An Uncertain Control Framework of Cloud Model Baohua Cao1 , Deyi Li2 , Kun Qin3 , Guisheng Chen2 , Yuchao Liu4 , and Peng Han5 1 Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA 2 The Institute of Beijing Electronic System Engineering, Beijing, China 3 School of Remote Sensing Information Engineering, Wuhan University, China 4 Department of Computer Science and Technology, Tsinghua University, China 5 Chongqing Academy of Science and Technology, ChongQing, China
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Abstract. The mathematical representation of a concept with uncertainty is one of the foundations of Artificial Intelligence. Uncertain Control has been the core in VSC systems and nonlinear control systems, as the representation of Uncertainty is required. Cloud Model represents the uncertainty with expectation Ex, entropy En and Hyper-entropy He by combining Fuzziness and Randomness together. Randomness and fuzziness make uncertain control be a difficult problem, hence we propose an uncertain control framework of Cloud Model called UCF-CM to solve it. UCF-CM tunes the parameters of Ex, En and He with Cloud, Cloud Controller and Cloud Adapter to generate self-adaptive control in dealing with uncertainties. Finally, an experience of a representative application with UCF-CM is implemented by controlling the growing process of artificial plants to verify the validity and feasibility. Keywords: Uncertainty, Uncertain Control, Cloud Model, UCF-CM
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Introduction
Uncertain control has been a solid challenge in the domain of Artificial Intelligence and also been the core issue in VSC (variable structure control) systems and nonlinear control systems [1]. VSC design trends to use parameters tuning with a variable structure control to improve the control system against disturbances and uncertainties. However, in VSC, the lack of a priori knowledge on the target control signal leads the designer to seek for alternative methods predicting the error on the control [2, 3]. Nonlinear control system alters its parameters to adapt to a changing environment. The changes in environment can represent variations in process dynamics or changes in the characteristics of the disturbances. There are, however, many situations where the changes in process
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An Uncertain Control Framework of Cloud Model
dynamics are so large that a constant linear feedback controller will not work satisfactorily. Cloud model is proposed by Deyi Li in 1995 to represent the uncertainty transition between qualitative concept and quantitative description [4]. Cloud Model has three digital characteristics: Expected value (Ex), Entropy (En) and Hyper-Entropy (He), which well integrate the fuzziness and randomness of spatial concepts in unified way. In the discourse universe, Ex is the position corresponding to the center of the cloud gravity, whose elements are fully compatible with the spatial linguistic concept; En is a measure of the concept coverage, i.e. a measure of the spatial fuzziness, which indicates how many elements could be accepted to the spatial linguistic concept; He is a measure of the dispersion on the cloud drops, which can also be considered as the entropy of En. With Cloud Model, an uncertain control framework called UCF-CM is proposed by us. UCF-CM is a framework to handle specific uncertain control problems with parameters tuning of Ex, En and He defined in Cloud Model. To verify its validity and feasibility, a representative application of UCF-CM is implemented. Controlling the growing process of artificial plants is a challenging topic as it is too difficult to take effective control in artificial plant’ generating course for the uncertain randomness and fuzziness during its variation process. Simulating the natural growing process of artificial plants has always been the challenge in Artificial Life, the challengeable domain of Artificial Intelligence. UCF-CM is used to have a try to control the uncertain variation of artificial plants to show its validity and feasibility. The paper is organized as follows. The next section introduces Cloud Model’s definition and its three parameters. Section 3 proposes the UCF-CM framework and section 4 details the application study. The last section summarizes our discovery and presents future research goals.
