Modeling and control of electromechanical systems

Modeling and control of electromechanical systems Carles Batlle, Arnau Dòria Cerezo IOC-DT-P-2005-11 Juliol 2005 Modeling and control of electromechanical systems Carles Batlle(1,2,3) and Arnau D`oria(1,2,4) (carles.batlle)(arnau.doria)@upc.edu (1) Escola Polit`ecnica Superior d’Enginyeria de Vilanova i la Geltr´u (2) Institut d’Organitzaci´o i Control de Sistemes Industrials (3) Departament de Matem`atica Aplicada IV (4) Departament d’Enginyeria El`ectrica Universitat Polit` ecnica de Catalunya Lectures for the II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems Bertinoro, FC, Italy, July 18-22 2005 Contents 1 Preamble 1 2 Electromechanical energy conversion 2 2.1 Network description of systems . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Energy storing elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Electric capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Magnetic stationary system . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Elementary electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 DC motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 Co-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Port Hamiltonian models of electromechanical systems 12 3.1 Dirac structures and port Hamiltonian dissipative systems: basic results . . 12 3.2 General electromechanical systems . . . . . . . . . . . . . . . . . . . . . . 13 4 Power converters 15 4.1 Variable structure systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Second order power converters . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 SSA and GSSA for second order power converters . . . . . . . . . . . . . . 21 5 A system for storing excess energy in an electrical vehicle transportation network 24 6 Energy based control 28 6.1 Control as interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 Casimir functions and the dissipation obstacle . . . . . . . . . . . . . . . . 33 7 IDA control method: basic ideas and examples 35 7.1 Magnetic levitation system . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.2 Boost converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.3 DC motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A Solving quasilinear PDEs 41 B Coordinate transformations for ac machines 44 1 Preamble These lectures are devoted to modeling and control of electromechanical systems in the framework of port Hamiltonian dissipative systems (PHDS), which in the simplest, explicit version are of the form ˙x= (J(x)−R(x))(∇H(x))T+g(x)u, where x∈Rn,Jis antisymmetric, Ris symmetric and positive semi-definite and u∈Rm is the control. The function H(x) is the Hamiltonian, or energy, of the system. The natural outputs in this formulation are y=gT(x)(∇H(x))T. Port Hamiltonian dissipative systems enjoy nice properties when interconnected, and yield themselves to passive control strategies quite naturally. The material presented in the these notes covers the sessions Modeling of electromechanical systems,Passive control theory I and Passive control theory II of the II EURON/GEOPLEX Summer School on Modeling and Control of Complex Dynamical Systems. A basic knowledge of port based modeling of physical systems, namely bond graph theory, and of the port Hamiltonian formulation, as presented in Breedveld’s and Maschke’s lectures in this summer school, is assumed. Nevertheless, some background material is included when needed. Throughout these notes, symbols like ∂xHor ∇Hrepresent the gradient of an scalar function H, and whether they are to be taken as column or row vectors should be clear from the particular context. We start with a general description of what an electromechanical system is from a network modeling point of view. Assuming no dissipation, the energy function is introduced and Maxwell’s reciprocity relations are deduced from the principle of conservation of energy. Dissipation is introduced by ending some of the open ports with resistive elements. As examples, we consider an electrical capacitor with moving plates, an elementary electromagnet, a solenoid transducer and a simple DC motor. Next, a general formulation in terms of PHDS is introduced (with a slight modification for the DC motor), and some of the previous electromechanical systems are rewritten in this formalism. Control of electromechanical systems, notably electrical machines, generally requires a device capable of delivering a variable voltage (or current, in some cases). This is the realm of power electronics, which uses solid state devices (for instance, sets of IGBT — Insulated Gate Bipolar Transistor), to switch between several topologies of special electric circuits, known as power converters. A series of topology changes performs the trick of temporally storing the energy into the converter and then delivering it in the required form to the load. Power converters can also be given a PHDS form; they are variable structure systems (VSS), and this is reflected in the fact that, generally, both Jand g, and even Rin some cases, depend on a set of discrete variables {Si}, which describe which switches are open or closed. In feedback control, these discrete variables are made dependent on the state of the system to accomplish the power conversion goal. It turns out that averaged models of VSS are useful for simulation and control design purposes, and they inherit the PHDS form from the original VSS. For completeness, and also for later use in the last example, we present a PHDS description of the lowest order power converters, both in VSS and averaged forms. 