995. [] B. Oksendal, Stochastic Differential Equations, An introduction with applications, Springer-Verlage, (. [3] C. Penski, A new numerical method for SDEs and its application in circuit simulation, Com. and Appl. Math, (, [4] B. L. S. Prakasa ao, Asymptotic theory for nonlinear least square estimator for diffusion processes, math. oper.forsch.stat, (983.

9 Stochastic perspective and parameter estimation...6 (5 No., [5] T. K. awat and H. Parthasarathy, Modeling of an C circuit using a SDEs, Int. J. SC. Tech, vol. 3, No., (8. [6] Q. Su and K. Strunz, Stochastic circuit modeling with Hermite polynomial chaos, In electronic Letters, Vol. 4, No., (5.