# Stable adaptive control of a multivariable nonlinear process

#### Descrição do Produto

2011 8th International Multi-Conference on Systems, Signals & Devices

Stable adaptive control of a multivariable nonlinear process

Choueikh Safa, Salim Hadj Saїd and Faouzi M’Sahli Department of Electrical Engineering, National Engineering School of Monastir, 5019, Monastir, Tunisia ABSTRACT In this paper, we apply an adaptive control algorithm to a nonlinear multivariable process. Such controller is based on the multiple models approach. As the design of the control law requires the knowledge of the dynamical model of the system, we deal firstly with the identification of the system parameters using the recursive least squares and the retro propagation of the gradient algorithms. Then, we focus on the application of the multiple model approach. So, we decomposed the nonlinear model of the system in sub-systems and we adopted a proper criterion of commutation between the various models. The global control consists in the interpolation between the elementary control extracted from each model. The resulting controller is applied to a multivariable process to solve a tracking problem of the water levels into a twin tank process. The control strategy ensures the stability of the closed loop system and guarantees a good behavior when tracking a reference trajectory.

Index Terms— Adaptive control, Switching, neural networks, multiple model, twin tank process 1. INTRODUCTION The study of nonlinear multivariable systems has been the subject of several research tasks. This is due to the fact that most of industrial processes are nonlinear and have many variables to be controlled. Indeed, the nonlinear systems are complex. Among the powerful techniques used to control such processes we considered the multiple model approach. This technique is based on the strategy “divid to reign” , . Indeed, the system is represented by a set of models. Each model is valid in a zone of the space of the system operation. This technique is based mainly on two rules. The first consists of the determination of the process library. The second is related to the moments of commutation between the various models , . The control of a multiple model consists of taking the control from each model of the library. This strategy of control is based on the definition of a base containing many couples (Model-Regulator). The parameters of each regulator are determined from the desired performances and depending on the model parameters. The determination of the commutation

moments between the different elementary control is a delicate stage and must be precised , . The design and the implementation of the control of any system require the knowledge of their dynamical model. So, the parameters must be well identified because they influence the quality of control. Generally, the nonlinear approximation of the parameters of the systems is rather difficult. Recent research showed that the neural networks are an effective tool of the approximation of the nonlinear systems , , . Indeed, the neural networks constitue an intelligent tool based on the digital techniques of training. There exist different methods of training such as the rule of Hebb, the genetic algorithms, Sollis and Wets algoritm and retropropagation of the gradient of the error algorithm . In our work, we will implement the retropropagation algorithm due to its simplicity and fiability . The adaptive control adopted here ensures the stability and the tracking of the reference signal. In this paper, we applied an adaptive control on a multiple- model to a multivariable nonlinear process represented by the twin tank process , , . The paper is organized as follows: In section 2, the system under consideration is described. In section 3, we used the multiple model technique. In section 4, we represented the adaptive controllers. In section 5, we defined the various stages of the algorithm of adaptive controller. In section 6, we showed the convergence or the stability of the control. In section 7, we applied the resulting total control from the process. Simulations results are reported in section 5. Conclusions are given in last section. 2. TWIN TANK PROCESS The process considered in this work is a hydraulic system with two coupled tanks . This system comprises two vertical cylindrical tanks connected by a drainage canal ordered by a pump system. Each tank is equipped by two drainage canals, each channel is equipped by a pump. (k3, k1) and (k1, k2) represent couples of the pumps corresponding respectively to the first and the second tank. The entry of this system is composed by two flows Q1 and Q2. The out puts of system includes the levels of liquids in h1, h2 the two tanks. Two motor-driven pumps (Mcc1, Mcc2) are used to transport water of a basin of recovery towards the two tanks. These two motor-driven pumps

are not responsible for pump water out of the tanks. The hydraulic system is governed by the following nonlinear differential equation:

k1 ⎧ ∂h1 ⎪ ∂t = − S . h1 − h2 .sign(h1 − h2 ) ⎪ 1 ⎪ + Q1 ⎪ S ⎪ ⎨ ⎪ ∂h k ⎪ 2 = 1 h1 − h2 . sign(h1 − h2 ) ⎪ ∂t S ⎪ 1 k − 2 h2 + Q2 ⎪ S S ⎩

(1)