1 We conclude the modeling part of these lectures with a rather complex example, showing the interconnection of subsystems from several domains, namely an arrangement to temporally store the surplus energy in a section of a metropolitan transportation system based on dc motor vehicles, using either arrays of supercapacitors or an electric powered flywheel. The second part of the lectures addresses control of PHD systems. We first present the idea of control as power connection of a plant and a controller, establish the energy balance equation, introduce Casimir variables and present the dissipation obstacle which precludes the application of the simplest version of passivity based control for PHDS to many electrical and electromechanical systems. Next we discuss how to circumvent this obstacle and present the basic ideas of Interconnection and Damping Assignment (IDA) passivity-based control of PHD systems. We apply the method to a magnetic levitation system, a power converter and a dc motor. Appendix A reviews the basic ideas of the method of characteristics, useful for solving some of the partial differential equations that appear when implementing the IDA control scheme, while appendix B contains some technical discussions related to alternating current machines. Some proposed exercises are scattered around the text. Most of them are straightforward but some may require consulting the cited literature for hints. The reader is encouraged to model the systems and controls proposed with 20sim. 2 Electromechanical energy conversion 2.1 Network description of systems The network description of systems is based on the separation of a system into two parts: the energy storing elements (and eventually, energy dissipating elements) and the network, which just shuffles energy around. External ports can also be included, in order to deliver or extract energy to the system and connect it to other systems. Figure 1 displays all these elements. Energy is exchanged through power ports connected by bonds. A couple of variables, an effort eand a flow f, are assigned to each bond. Their product has dimensions of power. The essential characteristic of the network is that it is power continuous: N X i=1 eifi= 0, N =n+m, (1) while the energy-storing elements impose constitutive relations between efforts and flows. A first formalization of the network description idea is provided by bond graph theory, also discussed in detail elsewhere in this school. A trivial example of network description is lumped circuit theory. The efforts eare voltage drops, while the flows fare currents. The network imposes Kirchoff laws, which are power continuous (Tellegen’s theorem). 2 e1 en en+m en+1 fn+1 f1fn fn+m Energy-storing elements Network Figure 1: Conceptual elements in a network-based description of physical systems. 2.2 Energy storing elements Consider a system with mpower ports. The energy supplied by the ports between t0and tis given by the field energy increase Wf(t)−Wf(t0) = Zt t0 m X i=1 ei(τ)fi(τ) dτ. (2) We assume that the state of the system can be described by nstate variables xi, which, using vector notation, evolve, with the efforts as input signals, according to1 ˙x=G(x) + g(x)e, (3) where g(x)∈Rn×mindicates how the inputs connect to the variation of the state variables and Gdescribes the internal dynamics of the system. We also suppose that the flows can be computed from the state as f=gT(x)φ(x).(4) In compact form, we have Wf(t)−Wf(t0) = Zt t0he(τ), f(τ)idτ=Zt t0he(τ), gT(x(τ))φ(x(τ))idτ =Zt t0hg(x(τ))e(τ), φ(x(τ))idτ=Zt t0h˙x(τ)−G(x(τ)), φ(x(τ))idτ. Now we assume that the term G(x) is actually of the form G(x) = J(x)φf(x), with JT=−J. This can be justified from the general theory of Dirac structures and port Hamiltonian systems, but we can make plausible its introduction by noticing that it yields a term of the form −hJ(x(τ))φ(x(τ)), φ(x(τ))i 1Notice that, from a basic bond graph point of view, some of the inputs may actually be flows; however, to reduce notation, we do not distinguish here between I-like and C-like ports. In fact, by the use of gyrators, only one of the two classes of ports are actually needed. The final result we want to deduce is independent of this distinction. 3 which vanishes due to the skew-symmetry of Jand hence does not contribute to the total energy variation of the system, as it fits to a shuffling around of energy among the internal degrees of freedom. We get thus Wf(t)−Wf(t0) = Zt t0h˙x(τ), φ(x(τ))idτ=Zγ(x0,x) φ(z) dz(5) where γ(x0, x) is any curve connecting x0=x(t0) and x=x(t) in state space. We will now assume that this system is conservative. This means that the variation of energy due to the power ports input only depends on the initial and final states, x0 and x, and not on the particular curve in state space used to go from the former to the later. Otherwise, it would be possible to go from x0to xusing a curve requiring less energy input and return from xto x0on a (reversed) curve with larger energy variation, yielding a system in the same initial state and a net surplus of energy, thus violating the First Principle of Thermodynamics. As it is well known, independence with respect to the curve means that the 1-form φ(x) dxis exact,i.e. in components φi(x) = ∂Φ(x) ∂xi , i = 1,...,n, (6) where Φ is a state space function. Actually, from (5) it follows immediately that then Wf is also a state space function and Wf(x)−Wf(x0) = Φ(x). In the sequel we will set t0= 0, x0= 0 and Wf(0) = 0 (there is no problem in doing so for time-invariant systems) and hence Wf(x) = Zγ(x) φ(z) dz, (7) where γ(x) is now any curve connecting the state space origin (or any other selected point) with x. The function Wf(x) is thus the energy necessary to bring the system to state x, and it is the same no matter how the power is injected through the ports. It is thus the energy of the system in state x; it coincides with the Hamiltonian function H(x), and we will use the two notations indistinctively. From (6) it follows, assuming continuity of the second order derivatives of Φ, that ∂φi ∂xj (x) = ∂φj ∂xi (x), i, j = 1,...,n. (8) These are known as Maxwell’s reciprocity relations for the constitutive equations of a conservative field. It allows, using derivation with respect to xifollowed by integration with respect to xj, to obtain φi(x) if φj(x) is known. Also, from (6), one obtains that all the constitutive relations can be computed if the energy is known: φi(x) = ∂Wf(x) ∂xi .