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Lihong (or initial) Xu ASABE Member?
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Michigan State University East Lansing, MI 48824 USA
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Tongji University 1239#,Siping Rd., Shanghai 200092
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073022 2007 ASABE Annual Meeting Paper
The authors are solely responsible for the content of this technical presentation. The technical presentation does not necessarily reflect the official position of the American Society of Agricultural and Biological Engineers (ASABE), and its printing and distribution does not constitute an endorsement of views which may be expressed. Technical presentations are not subject to the formal peer review process by ASABE editorial committees; therefore, they are not to be presented as refereed publications. Citation of this work should state that it is from an ASABE meeting paper. EXAMPLE: Author's Last Name, Initials. 2007. Title of Presentation. ASABE Paper No. 07xxxx. St. Joseph, Mich.: ASABE. For information about securing permission to reprint or reproduce a technical presentation, please contact ASABE at [email protected] or 269-429-0300 (2950 Niles Road, St. Joseph, MI 49085-9659 USA).
An ASABE Meeting Presentation
Paper Number: 073022
A Multi-Objective Compatible Control (MOCC) Algorithm for the Greenhouse Energy-Saving Control
Lihong Xu, ProfessorMichigan State University, Department of Electrical and Computer Engineering, Michigan
State University, East Lansing, MI 48824, [email protected]
Qingsong Hu, DoctorTongji University, 1239#,Siping Rd., Shanghai 200092, China, [email protected].
Written for presentation at the2007 ASABE Annual International Meeting
Sponsored by ASABEMinneapolis Convention Center
Minneapolis, Minnesota17 - 20 June 2007
Abstract. Energy-saving control is very important in the greenhouse environment control. However, the greenhouse is a very complex system and the users have multiple objectives, there are few suitable control methods to realize this by now. In this paper, a new two-layer multi-objective compatible control algorithm is proposed for the greenhouse control problems with two conflicting control objectives, control error and energy consumption. The first layer is devoted to obtaining a user's desired controlled objectives region, assured to be not only achievable but also Pareto-optimal. The second layer is devoted to designing an effective controller by optimizing the most important controlled objective --- energy consumption, subject to system constraints from the controlled objectives region in the first layer. This control algorithm provides an effective robust controller design method for the complex greenhouse energy-saving control with precise models and uncertain initial conditions. Simulations illustrate that the two-layer multi-objective compatible control (MOCC) algorithm has some advantages on energy-saving over traditional multi-objective control methods.
Keywords. Greenhouse, Energy-saving, Multi-objective Control, Compatible
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The authors are solely responsible for the content of this technical presentation. The technical presentation does not necessarily reflect the official position of the American Society of Agricultural and Biological Engineers (ASABE), and its printing and distribution does not constitute an endorsement of views which may be expressed. Technical presentations are not subject to the formal peer review process by ASABE editorial committees; therefore, they are not to be presented as refereed publications. Citation of this work should state that it is from an ASABE meeting paper. EXAMPLE: Author's Last Name, Initials. 2007. Title of Presentation. ASABE Paper No. 07xxxx. St. Joseph, Mich.: ASABE. For information about securing permission to reprint or reproduce a technical presentation, please contact ASABE at [email protected] or 269-429-0300 (2950 Niles Road, St. Joseph, MI 49085-9659 USA).
IntroductionIn greenhouse control systems, control error often conficts with energy consumption. Low control error generally means high energy consumption. In the past ten years, we have been studying the greenhouse environment control problem and have gained a considerable understanding of greenhouse dynamics. In a greenhouse, we must keep the temperature and humidity in certain range that is suitable for the plants. However, we are simultaneously required to minimize energy consumption to reduce the cost. The control means include ventilation, heating and spraying, of which heating and spraying are high-energy-consumption methods. In winter, we can improve the temperature by heating and decrease the humidity by heating and ventilating. With the traditional control strategy, we could maintain the temperature and humidity at a very precise point, but the high energy consumption and expensive cost of this strategy would make the greenhouse unprofitable, which implies that this control strategy would not to be chosen by any users.
To illustrate the MOCC algotithm, we firstly abstract the normal greenhouse multi-objective model as follows:
(1a)
(1b)
where denotes the plant states, and are the control outputs and inputs, subject to constraints . The aim of control is making . We aim to design controller
（2）and to minimize two conflicting objectives: control precision, , and control energy consumption, , defined as:
（3a） （3b）
where is the control horizon and are initial states values.
