Differentiation and Passivity for Control of Brayton-Moser Systems

Transcription

1 1 Differentiation and Passivity for Control of Brayton-Moser Systems Krishna Chaitanya Kosaraju, Michele Cucuzzella, Ramkrishna Pasumarthy and Jacquelien M. A. Scherpen arxiv: v1 [cs.sy] 7 Nov 2018 Abstract This paper deals with a class of Resistive-Inductive- Capacitive (RLC circuits and switched RLC (s RLC circuits modeled in Brayton Moser framework. For this class of systems, new passivity properties using a Krasovskii s type Lyapunov function as storage function are presented. Consequently, the supply-rate is a function of the system states, inputs and their first time-derivatives. Moreover, after showing the integrability property of the port-variables, two simple control methodologies called output shaping and input shaping are proposed for regulating the voltage in RLC and s RLC circuits. Global asymptotic convergence to the desired operating point is theoretically proved for both proposed control methodologies. Moreover, robustness with respect to load uncertainty is ensured by the input shaping methodology. The applicability of the proposed methodologies is illustrated by designing voltage controllers for DC-DC converters and DC networks. Index Terms Brayton-Moser systems, passivity-based control, RLC circuits, power converters, DC networks. I. INTRODUCTION In the recent years, passivity theory has gained renewed attention because of its advantages and practicality in modeling of multi-domain systems and constructive control techniques [1], [2]. In general, a system is passive if there exists a (bounded from below storage function S(x : R n R + satisfying S(x(t S(x(0 t 0 u ydt (1 where x R n is the system state, u,y R m are the input and the output, also called port-variables and the product u y is commonly known as supply-rate [3], [4]. Naturally, one can interpret the storage function as the total system energy, and the supply rate as power supplied to the system. Consequently, inequality (1 implies that the newly stored energy is never greater than the supplied one. In order to analyze the passivity properties of a nonlinear system, one requires to be artful in designing the storage function. For this reason, it is helpful to recast the system dynamics into a known framework, such as the port-hamiltonian (ph framework [5], where the storage function, also called Hamiltonian function, generally depends on the system energy. K. C. Kosaraju, M. Cucuzzella and J. M. A. Scherpen are with the Jan C. Wilems Center for Systems and Control, ENTEG, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, the Netherlands. R. Pasumarthy is with Department of electrical engineering, Indian Institute of Technology- Madras, Chennai-36, India. ( {k.c.kosaraju, m.cucuzzella, j.m.a.scherpen@rug.nl, ramkrishna@ee.iitm.ac.in This work is supported by the EU Project MatchIT (project number: Another well known framework that has been extensively explored for modeling of Resistive-Inductive-Capacitive (RLC circuits, is the Brayton-Moser (BM framework [6], where the storage function relies on the system power. For a more indepth analysis on the geometric properties of ph and BM systems, we refer to [7]. In modern times, switched power-converters play a prominent role in DC networks. Conventional power-converter consists of (passive subsystems interconnected with switches, which acts as an energy transferring devices between these subsystems. Switching elements are complex hybrid devices which operate as the controller that regulates the energy transfer between the subsystems. Considering they perform in a high-frequency range, one usually approximate them with their average PWM models. Typically, a subsystem consists of, passive elements such as inductors, capacitors, and resistors, power sources and loads. In this paper, we consider a large class of systems called switched RLC circuits, to which we refer to as s RLC circuits. This class models the majority of the existing power converters such as buck, boost, buck-boost, and Cúk systems. Although the analysis of s RLC circuits has received a significant amount of attention (see [8] [10] and the references therein, we notice that results based on the passivity properties of the open-loop system are still lacking (in Remark 3, we show that the passivity properties derived by using the system energy as storage function are limited to a small class of systems. On the other hand, a significant number of results have been published relying on Passivity-Based Control (PBC. The main idea in any PBC technique is to passify the closed-loop system such that the storage function has the minimum at a desired operating point [2]. There are multiple ways to achieve this, the most commonly used methodology is the Interconnection and Damping Assignment (IDA-PBC technique [11], and the main goal is to assign a desired structure to the closed-loop system [12] [14]. Another recent technique, relevant to this work, is called Proportional- Integral-Derivative (PID-PBC [15], [16], and the main goal is to find a (cyclo-passive map with a suitable output, such that the (integrated output port-variable can be used to shape the closed-loop storage function [17] [19]. Existing results on passivity based control of s RLC circuits are predominantly based on IDA-PBC technique [14], [20] [22]. However, these results are only local. In [23], the authors also proposed tuning rules using the Brayton-Moser stability theorems [6], [24].

