Novel stability and passivity analysis for three types of nonlinear LRC circuits

Article History: Received 08 January 2021 Accepted 09 May 2021 Available 31 July 2021 In this paper, the global asymptotic stability and strong passivity of three types of nonlinear LRC circuits are investigated by utilizing the Lyapunov’s direct method. The stability conditions are obtained by constructing appropriate energy (or Lyapunov) function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. Many specialists construct Lyapunov functions by using some properties of the functions with much trial and errors or for a system they choose candidate Lyapunov functions. So, for a given system the Lyapunov function is not unique. But we insist that the Lyapunov (energy) function is unique for a given physical system. Thus, this study clarifies Lyapunov stability with suitable tools and also improves some previous studies. Our approach is constructing energy function for a given nonlinear system that based on the power-energy relationship of the system. Hence for a dynamical system, the derivative of the Lyapunov function is equal to the negative value of the dissipative power in the system. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The provided results are central importance for the stability analysis of nonlinear systems. Some simulation results are also given successfully that verify the theoretical predictions.


Introduction
In history, modeling and stability analysis of nonlinear systems are the most important and popular problems in control theory. Since almost all systems are nonlinear in nature [1], a number of promising studies have been analyzed in the literature. Many researchers as Lagrange, Hamilton, Poincare and Lyapunov are focused on the modelling problem to analyze the dynamic behavior of systems [1,2]. The aforementioned methods are based on the energy utilization of the related systems. However, since all systems are not in linear forms, certain mathematical solutions are not available to solve these issues. Furthermore, closed-form expressions for the solutions of the linear systems are not possible to solve nonlinear systems. Nevertheless, it is important to be able to make some assumptions about the conduct of a nonlinear system called qualitative analysis.
The stability of the equilibrium point was first examined by Lagrange; however, the Lagrange principle was only suitable for the Lagrange systems (conservative systems) [1], but engineering systems usually have damping [3]. Then, the stability theory of motion derived from the concepts of Lagrange's principle and Poincare's regular solution (Lyapunov stable motion) was developed by Lyapunov [2]. Hamiltonian and Lagrangian systems comply with conservative systems (exact differential equations), but Lyapunov stability theory can be applicable to arbitrary differential equations. Thus, the Lyapunov direct method is *Corresponding Author 227 the most common and efficient tool for stability analysis [4][5][6][7]. In this context, many elegant studies on the qualitative behavior of systems can be recorded in the literature. Most experiments are carried out on the axiomatization of the stability principle. The problem of stability of the solar system attracted a great deal of early interest. Then, Lyapunov used his second (direct) method that there is no need to solve the differential equations explicitly to investigate the stability of the given systems. The Lyapunov's direct method is still recognized as an effective tool to study the stability theory of dynamical systems such as: the global asymptotic stability of the electrical RLC circuit [8], neural networks with time varying delays [9,10], power systems analysis [11], robot manipulators [12], dissipativity analysis of discretetime neural networks [13], global robust passivity analysis [14], dissipativity and passivity analysis of neural networks [15]. This method is the best way to determine the asymptotic stability or asymptotic controllability of nonlinear systems. The central notion is that the energy of the system diminishes along suitably chosen paths, such that the system attains a minimal energy configuration at the invariant equilibrium. Here, this result has been presented both mathematically and through simulation.
Lyapunov theory is based on the Torricelli principle [16]. Therefore, the storage energy of the dynamic system decreases over time along the trajectories of the system. So, the direct method provides the opportunity to examine the stability of the equilibrium points with minimum energy. This meaning (diminishing of energy) tends us to the passivity of the systems. Passivity, which is the basic feature of the dynamic systems theory [17][18][19][20], can now be debated. LRC circuits, viscoelastic systems and thermodynamic systems are typical examples of dissipative systems with the external sources. The terminology of dissipativity is a generalization of the concept of passivity [21]. Apparently, a dissipative system is not a conservative system. The main point of passivity theory is that the systems are internally stable [22,23]. Storage functions are bounded [21], and this result has been proved mathematically in the proof of Theorem 4. Thus, some new passivity results with Gronwall's inequality [3] can be shown to define the strict passivity or boundedness of the systems involved. Natural (real) energy functions of the dynamic systems empower the Lyapunov's direct process implementations more than the Lyapunov candidate functions. Thus, each energy function used in this study is constructed from the physical meaning of the given system and its time derivative (directional) is equal to the negative value of the dissipated power in the system. For example, for any unforced dissipative system, the time derivative of the energy (Lyapunov) function E(t) along the system orbits gives where R i is the damping term (or resistance) and I i is the velocity (or current) of the ith component of the system.
The above arguments are not clear in the related literature. Hence, many specialists chose candidate Lyapunov functions or consider some Lyapunov functions or construct Lyapunov functions with much trial and error for their systems [1,11,24,25] without any physical meaning. Generally, these tools make Lyapunov stability very complex (see [24,25]). Because, still there is an idea in the literature, constructing Lyapunov functions for nonlinear systems is a difficult task [26,27]. But, for the first and second order ordinary differential equations we highly simplified Lyapunov stability theory with LRC circuit systems. Hence, the interested knows how to construct the energy (Lyapunov) function and checks the result of the time derivative (directional) of the energy function with (1). This approach also improves some well-known studies. These improvements will take place in section 4. In addition, [6] does not involve the passivity analysis, the notion of power -energy relationship constructing Lyapunov functions, and equation (1) and its implications. The present work includes these and some improvements.
The rest of this paper is organized as follows. Section 2 presents some definitions and auxiliary results. Section 3 deals with the main results. Section 4 deals with discussion. Section 5 closes the paper with a short conclusion.

