Passivity Preserving Model Reduction and Selection of Spectral Zeros M A R Y A M S A A D V A N D I

Transcription

1 Passivity Preserving Model Reduction and Selection of Spectral Zeros M A R Y A M S A A D V A N D I Master of Science Thesis Stockholm, Sweden 28

2 Passivity Preserving Model Reduction and Selection of Spectral Zeros M A R Y A M S A A D V A N D I Master s Thesis in Numerical Analysis (3 ECTS credits) at the Scientific Computing International Master Program Royal Institute of Technology year 28 Supervisor at CSC was Axel Ruhe Examiner was Michael Hanke TRITA-CSC-E 28:15 ISRN-KTH/CSC/E--8/15--SE ISSN Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-1 44 Stockholm, Sweden URL:

3 Abstract In this work we will show projection methods, developed by Sorensen and Antoulas, for model order reduction. The algorithms are designed for passivity preserving model reduction of linear time invariant systems. The algorithms are based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. We show a (numerical) problem which might occur during application of the methods and discuss ways to deal with it. We also discuss which spectral zeros we should take to have a better approximation.

4 Referat Att beräkna passiva reducerade modeller, val av spektrala nollställen Sammanfattning: Vi studerar två projektionsmetoder för modellreduktion som utvecklats av Antoulas och Sorensen. De är avsedda för passivitetsbevarande modellreduktion för linjära tidsinvarianta system. De bygger på interpolation i utvalda spektrala nollställen hos den ursprungliga överföringsfunktionen, så att den reducerade överföringsfunktionen har de utvalda rötterna som nollställen. Vi visar på ett numeriskt problem som uppstår vid denna beräkning. Vi diskuterar även vilka spektralnollställen vi skall välja för att få bästa approximation.

5 Acknowledgments This research has been divided between Kungliga Tekniska Högskolan (KTH) at Stockholm, Sweden and NXP Semiconductors/Croporate I&T/DTF/DM/PDM at Eindhoven, Netherlands. This study has been financed by NXP Semiconductors. I would like to appreciate it gratefully. Professor Axel Ruhe was my supervisor at KTH, Dr. Jan ter Maten and Dr. Joost Rommes were my industrial supervisors. I would like to appreciate their support and contributions. I would like to express my gratitude to Dr. Lennart Edsberg the coordinator of the scientific computing program at KTH. I would like to thank my colleagues and friends at NXP. Last but not the least; I would like to thank especially my husband, Kasra Mohaghegh, and my parents because of their permanent love and support.

6 Contents Contents 1 Introduction 1 2 Circuits Introduction Electric Circuits Kirchhoff s Laws Kirchhoff s Current Law (KCL) Kirchhoff s Voltage Law (KVL) Branch Constitutive Relations (BCR) Circuit components Resistive components Reactive Components Controlled Components Circuit Equations Introduction Incidence matrix Nodal Analysis (NA) Modified Nodal Analysis (MNA) Analysis Of Circuit Equations 17

7 4.1 Introduction Direct Current Analysis (DC) Small Signal (Alternating Current) Analysis (AC) Transient Analysis (TR) Pole-zero Analysis (PZ) System Poles and Zeros Transfer Function Backward Differential Formula Method (BDF) Newton-Raphson Method Differential Algebraic Equation (DAE) Introduction Theory of Differential Algebraic Equations Initial Value Problem and Solvability Stability Index of DAEs Semi-Explicit DAE Dynamical Systems and Passivity Preserving MOR Introduction Dynamical System Model Reduction via Projection Matrices Passive Systems Spectral Zeros Spectral Zeros and Generalized Eigenvalue Problem Passivity Preserving Model Reduction Projection Method Model Reduction by Projection (Sorensen) Model Reduction by Projection (Antoulas)

8 7 Numerical Results Introduction Choosing the Spectral Zeros Preserving Real Spectral Zeros Common Poles and Spectral Zeros Effect of Real Spectral Zeros Reducing the Descriptor System (E I) Conclusions 63 Bibliography 65

9 Chapter 1 Introduction This thesis is concerned with linear time invariant (LTI) systems which result in circuit simulation. The tendency to analyze and design systems of ever increasing complexity is becoming more and more a dominating factor in progress of chip design. Along with this tendency, the complexity of the mathematical models increases both in structure and dimension. Complex models are more difficult to analyze, and due to this it is also more difficult to develop control algorithms. Therefore Model Order Reduction is of utmost importance [1]. One of the most important targets is that the reduction procedure preserves the stability and passivity of the original system [2, 25]. To acquire this goal we need some information about circuits. In chapter 2 we introduce the electric circuits and Kirchhoff s laws. Circuit components and some of their features are explained as well [8, 16]. Nodal analysis, a way to define the circuit equations, is studied in chapter 3 [22, 27]. In chapter 4 we discuss the analysis of circuit equations and their usages. Definition of poles and zeros are explained. The transfer function is introduced in this chapter as well. The transfer function is one of the most important concepts which we use in this thesis. For time-integration, the backward differential formula method is explained [22]. Most of the circuit equations are differential algebraic equations (DAEs). DAEs and their stability are studied in chapter 5 [3]. The main contribution of this work starts in chapter 6. In this chapter the LTI systems are introduced. We want to reduce the system by projection methods which are presented by Antoulas and Sorensen. For reduction by projection methods, we construct the projection matrices via interpolation of the spectral zeros. The concept of spectral zeros and their computation is explained completely in this chapter [2, 15, 19, 25]. In chapter 7 we study some examples and apply the projection method. Finally in chapter 8 we conclude which spectral zeros have effect in low and high frequency in reduced model. 1

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11 Chapter 2 Circuits 2.1 Introduction In this chapter a brief introduction to electrical circuits will be given. Kirchhoff s voltage law, Kirchhoff s current law and various circuit components will be discussed. Also we define the circuit components and show their symbols and related equations. 2.2 Electric Circuits In this report electric circuits are defined as a graph with nodes and branches. The branches connect the nodes to each other. i k represents the current through the branch k and v j denotes the voltage of node j. The electric properties of some branches like a voltage source require a concrete direction due to the difference between their positive and negative end nodes. Figure 2.1 shows a RCL-circuit including 3 nodes and 4 branches. There are two kinds of equations that describe the circuit : 1 R 2 e C L + Figure 2.1. A RCL circuit with 3 nodes and 4 branches. 3

