Controller Design for Processes with Unknown Dynamics and Input

Article pubs.acs.org/IECR

Controller Design for Processes with Unknown Dynamics and Input Nonlinearity with Applications to Bioreactors Zahra Shamsi and Mohammad Shahrokhi* Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11155-9465 Azadi Avenue, Tehran, Iran ABSTRACT: An efficient approach for controlling processes with completely unknown dynamics subject to input nonlinearity and in the absence of state measurements is presented. To handle the system input nonlinearity, this constrain has been included in the system unknown dynamics and a new input has been defined. An unknown Lyapunov function has been considered, and the terms appearing in its time derivative are estimated by neural networks and the stability of the closed-loop system has been established. The effectiveness of the proposed control scheme has been demonstrated by applying the scheme to different bioreactors with unknown dynamics under input saturation. The performances of designed controllers in set-point tracking, load rejection, and model mismatch have been evaluated via a simulation study.

1. INTRODUCTION The goal of this work is development of a stable controller for processes with unknown dynamics subjected to input nonlinearity, which has not been considered before in the literature. During the last decades, we have experienced a wide range of researches concentrated on the problem of controlling highly uncertain and possibly unknown nonlinear dynamical systems by using artificial intelligent techniques such as neural networks and fuzzy logic systems. The driving force behind this impressive activity is the capability of artificial estimators in approximating the nonlinear behavior of nonlinear functions.1−3 However, each of the proposed methods has certain limitations. For controlling different classes of nonlinear unknown dynamical systems, many different methods have been proposed in the literature, but none of them considers a general form of input dynamic nonlinearity. Several works have been published regarding the integration of a sliding mode control technique with the artificial intelligent estimator to improve robustness of the control scheme;4−7 however, these methods are only applicable to systems in a special form. In some other researches the backstepping control design is coupled with fuzzy systems to estimate the unknown states. Similarly, several stable fuzzy adaptive backstepping control schemes were proposed for uncertain nonlinear systems in the strict feedback form.8,9 Feedback linearization control has been also combined with artificial intelligent methods in some researches which are only suitable for systems in the normal form, while many of the actual systems are not transferable to the normal form.10,11 A more general control method is suggested which is motivated by the symmetric matrix decomposition; however, it is not suitable for a general form of system nonlinearity and only applicable to unknown nonlinear dynamics with symmetric gain matrix.12 For controlling affine systems in a more general form, a new © 2016 American Chemical Society

approach has been proposed by using adaptive neural networks to design a stable adaptive control scheme for completely unknown nonlinear affine dynamics.13 In the design of standard adaptive nonlinear control schemes proposed in the literature, such as feedback linearization, sliding mode control, and backstepping technique, generally it is assumed that the process model structure is known and only model parameters are unknown. Moreover for controller implementation, the system states are required which are not available in most of practical applications. In addition the control input is usually constrained and has a nonlinear dynamic such as backlash, saturation, and dead-zone which can lead to instability or performance deterioration.14−17 The advantage of the proposed neurocontroller over the standard nonlinear control strategies proposed in the literature is its capability in handling these limitations. The main objective of this work is controlling systems with completely unknown dynamics subject to input nonlinearity in the absence of full state measurements which has not been addressed before in the literature. To the best of authors’ knowledge, the control scheme proposed by Rovithakis et al.13 considers the most general form of unknown dynamics; therefore in this work their approach has been extended to cover the cases where the system with unknown dynamics is subject to input nonlinearity and all state measurements are not available. Because of the well-known capability of the neural network in approximating the unknown functions, they have been used to estimate the unknown process dynamics in this work. To handle the input nonlinearity, this nonlinearity has been included in the Received: December 26, 2015 Accepted: February 9, 2016 Published: February 10, 2016 2584