(9) Notice that we are adopting an all-input power convention, i.e. power is assumed to flow into the system if it is positive. In the electrical engineering literature, mechanical ports are treated with an output convention, so that mechanical efforts (force or torque) are defined as those done by the system. They are then computed from Wfby taking minus derivative with respect to the corresponding displacement. 4 000 000 000 000 000 111 111 111 111 111 +q−q zero Ezero Econstant E x F Nonlinear dielectric Figure 2: A nonlinear capacitor. 2.3 Electric capacitor Figure 2 shows a general plane capacitor, where the dielectric has a nonlinear constitutive relation. As usual, we assume that the transversal dimensions of the plates are much bigger than the plate’s separation x, so that the electric field is zero outside the capacitor and constant and perpendicular to the plates inside of it. In the vacuum or in linear isotropic dielectrics, it can be shown that the voltages of a set of conductors depend linearly on the charges. Here we assume a more general situation where the voltage difference depends nonlinearly on q. To make contact with the linear case, we introduce a capacitance-like function C(x, q) so that v(x, q) = q C(x, q). This is a two-port system, one mechanical and the other electrical. We are given v(x, q) and want to compute F(x, q). According to our general discussion, Maxwell reciprocity condition tells us that ∂F ∂q =∂v ∂x =q∂ ∂x 1 C(x, q)=−q C2(x, q)∂xC(x, q). Now we have to integrate this with respect to q. An integration by parts helps to make contact with the linear case: F(x, q) = Zq 0 ξ∂ ∂x 1 C(x, ξ)dξ =q2 2 ∂ ∂x 1 C(x, q)−Zq 0 ξ2 2 ∂2 ∂ξ∂x 1 C(x, ξ)dξ. For a linear system, C(x, q) = C(x), and F(x, q) = qE(x, q) 2 5 where E(q, x) = q∂ ∂x 1 C(x, q)=∂ ∂x q C(x, q)=∂ ∂xv(x, q) is the electric field seen by the +plate when its charge is qand the separation is x. Furthermore, the field energy is given by Wf(x, q) = Zq 0 ξ C(x)dξ=1 2 q2 C(x), which is the well-known elementary result. 2.4 Magnetic stationary system Figure 3 shows two coils coupled by the magnetic field they generate. Although this is not an electromechanical system (there are no moving parts) it allows us to introduce several concepts which will be used in the next example. The current of any of the coils produces a magnetic field which goes through the other coil. Variations in any of the currents change the flux of the magnetic induction field and this, according to Lenz’s law, originates an induced voltage in both coils. The iron core is introduced to “bend” the magnetic field lines so that the coupling is tighter. The coils have N1and N2turns, respectively. Any turn of any of the two coils has three contributions to the magnetic flux that goes through it: •the flux due to the lines of the induction magnetic field generated by the current of the same coil and which close through the iron core, •the flux due to the lines of the induction magnetic field generated by the current of the same coil and which do not close through the iron core, and •the flux due to the lines of the induction magnetic field generated by the current of the other coil and which close through the iron core. This can be written as Φ1= Φl1+ Φm1+ Φm2 Φ2= Φl2+ Φm2+ Φm1 + + − − v1v2 i1i2 Φl1Φl2 Φm1Φm2 N1N2 Figure 3: Magnetically coupled coils. 6 Let Hbe a smooth function on X. The port controlled Hamiltonian system corresponding to (X,V,D,H) is defined by (−˙x, ∂xH(x), f, e)∈ D(x). It follows from the self-duality of the Dirac structure that ˙ H(x) = (e, f), which expresses the energy balance. Assuming that F(x) is invertible for all x, one has ˙x −f=F−1(x)E(x)∂xH(x) e. It can be seen from FET+EFT= 0 that F−1Emust be skew-symmetric. Hence ˙x −f=J(x)g(x) −gT(x)Jf(x)∂xH(x) e. where J(x), Jf(x) are skew-symmetric. EXERCISE Show that FET+EFT= 0 implies that F−1Eis skew-symmetric. Dissipation may be included by terminating some of the open ports. Replacing m→ m+mrand setting er=−R(x)frwith RT(x) = R(x)≥0, one gets ˙ H(x) = (e, f) + (er, fr) = (e, f)−fT rR(x)fr≤(e, f). Again, in local coordinates the system can be expressed as ˙x= (J(x)−R(x))∂xH(x) + g(x)e f=gT(x)∂xH(x)−Jf(x)e. This is an explicit port Hamiltonian system with dissipation (PHD). In general, one can interconnect several PHD with open ports using Dirac structures, and the result is again a PHD, although not necessarily in explicit form. As said before, all this can be generalized [35] to distributed parameter systems, described by partial differential equations, for which numerical spatial discretization schemes have also been developed [17]. 3.2 General electromechanical systems As explained before, an electromechanical system exchanges energy between a mechanical and an electrical part by means of geometry variations. For a wide class of systems, the total energy can be written as H(λ, p, θ) = 1 2λTL−1(θ)λ+1 2pTJ−1 mp, (21) we λare the generalized electrical energy variables (they may be charges or magnetic fluxes), pare the generalized mechanical momenta (linear or angular, or associated to any other generalized coordinate), and θare the generalized geometric coordinates. In a minimal formulation, the generalized inertia matrix Jmmay be assumed to be independent of θ; if not, the computation of the torque would yield non-electrical terms coming from 13 non-inertial effects, which are better accounted for by attaching the corresponding varying geometry mechanical subsystem to the mechanical port. One has ˙ λ+Rei=Bv, ˙p=−Rmω−Te(λ, θ) + Tm, ˙ θ=J−1 mp(22) where Bis a matrix indicating how the input voltages are connected to the electrical devices, Teis the electrical torque, Tmis the external applied mechanical torque and i=L−1(θ)λ=∂λH, ω =J−1 mp=∂pH(23) are the electrical currents and mechanical velocities ω=˙ θ. We have, according