Methods of Computational Intelligence for Nonlinear Control Systems

Transcription

1 ICCAS June -, KINTEX, Geonggi-Do, Korea Methods o Computational Intelligence or Nonlinear Control Sstems Bogdan M. Wilamowski Auburn Universit, Alabama Microelectronic, Science and Technolog Center, Broun Hall, (Tel : ; wilam@ieee.org) Abstract: Conventional controllers like PID and man advanced control methods are useul to control linear processes. In practice, most processes are nonlinear. Nonlinear control is one o the biggest challenges in modern control theor. While linear control sstem theor has been well developed, it is the nonlinear control problems that cause most headaches. Nonlinear processes are diicult to control because there can be so man variations o the nonlinear behavior. Traditionall, a nonlinear process has to be linearized irst beore an automatic controller can be eectivel applied. This is tpicall achieved b adding a reverse nonlinear unction to compensate or the nonlinear behavior so the overall process input-put relationship becomes somewhat linear. The issue becomes more complicated i a nonlinear characteristic o the sstem changes with time and there is a need or an adaptive change o the nonlinear behavior. These adaptive sstems are best handled with methods o computational intelligence such as neural networks and uzz sstems. The problem is that development o neural or uzz sstems are not trivial. This presentation will ocus on several methods o developing close to optimal architectures and on inding eicient learning algorithms. The problem becomes even more comple i the methods o computational intelligence have to be implemented in hardware. Kewords: Computational intelligence, neural networks, uzz sstems.. INTRODUCTION It is relativel eas to control linear sstems. Unortunatel, in practice most sstems are nonlinear. Some o them can be linearized and use well developed linear control theor, but in man cases a special nonlinear control sstem has to be developed. Nonlinear control is one o the biggest challenges in modern control theor. Traditionall, a nonlinear process has to be linearized irst beore an automatic controller can be eectivel applied. This is tpicall achieved b adding a reverse nonlinear unction to compensate or the nonlinear behavior so the overall process o the input-put relationship becomes somewhat linear. In man cases nonlinear characteristics o the sstem change with time and there is a need or an adaptive change o the nonlinear behavior. These adaptive sstems are best handled with neural networks and uzz sstems [][]. In this presentation various uzz and neural network architectures will be studied and compared. An dnamic nonlinear sstem can be described b the set o nonlinear state equations: = = (,, Ln,,, L n) (,, L,,, L ) L n (,, L,,, L ) = n n n n () Implementation o analog integrators on silicon chips is relativel simple. Nonlinear terms with multiple inputs are more diicult to implement. These nonlinear blocks can be developed as universal elements using neural networks or uzz sstems (Fig. ). In both cases, uzz and neural sstems, these nonlinear terms can work in analog mode while the can be digitall tuned. In the case o neural networks onl weights need to be digitall controlled. In the case o the uzz sstems parameters o uzziiers and deuziiers, the n have to be digitall adjusted. In both cases signals can alwas be in analog orm. This analog tpe o signal processing is especiall important in sstems where large signal latenc is not acceptable. n Neural Network or Fuzz Sstem ( t) ( t) ( t) Fig.. Block diagram o nonlinear dnamic sstem using neural networks or uzz sstems.. FUZZY SYSTEMS There are two most commonl used approaches or development o Fuzz sstems. Fig. shows architecture proposed b Mamdani [] and Fig. shows architecture proposed b Takagi, Sugeno, and Kang [][]. There were also attempts to present a uzz sstem in a orm o neural network. One o such networks is shown in Fig. [6]. Note that onl architecture resembles neural networks because cells there perorm dierent unctions than neurons, such as signal multiplication or division. X Y MIN operators MAX operators Deuzziier Fuzz rules Fig.. Block diagram o a Mamdani tpe uzz controller. n P-