2 2 A. Motivation and Main Contributions In this work, we present an approach similar to the PID-PBC technique for RLC and s-rlc circuits, which is originally introduced in [25] [27]. Analogous to PID-PBC, the key step in this methodology is to find a passive map with integrable output port-variable. The fundamental instrument in finding a new passive map always lies in finding a new storage function. In [26], [27], the authors observed that a class of RLC circuits have contraction properties [28]. The contraction theory primarily originated by analyzing Krasovskii s type Lyapunov functions [29], that are usually a quadratic function of velocities rather than states. This motivated the authors to propose the following storage function S(x,ẋ = 1 2ẋ M(xẋ, (2 where M(x > 0 R n n. It is well-established that the integrated output port-variable can be used to shape the closed-loop storage function. However, the above proposed storage function has resulted in a passivity property whose supply-rate is a function of the system state x and input u, and also their first time derivatives ẋ, u. This allowed us to introduce a radically new idea in PBC methodology, which we refer to as input shaping. In this, we use the integrated input port-variable to shape the closed-loop storage function. Furthermore, this choice of storage function has the following advantages: (i The supply-rate is a function of the first time derivative of the system state and input, avoiding the so-called dissipation obstacle problem [2]. (ii There are no parametric constraints that usually appear in Brayton-Moser framework [30]. (iii The port-variables are integrable. In this work, we consider a class of RLC and s-rlc circuits. Below, we list the main contributions: (i The use of a storage function similar to (2 for s RLC circuits leads to a new passive map. (ii The port-variables associated with this new passive map have integrability properties. We then use the integrated port-variables to shape the closed-loop storage function and propose two simple control techniques: outputshaping and input-shaping. The techniques are utilized for regulating the voltage in RLC and s RLC circuits. (iii The input shaping technique is robust to load uncertainty and requires less assumptions on the system parameters/structure in comparison with the output shaping technique. The proposed techniques are finally illustrated with application to buck, boost, buck-boost, Cúk DC-DC converters and DC networks, which are attracting growing interest and receiving much research attention [31] [35]. Simulation results show excellent performance. B. Outline This paper is outlined as follows. In Section II, we present BM based modeling and passivity properties of RLC and s RLC circuits. The control objective and the required assumptions are made explicit in Section III. In Section I, we present the new passivity property for the RLC and s RLC circuits. Furthermore, using the derived passive maps we propose two control techniques, output shaping and input shaping. In Section and the Appendix, we illustrate the proposed techniques using buck, boost, buck-boost and Cúk DC-DC converters. Moreover, in Section I, we extend this technique to DC networks (with interconnected buck and boost converters. Finally, we conclude and present some possible future directions in Section II. C. Notation Let x R n and y R m. Given a mapping f(x,y : R n R m R, the symbol x f(x,y and y f(x,y denotes the partial derivative of f(x, y with respect to x and y respectively. Let K R n n, then K > 0 and K 0 denote K is symmetric positive definite and symmetric positive semidefinite respectively. Assume K > 0, then x K = x Kx and K s denotes the spectral norm of K. Let Q 1 and Q 2 denote square matrices of order m and n respectively. Then diag{q 1,Q 2 denotes block-diagonal matrix of order m+n with block entries Q 1 and Q 2. Given p R n and q R n, denotes the so-called Hadamard product (also known as Schur product, i.e., (p q R n with (p q i := p i q i,i = 1,...,n. Moreover, [p] := diag{p 1,...,p n. II. PRELIMINARIES In this section, we briefly outline the Brayton-Moser formulation of RLC circuits and extend it to include an ideal switching element. A. Non-Switched Electrical Circuits Consider the class of topologically complete RLC circuits [23] with σ inductors, ρ capacitors and m (currentcontrolled voltage sources u s R m connected in series with inductors. In [24], Brayton and Moser show that the dynamics of this class of systems can be represented as Lİ = IP(I, Bu s C = P(I,, where L R σ σ and C R ρ ρ are symmetric matrices with the inductances and capacitances as entries, respectively. The state variablesi R σ and R ρ denote the currents through the σ inductors and the voltages across the ρ capacitors, respectively. The matrix B R σ m is the input matrix with full column rank and P(I, : R σ ρ R represents the so-called mixed-potential function, given by, (3 P(I, = I Γ +P R (I P G (, (4 where Γ R σ ρ captures the power circulating across the dynamic elements. The resistive content P R (I : R σ R and the resistive co-content P G ( : R ρ R capture the power dissipated in the resistors connected in series to the inductors and in parallel to the capacitors, respectively.

3 3 Remark 1 (Current sources. For simplicity, in (3 we have not included current sources. However, the results presented in this note can also be developed for current sources in a straight forward manner. Modeling of RLC circuits in this framework is typically noted as the Brayton-Moser (BM formulation, and system (3 can compactly be written as Qẋ = x P + Bu s, (5 where x = ( I,, Q = diag{ L,C and B = ( B O, O R m ρ being a zero-matrix. To permit the controller design in the following sections, we introduce the following assumptions: Assumption 1 (Inductance and capacitance matrices. Matrices L and C are constant and symmetric positive-definite. We note that the L and C being symmetric rather than diagonal captures mutual inductances and capacitances, respectively. Assumption 2 (Resistive content and co-content. The resistive content and co-content of current controlled resistors R and voltage controlled resistors G are quadratic in I and respectively, i.e., P R (I = 1 2 I RI, P G ( = 1 2 G, (6 where R 0 R σ σ and G 0 R ρ ρ. With Assumptions 1 and 2, one can show that system (3 is passive with respect to the power-conjugate port-variables u s, B I and the total energy stored in the network as storage function. Alternatively, in [30, Theorem 1] it is shown, under some assumptions, that system (3 is passive with respect to the port-variables u s, B I and the so-called transformed mixed-potential function as storage function. However, this alternative passivity property requires that the system parameters satisfy the condition C 1/2 