12 CHAPTER 2. CIRCUITS Equations that reflect the topology of the circuit. Equations that reflect the properties of the circuit elements. First the equations are described that reflect the topology of the circuit. 2.3 Kirchhoff s Laws The equations that reflect the topology of the circuit do not depend on the type of branches, but describe the way in which the branches are connected. These equations are given by Kirchhof s laws: Kirchhoff s Current Law (KCL): current is not stored in any loop and the algebraic sum of the current at each node is zero. Kirchhoff s Voltage Law (KVL)or Kirchhoff s loop rule: This rule is a result of electrostatic field being conservative. It states that the total voltage around a closed loop must be zero Kirchhoff s Current Law (KCL) The sum of all incoming currents is equal to the sum of all outgoing currents: i k = (2.1) k node Kirchhoff s Voltage Law (KVL) The sum of all branch voltage through each closed circuit is equal to zero. v k = (2.2) k loop Note that the v k is the potential difference between the two nodes the branch connects: v k = v + k v k (2.3) 2.4 Branch Constitutive Relations (BCR) The branch constitutive relation (BCR) shows the electrical features of branches and the two Kirchhoff s laws and complete the description of the circuit together. The branch equations 4

13 2.5. CIRCUIT COMPONENTS can contain branch variables, such as current through a branch, and expressions containing branch variables. In general a branch equation only contains branch variables of the branch concerned. However it is possible that branch variables associated with other branches are included. These branch variables, called controlling variables, such as a voltage-controlled current source, make the branch a controlled branch. 2.5 Circuit components In this section circuit components and BCRs of circuit elements are introduced. There are two types circuit components: Resistive components Reactive components and each type has specific properties Resistive components Resistor components are defined by the algebraic branch equation x i = f(t, x), (2.4) where x i R is the circuit variable concerned, x R n is a vector containing all circuit variables and f : R R n R is a function depending on one or more circuit variables. Resistor In figure 2.2 a resistor is shown: Resistors are characterized by a relation between their current + Figure 2.2. A Resistor. and voltage by the Ohm s law. Ohm s law is V = IR. (2.5) 5

14 CHAPTER 2. CIRCUITS The general BCR of resistor, in the linear case, is The BCR is given by i R = V R. (2.6) i R = i(v R ), (2.7) covering linear and non-linear resistors, where v R is the potential difference between v + and v at the two nodes that are connected by resistor. Because in linear resistors the currents are explicitly known in terms of the voltage, they are called current-defined and voltagecontrolled. Independent Current Source The symbol of the current source is: In circuit theory, an ideal current source is a circuit + A Figure 2.3. An Independent Current Source. element where the current through it is independent of the voltage across it. If the current through an ideal current source can be specified independently of any other variable in a circuit, it is called an independent current source. Conversely if the current through an ideal current source is determined by some other voltage or current in a circuit it is called dependent or controlled current source. The BCR of an independent current source is: i I = I(t) and v I = any value where v I will be implicity determined by the system of equations. Independent Voltage Source In circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. However, it may be a function of time. If the voltage across an ideal voltage source can be specified independently of any other variable in a circuit, it is called an independent voltage source. The symbol of voltage source is shown in figure 2.4 by the following: An independent voltage source is given by the BCR: v V = V (t) and I V is implicity defined by the system. I V = any value. 6

15 2.5. CIRCUIT COMPONENTS + e Figure 2.4. An Independent Voltage Source Reactive Components The reactive components are determined by the differential equation: x i = d f(t, x), (2.8) dt where x is the number of the unknowns such as voltage node or current through the component. Note that the notation can be used. Capacitor ẋ = dx dt (2.9) The capacitor s capacitance (C) is a measure of the amount of charge (q) stored on each plate for a given potential difference or voltage (v) which appears between the plates: C = q v (2.1) or q = Cv The capacitor is given by the following symbol in figure 2.5: The current I through the capac- + Figure 2.5. A Capacitor. itor is the rate at which charge q is forced through the capacitor ( d dt q c). The charge-voltage relationship for a capacitor (may be nonlinear)is given by: 7

16 CHAPTER 2. CIRCUITS Therefore the BCR of a linear capacitor is q c = q(v c ). or i c = d dt q c = C dv c dt i c = q = Cv c with constant capacitance C. Inductor An inductor is a passive electrical device employed in electrical circuits for its property of inductance. While a capacitor contrasts changes in voltage, an inductor contrasts changes in current. An inductor is characterized by a relationship between its current (i L ) and the + Figure 2.6. A Inductor. magnetic flux (Φ L ). The inductor law is Φ L = Φ(i L ). Hence, the magnetic flux is related to the voltage through the inductor by v L = Φ L. As it is known that Φ L = Li then it follows that, d dt Φ L = L di dt Therefore, the BCR for the inductor becomes Φ = L i where L is the inductance. v L = L i L, (2.11) 8

17 2.6. CONTROLLED COMPONENTS 2.6 Controlled Components If the current through an ideal current source is determined by some other voltage or current in a circuit, it is called a dependent or controlled current source. Also, if the voltage across an ideal voltage source is determined by some other voltage or current in a circuit, it is called a dependent or controlled voltage source. 9

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19 Chapter 3 Circuit Equations 3.1 Introduction In this chapter we studied the relations between the incidence matrix and the Kirchhoff s lows. The nodal analysis for a test circuit is introduced. This chapter ends with Modified Nodal Analysis (MNA) for a test example. 3.2 Incidence matrix The incidence matrix is defined by A R n b (n corresponds to the number of nodes and b corresponds to the number of branches), where a ij = 1 branch j is incident at node i and the current direction is pointing toward node i. branch j is not incident at node i. 1 branch j is incident at node i and the current direction is pointing away from node i. The circuit considered here is a directed graph and each branch is connected to two distinct nodes. Every column of incidence matrix A has exactly two nonzero elements, a 1 and a 1 (and the rest of them are zeros). One of the important properties of the incidence matrix is time independency. Figure 3.1 shows the current of the figure 2.1 and their directions. 11