2 ICCAS X Rule selection cells min operations Normalization weighted sum June -, KINTEX, Geonggi-Do, Korea Let us assume that the required nonlinear unction has a shape as shown in Fig.. and let us compare results obtained with dierent uzziication methods used assuming Mamdani tpe o architecture. Results are shown in Fig. 6 to 8 and Fig.. Y. Fig.. TSK (Takagi-Sugeno-Kang) uzz architecture. -. X uzziication multiplication Π Π sum all weights equal epected values division - Π Fig. 6. Results or Mamdani architecture with uzziiers with triangular membership unctions. z Π all weights equal. Fig.. Fuzz neural networks [6]. Fuzziication In uzz sstems, at irst all analog inputs are converted b uzziiers into sets o uzz variables. For each analog input, several uzz variables, with values between and, are generated. Various tpes o uzziication methods can be used as triangular, trapezoidal, or Gausian. One ma consider miing these techniques. The simplest implementation is with triangular membership unctions and in most practical cases acceptable results are obtained with this simplest approach. For higher accurac, more membership unctions should be used. However, ver dense unctions can lead to requent controller action (also known as hunting ), and sometimes this ma lead to sstem instabilit Fig 7. Results or Mamdani architecture with uzziiers with trapezoidal membership unctions Fig. 8. Results or Mamdani architecture with uzziiers with Gaussian membership unctions... Rule evaluation with uzz logic Fig.. Desired nonlinear control surace Once inputs are converted to uzz variables these variables are processed b uzz logic blocks with MIN and P-

3 ICCAS MAX operators. The uzz logic is similar to Boolean logic but instead o AND operators, MIN operators are used and in place o OR operators, MAX operators are implemented. Interestingl uzz logic has a more general nature and it works equall well as Boolean logic. Fig. 9 shows uzz logic operations on zero-one Boolean variables (Fig. 9) and on uzz variables (Fig. 9). MIN MIN A B A B....7 MAX A B MAX A B union intersection Fig. 9. Comparison o Boolean and Fuzz logic. The Mamdani [] concept (see Fig. ) ollows the rule o ROM and PLA digital structures where AND operators are selecting speciied addresses and then OR operators are used to ind the put bits rom the inormation stored at these addresses. In the case o the uzz sstems AND and OR operators are replaced b MIN and MAX operators respectivel. More recentl Mamdani architecture was replaced b TSK (Takagi, Sugeno, Kang) [][] architecture where the deuzziication block was replaced with normalization and weighted average. The TSK structure, as shown in Fig., also does not require MAX operators, but a weighted average is applied directl to regions selected b MIN operators. What makes the TSK sstem reall simple is that the put weights are proportional to the average unction values at the selected regions b MIN operators. June -, KINTEX, Geonggi-Do, Korea control surace. Fig. shows the surace obtained with product encoding, which is smoother than the surace o Fig. 6 which is obtained with a MIN encoding. Eactl the same surace can be obtained with TSK architecture (Fig. ) when product encoding is used instead o MIN operators Fig. Results or Mamdani architecture with uzziiers with triangular membership unctions and product encoding instead MIN encoding Fig. Results or uzz neural network architecture o Fig... Deuzziication Fig.. Results or TSK architecture with uzziiers with trapezoidal membership unctions. When uzz neural networks are used (Fig.) MIN operators are replaced b product operators (signals are multiplied). Fuzz sstems with product encoding are more diicult to implement but the generate a slightl smoother As a result o uzz rules o Fig. a new set o uzz variables is generated, which later has to be converted to an analog put value. The rightmost block o the diagram represents deuzziication, where the put analog variable is retrieved rom a set o put uzz variables. Several more or less complicated deuzziication schemes are used. The most common is the centroid tpe o deuzziication. There were man attempts to urther improve uzz controllers b replacing uzziiers and MIN operators b other weighted sum approaches and RBF (Radial Base Function) networks [6]. These areas o research are known as uzz-neuro sstems and the resulting architectures are more close to neural networks than to uzz sstems.. NEURAL NETWORKS A single neuron can divide input space b line, plane, or hperplane, depending on the problem dimensionalit. In order to select just one region in n-dimensional input space, more than n+ neurons should be used. For eample, to P-