G 1 Γ L 1/2 s < 1. We refer to [30] for more details. B. (Average Switched Electrical Circuits We now consider the class of RLC circuits including an ideal switch (s RLC. Let u d {0,1 and s R m denote the state of the switching element, i.e., open or closed, and the (current-controlled voltage sources, respectively. To describe the dynamics of an s RLC circuit we adopt the Brayton-Moser formulation (3 by considering a switching-state dependent mixed-potential function P(u d,i, : {0,1 R σ R ρ R and input-matrix B(u d : {0,1 R σ m, i.e., P(u d,i, = u d P 1 (I,+(1 u d P 0 (I, B(u d = u d B 1 +(1 u d B 0, where P 1 (I,, B 1 and P 0 (I,, B 0 represent the mixedpotential function and the input matrix of the s RLC circuit when u d = 1 and u d = 0, respectively. Under the condition that the Pulse Width Modulation (PWM frequency is sufficiently high, the state of the system can be replaced by We use the expression power-conjugate to indicate that the product of input and output has units of power. (7 Table I DESCRIPTION OF THE USED SYMBOLS I L C G R State variables Inductor current Capacitor voltage Parameters Inductance Capacitance Conductance Resistance Inputs u s Control input (RLC circuits u Control input (s RLC circuits s oltage source (s RLC circuits the average state representing the average inductor currents and capacitor voltages, while the switching control input is replaced by the so called duty cycle of the converter. For the sake of notational simplicity, from now let I, and u [0,1] denote the average signals of I, and u d, respectively, throughout the rest of the paper. Remark 2 (Resistive content and co-content structure. For s RLC circuits, where the content and co-content structure is not affected by the switched signal, model (7 can further be simplified by rewriting the mixed-potential function in (7 as follows P(u,I, = I Γ(u +P R (I P G (, (8 where the mapping Γ(u : [0,1] R σ ρ is defined as Γ(u = uγ 1 +(1 uγ 0, (9 and Γ 1 and Γ 0 captures the interconnection of the storage elements when u d = 1 and u d = 0, respectively. Then, the average behavior of an s RLC electrical circuit can be represented by the following Brayton-Moser equations L I = I P(u,I, B(u s C = P(u,I,. (10 If the content and co-content structure is not affected by the switching signal (see Remark 2, system (10 can be written as Lİ = RI +Γ(u B(u s C = Γ (ui G. (11 We assume that the resistive content and co-content structure is not affected by the switching signal throughout the rest of the paper. The main symbols used in (3 (11 are described in Table I. Remark 3 (Total energy as storage function. The RLC circuit (3 is passive with respect to the storage function S(I, = 1 2 I LI C, (12

4 4 and port-variables u s and B I. However, for the s RLC circuit (11, the first time derivative of (12 along the solutions to (11 is Ṡ = I L I + C = I (RI +Γ(u B(u s + ( Γ (ui G = I RI G +I B(u s I B(u s = I B 0 s +ui (B 1 B 0 s. We can infer passivity properties from energy as storage function if and only if B 0 = 0 and B 1 0. Yet, this is e.g. not true if we consider the model of the boost converter (see Section -B. Moreover, even if B 0 = 0 and B 1 0, one can note that the supply rate ui B 1 s is generally not equal to zero at the desired operating point, proving the existence of the so-called dissipation obstacle problem [2]. III. PROBLEM FORMULATION The objective of this paper is to propose a new passivitybased control methodology for regulating the voltage in RLC and s RLC circuits. Before formulating the control objective and in order to permit the controller design in the next sections, we first make the following assumption on the available information of systems (3 and (11: Assumption 3 (Available information. The state variables I and are measurable. The system parameters L,C,R and G are unknown constants. The voltage source s in (11 is assumed to be a known constant different from zero ( s 0. Secondly, in order to formulate the control objective aiming at voltage regulation, we introduce the following two assumptions on the existence of a desired reference voltage for both RLC and s RLC circuits: Assumption 4 (Feasibility for RLC circuits. There exists a constant desired reference voltage > 0 and a constant control input u s such that a steady state solution (I, to system (3 satisfies 0 = Γ +RI Bu s 0 = Γ I G. (13 Assumption 5 (Feasibility for s RLC circuits. There exists a constant desired reference voltage > 0 and a constant control input u (0, 1 such that a steady state solution (I, to system (11 satisfies 0 = Γ(u +RI B(u s 0 = Γ (ui G. Now, we use (8 to rewrite system (11 as ] [ ] [ Γ0 B = 0 s (Γ1 Γ 0 (B 1 B 0 s Γ + 0 I G (Γ 1 Γ 0 I [ L İ C (14 Consequently, we introduce the following assumption for controllability purposes: Note that, when needed, we also assume that I and are available. ] u. Assumption 6 (Non-zero input-matrix. There exists an element in the column vector [ ] (Γ1 Γ 0 (B 1 B 0 s (Γ 1 Γ 0 I, that is not zero, for all (I, R σ ρ and any t 0. The control objective can now be formulated explicitly: Objective 1 (oltage regulation. lim (t =. (15 t Remark 4 (Robustness to load parametrs. In power networks it is generally desired that Objective 1 is achieved independently from the load impedance. I. THE PROPOSED CONTROL APPROACHES In this section, we present a new passivity property (akin to differential passivity [36] for the considered RLC circuits (3. Then, we extend this property to the s RLC circuits (11. A. New passivity properties The following passive maps for (a sub-class of RLC circuits are first presented in [26], where the authors use the following Krasovskii-type storage function S( I, = 1 2 I Lİ C. (16 The use of such storage function helps in relaxing the parametric constraints in [30, Theorem 1]. However, this implies that the port-variables depend on the first time derivative of the states and input. For example, in the RLC circuits, the passive maps is with respect to the first time derivative of the voltage source and the first time derivative of the current. The storage function (16 is in terms of I and. Therefore, we consider the following extended-dynamics of system (3 Lİ = Γ +RI Bu s C = Γ I G LÏ = Γ +R I Bυ s C = Γ I G u s = υ s, (17a (17b (17c (17d (17e where υ s R m. Then, the following result can be established. Proposition 1 (Passivity of RLC circuit. Let Assumptions 1 and 2 hold. System (17 is passive with respect to the storage functions (16 and the port-variables y s = B I and υs. Proof. The first time derivative of the storage function (16 along the trajectories of (17 is Ṡ = I ( Γ +R I B u s + ( Γ I G = I R I G + I B u s u s y s = υ s y s. (18 These dynamics are differentially ( extended with respect to time. In the new system, the state variables are I,, I,,u s and the input is υ s.