Then the multi-objective control problem (1) － (3) can be translated into the following multi-objective optimization problem:
,
1）
2
2）3）4）
In greenhouse control system, control error and control energy consumption always lie within an acceptable range; here we denote by and the maximum acceptable values—which are
called the practical objective constraint condition . Note that and are only the worst
acceptable values and not our control objectives. The worst acceptable values and only ensure that the plants survive, but not flourish; our aim is to construct a suitable environment for the plants to grow and not only one in which they can survive.
There have existed two main traditional methods to solve the above-mentioned multi-objective problem. One is the trade-off weight (Masaaki (1997)) method, which translates this multi-objective problem into a single-objective one with a trade-off weight (Masaaki (1997), Rangan(1997), Eisenhart(2003), Iskander(2003), Zhao(2003)). The major advantage of this method is that the translated single-objective control problem is very easy to deal with, but the disadvantage is that the single objective optimization isn’t equivalent to the original multi-objective control problem. The control result will be strongly associated with the trade-off weight chosen, and the controlled objective may not be satisfiable if we provide a bad weight. In addition, the selection of weights is very difficult for users and engineers in the practical problem. Another approach is the constraints method, which optimizes the most important control objective and translates the others into system constraints (Scherer(1995), Scherer(1997), Sznaier(2000)). For example, we can regard（3b）as the performance index and regard （ 3a ）as a constraint. The advantage of this approach is to satisfy all controlled objectives through constraints; however, the constraint bounds are very difficult for users or control engineers to determine suitably in a practical problem. Bounds that are too tight may bar the existence of a feasible solution for the optimization problem, while too loose bounds may make the optimization problem lose practical significance.
Multi-objective Compatible Control (MOCC) StrategySince a traditional multi-objective control method cannot ensure the existence of a feasible controller in advance, we have adopted a multi-objective coordinated control system in the greenhouse. When these objectives conflict with each other, it is impractical to fix all the objectives at some given optimal “points”. To ensure the existence of a feasible controller, we are willing to “back off” on our desire that all the controlled objectives are at their optimal values and relax these “point” controlled objectives to some suboptimal “interval” ones or to some “region” ones generally—we call them “compatible objective region”. For example, in the summer night, we regulate the greenhouse temperature objective to instead of exactly at and the humidity objective to instead of exactly . According to the experts, this greenhouse environment is also suitable for the plants. Then we design a controller by optimizing the energy consumption objective. This compatible control system can obtain better economic benefit than before and has gained considerable attention from users (Junhui(2003)).
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Based on the successful application, we have generalized a common compatible control theory framework (Lihong (2006)) from the practical experience obtained with the compatible objective region used in the greenhouse. We call this method “multi-objective compatible control,” as shown in Fig. 1, and it has a two-layer framework. We will first sketch the main ideas. In this paper, the uncertainty is all reflected as uncertainty of initial conditions, as will be described more fully in Section 3.
The first layer is compatible optimization and has the following two requirements:
1. Obtain a compatible (multi-dimensional) controlled objective region,
2. The compatible controlled objective region must meet certain optimality requirements (we adopt Pareto-optimal here) and meet users’ requirements;
The second layer is the compatible control layer and is devoted to satisfying the following requirements:
1. Design a real-time controller to control the system to remain within the (multi-dimensional) objective region determined in the first layer,
2. Optimize further the objective that is most critical to the user to optimize, rather than simply to keep within a specified region.
Fig.1. Two-layer compatible control framework Fig.2. Space with two conflicting objectives
In Fig. 2, the shaded area in rectangle AM is the space of objectives in which control solutions exist; the cross-hatched area in rectangle AL is the area that meets the constraints of the practical problem; we shall call it the objective region with feasible control solutions; the bold curve is the Pareto optimal front of the objective space with control solutions.
The “optimal point” A is not in the objective space with control solutions when the objectives conflict. So A is not an optimal point objective that can be attained. To guarantee the existence of a solution, the point objective A will be expanded to a region objective AB, where AB is a rectangular region. To ensure the existence of a solution (i.e., a compatible solution), B must be in the region that includes a feasible control solution (the cross-hatched area in Fig. 2). Now the problem is where B should be placed to allow meeting the users’ needs.
