Differentially passive circuits that switch and oscillate

Differentially passive circuits that switch and oscillate F.A. Miranda-Villatoro ∗ F. Forni ∗ R. Sepulchre ∗ ∗ Department of engineering, University of Cambridge, UK arXiv:1804.04626v1 [cs.SY] 12 Apr 2018 (e-mail: [email protected], [email protected], [email protected]). Abstract: The concept of passivity is central to analyze circuits as interconnections of passive components. We illustrate that when used differentially, the same concept leads to an interconnection theory for electrical circuits that switch and oscillate as interconnections of passive components with operational amplifiers (op-amps). The approach builds on recent results on dominance and p-passivity aimed at generalizing dissipativity theory to the analysis of non-equilibrium nonlinear systems. Our paper shows how those results apply to basic and well-known nonlinear circuit architectures. They illustrate the potential of dissipativity theory to design and analyze switching and oscillating circuits quantitatively, very much like their linear counterparts. Keywords: switches, oscillators, differential passivity 1. INTRODUCTION The concept of passivity originates in circuit theory. It characterizes circuit elements that can possibly store and dissipate the energy provided by the environment, but not the other way around. Passivity is inherently an interconnection concept: passive interconnections of passive components model passive circuits. Dissipativity theory, the system theoretic generalization of passivity theory, has become a cornerstone of system theory. It provides an interconnection theory to design and analyze stable dynamical systems. Such systems dissipate the energy stored internally and provided externally. In short, dissipativity theory is an interconnection theory for Lyapunov stability analysis. In recent years, many researchers have pointed to the relevance of studying stability incrementally or differentially when addressing questions that go beyond the stability analysis of isolated equilibria. Differential stability concepts include contraction theory (Lohmiller and Slotine, 1998), convergence theory (Pavlov et al., 2005), or differential Lyapunov theory (Forni and Sepulchre, 2014). They have proven relevant in a number of areas, most prominently in questions pertaining to nonlinear observers (Aghannan and Rouchon, 2003), oscillator synchronization (Stan and Sepulchre, 2007), or regulation theory (Jouffroy and Fossen, 2010). Differential dissipativity is to differential stability what dissipativity is to Lyapunov stability. It was introduced in the recent papers (Forni et al., 2013; Forni and Sepulchre, 2013; van Der Schaft, 2013). The present paper aims at illustrating the potential of differential passivity as an interconnection theory of circuits that can switch an oscillate. In contrast with classical dissipativity theory, such a theory must cope with the stability analysis of dynamical systems that posses multiple equilibria or limit cycles. It is based ⋆ The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645. on the concept of p-dominance and p-passivity recently introduced in Forni and Sepulchre (2017a,b). Intuitively, the attractors of a p-dominant system are the attractors of a p-dimensional system: a unique equilibrium for p = 0, but possibly multiple equilibria for p = 1, and limit cycles for p = 2. We interpret classical differential dissipativity theory as an interconnection theory for 0-dominance, that is, differential stability. The extension to p = 1 and p = 2 is motivated by the analysis of multistability or limit cycles in interconnected systems. Nonlinear circuit theory provides a realm of switching and oscillatory behaviors designed from the simple elements of linear circuit theory interconnected with operational amplifiers (op-amp) (Clayton and Winder, 2003). Our aim in the present paper is to illustrate that those building blocks are the natural building blocks of p-passive circuits. System theoretic tools have lacked so far for the quantitative analysis and design of such circuits. Their analysis normally rests on simplifying assumptions, time-scale separation arguments leading to asymptotic analysis, or reductions to two-dimensional phase portraits. In contrast, we are aiming at quantitative and computationally tractable certificates such as