5 5 See [26] for further details. Before presenting a similar passive map for s RLC circuits (11, we consider the following extended dynamics of (11 Lİ = RI +Γ(u B(u s (19a C = Γ (ui G (19b LÏ = R I +Γ(u +((Γ 1 Γ 0 (B 1 B 0 s υ (19c C = Γ (u I +(Γ 1 Γ 0 Iυ G (19d u = υ, (19e where we use the expressions of B(u and Γ(u given by (7 and (9 respectively, and υ R. Then, the following result can be established. Proposition 2 (Passivity of s RLC circuit. Let Assumptions 1 and 2 hold. System (19 is passive with respect to the storage functions (16 and the port-variables υ and y = ( (Γ 1 Γ 0 I I (Γ 1 Γ 0 I (B 0 B 1 s. (20 Proof. The time derivative of the storage function (16 along the trajectories of (19 is Ṡ = I ( ( (1 uγ0 +uγ 1 + u(γ1 Γ 0 +R I u(b 1 B 0 s + ( ( (1 uγ0 +uγ 1 I + u(γ1 Γ 0 I G = I R I G + uy uy = υy. (21 Remark 5 (Extended dynamics. Note that the storage function (16 depends on I and. This allow one to elucidate the storage function (16 as the energy in their tangent spaces. Hence, the new passivity properties presented in Propositions 1 and 2 also considers the ( dynamics evolving in their tangent spaces. In this way, I,, I,,u ( s and I,, I,,u are now states of the corresponding extended systems. This furthermore allowed us to define new input port-variables υ s andυ by appending a new dynamics foru s andu, respectively (see equations (17e, (19e. Using the passive maps presented in Propositions 1 and 2, we propose in the next subsections two possible control methodologies for both RLC and s RLC circuits. B. Output Shaping The first methodology, which we call output shaping, relies on the integrability property of the output port-variable. More precisely, we use the integrated output port-variable to shape the closed-loop storage function. This methodology was first introduced in [26] (for a parallel RLC circuit. Here we extend it to a large class of RLC and s-rlc circuits. We first present the output shaping methodology for the case of RLC circuits (3. Then, we extend it to the case of s RLC circuits (11. Theorem 1 (Output shaping for RLC circuit. Let assumptions 1 4 hold. Consider system (17 with control input υ s given by υ s = ( µ s k i B ( I Ī k d y s, (22 where y s = B I is defined in Proposition 1, kd, k i > 0 are free tuning parameters, and µ s R m. The following statements hold: (a System (17 in closed-loop with control (22 defines a passive map µ s y s. (b Let µ s be equal to zero. Moreover, if any of the following hold (i R > 0 and G > 0 (ii G > 0 and Γ has full column rank then the solution to the closed-loop system asymptotically converges to the set I,, I,,u s = 0,İ = 0, u s = 0,B ( I I = 0. (23 Proof. Consider the storage function (16. We use the integrated output port-variable to shape the desired closed-loop storage function, i.e., S d = S B (I I 2 k i. (24 Then, the time derivative of S d along the trajectories of system (17 controlled by (22 is Ṡ d = İ R I G +ys ( υs +k i B (I I (25a = İ R I G k d ys y s +µ s y s (25b µ s y s. (25c In (25a we use the controller (22. This concludes the proof of part (a. For part (b-i, let µ s be equal to zero. Then, from (25b, there exists a forward invariant set Π and by Lasalle s invariance principle the solutions that start in Π converge to the largest invariant set contained in Π I,, I,,u s : I = 0, = 0. (26 From (17c it follows that Bυ s = 0, i.e., υ s = 0 (B has full column rank. Moreover, from (22 it follows thatb (I I = 0, concludes the proof of part (b-i. For part (b-ii, the solutions that start in the forward invariant set Π converge to the largest invariant set contained in Π I,, I,,u s : R I = 0, = 0,y s = 0. (27 On this invariant set, from (17d we obtain Γ I = 0, which implies I = 0 (Γ has full column rank. This further implies that the solutions that start in Π also converge to the set (26. The rest of the proof follows from the proof of part (b-i. Remark 6 (Alternative controller to (22. The controller (22 needs the information of the first time derivative of the inductor current. This can be avoided by rewriting (22 as follows: u s := ( k i φ k d B I φ = B ( I I. (28 By using the storage function (24, the same results of Theorem 1 can be established analogously. Moreover, note that we

6 6 are able to rewrite the controller in such a way, due to the integrability assumption for the port-variables. We now extend this methodology to the case of s RLC circuits (11. One possible problem in extending this methodology to the case of s RLC networks is the integrability of the output port-variable. In the case the output is not integrable, the following assumption helps in finding a new integrable output. It is worth to mention that we relax this assumption in Section I-C where the input shaping methodology is proposed. Assumption 7 (Integrating factor. Consider the output portvariable y defined in Proposition 2 (see equation (20. Let m(i, : R σ R ρ R be different from zero. There exist γ(i, : R σ R ρ R andm(i, such that γ = m(i,y. The following lemma provides a new passive map with integrable output port-variables for system (19. Lemma 1 (Integrable output. Let Assumption 1, 2 and 7 hold. υ System (19 is passive with port-variables and γ = m(i, m(i,y. Proof. Multiply and divide equation (21 by m(i,. Then, we obtain with γ = m(i,y. Ṡ υy = υ m(i, m(i, y = υ m(i, γ, Theorem 2 (Output shaping for s RLC circuit. Let Assumptions 1 3 and 5 7 hold. Consider system (19 with control input υ given by υ := m(i,(µ k i (γ γ k d γ, (29 where γ = m(i,y with y defined in equation (20, γ = γ(i,, k i,k d > 0 are free tuning parameters, and µ R. The following statements hold: (a System (19 in closed-loop with control (29 defines a passive map µ γ. (b Let µ be equal to zero. Moreover, if any of the following hold (i R > 0 and G > 0 (ii G > 0, Γ (u has full column rank, and (Γ 1 Γ 0 (B 1 B 0 s 0. (30 then the solution to the closed-loop system asymptotically converges to the set I,, I,,u = 0, İ = 0, u = 0,γ = γ (31 Proof. Consider the storage function (16. We use the integrated output port-variable γ from Lemma 1 to shape the desired closed-loop storage function, i.e., S d = S k i(γ γ 2. (32 Then, the first time derivative