20 CHAPTER 3. CIRCUIT EQUATIONS 1 R 2 e C L + Figure 3.1. Define a current direction in a RCL circuit. and incidence matrix of the figure 3.1 becomes: A = b 1 b 2 b 3 b n n 1 n 2 For a circuit with n nodes and b branches the incidence matrix has rank(a) = n 1. The values v n R 3, i b R 4 contain the nodal voltages and branch currents, respectively. Both of them are arranged in the same order as rows and columns of A. Simple algebra shows that at any time, which is KCL (2.1), and according to KVL (2.2) Ai b =, (3.1) A T v n = v b, (3.2) at any time and v b R 4 contains the branch voltages. The formulations (3.1) and (3.2) are re-formulations of Kirchhoff s laws. 3.3 Nodal Analysis (NA) According to Kirchhoff s laws for each node at least two equations need to be written: First the current through the component i component, second the voltage across the component v component = v n1 v n2. The current through a component can be non-linear and depends on the electric properties. Note that writing these equations using the branch constitutive 12

21 3.3. NODAL ANALYSIS (NA) relation (BCR) with the Kirchhof s laws is called nodal analysis. The equation for the k-th branch is: i k = d dt q(t, v b) + j(t, v b ) (3.3) with i k is the current through k-th branch, v b R b contains the branch voltages, q : R R b R a function that represents reactive components and jr R b R a function that represents resistive components. Now rewrite the nodal analysis equation in matrix-vector for all currents: i b = d dt q(t, v b) + j(t, v b ) (3.4) where q, j : R R b R b. The system (3.4) can not be solved because it contains 2b unknowns (b unknowns belong to i b and b unknowns relate to v b ) and b equations. First left-multiply system (3.4) with the incidence matrix A: Ai b = d dt A q(t, v b) + A j(t, v b ). The system of equations draw up by KCL (3.1) and KVL (3.2) becomes: = d dt A q(t,at v n ) + A j(t,a T v n ). (3.5) According to section 3.2, rank(a) = n 1 so the rows of A are not linearly independent. For solving the systems one of the unknowns should be chosen as a ground node. The related row to the ground node in matrix A is omitted and the ground node itself is omitted from the vector of unknowns v n. Hence, v n is reduced to ˆv n, so the system has n 1 equations and n 1 unknowns: d dtâ q(t,ât ˆv n + v k Ae k ) + Â j(t,ât ˆv n + v k Ae k ) =, (3.6) where e k is a k-th unit vector in R n. If v k is ground node then v k Ae k = Now, Â( d dt q(t,ât ˆv n ) + j(t,ât ˆv n )) =, Â and by defining q(t, x) = Â q(t,ât ˆv n ) j(t, x) = Â j(t,ât ˆv n ) 13

22 CHAPTER 3. CIRCUIT EQUATIONS we have d q(t, x) + j(t, x) =. (3.7) dt 3.4 Modified Nodal Analysis (MNA) Except the nodal voltage, other unknowns exist that must be found. For this purpose, by applying KCL law write the equations for each nodes same as before: Ai b =. Then replace i k of the voltage-controlled components by their BCR. Substitute the BCR equations of voltage-controlled components into the KCL equations. Treat the currents i k of the current-controlled components as unknowns additionally to the nodal voltages. At the end add the voltage-current relations for all current-controlled components, that define implicity the i k. This procedure is called Modified Nodal Analysis (MNA). Apply an example to explain MNA. The circuit in Figure 3.1 and its incidence matrix A are considered, then to continue step by step: A = and current vector i b is: i b = ( i e i R i C i L ) T then using Ai b = and give following equation: i e + i C i L = i e i R = i R i C + i L =, (3.8) which are exactly the equations of the KCL for all nodes. Now replace the currents by their BCR. i e + C d dt (v v 2 ) i L = i e + v 1 v 2 R = (3.9) v 2 v 1 R + C d dt (v 2 v ) + i L = The next step is to add the voltage-current relations: { v1 v = e v 2 v = L d dt i L. (3.1) 14

23 3.4. MODIFIED NODAL ANALYSIS (MNA) The matrix form for the above system is Cv Cv 2 d dt Cv + Cv 2 + Li L i e i L 1 R v 1 1 R v 2 + i e 1 R v R v 2 + i L v + v 1 e v + v 2 = (3.11) The node is grounded, so v = and the first row in above matrices can be omitted. The result is rewritten in general form (3.7) and d dt Cv 2 Li L + 1 R v 1 1 R v 2 + i e 1 R v R v 2 + i L v 1 e v 2 = q(t, x) = Cv 2 Li L, j(t, x) = 1 R v 1 1 R v 2 + i e 1 R v R v 2 + i L v 1 e v 2, and x = v 1 v 2 i e i L. 15

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25 Chapter 4 Analysis Of Circuit Equations 4.1 Introduction In this chapter the following methods of circuit analysis are described: DC analysis: computes an equilibrium steady state of the circuit. AC analysis: computes the linearized effect of a sinusoidal input source in the circuit. TR analysis: studies time domain behavior of the circuit, and PZ analysis: gives information about the stability of circuit. The circuit equations have the following form: f(t, x, x ) = d q(t, x) + j(t, x) =. (4.1) dt Equation (4.1) is called a differential algebraic equation or DAE because it contains differential equations related to capacitors and/or inductors and algebraic equations corresponding to resistors. 4.2 Direct Current Analysis (DC) DC analysis is the very basic analysis in a circuit simulation and concerns to the steady state solution of a circuit. So all time derivation are zero and the time dependent expressions are constant. Hence the system is i(v DC = ). If the circuit is linear, then the circuit equations for DC analysis form a system of linear equations, which needs only a method to descibe the circuit into the equations and linear solver to solve the system of equations. MNA is the easiest method to transfer the circuit in terms of equations and the system of these linear 17