4 ICCAS separate a rectangular pattern, neurons are required, as is shown in Fig.. I more input clusters should be selected then the number o neurons in the hidden laer should be properl multiplied. I the number o neurons in the hidden laer is not limited, then all classiication problems can be solved using the three laer network. With the concept shown on Fig. uzziiers and MIN operators used or region selection can be replaced b simple neural network architecture. Let us analze Fig. where a two-dimensional input space was divided b si neurons horizontall and b si neurons verticall. The corresponding neural network is shown in Fig.. Each neuron is connected onl to one input. For each neuron input, weight is equal to + and the threshold is equal to the value o the crossing point on the or ais. Neurons in the second laers have two connections to lower boundar neurons with weights o + and two connections to upper boundar neurons with weights o -. Thresholds or all these neurons in the second laer are set to. Onl three o them are drawn on Fig.. > > > >. all weights equal to + > > > > Fig.. Separation o the rectangular area on a two dimensional input space and desired neural network to ulill this task. u v w z e d c b u, v w z a C Fig.. Two-dimensional input plane separated verticall and horizontall b si neurons in each direction. Weights in the last laer have values corresponding to the epected unction values in selected areas. All neurons in Fig. have a unipolar activation unction and i the sstem is properl designed, then or an input vector in certain areas onl the neuron o this area produces + while all remaining neurons have zero values. In the case o when the input vector is close to a boundar between two or more regions, then all participating neurons are producing ractional values A + + B AND e d c b a June -, KINTEX, Geonggi-Do, Korea and the sstem put is generated as a weighted sum. For proper operation it is important that the sum o all puts o the second laer must be equal to +. In order to assure the above condition, an additional normalization block can be introduced, in a similar wa as it is done in TSK uzz sstems as shown in Fig.. all weights equal thresholds are set b values a to z a b c d e u v w z all thresholds are equal to + - A B C weights are equal to the average o epected value in the selected region Fig.. Simple neural networks perorming the unction o TSK uzz sstem. It was shown above that a simple neural network o Fig. can replace a uzz sstem. All parameters o this network are directl derived rom requirements speciied or a uzz sstem and there is no need or a training process. The most commonl used learning algorithm such as EBP Error Back Propagation or LM - Levenberg-Marquar, were developed or laer b laer tpe eedorward neural networks. Note that eedorward networks can become much more powerul i weight connections across laers are also allowed. Unortunatel onl ver ew sotware packages are capable o training ull connected neural networks. Since it is not eas to train neural networks, several special neural networks were developed where no training or limited training is onl required. In the ollowing section these special neural network architectures are shortl described. More detailed inormation, with network diagrams, can be ound at []... Functional link and polnomial networks One laer neural networks are relativel eas to train, but these networks can solve onl linearl separated problems. One possible solution or nonlinear problems was presented b Pao [7] using the unctional link network, where additional inputs to one laer networks are created b arbitrar selection o nonlinear terms. I these nonlinear terms are generated using a polnomial unction then this network is known as a polnomial network. These networks are eas to train, but it is usuall not known which nonlinear terms are best suited or speciic problems. Polnomial networks have a more generalized approach, but with an increase in the dimensionalit o the problem the number o polnomial terms grow eponentiall and these networks become P-