to our discussion on electromechanical energy conversion, Te=∂H ∂θ =−1 2λTL−1∂θLL−1λ, (24) where ∂θL−1=−L−1∂θLL−1has been used. Using (23) and (24), equations (22) can be written in an explicit port Hamiltonian form as ˙x= −Re0 0 0−Rm−1 0 1 0  ∂xH+  B0 0 0 0 1  v Tm(25) where x= (λ p θ)T. This general model includes many of the classical electrical machines, as well as linear motors and levitating systems. EXERCISE Write the port Hamiltonian model (25) for the electromagnet and for the variable geometry capacitor. For the dc motor, the above formulation has to be modified slightly due to the different role played by the geometric coordinates (the angle in this case). Direct inspection of equations (17) shows that they can be given also a PHDS form, with Hamiltonian variables xT=λfλaJmωr,(26) Hamiltonian function H(x) = 1 2λL−1λT+1 2Jm ω2 r, L =Lf0 0LA,(27) interconnection and damping matrices J=  0 0 0 0 0 −LAf if 0LAf if0 ,R=  rf0 0 0ra0 0 0 Br ,(28) and port matrix g=I3, with inputs uT=VfVa−TL.(29) 14 - + - + ra rfLA Lf ia if Va VfLAf ωrif iT ωr Figure 8: Shunt-connected dc machine. The input voltages Vfand Vacan be obtained from the same source in several ways. Figure (8) shows a shunt-connected dc machine, for which Va=Vfand iT=ia+if. Figure (9) displays a series-connected dc machine, for which one has ia=ifand VT=Va+Vf. EXERCISE Obtain the port Hamiltonian models of the shunt-connected and of the series-connected dc machines. The later is in fact an implicit port Hamiltonian model. Check this by writing the corresponding bond graph and identifying the differential causality assignment. Alternating current machines can also be written in PHDS form. However, several coordinate transformations are used in the electrical engineering literature to simplify the complex, geometry dependent constitutive relations involved in most of the cases. It turns out that, after carrying out those transformations, the system is still in PHDS form, with different, and nontrivial, interconnection matrices. We will not pursue this here. The interested reader is referred to [3]. Appendix B contains a brief account of Park’s transformation from the point of view of bond graph theory and PHDS theory. 4 Power converters Power converters are of great value in many growing application areas such as the control of electromechanical devices, portable battery-operated equipment or uninterruptible power sources. In particular, dc-to-dc power converters are used to reduce or elevate a given dc voltage, storing the energy in intermediate ac elements and delivering it as needed. The essential trick is the use of switches, which are operated in a periodic manner and which make the system to alternate between several dynamics. Generally, the individual dynamics are linear and can be solved analytically; the action of the switches yields a nonlinear dynamics which can display a rich behavior (see [14] and references therein). As power converters are variable structure systems, we present first a general theory for the later, and will retake concrete examples of the former after that. 15 - + - + - + ra rfLA Lf ia if Va VfLAf ωrifVT ωr Figure 9: Series-connected dc machine. 4.1 Variable structure systems Assume a VSS system such that the change in the state variables is small over the time length of an structure change, or such that one is not interested about the fine details of the variation. Then one may try to formulate a dynamical system for the time average of the state variables (state space averaging, or SSA) hxi(t) = 1 TZt t−T x(τ) dτ, (30) where Tis the period, assumed constant, of a cycle of structure variations. Let our VSS system be described in explicit port Hamiltonian form ˙x= [J(S, x)−R(S, x)] (∇H(x))T+g(S, x)u, (31) where Sis a (multi)-index, with values on a finite, discrete set, enumerating the different structure topologies. For notational simplicity, we will assume from now on that we have a single index (corresponding to a single switch, or a set of switches with a single degree of freedom) and that S∈ {0,1}. Hence, we have two possible dynamics, which we denote as S= 0 ⇒˙x= (J0(x)−R0(x))(∇H(x))T+g0(x)u, S= 1 ⇒˙x= (J1(x)−R1(x))(∇H(x))T+g1(x)u. (32) Note that controlling the system means choosing the value of Sas a function of the state variables, and that uis, in most cases, just a constant external input. From (30) we have d dthxi(t) = x(t)−x(t−T) T.(33) Now the central assumption of the SSA approximation method is that for a given structure we can substitute x(t) by hxi(t) in the right-hand side of the dynamical equations, 16 so that (32) become S= 0 ⇒˙x≈(J0(hxi)−R0(hxi))(∇H(hxi))T+g0(hxi)u, S= 1 ⇒˙x≈(J1(hxi)−R1(hxi))(∇H(hxi))T+g1(hxi)u. (34) The rationale behind this approximation is that hxidoes not have time to change too much during a cycle of structure changes. We assume also that the length of time in a given cycle when the system is in a given topology is determined by a function of the state variables or, in our approximation, a function of the averages, t0(hxi), t1(hxi), with t0+t1=T. Since we are considering the right-hand sides in (34) constant over the time scale of T, we can integrate the equations to get2 x(t) = x(t−T) + t0(hxi)(J0(hxi)−R0(hxi))(∇H(hxi))T+g0(hxi)u +t1(hxi)(J1(hxi)−R1(hxi))(∇H(hxi))T+g1(hxi)u. Using (33) we get the SSA equations for the variable hxi: d dthxi=d0(hxi)(J0(hxi)−R0(hxi))(∇H(hxi))T+g0(hxi)u +d1(hxi)(J1(hxi)−R1(hxi))(∇H(hxi))T+g1(hxi)u,(35) where d0,1(hxi) = t0,1(hxi) T,(36) with d0+d1= 1. In the power converter literature d1(or d0, depending on the switch configuration) is referred to as the duty cycle. One can expect the SSA approximation to give poor results, as compared with the exact VSS model, for cases where Tis not small with respect to the time scale of the changes of the state variables that we want to take into account. The GSSA approximation tries to solve this, and capture the fine detail of the state evolution, by considering a full Fourier series, and eventually truncating it, instead of just the “dc” term which appears in (30). Thus, one defines hxik(t) = 1 TZt t−T x(τ)e−jkωτ dτ, (37) with ω= 2π/T and k∈Z. The time functions hxikare known as index-kaverages or k-phasors. Notice that hxi0 is just hxi. Under standard assumptions about x(t), one gets, for τ∈[t−T, t] with tfixed, x(τ) = +∞ X k=−∞hxik(t)ejkωτ .