Where: −1

⎛ (1 − z ) 0 ⎞ A( z −1 ) = ⎜ ⎟ (1 − z −1 ) ⎠ ⎝ 0

⎛ Te ⎜S B( z −1 ) = ⎜ ⎜ 0 ⎜ ⎝

⎞ 0⎟ ⎟ Te ⎟ ⎟ S⎠

iiiThe system has a globally uniformly asymptotically stable zero-dynamics. ivThe nonlinear term f ( h, Q ) is globally bounded , i.e,

f

≤ M and the bound M is known

3. MULTIPLE MODEL APPROACH

(2)

By applying the principal approach of the multiple model for this process, one can find that the library its contains linear models and nonlinear models. To control the water level h1 into tank1, one will elaborate a library of linear and nonlinear models. In the same way with the second water level h2. Finally, we obtain two water levels which are stable and pursuit the same trajectory of reference. For the first tank, the linear model is given by:

⎛T ⎞ h1lin (k ) = h1 (k − 1) + ⎜ e ⎟ * Q1 (k − 1) ⎝S⎠

The system can be put in the following form:

A( z −1 ) h(k ) = B( z −1 ) Q (k − 1) + f (h, Q )

Assumption: iThe system orders na , nb and the relative degree k are known a priori. iiThe linear parameter matrices forming

A( z −1 ) , B( z −1 ) lie in a compact region ∑ , and B (0) is nonsingular.

The descritisation of the system by using the method of finished differences where Te is the period of sampling, gives:

Te ⎧ ⎪h2 (k ) = h2 (k − 1) + ( S ) Q1 ( k − 1) + .. ⎪ ⎪+ ( k1Te ) h (k − 1) − h (k − 1) *.. 1 2 ⎪ S ⎪ ⎪sign(h1 (k − 1) − h2 (k − 1)) ⎪ ⎨ ⎪ T ⎪h2 (k ) = h2 (k − 1) + ( e ) Q1 ( k − 1) + .. S ⎪ ⎪ k1Te ) h1 (k − 1) − h2 (k − 1) *.. ⎪+( S ⎪ ⎪⎩sign(h1 (k − 1) − h2 (k − 1))

⎛ − h1 (k − 1) − h2 (k − 1) * ⎞ ⎜ ⎟ ⎜ sign(h1 (k − 1) − h2 (k − 1)) ⎟ kT ⎜ ⎟ f ( h, Q ) = ( 1 e ) ⎜ ⎟ S ⎜ h (k − 1) − h (k − 1) * ⎟ 2 ⎜ 1 ⎟ ⎜ sign(h (k − 1) − h (k − 1)) ⎟ 1 2 ⎝ ⎠

(4)

Equation (4) can be written as follows:

(3)

h1lin (k ) = θ1T (k ) w1 (k )

(5)

With:

θ1T = ⎡⎣ a0(1) b0(1) ⎤⎦ T w1 (k ) = [ h1 (k − 1) Q1 (k − 1) ]

(6) (7)

This model represents the linear part of the level h 1.

θ1

and w1 represent the vector of the parameters and the vector of observation respectively. The nonlinear part of the model of the level h1 is described by the following equation:

⎛kT ⎞ h1nl (k ) = − ⎜ 1 e ⎟ h1 (k − 1) − h2 (k − 1) *.. (8) ⎝ S ⎠ sign (h1 (k − 1) − h2 (k − 1)) Equation (11) can be written in the following form:

h1Nlin (k ) = f1 (h1 , h2 ) With f1 is a nonlinear function.

(9)

Thus, the linear model of the first tank is described by the following equation: T φ1lin (k ) =θ 1 w1 (k ) (10) Where θ 1 is the vector of the parameters and w1 is

J i (k ) = ∑

(11)

(

2

ai (l ) ei (l ) − 4 M 2

)

2 (1 + w(l − k )T w(l − k ) )

k

+c

∑ (1 − a (l ) ) e (l ) i

2

i

(20)

i = 1, 2

l = k − N +1

Where:

1

The model of the second tank is given by:

⎛T ⎞ h2 lin (k ) = h2 (k − 1) + ⎜ e ⎟ Q2 (k − 1) ⎝S⎠

k

l =1

given by equation (7). The nonlinear model is represented as follows: T φ2 NL (k ) = θ 1 w1 (k ) +  f1 (h1 , h2 )  f represent the estimated nonlinear function.

