Sliding Mode Control Part 16 pot

Sliding mode control with a variable corrective control gain using Neural Networks The method in this subsection applies an NN to produce the gain of the corrective control of SMC.. Fur

( )( )

p plant

where V NN( )t is the Lyapunov function of the NN of our method, and V SMC( )t is the

Lyapunov function of SMC of our method

For ( )VNN t , we assume that it can be approximated as

where ΔV NN( )k is the derivation of a discrete-time Lyapunov function, and TΔ is a

sampling time According to (Yasser et al., 2006 b), ΔV NN( )k can be guaranteed to be

negative definite if the learning parameter c satisfies the following conditions

20

q

c n

for the weights between the input layer and the hidden layer, m k Furthermore, if the iq( )

conditions in (26) and (27) are satisfied, the negativity of VNN( )t can also be increased by

reducing TΔ in (25)

For V SMC( )t , it is defined as

2 ( )( )2( ) ( ) ( )

p

y SMC

Then we the following assumption

Assumption 1: The sliding surface in (13) can approximate the sliding surface in (3) (Yasser

Sliding Mode Control Using Neural Networks 515

where k′ y p is a positive constant Following the stability analysis method in (Phuah et al.,

2005 a), we apply (1)—(3), (7), (14), (15), (29) and (30) to (28) and assume that (15) can

approximate (4) Thus, VSMC( )t can be described as

( ) ( ) ( )( ) ( )ˆ

n = in (18), μ= in (19), 2 c =0.001 in (21) and (22), J plant= + in (21), and 1 Δ =T 0.01 in

(25) are all fixed The switching speed for the corrective control of SMC is set to 0.02

seconds We assume a first-order reference model in (10) with parameters A = − m 10,

10

m

B = − , and C = m 1

Fig 1 and Fig 2 show the outputs of the reference model y t m( ) and the plant output y t p( )

using the conventional method of SMC with an NN and a sign function These figures show

that the plant output y t can follow the output of the reference model p( ) y t m( ) closely but

not smoothly, as chattering occurs as seen in Fig 2

Fig 3 and Fig 4 show the outputs of the reference model y t m( ) and the plant output y t p( )

using our proposed method It can be seen that the plant output y t can follow the output p( )

of the reference model y t m( ) closely and smoothly, as chattering has been eliminated as seen in Fig 4

Fig 2 Magnified upper parts of the curves in Fig 1

4 Sliding mode control with a variable corrective control gain using Neural Networks

The method in this subsection applies an NN to produce the gain of the corrective control of SMC Furthermore, the output of the switching function the corrective control of SMC is applied for the learning and training of the NN There is no equivalent control of SMC is used in this second method

Sliding Mode Control Using Neural Networks 517

Fig 4 Magnified upper parts of the curves in Fig 3

4.1 A variable corrective control gain using Neural Networks for chattering elimination

Using NN to produce a variable gain for a corrective control gain of SMC, instead of using a

fixed gain in the conventional SMC, can eliminate the chattering The switching function of

the corrective control is used in the sliding mode backpropagation algorithm to adjust the

weight of the NN This method of SMC does not use any equivalent control of (7) in its

control law For the SISO nonlinear plant with BIBO described in (1), the control input of

SMC with a variable corrective control gain using NN is given as

( )( ) p ( ) p( )

where u cV( )t is the corrective control with variable gain using NN, and k y V p ( )t is the

variable gain produced by NN described as

=

where α is a positive constant, u NNV( )t is a continuous-time output of the NN, u NNV( )k is

a discrete-time output of the NN, ⋅ is an absolute function, and f ZOH( )⋅ is a zero-order

hold function

As in subsection 3.2, we implement a sampler in front of the NN with an appropriate

sampling period to obtain the discrete-time input of the NN, and a zero-order hold is

implemented to transform the discrete-time output u NNV( )k of the NN back to the

continuous-time output u NNV( )t of the NN

The input ( )i k of the NN is given as in (16), and the dynamics of the NN are given as

