Nonlinear robust temperature–humidity control in livestock buildings

Computers and Electronics in Agriculture 49 (2005) 357–376

Nonlinear robust temperature–humidity control in livestock buildings A.G. Soldatos a , K.G. Arvanitis b,∗ , P.I. Daskalov c , G.D. Pasgianos a , N.A. Sigrimis b a

b

National Technical University of Athens, Department of Electrical and Computer Engineering, Division of Signals, Systems and Robotics, Zographou, 15773 Athens, Greece Agricultural University of Athens, Department of Agricultural Engineering, Laboratory of Agricultural Machinery and Automation, Iera Odos 75, Botanikos, 11855 Athens, Greece c Department of Automatics, University of Rousse, 8 Studentska Street 7017, Bulgaria Received 16 January 2005; received in revised form 1 August 2005; accepted 2 August 2005

Abstract The physical environment of farm animals inside livestock buildings is primarily characterised by hygro-thermal parameters and air quality. These parameters are influenced by the interaction with the outdoor situation on one hand, and the livestock, the ventilation system and the building on the other hand. Livestock housing must ultimately provide environmental conditions favorable to the preservation of animal health and welfare and, consequently, to the efficiency of animal production. To reach the best environmental conditions inside a building, it is necessary to apply advanced control systems and effective control techniques. In this paper, such a new control technique is presented. In particular, the use of robust nonlinear feedback control in conjunction with feedforward action, in order to assure arbitrarily small deviations in the desired temperature and humidity values in an animal building, is described. This method results in a controller whose characteristics depend on knowledge of the bounds of the uncertain elements in the system description. The method is illustrated by simulation studies for a contemporary pig house during summer and winter operation. © 2005 Elsevier B.V. All rights reserved. Keywords: Robust control; Temperature control; Humidity control; Livestock buildings

∗

Corresponding author. E-mail addresses: [email protected] (A.G. Soldatos), [email protected] (K.G. Arvanitis), [email protected] (P.I. Daskalov), [email protected] (G.D. Pasgianos), [email protected] (N.A. Sigrimis). 0168-1699/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.compag.2005.08.008

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1. Introduction Indoor climate of livestock buildings is of particular importance for the well-being and health of breeding animals in them, as well as for animal production performance (daily weight gain, milk yield, etc.). The quality of microclimatic conditions inside livestock buildings is the result of the complex influence of many factors. The housed animals are of primary importance; but building, technological equipment and external climatic conditions are rather important as well. Intensive livestock production under changing climate conditions requires advanced control systems and effective control techniques for the regulation of the microclimate conditions. The main purpose of an environmental control system for livestock buildings is to maintain different variables, such as temperature, humidity, and contaminant concentrations, at optimum levels for animals and humans (workers) by delivering outside airflow and supplemental heat when needed. Most technical equipment for environmental control in animal houses covers only ventilation and heating systems (Chao et al., 2000; Gates et al., 2001), but in some cases, equipment for air cleaning, cooling and humidifying is used. In recent years, new ventilation systems have also been built that are equipped with heat recovery systems. Conventional staged ventilation systems are commonly used in agriculture to maintain interior environments at or near the desired conditions for livestock housing. These systems utilize a series of discrete stages of ventilation and heating in order to compensate for deviations from a desired set-point. A number of different strategies are used for both heating and ventilation control in livestock buildings (Zhang and Barber, 1995). The most effective designs utilize the concepts of automatic feedback control based on internal air temperature, measured at one position within the building’s volume (Timmons et al., 1995; Seedorf et al., 1998), and on the better equipped production units, the ventilation rate (Berckmans and Goedseels, 1986; Pedersen et al., 1998; Vranken, 1999). At present, climate controllers make use of set-points of environmental variables, which are assumed to be optimal for an “average animal” (Reece et al., 1985; Black et al., 1986; van’t Klooster et al., 1995). These set-points have been derived from a combination of small-scale laboratory experiments and other less rigorous field trials (Pedersen and Thomsen, 2000). Accurate set-point values for different domestic animals were presented by CIGR (CIGR, 1984). These set-points are fixed and for some atmospheric environment conditions cannot be obtained by the microenvironment controllers. In spite of the application of modern techniques, the desired performance is not always achieved (Geers et al., 1984a,b; Parmar et al., 1992). The poor performance is mainly because of the oversimplification of the complex interactions between the animals and their environment. Thus, the livestock farmer may not be able to satisfy his customer’s specifications for livestock products or to meet increasingly stringent regulations on farming methods that aim to diminish the environmental impact of livestock production or provide a higher standard of animal welfare (Wathes et al., 2001; Aerts et al., 2003). Imprecise control of ventilation rates is a major cause of production losses and ventilation-related health problems in modern livestock buildings (Vranken et al., 1992; Taylor et al., 2004). Recent research leads to the application of more sophisticated methods for the design and control of microclimate systems. It has been shown, e.g. Sigrimis et al. (2000) and Gates et al. (2001), that conventional staged ventilation methods can be replaced by a fuzzy inference technique, while Taylor et al. (2004) suggested that mechanically ventilated agricultural