26 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS equations can be solved by LU decomposition, for instance [16]. Steady-state x DC is time independent and defined by: Equation (4.1) is time independent and becomes: ẋ DC =. (4.2) f DC (x,) = (4.3) t q DC(x) + x q DC(x)ẋ + j DC (x) = (time derivations are zero) j DC = (4.4) The latter system is nonlinear and can be solved by an iterative method such as the Newton- Raphson method. Again we write the equation for Figure 3.1 and consider v o = : i e + v 1 v 2 R = v 2 v 1 R + C d dt v 2 + i L = v 1 = e v 2 = L d dt i L. (4.5) In DC analysis all time derivatives are zero and the time independent expressions are constant: Then the system (4.5) becomes C d dt v k =, L d dt i L =. and the matrix form is: i e + v 1 v 2 R = v 2 v 1 R + i L = v 1 = e v 2 = (4.6) 1 R 1 R The solution of the linear system is: 1 R 1 1 R v 1 v 2 i e i L = e x DC = (v 1,v, i e,i L ) T = (e,, e R, e R )T. 18. (4.7)

27 4.3. SMALL SIGNAL (ALTERNATING CURRENT) ANALYSIS (AC) 4.3 Small Signal (Alternating Current) Analysis (AC) AC analysis determines the perturbation of the solution of the circuit due to small sinusoidal sources, in adding a small time varying signal e(t). Time independent elements such as capacitors and inductors are described as well. Start with considering equation (3.7): d q(t, x) + j(t, x) =. dt AC analysis is used to initial value (steady-state) x(t) = x DC which is the solution of a time independent system (4.3): f DC (x,) =. The AC analysis problem is defined by adding the small signal e(t) to system (3.7): f AC (t, x) = d q(t, x) + j(t, x) = e(t), (4.8) dt where x AC = x DC + x(t). We apply the Taylor expansion for function q and j at point x DC by assuming that e(t) is independent of x: d dt (q(x DC) + q x x(t) + O( x 2 )) + j(x DC ) + j xdc x x(t) + O( x 2 ) = e(t) (4.9) xdc where and q DC is constant d dt q(x DC) = q x is time independent d xdc dt ( q x x(t)) = q xdc x ẋ(t) xdc since x DC is solution of the system j(x) =, we have j(x DC ) =. Note that this whole contribution can be treated as an O( x 2 ) effect. Then equation (4.9) is reduced to: We choose this notation for two Jacobian matrices: q x ẋ(t) + j xdc x x(t) = e(t). (4.1) xdc C = q x xdc G = j. xdc x (4.11) 19

28 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS Substituting C and G in equation (4.1) gives: Cẋ(t) + Gx(t) = e(t), (4.12) which is a system of linear ordinary differential algebraic equations that describes the small signal e(t). The behavior of the solution can also be studied in frequency domain. By defining x(t) = X exp(iωt) and e(t) = E exp(iωt), where X and E are time independent vectors and ω is common angular frequency, the system (4.12) becomes Since exp(iωt), it follows that C d dtx exp(iωt) + GX exp(iωt) = E exp(iωt) CiωX exp(iωt) + GX exp(iωt) = E exp(iωt) For small signal e(t) = E exp(iωt) with E 1 it follows that (iωc + G)X = E. (4.13) X = (iωc + G) 1 E. Small signal analysis is also called Alternating Current analysis because the small signal added to the vector e(t) is a sinewave and it can be interpreted as an alternating current [8, 22]. 4.4 Transient Analysis (TR) Since the solution of the system (3.7) is time dependent, one has to integrate it in a time interval [, T]. This is called Transient Analysis. So we divide the time interval in to the small intervals [,t 1,t 2,...,T]. At each time interval [t k 1,t k ] the differential equations will be transformed by a numerical integration algorithm into algebraic equation. Therefore a system of non-linear algebraic equations should be solved. The solution at t = is determined by the DC-solution and the solution of the transient analysis then can be found iteratively. This numerical integration can be implicit as well as explicit. For circuit simulation it is preferred to use an implicit method, because of the algebraic equations and the possible stiff behavior, like the Euler backward method, which is a backward differential formula (BDF) method. We consider d dt q(t, x k) 1 t (q(t, x k) q(t, x k 1 )) (4.14) where t = t k t k 1 is the time step. Substituting (4.14) in (3.7): q(t, x k ) = q(t, x k 1 ) tj(t, x k ) (4.15) where q(t, x k 1 ) is known at t k. For more accurate approximation, the time step needs to be chosen small enough. 2

29 4.5. POLE-ZERO ANALYSIS (PZ) 4.5 Pole-zero Analysis (PZ) Pole-zero analysis is used in electrical engineering to analyze the stability of the electrical circuit. For example if the circuit is designed to be an oscillator, pole-zero analysis is one of the ways to verify that the circuit indeed oscillates. As the circuit becomes complicated nowadays, there is an urgent need for fast and accurate algorithms. Here we just introduce the pole-zero analysis and the two different strategies for dealing with the problem System Poles and Zeros The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable s = σ + jω, that is H(s) = b ms m + b m 1 s m b 1 s + b a n s n + a n 1 s n a 1 s + a (4.16) It is often convenient to factor the polynomials in the numerator and denominator, and to write the transfer function in terms of those factors: H(s) = N(s) D(s) = K (s z 1)(s z 2 )... (s z m 1 )(s z m ) (s p 1 )(s p 2 )... (s p n 1 )(s p n ) (4.17) where the numerator and denominator polynomials, N(s) and D(s), have real coefficients defined by the system s differential equation and K = b m /a n. As written in equation (4.17) the z i s are the roots of the equation N(s) = and are defined to be the system zeros and p i s are the roots of the equation D(s) = and are defined to be the system poles. In equation (4.17) the factors in the numerator and denominator are written so that when s = z i the denominator N(s) = and the transfer function vanishes: lim H(s) =, (4.18) s z i and similarly when s = p i the denominator polynomial D(s) = and the value of the transfer function becomes unbounded, lim H(s) =. (4.19) s p i All of the coefficients of polynomials N(s) and D(s) are real, therefore the poles and zeros must be either purely real, or appear in complex conjugate pairs. In general for the poles, either p i = σ i, or p i,p i+1 = σ i ± jω. The existence of a single pole without a corresponding conjugate pole would generate complex coefficients in the polynomial D(s). Similary, the system zeros are either real or appear in complex conjugate pairs. 21

