Abstract— The control of dynamical systems with inherent non-linear characteristics has motivated research in non-linear control theory. Two main approaches to dealing with uncertainties in control systems design are adaptive and robust control. In this paper, discrete direct adaptive control of unknown non-linear SISO systems is considered. The controller is implemented using a fuzzy neural network. The control concept is tested on a laboratory pilot plant and compared to a standard discrete PID Takahashi controller.

I. INTRODUCTION

ost control strategies assume that a time-invariant nominal model of a process is known and that there is

no difference between the nominal and actual model. However, problems arise relating to difference between the nominal and actual model and the parameters of the actual model varying during the process. Two methods used in dealing with uncertainties in control systems design are adaptive and robust control. Adaptive control mainly deals with “large” uncertainties whereas robust control mainly deals with “small” uncertainties [1].

In adaptive control, it is common to assume the unknown dynamics have a known structure with unknown parameters entering linearly in the dynamics. The linear parameterization of unknown dynamics poses serious obstacles in adopting adaptive control algorithms in practical applications, because it is difficult to fix the structure of the unknown nonlinearities. This fact has been the motivating factor behind the interest in on-line function approximators to estimate and learn the unknown function. The most common function approximators used in adaptive control are artificial neural network and fuzzy logic structures. On-line control algorithms that do not require knowledge of the system dynamics (except its dimension and relative degree) have been made possible by employing artificial neural networks in the feedback loop [2]. The ability of neural networks to approximate uniformly

E. K. Nyarko is with the Department of Automation and Process Control, Faculty of Electrical Engineering, University of Osijek, 31000 Osijek, Croatia (phone: +385-31-224750; fax: +385-31-224705; e-mail: nyarko@etfos.hr).

N. Peri is with the Department of Control and Computer Engineering in Automation, Faculty of Electrical Engineering and Computing, University of Zagreb, 10000 Zagreb, Croatia (e-mail: nedjeljko.peric@fer.hr).

I. Petrovi is with the Department of Control and Computer Engineering in Automation, Faculty of Electrical Engineering and Computing University of Zagreb, 10000 Zagreb, Croatia (e-mail: ivan.petrovic@fer.hr).

continuous functions has been proven in several articles [3, 4, 5]. An important aspect of neural network control applications is the difference between approximation theory results and what is achievable in on-line adaptive schemes. First and most importantly, in on-line applications the neural network weights are updated based on input-output matching, whereas in direct adaptive control situations the update of the network parameters is driven by a tracking error, which by its definition does not contain any input type information. In addition to this aspect, the adaptive update must yield boundedness of all signals in the closed loop.

In the case of sharp nonlinearities, the discrete time adaptive control problem is much harder than in the continuous-time case. Hence, most results have assumed sector bounded nonlinearities. In [6], sector boundedness is not assumed, but on the other hand the nonlinearity is assumed to be fully known (and invertible). The non-sector boundedness issue has been addressed directly in [7] where Lyapunov analysis was used to show stability of a first-order scalar system.

In this paper, attention is focussed on implementing the results in [8] which is in turn an expansion of the results in [7]. In [8], an adaptive tracking controller is designed by choosing the control signal as a parameter estimate multiplied by some basis function, and a least-squares estimator is employed to drive the parameter error in the control law to zero.

This paper is organised as follows. A description of fuzzy-neural networks and their application in non-linear modelling is given in section II. The adaptive control algorithm is presented in section III. In section IV, a short description of the laboratory pilot plant as well as the implementation of the adaptive algorithm and a discrete PID Takahashi controller on the process is also given. Section V gives an analysis of the results obtained and respective conclusions.

II. FUZZY NEURAL NETWORKS

The similarities between neural networks, with their abilities to learn how to approximate any continuous non-linear function, and fuzzy systems have been investigated, leading to the development of neurofuzzy modelling [9,10].

Fig. 1 shows the structure of the fuzzy neural network used in this paper. The fuzzy neural network consists of only one layer of basis (neuron) functions and one set of weights.