of S d along the trajectories of system (19 controlled by (29 is Ṡ d = I Rİ G υ + m(i, γ +k i(γ γ γ (33a = I Rİ G k d γ 2 +µ γ. µ γ (33b (33c In (33a we use Proposition 2, Lemma 1, and the controller (29. This concludes the proof of part (a. For part (b-i, let µ be equal to zero. Then, from (33b, there exists a forward invariant set Π and by Lasalle s invariance principle the solutions that start in Π converge to the largest invariant set contained in Π I,, I,,u s : I = 0, = 0, γ = 0. (34 On this invariant set, from (19c and (19d it follows that [ ] (Γ1 Γ 0 (B 1 B 0 s υ = 0. (Γ 1 Γ 0 I From Assumption 6 we have υ = 0, which implies u = 0. Moreover, from (29 it follows that γ = γ, concluding the proof of part (b-i. For part (b-ii, when only G is positive definite, the solutions that start in the forward invariant set Π converge to the largest invariant set contained in Π I,, I,,u s : R I = 0, = 0, γ = 0. (35 On this invariant set, from (19c we obtain υ = 0. Moreover from (19d, this implies Γ (u I = 0, which implies I = 0 (Γ (u has full column rank. This further implies that the solutions that start in Π also converge to the set (34. The rest of the proof follows from the proof of part (b-i. Remark 7 (Limit set. The limit sets (23 and (31 defined in Proposition 1 and 2 allow one to only comment that the integrated output port-variables have converges to their desired values while velocities of all the state-variables converges to zero. However, for the considered examples of buck, boost, buck-boost, and Cúk dc-dc converters (in Section and Appendix, we show that the solutions of the closed-loop system converges asymptotically to their desired operating points. Furthermore, in input-shaping methodology presented in next subsection, under some mild assumptions we are able to show that the solutions of the closed-loop system converges asymptotically to their desired operating points. Remark 8 (Limitations of output shaping. In practical cases, the input resistance R of a RLC circuit can be negligible. As a consequence, the asymptotic stability presented in Theorem 2 relies on satisfying condition (30. For a buck converter (see Section -A, satisfying condition (30 is equivalent to require s 0, which is in practice satisfied (see also Assumption 3. Yet, for a boost converter (see Section -B, satisfying condition (30 is equivalent to require 0, which in principle could be not always satisfied. Moreover, the output shaping control methodology relies on finding aγ that satisfies γ = m(i,y, with m(i, 0. This may not always be possible (see Table II further on. Finally, designing a controller by using the output shaping methodology requires the

7 7 information of I, which often depends on the load parameters. Consequently, the proposed output shaping methodology is highly sensitive to unmodelled changes of the load. C. Input Shaping The second methodology, which we call input shaping, relies on the integrability property of the input port-variableυ s or υ in Proposition 1 and Proposition 2 respectively. Similarly to the output shaping technique, we use the integrated input port-variable to shape the closed-loop storage function such that it has a minimum at the desired operating point (see Control objective 1. This methodology has the following advantages: (i the integrability property of the output-port variable (see Assumption 7 and Lemma 1 is no longer needed; (ii the knowledge of ū s or ū, defined in (13 and (14, does not usually require the information of the load parameters (see the examples in Section -A, -B, making the input shaping control methodology robust with respect to the load uncertainty. (iii condition (30 is not required anymore. Moreover, we show that the trajectories converge to the desired operating point. Now, we present the input shaping methodology for the considered RLC circuits (3. Theorem 3 (Input shaping for RLC circuits. Let Assumptions 1 4 hold. Consider system (17 with control input υ s given by υ s = 1 k d (µ s k i (u s ū s y s, (36 where y s = B I is defined in Proposition 1, kd, k i > 0 are free tuning parameters, and µ s R m. The following statements hold: (a System (17 in closed-loop with control (36 defines a passive map µ s u s (note that u s is a state of the extended system (17. (b Let µ s be equal to zero. Moreover, if any of the following hold (i R > 0 and G > 0 (ii R > 0 and Γ has full column rank (iii G > 0 and Γ has full column rank then the solution to the closed-loop system asymptotically converges to the set I,, I,,u s = 0, I = 0, us = 0,u s = u s. (37 (c Furthermore, if the matrix A s = [ ] R Γ Γ G (38 is full-rank, then the solution to the closed-loop system asymptotically converges to the desired operating point ( I,,u s. Proof. Consider the storage function (16. We use the integrated input port-variable to shape the desired closed-loop storage function, i.e, S d = S u s ū s 2 k i. (39 Then, the first time derivative of S d along the trajectories of system (17 controlled by (36 is Ṡ d = I R I G + u s (y s +k i (u s u s (40a = I R I G k d u s u s +µ s u s (40b µ s u s. (40c In (40a we use Proposition 1 and the controller (36. This concludes the proof of part (a. For part (b-i, let µ s be equal to zero. Then, from (40b, there exists a forward invariant set Π and by Lasalle s invariance principle the solutions that start in Π converge to the largest invariant set contained in Π I,, I,,u s : I = 0, = 0, u s = 0. (41 On this invariant set, I = 0 and us = 0 further imply that y s = 0 andυ s = 0 respectively. Moreover, from (36 it follows that u s = ū s, concluding the proof of part (b-i. For part (b-ii and (b-iii, the solutions that start in the forward invariant set Π converge to the largest invariant set contained in Π I,, I,,u s : R I = 0,G = 0, u s = 0. (42 On this set, from (17c and (17d we get Γ I = 0, Γ = 0 respectively. If, either R > 0 and Γ has full column rank or G > 0 and Γ has full column rank, then I = 0, = 0. This further implies that the solutions that start in Π converge to the set (41. The rest of the proof follows from the proof of part (b. For part (c, consider the steady state solutions given by the set (41. On