Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
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Control oriented modeling and simulation of the sawtooth instability in nuclear fusion tokamak plasmas G. Witvoet1,2 , E. Westerhof2 , M. Steinbuch1 , N.J. Doelman3 and M.R. de Baar2 TEXTOR shot 106478 Temperature
Abstract— Tokamak plasmas in nuclear fusion are subject to various instabilities. A clear example is the sawtooth instability, which has both positive and negative effects on the plasma. To optimize between these effects control of the sawtooth period is necessary. This paper presents a simple control oriented model, from current drive (input) to sawtooth period (output), which can mimic the most relevant aspects of the sawtooth instability. It also shows some simulation results, including a static inputoutput map and a comparison with experimental data.
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I. I NTRODUCTION Nuclear fusion is a very promising energy source for the future, as it is clean, safe and virtually inexhaustible [1], [2]. The energy is formed by fusing together two small nuclei (deuterium and tritium), producing a helium nucleus and a neutron. This reaction requires very high temperatures (in the order of 108 K), at which fuels and reactants form a plasma, a mixture of charged nuclei and electrons. This plasma is confined using magnetic fields; the most common fusion reactor using this principle is the tokamak [3]. Nevertheless, nuclear fusion still faces large scientific and technological challenges. For example, the transport of fuel and helium ash to and from the plasma center is far from trivial. Moreover, tokamak plasmas are subject to various instabilities which can deteriorate their performance [4]. Control engineering can play a vital role in overcoming these phenomena to ameliorate the condition of the plasma. An important plasma instability, observed in tokamaks worldwide, is the relaxation oscillation in the center of the plasma called the sawtooth instability [5]–[7]. Various measurements, like the temperature measurement in Fig. 1, show a repetitive rising and crashing, creating a signal with sawteeth. The crashes have a mixing effect on the plasma, i.e. they yield a very fast transport of energy and particles from the center to the outside. This limits the plasma temperature, while large sawteeth can also trigger other (undesired) plasma instabilities [8]. On the other hand the mixing mechanism provides a way to remove helium ash from the center, which will facilitate control of the helium density to prevent choking of the fusion reaction. The sawtooth instability is a magnetic phenomenon, for which two physical models are generally accepted in fusion 1 Dept. of Mechanical Eng., Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands,
[email protected] 2 FOM-Institute for Plasma Physics Rijnhuizen, Association EURATOMFOM, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands,
[email protected] 3 TNO Science & Industry, Precision Motion Systems, P.O. Box 155, 2600 AD Delft,
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
Fig. 1: Example of a central temperature measurement with a clear sawtooth instability.
physics [7], [9]. Roughly speaking, Kadomtsev’s model [10] describes what happens at a sawtooth crash, while Porcelli’s model [11] predicts when this crash will happen. These models can be embedded into so called numerical transport codes, which are very extensive, thorough and not suited for control purposes [12]. In contrast, this paper presents a straightforward set of equations and a state dependent reset criterion based on Kadomtsev’s and Porcelli’s model, yielding a simple 1-D control oriented model which can mimic relevant aspects of the sawtooth behavior. To optimize between the positive and negative effects of sawteeth, control of their size and length is needed. Controlling the sawtooth period, which is closely related to the crash size, is an important step in this direction [12]. Forcing a certain period onto the plasma influences the helium transport from the center and avoids other instabilities to occur. Various studies on different tokamaks, like TCV [12] and ASDEX [13], have shown that this period can be influenced using a spatially localized Electron Cyclotron Current Drive (ECCD). However, so far this ECCD has never been used in a feedback loop. As a first step in this direction, this paper will therefore define the additional current drive as the input for the system, and will determine static inputoutput maps to investigate its influence on the period. We will see that this indeed matches reality quite well. The input-output model presented in this paper is focussed on the TEXTOR tokamak, where future experiments for model validation and controller testing will be performed. TEXTOR is equipped with an ECCD system and various Electron Cyclotron Emission (ECE) channels to measure the electron temperature inside the plasma. Hence, the electron temperature evolution is also included in our model and this temperature is defined as the system output. This paper is organized as follows. Section II gives a short introduction on tokamak geometry and terminology,
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~φ B
~ B ~θ B
namely perpendicular to the flux surfaces. For circular (possibly eccentric) cross-sections this dimension corresponds to the radial direction, hence quantities like temperature and pressure only depend on r. Due to the toroidal and poloidal axisymmetry we can ∂ ∂ ~ only has = ∂θ = 0. The magnetic field B assume that ∂φ a toroidal Bφ and poloidal Bθ component (see Fig. 2). For large aspect-ratio tokamaks we can assume that Bθ is constant along a flux surface, so Bθ = Bθ (r, t), while Bφ is constant over r and t. Section III-D forms a small exception, where we need Bφ ∝ R1 to locate the ECE channels.