(38) If the hxik(t) are computed with (37) for a given t, then (38) just reproduces x(τ) periodically outside [t−T, t], so it does not yield xoutside of [t−T, t] if xis not T-periodic. 2We also assume that udoes not vary over this time scale; in fact uis constant in many applications. 17 However, the idea of GSSA is to let tvary in (37) so that we really have a kind of “moving” Fourier series: x(τ) = +∞ X k=−∞hxik(t)ejkωτ ,∀τ. (39) A more mathematically advanced discussion is presented in [37]. In order to obtain a dynamical GSSA model we need the following two essential properties: •Derivation. Writing (37) as hxik(t) = 1 TZT 0 x(τ+t−T)e−jkω(τ+t−T)dτ, (40) one immediately gets d dthxik(t) = dx dtk (t)−jkωhxik(t).(41) •Convolution. If xand yare two signals, then hxyik= +∞ X l=−∞hxik−lhyil.(42) In particular, considering a first-harmonic GSSA approximation, one gets hxyi0=hxi0hyi0+hxi−1hyi1+hxi1hyi−1, hxyi1=hxi0hyi1+hxi1hyi0, hxyi−1=hxi0hyi−1+hxi−1hyi0.(43) Using (41) and (31) one gets d dthxik=dx dtk−jkωhxik =[J(S, x)−R(S, x)] (∇H(x))T+g(S, x)uk−jkωhxik.(44) Assuming that the structure matrices Jand R, the Hamiltonian H, and the interconnection matrix ghave a series expansion in their variables, the convolution formula (42) can be used and an (infinite) dimensional system for the hxikcan be obtained. Notice that, if we restrict ourselves to the dc terms (and without taking into consideration the contributions of the higher order harmonics to the dc averages), then (44) boils down to (35) since, under these assumptions, the zero-order average of a product is the product of the zero-order averages. 18 − + vC iL vL iC v1 i1 v2 i2vR iR E iE s1 s2R L C Figure 10: A functional description of the boost converter. 4.2 Second order power converters Figure 10 shows a functional model of the boost (or elevator) converter (the detailed electronics of how the switches are implemented is not shown). The switches s1and s2 are complementary: when s1is closed (s1= 1), s2is open (s2= 0), and viceversa. Thus, the different circuit topologies can be described with a single boolean variable S=s2. The port Hamiltonian modeling of electric circuits can be done in a systematic way using tools from graph theory [7], but since we are dealing here with a circuit of very small size we will adopt a more pedestrian approach and concentrate on the problems presented by the switches, using the ideas of [13]. A more in-deep conceptual analysis of the switches can be found in [16]. The Hamiltonian dynamical variables of the boost converter are the magnetic flux at the coil, φL, and the charge of the capacitor, qC. Hence we have two one-dimensional Hamiltonian subsystems, with a global Hamiltonian H=HC+HL, dqC dt=iC, vC=∂H ∂qC ,(45) and dφL dt=vL, iL=∂H ∂φL ,(46) connected by Kirchoff’s laws iL=i1+i2 i1=iC+iR v2+vL=E vC+v1=v2 vC=vR iE+iL= 0 (47) Here we treat the switches as ports, with their correspondent effort and flow variables. For the time being we do not terminate the resistive port, i.e. we do not use vR=RiR. EXERCISE Write the EF-representation [10] of the Dirac structure associated to (47). 19 Using (45) and (46), the first four equations of (47) can be written as ∂H ∂φL =i1+i2 i1=dqC dt+iR v2+dφL dt=E ∂H ∂qC +v1=v2(48) The second and third equations in (48) yield a Hamiltonian system with four inputs and J=R= 02×2: d dtqC φL= 0(∇H)T+1 0 −1 0 0−101    i1 v2 iR E    .(49) Next we will use the constraints imposed by the switches to absorb the ports s1and s2into the Hamiltonian structure: •S= 0 ⇒s1= 1, s2= 0 ⇒v1= 0, i2= 0, •S= 1 ⇒s1= 0, s2= 1 ⇒i1= 0, v2= 0. Hence, when S= 1 we already have the values of the port variables i1,v2in (49), while if S= 0, using the first and fourth equations in (48), i1=∂H ∂φL , v2=∂H ∂qC . We can put together both results as i1= (1 −S)∂H ∂φL , v2= (1 −S)∂H ∂qC .(50) Now d dtqC φL=1 0 −1 0 0−101    (1 −S)∂H/∂φL (1 −S)∂H/∂qC iR E     =0 1 −S −(1 −S) 0 ∂H/∂qC ∂H/∂φL+−1 0 0 1 iR E,(51) which is a port Hamiltonian system with outputs y=−1 0 0 1 T∂H/∂qC ∂H/∂φL=−vC iL=−vR −iE. 20 − + vC iL vL iC v1 i1 v2 i2vR iR E iE s1 s2R L C Figure 11: The buck converter. Finally, we terminate the resistive port using iR=vR R=vC R=1 R ∂H ∂qC and get our final port Hamiltonian representation of the boost converter d dtqC φL= 0 1 −S −(1 −S) 0 −1/R 0 0 0 ∂H/∂qC ∂H/∂φL+0 1E, (52) with natural output y=0 1T∂H/∂qC ∂H/∂φL=iL=−iE. Notice that the interconnection structure Jis modulated by the boolean variable S. Designing a control for this system means choosing Sas a function of the state variables. EXERCISE Figure 11 shows an scheme of the buck (or step-down) power converter. Show that the final port Hamiltonian structure is d dtqC φL= 0 1 −1 0 −1/R 0 0 0 ∂H/∂qC ∂H/∂φL+0 1−SE. 4.3 SSA and GSSA for second order power converters Figure 12 shows the functional scheme of the buck-boost, which is the remaining elemental second order dc-dc converter. The VSS port Hamiltonian models for the three converters presented, with the resistive port left open, can be written in an unified way as ˙x=0α−βS −(α−βS) 0 (∇H(x))T+−1 0 0 1 −γS iR E,(53) where the state variables are x= (qCφL)Tand the Hamiltonian function is H(qc, φL) = 1 2Cq2 C+1 2Lφ2 L,(54) with iL=∂φLH,vC=∂qCH. The parameters corresponding to the different converters are given in Table 1. The variable S= 0,1 represents the state of switch 2 (0 closed and 21 − + vC iLvL iC v1 i1 v2 i2 vR iR E iE s1s2 R LC Figure 12: The buck-boost converter. Converter α β γ buck 1 0 1 boost 1 1 0 buck-boost 0 1 1 Table 1: Parameter values for the unified description of the second-order power converters. 