S h k m k

=

where ( )i k i is the input to the i -th neuron in the input layer ( i= "1, ,n Vi),h V q( )k is the

input to the q -th neuron in the hidden layer ( q= "1, ,n Vq),o k V( ) is the input to the single

neuron in the output layer, n V i and n are the number of neurons in the input layer and Vq

the hidden layer, respectively, m Viq( )k are the weights between the input layer and the

hidden layer, m Vqj( )k are the weights between the hidden layer and the output layer, and

1( )

S ⋅ is a sigmoid function The sigmoid function is chosen as in (19)

4.2 Sliding mode backpropagation for Neural Networks training

In the sliding mode backpropagation, the objective of the NN training is to minimize the

error function E k y p( ) described in (20) The NN training is done by adapting m Viq( )k and

( )( )

Sliding Mode Control Using Neural Networks 519

where c is the learning parameter, and J V plant is described as

( )

( )( )

where V NNV( )t is the Lyapunov function of the NN of our method, and V SMCV( )t is the

Lyapunov function of SMC of our method

where ΔV NNV( )k is the derivation of a discrete-time Lyapunov function, and TΔ is a

sampling time According to (Yasser et al., 2006 b), ΔV NNV( )k can be guaranteed to be

negative definite if the learning parameter c satisfies the following conditions

20

Vq

c n

for the weights between the input layer and the hidden layer, m V iq( )k Furthermore, if the

conditions in (44) and (45) are satisfied, the negativity of VVNN( )t can also be increased by

reducing TΔ in (43)

For V SMCV( )t , it is defined as

2( )( )2( ) ( ) ( )

p V

y SMC

where k′ y pV is a positive constant Based on the stability analysis method in subsection 3.3,

we apply (1)—(3), (34), (35), (29) and (30) to (28) Thus, VSMC( )t can be described as

( ) ( ) ( )( ) ( )ˆ

k t = c B k t VSMC( )t in (48) is negative definite if k y V p ( )t produced by the

NN is large enough The reaching condition (Phuah et al., 2005 a) can be achieved if

and (40), J plant= + in (41), and 1 Δ =T 0.01 in (43) are all fixed The switching speed for the

corrective control of SMC is set to 0.02 seconds We assume a first-order reference model in

(10) with parameters A = − m 10, B = − m 10, and C = m 1

Fig 5 and Fig 6 show the outputs of the reference model y t m( ) and the plant output y t p( )

using our proposed method It can be seen that the plant output ( )y t can follow the output p

of the reference model y t m( ) closely and smoothly, as chattering has been eliminated as

seen in Fig 6

5 Conclusion

In this chapter, we proposed two new SMC strategies using NN for SISO nonlinear systems

with BIBO has been proposed to deal with the problem of eliminating the chattering effect

In the first method, to eliminate the chattering effect, it applied a method using a simplified

distance function Furthermore, we also proposed the application of an NN using the

backpropagation algorithm to construct the equivalent control input of SMC

The second method of this paper applied an NN to produce the gain of the corrective

control of SMC Furthermore, the output of the switching function the corrective control of

Sliding Mode Control Using Neural Networks 521 SMC was applied for the learning and training of the NN There was no equivalent control

of SMC used in this second method The weights of the NN were adjusted using a sliding mode backpropagation algorithm, that was a backpropagation algorithm using the switching function of SMC for its plant sensitivity Thus, this second method did not use the equivalent control law of SMC, instead it used a variable corrective control gain produced

by the NN for the SMC

Brief stability analysis was carried out for the two methods, and the effectiveness of our control methods was confirmed through computer simulations

6 References

Ertugrul, M & Kaynak, O (2000) Neuro-sliding mode control of robotic manipulators

Mechatronics, Vol 10, page numbers 239–263, ISSN: 0957-4158

Hussain, M.A & Ho, P.Y (2004) Adaptive sliding mode control with neural network based

hybrid models Journal of Process Control, Vol 14, page numbers 157—176, ISSN: 0959-1524