30 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS Transfer Function As already mentioned in section (4.1) the way which the system (4.1) is solved depends on the kind of analysis (DC-analysis, AC-analysis, Transient analysis and Pole-zero analysis). To compute the dynamic response of a circuit variable or expression to small pulse excitations by an independent source we use a pole-zero analysis. It is explained by the transfer function. The transfer function is defined by its pole-zero representation. Pole-zero analysis provides with the pole-zero representation stability properties of the circuit. The circuit transfer function H(s) approaches the response of a linear circuit to source variations in the Laplace (frequency) domain: H(s) = L(zero stateresponse)(s) L(sourcevariation)(s) (4.2) where L(f)(s) is the Laplace transform of function f (defined in the time domain) and s is the (complex) variable in the frequency domain. The zero state response represents the response to the stationary solution. The zero state response independents on the initial condition (solution), only depends on the excitation. Starting from a linearization around the operating point, the time domain formulation is as follows: { C d dtx(t) + Gx(t) = e(t) x() = (4.21) where e(t) models the excitation, C and G are defined in (4.11). Because not all properties can be computed in the time domain, the problem is transformed to the frequency domain by applying a Laplace transform (sc + G)X(s) = E(s), (4.22) where X(s), E(s) are the Laplace transforms of the variables x, e and s is the variable in the frequency domain. The response of the circuit to a variation of the excitation is given by the transfer function H(s) = X(s) E(s). (4.23) Hence, H(s) = (sc + G) 1. (4.24) 4.6 Backward Differential Formula Method (BDF) One of the best solutions for solving the DAE is a combination of an implicit integration method and a nonlinear solver. Instead of using one step methods, it is possible to use a 22

31 4.6. BACKWARD DIFFERENTIAL FORMULA METHOD (BDF) Linear Multistep Method (LMM). The integration method can be chosen from the class of LMM. A linear k-step method can be used to compute the solution of q(t i, x i ) by using k 1 q(t i+k, x i+k ) α j q(t i+j, x i+j ) + t j= k j= β j d dt q(t i+j, x i+j ) (4.25) where t = t i t i, k N, α,α 1,...α k 1 R, β,β 1,... β k R and α 2 + β2 > (It is guarantees The formula is a real k-th step method and it is not k -method with k < k ). Now, if k = α = β = 1 and β 1 = equation (4.25) becomes: q(t i+1, x i+1 ) = q(t i, x i ) + t d dt q(t i, x i ) This is an Euler Forward scheme. If k = α = β 1 = 1 and β = the equation (4.25) is reduced to Euler Backward scheme q(t i+1, x i+1 ) = q(t i, x i ) + t d dt q(t i+1, x i+1 ). We pursue the argument leading to the backward Euler method to derive the family of Backward Differential formulas (BDF). Using the (i + 1)-th iteration of d dtq(t, x) of the equation (3.7) with α k = 1 can be approximated by: d dt q(t i+1, x i+1 ) Then we define b i+k as: 1 β k t k α j q(t i+j, x i+j ) 1 k 1 d β j β k dt q(t i+j, x i+j ). (4.26) j= j= b i+k = 1 k 1 β k t j= Substitution b i+k in (4.26) gives: α j q(t i+j, x i+j ) 1 k 1 β k j= β j d dt q(t i+j, x i+j ). and hence equation (3.7) can be written as: d dt q(t i+1, x i+1 ) b i+k + 1 β k t q(t i+k, x i+k ) b i+k + 1 β k t q(t i+k, x i+k ) + j(t i+k, x i+k ) =. (4.27) A Newton method can be used to compute x i+k. In a Newton method at each time step a nonlinear equation must be solved. For solving this equation it needs the Jacobian of the equation (4.27), which is given by 23

32 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS 1 C(t, x) + G(t, x). β k t Newton-Raphson Method Application of the MNA to a circuit and the usage of a suitable time integration method, will lead to a system of nonlinear algebraic equations for each discretisation point t i. Consider the equation f(x) = (4.28) with x = x i is the vector of the unknown variables (can be voltages or currents) at the moment t i. The Newton-Raphson method is usually used in simulation programs because of its efficiency. It starts with initial value (solution) x. The i-th iteration of the Newton- Raphson is x i+1 = x i (Jf(x i )) 1 f(x i ), (4.29) where Jf(x i ) = f xi is the Jacobian matrix of f computed at x i. x Convergence of Newton-Rephson Method We want to state a convergence theorem of Newton-Rephson s method, so at first we recall a following lemma. Lemma 4.1. If the Jacobian matrix Jf(x) exists for all x in a convex region C R n, and if a constant γ exists with then for all x, y C the estimate Jf(x) Jf(y) γ x y for all x, y C, f(x) f(y) Jf(y)(x y) γ x y 2 2 holds. 1 Recall that a set M R n is convex if x, y M implies that the line segment [x, y] := z = λx + (1 λ)y λ 1 is contained within M. Now we can show that Newton-Rephson s method is quadratically convergent. 1 Proof [26], page