1 open), and switch 1 is complementary to switch 2. The natural Hamiltonian outputs of the system are y=−1 0 0 1 −γS T (∇H(x))T=−vC (1 −γS)iL=−vR −(1 −γS)iE.(55) The resistive port can be terminated (i.e. one can use vR=RiR) and then the structure matrix in (53) gets a dissipative part: ˙x=−1/R α −βS −(α−βS) 0 (∇H(x))T+0 1−γS E. (56) Applying (35) to (56), one immediately gets d dthxi=d0(hxi)−1/R α −α0(∇H(hxi))T+0 1E +d1(hxi) −1/R α −β −(α−β) 0 (∇H(hxi))T+0 1−γE =−1/R α −βd1(hxi) −(α−βd1(hxi)) 0 (∇H(hxi))T+0 1−γd1(hxi)E,(57) where d0+d1= 1 has been used. Since hSi= 0 ·d0+ 1 ·d1, one finally gets d dthxi=−1/R α −βhSi −(α−βhSi) 0 (∇H(hxi))T+0 1−γhSiE, (58) 22 Figure 17: 20sim model with an electromechanical subsystem attached. 1. computations are far from obvious. 2. very complex controls are needed to quench the large set of undesirable signals, and the result is very inefficient, with a lot of energy being consumed and always on the verge of instability (a typical example is provided by bipedal walking machines; see Stramigioli’s lectures in this summer school). Most of the problem stem from the fact that no information about the structure of the system is used. A change of control paradigm is needed: control systems as energy exchanging entities We present the basic ideas involved in this energy-based approach to control. We follow the pedagogical account in [28]; complete proofs are presented in [27]. The map u7→ yis passive if there exists a state function H(x), bounded from below, and a nonnegative function d(t)≥0 such that Zt 0 uT(s)y(s) ds |{z } energy supplied to the system =H(x(t)) −H(x(0)) |{z } stored energy +d(t) |{z} dissipated . The simplest example of a passive system is probably the forced mass-spring-damper arrangement of Figure 18. One has Zt 0 F(s)v(s) ds=Zt 0 (m˙v(s) + kq(s) + λv(s))v(s) ds =1 2mv2(s) + 1 2kq2(s) t 0 +λZt 0 v2(s) ds =H(x(t)) −H(x(0)) + λZt 0 v2(s) ds. 29 q F k λ m Figure 18: Example of a mechanical passive system. If x∗is a global minimum of H(x) and d(t)>0, and we set u= 0, H(x(t)) will decrease in time and the system will reach x∗asymptotically. The rate of convergence can be increased if we actually extract energy from the system with u=−Kdi y with KT di =Kdi >0. However, the minimum of the natural energy Hof the system is not a very interesting point in most engineering problems. The key idea of passivity based control (PBC) is as follows: use feedback u(t) = β(x(t)) + v(t) so that the closed-loop system is again a passive system, with energy function Hd, with respect to v7→ y, and such that Hdhas the global minimum at the desired point. Passivity for the closed-loop system is far from obvious: physically, the controller is injecting energy into the system. PBC is robust with respect to unmodeled dissipation, and has built-in safety: even if we don’t know Hexactly, if passivity is preserved the system will stop somewhere instead of running away and finally blowing up. If −Zt 0 βT(x(s))y(s) ds=Ha(x(t)) (66) then the closed-loop system has energy function Hd(x) = H(x) + Ha(x). One has the following energy balance equation (EBE), which yields an interpretation to PBC: Hd(x(t)) |{z } closed-loop energy =H(x(t)) |{z } stored energy −Zt 0 βT(x(s))y(s) ds |{z } supplied energy . For (f, g, h) systems, the EBE is equivalent to the PDE −βT(x)h(x) = ∂Ha ∂x (x)T (f(x) + g(x)β(x)).(67) As an example, consider the electrical system in Figure 19. x=q φstate u=Vcontrol y=i=φ Loutput 30 C R V i L Figure 19: Example of electrical passive system. ˙x=x2/L −x1/C −x2R/L +0 1V(t) The map V7→ iis passive with energy function H(x) = 1 2Cx2 1+1 2Lx2 2 and dissipation d(t) = Rt 0 R L2φ2(s) ds. Notice that the natural minimum is (0,0), but forced equilibrium points are of the form (x∗ 1,0). The PDE (67) is in this case x2 L ∂Ha ∂x1−x1 C+R Lx2−β(x)∂Ha ∂x2 =−x2 Lβ(x). Since x∗ 2= 0 is already a minimum of H, we only have to shape the energy in x1. Hence, we can take Ha=Ha(x1) and the above PDE boils down to β(x1) = −∂Ha ∂x1 (x1) i.e. it defines the closed-loop control. Then we are free to choose Haso that Hdhas the minimum at x∗ 1. The simplest solution is Ha(x1) = 1 2Ca x2 1−1 C+1 Cax∗ 1x1+K where Cais a design parameter. The closed-loop energy Hdcan then be computed and it is seen that it has a minimum at (x∗ 1,0) if Ca>−C. Finally, one can compute the control as u=−∂Ha ∂x1 (x1) = −x1 Ca +1 C+1 Cax∗ 1. This control is an energy-balancing PBC that stabilizes x∗under the stated parameter restrictions. EXERCISE Show that the total energy supplied by the controller is finite. Consider now the slightly different circuit of Figure 20. With the same states, energy, input and outputs than the preceding system, the equations of motion are now ˙x1=−1 RC x1+1 Lx2, ˙x2=−1 Cx1+V(t). 31 C R V L i Figure 20: Another electrical passive system. Σ ΣcΣl yc uc y u Figure 21: Network interpretation of control. Only the dissipation structure has changed, but the admissible equilibria are of the form x∗ 1=CV ∗, x∗ 2=L RV∗ for any constant V∗. The power delivered by the source, Vx2 L, is nonzero at any equilibrium point except for the trivial one. Hence, the source has to provide an infinite amount of energy to keep any nontrivial equilibrium point, a task which is clearly not feasible. This situation will reappear later when we discuss invariant and Casimir functions. Notice that pure mechanical systems are free of this problem, since any equilibrium point has velocities equal to zero and hence no power is necessary to keep the system at the equilibrium point. 