Since the selection of B would ideally optimize certain of the users’ requirements, B should be a point on the Pareto optimal front, and included in rectangle AL (the bold Pareto front in Fig. 2),
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in order not to be dominated by a better choice of B. However, it is difficult in a practical problem to determine the theoretical Pareto optimal front, so a practical B must be chosen to approach some point of the theoretical Pareto optimal front as closely as possible.
To determine the position of B, we must have two steps in the first layer algorithm: the first step is find the Pareto front of the objective space with control solutions—that is, to find multiple, uniformly distributed points on (or approximating) the Pareto front; the second step is, according to the requirements of the users, to select one point B on the Pareto front that best defines the users’ objective region (that is, the users’ desired region for keeping the objectives within).
The second layer aims to design a compatible control system to realize the compatible multiple objectives from the first layer. In the controller design, we will not only realize these interval objectives, but also further optimize the objective that users are most concerned to minimize.
The discussion above sketches the main ideas of our compatible control methodology. The detailed algorithm will be introduced in the next section.
Energy-saving Multi-objective Compatible Control AlgorithmSupposing the system model to be precise, an open-loop control method is adopted here. To conveniently illustrate our compatible control algorithm, we take a linear discrete-time system as our controlled model
(4)
where denotes the plant states, and are the control outputs and inputs with constraints . Because of different practical situations, the initial conditions may be different. We suppose that the initial conditions lie in some given compact set, i.e., , and the state-space matrices are
, ,
We shall denote this kind of control problem with uncertain initial states as the I.C.-( ) problem (namely, control under uncertain initial state).The control horizon is set and . We aim to minimize the following two control performance indexes (control error, , and energy consumption, ).
, (5)
Assumption1 The largest objective space (see M in Fig.2) is , .
Assumption2 The control error, , and energy consumption, of the practical problem have the constraints (see L in Fig.2)
, (6a)
If we normalize and in the intervals and , then the constraints are translated into the following
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, (6b)
To reduce computation and ensure control performance, we enforce
(7)
Since the control precision, , conflicts with the energy consumption, , it is very difficult or even impossible for us to achieve the optimal control objectives simultaneously.
The following subsection will show how to control the plant (4) for the I.C.-( ) problem using our two-layer compatible control algorithm.
The first layer —the compatible optimization layer
The aim of the compatible optimization layer is to find a compatible and relatively optimal objective region. To achieve this, first we should have a method to compare points in the multi-objective space, judging them to be better or worse, or sometimes, neither. It is easy to compare points in a single objective problem. However, it is not so direct in multi-objective problems. In this paper, we adopt Pareto non-domination as the comparison method. To introduce the Pareto optimization definition, we first present some background information.
A. PreliminaryWe illustrate the domination definition with Fig. 3. Solution 1 dominates solution 2 because solution 1 is less than solution 2 in both and . Solution 2 doesn’t dominate solution 3, because solution 2 is less than solution 3 only in , while solution 3 is less than solution 2 in . By the non-dominated definition, the six individuals are divided into three ranks—all non-dominated individuals are in rank 1; then they are removed and all individuals now not dominated are in rank 2, etc. For the example, ,
, .
Fig.3. Three fronts according to non-domination Fig.4. The crowding distance calculation
relationship
The next step will be to seek a Pareto-optimal set—i.e., a set composed of many non-dominated solutions. These solutions must have the following characteristics:
1. Solutions are as close to the actual (ideal, but unknown) Pareto-optimal front as possible;
2. Solutions are as uniformly distributed as possible along the front.
It is very difficult to obtain these Pareto solutions by traditional optimization method. Genetic algorithms (GA) or genetic programming (GP) are evolutionary computing methods that have been widely applied. ( Firpi(2005)). Since 1990, multi-objective optimization methods based on
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evolutionary computation have received intense development (Kalyanmoy(2002)) especially the multi-objective genetic algorithms (MOGA). Many good methods have resulted, such as TDGA(Kita(1996)) 、 PAES(Knowles(2000)) 、 NSGA(Goldberg(1990)) 、 and NSGA-II(De b(2000)). They appear to be very promising ways to approximate Pareto fronts. NSGA-II is an excellent MOGA method. With the help of NSGA-II, we can get multiple, well-distributed points approximating the Pareto front.