this set, the system dynamics (3 evaluate to [ ] R Γ I Γ G][ = [ ] Bus 0 (43 Since, we assume that A s in (38 is invertible, given an u s, there exists an unique (I, satisfying (13 and (43. Hence I = I, =. Next, we extend the input shaping methodology to the case of s RLC circuits (11. Theorem 4 (Input shaping for s RLC circuits. Let Assumptions 1 3, 5 and 6 hold. Consider system (19 with control input υ given by υ := 1 k d (µ k i (u ū y, (44 where y (20 is defined in Proposition 2, k i,k d > 0 are free tuning parameters, and µ R. The following statements hold: (a System (19 in closed-loop with control (44 defines a passive map µ u (note that u is a state of the extended system (19. (b Let µ be equal to zero. Moreover, if any of the following hold (i R > 0 and G > 0 (ii R > 0 and Γ(u has full column rank (iii G > 0 and Γ (u has full column rank then the solution to the closed-loop system asymptotically converges to the set I,, I,,u = 0, I = 0, u = 0,u = u. (45

8 8 u s Figure 1. Electrical scheme of the buck converter. (c Furthermore, if the matrix [ ] R Γ(u A = Γ (u G I L + C G (46 is full-rank, then the solution to the closed-loop system asymptotically converges to the desired operating point ( I,,u. Proof. Consider the storage function (16. We use the integrated input port-variable to shape the desired closed-loop storage function, i.e., S d = S k i(u ū 2. (47 By using the storage function (47, the claimed results in this theorem can be established by following a similar procedure in Theorem 3. Remark 9 (Robustness. Observe that the controllers (36 and (44 proposed in Theorem 3 and 4, respectively, requires the steady state value of input. In the case when R = 0, from the first line of the steady-state equations (13 and (14, one can notice that the value of u andu s requires solely the knowledge of the desired voltage i.e.,. This implies that the controllers does not require the information of Load parameter G, hence robust to its uncertainties. In input shaping methodology, under conditions on Γ, we can allow for R 0 or G 0. However, in output shaping the result is exclusive to R 0. Furthermore, using some conditions on steady state equations, we show that the solutions to the closed-loop system in input shaping asymptotically converge to the desired operating point. Nevertheless, this is not easy to ascertain for output shaping.. APPLICATION TO DC-DC POWER CONERTERS In this section, we use the control methodologies proposed in the previous section for regulating the output voltage of the most widespread DC-DC power converters : the buck and the boost converters, respectively. A. Buck converter Consider the electrical scheme of the buck converter in Figure 1, where the diode is assumed to be ideal. Then, by Buck and boost converters describe in form and function a large family of DC-DC power converters and, therefore, we will use them to exemplify the theory throughout this paper. However, in Appendices A and B we also study other common types of DC-DC power converters: the buck-boost and Cúk converters applying the Kirchhoff s current (KCL and voltage (KL laws, the average governing dynamic equations of the buck converter are the following: Lİ = u s, u [0,1] C = I G. (48 Equivalently, system (48 can be obtained from (11 withγ 0 = Γ 1 = 1, B 0 = 0, B 1 = 1 and R = 0. Using Proposition 2, we can establish the following passivity property. Lemma 2 (Passivity of buck Converter. Let Assumptions 1 and 2 hold. System (48 is passive with respect to the storage function (16 and the port-variables u and I s. We now use this passive map and exploit the output shaping and input shaping control methodologies presented in Theorems 2 and 4 respectively, in order to design a voltage controller. The following results are now established. Corollary 1 (Output shaping of buck DC-DC converters. Let Assumptions 1 3, and 5 7 hold. Consider system (48 with the dynamic controller u = s (k i ( I I +kd I, (49 where k i,k d > 0 are free tuning parameters. Then, the solution (I,, u to the closed-loop system asymptotically converges to the desired steady-state ( I,,u. Proof. First, in order to satisfy Assumption 7, we select m = 1, γ = I s, and γ = I s. Note that R = 0. Therefore, Theorem 2 can be used if the system (48 satisfies (30, i.e., s 0, which holds by Assumption 3. Following the steps in Theorem 2 with the proposed γ and m, we can show that the solutions of the closed-loop system converge to the set { Π (I,,u : I = 0, = 0. (50 On this invariant set from (48, by differentiating the first line of (48 we get u = 0. Moreover, from (49 it follows that I = I which further implies = and u = u. Corollary 2 (Input shaping for buck DC-DC converter. Let Assumptions 1 3, 5 and 6 hold. Consider system (48 with the dynamic controller u = 1 k d (k i (u u+ s I, (51 where k i,k d > 0 are free tuning parameters. Then, the solution (I,, u to the closed-loop system asymptotically converges to the desired steady-state ( I,,u. Proof. Note that R = 0. To use the input shaping controller (44 proposed in Theorem 4, we need to show that Γ(u is full rank. In the case of buck converter Γ 0 = Γ 1 = 1, hence Γ(u = 1. The rest follows from the proof of the Theorem 4. Remark 10 (Alternative proof to Corollary 2. On the other hand, one can also infer asymptotic stability of Corollary 2, from the time derivative of the storage function (47 along the trajectories of the closed-loop system (48 and (51, i.e., Ṡ d = G 2 k d u 2. (52