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Cloud Model
As a result of random, fuzziness, incompleteness and disagreement of description, uncertainty study has been the core in artificial intelligence [4, 5]. Cloud model theory is proposed to research the uncertainty [4] based on three parameters: Ex, En and He. Membership cloud employs expectation Ex, entropy En and hyperentropy He to describe a specific concept. Ex is the expectation. En represents a granularity and it reflects the range of domain space. He describes the uncertain measurement of entropy. It can be used to express the relationship between randomness and fuzziness [5]. 2.1
Definition
Definition 1: Membership Cloud. Let U denote a quantitative domain composed by precise numerical variables; C is a qualitative concept on U. If the quantitative value x ∈ U is a random realization of qualitative concept C, then the x ’s
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confirmation on C can be denoted µ(x) ∈ [0, 1] , which is a random number with stable tendency. µ : U → [0, 1], x ∈ U, x → µ(x) The distribution of x on U is called Cloud, x is called a Cloud Droplet. The cloud is from a series of cloud drops. In the rocess of the formation of clouds, a cloud droplet is a realization of qualitative concept through numeric measurement. Definition 2 [4,5]: Normal Cloud. Let U denote a quantitative domain composed of precise numerical variables; C is a qualitative concept on U. If the quantitative value x ∈ U is a random realization of qualitative concept C, x ∼ N (Ex, En′2 ), En′ ∼ N (En, He2 ), and the certainty degree of x on C is 2 µ = exp(− (x−Ex) 2En′2 ) The distribution of x on U is called Normal Cloud. 2.2
Algorithm
Given three digital characteristics Ex, En, and He, a set of cloud drops could be generated by the following algorithm [6,7]: Algorithm: Cloud (Ex, En, He, N) Input: the expectation of cloud Ex, the entropy of cloud En, the hyper entropy of cloud He, the number of drops N. Output: a normal cloud with digital characteristics Ex, En, and He. step1: Generate a normal random number En’ with the expectation En and stand deviation He, En′ ∼ N (En, He2 ). step2: Generate a normal random number xi with the expectation Ex and stand deviation En’, x ∼ N (Ex, En′2 ). step3: Compute the µi of the cloud droplet xi . step4: Generate cloud droplet (xi , µi ) step5: Repeat step 1-4, until i = N. Figure 1 shows some samples of cloud generated by the algorithm.
Fig. 1. Clouds
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An Uncertain Control Framework of Cloud Model
UCF-CM
With Cloud Model, an uncertain control framework is designed for handling uncertain control problems. Figure 2 shows the design of UCF-CM.
Fig. 2. UCF-CM : An Uncertain Control Framework of Cloud Model
UCF-CM is composed of six parts: input, cloud, variable, cloud controller, cloud adapter, output. Here is the list of description for the six modules. Input: Initialized values of the three parameters of Cloud Model are required for the input. Besides the parameters, pre-defined rules for uncertain control are also required. They could be taken as part of the input. Cloud: With the input, Cloud module will calculate a new variable with Ex, En, and He. The algorithm of Cloud Model is used for the generation. In this model, the change of He can affect the value of En, while the change of En can affect the value of variable. Variable: Variable deposits the variable generated by cloud model and delivers it out as the input of cloud controller. Cloud Controller: Cloud controller is composed of three components: ruler, evaluator and controller. Ruler provides kinds of pre-defined rules for evaluator
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and controller. Some rules are used to evaluate the validity of the newly generated variable while others are used to control the adaptive change of parameters of the cloud model. Evaluator executes the evaluation for the variable to judge the situation that whether the variable meets the requirement and provide guidance for the controller. Controller implements the adjustment operations to control the change of the parameters. Cloud Adapter: Cloud adapter takes the actions to change the three parameters defined in Cloud Model. Three adapters are included for Ex, En, and He separately. Cloud adapter calls the cloud model to execute the next calculation. Output: Finally, the variable generated by the cloud model, which meets the evaluated conditions and needs no further control actions, is on the output for further usage. In the overview of UCF-CM, three parts are the core: Cloud, Cloud Controller, and Cloud Adapter. Cloud calculates the value of a variable, Cloud Controller provides decisions for the change of controlled parameters, and Cloud Adapter implements the change. Meanwhile, the UCF-CM can also support a repeatable control to enable the output of X injects into the input. With this support, circle control can be implemented automatically by the system. Ruler in the Cloud Control decides the breakout. The extended framework can be illustrated by the following experiment. With more flexible and accurate control rules, UCF-CM can make more strong uncertain control on the variables. Furthermore, self-learning mechanism can be integrated into the UCF-CM for a stronger and unbelievable ability for uncertain control. This topic of research can be pioneered in future.