To get an estimation of the density of solutions surrounding a particular point in the population, it calculates the average distance from the two points on either side of this point along each of the objectives. This quantity, , serves as an estimate of the size of the largest cuboids enclosing point without including any other point in the population (Deb calls this the crowding distance). In Fig. 4, the crowding distance of the solution in its front is the average side-length of the cuboids (shown with a box).
Assuming that every individual in the population has two attributes:1. Non-domination rank ( )2. Local crowding distance ( )
The non-dominated sorting principle used to select solutions is:
If ( ) or (( ) and )
Then will be selected over .
B. Pareto-Band of the I.C.-( ) problem
The initial state of the multi-objective energy-saving problem I.C.-( ), defined by (4)-(7), is uncertain. First, we shall examine the characteristics of the Pareto front of the I.C.-( ) problem.
According to system equation (4), we have
……
,
（8）So objectives (5) can be expressed as
（9）So the objectives (9) are functions of the variables and the initial state parameters .
For illustration, we choose five arbitrary initial states in the initial state region : [10.9, 10.8], [10.8, 10.9], [10.7, 11], [11, 10], [11, 9.5]; for each, we calculate and graph the corresponding Pareto front by optimizing (9) using the NSGA-II algorithm (see Fig.5).
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If we add the curves of Fig.5 to Fig.2, we obtain Fig.6. From this we can observe that for every fixed initial point , there is a corresponding Pareto front in the shaded area of Fig.2. So across all the initial points in , we obtain a broad band of Pareto fronts of the shaded area of Fig.2 (see Fig.6), which we shall call the Pareto band.
60 70 80 90 100 110 1200
2
4
6
8
h1
h 2
1----interval initial region2---[10.9,10.8]3-----[10.8,10.9]4---[10.7,11]5----[11,10]6----[11,9.5]
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5
6
Fig.5 Pareto fronts for some example fixed x(0)’s, namely, Fig.6. Illumination of the
[11,10],[11,9.5], [10.7,11],[10.8,10.9] and [10.9,10.8]. Pareto band
Definition 2.2 (Pareto band of I.C.-( ) problem)
In the I.C.-( ) problem, the band–shaped Pareto front corresponding with initial state region is called the Pareto band of the I.C.-( ) problem.
C. Robust multi-objective compatible optimization algorithm in the first layer
To assure that the control system is robust for the uncertain initial state in the I.C.-( ) problem, where should B be placed? We analyze as follows.
To assure the robustness for the uncertain initial state in the second layer controller design, the optimal and compatible rectangular objective region AB, which was got in the first layer and would be taken as the control objective in the second layer, should be the largest and most conservative of all the rectangles of objectives corresponding to ANY initial state in , so B should lie on the upper-right-hand boundary of the Pareto band of the I.C.-( ) problem.
The further problem is how to acquire the upper-right boundary of the Pareto band of the I.C.-( ) problem.
Further analysis reveals that the boundary can be acquired approximately by calculating the Pareto front of the following multi-objective problem
， （11）
This conclusion is justified as follows:
Assume there is some point T, in a Pareto front that corresponds with an initial state in , and that point T is located to the upper-right of curve , then according to Definition 3.1, there is at least one point Q in curve that makes the objective functions satisfy
，
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But in calculation of the Pareto front of the multi-objective problem （11）, we have taken
，as the fitness function, where includes the entire range of and . So
there exists no such point such that ， , as it would conflict with the assumption.
The multi-objective problem （ 11 ） is a max-min optimization problem. That is, in each generation of the multi-objective GA, we optimize (minimize) the maxima of the objective functions and over region so as to guarantee that the result is effective for all the initial points in region (although that is certainly conservative). The advantage of this approach is that it assures the existence of the controller despite the uncertainty of the initial state. The disadvantage is that the objective region AB (see Fig. 6) is made somewhat larger because of the conservative condition imposed.
Based on the above, and combined with the non-dominated sorting principle of NSGA-II, we get the computational results as a curve are shown in Fig.7.
The result shows that solutions converge to one Pareto front curve—the upper-right-hand boundary of the Pareto band associated with initial region —and the result is excellent. To make it convenient to compare with the Pareto fronts for some fixed , we also show it as curve 1 in Fig.5. There we can see that it is indeed the upper-right-hand boundary of the Pareto band of the I.C.-( ) problem.