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buildings can be redesigned using proportional-integral-plus (PIP) control. Aerts et al. (2003) discuss an application of modern process control for the creation of an integrated management system for poultry production. An alternative to the fixed operating set-points is to consider conditions that depend on ventilation, food intake level and environmental factors (Boon, 1981; Daskalov et al., 1997). Energy savings under dynamic set-point changing and environmental control in pig buildings using the low critical temperature (LCT) as set-point was demonstrated in Boon (1981). Daskalov et al. (1997) showed that the LCT of weaned pigs can be reduced significantly when floor heating is used, and at the same time, the microclimate system energy efficiency can be raised considerably. These investigations demonstrate the need for new methods of microclimate control design using sophisticated microprocessor devices for the creation of effective environment regulatory systems. Most environmental studies on animal buildings have been focusing on the effects of indoor temperature. Environmental control has also been primarily based on indoor temperature, relaying on a constant minimum ventilation rate (MVR) for relative humidity control. Such a practice stems from the impracticality of physically measuring all the environmental parameters to produce a comprehensive thermal index quantifying the microenvironment. Other factors, like the corrosive environment of livestock facilities, affect the accuracy of humidity sensors to varying degrees and they can even fail after a relatively short barn exposure. Consequently, incorporation of relative humidity data into the environmental control strategy for livestock buildings is limited by the availability of relative humidity sensors. These sensors should be economical and reliable, require low maintenance, and provide an easy interface with automatic controllers. Hence, humidity control has not been widely implemented and the majority of current animal production facilities use temperature-only control strategies. Although the inside temperature can be controlled near the seemingly optimum values, health- and behaviour-related problems still may occur that frequently suppress animal performance (Geers et al., 1984a,b). This is particularly true when the current temperature control strategies achieve poor performance in maintaining an adequate relative humidity level inside the animal building. In particular, underestimation of MVR will result in high relative humidity and contaminant concentration while an overestimated MVR results in higher energy costs for the ventilation and supplemental heating. High relative humidity can reduce animal performance and affect heat dissipation of animals at elevated temperatures. On the other hand, low relative humidity can lead to respiratory disorders and high dust levels. Relative humidity is also generally used as an indicator of the air quality in the building and it is assumed that proper humidity control will also provide acceptable gas concentrations. Therefore, humidity control has long been recognized as one of the most important approaches to improve the environment of livestock buildings. Several attempts in developing advanced relative humidity sensors and practical temperature–humidity control strategies for animal houses have been reported in the literature (Vansteelant et al., 1988; Zhang, 1993; Hao and Leonard, 1995; Lemay et al., 1998; Lambert et al., 1999, 2001; Guo et al., 2001). The promising performance of new relative humidity sensors (Lemay et al., 1998) makes temperature–humidity control possible under barn conditions. At the same time, there is interest for practical controllers that meet the performance requirements using adjustments based on expert-system considerations and ad hoc compensators (see Lambert et al., 2001; Guo et al., 2001, and the references cited therein). However, the fact

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that temperature and humidity inside animal buildings are highly coupled through nonlinear thermodynamic laws, and that the actuators (windows and regulation valves) are usually subject to changing characteristics (the gain is largely perturbed by cross product terms with disturbances, such as wind velocity, outside temperature, etc.) has not been treated as yet explicitly and analytically to provide a robust control scheme for temperature–humidity regulation in livestock buildings. The objective of this paper is to provide a means of combining biological and physical models to simultaneously control the coupled temperature and humidity of air in animal buildings. Emphasis is placed on the uncertainties and the robustness of the proposed controllers. In the control of incompletely known systems, one of the main research avenues relies upon statistical considerations. In general, the uncertain quantities involved are assumed to behave according to some probabilistic patterns, and the desired behaviour is given by a similar description. A second approach is based on fuzzy logic considerations and is a topic of current important developments. Another stream of research uses the deterministic approach, which will be shown here to be well suited for temperature–humidity regulation in animal buildings. Typical system representations used for control purposes contain uncertain elements, which result from imprecise knowledge of inputs, unknown parameters pertinent to the system description and unmodeled dynamics. These “disturbances”, internal or external to the system, may adversely affect its performance. Hence the task is to control the system so as to achieve the desired objective in the presence of these uncertainties. Classical control methods typically involve repeated tuning of parameters in single-input single-output systems, while considering conflicting design objectives such as the gain margin and the closed-loop bandwidth until an acceptable controller is found. In many cases, these classical control techniques lead to a perfectly satisfactory solution and more powerful tools hardly seem necessary. Difficulties arise when the uncertainties come into play, the plant dynamics are complex and the performance specifications are particularly stringent. Then, even if a solution is eventually found, the process is likely to be expensive and time consuming for the designer. Furthermore, by obtaining a controller with “classical” methods, one cannot usually guarantee that it will perform as expected under poorly modelled dynamics and unknown disturbances. Robust control design deals with the question of how to develop such controllers for system models with uncertainty. Here, we use a deterministic nonlinear robust control approach for an animal building whose mathematical representation involves “uncertain” inputs and measuring errors. In particular, we consider unknown but bounded errors when measuring the outside temperature, humidity and wind velocity as well as an uncertain contribution from the evaporation of floor water inside the building. Unknown parameters and other uncertainties may be easily incorporated at the expense of more control requirements. The control technique described in the paper results in a feedback controller whose characteristics depend on knowledge of the bounds of the uncertain elements in the system description. The proposed nonlinear robust controller assures arbitrarily small deviations in the desired temperature and humidity values inside the livestock building and can be readily implemented, since all pertinent information is known a priori, except the inside temperature and humidity which can be obtained during operation. Simulation results in the case of temperature/humidity control of a pig house, obtained after some preliminary identification tests to identify hygro-thermal