Preliminaries
A commonly used model for an autonomous nonlinear system is where t ∈ + ( + = [0, ∞)) denotes time, x ∈ n denotes the state of the system, while u ∈ m is called the input or the control function. However, f : n × m → n satisfies Lipschitz condition. The state vector x(t) ∈ D in which D n is a domain that contains the origin x = 0. We assume that (2) is well posed, that is, there exists a unique solution x : [0, ∞) → n for every initial data x(0) = x 0 ∈ n , and x depends continuously on x 0 according to the normed topology on n . Let f (0, 0) = 0, f (x, 0) = 0 for x = 0, and ||.|| is the Euclidean norm on n . Further, assume that u is an admissible real valued input function so that where K is a positive constant. A state x ∈ n is an equilibrium of (2) if f (x, 0) = 0. A system or machine attains its minimum of energy at the equilibrium points.
We shall now need some basic definitions on the properties of the Lyapunov functions.
is said to be positive definite, decrescent and radially unbounded function if there exist functions α and β of class κ are such that (iv) Furthermore, assume that the set S = {x ∈ n : E (x) = 0}, contains no invariant set other than the set {0}. Now, to motivate the definitions of passivity we can use electrical circuits. Inflow power of a simple resistive system is always nonnegative with the voltage u(t) as input and the current y(t) as output, that is, if uy ≥ 0 for all u, y ∈ , and for a multiport network we have u T y = m i=1 u i y i ≥ 0. Therefore, for general nonlinear systems we can state some new passivity properties. These new properties are of interest in circuit and control theory [24] and has applications in mathematical control theory with Gronwall's inequality [3].

Moreover, it is lossless if
and strictly passive if for some positive definite function Ψ, where y = x , and r(t) = r(u(t), y(t)) = m i=1 u i y i is the supply rate function of (2) defined on m × n , and satisfies t 0 |r(s)|ds < ∞, Proof. See [24].

Main results
The matter under discussion is the stability of the origin (0, 0) and the passivity of the following nonlinear resistive, inductive, and capacitive LRC circuits for one input variable (m = 1) and two state variables (n = 2). In this work, the inputs of the systems (circuits) are bounded and admissible continuous functions in t. In addition, a nonlinear resistor can be both current controlled (R(i)) and voltage controlled (R(v)) element. A nonlinear inductor is current controlled (L(i)) element while a nonlinear capacitor is said to be voltage controlled (C(v)) element. The internal resistance of a current source is infinite while that of a voltage source is finite and especially chosen to be small.