Experimental Investigations of a Direct Adaptive Neurofuzzy Controller

Emmanuel K. Nyarko, Nedjeljko Peri and Ivan Petrovi

M

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TC3.6

0-7803-9354-6/05/$20.00 ©2005 IEEE 1051

Fig. 1. The structure of the fuzzy neural network

This network is functionally equivalent to radial basis functions networks since the response is obtained as a linear combination of the responses from localized basis functions.

The input layer of the network has a total of n(x)neurons. Fuzzification is performed in the middle layer. The number of neurons in the middle layer (a total of n(x) Rneurons) is determined by the number of inputs n(x) and the number of fuzzy rules R.

The number of neurons in the output layer is n(y). This layer performs the inference and defuzzification processes.

Mathematically, the fuzzy neural network can be defined as:

n(x)

1 1,,

i

R

jijijll xy ,1 l n(y), (1)

or in matrix form:

xy , (2)

where:

n(x)

1

x

xx – input vector of dimension n(x);

n(y)

1

y

yy – output vector of dimension n(y);

R

1

– vector of basis functions (fuzzy

rules), dimension R;

n(x),,n(y)n(x),1,n(y),,n(y),1,n(y)1,,n(y)1,1,n(y)

n(x),,n(x),1,,,,1,1,,1,1,

n(x),,1n(x),1,1,,1,1,11,,11,1,1

RiRiR

RlliRlilRll

RiRiR

– three dimensional weight area of dimension (y) n(x) R).

This fuzzy neural network belongs to the class of models known as generalized linear models. The basis functions are non-linear functions of their input, whereas the model itself is linear in the parameters l,j,i. The main advantage is that, given sets of training samples and basis functions, the optimal set of weights can actually be found. On the other hand, when employing the multi-layer perceptron structure with at least one hidden layer, optimal weights cannot be guaranteed.

The basis functions are chosen to have bounded and compact support and form a partition of unity:

1;0)(, ji , 1)(1 ,

ir

j ji , M where M is

the input space of the individual input argument and )(, ji is the j'th basis function in the i'th basis function

vector. These requirements are satisfied by, for instance, B-splines.

Assuming that only the input and output of a single input single output (SISO) non-linear system can be measured then it can be defined using an input-output model [11]:

y(k+1) = f(y(k),..., y(k-n+1), u(k-1),..., u(k-m)) + g(y(k),..., y(k-n+1), u(k-1),..., u(k-m))·u(k), (3)

where f and g are non-linear vector functions.

If the order as well as the relative degree of the system is known (see [12]), then as suggested in [13], the state variables can be chosen as:

,1

,,

,1

2

21

1

11

kukx

mkukxkykx

nkykx

m

n (4)

giving the state space model:

x11(k + 1) = x12(k),

x1n(k + 1) = f(x(k)) + g(x(k)) u(k), x21(k + 1) = x22(k), (5)

x2m(k + 1) = u(k), y(k) = x1n(k).

Assuming the non-linear single input single output system (3) can be modelled using fuzzy neural networks then in practice the output prediction can be obtained from the input and output samples in the training samples

1,1,1

1,R,1

+y1x1

1

j

R

.

.

.

.

1,j,1

Fuzzification Inference and Defuzzification

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y (k + 1) = Tf (x(k)) + T

g (x(k)) u(k), (6) where:

x(k) – vector of state variables; – vector of basis functions (fuzzy rules), in this case B - spline functions;

Tf , T

g – weight vectors.

Comparing equations (3) and (6), it can be noticed that the unknown non-linear functions f(x(k)) and g(x(k)) are modelled by two fuzzy neural networks, T

f (x(k)) and Tg (x(k)) respectively.

III. DIRECT ADAPTIVE CONTROL USING FUZZY NEURALNETWORK

The idea of direct adaptive control is to find the controller parameters from the available input output data when the process model itself is unknown. Fig. 2 shows an adaptive control system using fuzzy neural network for the system (3). An adaptive tracking controller is designed which employs neurofuzzy basis functions to compensate for the nonlinearities.