33 4.6. BACKWARD DIFFERENTIAL FORMULA METHOD (BDF) Theorem 4.1. Let C R n be a given open set. Further, let C be a convex set with C C, and let f : C R n be a function which is differentiable for all x C and continuous for all x C. For x C let positive constants r,α,β,γ,h be given with the following properties: and let f(x) have the following properties S r (x ) := x x x < r C, h := αβγ 2 < 1, α r := (1 h), Jf(x) Jf(y) γ x y for all x, y C. Jf(x) 1 exists and satisfies Jf(x) 1 β for all x C. Jf(x) 1 f(x ) α. then 1. beginning at x, each point x i+1 := x i Jf(x i ) 1 f(x i ), i =,1,..., is well defined and satisfies x i S r (x ) for all i. 2. lim i x i = ξ exists and satisfies ξ S r (x ) and f(ξ) =. 3. for all i 1 x i ξ α h2i 1 1 h 2i. since < h < 1, Newton-Raphson s method is at least quadratically convergent. 2 We give a short proof of the quadratically convergence. By applying (4.29) we get because Jf(x i ) 1 exists and f(ξ) =, we get x i+1 ξ = x i Jf(x i ) 1 f(x i ) ξ, x i+1 ξ Jf(x i ) 1 Jf(x i )(x i ξ) f(x i ) + f(ξ). Applying lemma 4.1 and quadratically convergence is shown: 2 Proof [26], page 27 x i+1 ξ βγ 2 x i ξ 2. 25

34 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS It will be applied to the system given in figure 7.1 for explaining the Newton-Rephson method. One of the resistors is defined by variable resistance and the current through it is described by function i 1 (v) = v 2 + v. This will lead to a nonlinear system. 1 R 1 i 1 (v) = v 2 + v 2 + e R 2 R 3 Figure 4.1. A Non-linear Circuit. The equations are: 1 R R 3 1 R R 3 1 2v 2 v 1 1 v R R 3 2v 2 v 1 1 v R R v v 1 v 2 i e = e. If node is grounded (v = ), the system becomes:. 2v 2 v 1 1 v v 1 + 2v 2 1 v R R 3 1 v 1 v 2 i e = e Now writing in equation form f(x) = f(x) = v2 2 v v 1v 2 + v 2 v 1 i e v1 2 v v v 1v 2 + v 2 v 2 1 v 1 e 26 R 2 + v 2 R 3 = (4.3)

35 4.6. BACKWARD DIFFERENTIAL FORMULA METHOD (BDF) and the its Jacobian matrix becomes Jf(x) = 2v 1 + 2v 2 1 2v 2 + 2v v 2 2v 1 1 2v 2 + 2v R R 3 1 and the unknowns are x = v 1 v 2 i e. It is assumed R 2 = 1Ω, R 3 = 5Ω and the voltage across the voltage source is e = 1V. As an initial value take the DC solution x = ( 1, 8.8, 2.64). Then the functional matrix looks like: 2v 2 v 1 1 v v 1 + 2v 2 1 v , therefore the equation (4.3) becomes f(x) = v2 2 v v 1v 2 + v 2 v 1 i e v1 2 v v v 1v 2 + v 2 v v 2 5 v 1 1 = The k-th iteration of the Newton-Raphson process can be described as equation (4.29). For this example, the system of equation will converge after 6 iterations steps. The values for x at the each iterations steps are (see Table 4.1): Table 4.1. The value for x at each steps #iteration v v i e The error x i x, where x is the exact solution, is given in Table

36 CHAPTER 4. ANALYSIS OF CIRCUIT EQUATIONS Table 4.2. The error x i x where x is the exact solution at each steps #iteration v 1 v e e-14 i e e-3.222e-14 The quadratic convergence is clearly seen in the table

37 Chapter 5 Differential Algebraic Equation (DAE) 5.1 Introduction In this chapter the Differential Algebraic Equation (DAE) is explained. Their solvability and stability are also studied. Finally the definition of the index of a DAEs is defined. More information about the dynamical systems can be found in [17, 18]. 5.2 Theory of Differential Algebraic Equations Initial Value Problem and Solvability We consider a certain electrical circuit system which is defined by the following differential algebraic equation: d dtq(t, x) + j(t, x) = x() = x, (5.1) where x R b is a state vector and q, x : [,T] R b R b (which can be find by Modified Nodal Analysis (section (3.4))). The solution x(t) of (5.1) describes the dynamic behavior of the system for a known initial value, the steady state for instance. Ordinary differential equation (ODE) is a special case of the DAEs { ẋ = f(t, x) x() = x, (5.2) where f : [,T] R b R b. In this case TR analysis can compute the solution. According to 29

38 CHAPTER 5. DIFFERENTIAL ALGEBRAIC EQUATION (DAE) the uniqueness theorem the IVP (5.2) with f(t, x) Lip(I Ω) 1 for some domain (I Ω) containing (t, x ) has at most one solution [11]. Here there is a problem in solving DAEs. Most of the DAEs can not be represented by ODEs. We consider system (5.1) and apply the expansion of derivatives for function q: Note that with d dx q(t, x)ẋ + q(t, x) + j(t, x) =. (5.3) t equation (5.3) becomes C(t, x) = d q(t, x), dx C(t, x)ẋ + q(t, x) + j(t, x) =. t For solving this equation we need to discuss about C(t, x); if C(t, x) is nonsingular and invertible for all x, the DAE can be changed to an ODE and the system is solved. But in many cases C(t, x) is singular. The reason of this singularity are the algebraic equations. So the solution has to satisfy a number of algebraic equation also in t = a proper initial solution has to satisfy the algebraic equations which initial solution is called consistent. This means that all initial values are not consistent. If the initial solution is equal to the steady state, it also satisfies the algebraic equations. Hence, for solving the DAEs and to find the best approximation of solution in close form, we should choose the accurate and efficient tools to approximate the solution. The time is discretized in small time interval, while for each time interval the DAE is approximated by a numerical integration scheme [27]. Theorem 5.1. The DAE system (4.12) Cẋ(t) + Gx(t) = e(t) is linear both in x and ẋ is solvable if and only if the matrix pencil λc + G is regular Stability We like to have a numerical method with the property that the numerical solution is close to the exact solution. That means beside the solvability of the problem, stability also is important and necessary. 1 Lipschitz Continuity: The vector field f(t, x) is Lipschitz continuous on I Ω if a constant L exists such that for all x, y Ω and all t I: f(t, y) f(t, x) 2 L y x 2. If f is Lipschitz continuous on (I Ω) we denote this as f Lip(I Ω) [11] 2 You can find the proof [17] 3