9 9 I L s u Figure 2. Electrical scheme of the boost converter. + Table II PASSIE MAPS FOR BOOST-CONERTER m(i, Passive map γ(i, 1 u I I I I 2 1 I C 2 u d I dt I 2 u d dt I ( 2 +I 2 u d ( I dt tan 1 (I u d ( I dt ln G I I ( I tan 1 ( I ln Then, from (52, there exist a forward invariant set Π and by Lasalle s invariance principle the solutions that start in Π converge to the largest invariant set contained in { Π (I,,u : u = 0, = 0. (53 On this invariant set from (48, by differentiating the second line of (48 we get I = 0. Moreover, from (51 it follows that u = u which further implies = and I = I = G. B. Boost converter Consider now the electrical scheme of the boost converter in Figure 2, where the diode is again assumed to be ideal. The average governing dynamic equations of the boost converter are the following: L I = (1 u s, u [0,1] C = (1 ui G. (54 Also in this case, system (54 can be equivalently obtained from (10 with Γ 0 = 1, Γ 1 = 0, B 0 = B 1 = 1 and R = 0. By using Proposition 2, the following passivity property is established. Lemma 3 (Passivity of boost converter. Let Assumptions 1 and 2 hold. System (54 is passive with respect to the storage function (16 and the port-variables u and I I. Remark 11 (Integrable outputs for boost converter. Note that the output port-variable I I is not integrable, but one can still find an integrable output using Lemma 1. For example, the choice m(i, = 1 defines the passive map I2 ui 2 d. See Table II, for more passivity properties dt I with integrable output port-variables. The following results are now established. Corollary 3 (Output shaping for boost converter. Let Assumptions 1 3 and 5 7 hold. Consider system (48 with the dynamic controller u = 1 2 ( k i ( I I d +k d dt I, (55 where k i,k d > 0 are free tuning parameters. Then, the solution (I,, u to the closed-loop system asymptotically converges to the desired steady-state ( I,,u. Proof. Note that R = 0. Therefore, Theorem 2 can be used if system (54 satisfies (30, i.e., 0, which is an assumption in the corollary statement. Following the steps in Theorem 2 with γ = I, m = 1 2 and γ = I, we can show that the solutions of the closed-loop system converge to the set { Π (I,,u : = 0, I = 0. (56 By differentiating the first line of (54 we get u = 0. Moreover, from (55 it follows that γ = γ. From the steady state equations of system (54, the second line of (54 results in u = 1 G I = 1 G1 γ = 1 G = u, (57 I which further implies = and I = I. Corollary 4 (Input shaping for boost converter. Let Assumptions 1 3, 5 and 6 hold. Consider system (54 with the dynamic controller u := 1 (k i (u ū+( I I, (58 k d where k i,k d > 0 are free tuning parameters. Then, the solution (I,, u to the closed-loop system asymptotically converges to the desired steady-state ( I,,u. Proof. Note that R = 0. To use the input shaping controller (44, proposed in Theorem 4, we need to show that Γ(u is full rank. In the case of boost converter Γ 0 = 1, Γ 1 = 0, hence Γ(u = 1 u. The full rank condition of Γ(u = 1 u is satisfied by Assumption 5. The rest follows from the proof of the Theorem 4. C. Simulations In this subsection, we present the simulation analysis of the proposed techniques. The system parameters for the buck and the boost converters are reported in Table III. Moreover the tuning values for the controllers are given in Table I. Input shaping and the output shaping techniques for the buck and boost are simulated for a duration of 5 seconds in total. The time evolution of the voltage, current and duty-ratio are depicted in Figures?? 5. To verify the controller s robustness against load-uncertainty, at time t = 1 second, the value of the load has been changed from G to G+ G (see Table III. From Figures 3 and 5, we observe that the load voltage settles to the desired value after a minor perturbation. Therefore, as pointed out in Remark 9, control techniques derived using

10 10 Table III SYSTEM PARAMETERS Converter L(mH C(mF G(S G (S s( ( buck boost Table I TUNING PARAMETERS Converter Output Shaping Input Shaping k i (10 7 k d (10 3 k i (10 7 k d (10 6 Buck Boost input shaping technique are robust to uncertainties in load G. We hence conclude with the following remark. Remark 12 (Knowledge of the load parameter. Controllers (49 and (55 depend on the load parameter. Hence, the output shaping methodology is not robust with respect to load uncertainty (see Figures?? and 4. However, controller (51 and (58 are independent from the load parameter. Hence, the input shaping methodology is robust with respect to load uncertainty (see Figures 3 and 5. I. APPLICATION TO DC NETWORKS In this work we consider a typical DC microgrid of which a schematic electrical diagram is provided in Figure 6, including buck and boost DC-DC power converters. In the following we adopt the subscripts α or β in order to refer to the buck or boost type converter, respectively. The network consists of n α buck converters and n β boost converters, such that the total number of converters is n α + n β = n. The overall network is represented by a connected and undirected graph G = ( α β,e, where α = {1,...,n α is the set of the buck converters, β = {n α +1,...,n is the set of the boost ( I (A u time (s time (s time (s Figure 4. (Output shaping for boost converter From the top: time evolution of voltage, current and duty cycle considering a load variation at the time instant t = 1 s (Parameters: L = 1.12mH, C = 6.8mF, s = 280, G = 0.04S, G = 0.02S, = 380, k i = , k d = u ( I (A time (s time (s 0.26 time (s u ( I (A time (s time (s time (s Figure 3. (Input shaping for buck converter From the top: time evolution of voltage, current and duty cycle considering a load variation G at the time instant t = 1 s (Parameters: L = 1mH, C = 1mF, s = 400, G = 0.04S, G = 0.02S, = 380, k i = , k d = , u = / s. Figure 5. (Input shaping for boost converter From the top: time evolution of voltage, current and duty cycle considering a load variation at the time instant t = 1 s (Parameters: L = 1.12mH, C = 6.8mF, s = 280, G = 0.04S, G = 0.02S, = 380, k i = , k d = , u = 1 s/. converters and E = {1,...,m is the set of the distribution lines interconnecting the n converters. The network topology is represented by its corresponding incidence matrix D R n m. The ends of edge k are arbitrarily labeled with a + and a, and the entries of D are given by +1 if i is the positive end of k D ik = 1 if i is the negative end of k 0 otherwise. According to (48, the average dynamic equations of the