poloidal angle θ
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r
Fig. 2: Illustration of magnetic fields inside a tokamak.
The most important variables in sawtooth modeling are magnetic ones: the safety factor q and the magnetic shear s. The safety factor q (its name comes from the fact that it is associated with operational limits) can be seen as the ‘pitch’ of the magnetic field, i.e. the ratio between Bφ and Bθ . The magnetic shear is its radial derivative. For large aspect-ratio tokamaks they are defined as [3]
~ B
Fig. 3: Illustration of nested magnetic flux surfaces.
q(r) =
describes the TEXTOR environment, and discusses the sawtooth control feedback loop. The sawtooth model is explained in Section III, together with some important parameter choices. Some simulation results with this model, together with some experimental data, are shown in Section IV. Section V finally summarizes the conclusions and discusses future research towards real-time sawtooth control. II. T OKAMAK DESCRIPTION AND TERMINOLOGY Most of the nuclear fusion research focusses on the use of a tokamak [3], a toroidal machine which confines the hot plasma using strong magnetic fields. In this paper we focus on circular cross-section tokamaks, like TEXTOR, whose geometry can be described in polar-toroidal coordinates, with a radius r, poloidal angle θ and toroidal angle φ (see Fig. 2). We use R to indicate a distance to the center of the torus, the major radius R0 is the distance from the center of the torus to the plasma center, and the minor radius a is the distance from the plasma center to the vessel wall. We assume a large aspect-ratio tokamak, i.e. R0 ≫ a, which will simplify equations later on. A. Magnetic topology ~ in a tokamak is such that there The magnetic field B are surfaces of constant magnetic flux. These flux surfaces form nested toroidal tubes, such as illustrated in Fig. 3. A magnetic field line always remains on the same flux surface as it goes around the tokamak, hence there is no magnetic field perpendicular to the flux surface [3]. Transport of energy and particles is extremely fast along a flux surface, so quantities like temperature and pressure are constant along such a surface. Even for more complicated non-circular plasma shapes this allows a 1-D description of the tokamak topology in the single remaining dimension,
s(r) =
rBφ R0 Bθ r ∂Bθ r ∂q =1− . q ∂r Bθ ∂r
(1) (2)
Both q and s are uniquely determined by the poloidal field Bθ . Consequently, q and s are constant along a flux surface. The flux surface where the safety factor q = 1 is associated with the sawtooth instability [3], as it is the surface where sawtooth oscillations are generally observed. Hence, this q = 1 surface plays an important role in our model, as explained in Section III-C. B. The TEXTOR environment On TEXTOR we choose to measure the sawtooth behavior with Electron Cyclotron Emission (ECE) temperature measurements, while we perturb the magnetic field locally using Electron Cyclotron Current Drive (ECCD). The ECCD installation is accurately described in [14]. It involves a 140GHz gyrotron which can generate an electro-magnetic beam of up to 800kW, an optical transmission line, and a mechanically steerable mirror at the end of the transmission. This mirror is actively controlled [15], such that the ECCD power can be deposited locally at a desired radial position inside the plasma, generating extra current at the same position (with a deposition width of a few cm). The actual distribution of the current drive is further discussed in Section III-D. The ECCD installation is also equipped with an inlineECE receiver, which measures the emission from the plasma along the same line-of-sight as the ECCD power is deposited. More details can be found in [16]. This emission is a measure for the electron temperature Te inside the plasma, in which sawteeth are clearly visible. Fig. 1 is an example of such a measurement. The angle of the mirror, together with the toroidal field Bφ , determines the radial location of the measurement channels inside the tokamak, which is also further discussed in Section III-D.