recursive least squares algorithm. However, the algorithm of retro-propagation is used to estimate nonlinear model. The following step we will determine the criterion of validity. For our case, the criterion of commutation is given by the following equation :

(12)

⎧ e1 (k ) = φ1lin (k ) − h1lin (k ) ⎨ ⎩ e2 (k ) = φ2 NL (k ) − h2 nl (k )

(21)

And (13)

⎪⎧1 if ei (k ) > 2Δ ai (k ) = ⎨ ⎪⎩0 otherwise

(14)

With N is an entity, c is a positive constant, Δ is the

(15)

greatest value of ei ( k ) . One notes that this criterion is

This model represents the linear part of the characteristic equation of the second level. With θ 2 and w2 represent the vector of the parameters

composed of two parts. First part is used to control the growth rates of the signals. The second part calculates the average quadratic error between the estimated model and the real system. The principle of commutation is explained as follows. At every moment, the model which has the minimal criterion is responsible for the management of the control of the process.

The model (12) can be written in the following form:

h2 lin (k ) = θ 2T w2 (k ) With:

θ 2T = ⎡⎣ a0(2) b0(2) ⎤⎦ T w2 = [ h2 (k − 1) Q2 (k − 1)]

and the vector of observation respectively. The nonlinear part is described by the following equation:

⎛kT ⎞ h2 nl (k ) = ⎜ 1 e ⎟ h1 (k − 1) − h2 (k − 1) *.. ⎝ S ⎠ sign (h1 (k − 1) − h2 (k − 1))

(16)

Equation (16) can be written in the following form:

h2 Nl (k ) = f 2 (h1 , h2 ) (17) Where f 2 is a nonlinear function. Thus the library corresponding to the second tank is also formed by a linear model and a nonlinear one. The linear model is described as follows:

φ2lin (k ) = θ (k ) w2 (k ) (18) Where θ 2 and w2 represent the vector of the T 2

estimated parameters and the vector of observation respectively. The nonlinear model is given by the following equation: T φ2 NL (k ) = θ 2 (k ) w2 (k ) + f 2 (h1 , h2 ) f represent the estimated nonlinear function.

(19)

2

The library of the system is being elaborated. Now we determine the moments of validity corresponding to the linear model. To estimate the linear model, we used the

(22)

4. ADAPTIVE CONTROLLERS The multi-model control is generated by an interpolation between the elementary controls of each model. We limit our study to elementary controls relating to the model of the first tank. By the linear estimate model (10) or (18), we have the linear adaptive controller C1 : T θ 1 w1 (k ) = ⎡⎣ yref (k ) − h1 (k ) ⎤⎦

(23)

By the nonlinear estimate model (11) or (19), we have the neural network based nonlinear adaptive controller C2: T θ 1 w1 (k ) + f (h1 , h2 ) = ⎡⎣ yref (k ) − h1 (k ) ⎤⎦

(24)

The same principle of the controller to the elementary control of the second tank. Following these structures of control and the criterion of commutation we obtain the control using the multimodel approach.

Moreover;

5. ADAPTIVE CONTROL ALGORITHM The adaptive control algorithm proposed in this paper is composed of the identification algorithm for controller parameters, the linear adaptive controller, the neural network nonlinear adaptive controller and the switching mechanism. It can be summarized as flows: Step 1: Measure h(t) and construct datum vector w(k-1).

Step 2: Estimate the controller parameters θ * (k ) using (10) - (11) or (18)-(19). Step 3: Calculate the model error e1 ( k ) and e2 (k ) by (21), and calculate J1 ( k ) and J 2 ( k ) by (20). Step 4: Compare J1 ( k ) and J 2 ( k ) , and choose the controller C* ; (23) or (24), corresponding to the smaller J * ( k ) .

N →∞

lim

k →∞

t =k

the controller C* to be applied to the system.