those used in the theory of linear time-invariant systems. Such tools have made the success of robustness and performance analysis of linear time-invariant systems. A pillar of passivity theory is the passivity theorem, which states that the negative feedback interconnection of two passive systems is passive. It is also well-known that switches and oscillatory circuits require both positive and negative feedback interconnections of op-amps with passive elements. We stress in the present paper that such interconnections fall in the category of the p-passivity theorem, which states that the negative feedback interconnection of a p1 -passive with a p2 -passive circuit is p1 + p2 passive. The differential analysis in this paper assumes smooth systems. A companion paper shows that an analog framework exists for non-smooth systems. To account for the lack of differentiability, differential concepts have then to be replaced by incremental concepts. Many of the nonlinear circuits discussed in the present paper have a non-smooth analog that falls in the category of linear complementarity systems studied in Miranda-Villatoro et al. (2018). The rest of the paper is organized as follows. Section 2 provides a brief summary of the concepts of dominance and ppassivity. Section 3 revisits standard properties of the operational amplifier its differential passivity properties. Section 4 revisits the basic architectures of circuits that switch and oscillate, analyzing those systems as both positive and negative feedback interconnections of operational amplifiers with passive linear circuits. The discussion in Section 5 suggests that those architectures are robust and amenable to regulation. 2. DOMINANCE AND DIFFERENTIAL PASSIVITY We consider the nonlinear system ẋ = f (x) (1) n where x ∈ R and f is a smooth vector field. The prolonged system consists of (1) augmented with the linearized ˙ = ∂f (x)δx, where ∂f (x) denotes the Jacobian equation δx linearization of f . By construction δx ∈ Rn (identified with the tangent space of Rn ). An important notion for this paper is the inertia (p, 0, n − p) of a symmetric matrix, meaning that the matrix has p eigenvalues in the open left half-plane, 0 eigenvalues on the imaginary axis, and n − p eigenvalues in the right half-plane. The following definition is taken from (Forni and Sepulchre, 2017a). Definition 1. A nonlinear system (1) is p-dominant with rate λ ≥ 0 if there exist a constant symmetric matrix P with inertia (p, 0, n − p) and ε ≥ 0 for which the prolonged system satisfies    ⊤  ˙ ˙ 0 P δx δx ≤0 (2) P 2λP + εI δx δx for all (x, δx) ∈ Rn × Rn The property is strict if ε > 0. y Solving (2) is equivalent to finding a uniform solution P to the linear matrix inequalities ∂f (x)⊤ P + P ∂f (x) + 2λP ≤ −ǫI for all x ∈ Rn . For a linear system ẋ = Ax, the inequality reduces to (A + λI)⊤ P + P (A + λI) ≤ −ǫI, which is feasible for linear systems whose eigenmodes can be split into p dominant modes and n − p transient modes, separated by the rate λ, (Forni and Sepulchre, 2017b, Proposition 1). For nonlinear systems p-dominance captures the property that the asymptotic behavior of the system is p-dimensional. This intuitive characterization is made precise in the following result (Forni and Sepulchre, 2017a, Corollary 1). Theorem 1. Let (1) be a strictly p-dominant system with rate λ ≥ 0. Then, every bounded solution of (1) asymptotically converge to • a unique fixed point if p = 0; • a fixed point if p = 1; • a simple attractor if p = 2, i.e. a fixed point, a set of fixed points and their connected arcs, or a limit cycle. In what follows we will study p-dominant systems as interconnections of open systems. We consider open systems of the form ẋ = f (x) + Bu , y = Cx (3) where u ∈ Rm and y ∈ Rq define the input and the output to the system, respectively. B and C are matrices of appropriate dimension. The prolonged system to (3) is obtained by augmenting (3) with the linearized equations ˙ = ∂f (x)δx + Bδu, δy = Cδx. The following definition δx is taken from (Forni and Sepulchre, 2017a). Definition 2. A nonlinear system (3) is p-passive from u to y with rate λ ≥ 0 if there exist a constant symmetric matrix P with inertia (p, 0, n − p) and ε ≥ 0 for which the prolonged system satisfies  ⊤        ⊤  ˙ ˙ δy 0 I 0 P δy δx δx (4) ≤ I 0 δu P 2λP + εI δu δx δx for all (x, δx) ∈ Rn × Rn and all (u, δu) ∈ Rm × Rm . The property is strict if ε > 0. y The concept of p-passivity is related to p-dominance in the same way as passivity is related to stability. Differential (Forni and Sepulchre, 2013) or incremental (Pavlov and L, 2008) passivity are synonyms of 0-passivity. For a static differentiable nonlinearity y = ϕ(u), 0-passivity simply means monotonicity, that is positivity of its derivative: if ∂ϕ(u) ≥ 0, then δy T δu = (∂ϕ(u)δu)⊤ δu ≥ 0 for all δu. The following p-passivity theorem is the natural extension of the classical passivity theorem. It is taken from (Forni and Sepulchre, 2017a, Theorem 4). Theorem 2. Let Σ1 and Σ2 be (strictly) p1 and p2 passive, respectively, from input ui to output yi , i ∈ {1, 2}, both with rate λ ≥ 0, Then, the negative feedback interconnection u1 = −y2 + v1 , u2 = y1 + v2 of Σ1 and Σ2 is (strictly) (p1 + p2 )-passive from v = (v1 , v2 ) to y = (y1 , y2 ), with rate λ. The interconnection is (strictly) (p1 + p2 )- dominant. We observe that negative feedback preserves p-passivity only if the two components share a common rate λ. For linear systems of the form ẋ = Ax + Bu, y = Cx, ppassivity has a useful frequency domain characterization in terms of the shifted transfer function G(s − λ) = C(sI − (A + λI))−1 B, as shown by the next theorem from Miranda-Villatoro et al. (2017). Theorem 3. A linear system is p-passive with rate λ if and only if the following two conditions hold, (1) ℜ {G(jω − λ)} ≥ 0, for all, ω ∈ R ∪ {+∞}. (2) G(s − λ) has p poles on C+ . The property is strict if G(s− λ) has p poles in the interior of C+ 3. THE OPERATIONAL AMPLIFIER IS 0-PASSIVE Figure 1 represents a classical model of operational amplifiers (Karki, 2000; Noseek, 2009): the RC network is a circuit realization of a first order linear model in parallel with a voltage-controlled current source αVE , where α ∈ (0, +∞), (Noseek, 2009). The RC network models the finite bandwidth property of a real device and it is connected to a static nonlinear element i = ϕ(V ) (5) Fig. 1. First order op-amp model with saturation: a) internal structure, b) symbolic representation. to account for the bounded range of V0 = x. The static nonlinear element is a smooth and odd nonlinearity modelled for instance as follows: i = ϕ(V ) := η sinh(βV ) (6) where η > 0 and β > 0 are suitable parameters. The results of this paper hold for any stiffening nonlinearity ϕ satisfying the following assumption (Stan and Sepulchre, 2007). Assumption 4. The static nonlinearity ϕ : R → R is an odd function such that ∂ϕ(y) ∂y ∈ [0, +∞). Furthermore, for any k > 0, there exists a r > 0 such that yϕ(y) − ky 2 > 0, for all |y| > r . (7) For example, ϕ(y) = y 2n+1 , for n ∈ N, ϕ(y) = arctanh(y) and ϕ(y) = sinh(y) all satisfy Assumption 4. The firstorder model of the op-amp in Figure 1 has the state-space model  ẋ = − 1 x − 1 ϕ (x) + α V E Σop : (8) R C C0 C0 V = x 0 0 0 which, notably, admits the block diagram representation of the Lur’e system in Figure 2. The transfer function of of behaviors, enabled by the interconnection of the op-amp with suitable linear networks. The circuits in this paper will only include interconnections of op-amps with linear stable networks ż = Az + Bu, y = Cz . (10) where z ∈ Rn , u ∈ R, y ∈ R are the state, input, and output of a generic linear network, respectively. A, B and C are constant matrices of appropriate dimensions. Such interconnections always lead to bounded behaviors: Theorem 5. Suppose that A is a Hurwitz matrix. Under Assumption 4, the trajectories of the system defined by (8), (10), and the interconnection rule VE = ±Cz + Vr u = V0 (11) are all bounded, for any constant voltage Vr ∈ R. Proof. Let V : R × Rn → R be the positive definite function 1 1 V (x, z) = x2 + z ⊤ P z, (12) 2 2 where P = P ⊤ > 0 satisfies A⊤ P + P A ≤ −Q, for some Q = Q⊤ > 0. Then, taking η = R01C0 , β = C10 , ρ = αkCk + 2kP Bk, ε > 0, and µQ given by the smallest eigenvalue of Q, we have V̇ (x, z) = −ηx2 − z ⊤ Qz − βxϕ(x) ± αxCz + 2z ⊤ P Bx ≤ −ηx2 − µQ kzk2 − βxϕ(x) + ρ|x|kzk ρ ≤ −ηx2 − µQ kzk2 − βxϕ(x) + |x|2 + ρεkzk2 . ε µ Setting ε = 2ρQ yields ρ µQ kzk2 − βxϕ(x) + |x|2 . V̇ (x, z) ≤ −ηx2 − 2 ε From (7), there exists r > 0 such that, for all |x| > µ r, V̇ (x, z) ≤ −ηx2 − 2Q kzk2. Boundedness of solutions follows.  In the next sections we will design particular interconnections based on Theorems 1 and 2. 