6.1 Control as interconnection Now we would like to have a physical interpretation of PBC . To be precise, we would like to think of the controller as a system exchanging energy with the plant. Consider two systems, Σ and Σc, exchanging energy through an interconnection network given by Σl, as depicted in Figure 21. The condition for the interconnection to be power continuous is uT c(t)yc(t) + uT(t)y(t) = 0 ∀t. As an example, consider the typical negative feedback interconnection displayed in Figure 22. The interconnection is given by uc=y u=−yc and is clearly power continuous. Suppose now that we add some extra inputs u→u+v,uc→uc+vcto the interconnected system. Then it is easy to show the following. Let Σ and Σchave state variables xand ξ. If Σ and Σcare passive with energy functions H(x) and Hc(ξ) and Σlis power preserving, then the map (v, vc)7→ (y, yc) is passive for the interconnected system, with energy function Hd(x, ξ) = H(x) + Hc(ξ). Or, in short, 32 − Plant Controller ycuc yu Figure 22: Typical negative feedback interconnection. power continuous interconnection of passive systems yields passive systems. Now we have a passive system with energy function Hd(x, ξ) = H(x)+Hc(ξ), but this is not very useful unless we get an energy function depending only on x. To solve this, we restrict the dynamics to a submanifold of the (x, ξ) space parameterized by x: ΩK={(x, ξ) ; ξ=F(x) + K}, and dynamically invariant: ∂F ∂x T ˙x−˙ ξ!ξ=F(x)+K = 0. Instead of solving this in general, it is convenient to formulate the problem for PHD systems. 6.2 Casimir functions and the dissipation obstacle Consider a PHD system in explicit form given by ˙x= (J(x)−R(x))∂H ∂x (x) + g(x)u, y=gT(x)∂H ∂x (x), with JT(x) = −J(x), and RT(x) = R(x)≥0. A PHD is passive with respect to the pair (u, y): Zt 0 uTy=H(x(t)) −H(x(0)) + Zt 0∂H ∂x T R(x)∂H ∂x . More precise results about the possibility of obtaining invariant manifolds expressing the controller variables in terms of the variables of the system can be formulated if both system and controller are PHDS. Let thus Σ : ˙x= (J(x)−R(x))∂H ∂x (x) + g(x)u y=gT(x)∂H ∂x (x) define the plant and Σc:(˙ ξ= (Jc(ξ)−Rc(ξ))∂Hc ∂ξ (ξ) + gc(ξ)uc yc=gT c(ξ)∂Hc ∂ξ (ξ) 33 define the controller. With the power preserving, standard negative feedback interconnection u=−yc,uc=y, one gets ˙x ˙ ξ=J(x)−R(x)−g(x)gT c(ξ) gc(ξ)gT(x)Jc(ξ)−Rc(ξ) ∂Hd ∂x ∂Hd ∂ξ  where Hd(x, ξ) = H(x) + Hc(ξ). Let us look now for invariant manifolds of the form CK(x, ξ) = F(x)−ξ+K. Condition ˙ Ck= 0 yields ∂F ∂x T| −IJ−R−ggT c gcgTJc−Rc ∂Hd ∂x ∂Hd ∂ξ = 0. Since we want to keep the freedom to choose Hc, we demand that the above equation is satisfied on CKfor every Hamiltonian, i.e. we impose on Fthe following system of PDEs: ∂F ∂x T| −IJ−R−ggT c gcgTJc−Rc= 0. Functions CK(x, ξ) such that Fsatisfies the above PDE on CK= 0 are called Casimir functions. They are invariants associated to the structure of the system (J, R, g, Jc, Rc, gc), independently of the Hamiltonian function. One can show [27] that the PDE for Fhas solution iff, on CK= 0, 1. ∂F ∂x TJ∂F ∂x =Jc, 2. R∂F ∂x = 0, 3. Rc= 0, 4. ∂F ∂x TJ=gcgT. Conditions 2 and 3 are easy to understand: essentially, no Casimir functions exist in presence of dissipation. Given the structure of the PDE, Rc= 0 is unavoidable, but we can have an effective R= 0 just by demanding that the coordinates on which the Casimir depends do not have dissipation, and hence condition 2. If the preceding conditions are fulfilled, an easy computation shows that the dynamics on CKis given by ˙x= (J(x)−R(x))∂Hd ∂x with Hd(x) = H(x) + Hc(F(x) + K). Notice that, due to condition 2, R(x)∂Hc ∂x (F(x) + K) = R(x)∂F ∂x |{z } =0 ∂Hc ∂ξ (F(x) + K) = 0, so we can say that, in energy-balancing PBC, dissipation is only admissible for those coordinates which do not require energy shaping. 34 For regulation problems in mechanical systems, where the state consists of positions and velocities, dissipation only appear associated to the later, while energy shaping is necessary only in the position part, since the kinetic energy already has the minimum where we want it (that is, at velocity equal to zero). Hence, the dissipation obstacle is always absent for mechanical regulation problems. For the first of the two simple RLC circuits considered previously, dissipation appears in a coordinate, x2, which already has the minimum at the desired point. For the second one, the minimum of the energy has to be moved for both coordinates, and hence the dissipation obstacle is unavoidable. 7 IDA control method: basic ideas and examples The previous Section has exposed some shortcomings of the passivity based control of PHDS by means of control-as-interconnection: •Nonlinear PDE for the Casimir function. •Dissipation obstacle. One can get a method with more freedom if not only the energy function is changed but also the interconnection (J) and the dissipation (R), i.e. one aims at a closed-loop system of the form ˙x= (Jd(x)−Rd(x))∂Hd ∂x (x), where JT d(x) = −Jd(x), RT d(x) = Rd(x)>0, and x∗a minimum of Hd(x). One has the following fundamental result [27]: If one can find a (vector) function K(x), a function β(x), an antisymmetric matrix Ja(x), and a symmetric, semipositive definite matrix Ra(x) such that (J(x) + Ja(x)−R(x)−Ra(x))K(x) = −(Ja(x)−Ra(x))∂H ∂x (x) + g(x)β(x),(68) with Kthe gradient of a scalar function, K(x) = ∂Ha ∂x (x), then the closed loop dynamics with u=β(x) is a Hamiltonian system with Hd=H+Ha,Jd=J+Ja,Rd=R+Ra. One can then try to impose conditions on Ha(or on K) so that x∗is an asymptotically stable point of the dynamics. This is called IDA-PBC (from Interconnection and Damping Assignment Passivity Based Control). Notice that (68) comes from matching the original system in closed loop to the desired dynamics: (J(x)−R(x))∂xH(x) + g(x)β(x) = (Jd(x)−Rd(x))∂xHd(x)).