2 (1 + w (k − 1) w(k − 1) )

(

2 (1 + wT (k − 1) w(k − 1) )

(25)

k →∞

Proof: According to the recursive algorithm of least squares, we define θ 1 (k ) as flows:

According to the expression of the error e1 ( k ) , along with (iii) in assumptions 1, there exist positive constants c1 , c2 , c3 and c4 , such that:

hi (k ) ≤ c1 + c2 max 0 ≤τ ≤t e1i (k ) i = 1, 2 1≤ i ≤ n

Qi (k − 1) ≤ c3 + c4 max 0≤τ ≤k hi (τ ) 1 ≤i ≤ n

w(k − 1) = [ h1 (k − 1) h2 (k − 1) Q1 (k − 1) Q2 (k − 1)] ; T

From (24), the boundedness of the input and output signals are determined by the boundedness of e1 ( k ) . Now assume that e1 ( k ) is unbounded. Then by (22)

k > T , e1 (k ) > Δ and

there exist T > 0 ; when

a1 (k ) = 1 , i.e the numerator in Eq (20) is a positive increasing

{ e (k ) } such

sequence

that

n

1

lim e1 (kn ) = ∞ .

kn →∞

kn →∞

a (k ) w (k − 1) e (k ) θ 1 (k ) =θ 1 (k − 1) + 1 T 1 1 + w1 (k − 1) w1 (k − 1)

Since:

a1 ( k n )

2

a1 (k ) e1 (k ) − 4M 2

)

2 (1 + wT (k − 1) w(k − 1) )

=

a1 ( k n )

n

(

a1 (k ) = 1 for e1 (k ) > Δ and is 0 otherwise ;

2

=

( e (k 1

2 1+

)

2

− 4M

n

2

)

2

− 4M

w ( k n − 1)

a1 ( k n )

( e (k 1

)

n

2

2

2 (1 + ( w ( k n − 1) +

Since

is nonincreasing sequences.

1

)

2 (1 + w ( k n − 1) w ( k n − 1) )

described in Chen and Narendra (2001), it

(

( e (k T

Adopt the similar approach as what is

Hence, θ 1 ( k ) is bounded.

(26)

Since: such that lim e1 ( kn ) = ∞ T 1

1

→0

real scaler sequence, and there exist a monotony

lim e(k ) = lim y (k ) − yref (k ) < ε

{ θ (k ) }

)

w(k − 1) ≤ c5 + c6 max 0≤τ ≤ k e1 (τ )

Theorem , : For the system (1) with the adaptive control algorithm (10)-(17), the input and output signals in the closed loop system are bounded. Moreover, by properly choosing the structure and parameters a neural network, for a predefined arbitrary small positive number ε , the tracking error of the system satisfy:

θ 1 (k ) ≤ θ 1 (k − 1) −

2

a1 (k ) e1 (k ) − 4 M 2

0 (27) 8c62

But this contradicts (22), and hence the assumption that e1 (k ) is unbounded is false. Consequently, the input and output signals are bounded when the linear adaptive controller is used alone. Second, According to the expression of the error e1 ( k ) , and along with (iii) in assumptions 1, there exist positive constants d1 , d 2 , d3 , d 4 such that : 1≤ i ≤ n

Qi (k − 1) ≤ d3 + d 4 max 0 ≤τ ≤ k hi (τ ) Therefore, similarly there exist positive constants d5 , d 6 such that:

w(k − 1) ≤ d 5 + d 6 max 0≤τ ≤ k e2 (τ )

(28)

By (24), the second term in (17) is always bounded, so J1 (k ) is bounded by employing (20). For J 2 ( k ) there can be two cases: (i): J 2 ( k ) is bounded. By the switching rule (20), it

lim

k →∞

(

2

) →0

the model error e(k ) = e1 (k ) or e2 (k ) satisfies:

lim

k →∞

(29)

2 (1 + wT (k − 1) w(k − 1) )

Therefore,

(

2

a ( k ) e( k ) − 4M 2

)

2 (1 + wT (k − 1) w(k − 1) )

of

→0

k →∞

j = 1, 2

Then by the switching rule (20) and (21), when k → ∞ , the system chooses the controller corresponding to the smaller model error as the control input of the system, so from (21), the tracking error of the system, is equivalent to the nonlinear model error, that is to say, the linear model error. For the nonlinear model error, from (21), we have

e2 (k ) = f (h1 , h2 ) − f (h1 , h2 )

(32)

ε < lim e1 (k ) , f (h1 , h2 ) − f (h1 , h2 ) < ε

1≤i ≤ n

a2 (k ) e2 (k ) − 4 M 2

lim e j (k ) ≤ 2M

When properly choose the structure and parameters of a neural network, for a predefined arbitrary small positive number.