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Taking the curl of this expression and substituting Maxwell’s equations (3) we then obtain ~ ∂B ~ ×B ~ − η J~CD . ~ × η ∇ (6) = ∇ − ∂t µ0
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Due to the tokamak axisymmetry and the large-aspect ratio assumption (i.e. R ≫ a ≥ r), as discussed in Section II-A, we can reduce (6) to its poloidal component: ∂ ∂ η ∂ Bθ + r Bθ − ηJCD . (7) Bθ = ∂t ∂r µ0 r ∂r
Fig. 4: The foreseen sawtooth control feedback loop.
C. The feedback loop The sawtooth control problem is summarized in the feedback loop of Fig. 4. The sawtooth behavior of the plasma (i.e. the plant) is influenced by the gyrotron power P and mirror angle ϑ, and can be observed via the six ECE temperature measurements. A sawtooth recognition algorithm derives the actual period τs from these signals. The controller uses this to calculate the input signals Pref and ϑref . These form the inputs for lower level control loops on the gyrotron and the mirror [15], which are the actuators for our sawtooth plant. The plasma and ECE blocks of Fig. 4 are modeled in the next section, creating an important simulation environment (see Section IV) for future controller design. The sawtooth recognition (i.e. period detection) can simply be embedded in this model, but for future experiments a real-time algorithm should be created. This, together with the actual controller design, are the main topics for future research. III. S AWTOOTH MODELING
A. Inter-crash evolution The sawtooth instability is believed to be a magnetic instability, depending on the poloidal field Bθ , which is a function of space r and time t. The evolution of Bθ (r, t) between sawtooth crashes follows from Maxwell’s equations = µ0 J~ ~ ~ ×E ~ = − ∂B ∇ ∂t and Ohm’s law, which is given by ~ = η J~ind . E
(3a) (3b)
(4)
~ is the Here µ0 is the magnetic permeability in vacuum, E ~ toroidal electric field, B is the total magnetic field, η is the plasma resistivity and J~ind is the current density induced by ~ The total plasma current J~ consists of this induced current E. and non-inductive currents J~CD driven by other mechanisms like ECCD. Hence J~ = J~ind + J~CD so that (4) becomes ~ = η J~ − η J~CD . E
η = 1.65 · 10−9 ln Λ Zeff Te−3/2 ,
(8)
with Te in keV [3]. We assume that the Coulomb logarithm ln Λ ≈ 17 and the effective ion-charge Zeff ≈ 1.5 on TEXTOR. Finally, the driven current density JCD represents the ECCD input to the plasma and thus forms the control input for our model. We can define two boundary conditions for (7). In the center of the plasma we have a pure toroidal field line, so Bθ (0, t) = 0 at r = 0. The poloidal field at the edge r = a is defined by the total plasma current Ip (Amp`eres law): ˛ ˆ 2π aBθ (a) dθ = 2πaBθ (a), (9) µ0 Ip = Bθ ds = 0
We will model the sawtooth instability in a hybrid manner. The crashes are modeled as discrete state transitions, while between crashes the states evolve according to partial differential equations (PDE). The relevant states are poloidal magnetic field Bθ [T] and electron temperature Te [eV1 ].
~ ×B ~ ∇
This PDE is known as the diffusion equation of the poloidal magnetic field Bθ (r, t). The plasma resistivity η in this equation depends on the electron temperature Te . A widely acknowledged and commonly used relation is
(5)
1 In plasma physics the electronvolt (eV) is used as a unit of temperature: 1eV = 11.6 · 103 K.
so that we have Bθ (a, t) =
µ0 Ip 2πa
at r = a.