11 11 u i Ii L i R lk Ilk L lk Ij L j si C i i G i G j C j j u j sj + + Figure 6. The considered electrical diagram of a (Kron reduced DC network representing node i α and node j β interconnected by line k E. buck converter i α become and L i Ii = i u i si, u i [0,1] C i i = I i G i i k E i I lk, (59 P(u,I, = I Γ(u I l R l I l 1 2 α G α α 1 2 β G β β, (64 where E i E is the set of the distribution lines incident to the node i, and I lk denotes the current through the distribution lines. On the other hand, according to (54, the average dynamic equations of the boost converter i β become L i I i = (1 u i i si, u i [0,1] C i i = (1 u i I i G i i k E i I lk. (60 Let = [ α, β ], with α = [ 1,..., nα ] and β = [ nα+1,..., n ]. Analogously, let I α = [I 1,...,I nα ] and I β = [I nα+1,...,i n ]. The dynamic of the current I lk from node i to node j i, i,j α β, is given by L lk Ilk = ( i j +R lk I lk. (61 To study the interconnected DC network we write systems (59-(61 compactly for all buses i α β L αiα = α u α sα L βiβ = (1 nβ u β β sβ L lil = D T +R l I l C α α = I α G α α +D α I l C β β = (1 nβ u β I β G β β +D β I l, (62a (62b (62c (62d (62e where I α, α, sα,u α R nα, I β, β, sβ,u β R n β, I l R m. Moreover,, according to Assumptions 1 and 2, L α,l β,l l,c α,c β,r l,g α,g β, are positive definite diagonal matrices of appropriate dimensions, e.g. L α = diag(l 1,...,L nα, 1 nβ R n β denotes the vector consisting of all ones. The matrices D α R nα m and D β R nβ m are obtained by collecting from D the rows indexed by α and β, respectively. Let I = [Iα,I β,i l ], u = [u α,u β ], s = [sα, sβ ], L = diag(l α,l β,l l and C = diag(c α,c β. We notice that system (62 can be expressed in the BM formulation (10 with B(u = diag(u α 0 nα n β 0 n β n α I nβ 0 m nα 0 m n β, (63 where Γ R (n+m n is given by I nα 0 nα n β Γ(u = 0 n β n α I nβ diag(u β, (65 D T α I being the identity matrix. Upon using the storage function in (16, we state the following passivity property for the considered DC network equations (62. Lemma 4 (Passivity of DC Networks. Let Assumptions 1 and 2 hold. System (62 is passive with respect to the storage function (16 and the port-variables u and [ ] I y DC = α sα I β β (66 β I β Next, with the help of the above passive map and by following a similar approach proposed in Corollaries 2 and 4, we present decentralized input-shaping control schemes for the DC-Network (62. Proposition 3 (Input shaping for DC Networks. Let Assumptions 1 3, 5 and 6 hold. Consider system (62 with the dynamic controller D T β u = K 1 d (K i (u ū+y DC, (67 where K i,k d > 0 R nα n β are free tuning parameters and y DC is given by (66. If R l > 0 and G > 0, then the solution (I,, u to the closed-loop system asymptotically converges to the desired steady-state ( I,,u. Proof. Consider the storage function (16. We use the integrated input port-variable to shape the desired closed-loop storage function, i.e., S d = S (u ū K i (u ū. (68 Then, the first time derivative of S d along the trajectories of system (62 controlled by (67 is Ṡ d = Il R lil G + u y DC + u K i (u u (69a = Il R I l G u K d u (69b In (69a we use Lemma 4, the controller (67. Then, from (69b there exists a forward invariant set Π and by Lasalle s

12 12 Table NETWORK PARAMETERS Node L i (mh C i (mf si ( i ( G (S G (S Table I LINE PARAMETERS Line R lk (mω L lk (µh ( time (s I 1 (1 u 2 I 2 I 3 (1 u 4 I 4 Node (buck1 {{ I 1 G 1 1 Node (boost4 {{ 1 L l4 R (1 u 4I 4 G 4 4 l4 4 I (A R l1 L l1 L l3 R l3 0 time (s L l2 2 (1 u R l2 3 2I 2 G 2 2 {{ Node (boost2 I 3 G 3 3 {{ Node (buck3 Figure 7. Scheme of the considered (Kron reduced microgrid with 4 power converters. The dashed lines represent the communication network. invariance principle the solutions that start in Π converge to the largest invariant set contained in Π I,, I,,u : I l = 0, = 0, u = 0. (70 On this invariant set, by differentiating (62d and (62e we get I = 0. Moreover, from (67 it follows that u = u which further implies = and I = I. The input shaping control strategy is now assessed in simulation, considering a DC network comprising four power converters (i.e., two buck and two boost interconnected as shown in Figure 7. The parameters of each converter and the line parameters are reported in Tables and I, respectively. The controller gains for the buck converters are k dα = and k iα = , while for the boost converters are k dβ = and k iβ = The most significant electrical signals of the simulation results are shown in Figure 8. Moreover, the simulations are conducted for a total of 5 seconds and further, at t = 1 second, the load has been changed from G to G+ G (see Table. One can appreciate that the voltage signals (in Figure 8 settles back to the desired values after a minor perturbation. II. CONCLUSIONS AND FUTURE WORKS In this paper, we have presented new passivity properties for a class of RLC and s RLC circuits that are modeled using a Brayton-Moser formulation. We use these new passivity properties to propose two control methodologies. The key observations are: u u 1 u 2 u 3 u 4 0 time (s Figure 8. (Input shaping for the DC network From the top: time evolution of the voltage of each node, current generated by each converter and duty cycle of each converter, considering a load variation at the time instant t = 1 s. (i The obtained passive map in Propositions 1 and 2, consider only the dynamics evolving in the tangent space of the system. In this way, the storage function (16 can be elucidated as the energy in the tangent space. (ii In Table II we have summerized the passivity properties derived in the port-hamiltonian, Brayton-Moser and proposed frameworks, respectively. As shown in Remark 3, in order to infer new passive maps for s RLC circuits using the total energy as storage function, matrices B 0 and B 1 need to be zero and non-zero, respectively. (iii The output shaping methodology exploits the integrability property of the output port-variable of the open-loop system. The input shaping technique instead exploits the integrability property of the input port-variable. (iv The controllers designed using the input shaping methodology are robust to load uncertainty. Possible future directions include to investigate the relationship between the proposed methodologies and similar passivity properties (e.g. differential passivity, incremental passivity and equilibrium independent passivity [37], incorporate nonlinear loads (e.g. constant power loads and develop distributed control schemes (e.g. for achieving power sharing.