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parameters of the livestock building (Daskalov, 1997), show the effectiveness and good performance of the proposed control scheme, which provides smooth set-point tracking with arbitrarily small attenuation of the effects of unknown disturbances, and direct cancellation of external disturbances, which can be measured with relative ease. The proposed method is applicable to any air-conditioning system and is expected to gain wide acceptance in modern SCADA systems with extended computational capabilities. The proposed method is currently implemented in MACQU systems (Sigrimis et al., 2000) to be placed in field operation.

2. A brief review of nonlinear robust control We begin our analysis by first considering a brief overview of the particular nonlinear robust controller design method, which will be the theoretical basis of this study on temperature and humidity control of livestock buildings. Here, and in the sections that follow, the time argument t is omitted when no confusion is likely to arise. Consider the dynamical system: dx(t) = [A +
A(r(t))] x(t) + [B +
B(s(t))] u(t) + Cv(t) dt x(to ) = xo

(1) (2)

where x ∈ Rn , u ∈ Rm , v ∈ Rl , r ∈ Rp , s ∈ Rq , while A ∈ Rn × m , B ∈ Rn × m , C ∈ Rn × 1 are known constant matrices, and
A(.): Rp → Rn × m ,
B(.): Rl → Rn × m are known continuous functions. Uncertainties in the system matrix, input matrix, and input, respectively, are modeled by the unknown Lebesgue measurable functions r(.): R → P, s(.): R → S, v(.): R → V, where P, S, V are known compact subsets of the appropriate spaces. Thus, the only information concerning the unknown elements r(t), s(t), v(t) resides in their bounding sets P, S, V. Concerning system (1) and (2), we assume that the uncertainties belong to the range space of the input matrix B or, more precisely, that there exist continuous functions D(.): Rp → Rn × m , E(.): Rp → Rn × m and constant matrix F ∈ Rm × l such that the so-called matching conditions (Leitmann, 1980) are met; namely,
A(r) = BD(r),

∀r ∈ P

(3a)


B(s) = BE(s),

∀s ∈ S

(3b)

C = BF

(3c)

and max E(s) < 1 s∈S

The matching conditions embodied in Eqs. (3a)–(3c) assure that the range space of the input matrix B contains that of the uncertain quantities
A(r),
A(r) and C. Thus, in principle, there is an input that can cancel the possible uncertainties. Furthermore, we assume that the matrix pair (A, B) is stabilizable; that is, there exists a constant matrix K ∈ Rm × n ¯ = A + BK is stable. Of course, (A, B) is stabilizable if it is controllable (Chen, such that A

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1970). Before proceeding, we state two desirable properties to be achieved, for the uncertain system (1) and (2) by appropriate choice of control u(t). 2.1. Property P1: Uniform boundedness Given xo ∈ Rn , there is a positive d(xo ) < ∞ such that, for all solutions x(.): [to , t1 ) → Rn , x(to ) = xo , x(t) < d(xo ),

∀t ∈ [to , t1 )

2.2. Property P2: Uniform ultimate boundedness Given xo ∈ Rn and J = {x ∈ Rn | ||x|| ≤ δ > 0}, there is a non-negative T(xo , J) < ∞, such that, for all solutions: x(t) ∈ J,

∀t ≥ to + T (xo , J)

Loosely speaking, uniform boundedness implies that every solution emanating from initial state xo remains within a bounded neighborhood whose radius may depend on xo . Uniform ultimate boundedness implies that every solution starting at xo will enter and remain within a neighborhood of prescribed radius δ, after a finite time which may depend on xo and δ. These two properties, sometimes stated in a slightly different but equivalent form, are the main conditions of practical stability (Leitmann, 1980). Now, consider a control law of the form (see Leitmann, 1980; Leitmann et al., 1986, and the references therein): u = Kx + pe (x)