It is noted that the system output y(k) = x1n(k) has to follow the uniformly bounded reference sequence

lky kr 0 i.e. kykyy rrr max,, .The control error is defined as:

e(k) = yr(k) y(k). (7) The system state is consequently redefined as the

(delayed) control error, i.e.

.1

,,

,1

1

1

kukx

mkukxkekx

nkekx

mn

n

n (8)

With such a choice of system variables, the system is considered to be the (scaled) control error system and equation (6) is now of the form:

e(k + 1) = Tf1

(x(k)) + Tg1

(x(k)) u(k). (9)

It should be noted that the weight vectors Tf1

andTg1

differ from those in (6) because they represent other

Fuzzy Neural

Network +

-

z-n

z -1

z -m

z -1

.

.

.

.

.

.

Proces

z -1

u(k) y(k+1)

yr(k) x(k)

Fig. 2. Adaptive control system using fuzzy neural network

non-linear vector functions1.The control law is to be defined such that it drives the

control error to 0, i.e., the adaptive controller adapts on-line such that:

0)1()1( kxke n .

The parameter estimates of the fuzzy neural network model are tuned such that the network compensates for the system causing the resulting closed loop control error to behave like a first order system, i.e.:

10),()1( akaxkx nn . (10)

Comparing (8) and (9):

xn(k + 1) = Tf1

(x(k)) + Tg1

(x(k)) u(k). (11)

If the control law is defined as:

u(k) = [ * Tg1

(x(k))]-1 [ * Tf1

(x(k)) + axn(k)], (12)

where * Tf1

and * Tg1

represent the approximations of Tf

and Tg respectively and (12) is substituted in (11):

xn(k + 1) = Tf1

(x(k)) + Tg1

(x(k)) [ * Tg1

(x(k))]-1

[ * Tf1

(x(k)) + axn(k)]. (13)

It can be noted that the choice of the non-linear vector function g1, (i.e. T

g1(x(k)) ) poses a problem especially

when it is not easily invertible. In order to avoid further complications, it is chosen to approximate the function with an easily invertible function (unity): g1 = T

g1(x(k)) 1.

Hence equation (13) becomes: xn(k + 1) = T

f1(x(k)) + [ * T

f1(x(k)) + axn(k)],

and when Tf1

* Tf1

:

xn(k + 1) axn(k).

___________________________________1 Due to the choice of system variables the non-linear control error

system is defined as: e(k+1) = f1(e(k),..., e(k-n+1), u(k-1),..., u(k-m))

+ g1(e(k),..., e(k-n+1), u(k-1),..., u(k-m))·u(k).

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The constant a is used to tune the control system. The (scaled) control error system is now a generalized

linear model:

e(k + 1) = T (k) + u(k), (14)

where T= Tf1

and (k)= (x(k)).

The control law is thus defined as:

u(k) = *(k)T (x(k)) + axn(k), (15) where *T = * T

f1.

The prediction error is then defined as the difference between the estimate and the actual measurement (of the control error):

(k) = e (k) e(k) = nx (k) xn(k)

= *(k 1)T (k 1) + u(k 1) xn(k). (16)

In order to deal with the prediction error the recursive Gauss-Newton least-squares estimator is used in updating the fuzzy neural network parameters.

The control problem of the Direct Adaptive Control Algorithm can be formulated as: how to stabilize the control error xn and the prediction error to a small neighbourhood of 0 while maintaining bounded input and bounded states.

Using the following assumptions: - the non-linear function f1 is sector-bounded by a known

bound i.e. | f1(x(k))| 1 + 2||x||,- the constant a used in tuning the control has values

0 a < 1, - the basis functions )( are uniformly bounded by 1,

Bendtsen applies Lyapunov-like analysis to show stability of the closed loop [8]. He uses the Key Technical Lemma suggested in [7] to show that the prediction error converges to zero. He also shows that the norm of the error state depends on the prediction error. Since the prediction error converges to zero independently of the control error, a conceptual equivalence to the separation principle in observer-based control is provided.