39 5.2. THEORY OF DIFFERENTIAL ALGEBRAIC EQUATIONS Definition 5.1. Consider the perturbation IVP of (5.1) with initial value ˆx and solution ˆx(t). The system is stable if: ǫ > δ > ˆx x < δ t ˆx(t) x(t) < ǫ Stability guarantees if the initial value is changed the difference between the approximated solution and the exact solution also is small. In electrical circuits, the time behavior is only caused by sources function. This means that the system (5.1) can be changed to: d dtq(x) + j(x) + u(t) = x() = x, where u(t) is an input function and time dependent function. To check local stability around the initial value is easier than the global stability. A linear time invariant system is stable if the Jacobian matrix is stable 3. (5.4) Theorem 5.2. Let x be the steady state of (5.4), with j(x ) =. Consider the linearised homogeneous system around x Cẋ + Gx = (5.5) where C = x q(x ) and G = x j(x ). This system is stable if all roots of the next equation have strict negative real part: det(λc + G) =. If G is invertible and G 1 C is stable matrix, then this condition has been satisfied. If (5.5) is stable, then the nonlinear system is locally stable around x Index of DAEs To distinguish the degree in solving DAEs we associate an index. We consider the following DAE: F(t, y, ẏ) =. (5.6) This system contains an algebraic and differential parts. As described in section (5.2.1) if F ẏ is nonsingular and invertible then the system (5.6) can be changed to an ODE system. Although, if F show the dynamics of an electrical circuit, it is not the case, but DAE system 3 A square matrix is said to be a stable matrix if every eigenvalue of has negative real part. 4 You can find the proof [18] 31

40 CHAPTER 5. DIFFERENTIAL ALGEBRAIC EQUATION (DAE) can be transferred into the ODE system by differentiating the DAE system and substitute the algebraic equations by extra derived differential equations. Definition 5.2. For general DAE system (5.6), the index along a solution y(t) is a minimum number of differentiations of the system which would be required to solve for y uniquely in terms of y and t (i.e., to define an ODE for y). Thus, the index is defined in terms of the overdetermined system F(t, y, y ) =, df dt (t, y, y, y ) =,. d p F dt p (t, y, y,..., y (p+1) ) = (5.7) to be the smallest integer p so that y in (5.7) can be solved for in terms of y and t [3]. In practice, differentiation of the system as in (5.7) is rarely done in a computation. Nevertheless, such a definition is very useful in understanding the underlying mathematical structure of the DAE system, and hence in selecting an appropriate numerical method. Theorem 5.3. If the pencil matrix λc + G is regular (invertible), there exist nonsingular matrices P and Q such that: PCQ = [ ] I, PGQ = N [ ] A I l (for some l) where the matrix N consists of nilpotent Jordan blocks N i, in other words N = diag(n 1,...,N k ) for some k, with N i, i = 1,...,k given by: 1 1 N i = (or possibility N i = ) and A consists of Jordan blocks with nonzero eigenvalue. The nilpotency index µ is defined as: µ = min{k N,N k = }. If C is nonsingular, we define µ =, because then N is empty. The nilpotency index is also called the local index of the DAE (5.1)[7]. 32

41 5.2. THEORY OF DIFFERENTIAL ALGEBRAIC EQUATIONS Semi-Explicit DAE Most applications of either linear constant coefficient or nonlinear DAE s have led to linear time varying DAE s (5.5) Cẋ + Gx = where C singular. It shows the behavior which distinguishes general DAE s from linear constant coefficient DAE s. System (5.5) is the general or fully-implicit linear time varying DAE [7]. The general (or fully-implicit) nonlinear DAE is F(t, x, ẋ) =. Depending on the application, we sometimes refer to a system as semi-explicit if it is in the form: where Fẋ is nonsingular. { F(ẋ, x, y,t) = G(x, y, t) =, The advantage of the semi-explicit form is that distinguish the differential equations from the algebraic equations. The semi-explicit form of (5.1) is: { ẋ = j(t, x) = y q(t, x). 33

42

43 Chapter 6 Dynamical Systems and Passivity Preserving MOR 6.1 Introduction This chapter begins with introducing dynamical systems and their transfer functions (Section 6.2), [23]. In Section 6.3 we give some information about the reduction model by projection matrices In next two sections we define the passivity of the system and how to preserving passivity after reduction. In Section 6.5 the spectral zeros and the method for computing them are introduced. In the following we describe the projection method for reducing the system. Hence we introduce two methods for finding the projection matrices, which are discussed by Sorensen [25] and Antoulas [2]. These two approaches are based on a projection method by selecting spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. 6.2 Dynamical System This chapter is concerned with dynamical systems = (E,A,B,C,D) of the form { Eẋ(t) = Ax(t) + Bu(t) y(t) = C x(t) + Du(t), (6.1) where A,E R n n, E may be singular (we assume E is symmetric and positive (semi) definite), B R n m, C R n p, D R p m, x(t) R n, y(t) R p and u(t) R m. The matrix E is called descriptor matrix, the matrix A is called state space matrix, the matrices B and C are called input and output map, respectively, and D is direct transmission map. The vectors u(t) and x(t) are called input and state vector, respectively, and y(t) is called the output of the system. The dimension n of the state is defined as the complexity of the system. These systems have been shown in circuit simulation for instance and in this application the system 35