DOI: 10.1021/acs.iecr.5b04928 Ind. Eng. Chem. Res. 2016, 55, 2584−2593

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Industrial & Engineering Chemistry Research ⎡ z1̇ ⎤ ⎢ ⎥ ⎡0⎤ ⎢ z 2̇ ⎥ ⎢ ⎥ ⎡ ⎤ · f (z) + G(z)N (zn + 1) ⋮ ⎢ ⎥ ⎥ + ⎢ ⎥v z = ⎢⋮ ⎥ = ⎢ ⎣ ⎦ ⎢0⎥ 0 ⎢ zṅ ⎥ ⎢⎣ ⎥⎦ 1 ⎢ ⎥ ⎣ zṅ + 1⎦

system dynamics. The time derivative of an unknown Lyapunov function is estimated by the neural network and stability of the closed-loop system has been established by using the Lyapunov stability theorem. If the system states are not available, by taking advantage of the system observability, the network regressors are selected to be only functions of system outputs, their derivatives and tracking error. Effectiveness and robustness of the proposed control scheme has been shown by applying it to bioreactors through simulation study.

By defining

2. PROBLEM FORMULATION Consider the affine nonlinear dynamical system under input nonlinearity in the following form: z ̇ = f (z ) + G (z )U

(1)

y = h(z)

(2)

U = N (u)

(3)

(11)

⎡ f (z) + G(z)N (zn + 1)⎤ ⎥ f ̅ (z ̅ ) = ⎢ ⎣ ⎦ 0

(12)

⎡0⎤ ⎢ ⎥ ⋮ G̅ (z ̅ ) = ⎢ ⎥ ⎢0⎥ ⎢⎣ ⎥⎦ 1

(13)

The modified system equations can be written as ·

where z ∈ Rn is the system state which is assumed to be available, U ∈ R is control input, y is the process output, and f, h, N, G are continuous, locally Lipschitz, vector fields. N(u) indicates the input nonlinearity such as the saturation function. The control objective is to force the process output to follow a given bounded reference trajectory yr(t), using a bounded control action. It is assumed that ẏr(t) is also bounded. By substituting eq 3 into eq 1 we have z ̇ = f (z) + G(z)N (u)

z = f ̅ (z ̅ ) + G̅ (z ̅ )v

(14)

y = h ̅ (z ̅ )

(15)

where f ̅ (z ̅ )and G̅ (z ̅ ) are unknown, continuous, and locally Lipschitz functions. The adaptive neuro-control strategy can be developed based on the above equations.13 Figure 1 illustrates the proposed control system diagram.

(4)

It is assumed that, N(u) and G(z) are unknown continuous locally Lipschitz functions. Therefore, the product of G(z)N(u) is also continuous and locally Lipschitz. A new function, G1, can be defined which includes the input nonlinearity of the system as given below:

G1(z , u) = G(z)N (u)

Figure 1. Schematic diagram of the proposed control strategy.

(5)

The system equation can be rewritten in the following form: z ̇ = f (z) + G1(z , u)

3. NEURO-CONTROLLER DESIGN Define the tracking error as e = y − yr

(6)

If a new function is defined as f1 (z , u) = f (z) + G1(z , u)

From eq 14−16, the error dynamic equation can be obtained as

(7)

then the system is described by z ̇ = f1 (z , u)

ė = (8)

zṅ + 1 = u ̇ = v

∂h ̅ (z ̅ ) ∂h ̅ (z ̅ ) G̅ (z ̅ )u − yṙ (t) f ̅ (z ̅ ) + ∂z ̅ ∂z ̅

(17)

It is assumed that the solution of eq 17 can be forced to be uniformly ultimately bounded with respect to an arbitrarily small neighborhood of e = 0. Therefore, there exists a radially unbounded Lyapunov function V(e):Rn → R+ and a control input u0(e,z)̅ such that ⎤ ∂V (e) ⎡ ∂h ̅ (z ̅ ) ∂h ̅ (z ̅ ) f ̅ (z ̅ ) + ⎢ · G̅ (z ̅ )u 0(e , z ̅ ) − yṙ ⎥ ≤ 0, ∂e ⎣ ∂z ̅ ⎦ ∂z

As can be seen from the above equation, the new system is not affine. For using the adaptive neural network control strategy, the system should be in the affine form.13 For this purpose, a new state is added to the system equation. This new state is the actual process input, and the time derivative of this state is defined as the virtual input. Therefore, we have the following equations:

zn + 1 = u

(16)

(9)

(18)

∀e∈ε

where ε is a set defined as ε = {e ∈ R :|e| ≥ e0 > 0} and e0 is an arbitrarily small positive constant. Since f ̅ (z ̅ ) and G̅ (z ̅ ) are unknown functions, V(e) is also unknown. Neural networks are used to estimate the unknown n

(10)

Using the above equations, the augmented system equation is written as 2585


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outputs and these changes are sensed via tracking errors, leading to adaptation of the network weights. The following control law is considered:

parts of time derivative of the unknown Lyapunov function. Functions A, B and C are defined as13 A (e , z ̅ ) =

∂V(e) ∂h ̅ (z ̅ ) f ̅ (z ̅ ) ∂e ∂z ̅

∂V(e) ∂h ̅ (z ̅ ) G̅ (z ̅ ) ∂e ∂z ̅ ∂V (e) C(e) = ∂e B (e , z ̅ ) =

⎡ a(e , z ̅ , Wa , Wc , y ̇ ) + γ(|e|) ⎤ r ⎥bT(e , z ̅ , Wb) u = −⎢ 2 b ( e , z , W ) | | ⎣ ⎦ ̅ b

(19)

where γ(|e|) is a positive bounded and invertible function ∀e ∈ Ω ⊂ Rn and a and b are defined as

(20)

a(e , z ̅ , Wa , Wc , yṙ ) = WaTSa(e , z ̅ ) − WcTSc(e)yṙ

(21)

Without loss of generality, linear in weight neural networks are used to estimate the above functions. These networks have simple structures and can be designed easily. Therefore, A, B, and C are estimated by neural networks as given below:18 A(e , z ̅ ) = W a*TSa(e , z ̅ ) + ωa(e)

(22)

B(e , z ̅ ) = W b*TS b(e, z ̅ ) + ω b(e)

(23)

C(e) = W c*TSc(e) + ωc(e)

(24)

(26)

W͠ b = Wb − W b*

(27)

W͠ c = Wc − W c*

(28)

where Wa*, Wb*, Wc* are the optimal values of the network weights. The following adaptation laws for updating the neural network weights are considered: Wȧ = Pa{−kaWa + kSa(e , z ̅ )}

(29)

Ẇ b = −k bWb + kS b(e , z ̅ )u

(30)

Wċ = Pc{−kcWc − kSc(e)yṙ }

(31)

4. SIMULATION RESULTS Chemical and biochemical reactors are good candidates for utilizing the proposed control strategy. These reactors are widely used in food, pharmaceutical, and environmental industries, and their control is an important subject in the literature. Because of the complex and time varying nature of the bioreactors, their dynamics is mostly unknown. Ideally, a controller should be able to ensure the required performance for a whole variety of fermentation (different microorganism) and different process scales. The main control challenge roots in the fact that the dynamics of the system depend on the particular process type and reactor scale. Moreover, these dynamics are strongly timevarying due to gradual changes in the process operating

where ka, kb, kc > 0 are design constants and Pa and Pc denote the projection operators with respect to the convex sets >a and >c 19 defined below: >a = {Wa ∈ RLa: |Wa| ≤ Ma} Lc

>c = {Wc ∈ R : |Wc| ≤ Mc}

(35)

b(e , z ̅ , Wa) = WbTS b(e , z ̅ ) (36) As shown in ref 13, control law 34 with update laws 29−31, guarantee the uniform ultimate boundedness of the error e with respect to the a compact set, as well as the boundedness of all n other system signals, if |b(e,z,W ̅ b)| ≠ 0, ∀|e| ∈ Ω ⊂ R . For details, the reader is referred to ref 13. Using a resetting method suggested in ref 13 keeps the value of b away from zero. Therefore, stability of the closed loop system is established.13 Regarding the tuning parameters of the proposed neurocontroller, it should be mentioned that there are three groups of tuning parameters. The first group of parameters are gains of adaptation laws, ka, kb, kc, k, in eqs 29−30, which are positive constants as mentioned before. Since the adaptation laws are gradient type, changing the gain has not a significant effect on the controller performance and they can be set to some positive constants. The second group corresponds to initial values of the neural network weights. Since the network is adaptive and weights are updating, the effects of initial values of the weights are damped in a short period of time. Therefore, these initial values have also no significant effect on the controller performance and can be chosen arbitrarily. The last and the most important design parameters are functionality of the regressors. The regressors contain smooth monotonic increasing functions, which are usually sigmoid activated functions.13 Parameters of these sigmoid functions are selected in such a way that the regressor is a monotone and smoothly increasing function in the range of its variable. A combination of regressors should be also selected such that comprehensive and none repetitive information on all available signals is captured.13,18 Remark: It was assumed that the system states which are used in regressors are available. If the system is observable and only the system outputs are measured, we can still apply the proposed control scheme by replacing the system state in the regressor by the system outputs and their derivatives due to system observability property. In the first case study, two different simulation results are given. In the first one, it is assumed that both system states are measured while in the second one, it is assumed that only one state is measured.

In eqs 22−24, A, B, and C are neural network outputs, z̅ and e are the network inputs. Wa, Wb, Wc are vectors of synaptic weights and Sa, Sb, Sc are matrices of regressors. ωa, ωb, and ωc are network errors. To derive adaptive laws for updating the neural network weights and the control action, the following Lyapunov function is considered:13 1 1 1 L = kV (e) + |W͠ a|2 + | W͠ b|2 + |W͠ c|2 (25) 2 2 2 where V(e) is the unknown Lyapunov function, k > 0 is a design constant and W̃ c,W̃ b,W̃ a are the network weights errors given below:

W͠ a = Wa − W a*

(34)

(32) (33)

Ma and Mc can be chosen as sufficiently large positive constants. The neuro-controller weights, Ẇ a, Ẇ b and Ẇ c, are updated in an online manner according to the adaptive laws given by eqs 29−31. As can be seen from these equations, the network weights are updated as long as the tracking errors are nonzero. The changes in the system dynamics are reflected in the system 2586


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Industrial & Engineering Chemistry Research conditions. On the other hand, in bioreactor control, usually the control input is the dilution rate which is subject to the saturation constraint. These processes are classified as systems with unknown dynamics subject to input nonlinearity. The control objective in bioreactors could be biomass production or producing a special kind of product. Industrial experience at Applikon BV, where controllers are developed for different types of fermentation processes, shows that a controller with fixed parameters cannot control the bioreactor throughout the entire process run; therefore adaptive control has been chosen to handle the time-varying nature of the process. To show the effectiveness of the proposed control scheme, simulation studies have been performed for two different cases. In the first case, the reactor has two states and one output, while in the second case the reactor has four states and one output which is a nonlinear function of the states. 4.1. First Case Study. (a). System States Are Measured. Consider a continuous stirred tank bioreactor where a single population of micro-organisms is cultivated on a single limiting substrate. Substrate is fed into the culture with concentration sR.The process dynamics are given by the following equations:20 x1̇ = x1(μ(s1) − D1) s1̇ = D1(sR − s1) −

y=x

xs K1 + s

⎡0⎤ ⎢ ⎥ G̅ (z ̅ ) = ⎢ 0 ⎥ ⎣1 ⎦

(50)

h ̅ (z ̅ ) = z1

(51)

·

z = f ̅ (z ̅ ) + G̅ (z ̅ )v

(52)

y = h ̅ (z ̅ )

(53)

Control law 34 and adaptive laws 29−31 are applied with the following functions and parameters:

(41)

The control objective is tracking the trajectory of yr(t). D is subject to saturation constraint. Therefore, it can be substituted by Sat(u) in eqs 43 and 44, where u is the control input signal. After substitution for D, the system equations become

s ̇ = Sat(u)(1 − s) −

(49)

In terms of new state variables, the system equations can be written as follows:

(42)

⎞ ⎛ s x ̇ = x⎜ − Sat(u)⎟ ⎠ ⎝ K1 + s

⎡ ⎞ ⎤ ⎛ ⎢ z1⎜ z 2 − Sat(z 3)⎟ ⎥ ⎢ ⎠ ⎥ ⎝ K1 + z 2 ⎥ f ̅ (z ̅ ) = ⎢⎢ z1z 2 ⎥ Sat(z 3)(1 − z 2) − ⎢ K1 + z 2 ⎥ ⎢ ⎥ ⎣ ⎦ 0

(38)

(40)

xs K1 + s

(47)

(48)

where μm is the maximum growth rate constant and Ks is a saturation constant. To make the variables dimensionless, YsR, sR, μm, 1/μm are used as the units of x1, s1, D1, time, respectively.20 Therefore, the dimensionless system model becomes

s ̇ = D(1 − s) −

z3 = u ⎡ ⎞ ⎤ ⎛ ⎢ z1⎜ z 2 − Sat(z 3)⎟ ⎥ ⎡ z1 ⎤ ⎢ ⎠ ⎥ ⎡0⎤ ⎝ K1 + z 2 · ⎢z ⎥ ⎢ ⎥ ⎢ ⎥ z = ⎢ 2⎥ = ⎢ z1z 2 ⎥ + ⎢ 0 ⎥v − − Sat( z )(1 z ) 3 2 ⎢⎣ z 3 ⎥⎦ ⎢ K1 + z 2 ⎥ ⎣ 1 ⎦ ⎢ ⎥ ⎣ ⎦ 0

where x1 is the biomass concentration, s1 is the substrate concentration, μ is the specific growth rate function, and Y is the yield coefficient. D1 is the dilution rate which could vary between 0 and 1.21 Many analytical expressions for the function μ have been proposed empirically or experimentally. The most classical function is the Monod model: μ s1 μ= m Ks + s1 (39)

⎞ ⎛ s x ̇ = x⎜ − D⎟ ⎠ ⎝ K1 + s

(46)

where ż3 = u̇ = v. Define f ̅ (z ̅ ), G̅ (z ̅ ) and h ̅ (z ̅ ) as given below:

(37)

x1μ(s1) Y

z2 = s

(43)

γ(|e|) = 0.4|e|

(54)

⎡ s (e)2 ⎤ ⎥ ⎢ 1 ⎢ s (e)s (z )⎥ 1 2 1 ⎥ Sa(e , z ̅ ) = ⎢ ⎢ s1(e)s2(z 2)⎥ ⎥ ⎢ ⎢⎣ s1(e)s3(z 3) ⎥⎦

(55)

⎡ s1(e)s3(z1) ⎤ ⎢ ⎥ S b(e , z ̅ ) = ⎢ s1(e)s3(z 2)⎥ ⎢ ⎥ ⎢⎣ s1(e)s3(z 3)⎥⎦

(56)

⎡ s1(e) ⎤ ⎥ Sc(e) = ⎢ ⎢⎣ s (e)2 ⎥⎦ 1

(57)

where

(44)

s1(x) =

3 − 1.5 1 + exp( −x)

(58)

To put the system equations in the desired general form, the following states are defined: z1 = x (45)

s 2 (x ) =

2 −1 1 + exp( −x)

(59)

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Industrial & Engineering Chemistry Research s3(x) =

1 − 0.5 1 + exp( −x)

(60)

The initial conditions of the neural network weights are WaT(0) = [1 1 −1 −1 ]

(61)

WbT(0) = [−300 −3000 −300]

(62)

WcT(0) = [1 1]

(63)

Two different set point trajectories are considered, namely sine and step changes. Sine Set-Point Tracking. In this simulation, it is desired to track the following bounded reference trajectory: yr = 0.04 sin(0.07t + 2.5) + 0.65

(64)

Variations of biomass and substrate concentrations and dilution rate are shown in Figures 2−4, respectively. As can be seen from Figure 2, the desired trajectory is tracked very well.

Figure 4. Dilution rate variations versus time for sine set-point tracking.

0.96. To test the performance of the designed controller in the case of model mismatch at t = 70, system equations have switched to the following equations which are based on the Haldane kinetic model20 as given below: ⎛ 1 x ̇ = x⎜⎜ K1 ⎝1 + s + s ̇ = D(1 − s) −

s K2

⎞ − D⎟⎟ ⎠

(65)

x 1+

K1 s

+

s K2

(66)

Variations of biomass concentration and dilution rate are shown in Figure 5 and Figure 6, respectively. As can be seen

Figure 2. Biomass concentration variations versus time for sine set-point tracking.

Figure 5. Biomass concentration variations versus time for step setpoint tracking in the presence of model mismatch.

from Figure 5, the desired set-point is tracked well and the effect of model mismatch on the biomass concentration has been damped quickly. (b). System Output Is Measured. In this simulation, for the previous reactor, it is assumed that only system output is measured and therefore the regressors given by eqs 55−57 are modified as follows:

Figure 3. Substrate concentration variations versus time for sine setpoint tracking.

Step Set-Point Tracking. In this simulation, it is desired to change the dimensionless biomass concentration from 0.81 to 2588


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Industrial & Engineering Chemistry Research ⎡ s1(e) ⎤ ⎥ Sc(e) = ⎢ ⎢⎣ s (e)2 ⎥⎦ 1

(69)

It should be noted that z3 is not the system state and is the augmented state defined by eq 9 and therefore it is available. The results for sine and step set-point trajectories are shown in Figures 7 and 8.

Figure 6. Dilution rate variations versus time for step set-point tracking in the presence of model mismatch.

⎡ s (e)2 ⎤ ⎢ 1 ⎥ ⎢ s (e)s (y)3 ⎥ 1 2 ⎥ Sa(e , z ̅ ) = ⎢ ⎢ s (e)s (y) ⎥ ⎢ 1 2 ⎥ ⎢⎣ s1(e)s3(z 3)⎥⎦

⎡ s1(e)s3(z 3)⎤ ⎢ ⎥ S b(e , z ̅ ) = ⎢ s1(e)s3(y) ⎥ ⎢ ⎥ ⎢⎣ s1(e)s3(z 3)⎥⎦

(67)

Figure 8. Biomass concentration variations versus time for step set-point tracking and output measurement in the presence of model mismatch.

As can be seen from these figures, performances for the case where output is measured are almost the same as those for the case where system states are measured. To test the performance of the proposed scheme in the presence of measurement noise, a white noise with noise-to-signal

(68)

Figure 7. Biomass concentration variations versus time for sine set-point tracking and output measurement. 2589


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Industrial & Engineering Chemistry Research ratio equal to 10% has been added to the process output and the same simulation is conducted. The results for biomass concentration variation and dilution rate variation are shown in Figure 9

Figure 11. Biomass concentration variations versus time for step setpoint tracking in the presence of model mismatch under PID controller. Figure 9. Biomass concentration variations versus time for step setpoint tracking and output measurement in the presence of model mismatch and measurement noise.

Figure 12. Biomass concentration variations versus time for step set-point tracking and output measurement in the presence of model mismatch and measurement noise under PID controller.

Figure 10. Dilution rate variations versus time for step set-point tracking and output measurement in the presence of model mismatch and measurement noise.

and Figure 10. As can be seen the proposed controller can handle model mismatch and measurement noise satisfactorily. To compare the performance of the designed neuro-controller with that of a conventional PID controller, the Matlab autotuner is used to tune the PID parameters. The tuned PID controller has been applied to the bioreactor under the same conditions. The result is shown in Figure 11. Comparing performance of the PID controller with that of the proposed neuro-controller shown in Figure 8 indicates that response of the proposed controller is faster with almost no overshoot for the set-point tracking. The results also indicate that both controllers are robust against the system uncertainty and PID has a faster response in damping the system dynamic changes. To check the effect of measurement noise, the same level of noise is added to the process output and the tuned PID controller is tested under the same condition used for the proposed controller. Simulation results are shown in Figures 12 and 13. Comparing these results with those of the

Figure 13. Dilution variations versus time for step set-point tracking and output measurement in the presence of model mismatch and measurement noise under PID controller.

proposed controller (Figures 9 and 10) reveals that the proposed control scheme has a much better performance in the case of 2590


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Industrial & Engineering Chemistry Research measurement noise. It should be also noticed that the closedloop stability cannot be proved for the PID controller while it is established for the proposed neuro-controller. 4.2. Second Case Study. In the second case, a dynamical model of an anaerobic digestion bioreactor in which the methane gas production rate is the desired output has been considered. In this bioreactor, due to the high complexity of the process dynamic, control of the methane gas production has been remained as a challenging problem in practice. The system equations are expressed as follows:22 s ̇ = −k3μ1x1 + u(s1,0 − s1)

(70)

x1̇ = μ1x1 − αux1

(71)

s2̇ = −k1μ2 x 2 + k 4μ1x1 + u(s2,0 − s2)

(72)

x 2̇ = μ2 x 2 − αux 2

(73)

y = k 2μ2 x 2

(74)

(76)

mmol mmol , k 2 = 453 , g g mmol mmol k 3 = 42.14 , k4 = 116.5 g g

k1 = 268

μm,1 = 1.2d−1,

(80)

s1(x) =

2 −1 1 + exp( −0.05x)

(81)

s 2 (x ) =

2 −1 1 + exp( −0.05x)

(82)

s3(x) =

1 − 0.5 1 + exp( −0.05x)

(83)

s4(x) =

1 − 0.5 1 + exp( −0.05x)

(84)

WaT(0) = [0.1 0.1 0.1 0.1 0.1 0.1 ]

(85)

WbT(0) = [0.002 0.002 0.02 0.02 0.02 0.02 ]

(86)

WcT(0) = [1 1]

(87)

Sine Set-Point Tracking. In this simulation, it is desired to track the following bounded reference trajectory.

The model parameters are

α = 0.5,

⎡ s1(e) ⎤ ⎥ Sc(e) = ⎢ ⎢⎣ s (e)2 ⎥⎦ 1

The initial conditions of neural network weights are as follows:

μm,2 s2 Ks ,2 + s2 + KIs22

(79)

where

where s1, x1, s2, x2, and y represent the organic matter concentration (g/L), the acidogenic bacteria concentration (g/L), the volatile fatty acid concentration (mmol/L), the methanogenic bacteria concentration (g/L), and the gaseous outflow rate of methane (mmol/d), respectively. The reactor is a fixed bed type in which the fixed biomass is characterized by a death/detachment rate proportional to the dilution rate and the reactor is well mixed respect to substrates.22 The two specific growth rates are expressed as follows: μm,1s1 μ1 = K s ,1 + s1 (75) μ2 =

⎡ s1(e)s4(z1) ⎤ ⎢ ⎥ ⎢ s1(e)s4(z 2) ⎥ ⎢ ⎥ ⎢ s2(e)s3(z 3)⎥ ⎥ S b(e , z ̅ , y) = ⎢ ⎢ s1(e)s4(z4) ⎥ ⎢ ⎥ ⎢ s1(e)s4(z5) ⎥ ⎢ ⎥ ⎣ s2(e)s3(y) ⎦

yr (t ) = 10 sin(0.2t + 1.5) + 200

(88)

Variations of process output and dilution rates are shown in Figure 14 and Figure 15, respectively. As can be seen the desired trajectory has been tracked very well.

g K s,1 = 8.85 , L

μm,2 = 0.74d−1

K s,2 = 23.2 mmol,

KI = 0.0039 mmol−1

The design parameters and functions used in the adaptive controller 34 and adaptive laws 29−31 are

γ(|e|) = 0.01 |e|2

(77)

⎡ s1(e)s3(z5) ⎤ ⎥ ⎢ ⎢ s2(e)s3(z1) ⎥ ⎥ ⎢ ⎢ s2(e)s3(z 2)⎥ ⎥ Sa(e , z ̅ , y) = ⎢ ⎢ s2(e)s3(z 3)⎥ ⎥ ⎢ ⎢ s2(e)s3(z4) ⎥ ⎥ ⎢ ⎣ s2(e)s3(y) ⎦

(78)

Figure 14. Methane rate variations versus time for sine set-point tracking. 2591


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Figure 17. Dilution rate variations versus time for sine set-point tracking in the presence of external loads.

Figure 15. Dilution rate variations versus time for sine set-point tracking.

In order to evaluate the robustness of the designed controller for load rejection, in tracking sine set-point, after time 30 days values of s1,0 and s2,0 have been changed from 100 g/L to different values as given in Figure 16. The performance of the control

Figure 18. Methane rate variations versus time for step set-point tracking in the presence of model mismatch.

Figure 16. Methane rate variations versus time for sine set-point tracking in the presence of external loads.

scheme is shown Figure 16. Variation of dilution rate is illustrated in Figure 17. As can be seen from Figure 16, the effect of load has been rejected satisfactorily. Step Set-Point Tracking. In this simulation, it is desired to change the methane rate from 165 (mmol/d) to 205 (mmol/d). To test the performance of designed controller in the case of model mismatch, at t = 5 days the maximum specific growth rate for one type of biomass and parameter α have been changed in two different directions (increasing and decreasing). It should be mentioned that dynamical behavior of the system is more sensitive to variation of maximum specific growth rate respect to the other model parameters. Variations of process output and dilution rate are shown in Figure 18 and Figure 19, respectively.

Figure 19. Dilution rate variations versus time for step set-point tracking in the presence of model mismatch.

input nonlinearity has been presented. On the basis of the modified system equation and estimating the derivative of an unknown Lyapunov function, an adaptive controller has been designed. The developed control approach guarantees the

5. CONCLUSION In this work a new adaptive control approach for controlling processes with completely unknown nonlinear dynamics under 2592


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(18) Rovithakis, G. a; Chalkiadakis, I.; Zervakis, M. E. High-Order Neural Network Structure Selection for Function Approximation Applications Using Genetic Algorithms. IEEE Trans. Syst. Man. Cybern. B. Cybern 2004, 34, 150−158. (19) Wang, W.; Leu, Y.; Hsu, C. Robust Adaptive Fuzzy-Neural Control of Nonlinear Dynamical Systems Using Generalized Projection Update Law and Variable Structure Controller. Syst. Man, Cybern. Part B Cybern. IEEE Trans. 2001, 31, 140−147. (20) Wang, H.; Krstic, M.; Bastin, G. Optimizing Bioreactors by Extremum Seeking. J. Adapt. Control Signal 1999, 13, 651−669. (21) Parolini, D.; Carcano, S. A Model for Cell Growth in Batch Bioreactors. Master Thesis, Polytechnic University of Milan, Milan, Italy, 2010. (22) Marcos, N. I.; Guay, M.; Dochain, D.; Zhang, T. Adaptive Extremum-Seeking Control of a Continuous Stirred Tank Bioreactor with Haldane’s Kinetics. J. Process Control 2004, 14, 317−328.

uniform ultimate boundedness of the tracking error. The effectiveness of the proposed controller has been demonstrated via a simulation study. If the system states are not available, the regressors used in the adaptive laws can be expressed as functions of system output and its derivatives for the observable processes. Therefore, the proposed control scheme can be used for controlling the observable nonlinear systems with unknown dynamics subject to input nonlinearity in the absence of state measurements. Control of such processes has not been addressed in the literature before.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +98 21 66165419. Fax: +98 21 66022853. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Controller Design for Processes with Unknown Dynamics and Input

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