(4)

such that, given ε > 0, there holds:    −BT Px    · ρ(x), if BT Px ≥ ε  T B Px pε (x) = T     −B Px · ρ(x), if  BT Px < ε ε

(5)

where P ∈ Rn × m is the symmetric, positive-definite solution of the Lyapunov equation: ¯ +A ¯ TP + Q = 0 PA

(6)

for given symmetric, positive definite:

Q ∈ Rn × m ,

and

ρ(x) = [1 − max E(s)]−1 [max D(r)x + max E(s)Kx + max Fv] s∈S

r∈R

s∈S

v∈V

(7)

The general underlying idea is to simplify the analysis of a complex system by considering a scalar function of its state, which goes, as the system evolves, to a minimum. This minimum corresponds to the equilibrium point of the system in such a way that they both occur simultaneously. That is, as the function of the state reaches its minimum, the system must go to the equilibrium point. A function of this type, which allows one to deduce stability, is termed a Lyapunov function (Chen, 1970). The design of the above controller is based

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on the constructive use of Lyapunov stability theory. Roughly speaking, one considers a Lyapunov function for the system without the uncertainties. This function is employed as a candidate Lyapunov function for the actual uncertain system. The proposed stabilizing control guarantees a negative derivative, and therefore, reduction of the Lyapunov function outside a certain neighborhood of the desired state along all possible solutions in the presence of uncertainties. Now, provided the stated assumptions are met, control (4) guarantees practical stability for uncertain system (1) and (2) and, in particular, Properties P1 and P2 for every possible realization of uncertain elements r(.), s(.), v(.), e.g. see (Leitmann, 1980) and the references cited therein, where one can also find explicit expressions for d(xo ) and T(xo , J). The feedback control (4) is nonlinear; in particular, it is composed of a linear and a saturation control. In general, the addition of saturation control (5) improves performance by decreasing the size of the region of ultimate boundedness or even by stabilizing a response that may be unstable in the presence of only linear control (Leitmann et al., 1986). Here, it should be noted that δ = δ(ε) and can be made arbitrarily small by choice of ε; namely, decreasing of ε results in a decrease of the radius of the ultimate boundedness set. By letting ε go to zero, one could achieve asymptotic stability while the control (5) becomes discontinuous. Here, we prefer to have ε > 0, since discontinuous control is not realizable in practice (chattering control) and, in addition, leads to complications in the analysis; in particular, it forces one to use the notion of “generalized solutions”.

3. Mathematical modelling of the hygro-thermal regime in livestock buildings The dynamic model of the energy and mass balance of the air in an animal building is shown to be highly nonlinear. A simple heating–cooling ventilating model can be obtained by considering the differential equations, which govern sensible and latent heat, as well as water balances on the interior volume (Daskalov, 1997). These dynamical equations are as follows: ρCp VT ρVH

dTin (t) = Qan + Qheater − Qv − Qc − Qe − Qfog dt

dwin (t) = Wan + Wfog − Wair,in + Wair,out + We dt

(8a) (8b)

where ρ is the air density (1.2 g m−3 ), Cp the specific heat of air (1006 J (kg K)−1 ), Tin indoor air temperature (◦ C), Qan the sensible heat production of the animals (kW), Qheater the heat emanated by the floor heating system (kW), Qv the heat loss due to ventilation (kW), Qc the conductive heat losses through the walls, ceiling and the floor of the building (kW), Qe the heat loss for evaporation of the water from the floor (kW), Qfog the heat loss due to the fog system (kW), win the moisture content of air inside the building (water vapor mass ratio, g H2 O kg−1 of dry air), Wan the water–vapor production of the animals (kg/h), Wfog the water capacity of the fog system (kg/h) used for air cooling, Wair,in the rate of moisture transfer from the inside air leaving the building (kg/h), Wair,out the rate of moisture transfer from the outside air entering the building (kg/h), We the rate of moisture transfer

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from the evaporation of water from the floor (kg/h) and VT and VH are the temperature and humidity active mixing volumes (m3 ), respectively (Young and Lees, 1993; Young et al., 2000). Short-circuiting and stagnant zones exist in ventilated spaces and the active mixing volume is typically significantly less than the calculated total volume. The active mixing volume of a ventilated space may easily be as small as 60–70% of the geometric volume. This, of course, means that indoor air temperature and humidity are unlikely to be uniform throughout the air space. Moreover, in a model with only one state variable, i.e. the temperature, the effective heat capacity must be usually assumed to be larger than that determined by just the air volume, to encompass some of the heat capacity of construction materials. Similarly, the effective volume for humidity may be smaller or larger than the geometric one, depending on the degree of mixing and other effects, such as air and humidity losses. The animal-heat production Qan and the animal water-vapor production Wan are, in general, nonlinear functions of the inside air temperature Tin , the number of animals Nan , and their mass Man , and are given by: Qan = g1 (Tin , Nan , Man ),