hi (k ) ≤ d1 + d 2 max 0 ≤τ ≤ k e2i (τ ) i = 1, 2

follows that:

the model error e j ( k ) , j = 1, 2 , satisfies:

the

system,

(30)

Where:

⎧⎪1 si e(k ) > 2 M a( k ) = ⎨ (31) ⎪⎩0 otherwise (ii) : J 2 ( k ) is unbounded. Since J1 ( k ) is bounded, ther exists a constant k 0 such that J1 (k ) ≤ J 2 (k ) , ∀ k ≥ k0 Therfore when k ≥ k0 + 1 , by the switching mechanism, the model e(k ) = e1 (k ) and also satisfies Eq (26). From (26), (27) and (29), using the same lines as the above, it follows that w( k − 1) is bounded, i.e, the input and output signals in the closed-loop switching system are bounded.

k →∞

Can be achieved. Thus, when k → + ∞ the nonlinear model error can be less than the linear model error, consequently, the tracking error of the system will be e2 (k ) , which satisfy:

lim e(k ) = lim e2 (k ) < ε k →∞

k →∞

7. SIMULATION RESULTS In this section, we present the simulation results coping with application of the adaptive multi-model controller to the twin tank process. The parameters of hydraulic system are : section of each tank = 155.44 cm2, the gain of motor driven kp= 0.59 m3/s.v , characteristics of pumps k1= 23.45, k2 = 17.62, high level hmax= 0.25m, input voltage umax= 10v and the constant of gravity g= 9.8 m/sec2. We present like result of simulation, elementary flows taken from each model respectively of the library of the first and the second and we present the criteria of commutation and the water level hi corresponding to tanks is by applying the order multi-model.

7.1. Results of the first tank In figure 1, we presented the flow extracted by the linear model (continuous line) and the flow of the nonlinear model (discontinuous line).

flow of the nonlinear model

0.25

7.2. Results of the second tank

flow of the linear model

flows (L/s)

0.2

0.15

flow of the nonlinear model flow of the linear model

0.25

0.1

0.05

0

0

10

20

30

40

50 times (s )

60

70

80

90

100

flows (L/s)

0.2

0.15

0.1

Fig1: The two basic flows for the first tank.

0.05

0

3 1- Applied the nonlinear control 2- Applied the linear control

0

10

20

30

40

50 times (s)

60

70

80

90

100

Fig 5 : The two basic flows for the second tank.

2.5

2

In figure 5, we presented the flow extracted by the linear model (continuous line) and the flow of the nonlinear model (discontinuous line).

1.5

1

0.5

3 0

0

10

20

30

40

50 times (s)

60

70

80

90

1- Applied the linear model 2- Applied the nonlinear model

100

2.5

Fig 2 : Switching criterion 2

In figure 2, we presented the moments of commutations. If the criterion is equal to one (1) then the control extracted of the linear model is applied. Whereas the control extrated of the nonlinear model is intervened only when the criterion is equal to (2).

1.5

1

0.5

0

0

10

Signal of the first tank h1 Signal of refernce

40

50 times(s)

60

70

80

90

100

In figure 6, we presented the moments of commutations. If the criterion is equal to one (1) then the control extracted of the linear model is applied. Whereas the control extrated of the nonlinear model is intervened only when the criterion is equal to (2).

15

10

5

0

10

20

30

40

50 times(s)

60

70

80

90

Fig 3 : Tracking of the water level h1 of the reference signal. We chose a dynamic signal of reference to check the good tracks of the water level. In figure 3, we show that the level of water of the first tank tracks well of the reference signal. Quadratic error

The water level h2 The reference signal

20

15

10

5

0

3

0

10

20

30

40

50 times(s)

60

70

80

90

100

Fig 7 : Tracking of the water level h2 of the reference signal.

2.5

2

1.5

1

0.5

0

25

100

The water level h2 (cm)

thewater level h1(cm)

20

eps1(%)

30

Fig 6 : Switching criterion.

25

0

20

0

10

20

30

40

50 times (s)

60

70

80

Fig 4 : The quadratic error.

90

100

2.5 quadratic error between yref and h2

2

eps2(%)

1.5

1

0.5

0

0

10

20

30

40

50 times(s)

60

70

80

90

100