On TEXTOR we choose to monitor the sawtooth with inline-ECE temperature measurements. Hence, to take the influence of sawteeth on the output signal into account, we need to include the evolution of the electron temperature Te in our model. In this paper we will approximate this with a simple energy transport PDE [17] 3 ∂ ~ ·ϕ (nTe ) = −∇ ~ + Sources. (10) 2 ∂t ~ e , with n the plasma Here the energy flux ϕ ~ = −nχ∇T density and χ the effective thermal diffusivity. Since the transport on a flux surface is extremely fast, we only consider the transport between flux surfaces, i.e. in radial direction. ∂ Te , so that we obtain The energy flux is then ϕ = −nχ ∂r 3 ∂ ∂ ∂ ∂ 1 nχ Te + nχ Te +Sources. (11) (nTe ) = 2 ∂t ∂r ∂r r ∂r The Sources term is a collection of various physical phenomena and heating mechanisms, which are assumed to be timeinvariant. Its radial distribution influences the temperature equilibrium to a large extent (see Section III-B), however precise modeling of this term is outside the scope of this paper. Moreover, the Sources term is assumed to be unaffected by the control input JCD . At the plasma center r = 0 the temperature has a global e maximum, so ∂T ∂r = 0 there. At the edge r = a the
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Sources χ n Te,eq
Remark 1 The energy transport (11) and the magnetic diffusion (7) are one-sidedly coupled via the resistivity η. Hence, Bθ depends on Te , while Te is unaffected by Bθ . Remark 2 The energy transport equation (11) generally has a much smaller time scale than the magnetic diffusion equation (7). However, we will see later on that the sawtooth period on TEXTOR has a comparable time scale, and as such the transient behavior of Te is still very important.
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Most of the parameters introduced in the above are spatially distributed, i.e. they depend on r, in which case we speak of parameter profiles. The specific profiles used in this paper are based on real experimental data and experience. We assume that these profiles are time-invariant. Remark 3 In the remainder of this paper we will use the term equilibrium to indicate a static time-invariant plasma condition, without sawtooth oscillations. Hence, the equilib∂ = 0. ria are the solutions of (7) and (11) where ∂t 1) Effective thermal diffusivity χ: It is generally observed that the thermal diffusivity increases strongly towards the edge of the plasma. To model this, we assume the following profile for the effective thermal diffusivity χ on TEXTOR: r 2 . (12) χ(r) = χ0 1 + α a Here we choose χ0 = χ(0) = 1m2 /s and α = 2. 2) Density n: The sawtooth oscillation influences the particle transport in the plasma center, thereby flattening the particle density n in this region [7]. However, for simplicity this effect is neglected in our model. To justify this we assume a density profile which is rather flat around r = 0. A reasonable quadratic density profile is then given by r 2 n(r) = (n0 − na ) 1 − + na , (13) a where n0 and na are the densities at the plasma center and edge respectively. For TEXTOR we choose n0 = 3·1019 m−3 and na = 0.1n0 . 3) Equilibrium profiles qeq and Te,eq : The Sources term in (11) should be chosen such that (7) and (11) yield realistic equilibrium profiles of q and Te , which are linked via the resistivity η in (8). Here we choose a commonly used quadratic equilibrium profile of q, namely r 2 qeq (r) = q0,eq + (qa − q0,eq ) , (14) a where qa = q(a) is completely determined by Ip and Bφ : qa =
aBφ 2πa2 Bφ = . RBθ (a) µ0 R0 Ip
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(15)
qa Furthermore, we assume that q0,eq = 1+q , such that there a is a q = 1 surface (see Section III-C) located at rq=1 = qaa .
Fig. 5: Shapes of the chosen parameter profiles
It can then be shown that the corresponding temperature equilibrium for this q-profile should be of the form r 2 −4/3 Te,eq (r) = T0,eq 1 + qa , (16) a where T0,eq is the central temperature, which is typically 2keV for TEXTOR. 4) Sources term: Given these profiles the Sources term e can now be found by setting ∂T ∂t = 0 in (11), i.e. ∂ ∂ ∂ 1 nχ Te,eq − nχ Te,eq . (17) Sources(r) = − ∂r ∂r r ∂r An explicit solution can be found by substituting the above profiles. The shapes of the profiles in (12), (13), (16) and (17) are illustrated in Fig. 5. Remark 4 The above parameter profiles are simplifications of reality. Note however, that this paper does not provide a comprehensive description of the physics of the sawtooth, but merely presents a simple model for control purposes. As such, these profiles are representative for the type of experiments that we consider on TEXTOR, i.e. they yield realistic Bθ and Te distributions and the corresponding time scales are of the appropriate order. As we will see in Section IV, this captures the essence of the sawtooth inputoutput behavior, which is necessary for control. C. Crash definitions The above system of PDEs (7) and (11), together with their boundary conditions and arbitrary initial conditions, will converge to a crashless equilibrium of constant Bθ (r) and Te (r). The crashes will be included into our model in a hybrid way, using a state dependent reset condition, thus yielding an impulsive dynamical system [18]. This impulsive behavior is very natural, as can be concluded from Porcelli’s widely accepted model of the sawtooth instability [11]. This comprehensive theoretical work models the sawtooth period by means of thresholds and triggering conditions for the sawtooth crashes. Under some plasma conditions, as we assume in this paper, this threshold can be written as a critical value of the shear s at the q = 1