Wan = g2 (Tin , Nan , Man )

The following experimentally obtained regression equations may be used, say, for 20 kg weaned piglets (ASHRAE, 1981; CIGR, 1984). Qan = g1 (Tin , Nan , Man ) = Nan × 0.096 × [0.8 − 1.85 × 10−7 (Tin + 10)4 ]

(9a)

Wan = g2 (Tin , Nan , Man ) = Nan × 0.001 × [0.26Tin2 − 6.46Tin + 81.6]

(9b)

where Nan is the number of animals and Man is the mass of animals (kg). Similar relations for other kinds of animals can be found in (CIGR, 1984, 2002). The heat losses due to ventilation are given by: Qv = ρVR Cp (Tin − Tout )

(10)

where Tout is the outdoor air temperature (◦ C). The total natural ventilation flow rate, VR (m3 h−1 ), can be represented as the sum of the measured, i.e. controlled, ventilation rate VR,b (m3 h−1 ), through the ridge vents and the non-controlled ventilation rate VR,n (m3 h−1 ), originating from the cross-ventilation and the infiltration through small cracks in the building shell. Therefore, VR = VR,b + VR,n The ventilation through the cracks, VR,n , depends, in general, on the outside wind velocity, vwind (m s−1 ) and thus Eq. (10) becomes: Qv = g3 (VR,b , vwind , Tin , Tout ) = ρ[VR,b + h(vwind )]Cp (Tin − Tout )

(11)

The conductive heat losses, Qc , through the building walls, ceiling and floor can be expressed as: Qc = g4 (Tin , Tout ) = UA(Tin − Tout ) where UA = 712 W/K is the overall (including radiation) heat transfer coefficient.

(12)

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The quantities Qe and We related to heat losses for the evaporation of floor water and the rate of moisture transfer for this evaporation have small but unknown influence, and may be considered as unknown disturbances. These are associated to the water contained in manure, urine, spilled food, etc. One writes: Qe = λWe

(13)

where λ is the latent heat of vaporization (2.257 kJ g−1 ). Moreover, heat losses due to the fogging system are related to its water capacity through the equation: Qfog = λWfog

(14)

The rates of moisture transfer from the outside air Wair,out and the inside air Wair,in may be written as: Wair,out = g5 (VR , wout ) = ρVR wout = ρ[VR,b + h(vwind )]wout , Wair,in = g6 (VR , win ) = ρVR win = ρ[VR,b + h(vwind )]win

(15)

where wout is the moisture content of air outside the building (water vapor mass ratio, g H2 O kg−1 of dry air). Upon substitution of Eqs. (9a) and (9b) and (11)–(15) into Eq. (8a) and (8b) one gets: dTin (t) = g7 (Tin , Nan , Qheater , VT , VR,b , vwind , Tout , We , Wfog ) dt dwin (t) = g8 (Tin , Nan , Wfog , VH , VR,b , vwind , win , wout , We ) dt where g7 (Tin , Nan , Qheater , VT , VR,b , vwind , Tout , We , Wfog ) Nan 0.096[0.8 − 1.85 × 10−7 (Tin + 10)4 ] + Qheater − UA(Tin − Tout ) − λWe − λWfog ρCp VT

= −

[VR,b + h(vwind )](Tin − Tout ) VT

g8 (Tin , Nan , Wfog , VH , VR,b , vwind , win , wout , We ) =

Nan 0.001(0.26Tin2 − 6.46Tin + 81.6) + Wfog + We ρVH −

[VR,b + h(vwind )](win − wout ) VH

Theoretical model uncertainties and approximations were circumvented in Daskalov (1997) using experimental ARMA modeling techniques. There, the coefficients in the

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derived equations were identified using sequential recursive algorithms and the mathematical description of a pig building for weaned piglets was found to be (Daskalov, 1997): dTin (t) = f1 (Tin , Tout , wout , vwind , VR,b , Qheater , Nan , Wfog , We ) dt 1 = {−Tin + ag1 Tout + ag2 wout + ag3 vwind + ag4 VR,b τT + ag5 [Qheater + Nan 0.096[0.8 − 1.85 × 10−7 (Tin + 10)4 ]] + ag6 [Wfog + Nan 0.001(0.26Tin2 − 6.46Tin + 81.6)] + (ag6 − λag5 )We



(16a) dwin (t) = f2 (win , Tout , wout , vwind , VR,b , Qheater , Nan , Wfog , We ) dt 1 = {−Win + bg1 Tout + bg2 wout + bg3 vwind + bg4 VR,b τw + bg5 [Qheater + Nan 0.096[0.8 − 1.85 × 10−7 (Tin + 10)4 ]] + bg6 [Wfog + Nan 0.001(0.26Tin2 − 6.46Tin + 81.6)] + (bg6 − λbg5 )We } (16b) where agi , bgi , i = 1, . . ., 6, τ T and τ w are the identified model coefficients.