From the practical constraints (6): ， , we learn the permissible maximum point L(0.66,0.32) and get a rectangle OL (see Fig. 7), where the point O(0,0) is the coordinate origin of the objective space. The objective region with feasible solutions is included in rectangle OL (seen in Fig.2). So users can select one point, assumed to be , from the part of the computational results curve in rectangle OL according to the practical requirement.
In the problem of this paper, we can use a rectangular region OB instead of AB because AB is included in OB (see Fig. 2). So it is important only how to determine point B.
At this time we have determined a compatible and Pareto optimal rectangular objective region OB.
Fig.7 The upper-right-hand boundary of the Pareto Fig.8 The control result of the second
band and the user’s selection of interval objective layer controller
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The second layer-- compatible control system design
The second layer aims to design multi-objective compatible control system. Assuming that we now take energy consumption as the objective that users are most concerned to minimize, and that we make the interval objective from the first layer into a constraint, we now do constrained, single-objective optimization of subject to the constraint. The online multi-objective compatible control algorithm is now defined as follows:
In order to show that the performance for the primary controlled objective (i.e., energy consumption) has been improved in the second layer design, the control result of the first layer controller at the same point B is shown in Fig. 9, and the difference of objective between the controllers of the first and second layers is quite apparent in Fig. 10.
From Fig. 10, compared with the controller obtained in the first layer at B, the energy consumption with control input obtained in the second layer has decreased from 2.338 to 1.5651. It indicates that our method not only ensured the robustness of the system but also obviously reduced energy consumption.
Fig.9 The control result of the first Fig. 10 Compare of objective between the
layer(to point B in fig.7) controllers of the first and second layer
Comparison with Weighted Trade off Method and -constraintsThe weighted tradeoff method combines the two control objectives with weights , and then optimizes the following comprehensive control objective with plant model (4)
， (12)
Here the initial conditions are , . Table I shows the calculation results with different weights.
Table 1. Results calculated with different weightsTrial numble Tradeoff Weight Objective Objective
1 0.33:0.67 0.7724 0.00842 0.5:0.5 0.7419 0.03153 0.67:0.33 0.6927 0.10404 0.75:0.25 0.6547 0.19825 0.8:0.2 0.6420 0.30516 0.83:0.17 0.6000 0.4123
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From Table I, we see the following results.1. Except for the 5th weight pair, there are no other pairs satisfying performance constraint (6).
2. The 4th and 5th pairs show that a small change of weights will lead to large differences in control performance, which indirectly illustrates that selecting a suitable weight in advance is very difficult.
3. In addition, when the optimization problem (12) is non-convex, many Pareto optimal solutions can’t be found by the weighted tradeoff method, which implies that it is very difficult or even impossible for users or control engineers to select suitable weights.
ConclusionsIn this paper we propose a new, two-layer energy-saving multi-objective compatible control (MOCC) algorithm. The first layer first obtains the Pareto front of the objective space with control solutions, and then one desired point B of the Pareto front is selected by users according to practical requirements. This yields a compatible and Pareto-optimal rectangular objective region OB. In the second layer, we first construct a performance index based on the most important control objective, which is subjected to the constraint of rectangular objective region OB from the first layer,and then optimize it to get the multi-objective compatible controller.
Simulations illustrate that our two-layer compatible control algorithm has the following advantages: For a conflicting multi-objective control problem, the first layer can effectively resolve the
objectives conflict problem and obtain an achievable “relatively-optimal” compatible controlled objective region.
It ensures the existence of an achievable controller before designing the control system.
The control system can meet users’ requirements because the system performance constraints are from the control objective region in the first layer, which can readily be selected by users according to practical requirements.
The compatible control system design method in the second layer not only ensures the achievability of all objectives, but also makes the most important objective (the energy consumption) better. Thus the aim of energy-saving control has been realized.
Realizes robust control for a precise plant model with uncertain initial conditions.
Compared with the weighted tradeoff method, the compatible control algorithm avoids the significant drawback of non-existence of an achievable controller, which is faced by the traditional approaches, and makes the whole control system more easily realized and able to meet users’ practical requirements.
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