Nonlinear resistor
In the following circuit [28] (Figure 1) there is a nonlinear resistive element which is specified by i 2 = f (v 2 ) and the remaining components R 1 , R 2 , L, C are positive scalars.
with i(t) = 0, is globally asymptotically stable and the circuit will be lossless at infinity if Proof. First, let write down the state equations of the above circuit: Then after some arrangement we obtain system (4). The natural energy function E 1 (t) = E 1 (v 1 , v 2 ) from the storage elements (capacitor and inductor) of this circuit is The energy function (E 1 : 2 → + ) satisfies The derivative of the Lyapunov function along the trajectories of system (4) gives By using system (4), we have For i(t) = 0, it follows that where R N L represents the resistance value of the nonlinear element. E 1 is verified by (1). The application of Theorem 1 shows that: E 1 ≤ 0 on 2 , E 1 (∞) = 0 and E 1 (v) → ∞ as ||v|| → ∞. Hence, all the motions of (4) are bounded (as illustrated in Figure 2a, 2b). The set S where E 1 = 0 is {0, 0}. This implies that {0, 0} is the only invariant subset of S, and the zero solution or equilibrium solution of (4) is globally asymptotically stable. It can be seen that (4) is a lossless system at infinity due to v(t) = 0. Hence, the system is zero-state observable.

Nonlinear inductor
The circuit shown in Figure 3 with i s (t) = 0 is globally asymptotically stable or the circuit is lossless at infinity.
Proof. First, let write down the node and v equations of the above circuit: Then after some rearrangement we obtain system (6). The native energy function for this circuit is The energy function (E 2 : 2 → + ) satisfies The derivative of the energy function E 2 along the trajectories of system (6) gives By using (6), we have For i s (t) = 0, it follows that E 2 is verified by (1). The application of Theorem 2 shows that: E 2 ≤ 0 on 2 , E 2 (∞) = 0 and E 2 (φ, v) → ∞ as φ 2 + v 2 → ∞. Hence, all the motions of (6) are bounded (as illustrated in Figure 4a, 4b). The set S where E 2 = 0 is {φ, 0} and from (6) this implies that {0, 0} is the only invariant subset of S, such that the zero solution or equilibrium solution of (6) is globally asymptotically stable. Thus, system (6) with its Lyapunov function satisfies all the assumptions of  Theorem 2. Therefore, (6) or the related circuit is lossless due to the trivial solution which occurs at infinity.

Nonlinear capacitor
A nonlinear capacitive element is present in the following circuit [29] and its voltage drop is no longer given by q/C, but is more accurately described by αq(t) + βq 3 (t), where α and β are constants. The remaining elements L and R are positive scalars. Let, q = x is the flow of the charge, α and β are real constants, and u is the applied voltage. Instead of the above equation will be discussed.
and it is globally asymptotically stable if β < 0 is replaced by β > 0.
Proof. Let (x(t), y(t)) be a solution of (8) for t ≥ 0. In the case of β < 0: Let β = −c (c > 0, α constant), the system has the equilibrium point (0, 0) and (± √ αc −1 , 0). In the other case (β > 0), the system has (0, 0) and (± αβ −1 , 0). Therefore, (0, 0) is the only invariant equilibrium point of (8). For the first case we investigate the stability of (8). The storage energy function from power-energy relationship can be constructed in the neighborhood of the equilibrium point (0, 0) as The energy function (E 3 : 2 → + ) satisfies E 3 is not radially unbounded. But, in the neighborhood of (0, 0), E 3 is positive definite, and we have (9) obeys (i) and (ii) of Definition 2. The derivative of the energy function E 3 along the trajectories of system (8) gives By using (8), we have For u(t) = 0, it follows that (10) is verified by (2) and system (8) is stable. . On the other hand, the integration of (10) from 0 to t(≥ 0) gives That is, E 3 is a decreasing function along the solution curve (x(t), y(t)), and (0, 0) is a minimum point of E 3 . The above inequality implies that the motion (x(t), y(t)) will stay in the neighborhood of the equilibrium point (0, 0) for t ≥ 0 provided that the initial point (x(0), y(0)) = (x 0 , y 0 ) is sufficiently near the point (0, 0). Hence, the origin is stable. In addition, for α = 1 and β < 0 we have the following inequalities.
This is precisely the most common definition of stability of a system which has an isolated equilibrium point (0, 0).
In the case where β > 0 : E 3 (x, y) ≤ 0 at all points (x, y) ∈ 2 . That is E 3 (t) is a decreasing function along any motion of (8), E 3 (∞) = 0 and E 3 (x) → ∞ as ||x|| → ∞.Hence, all the solutions of (8) are bounded.The set S, where E 3 = 0 is (x, 0), and (0, 0) is the only invariant subset of S. Thus, the application of Theorem 3 shows that the solution x(t) = 0 to (8) as t → ∞. Therefore, the system is zero-state observable. This also implies that there is no energy dissipation in the circuit at infinity (t = ∞); that is, the circuit will be lossless at infinity. This explanation is compatible with Figure 6a, 6b. Besides, when the value of R increased, the motion goes to the equilibrium point immediately.
The three strong passivity results or the boundedness of the motions (strict passivity) of (4), (6) and (8) with their input functions are the following.
Theorem 4. Suppose that all the conditions in Theorems 1, 2, and 3 are satisfied and also assume that (3) holds such that where K is a positive constant. Then, all the motions of (4), (6) and (8) with their forcing functions are bounded or the systems are strongly passive.
Hence, by Definition 3, the system (4) is strictly passive. Furthermore which also known as the dissipative inequality.
By (5), it follows that Integrating (11) from 0 to t(> 0), and using Theorem 4, it follows that Then, Gronwall's inequality [3] yields Using the foregoing procedure, the following results are obtained which determine the upper bounds of E 2 and E 3 , respectively. and Finally, the connection between (5) and (12), (7) and (13), and (9) and (14), respectively, give the following results: Thus, the energy functions E 1 , E 2 , and E 3 are bounded. This also implies that all the motions of (4), (6), and (8) are bounded in magnitude. Hence, the related systems (or circuits) are strongly passive.