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surface. Since both q and s are fully determined by the state Bθ in (7), this threshold forms a state dependent condition. As soon as s at q = 1 exceeds the value scrit the state is reset to a new value. Note that rq=1 , the location of the q = 1 surface, varies over time. The sawtooth crash itself is described by Kadomtsev’s model [10]. According to this model flux surfaces on either side of the q = 1 surface reconnect with each other, flattening the q-profile inside a certain mixing radius rmix > rq=1 , such that q = 1 in the whole region2 . As a result Te is also flattened in this region, with conservation of energy. Outside rmix the profiles are unaffected by sawtooth crashes. Hence, the sawtooth crash is defined by the reset Bθ (t− , r) for r ≥ rmix (18a) Bθ (t+ , r) = 1 rB for r < rmix φ R − Tˆe (t , r) for r ≥ rmix rmix n(r)Te (t− , r)r dr (18b) Te (t+ , r) = 0 ˆ for r < rmix rmix n(r)r dr
Remark 5 The temperature flattening yields sudden increases of Te in a small region outside the plasma center. This region will show reversed sawtooth crashes, as is also observed in tokamaks. The location where the crashes change sign is called the inversion radius, denoted by rinv . D. Translation to TEXTOR The PDEs in (7) and (11) and the state dependent reset condition in Section III-C are sufficient to mimic the sawtooth behavior. However, to simulate the TEXTOR environment we also need to incorporate the ECE diagnostics (sensors) and the ECCD input (actuator) into our model. 1) Poloidal reconstruction: Both ECE and ECCD make use of the gyration frequency of the electrons around magnetic field lines. This frequency depends on the toroidal field Bφ , which is inversely proportional to R. Consequently, each ECE or ECCD frequency corresponds to a unique position R in the tokamak. TEXTOR uses the second harmonic of the gyration frequency, hence we obtain Ri =
1 2eR0 B0 , · 2π fi me
(19)
where B0 = Bφ (R0 ) is the toroidal field at the plasma center, e and me are the electron charge and mass. On TEXTOR the ECCD frequency fECCD is 140GHz and the ECE frequencies fECE are 132.5, 135.5, 138.5, 141.5, 144.5, and 147.5GHz. The vertical position in the poloidal plane zi depends on the 2 Here we assume that r mix is fixed at rmix = 1.2 · req , where req is the location of the q = 1 surface at the equilibrium without sawteeth.
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Fig. 6: Poloidal cross-section of TEXTOR, showing the locations of various q-surfaces and the reconstructed surfaces of the ECE channels and ECCD location.
elevation angle ϑ of the steerable mirror. Since this mirror is located at R = 2.4m, we have that
0
where t− and t+ indicate the time t just before and after a crash. Note that (18) introduces singularities in Bθ and Te at r = rmix , which is also observed in tokamak experiments. The time between crashes defines the sawtooth period, which is indicated by τs .
147.5GHz ECE
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(20)
Finally, we take the so called Shafranov shift [3] into account, an outward displacement of the flux surfaces w.r.t. the plasma center, for which we take r 2 . (21) ∆(r) = ∆0 1 − a Here ∆0 is the shift at r = 0, which is typically around 4cm. The radial location ri of the ECE and ECCD can then 2 be derived using ri2 = zi2 + (Ri − R0 − ∆(ri )) and (19), (20) and (21). A graphical representation of this poloidal reconstruction is shown in Fig. 6. 2) Current drive: The ECCD input is in our model represented by JCD . Although ECCD is a very localized method, the final power deposition is actually distributed in radial direction. Consequently, we model JCD with a Gaussian distribution over r, hence (r − rECCD )2 , (22) JCD = J0 exp − w2 where w is the deposition width (typically 12mm). The peak value J0 can be determined using the total amount of driven current ICD , which is the surface integral of JCD . The final input parameters are thus the mirror angle ϑ, which determines rECCD , and the total current drive ICD , which is directly linked to the gyrotron power P . Remark 6 In a normal tokamak experiment the total plasma current Ip is kept constant by a control loop, which µ0 Ip is expressed by the boundary condition Bθ (a) = 2πa . The current drive JCD is therefore not additional current, but merely a redistribution of the plasma current density. IV. S IMULATION RESULTS To simulate our model (7) and (11) have been discretized over r in 200 steps, yielding a system of non-linear ODEs with 400 states in total. This has been implemented in Matlab
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Fig. 8: ECE measurements of TEXTOR shot 107906, where B0 = 2.45T, Ip = 350kA and ϑ = 2◦ . The actual output values of the ECE channels are irrelevant, since the channels were not calibrated.