4. Nonlinear robust controller design for temperature–humidity regulation In this study, two modes of operation of the environmental control system were considered: one for summer operation and the other for winter operation. To proceed with temperature/humidity controller design, we next perform an approximate linearization of Eqs. (16a) and (16b) around some desired operating point, whereupon the subscripts s and w indicate summer and winter operation, respectively. Then, we obtain: dx = Ax + Bu + C1 v1 + Cv dt

(17)

x(to ) = xo

(18)

where x = [ Tin v1 = [ Tout

T

win ] , u = us = [ VR,b wout

T

T

Wfog ] , or u = uw = [ VR,b

vwind ] , v = [ We

␦Tout

␦wout

␦vwind ]

Qheater ]

T

T

are the state, the summer or winter control inputs, the measurable disturbing inputs and the unknown but bounded disturbance vectors, respectively. Note that, as is customary, we

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use the same symbol for the absolute and the relative, with respect to the desired operating point, variables. The quantities ␦Tout , δwout , δvwind represent measuring errors of the corresponding disturbances. Moreover,   ∂Qan ∂Wan   −1 + a + a g5 g6   ∂f1 ∂f1 ∂Tin ∂Tin

 0    ∂Tin ∂win   τ T   A=  =    ∂f2 ∂Q ∂W ∂f2 

Tin = Tin, desired an an   bg5 + bg6

  −1 ∂Tin ∂Tin ∂Tin vwin win = win, desired τw τw u = u∗ (19)  ∂f 1  ∂VR,b B = Bs =   ∂f2 ∂VR,b or

∂f1 ∂Wfog ∂f2 ∂Wfog



∂f1  ∂VR,b B = Bw =   ∂f2 ∂VR,b

  ag4

  τT  =b  g4

Tin = Tin,desired

τw win = win,desired u = u∗

∂f1 ∂Qheater ∂f2 ∂Qheater

ag6  τT  bg6  τw

  ag4

  τT  =b  T = T g4

in in,desired

τ w win = win,desired

(20a)

ag5  τT  bg5  τw

(20b)

u = u∗ 

∂f1  ∂Tout C1 =   ∂f2 ∂Tout

∂f1 ∂wout ∂f2 ∂wout

∂f1 ∂vwind ∂f2 ∂vwind

  ag1

  τT  =b  T = T g1 in,desired

in

τ w win = win,desired

ag2 τT bg2 τw

ag3  τT  bg3  τw

(21)

u = u∗ and



∂f1  ∂We C=   ∂f2 ∂We 

∂f1 ∂Tout ∂f2 ∂Tout

ag6 − λag5  τT =  bg6 − λbg5 τH

∂f1 ∂wout ∂f2 ∂wout ag1 τT bg1 τw

ag2 τT bg2 τw

∂f1 ∂vwind ∂f2 ∂vwind



   T = T in,desired

in

win = win,desired

 ag3 τT   bg3  τw

u = u∗

(22)

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where u* is the control input required to drive the original, nonlinear, uncontrolled system at the desired operating point in the absence of any disturbances. To illustrate the utility of nonlinear control for the satisfaction of our objective, namely the regulation of indoor temperature and humidity, we employ the linear description derived above. We consider now the robust nonlinear feedback controller and the governing equations reported in Section 2. Eqs. (17) and (18) are of the form (1) and (2) with
A(r(t)) =
B(s(t)) = 0 and the additional term C1 v1 represents the effect of the measurable disturbing inputs that will be cancelled with feedforward action. Here, the uncertain inputs We , ␦Tout , ␦wout and ␦vwind are due to the unknown rate of moisture transferred by evaporated floor water, and the measuring errors of the external inputs Tout , wout , and vwind . In conformity with the controller theory summarized in Section 2, we assume v ∈ V , a known compact set. This assumption is reasonable, since maximum values of moisture transfer by floor water and maximum errors introduced by typical measuring instruments are known for the worst operating conditions. Thus, max max V = {v ∈ R4 : |v1 | ≤ Wemax , |v2 | ≤ ␦Tout , |v3 | ≤ ␦wmax out , |v4 | ≤ ␦vwind }

It is readily verified that the other assumptions of Section 2 are satisfied, i.e. the matching conditions Cs,w = Bs,w Fs,w and C1s,w = B1s,w F1s,w are met with F s,w = (Bs,w )−1 Cs,w and F 1s,w = (Bs,w )−1 C1s,w

(23)

for Bs,w invertible. Thus, given ε > 0, the control that assures practical stability (as defined in Section 2) of the dynamic system represented by Eqs. (17) and (18), for all disturbances v(.), whose values range in V, is: us,w = Kx + pe (x) − F 1 v1 + u∗s,w with pes,w (x) given by Eq. (5), i.e.    −BT P x     s,w s,w  · ρs,w (x), if BTs,w P s,w x ≥ ε  BT P s,w x s,w p␧s,w (x) =  −BT P s,w x     s,w  · ρs,w (x), if BTs,w P s,w x < ε ε for Ps,w determined from Eq. (6) for an appropriate choice of Q. From Eq. (7),   ρs,w (x) = max F s,w v v∈V