The control law stated in this paper requires that the nonlinearity does not grow too fast with the control error. Otherwise the linear update of the parameter vector cannot ‘keep up’ [8]. Bendtsen further modifies the algorithm so that the parameter update can keep up with nonlinearities growing faster than linearly. The basis functions are weighted as well as the parameter update. He then shows how three separate component Lyapunov function candidates can be constructed for the weighted parameter update, the weighted covariance estimate and the state equations, respectively. The sum of the component Lyapunov functions is then demonstrated to form a Lyapunov function for the whole closed loop, implying stabilization in finite time to some (arbitrarily small)

neighbourhood of 0. He also showed that including the tuning term, a, in the controller could give stability problems at small prediction errors, so the dead-beat controller (a = 0) is preferred.

IV. EXPERIMENTAL RESULTS

The process used in this work is a single input single output laboratory pilot plant (Fig. 3.) for the transport and heating of fluids (air in this case).

Fig. 3. Principle schematic diagram of the modelled plant

Air is sucked in from the ambient atmosphere with the aid of a motor or fan (A) through the entrance (B) and driven through an electrical heater grid (C) and then through a plastic tube (D) out in the atmosphere again. It has many similarities with a hair dryer. The input signals are voltages, Urq and Urt , that drive the motor and produce a current through the heater grid respectively. The output signals are also voltages that represent measurements of the entering air temperature, outgoing air temperature as well as the rate of flow of the outgoing air. With respect to the motor, an input voltage range of 0-8V gives an output air flow rate of 0 3,5l/s whereas an input voltage range of 0-8V to the heater grid gives an output air temperature of ambient temperature to about 140 C (depending on the output flow rate). The plant thus represents a loosely coupled system. However, the control problem is to control the outgoing air temperature.

The adaptive control system using fuzzy neural network is tested on the process. The three assumptions needed to ensure stability of the closed loop are satisfied because: - the limited output range of the process satisfies the assumption that the nonlinearity is sector-bounded, - a = 0 ensuring the control action is as fast as possible, - the basis functions used for the fuzzy neural network are B-Spline functions and are therefore bounded by 1.

The input values to the fuzzy neural network are chosen as two past states of the control error and one past state of the control value i.e. n = 2 and m =1 ( see (8) ). A third order B-Spline function with eight basis functions is used as basis functions for fuzzy neural network.

The adaptive control algorithm is implemented as an S-function in Matlab. With the aid of Simulink and Real Time WorkShop, the adaptive control algorithm is run on a computer in real time.

The communication between the process and the computer is handled by an acquisition card, PCL-818L, from

M

qmu, Tzu

qmi, Tzi

Urt

Urq

A

DC

B

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Advantech Co, Ltd. PLC-818L contains a 12-bit A/D converter with sampling rates up to 40kHz and a 12-bit D/A converter with 5 microseconds settling time. The range of the D/A converter is ±10V [13]. A sample rate of 5s was used.

Figs. 4. and 5. show the results obtained with the adaptive controller when the flow rate of the input air is kept constant (about 3 l/s). In order to check the performance of the adaptive controller in the prescence of process disturbance, the referent temperature is kept constant while the rate of flow of the input air is changed (Figs. 6. and 7.). The process disturbance is shown in Fig. 8.

The performance of the adaptive controller is compared to that of a discrete controller with the parameters tuned according to Takahashi PID tuning rules [14]. The parameters of the PID controller not only depend on the type of process but also on the operating point of the process. The parameters need to be determined off-line and beforehand. The parameters are determined for the operating point of around 40 C and 3 l/s.

The experiment is performed under the same conditions as in the previous case: first without disturbance (Figs. 9. and 10.) and then in the prescence of disturbance (Figs. 11. and 12.). The same reference and disturbance sequence and initial conditions are used.