Phuah, J.; Lu, J & Yahagi, T (2005) (a) Chattering free sliding mode control in magnetic

levitation system IEEJ Transactions on Electronics, Information, and Systems, Vol

125, No 4, page numbers 600—606, ISSN: 0385-4221

Phuah, J.; Lu, J.; Yasser, M.; & Yahagi, T (2005) (b) Neuro-sliding mode control for magnetic

levitation systems, Proceedings of the 2005 IEEE International Symposium on Circuits and Systems (ISCAS 2005), page numbers 5130—5133, ISBN: 0-7803-8835-6, Kobe, Japan, May 2005, IEEE, USA

Slotine, J.E & Sastry, S S (1983) Tracking control of nonlinear systems using sliding surface

with application to robotic manipulators International Journal of Control, Vol 38, page numbers 465—492, ISSN: 1366-5820

Topalov, A.V.; Cascella, G.L.; Giordano, V.; Cupertion, F & Kaynak, O (2007) Sliding mode

neuro-adaptive control of electric drives IEEE Transactions on Industrial Electronnics, Vol 54, page numbers 671—679, ISSN: 0278-0046

Utkin, V.I (1977) Variable structure systems with sliding mode IEEE Transactions on

Automatic Control, Vol 22, page numbers 212—222 , ISSN: 00189286

Yasser, M.; Trisanto, A.; Lu, J & Yahagi T.(2006) (a) Adaptive sliding mode control using

simple adaptive control for SISO nonlinear systems, Proceedings of the 2006 IEEE International Symposium on Circuits and Systems (ISCAS 2006), page numbers 2153—2156, ISBN: 0-7803-9390-2, Island of Kos, Greece, May 2006, IEEE, USA Yasser, M.; Trisanto, A.; Haggag, A.; Lu, J.; Sekiya, H & Yahagi, T (2006) (c) An adaptive

sliding mode control for a class of SISO nonlinear systems with bounded-input bounded-output and bounded nonlinearity, Proceedings of the 45th IEEE Conference on Decision and Control (45th CDC), page numbers 1599–1604, ISBN: 1-4244-0171-2, San Diego, USA, December 2006, IEEE, USA

Yasser, M., Trisanto, A., Lu, J & Yahagi, T (2006) (b) A method of simple adaptive control

for nonlinear systems using neural networks IEICE Transactions on Fundamentals, Vol E89-A, No 7, page numbers 2009—2018, ISSN: 1745-1337

Yasser, M., Trisanto, A., Haggag, A., Yahagi, T., Sekiya, H & Lu, J (2007) Sliding mode

control using neural networks for SISO nonlinear systems, Proceedings of The SICE Annual Conference 2007, International Conference on Instrumentation, Control and Information Technology (SICE2007), page numbers 980–984, ISBN: 978-4-907764-27-2, Takamatsu, Japan, September 2007, SICE Japan

Young, K.D.; Utkin, V.I & Ozguner, U (1999) A control engineer’s guide to sliding mode

control IEEE Transactions On Control System Technology, Vol 7, No 3, ISSN: 1063-6536

1 Introduction

This chapter includes contributions to the theory of on-line training of artificial neuralnetworks (ANN), considering the multilayer perceptrons (MLP) topology By on-line training,

we mean that the learning process is conducted while the signal processing is being executed

by the system, i.e., the neural network continuously adjusts its free parameters from thevariations in the incident signal in real time (Haykin, 1999)

An artificial neural network is a massively parallel distributed processor made up of simpleprocessing units, which have a natural tendency to store experimental knowledge and make

it available for use (Haykin, 1999) These units (also called neurons) are non-linear adaptabledevices, although very simple in terms of computing power and memory However, whenlinked, they have enormous potential for nonlinear mappings The learning algorithm is theprocedure used to do the learning process, whose function is to modify the synaptic weights

of the network in an orderly manner to achieve a desired goal of the project (Haykin, 1999).Although initially used only in problems of pattern recognition and signal processing andimage, today, the ANN are used to solve various problems in several areas of humanknowledge