44 CHAPTER 6. DYNAMICAL SYSTEMS AND PASSIVITY PRESERVING MOR is often passive 1. The transfer function G : C m C p, of (6.1), G(s) = C (se A) 1 B + D, can be obtained by applying the Laplace transform to (6.1) under the condition x()=. The transfer function relates outputs to inputs in the frequency domain via Y(s) = G(s)U(s) where Y(s) and U(s) are the Laplace transforms to y(t) and u(t), respectively 2. We want to reduce the original system to a reduced order model ˆ = (Ê,Â, ˆB,Ĉ,D) { Ê ˆx(t) = ˆx(t) + ˆBu(t) ŷ(t) = Ĉ ˆx(t) + Du(t), (6.2) where Â,Ê Rk k, ˆB R k m, Ĉ R k p, D R p m, ˆx(t) R k, ŷ(t) R p, u(t) R m and k n. It is important to produce a reduced model that preserves stability (which is discussed in more details in chapter 5) and passivity. Remark 6.1. Throughout the reminder of this chapter it is assumed that: m = p such that B R n p, C R p n and D R p p. A is a stable matrix i.e. Re(λ i ) < with λ i σ(a),i = 1,,n. The system is observable and controllable [29] and it is passive. 6.3 Model Reduction via Projection Matrices The reduction method in this thesis is based on a projection method. In section 6.7 we introduce two methods which are projection methods. In this section you can find a formulation of projection matrices. We develop a structure of a projection methods for linear time invariant (LTI) systems (6.1) { Eẋ(t) = Ax(t) + Bu(t) y(t) = C x(t) + Du(t), where A,E R n n, E may be singular (E is symmetric and positive (semi) definite), B R n m, C R n p, D R p m, x(t) R n, y(t) R p and u(t) R m. 1 Passivity condition is one of the important concepts and many researches have been studied it, [4, 5, 6, 9, 1, 14, 2, 21]. 2 see Subsection

45 6.4. PASSIVE SYSTEMS Now it is assumed that M and N are k-dimensional subspaces of R n. V and W are built for reducing the system by a projection method. So we construct V = {v 1,,v k } R n k where column vectors form a basis of M and W = {w 1,,w k } R n k where column vectors form a basis of N, (we are interested in W V = I k ). Assuming system ˆΣ is reduced model of original system Σ where k is order of ˆΣ. Therefore the ˆΣ is acquired as a projection of Σ on M and the residual of ˆΣ with respect to Σ is orthogonal to N. We suppose x is an approximate solution of Σ where x satisfies the above structure which means x is a projection of the solution on M and the residual is orthogonal to N. So we can define x = Vˆx, where ˆx R k and ẋ = V ˆx. Then the residual is Eẋ Ax Bu = EV ˆx AVˆx Bu. The residual is orthogonal to N W (EV ˆx AVˆx Bu) = W EV ˆx W AVˆx W Bu = The reduced model ˆΣ becomes: { Ê ˆx(t) = ˆx(t) + ˆBu(t) ŷ(t) = Ĉ ˆx(t) + Du(t), where  = W AV R k k, Ê = W EV R k k, ˆB = W B R k m, Ĉ = CV R k p, ˆx(t) = Vˆx R k and y = ŷ(t) R p [19]. 6.4 Passive Systems We can reduce the model by V and W which are constructed in the previous section 6.3. With arbitrary V and W, some features of the original system may not be preserved. One of these properties which we are interested in to preserve is passivity. When we want to reduce the system, we should preserve the passivity and stability. The matrix A is assumed to be stable which means all its eigenvalues are in the open left halfplane. Definition 6.1. A system is passive if it does not generate energy internally, and strictly passive if it consumes or dissipates input energy [25]. 37

46 CHAPTER 6. DYNAMICAL SYSTEMS AND PASSIVITY PRESERVING MOR In other words Σ is passive if or strictly passive if t Re u(τ) y(τ)dτ, t R, u L 2 (R) δ > t t Re u(τ) y(τ)dτ δre u(τ) u(τ)dτ, t R, u L 2 (R) The transfer function of system Σ is G(s) = C (se A) 1 B + D which shows the relation between input u(s) and output y(s) in frequency domain 3. Another more practical definition of passivity is in the following Definition 6.2. [25] The system Σ is passive iff the transfer function G(s) is positive real, which means that: 1. G(s) is analytic for Re(s) >, 2. G( s) = G(s), s C, 3. G(s) + (G(s)) for Re(s) > where (G(s)) = B (se A ) 1 C + D. Property 3 implies the existence of a stable rational matrix function K(s) R p p (with stable inverse) such that G(s) + (G( s)) = K(s)K ( s). We try to construct the V and W in such a way the transfer function of reduced model has these three properties. 6.5 Spectral Zeros Again we consider the Σ system (6.1) and its transfer function is 3 See Section 6.2 G(s) = C (se A) 1 B + D. 38

47 6.5. SPECTRAL ZEROS In section 6.4 we have seen that if Σ is passive then there exists a stable rational matrix function K(s) R p p (with stable inverse) i.e if we assume G(s) = n(s) d(s) then (G( s)) = n ( s) d ( s). Now we have G(s) + (G( s)) = n(s) d(s) + n ( s) d ( s) = n(s)d ( s)+d(s)n ( s) d(s)d ( s) (because numerator of a fraction is a polynomial) = r(s)r ( s) d(s)d ( s) = K(s)K ( s) This is the spectral factorization of G. K is a spectral factor of G. The zeros of K i.e. λ i, i = 1,,n such that det(k(λ i )) =, are the spectral zeros of G Spectral Zeros and Generalized Eigenvalue Problem We start this section with explaining a generalized eigenvalue problem which Sorensen used it [25]. It brings together the theory of positive real interpolation by Antoulas and invariant subspace method for interpolating the spectral zeros by Sorensen. The components of the generalized eigenvalue problem are constructed from those of the realization of an LTI system. We recall the system Σ (6.1) { Eẋ(t) = Ax(t) + Bu(t) and also recall its transfer function y(t) = C x(t) + Du(t), G(s) = C (se A) 1 B + D. Now we consider (G( s)) = B ( se A ) 1 C + D = B (se ( A )) 1 ( C) + D. Then we compute G + G, 4 G(s) + (G( s)) = (C (se A) 1 B + D) + (B (se ( A )) 1 ( C) + D ) = [ C B ][ (se A) 1 (se ( A )) 1 ][ B C ] + (D + D ) 4 Block wise inversion:» 1» A B A 1 + A 1 B(D CA 1 B) 1 CA 1 A 1 B(D CA 1 B) 1 = C D (D CA 1 B) 1 CA 1 (D CA 1 B) 1 39