5 ICCAS impractical... Counterpropagation networks Counterpropagation networks were originall proposed b Hecht-Nilsen [8]. This architecture requires several hidden neurons which are equal to the number o input patterns. When binar input patterns are considered, then the input weights can be eactl equal to the input patterns. Since or a given input pattern, onl one neuron in the irst laer ma have the value o one, and the remaining neurons have zero values, the weights in the put laer are equal to the required put pattern. The counterpropagation network is ver eas to design. The number o neurons in the hidden laer should be equal to the number o patterns. In the case o binar or normalized patterns the weights in the input laer should be equal to the input patterns. Weights in the put laer should be equal to the put patterns. A disadvantage o the counterpropagation network is that the number o neurons in the hidden laer must be equal to the number o training patterns and this number is sometimes ecessivel large... LVQ Learning Vector Quantization networks LVQ networks are derived rom counterpropagation networks b combining some patterns into clusters. B doing this the size o the network is reduced. In the LVQ network the irst laer detects subclasses. The second laer combines subclasses into a single class. The irst laer computes Euclidean distances between input patterns and stored patterns. A winning neuron is the one with the smallest distance in the input pattern... Cascade correlation architecture The cascade correlation architecture was proposed b Fahlman and Lebiere [9]. The process o network building starts with a one laer neural network and hidden neurons are added as needed. In each training step, the new hidden neuron weights are adjusted to maimize the magnitude o the correlation between this neuron put and the residual error signal on the network put. The put neurons are trained using a simple one-neuron training algorithm. Each hidden neuron is trained just once and then its weights are rozen. The network learning and building process is completed when satisactor results are obtained... RBF - Radial Basis Function networks The RBF network usuall has onl one hidden laer with special "neurons". These "neurons" respond onl to the input signals close to the stored pattern characteristic or each neuron. Note that the behavior o this "neuron" signiicantl diers rom the biological neuron. In this "neuron", ecitation is not a unction o the weighted sum o the input signals but the Euclidean distance between the input and stored pattern is computed..6. Sarajedini and Hecht-Nielsen network The Sarajedini and Hecht-Nielsen [] network o Fig. 6 is capable o calculating Euclidean distances between input pattern and stored pattern using onl a neuron with linear activation unction and inormation ab the square o the input vector length. The network is using the ollowing June -, KINTEX, Geonggi-Do, Korea analtical ormulas: w n n = + w net + -w w -w -w n w + Fig. 6. Sarajedini and Hecht-Nielsen neural network and obtained surace or -dim case.. R z z z n z n Fig. 7. Network with increased input dimensionalit b input pattern transormation and an eample o three cluster separation..7. Networks with increased dimensionalit The network shown in Fig. has a similar propert (and power) to RBF networks, but it uses onl traditional neurons with sigmoidal activation unctions [][]. One wa to generate this additional input is to use the ormula: + = R z n () This wa all input patterns are projected on a hper sphere with a radius R and round clusters could be separated b hper planes (traditional sigmoidal neurons). Fig. shows a separation o three clusters using three sigmoidal neurons. With this approach traditional neurons are gaining the capabilit o separating patterns b circle, sphere, or hpersphere..8. Comparison o neural network architectures One o the most diicult problems to solve with neural networks is the parit problem. This problem has a ver nonlinear character with multiple minimas and maimas []. Let us compare dierent neural network architectures to solve the parit-7 problem. This problem is so comple that the most common EBP algorithm is not able to solve it, unless, luckil, initial starting weights are used. In the case o the most popular neural networks with one hidden laer and with connections across laers there are at least 8 neurons required and 7*8+8= weights. See Fig. 8 or the () P-

6 ICCAS network architecture. all weights = all weights = June -, KINTEX, Geonggi-Do, Korea Note, that in the case o the parit problem (due to the smmetr o inputs) each o networks shown in Fig. 8 and Fig. 9 can be urther simpliied b adding an additional neuron with linear activation unction (summator) to the ront. This wa, the network o Fig. 9 can be simpliied to the architecture shown in Fig. 9 with 6 neurons and 7+=++++=7 weights. One ma conclude that the cascade network (Fig. 9) is the most powerul, since it would require a minimum number o elements. At the same time because o a long signal path (across man laers), the cascade architecture is more diicult to train and it is also more sensitive to the variation o weights. The ull connected network (Fig. 8) has a slightl larger number o neurons than cascade architecture (Fig. 9) but it is easier to train and in most cases this would be the preerred choice Fig.. Feedorward neural networks laered and ull connected ; -; -; -; -8; -6; -; -; ; ; ; 6; 8; ; ; ; 6 -.; -.; -6.; -.;.;.; 9.;. Fig.8. Parit-7 problem with eedorward bipolar neural network with one hidden laer with and with connections across laers weights = Fig.. Control surace obtained using ull connected neural network with one hidden neuron. weights = linear Fig. 9. Bipolar implementation o a ull connected cascade neural network or the parit-7 problem: ive neuron architecture and si neuron architecture In the case o when connections across laers are allowed, as shown in Fig. 8, the number o neurons in the hidden laer can be reduced rom 7 to 8 and the total number o weights is 8*8+8+8=7. When neurons are connected in a cascade, as shown in Fig. 9, then onl neurons are required and the total number o weights is *8+++++= Fig.. Control surace obtained using ull connected neural network with two hidden neurons..9. Training algorithms P-6

7 ICCAS Unortunatel most o the neural network sotware (like MATLAB Tool Bo) is not suitable or training ull connected neural networks. One eception is the SNNS (Stuttgart Neural Network Sstem) [], which can handle ull connected architectures, but the LM - Levenberg-Marquar [] algorithm is not implemented. The LM algorithm has currentl the best reputation o all the training algorithms. For most cases it converges within - iterations while the most popular EBP Error Back Propagation-algorithm requires hundreds or thousands times more iterations to reach a solution. An additional advantage o the LM algorithm is its ast convergence to the solution, while the EBP reaches the solution onl asmptoticall Fig.. Control surace obtained using ull connected neural network with three hidden neurons. June -, KINTEX, Geonggi-Do, Korea suitable or an neural network architecture (including cascade and ull connected networks) The MATLAB code can be downloaded rom [6]. In the current version o the sotware all neural network nodes have to be numbered sequentiall starting rom inputs to puts. The entire network architecture is described b a sequence o numbers. For eample the network shown in Fig is described b the sequence:,,,,, 6,,,,, 7,,,,, 8,,,,, 9,, 6, 7, 8 while the network shown in Fig. is described b the sequence:,,,,, 6,,,,, 7,,,,,, 6. In the numerical sequence the number o neurons is listed irst and then all input nodes, and then the number o the net neuron is given with all its associated inputs. The process is repeated until all neurons are listed. Note, that the sotware can handle ull or sparsel connected networks with arbitrar architectures, as long as the concept o a one directional signal low is preserved. Figures to show results obtained with dierent neural network architectures trained to the same required unction, which was used or uzz sstems (Fig. ). Note that when neural network approach is used then better results are obtained than in the case o uzz sstems. Neural networks require simple hardware and work aster. In [7] a practical comparison o various neural and uzz architectures, implemented on the HC Motorola microcontroller, were presented. Again neural sstems had shorter and aster assembl codes. V IN I M M M V X M I M I M I REF M o utp ut cu rren ts [µ A] W L positive put W L negative put Fig.. Control surace obtained using ull connected neural network with our hidden neurons input voltage [µa] Fig. 6. Simple VLSI implementation o neuron with a dierential pair: circuit diagram result o SPICE simulation. V IN M7 I M7 M M8 I M8 M M6 I M6 M I M I M.V.V V M M V. ua W/L 7,8 = W/L,6 = W/L, = W/L, = I REF ua -. - Fig.. Control surace obtained using neural network with our neurons connected in cascade. The author has developed a code or a LM algorithm that is 8uA ua A V.V.V.V.V.V ID(M) ID(M7) ID(M) ID(M6) ID(M8) Vin Fig. 7. VLSI implementation o uzziier block: P-7