(24)

(25)

The nonlinear control given by Eq. (24) consists of four components. The first two components constitute the robust feedback action, which is used to stabilize A, if necessary, and to suppress the unknown but bounded disturbances. The feedback design is based on the linearized dynamical equations according to Section 2 and is effective on both the nonlinear system and its linearized counterpart, obtained by linearizing (16a) and (16b) at the operating point considered. The third and fourth components constitute the feedforward action, which is employed to cancel the measurable disturbances and to drive the undisturbed nonlinear system at the desired operating point, where the linearization is effected. The control given

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by Eq. (24) is applied to the nonlinear system described by Eq. (16a) and (16b) and can be readily implemented, since all pertinent parameters are known a priori. Information about the state and the measurable disturbances to be cancelled can be obtained in real time during operation.

5. Simulation studies To illustrate the efficiency and good performance of the nonlinear robust control scheme, a series of simulation experiments is presented in this Section. We consider summer and winter operations separately. The nonlinear control design is based on the linearized equations around two desired operating points. The resulting controllers are then used with the nonlinear system near the operating points. The livestock building selected for simulation purposes is a pig house situated at a location where two modes of operation, summer and winter, are sufficient for normal breeding. A graphical representation of the building is illustrated in Fig. 1. The effective dimensions, a portion of the total geometric volume of the house, yield a 35 m × 12 m × 5 m = 2100 m3 active-mixing interior volume where the animals (Nan = 880) are contained. The desired inside temperature and humidity ranges were 16–28 ◦ C and 50–75%, respectively, and maintenance at pre-selected target values within the given ranges was preferable. The computation of the model coefficients of Eq. (16a) and (16b), listed in Table 1, can be found in Daskalov (1997). Substituting those coefficients in Eqs. (21) and (22), we get,   0.018 0.0166 −0.0169 C1 = and 0.00654 0.021 −0.0172   −0.00496 0.018 0.0166 −0.0169 C= −0.00265 0.00654 0.021 −0.0172

Fig. 1. General layout of the pig building: (1) arrangement for controlled ventilation; (2) air cooling by means of additional humidifying (fog system); (3) floor heating.

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Table 1 Numerical values of the coefficients of the mathematical model of the pig building Coefficient

Value

ag1 ag2 ag3 ag4 ag5 ag6 bg1 bg2 bg3 bg4 bg5 bg6 ␶T ␶w

0.62 0.57 ◦ C kg/g −0.58 ◦ C s/m −0.33 × 10−4 ◦ C h/m3 0.25 ◦ C/kW −0.014 ◦ C h/kg 0.199 g/kg ◦ C 0.64 −0.525 g s/kg m −0.32 × 10−4 g h/kg m3 0.137 g/kg kW 0.53324 × 10−2 g h/kg2 34.41 min 30.44 min

These matrices can be used for both summer and winter operations since their determination was independent of the operating point considered. 5.1. Summer operation In summer operation mode, we consider the following desired operating point: Tin,desired = 26 ◦ C, win,desired = 16.72 g/kg (RH = 70%). The average external weather conditions obtained from typical records for Tout , wout and vwind were assumed to be 38 ◦ C, 14.05 g/kg (RH = 30%) and 1 m s−1 , respectively. Using Table 1 and Eqs. (19), (20a) and (23), we get:     −5.278 0 −0.96 −406.86 −2 −6 As = 10 × , Bs = 10 × −1.204 −3.285 −1.053 175.178 

−0.976 −1.923 F 1s = 104 × −0.00213 0.00046  0.327 −0.976 4 F s = 10 × 0.00045 −0.00213

 1.674 , 0.0002 −1.923 0.00046

1.674 0.0002



¯ = As . Hence, the control The matrix As is stable, and therefore, we select K = 0 and A given by Eq. (24) becomes: us,w = pes (x) − F 1s v1 + u∗s



 1 0 where we use ε = 0.1 and Q = 103 × . The unknown but bounded disturbances We , 0 1 ␦Tout , ␦wout and ␦vwind are shown in Fig. 2 and they were produced based on their expected

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Fig. 2. Unknown disturbances We , ␦Tout , ␦wout and ␦vwind for summer operation.

max = characteristics using a random generator. From Fig. 2, we select Wemax = 10kg/h, ␦Tout max max −1 ◦ 1 C, ␦wout = 2g/kg and ␦vwind = 1m s . Then, using Eq. (25), ρs = max  F s v = 1177. v∈V

Fig. 3 shows the controlled and uncontrolled responses for the indoor temperature and humidity. We note that the controlled records satisfy the desired performance and there was considerable improvement over the uncontrolled situation in the amplitudes of the responses. Fig. 4 shows the required control effort, which appears to be within the performance capabilities of a contemporary animal building.