An important feature of ANN is its ability to generalize, i.e., the ability of the network toprovide answers in relation to standards unknown or not presented during the training phase.Among the factors that influence the generalization ability of ANN, we cite the networktopology and the type of algorithm used to train the network

The network topology refers to the number of inputs, outputs, number of layers, number

of neurons per layer and activation function From the work of Cybenko (1989), networkswith the MLP topology had widespread use, because they possessed the characteristic ofuniversal approximator of continuous functions Basically, an MLP network is subdividedinto the following layers: input layer, intermediate or hidden layer(s) and output layer Theoperation of an MLP network is synchronous, i.e., given an input vector, it is propagated

to the output by multiplying by the weights of each layer, applying the activation function(the model of each neuron of the network includes a non-linear activation function, being thenon-linearity differentiable at any point) and propagating this value to the next layer until theoutput layer is reached

Issues such as flexibility of the system to avoid biased solutions (under tting) and, conversely,

limiting the complexity of network topology, thus avoiding the variability of solutions

Ademir Nied and José de Oliveira

Department of Electrical Engineering, State University of Santa Catarina

(over tting), are inherent aspects to define the best topology for an MLP This balance between

bias and variance is known in the literature as “the dilemma between bias and variance”(German et al., 1992)

Several algorithms that seek to improve the generalization ability of MLP networks areproposed in the literature (Reed, 1993) Some algorithms use construction techniques,changing the network topology That is, from a super-sized network already trained, methods

of pruning are applied in order to determine the best topology considering the best balance

between bias and variance Other methods use restriction techniques of the weights values

of MLP networks without changing the original topology However, it is not always possible

to measure the complexity of a problem, which makes the choice of network topology anempirical process

Regarding the type of algorithm used for training MLP networks, the formulation of thebackpropagation algorithm (BP) (Rumelhart et al., 1986) enabled the training of fedforwardneural networks (FNN) The algorithm is based on the BP learning rule for error correctionand can be viewed as a generalization of the least mean square algorithm (LMS) (Widrow &Hoff, 1960), also known as delta rule

However, because the BP algorithm presents a slow convergence, dependent on initialconditions, and being able to stop the training process in regions of local minima wherethe gradients are zero, other methods of training appeared to correct or minimize thesedeficiencies, such as Momentum (Rumelhart et al., 1986), QuickProp (Fahlman, 1988), Rprop(Riedmiller & Braun, 1993), setting the learning rate (Silva & Almeida, 1990; Tollenaere, 1990),the conjugate gradient algorithm (Brent, 1991), the Levenberg-Marquardt algorithm (Hagan

& Menhaj, 1994; Parisi et al., 1996), the fast learning algorithm based on the gradient descent

in the space of neurons (Zhou & Si, 1998), the learning algorithm in real-time neural networkswith exponential rate of convergence (Zhao, 1996), and recently a generalization of the BPalgorithm, showing that the most common algorithms based on the BP algorithm are specialcases of the presented algorithm (Yu et al., 2002)

However, despite the previously mentioned methods accelerating the convergence of networktraining, they cannot avoid areas of local minima (Yu et al., 2002), i.e., regions where thegradients are zero because the derivative of the activation function has a value of zero ornear zero, even if the difference between the desired output and actual output of the neuron

is different from zero

Besides the problems mentioned above, it can be verified that the learning strategy of trainingalgorithms based on the principle of backpropagation is not protected against externaldisturbances associated with excitation signals (Efe & Kaynak, 2000; 2001)