8 ICCAS circuit diagram, result o SPICE simulations with ive membership unctions plotted. VLSI implementations In the case o neural networks, a sigmoidal tpe o activation unction can be implemented using a simple dierential pair (Fig 6). Positive or negative weights can be implemented b taking a signal to the net laer rom inverted and non inverted puts. One possible solution is to use current controlled weights in neural networks and current controlled parameters in uzz sstems. Thereore, in order to ull control nonlinear sstems, onl digitall controlled currents are required. The same dierential pairs with a unique coniguration, as shown in Fig. 7, ma act as uzziiers. Fig. 7 shows membership unctions implemented b the circuit o Fig. 7. Fig. 8 shows a simple solution o digitall programmed 6-bit weights. More detailed review o VLSI implementation o neural and uzz sstems are in [8]. V REF V REF V REF V REF I IN 8 6 LSB Fig. 8. Programmable current multipliplier b digital weights.. CONCLUSION Fuzz sstems and neural networks as two major methods o computational intelligence were described and compared. Fuzz sstems are easier to design, while neural networks require training (optimization). In practical implementations uzz sstems require more hardware and resulted control surace is not as smooth as in the case o neural networks. For eample, in the stud case the TSF uzz sstem with triangular membership unction (see Fig. ) required 6+6+6= 8 values to be stored. In the case o ull connected neural network with three hidden neurons (Fig. ), onl *+8= values had to be stored. One ma notice that neural networks require not onl less hardware, but also it generates superior control surace. In the case when onl design rules have to be used and optimization is not desired, neural networks can also replace uzz sstems as was shown in Fig.. REFERENCES [] W. Duch, J. Korbicz, L. Rutkowski, R. Tadeusiewicz Sieci Neuronowe Akademicka Oicna Wdawnicza EXIT, Warszawa. [] B. M.Wilamowski, Neural Networks and Fuzz MSB I OUT June -, KINTEX, Geonggi-Do, Korea Sstems, chapter in Mechatronics Handbook edited b Robert R. Bishop, CRC Press, pp. - to -6,. [] E. H. Mamdani, Application o Fuzz Algorithms or Control o Simple Dnamic Plant, IEEE Proceedings, Vol., No., pp. 8-88, 97. [] Sugeno and G. T. Kang, Structure Identiication o Fuzz Model, Fuzz Sets and Sstems, Vol. 8, No., pp. -, 988. [] T. Takagi and M. Sugeno, Fuzz Identiication o Sstems and Its Application to Modeling and Control, IEEE Transactions on Sstem, Man, Cbernetics, Vol., No., pp. 6-, 98. [6] D. Rutkowska, Y. Haashi Neuro-uzz sstems approaches Int. J. Advanced Computational Intelligence, vol., no, pp [7] Y. H. Pao, Adaptive Pattern Recognition and Neural Networks, Reading, Mass. Addison-Wesle Publishing Co. 989 [8] Hecht-Nielsen, R Counterpropagation networks Appl. Opt. 6(): [9] S.E Fahlman,.and C. Lebiere, The cascadecorrelation learning architecture nn D. S.Touretzk, Ed. Advances in Neural Inormation Processing Sstems, Morgan Kaumann, San Mateo, Cali., (99),-. [] A. Sarajedini, R. Hecht-Nielson, The best o both worlds: Casasent networks integrate multilaer perceptrons and radial basis unctions IJCNN 9. International Joint Conerence on Neural Networks 7- Jun 99 pp. 9-9 vol. [] Y. Ota and B. M. Wilamowski, "Input Data Transormation or Better Pattern Classiication with Fewer Neurons," proceedings o Word Congress on Neural Networks, San Diego, Caliornia, USA, vol., pp , June -9, 99. [] B. M. Wilamowski, and R. C. Jaeger, "Implementation o RBF Tpe Networks b MLP Networks," IEEE International Conerence on Neural Networks, Washington, DC, June -6, 996, pp [] B. Wilamowski, D. Hunter, "Solving Parit-n Problems with Feedorward Neural Network," Proc. o the IJCNN' International Joint Conerence on Neural Networks, pp. 6-, Portland, Oregon, Jul -,. [] Universit o Tübingen. [Online]. Available: [] Hagan, M. T. and Menhaj, M., Training eedorward networks with the Marquar algorithm, IEEE Transactions on Neural Networks, vol., no. 6, pp , 99. [6] Auburn Universit [Online]. Available: [7] B.M. Wilamowski and J. Binet, Do Fuzz Controllers Have Advantages over Neural Controllers in Microprocessor Implementation, ICRAM'99 -nd International Conerence on Recent Advances in Mechatronics -, Istanbul, Turke, pp. -7, Ma -6, 999. [8] B. M. Wilamowski, J.Y. Hung, and R. Gottiparth Digitall Tuned Analog VLSI Controllers ISIE IEEE International Smposium on Industrial Electronics, Dubrovnik Croatia, June 9-, P-8