Fig. 3. Controlled (

) and uncontrolled (

) responses (Tin , Win ) in summer operation.

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Fig. 4. Control records (VR,b , Wfog ) for summer operation.

5.2. Winter operation For this case, the desired operating point is Tin,desired = 22 ◦ C, win,desired = 13.26 g/kg (RH = 70%). The corresponding average external weather conditions for Tout , wout and vwind were assumed to be 2 ◦ C, 5.07 g/kg (RH = 95%) and 3.7 m s−1 , respectively. From Table 1 and Eqs. (19), (20b) and (23) we get:     −4.573 0 −0.96 7265.33 −2 −6 Aw = 10 × , Bw = 10 × −0.845 −3.285 −1.053 4500.66 

F 1w

1.011 −2.354 = 10 × 0.00038 −0.000083 4



0.0934 F w = 10 × −0.000080 4

1.011 0.00038

1.489 −0.000035 −2.354 −0.000083



1.489 −0.000035



¯ = Aw is appropriate, since Aw is stable. The application of The selection of K = 0 and A   1 0 the control given by Eq. (24) was done here using ε = 0.5 and Q = 10−2 × . The 0 1 disturbances We , ␦Tout , ␦wout and ␦vwind are illustrated in Fig. 5 and they were, again, promax = 1 ◦ C, ␦wmax = 2 g/kg duced randomly. From Fig. 5, we select Wemax = 10 kg/h, ␦Tout out max −1 and ␦vwind = 1 m s . Then, using Eq. (25), ρw = max  F w v = 31432. v∈V

Fig. 6 shows the indoor temperature and humidity time histories of both the controlled and uncontrolled situations. Again, there was considerable improvement in the controlled responses for both the amplitude variations and the frequency content. There was also very tight tracking of the desired operating point. Fig. 7 shows the required controls, which can be produced faithfully in modern animal buildings. Considering the simulation results obtained, a few observations can be made. The control design parameters, ε and Q, affect the final responses in different ways. Decreasing ε

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Fig. 5. Unknown disturbances We , ␦Tout , ␦wout and ␦vwind for winter operation.

increases the control effort in general and makes the deviations from the desired operating point even smaller (Leitmann, 1980; Leitmann et al., 1986). Increasing the elements of Q results in further improvement in the responses at the expense of larger controls. Breaking the symmetry of the elements of Q tends to favor one or the other of the two states accordingly, with consequent variations in the controls. These observations are based on a number of simulation runs, which were made before the final results were produced. While the control scheme of Section 2 allows for uncertainty in the system parameters, that is
A(r) = 0,
B(s) = 0, it does so at the expense of more control requirements. Another point of interest concerns the appearance of delays. The sensors and actuators, which determine the system’s

Fig. 6. Controlled (

) and uncontrolled (

) responses (Tin , Win ) in winter operation.

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Fig. 7. Control records (VR,b , Qheater ) for winter operation.

state and implement the control, possess their own dynamics, which were neglected in the mathematical model. In effect, the state x(t) required for the control is not available. Instead, what is available at time t is a retarded state x(t − td ), where td > 0 is a delay due to the neglected system dynamics. Provided this delay is sufficiently small, the assured practical stability is not vitiated, except for a possible deterioration in the response bounds, e.g. see Leitmann et al. (1986). Note that the effect of the delays in the feedforward part of the controller can be incorporated in the assumed unknown disturbances. A delay of td = 2 min was introduced in the state upon which the control is based, and as expected in view of the relatively low response frequencies, no discernible effect on the response was detected.

6. Conclusions In this paper, an effective way of regulating the indoor temperature and humidity in livestock buildings has been developed. The control method requires the combination of feedforward and feedback actions. The control scheme introduced here makes use of the advantages of nonlinear robust control, namely arbitrarily small attenuation of the effects of unknown disturbances, and direct cancellation of external disturbances, which can be measured with relative ease. In contradistinction to earlier attempts for temperature–humidity control, this control method does not require any knowledge of the unknown inputs other than their possible range of variation. In particular, it does not require an a priori statistical description of the uncertain inputs. This approach appears to be reasonable for the current problem since the maximum expected input-disturbance bounds are estimated by considering the worst cases of weather variations on record. On the other hand, there is no known reliable statistical description for changes in the weather conditions. Another point of interest concerns the use of nonlinear control. In e.g. Leitmann (1980) and the references cited therein, it was demonstrated that nonlinear control results in a smaller region of ultimate boundedness when compared with that obtained using linear feedback control. Also, as mentioned earlier, other internal or external disturbances, such as parametric uncertainties, may be easily incorporated, if desired. This addition is not always possible when using standard linear regulator design. Implementation of the presented approach is

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straightforward since the controller is relatively simple with minimal real-time information requirements.

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