The high performance of variable structure system control (Itkis, 1976) in dealing withuncertainties and imprecision have motivated the use of the sliding mode control (SMC)(Utkin, 1978) in training ANN (Parma et al., 1998a) This approach was chosen for threereasons: because it is a well established theory, it allows for the adjustment of parameters(weights) of the network, and it allows an analytical study of the gains involved in training.Thus, the problem of the training of MLP networks is treated and solved as a problem ofcontrol, inheriting characteristics of robustness and convergence inherent in systems that useSMC

The results presented in Efe & Kaynak (2000), Efe et al (2000) have shown that theconvergence properties of gradient-based training strategies widely used in ANN can beimproved by utilizing the SMC approach However, the method presented indirectly usesthe Variable Structure Systems (VSS) theory Some studies on the direct use of SMC strategy

are also reported in the literature In Sira-Ramirez & Colina-Morles (1995) the zero-level set

of the learning error variable in Adaline neural networks is regarded as a sliding surface inthe space of learning parameters A sliding mode trajectory can then be induced, in finitetime, on such a desired sliding manifold The proposed method was further extended in Yu

et al (1998) by introducing adaptive uncertainty bound dynamics of signals In Topalov et al.(2003), Topalov & Kaynak (2003) the sliding mode strategy for the learning of analog Adalinenetworks, proposed by Sira-Ramirez & Colina-Morles (1995), was extended to a more generalclass of multilayer networks with a scalar output

The first SMC learning algorithm for training multilayer perceptron (MLP) networks wasproposed by Parma et al (1998a) Besides the speed up achieved with the proposed algorithm,control theory is actually used to guide neural network learning as a system to be controlled

It also differs from the algorithms in Sira-Ramirez & Colina-Morles (1995), Yu et al (1998) andTopalov et al (2003), due to the use of separate sliding surfaces for each network layer Acomprehensive review of VSS and SMC can be seen in Hung et al (1993), and a survey aboutthe fusion of computationally intelligent methodologies and SMC can be found in Kaynak

et al (2001)

Although the methodology used by Parma et al (1998a) makes it possible to determine thelimits of parameters involved in the training of MLP networks, their complexity still makes itnecessary to use heuristic methods to determine the most appropriate gain to be used in order

to ensure the best network performance for a particular training

In this chapter, an algorithm for on-line ANN training based on SMC is presented The mainfeature of this procedure is the adaptability of the gain (learning rate), determined iterativelyfor every weight update, and obtained from only one sliding surface

To evaluate the algorithm, simulations were performed considering two distinct applications:function approximation and a neural-based stator flux observer of an induction motor(IM) The network topology was defined according to the best possible response with thefewest number of neurons in the hidden layer without compromising the ability of networkgeneralization The network used in the simulations has only one hidden layer, differing inthe number of neurons in this layer and the number of inputs and outputs of the network,which were chosen according to the application for the MLP

2 The On-line adaptive MLP training algorithm

This section presents the algorithm with adaptive gain for on-line training MLP networkswith multiple outputs that operates in quasi-sliding modes The term “quasi-sliding regime”was introduced by Miloslavjevic (1985) to express the fact that the extension to the case ofdiscrete time under the usual time for the continuous existence of a sliding regime, does notnecessarily guarantee chattering around the sliding surface in the same way that it occurs incontinuous time systems Moreover, in Sarpturk et al (1987) it was shown that the conditionproposed by Miloslavjevic (1985) for the existence of a quasi-sliding mode could cause thesystem to become unstable Now, let us specify how the quasi-sliding mode and the reachingcondition are understood in this paper

Definition 1. Let us define a quasi-sliding mode in the ε vicinity of a sliding hyperplane s(n) =0 for

a motion of the system such that

This definition is different from the one proposed in Gao et al (1995) since it does not require

the system state to cross the sliding plane s(n) =0 in each successive control step

The convergence of the system state to the sliding surface can be analyzed considering theconvergence of the series