0 2000 4000 6000 8000 10000 1200025

30

35

40

45

50

55

60

65

70

time [s]

tem

pera

ture

[C]

process output signal ( Tzi )refere nce signal ( Tziref )

Fig. 4. Performance of the fuzzy neural adaptive controller (tracking control)

0 2000 4000 6000 8000 10000 12000-8

-6

-4

-2

0

2

4

6

8

10

12

time [s]

perc

enta

ge e

rror

[%]

MSE = 0.003382

Fig. 5. Tracking error of the fuzzy neural adaptive control system

0 2000 4000 6000 8000 10000 1200025

30

35

40

45

50

55

time [s]

tem

pera

ture

[C]

process output signal ( Tzi )reference signal ( Tziref )

Fig. 6. Performance of the fuzzy neural adaptive controller in the presence of disturbance

0 2000 4000 6000 8000 10000 12000-25

-20

-15

-10

-5

0

5

10

time [s]

perc

enta

ge e

rror

[%]

MSE = 0.005032

Fig. 7. Steady state tracking error of the fuzzy neural adaptive control system

0 2000 4000 6000 8000 10000 12000

2

2.5

3

3.5

4

time [s]

inpu

t flo

w ra

te [l

/s]

Fig. 8. Disturbance signal

0 2000 4000 6000 8000 10000 1200025

30

35

40

45

50

55

60

65

70

time [s]

tem

pera

ture

[C]

process output signal ( Tzi )reference signal ( Tziref )

Fig. 9. Performance of the discrete PID controller (tracking control)

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0 2000 4000 6000 8000 10000 12000-3

-2

-1

0

1

2

3

4

5

6

7

time [s]

perc

enta

ge e

rror

[%]

MSE = 0.0093012

Fig. 10. Tracking error of the discrete PID control system

0 2000 4000 6000 8000 10000 1200025

30

35

40

45

50

time [s]

tem

pera

ture

[C]

process output signal ( Tzi )reference signal ( Tziref )

Fig. 11. Performance of the discrete PID controller in the presence of disturbance

0 2000 4000 6000 8000 10000 12000-25

-20

-15

-10

-5

0

5

10

time [s]

perc

enta

ge e

rror

[%]

MSE = 0.0023569

Fig. 12. Steady state tracking error of the discrete PID control system

Comparing Figs. 4. and 9., the adaptive controller is noticed to out perform the discrete PID controller. This is confirmed in Figs. 5. and 10., where it is seen that the mean squared error of the adaptive controller (MSE=0.003382) is about three times lower than the PID controller (MSE=0.009301).

The PID controller (MSE=0.0023569), on the other hand, is better than the fuzzy neural controller (MSE=0.005032), in the prescence of disturbance when

the reference is kept constant. (Figs. 6. and 11. and 7. and 12. respectively). This is to be expected since the parameters of the PID controller were determined specifically for that operating point before the experiment. In both situations, the closed loop system with the adaptive controller is noticed to be stable.

V. CONCLUDING REMARKS

In this paper a direct adaptive control law for stabilization of non-linear SISO systems has been treated. The control law was chosen as a generalised linear model estimate of the system, and a second order least squares estimator was used to adapt the parameters.

The adaptive controller is implemented in a control loop of a SISO laboratory plant in order to experimentally investigate the adaptive fuzzy neural controller concept. The performance of the adaptive fuzzy neural controller is compared to that of a a discrete PID Takahashi controller.

The adaptive control system is observed to be stable and the adaptive fuzzy neural controller comparable to the PID Takahashi controller. Thus, these results demonstrate the feasibility of the adaptive fuzzy neural controller concept.

REFERENCES

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[12] A. Isidori, Non-linear control systems, Springer, Berlin ; New York, 3rd edition, 1995.

[13] F. –C. Chen and H. K. Khalil, “Adaptive control of a class of non-linear discrete-time systems using neural networks”. IEEE Trans. Automat. Contr., vol. 40, 1995, pp. 791-801.

[14] R. Isermann. Digital Control Systems, Springer-Verlag, 1981.

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