4.2 p-Passivity and interconnections We consider the interconnection of op-amp (8) and linear networks of the form (10) typically in positive or negative feedback, as show in Figure 3. Passivity is a theory of Fig. 2. Open-loop operational amplifier model the linear part Σ0 G0 (s) = 1 C0 s+ 1 R0 C 0 (9) is a strictly 0-passive network (by Theorem 3) with rate λ ∈ [0, R01C0 ). The op-amp is thus given by the negative feedback interconnection of a strictly 0-passive linear system with a static 0-passive nonlinearity (under Assumption 4). Therefore, by Theorem 2, the closed loop is a strictly 0-passive system from VE to V0 with rate λ ∈ [0, R01C0 ). The same representation would hold if the RC circuit was replaced by any 0-passive network. 4. SWITCHES AND OSCILLATORS VIA FEEDBACK AMPLIFIERS 4.1 Feedback and boundedness The great versatility of the op-amp comes from its interconnection properties. The device allows for a wide range Fig. 3. Feedback loops of a circuit with an operational amplifier negative feedback. In order to apply Theorem 2 to positive feedback interconnections, we consider the reverted output ȳ = −y and interpret the positive feedback interconnection VE = +y + Vr as negative feedback interconnection of the reverted output: VE = −ȳ + Vr , u = V0 . (13) We note that the network in Figure 4 is strictly 0-passive from u to y = z and strictly 1-passive from u to ȳ = −z. a Indeed, define a = RR11R+R , b = Ra1C1 , and c = 1. Then the a C1 4.4 2-Passive circuits By Theorem 1, oscillatory circuits may arise from the interconnection of the op-amp with strictly 2-passive networks. As an illustration, consider the negative feedback of the op-amp with the RC network in Figure 6. The RC Fig. 4. RC network that is both 0-passive and 1-passive from different ports transfer functions from u to y reads cb G(s) = (14) s+a whereas the transfer function from u to ȳ reads cb Ḡ(s) = − (15) s+a By Theorem 3, G(s) is strictly 0-passive for any rate λ ∈ [0, a). On the other hand, Ḡ(s) is strictly 1-passive for any rate λ ∈ (a, +∞). 4.3 1-Passive circuits By Theorem 1, multistable circuits that switch among several fixed points may arise from the interconnection of the op-amp with strictly 1-passive networks. As an illustration, consider the positive feedback of the op-amp with the RC network in Figure 4. The RC network is Fig. 5. An op-amp in positive feedback with a passive network. strictly 1-passive from u to ȳ = −z with rate λ ∈ (a, +∞), and takes the role of Σup in Figure 3. The op-amp is 0passive with rate λ ∈ [0, R01C0 ). Thus, for 1 > a, (16) R0 C0 the two systems share a common interval for their λ rates 1 . By Theorem 2 the closed loop system in Figure 5 is strictly 1-passive from Vr to V0 with rate λ ∈ (a, R01C0 ). Theorem 5 guarantees boundedness of the closed-loop trajectories for any constant input Vr . Therefore, Theorem 1 guarantees asymptotic convergence of all trajectories to some fixed point. In particular, taking Vr = 0 for simplicity, the closed loop is bistable for ∂ϕ(x) αR1 1 + , (17) <− ∂x x=0 R0 R1 + Ra which guarantees the existence of three equilibrium points (two stable nodes and a saddle). 1 (16) requires that the op-amp dynamics is faster than the dynamics of the external network, as usual in applications. Fig. 6. RC oscillator circuit. network admits the state-space representation # " " # −2 1 0 1 1 1 1 −2 1 , B = 0 , C = [0 0 1] . A= R1 C1 0 1 −1 R1 C1 0 The transfer function G(s) from u to y reads G(s) = s3 + 1 (R1 C1 )3 6 5 2 R1 C1 s + (R1 C1 )2 s + 1 (R1 C1 )3 . (18) Denoting by pi the i-th pole of G and by βi = |ℜ{pi }| the magnitude of the real part of the poles of G (without loss of generality, we assume 0 ≤ β1 ≤ β2 < β3 ), the RC network  n is strictly 2-passive  from u to y with o rate λ ∈ max β2 , β1 +β32 +β3 , β3 . The network is thus constrained to a negative feedback interconnection, taking the role of Σdown in Figure 3. For   β1 + β2 + β3 1 (19) > max β2 , R0 C0 3 the op-amp and the RC network share a common interval for their λ rates. By Theorem 2 the closed loop system in Figure to V0 with rate  6 is nthus strictly 2-passive o n from Vr o β1 +β2 +β3 1 λ ∈ max β2 , , min β3 , R0 C0 . Theorem 5 3 guarantees boundedness of the closed loop trajectories for any constant input Vr , thus Theorem 1 guarantees that the trajectories of the four dimensional closed-loop circuit all converge to a simple attractor. For R1 = 3.3KΩ and C1 = 200µF , G(s) has poles p1 = −0.3, p2 = −2.35 and p3 = −4.92. Therefore, strict 2-passivity holds for λ ∈ (2.52, 4.92). For op-amp parameters α = 0.1, R0 = x 5 1M Ω, C0 = 15.9nF , ϕ(x) = ( 12 ) condition (19) holds. These specific parameters also ensure that the unique fixed point at the origin is unstable. Thus, by Theorem 1, every trajectory converges asymptotically to a limit cycle. The steady-state is an oscillation, as shown in Figure 7. 5. MIXED FEEDBACK AND MODULATION The mixed feedback amplifier is a classical device of nonlinear circuit theory (Chua et al., 1987). It combines positive and negative feedback around an operational amplifier to create nonlinear behaviors. We illustrate this flexibility with the simple system of Figure 8. The two linear networks correspond to two copies of the RC network in (2) G(s) is strictly 2-passive with rate λ ∈ for a1 > a2 .   a2 b2 −a1 b1 a2 −a1 , +∞ The conditions above reveal that mixed feedback allows for both 0-passivity and 2-passivity. The network behavior can be modulated from monostable to oscillatory via parameter variations. Figure 9 shows the degree of ppassivity of the closed loop for different values 0Ω < Ra , Rb < 3KΩ. Indeed, transitions from monostable to oscillatory regimes are obtained by the variation of one of the two resistances. As a final illustration, we consider Fig. 7. Output of the RC circuit of Figure 6, with R1 = 3.3KΩ and C1 = 200µF . Fig. 8. Mixed feedback with switches. Figure 4. The behavior of the closed loop is dictated by the interconnection pattern of the switches Sa and Sb . If Sa is closed and Sb is open then the network reduces to the one in Figure 5; the closed loop is 1-passive. If Sa is open and Sb is closed, then the closed loop is 0-passive. When both switches are closed, the feedback circuit is not necessarily 1-passive because the rates of the two networks may not be compatible. In fact, a suitable selection of the network parameters lead to richer behaviors. Take Sa and Sb both closed and define a1 = Ra1C1 , a2 = Rb1C2 , and bi = ai + Ri1Ci , i = 1, 2. With these data, the upper network is strictly 1-passive from V0 to −z1 and the lower network is strictly 0-passive from V0 to z2 , respectively with rates λup ∈ (b1 , +∞) and λdown ∈ [0, b2 ). A common rate λ can be found for b1 < min{b2 , R01C0 }, since the op-amp is 0passive with rate λop ∈ [0, R01C0 ). In this case, the feedback circuit is 1-passive, by Theorem 2. In contrast, Theorem 2 cannot be used for b1 > b2 because of the absence of a common rate. However, the aggregate transfer function reads (a2 − a1 )s + a2 b1 − a1 b2 G(s) = , (20) (s + b1 )(s + b2 ) which has positive real part if and only if there exist λ ∈ [0, b2 ) ∪ (b1 , +∞) such that (a2 − a1 )λ < a2 b2 − a1 b1 (21) (a2 − a1 )λ < a2 b1 − a1 b2 . (22) Indeed,  h −a1 b1 (1) G(s) is strictly 0-passive 2 with rate λ ∈ 0, a2ab22 −a 1 for a2 > a1 and 2 a2 b2 −a1 b1 a2 −a1 > 0; Indeed, since b1 > b2 , it follows that a1 b1 > a1 b2 and a2 b1 > a2 b2 . Hence, a2 b2 −a1 b1 < a2 b1 −a1 b2 and conditions (21) and (22) reduce Fig. 9. p-passivity of the circuit of Figure 8 with both switches closed, as a function of the resistors Ra and Rb . Gaps correspond to lack of p-passivity. parameters R1 = R2 = 3.3KΩ, Ra = Rb = 1KΩ, C1 = 100µF , C2 = 200µF , R0 = 1M Ω, C0 = 15.9nF and α = 1. For these parameters G(s) in (20) is strictly 2-passive. When both switches are closed the origin is the only equilibrium point and is unstable. Hence, by Theorems 1 and 5 we conclude the existence of a limit cycle. Figure 10 shows transitions among different behaviors, driven by the switches. 6. CONCLUSIONS We have analyzed basic examples of circuits that switch and oscillate as interconnections of linear circuits and operational amplifiers. The approach builds upon dominance theory and p-passivity. The saturated op-amp model guarantees boundedness of trajectories in closed loop and allows for positive and negative feedback interconnections with linear p-passive networks. Specific interconnections lead to 1- and 2-passive networks, leading to a tractable analysis of bistability and oscillations in possibly highdimensional models. The stability analysis in this paper is based on solving linear matrix inequalities very much as in the stability analysis of linear systems. 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