Discussion
The properties of energy function and its Lie derivative determine the criteria of Lyapunov stability theory. Thus, our natural approach in this paper may be applicable to all physical systems whatever the orders of the systems. Here, we will only improve the stability of some second order systems that relevant to our study. In this connection, the stability of the following differential equations (with their arguments) has been investigated in [25] and [30], respectively, (k > 0, constant), x 0 g(ξ)dξ; (a3) and x + x + p(t)g 1 (x) + q(t)g 2 (x) = 0, (b1)

Remark 1
There may be some objection regarding to the derivative of Lyapunov functions, but powerenergy relationship shows that this objection is unfounded. For example, consider a series LRC circuit which has b(t)g(q) voltage on a time varying nonlinear capacitor with q(t) charge that flows in the circuit. Let P C , W C be the power and energy of the capacitor. Then, we have the followings: where ds = q (t)dt. Then, Thus, for the construction of energy and power functions that associated with capacitors, we nicely apply the above loop. This may enable us to improve the stability of many systems, because the above algorithm eliminates to take the partial derivative of W C (t).
(A) The natural approach improves the result given in [25] such as: (a1) may represent a LRC circuit (dissipative) system with The natural energy function for (a4) must be V (t, x, y) = 1 2 Since, (a5) has been constructed from the powerenergy relationship of (a4). The Lie derivative of (a5) is (a6) is the dissipated power of (a4) and verified by (1). The difference between (a2) and (a5), and between (a3) and (a6) state our improvement.
(B) The natural approach improves the results given in [30] such as: The actual energy function for (b1) is where ds = x (t)dt, p > 0 and q > 0 are continuous functions on [0, ∞), g 1 , g 2 are continuous functions on , satisfying (A 1 ) of [30].
In fact, the coefficient of x in (b1) is 1, and (b5) is confirmed by (1) due to the suitable tool. The comparisons between (b2) and (b4), and between (b3) and (b5) give our improvement.
Further, the approach in this study also improves many results in the books [1] and [24] that based on Lyapunov approach. This list can be extended for the other related references that not cited here.

Conclusion
The energy of a system determines its behavior. In this context, this paper plays two important roles in the Lyapunov stability theory. First, it provides the construction of the energy function, which may obtain from the physical meaning of the given system. Second, it implies that the derivative of the energy function along the system trajectories is equal to the negative value of the dissipated power in the system. These further clarify the Lyapunov stability. Hence, one can nicely check the derivative of the energy function of a given physical system with (1). The proposed approach can be applicable to higher order differential systems. From now on, everyone involved in the subject will be able to find the same stability results for a system under consideration. We hope this present work will open new doors in the stability analysis of differential systems.