(69) This is called the matching equation. Notice that, using if necessary algebraic combinations, one can split the nrow equations in (69) into those that contain u=β(x) and those that not. The former are not, in fact, equations to be solved, as they give the desired controller; the remaining equations are the ones that have to be satisfied using the freedom in Ja,Raand Ha. Several techniques have been proposed in the literature [27, 29, 26] to solve (69. One idea is to fix Ja(x) and Ra(x). Pre-multiplying by a left annihilator of g(x) yields a linear PDE for Ha. After solving it, β(x) can be computed using (gT(x)g(x))−1. Alternatively, 35 my φ g ui Figure 23: A magnetic levitation system. one can fix the desired Hdand then solve the algebraic equations for Jaand Rathat are obtained. In between, one can fix parts of the interconnection and damping structures, and also part of the Hadependence with respect to some of the variables, and solve the resulting mixture of algebraic and PDE equations. On a more theoretical note, it can be shown [27] that •IDA-PBC generates all asymptotically stabilizing controllers for PCH systems. •If R∂xHa= 0, IDA-PBC is energy-balancing. •If, additionally, Ja=Ra= 0, one can think of IDA-PBC in terms of a Casimir function C(x, ξ) = −Ha(x)−ξ, a feedback interconnection modulated by β(x) and a controller with energy Hc(ξ) = −ξ. Here we are going to illustrate some of the basic techniques with concrete examples. Appendix A contains a very simple account of the method of characteristics, which is the basic tool for solving the kind of PDE that arises from (68). 7.1 Magnetic levitation system Figure 23 shows a very simplified model of a magnetic levitation system. The flux lines generated by the current at the coil close through the air gap and the iron ball. Since the air gap has a variable reluctance, the system tries to close it, and this counteracts the gravity. The equations of motion are ˙ φ=−Ri +u ˙y=v m˙v=Fm+mg with φ=L(y)ithe linkage flux, Rthe resistance of the coil, and Fmthe magnetic force, given by Fm=∂Wc ∂y , 36 where the magnetic coenergy is (we assume a linear magnetic system) Wc=1 2 ∂L ∂y i2. In general, Lis a complicated function of the air gap, y. A classical approximation for L for this kind of systems for small yis L(y) = k a+y with k,aconstants, which is essentially the relation used for our model of the electromagnet, without the leakage term. Taking x1=φ,x2=y,x3=mv, this can be written as a PCH ˙x=   000 001 0−1 0  −  R0 0 0 0 0 0 0 0   ∂H ∂x +  1 0 0 u with Hamiltonian H(x) = 1 2k(a+x2)x2 1+1 2mx2 3−mgx2. Note that the gravity term could also have been included as an external mechanical force. The gradient of the Hamiltonian is (∇H)T=  x1a+x2 k 1 2kx2 1−mg x3 m  . Given a desired y∗, the equilibrium point is x∗=  √2kmg y∗ 0 , with an equilibrium control u∗=R kx∗ 1(a+x∗ 2). We will first try to compute the IDA-PBC control without using the specific theory presented for general electromechanical systems. Taking first Ja=Ra= 0, the IDA-PBC equation (J−R)K(x) = gβ(x) yields in this case −RK1(x) = β(x) K3(x) = 0 −K2(x) = 0, and we see that Ha(x) = Ha(x1), which can be chosen so that Hd(x) = H(x) + Ha(x1) 37 has a critical point at x=x∗. Unfortunately ∂2Hd ∂x2(x) =   1 k(a+x2) + H′′ a(x1)x1 k0 x1 k0 0 0 0 1 m   has at least one negative eigenvalue no matter what Hawe choose, so x∗will not be asymptotically stable. The source of the problem is the lack of coupling between the mechanical and magnetic part in the interconnection matrix J. To solve this, we aim at Jd=  0 0 −α 0 0 1 α−1 0  ,i.e. Ja=  0 0 −α 0 0 0 α0 0  . Taking Ra= 0, the IDA-PBC equation now becomes −αK3−RK1(x) = α mx3+β(x) K3(x) = 0 αK1(x)−K2(x) = −α k(a+x2)x1, Now Ha=Ha(x1, x2). Using the second equation, the first equation yields the control u=β(x) = RK1−αx3 m, while the third equation is a PDE for Ha(x1, x2): α∂Ha ∂x1−∂Ha ∂x2 =−αx1(a+x2) k.(70) EXERCISE Solve (70) by the method of characteristics. It is better to give the initial condition as (0, s, Φ(s)) instead of (s, 0,Φ(s)). The way in which (68) has been solved in this case seems to be quite model-dependent. In [29] a more general method for systems of the form (25) with Tm=∂θV(θ) is proposed. The central idea is to aim for a Hdgiven by Hd(x) = 1 2(λ−µd(θ, p))TL−1(θ)(λ−µd(θ, p)) + Vd(θ) + 1 2Jp2, where µd(θ, p) is a kind of desired permanent magnet, and consider a Hamiltonian structure of the form Jd−Ra= −R α(x)β(x) −αT(x) 0 1 −βT(x)−1−ra(p) . It can be seen then that the method boils down to an algebraic equation for id=L−1(θ)µd(θ, p), namely 1 2iT d∂θL(θ)id+∂θVd(θ)−∂θV(θ)−(rm−ra(p)) p J= 0,(71) 38 TF : MTF : yabc K(θ, δ) yαβγ ydq T Figure 25: Bond graph of the dq-transfromation. 0 TF : TF : TF : 1 0 TF : TF : TF : 0 TF : TF : TF : 1 1 ea ea ea ea eb eb eb eb ec ec ec ec fa=fa1+fa2+fa3 fb=fb1+fb2+fb3 fc=fc1+fc2+fc3 fa1 fb1 fc1 fa2 fb2 fc2 fa3 fb3 fc3 t11 t12 t13 t21 t22 t23 t31 t32 t33 t11ea t12eb t13ec t21ea t22eb t23ec t31ea t32eb t33ec 1 t11 fa1 1 t12 fb1 1 t13 fc1 1 t21 fa2 1 t22 fb2 1 t23 fc2 1 t31 fa3 1 t32 fb3 1 t33 fc3 eα=t11ea+t12eb+t13ec eβ=t21ea+t22eb+t23ec eγ=t31ea+t32eb+t33ec fα=1 t11 fa1=1 t12 fb1=1 t13 fc1 fβ=1 t21 fa2=1 t22 fb2=1 t23 fc2 fγ=1 t31 fa3=1 t32 fb3=1 t33 fc3 Figure 26: Detailed bond graph of the Ttransformation. equilibrated case, from a 3-phase (abc) to a 2-phase (αβ) system. Then the K(δ, θ) transformation, which is also power continuous, can be implemented by means of a modulated transformer. Fig.(26) shows a completed bond graph of the first step of dq-transformation. Note that for an equilibrated 3-phase system the output of the γport is zero. For the second part of the transformation, the bond graph is shown in Fig.(27). Using e−J2φ= (eJ2φ)T one can readily check that the bond graph yields indeed the Ktransformation. Finally, the whole dq-transformation is depicted in Fig.(28), with a 3-inputs port (abc) and a 2-output port(dq). References [1] Banerjee, S., and G. Verghese (editors), Nonlinear Phenomena in Power Electronics, IEEE Press, 2001. 45 0 1 0 MTF : MTF : 1 MTF : MTF : eα eβ fα fβ sin(φ) −sin(φ) cos(φ) cos(φ) ed eq fd fq Figure 27: Bond graph of the modulated Ktransformation. 0 TF : TF : TF : 1 0 TF : TF : TF : 0 TF : TF : TF : 1 1 0 1 0 MTF : MTF : 1 MTF : MTF : ea eb ec fa fb fc t11 t12 t13 t21 t22 t23 t31 t32 t33 sin(φ) −sin(φ) cos(φ) cos(φ) ed eq fd fq Figure 28: Complete bond graph of the dq-transfromation for an equilibrated 3-phase system. 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