code and solved using ode15s; an additional event function was used to include the reset condition of Section III-C. A specific simulation result is shown in Fig. 7, whereas an actual TEXTOR measurement with the same settings is shown in Fig. 8.3 Both are performed without additional ECCD. Although the transients after a crash are somewhat different, there is a large similarity between the two regarding sawtooth period and inversion radius. In simulation the sawtooth period is 11.6ms, whereas the average period in the experiment is 11.2ms. Furthermore, both figures show a regular sawtooth behavior on the 132.5, 135.5 and 138.5GHz channels, and reversed sawteeth on the 141.5 and 144.5GHz channels. Hence, in both simulation and experiment the inversion radius rinv is located somewhere between the 138.5 and 141.5GHz channels.
The influence of the current drive on the sawtooth period was also simulated, of which Fig. 9 shows two specific results, with an ICD of 2kA and 4kA respectively. These figures depict static input-output maps from the current drive position rECCD to the steady state sawtooth period. In both cases current drive close to the plasma center decreases the sawtooth period, whereas ECCD with larger rECCD increases it. This is believed to be linked to the location of the q = 1 surface, i.e. ECCD inside rq=1 decreases and ECCD outside rq=1 increases the period. The results in Fig. 9 underline this to some extent, but also show that it depends on the total current drive ICD .4 It is clear that the period is very sensitive to rECCD . This holds especially for larger current drives like ICD = 4kA, when the sawtooth completely disappears for some rECCD (the grey region in Fig. 9(b)). This non-linearity forms a big challenge from a control point of view. Most importantly, the maps in Fig. 9 agree with the experimental results in [9]. A specific example is shown in Fig. 10, which confirms that there is a large similarity in both shape and size with the input-output map in Fig. 9.
3 The 147.5GHz ECE channel is omitted in both figures, since this channel was malfunctioning during shot 107906.
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analysis of the sawtooth period dynamics and linearization around certain operating points. Subsequently, it provides a platform for controller design of the sawtooth period. This design still faces some large challenges, which will be an important point of attention in future research. Moreover, for the actual implementation on TEXTOR the construction of e.g. a real-time crash detection algorithm is also necessary.
(φ = −3.5◦ ) (φ = −3.5◦ ) (φ = −3.5◦ ) (φ = −3.5◦ ) (φ = −5◦ )
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Fig. 10: Experimental results with ECCD on TEXTOR, taken from [9]. The sawtooth period τs is normalized to the period without current drive, the deposition location rECCD is normalized to the minor radius a.
This underlines that our model captures the essence of the sawtooth instability, i.e. it yields a realistic dependence of the sawtooth period on the current drive location. Hence, it is expected that this model is accurate and adequate enough for future controller design. Although our model does not match reality completely (which is apparent in e.g. the transient response in Fig. 7), the modeling errors are of an unimportant kind under the closed loop conditions of Fig. 4. Errors like these should be considered as uncertainties, implying that robustness for these uncertainties will be an important aspect in the eventual controller design. V. C ONCLUSIONS AND FUTURE WORK In this paper we have presented a relatively simple model for the sawtooth instability and its period, based on the magnetic diffusion and energy transport equations and a state dependent reset condition. Furthermore, the TEXTOR environment has been included. We have shown that this model can mimic those aspects of the sawtooth instability which are most relevant and necessary for control purposes, i.e. it yields a realistic period and inversion radius, and returns static input-output maps which match with experiments. As such, our model defines a good starting point for future control oriented analysis of the sawtooth instability, including
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