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Experimentation
To verify the validity and feasibility of UCF-CM, an experience with a representative application of controlling the growing process of artificial plants is designed and implemented. As much randomness and fuzziness exist in the whole course, although there are series of algorithms in generating artificial plants based on fractal technology at present, such as L-System, iterative function system, interpolates model, etc [8-11], no available solution could be claimed to enable the system to control the growing process of artificial plants. Generation of artificial plants is always implemented by fractal technology. Fractal is a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole[8,9]. This recursive nature is obvious in these examples-a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature [10,11]. To generate a simple artificial plant, at least four parameters are required: [α, β, sl, sr]. Here α, β, sl, sr separately stand for left partial angle, right partial angle, left rate of branch length, right rate of branch length. With the four parameters, a normal and simple fractal tree can be generated by computer with the fractal algorithm: With UCF-CM framework for uncertain
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An Uncertain Control Framework of Cloud Model
control and the fractal algorithm, an experiment is implemented for controlling the uncertain growing process of an artificial tree. Here is the updated UCF-CM for this experiment.
Fig. 3. UCF-CM for controlling the growing process of Artificial Trees
With the updated UCF-CM for artificial tree, series of detailed steps for this experiment is designed as follows: 1. Locate the starting generation point as P0(0,0), define the original trunk length L, get the coordinate P0(0,L), and draw the trunk. 2. Define the tree depth M and initialize values of Ex = [α, β, sl, sr], En = [Enα , Enβ , Ens l, Ens r], He = [Heα , Heβ , Hes l, Hes r]; 3. Define pre-rules for evaluator and controller. In this experiment, here five attributes of a tree are considered and designed: symmetry(S), tightness (T), hierarchy (H), dense (D) and ornament (O). S = 1 − 0.5( α−β π/2 + |sl − sr|) T =1− H=
α+β π 1 − |sl−0.618|+|sr−0.618| 2 BestDeep−M 0.5(1 − sl+sr 2 ) + 0.5( BestDeep ),BestDeep=12
D= O = 0.1S + 0.1(1 − |T − 0.618|) + 0.35H + 0.45D
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Once generated the parameter [α, β, sl, sr], the values of S,T,H,D,O can be calculated according to the pre-defined formulas. The expected values of the five attributes can be defined for the evaluation. Meanwhile, series of rules for evaluation of the branch can be defined. There are always compound rules which include many simple rules. 4. Calculate the values of S, T, H, D, O with the generated variables of [α, β, sl, sr]. 5. Control the change of variables of [α, β, sl, sr] with the evaluator and controller. If all the rules of the evaluation are satisfied, the variables of [α, β, sl, sr] will be finalized for generating a branch of the artificial tree. Otherwise, the rules of the controller will generate decisions for the change of the variables with the control of cloud model. 6. Generate a branch of the tree with [α, β, sl, sr], or keep tailoring the parameters with the cloud model until the output. 7. Repeat the steps from 4-6 until the depth of the tree satisfies the pre-defined value, or the time is over. Finally, an artificial tree will be generated when the whole process is completed. Through the relationship between Expected value (Ex), Entropy (En), Hyper-Entropy (He) and the fractal characteristic, the connection and difference are shown while controlling the growing process of the fractal course. An expected fractal tree will be generated under the UCF-CM control. With the UCF-CM control on the fractal growing process, various of artificial trees can be generated. Here are some samples of the result.
Fig. 4. Experiment result of UCF-CM
The trend of tree variation can be controlled by man and also can be mapped to a reference, such as the sun. Now suppose that the sun rises in the left of tree, and sinks in the right of the tree. The trend of the variation can be mapped to the trend of the sun’s situation as the figure shows.
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An Uncertain Control Framework of Cloud Model
Fig. 5. Uncertain control of UCF-CM on Artificial Plants
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Conclusion
Randomness and fuzziness make uncertain control be a difficult problem; therefore, we propose an uncertain control framework of Cloud Model called UCF-CM. UCF-CM tunes the parameters of Ex, En and He with Cloud, Cloud Controller and Cloud Adapter to generate self-adaptive control in dealing with uncertainties. Finally, an experience with a representative application of UCF-CM is implemented to verify its validity. Furthermore, integrating self-learning mechanism to the UCF-CM is on the schedule for next steps.
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