∞

∑

If the convergence of the series is guaranteed, then the system state will converge, at least

assimptotically, to the sliding surface s(n) =0

Consider Cauchy’s convergence principle (Kreyszig, 1993): The series s1+s2+ · · · + s n

converges if and only if, for a given value ε ∈ +, a value N can be found such that

| s n+1+s n+2+ · · · + s n +p |< ε for all n > N e p=1, 2,· · · A series is absolutely convergentif:

Remark: From Definition 2, crossing the plane s(n) =0 is allowed but not required

Theorem 1. Let s(n) : 2 →  , the sliding surface defined by s(n) = CX1(n) +X2(n), where

{ C, X1(n )} ∈ +and X2(n ) ∈  If X1(n) =E(n), being E(n) = 1

2∑m L

k=1e2k(n)defined as the instantaneous value of the total energy of the error of all the neurons of the output layer of an MLP, where e k(n) = d k(n ) − y k(n) is the error signal between the desired value and actual value at the output of the neuron k of the network output at iteration n, m L is the number of neurons in the output layer of the network, and X2(n) = X1 (n)−X1(n−1)

T is defined as the variation of X1(n)in a sample period of T, then, for the current state s(n)to converge to a vicinity ε of s(n) =0, it is necessary and

sufficient that the network meet the following:

Proof:Defining the absolute value of the sliding surface as follows

then, from (5) it holds that

| s(n+1)| < | s(n )| ⇒ sign(s(n+1))s(n+1) < sign(s(n))s(n)

As sign(s(n))sign(s(n)) =1, yields

sign(s(n))[sign(s(n))sign(s(n+1))s(n+1) − s(n )] <0

If sign(s(n+1)) =sign(s(n)), then sign(s(n))[s(n+1) − s(n )] <0 Replacing the definition

of s(n)as given by Theorem 1 yields

sign(s(n)) [CX1(n+1) +X2(n+1) − ( CX1(n) +X2(n ))] <0⇒ (6)

If sign(s(n+1)) = − sign(s(n)), then sign(s(n ))[− s(n+1) − s(n )] < 0 Replacing the

definition of s(n)as given by Theorem 1 yields

sign(s(n)) [CX1(n+1) +X2(n+1) +CX1(n) +X2(n )] >0⇒ (7)

To prove that the conditions of Theorem 1 are sufficient, two situations must be established:

• The sliding surface is not crossed during convergence In this situation, it holds that

sign(s(n))[s(n+1) − s(n )] = −| s(n+1)| − | s(n )| ⇒ (6)



From Theorem 1, it can be verified that (6) is responsible for the existence of a quasi-sliding

regime for s(n) =0, while (7) ensures the convergence of the network state trajectories to a

vicinity of the sliding surface s(n) = 0 One can also observe that the reference term from

the sliding surface signal sign(s(n))determines the external and internal limits of the range

of convergence for the following expressions:

C(X1(n+1) − X1(n)) +X2(n+1) − X2(n) (9)

C(X1(n+1) +X1(n)) +X2(n+1) +X2(n) (10)

To study the convergence of the sliding surface s(n) =CX1(n) +X2(n), the decomposition

of (9) and (10), with respect to a gainη, is necessary in order to obtain a set of equations for

these variables and, from the conditions defined by Theorem 1, to determine an interval in

due to a gainη, that can guarantee the convergence of the proposed method.

527

Sliding Mode Control Approach for Training On-line Neural Networks with Adaptive Learning Rate

Theorem 2. Let s(n) : 2 →  , the sliding surface defined by s(n) = CX1(n) +X2(n), where

{ C, X1(n )} ∈ +and X2(n ) ∈  If X1(n), X2(n) and T are defined as in Theorem 1, then, for the current state s(n)to converge to a vicinity ε of s(n) =0, it is necessary and sufficient that the

network meets the following:

From (16), (17), (18), (19) and considering the definition of X2(n)given by Theorem 1, one can

derive the terms of (9) taking into account that d k(n −1) =d k(n) =d k(n+1) =d k Thus, itholds: