Feedback Linearization Control Methods for Accurate Leaf Photosynthesis Measurements

Transcription

1 2017 American Control Conference Sheraton Seattle Hotel May 24 26, 2017, Seattle, USA Feedback Linearization Control Methods for Accurate Leaf Photosynthesis Measurements Philip E. Paré, Berkley J. Walker, and Justin McGrath* Abstract Accurate measurements of photosynthesis are vital for understanding the response of our planet to climate change and developing novel strategies for improving food production. Since photosynthesis is sensitive to a myriad of inputs, including temperature, these measurements require precise control to produce meaningful and accurate data. This paper develops a biophysical model of energy balance in a leaf and environmental control system that incorporates plant physiology and the biophysical relationship between a leaf and its environment. This model is then parameterized for a commonly-used device used for measuring leaf photosynthesis. Feedback linearization is applied to this model to design a controller for leaf temperature. The model is validated and the controller is then implemented on actual measurements. The result is a family of more efficient control algorithms built from first-order principles governing the exchange of matter and energy between a leaf and its environment. To the best of our knowledge, this is the first attempt at developing such a control algorithm. I. INTRODUCTION Measuring photosynthetic uptake of carbon dioxide in leaves is of critical importance in efforts to simulate the response of the biosphere to climate change and investigate novel routes to improve crop production [1]. Photosynthetic rates are typically measured using commercially-available systems such as the LI-COR 6400 XT (see Fig. 1) that are able to control multiple environmental factors such as light intensity, carbon dioxide concentration, and temperature. Despite having an extensive understanding of the rate equations governing the transfer of matter and energy between the leaf and environment (viz. radiative heat transfer, sensible heat transfer, photosynthesis, and leaf transpiration), measurement conditions are often controlled with algorithms that depend solely on the factor being controlled, and do not use information about interrelated factors, even though the instrument measures those factors. While these simple algorithms eventually reach steady state, compared to more sophisticated algorithms, developed herein, they often take longer to reach the equilibrium and have larger variance. In this paper we present a feedback linearized control algorithm based on the biophysical relationship between the leaf and its measurement environment. This is the first attempt we * Philip E. Paré is with the Coordinated Science Laboratory at the University of Illinois at Urbana-Champaign and can be reached at philip.e.pare@gmail.com. Berkley Walker is with the Institute of Plant Biochemistry and the Cluster of Excellence on Plant Science (CEPLAS) Heinrich-Heine University of Dsseldorf and can be reached at berkley.j.walker@gmail.com. Justin McGrath is with the Carl R. Woese Institute for Genomic Biology, University of Illinois, Urbana- Champaign and can be reached at jmcgrath@illinois.edu. This material is based on research partially sponsored by a postdoctoral fellowship from the Alexander von Humbol Foundation (BJW). are aware of that combines a biophysical representation of the leaf and its environment to produce a sophisticated control design. An improved algorithm is valuable for improving the ability to stably control the environment during photosynthesis measurements, increasing measurement precision in dynamic conditions, increasing measurement speed, and minimizing disturbance to leaf physiology that results from long measurement times. The concept of incorporating a biological model into control system design can be applied to any environmental factor that interacts with the leaf, but in this paper we focus on temperature. Temperature has long been recognized as an important factor impacting photosynthetic rates. As temperature changes, so do rates of key enzymes which drive photosynthesis [2], evaporative cooling of the leaf (transpiration), and conductance of water from the leaf through stomatal pores [3]. Additionally, the biophysical interaction of the temperature with plant leaves and canopies, which has a non-linear relationship, has been extensively studied and validated [4]. Given the importance of these factors to net photosynthesis, temperature must be controlled during a measurement in order to produce meaningful and reproducible results. Leaf temperature is dependent on changing rates of transpiration, sensible heat flux, radiative heating from incident light, and outside temperature. Leaf temperature is typically measured using a fine-wire thermocouple pressed to the underside of a leaf and controlled by modulating the temperature of a metal block. Air passes over the block and enters a measurements chamber, which has been clamped over the leaf in order to measure carbon dioxide exchange (see Fig. 1). As air is passed through the measurement chamber, rates of carbon dioxide exchange and transpiration (among other things) are measured from the flow rate and by means of infra-red gas analyzers. Modern gas exchange instruments include the ability to change incident light intensity, carbon dioxide concentration, and, to a limited degree, temperature. The relationship between leaf temperature and various physiological and biophysical factors will be explained in more detail in Section II. Feedback linearization, first introduced in [5], is a technique that uses feedback and coordinate transformations to change a nonlinear system into a linear system that is controllable. An in-depth treatment of the topic can be found in [6]. This technique has be used to design controllers for numerous different applications from chemical batch reactor control [7], to mobile robotics [8], to cell injection control [9], to quadrotors [10]. Given the nonlinear nature of the /$ AACC 801

2 units description r(t) mmol/m 2 /s radiation e(t) mol/s transpiration f(t) mol/s flow rate p(t) Pa pressure TABLE I: Table of measured disturbances Fig. 1: Commercially available leaf gas-exchange system (LI- COR 6400 XT). Shown are the entire system attached to a plant (left) and the measurement cuvette specifically (right). Shown also are an empty chamber where the sample leaf is inserted (upper-right) and the same cuvette clamped on to a leaf for a measurement (lower right). The fan boxes on either side of the cuvette are located over Peltier elements which change the temperature of the cuvette during the course of a measurement. models developed herein, feedback linearization is a wellsuited technique for controller design on this system. In Section II we present two different models of the system, with different levels of detail. There is a full model of the system that attempts to accurately represent all temperature-dependent energy flows in the leaf and measurement system and a simpler model that ignores or treats some of the variables as constants. In Section III we present the derivation of the input output feedback linearization controller. We evaluate the effectiveness of the models by comparing them to the measured leaf temperature in Section IV. In Section V we show the results of implementing the controller in real time for the LI-COR 6400 XT instrument. We conclude in Section VI by discussing future work. II. MODEL To model the change in the temperature of the leaf, T l, various aspects of the environment must be considered. The net energy balance a leaf can be described in terms of radiation, sensible heat flux, latent heat flux, and chemical potential [4]: dt l = (0.15R qr(t) + Re(T l, T w ) Q(T l, T a, g b (T a, t)) e(t)(v + v ap (100 T l 273)))/c p, (1) where T a is the temperature of the air; T w is the temperature of the wall; c p is a coefficient that combines the energy required to heat a given mass of leaf and the specific area of the leaf; R q is a conversion factor that converts the incident light from number of photons (units measured by the instrument) to watts; r(t) and e(t) are the radiation (mmol/m 2 /s) and the transpiration rate (mol/s), respectively (both are directly measured and will be modeled as measured disturbances); Re(T l, T w ) = e m s(f 1 T 4 w f 2 T 4 l ) (2) is the net radiation flux between the cuvette walls and the leaf, which are both black-body emitters of infrared wavelengths, and f 1 and f 2 are dimensionless coefficients that represent the amount of radiation emitted by the chamber walls that reaches the leaf (f 1 ) and the amount of radiation emitted by the leaf that reaches the chamber walls (f 2 ); v is a factor that converts transpiration rate to energy flux; v ap is the energy required to heat water; and Q(T l, T a, g b (T a, t)) = 2cg b (T a, t)(t l T a ) (3) is the sensible heat flux, with g b (T a, t) = λ a ( f(t)rta d l d c p(t) ) 1/2, where λ a is a constant that relates wind velocity to thermal conductance; f(t) is the air flow rate measured in mol/s; R is the universal gas constant; d l is the characteristic dimension of the leaf; d c is the characteristic dimension of the cuvette; and p(t) is the measured pressure (Pa). See Table I for the units and descriptions of the measured disturbances. See Table II for the values, units, and descriptions of the constants. Leaves absorb shortwave radiation from the light source. The amount of energy absorbed by the leaf is a function of the number and wavelength of photons. This is described by the first term in (1). Like all black bodies, leaves emit and absorb radiation, which is in the long-wave spectrum because of the low temperatures. Thus they emit radiation to the walls of the measurement chamber, and absorb radiation emitted by the chamber, purely as a function of temperature, as described in (2), which is the Stefan-Boltzman Law converted from a per area basis to a per mass basis. Heat flows between the leaf and the air via conduction as a function of temperature and the conductance between the two media, which depends on air flow rate and the dimensions of the exposed leaf within the chamber, as described in (3). As water evaporates within the leaf and escapes to the air, heat is also lost as the latent heat of evaporation, which depends on temperature and the amount of water lost, as described in the fourth term in (1). Photosynthesis and respiration absorb and release heat in chemical reactions, but the effect is small compared to the other terms and is ignored here. Lastly, the sum of the terms is divided by a heat capacity term in order to describe energy balance in units of K/s. The term accounts 802

3 value units description c p 816 J/kg/K combination of specific area of leaf and energy required to heat leaf* R q W/mmol conversion factor for incident light from photons to watts f 1 1 dimensionless accounts for the amount of radiation from the cuvette that arrives at the leaf f dimensionless accounts for the amount of radiation from the leaf that arrives at the cuvette v J/mmol converts transpiration rate to energy flux v ap J/K energy required to heat water* e m 0.95 dimensionless emissivity s 21 m 2 /kg specific leaf area λ a mol / s relative wind velocity to thermal conductance d l 0.81l r m 2 characteristic dimension of leaf (pg 107 of [4]) d c 8e-3 * 38.8e-3 * 2 m 3 characteristic dimension of the cuvette, measured by hand R J/K/mol universal gas constant l r 1/(5000π) m radius of the exposed leaf area in the sample cuvette R c J s/mol/k resistance to conductive heat flow between the block and an air buffer R c J s/mol/k resistance to conductive heat flow between the leaf air and cuvette material c 29.3 J/mol/K specific heat capacity of air c dimensionless mixing chamber 1 coefficient* ĉ K resistance term for energy flow from outside to inside of the instrument c dimensionless mixing chamber 2 coefficient* V c m 3 internal system air volume of the instrument measurement head TABLE II: Table of constants: * indicates learned from the data. for the mass per area of the leaf, and the energy per mass required to heat the leaf. In this implementation, the term was determined via parameterization. The majority of the parameters are physical constants or are characteristics of the measurement chamber, so they are appropriate for any plant species. Emissivity (e m ) and specific leaf area (s) can vary between leaves, but e m varies very little and a value of 0.95 is appropriate for most species. The value s does vary between species, but in the implementation used herein a generic value was used, and effective control was achieved. In addition to the energy balance of the leaf, energy flow through the gas-exchange system is modeled. The concepts of the model are appropriate for any system, but here the model has been parameterized specifically for the LI-COR 6400 XT. Heating elements, in this case two Peltier elements, are attached to the outside of the chamber and they heat a metal block (see Fig. 1), whose temperature we denote as T blk, and is the control input. The block is in contact with an air mixing chamber, whose temperature is represented by T blkb. The following heat flows, with units K/s, are defined to help facilitate the other variables. Air moves from the chamber with the metal block to a cuvette containing the leaf, where heat is exchanged between the air and the leaf, and the air and the walls of the cuvette (see Fig. 2). The temperature of the air around the leaf is denoted by T a and the temperature of the walls of the cuvette are denoted by T w : v 1 = 1 R c1 c (T blk T blkb ), (4) v 2 = c 2 γ(t)t a (T blkb T a ), (5) v 3 = 1 R c2 c (T a T w ), (6) v 4 = c 1 γ(t)t blkb (T in T blkb ), (7) where R c1 and R c2 are the resistance to conductive heat flow between the block and the air buffer and between the leaf air and the cuvette material, respectively; c is the specific heat capacity of air; c 1 and c 2 are cuvette mixing coefficients; T in is the room temperature (295 K), the temperature of the air that is entering in the flow; and γ(t) := Rf(t) p(t)v c, (8) where V c is the air volume of the instrument head. Transfer of heat from the block to the air within the chamber is dependent on temperature and a resistance term, and it is converted to K/s using the specific heat capacity of air as described by (4). Heat is transferred in a similar manner between the air in the leaf measurement chamber and the walls of the chamber as described by (6). Heat is transferred between the mixing chamber and the leaf measurement chamber by the flow of air between the chambers, and is dependent on temperature, air molar flow rate, pressure, the universal gas constant, the total volume of the chambers, and a mixing coefficient as described by (5) and (8). Heat is transferred in a similar manner between the air inlet and the mixing chamber as described by (7) (see Fig. 2). The rates of change of the rest of the state variables of the model are given by T blkb = v 1 + v 4 v 2, (9) dt a = v 2 v 3, (10) dt w = v 3. (11) 803

4 T in v 4 T blkb v 1 T blk v 2 T a T l T w v 3 T in ˆv 2,4 T blk ˆv 1 T a T l Fig. 2: A diagram of heat transfer as described in the full model (1), 9-(11) (flows shown in (4)-(7)) in the gas exchange system. Solid lines represent the sides of the measuring cuvette, dotted lines represent the boundary of the buffer air space and arrows represent fluxes of heat. Due to the complexity of the full model in (1), (9)-(11), we consider a simplified model for the controller design. Eliminating the buffer for the block, T blkb, by using T blk and T in in its place (effectively combining v 2 and v 4 into ˆv 2,4, and allowing T blk to directly affect T a via ˆv 1 ), and ignoring the wall temperature, T w (removing v 3 ), gives the simplified model (flows depicted in Figure 3): dt l = 1 c p (0.15R q r(t) + Re(T l, T a ) Q(T l, T a, g b (T a, t)) dt a e(t)(v + v ap (100 T l 273))), (12) = 1 R c1 c (T blk T a ) + ĉ 1 γ(t)(t in T a ), (13) }{{}}{{} ˆv 2,4 ˆv 1 where ĉ 1 is a resistance term for energy flow from the outside to the inside of the LI-COR and has units K. We set ĉ 1 = c 1 298, since the temperatures are usually around 298 K and we removed the second order temperature terms that appear in v 2 (and v 4 ). Note that all the variables and constants in the simplified model, (12)-(13), are measured or known. III. INPUT-OUTPUT LINEARIZATION We derive a feedback linearization controller by taking the derivative of the output until the control input appears. To simplify the controller design we treat the measured disturbances (see Table I) as constants, since they change very slowly with time. To avoid confusion we will drop the disturbances dependence on time in the following, that is, write f(t) as f. We will use the difference between our target leaf temperature Tl and the direct measurement of T l as our output: y = Tl T l. (14) Therefore, ẏ = T l = dt l, (15) which equals the negation of (12). Since ÿ = T l (16) and the control input T blk does not appear in (12), we take Fig. 3: A diagram of heat transfer as described in the simplified model (12)-(13) in the gas exchange system. another derivative using (12)-(13): T l = 1 ( ) (4e m s(f 1 Ta 3 Tblk c p R c1 c + A 0 f 2 Tl 3 T l ) 2cg b (T a, t) where ( ) + cg b (T a, t)(3 2Ta 1 Tblk T l ) R c1 c + A 0 + ev ap T l ), By letting A 0 = 1 R c1 c T a + ĉ 1 γ(t in T a ). (17) T blk = ψ 1 [e m s(4f 1 Ta 3 A 0 4f 2 Tl 3 T l ) 2cg b (T a, t) T l (18) + cg b (T a, t)a 0 (3 2Ta 1 T l ) + ev ap T l + c p u], where R c1 c ψ 1 = 4e m sf 1 Ta 3 + cg b (T a, t)(3 2Ta 1 T l ), (19) and plugging into (16) using u = T blk, we get ÿ = u, (20) with relative degree two. Since the relative degree is equal to the number of state equations in the simplified model (12)-(13), the system is controllable [6]. Note that for the normal temperature range (K [255, 355]) the first term of the denominator of (19) will always be nonzero and will dominate the second term. Therefore there will be no singularities in the controller. Define the state variable z = [ ] y, ẏ where y is from (14) and ẏ is from (15). Therefore, using (20), we have [ ] [ ż = z + u ] Define the feedback input as u = Kz; then the closed loop system is [ ] 0 1 ż = z. k 1 k 2 Therefore to stabilize the system we set giving poles of { 1 ± i}. K = [ 2 2 ], T l 804

5 Fig. 4: Comparison of the models in (1) (9)-(11) and (12)- (13) to the measured leaf temperature from the LI-COR 6400 XT. Fig. 5: Comparison of the controllers in (18)-(19) and the LI-COR 6400 XT driving T l from 25 C to 26 C. IV. MODEL VALIDATION In this section we validate the full model (1), (9)-(11), and the simplified model (12)-(13) by comparing them to the actual measured leaf temperature from the LI-COR 6400 XT. The observed dataset was acquired by setting the temperature of the block, allowing a short time for the system to react, and then setting a new temperature for the block. To test the models, r(t), e(t), f(t), p(t), and T blk (t), as recorded by the instrument, were plugged into (1), (9)-(11) and (12)-(13), and T l was calculated. Both models track the temperature of the leaf within 0.5 K (see Fig. 4). Note that the sharp spikes in the observed leaf temperature (depicted in green) are the result of bias in the temperature sensor, which occurs when block temperature is changing. The model does not account for this bias, so it does not produce the spikes. The simplified model tends to overestimate the change in leaf temperature as block temperature is changed, but the full model is less affected by this problem. This is likely due to the difference in the long-wave radiation portion of the models, since in the full model the temperature of the walls of the chamber is modeled, and thus acts as a heat storage buffer, absorbing and releasing heat via conduction, and slowing any change in leaf temperature. In the simplified model, wall temperature is not modeled, and hence it cannot act as a heat storage buffer. The simplified model, however, predicts leaf temperature well enough, as the controller based on the simplified model performs well in practice. This is illustrated in Section V. V. IMPLEMENTATION We implemented the controller in (18)-(19) on the LI-COR 6400 XT (see Fig. 1), using a Python program running on a PC. The program reads the measured variables including T l, T a, r(t), e(t), f(t), and p(t), and sends the control input T blk back to the machine. The inlet temperature, T in, is not measured by the instrument, so room temperature was read by a wall thermostat and hard-coded into the controller. Fig. 6: Comparison of the controllers in (18)-(19) and the LI-COR 6400 XT driving T l from 26 C to 25 C. Going from 25 C to 26 C, the controller in (18)-(19) takes approximately 75 seconds to near Tl (see Fig. 5, the solid red line) and then reaches steady state, hitting the target temperature exactly, shortly thereafter. The controller on the LI-COR 6400 XT takes over 200 seconds to get in the range of Tl and it does not hit the target temperature very well (see Fig. 5, the solid black line). When collecting numerous observations over the course of a day, more than halving the time required to reach a stable temperature can greatly increase the speed of measurements. The more precise control of temperature would also improve estimates of enzyme kinetics inferred from gas-exchange data, which are very temperature sensitive. Going from 26 C to 25 C, the performance of the controller in (18)-(19) is not quite as good as when increasing temperature, but it still outperforms the LI-6400 XT controller. As is shown in Figure 6, the controller in (18)- (19) gets in range of the target in a little over 100 seconds, converging shortly thereafter, while the LI-COR controller takes almost 300 seconds to get in range. Increasing and decreasing by 3 degrees, as is shown in Figures 7 and 8, the feedback controller still outperformed the 805

6 Fig. 7: Comparison of the controllers in (18)-(19) and the LI-COR 6400 XT driving T l from 23 C to 26 C. Fig. 8: Comparison of the controllers in (18)-(19) and the LI-COR 6400 XT driving T l from 26 C to 23 C. standard practices. However, the two controllers performed fairly similarly; we believe this is due to the limitations imposed by the instrument itself, that is, the elements can only heat up/cool down to certain temperatures. VI. CONCLUSION We developed a full biophysical model of a LI-COR 6400 XT and a leaf in the cuvette. We used a simplified version of this model to develop a feedback linearization controller of leaf temperature. We examined the effectiveness of the models by comparing to actual leaf-level measurements. We implemented the controller and significantly outperformed the current practices on smaller temperature changes and performed slightly better on larger changes. Temperature control was implemented here, but the approach could be used to control any of the biophysical properties of the environment, including carbon dioxide concentration and water vapor pressure. Furthermore, rather than controlling the environment in which the leaf is placed, it could be used to control rates of biological processes in the leaf itself, such as the transpiration rate and the photosynthetic rate. Similarly, the approach could be used for biological systems other than plants and it could be used in other instruments, such as greenhouses, growth cabinets, oxygen consumption chambers, and even large-scale experimental systems such as open-top chambers and Free Air gas Concentration Experiments (FACE) [11]. The technique could help reduce expenses in controlled-growth cabinets commonly used in plant science research or in commercial greenhouses where energy costs are the third largest expense [12]. Greenhouses would present an especially interesting challenge, since temperature must be controlled despite dynamic environmental conditions. For future work we would like to try more control techniques (e.g., stable-inversion or optimal control) and compare their performance to find the best algorithms for implementation at the leaf level for routine application to plants from different species and under different conditions. For future implementation we would like to add a sensor that reads the inlet temperature T in directly and feeds this signal to the controller. This would enable leaf temperature control during outdoor field measurements, which is often not possible because stable leaf temperature is practically unattainable in environments with variable ambient temperature. VII. ACKNOWLEDGEMENT The authors wish to thank Carolyn L. Beck and Usman A. Syed at University of Illinois at Urbana-Champaign for useful discussions which have contributed to this work. REFERENCES [1] S. P. 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