ISCCSP 2008, Malta, 12-14 March 2008
515
Control Relevant Identification for Controlling a Continuous-Stream Bioreactor with Unknown Dynamics Mehrsan Javan-Roshtkhari, Ahmad Ashoori
Soroor Javan-Roshtkhari
Control and Intelligent Processing Centre of Excellence, School of ECE, University of Tehran, Tehran, Iran {m.javan, a.ashoori}@ece.ut.ac.ir
College of Chemical Engineering, Iran University of Science and Technology, Tehran, Iran
[email protected]
Abstract— This paper presents an iterative procedure for control relevant identification for closed-loop system with unknown dynamics. At each step of iteration, an optimal controller is designed for the worst case model that obtained from identification task. The proposed framework uses simple model for identification task and an optimal H∞ controller will be designed. This method is also applied on a continuous stream bioreactor and the result was quite satisfactory. Keywords- omptimal control; identification; bioreactor
I. INTRODUCTION It is well known that the performance of a model-based controller is limited by the accuracy of the model used in designing the controller. In practice, models are increasingly identified from experimental data, and model error results from many factors such as model structure errors and the use of a finite number of noise-corrupted data. When identifying models from experimental data, it is important to remember that the experimental conditions (e.g., the input signal used to excite the system and the noise/disturbance conditions) have a large bearing on the accuracy of the identified model [1]. The identification of dynamic models out of experimental data has very often been motivated and supported by the presumed ability to use the resulting models as a basis for model-based control design. As such, control design is considered an important intended-application area for identified models. On the other hand, model-based control design is built upon the assumption that a reliable model of the plant under consideration is available. Without a model, no model-based control design. In the past twenty years identification and control design have shown a development in two separate directions with hardly any relationships [2]. The challenge to bring identification and control design more closely together, has led to a substantially increased attention for the problem area indicated by “identification for control” (from an identification point of view) or “experimentbased control design” (from a control-design point of view). Identification methods deliver a nominal model of a plant with unknown dynamics. The nominal model is just an approximation of the plant. Based on this nominal model, a
c 978-1-4244-1688-2/08/$25.00 2008 IEEE
controller has to be designed, assuming a certain level of accuracy (uncertainty) of the nominal model. The performance achieved by this controller when applied to the plant will be highly dependent on the nominal model and the assumed uncertainty. This work naturally led to the development of iterative approaches to control relevant identification where an identification experiment is first performed in closed-loop to acquire an initial model estimate and subsequent closed-loop experiments are performed, where in the kth iteration, the controller, Ck, operates in the control loop to give the new model estimate Pk+1, from which the new controller, Ck+1 is derived, and so on. The reader is referred to [3] for a recent review of progress in this area of iterative process controller design. In this paper we will first elucidate the problem of concern, and we will propose a framework for control relevant identification, then this method applied on a continuous stream bioreactor for controller design. This paper organised as follow, in Section 3, we will present a framework for handling the problem of control and identification, section 4 shows the result of the simulation. Finally, section 5 concludes the paper. II.
PERFORMANCE MEASURE
Suppose that P denotes the actual model of the system ( P0 ) and the identified model of the system ( Pˆ ). In general terms a performance function of a closed-loop configuration composed of plant ( P0 ) and controller C, is a system property, such as a step response, a sensitivity function, a complementary sensitivity etc. We formalize this control performance function as an element J ( P , C ) in Banach space χ . The control performance cost is then measured by the norm
J ( P, C )
χ
∀J ( P, C ) ∈ χ
(1)
and a corresponding control design method will provide a controller that minimizes this cost.
516
ISCCSP 2008, Malta, 12-14 March 2008
C p = min c J ( P, C )
(2)
χ
Step 1: perform an identification
In this paper, we used the infinity norm of the closed-loop transfer function.
Step 2: find the worst model parameters according to performance measure and the parameters variance
For the controller design, the minimization problem in (2) must be solved. Since the actual model is unknown, this problem cannot be solved simply. So the iterative schemes must be used. The basic principle behind the iterative schemes that have been proposed until now, is the exploration of the triangle inequality (3), in the sense that one aims at minimization of the right part (upper bound of the performance cost), by separate stages of minimization of either of the two terms “a” and “b” in (3) (Van Den Hof et al., 1995).
Step 3: design a controller and iterate
J ( P0 , C i −1 )
χ
≤
J ( Pˆi , C i −1 ) + J ( P0 , C i −1 ) − J ( Pˆi , C i −1 )
χ χ a
(3)
b
Simultaneous optimization of the upper bound (3) over
both Pˆ and C is intractable by common identification and control-design techniques. Instead, separate optimization over Pˆ (identification) and over C (control design) is performed. In general terms the model and the controller are obtained according to (indexes refer to step number in the iteration):
Pˆi = arg min J(P0 ,C i −1 ) − J(P,Ci −1 ) P
C i = arg min J(Pˆi ,C ) C
(4)
χ
(5)
χ
The identification method has to provide a model p that (asymptotically) minimizes the performance degradation ∞
). In the proposed iterative
scheme an identification criterion is chosen in which both
J ( P0 , Ci −1 ) − J ( Pˆi , Ci −1 ) and J ( P0 , Ci −1 ) − J ( Pˆi , Ci −1 ) are ∞
2
equal, where minimization of the 2-norm is used to obtain a reduction of the co-norm of the corresponding mismatch. This replacement is justified by the fact that an accurate L∞ approximation implies an accurate L2 approximation, provided that the error term is sufficiently smooth and small [8]. Identification of P can be obtained by applying the leastsquares identification. III.
A. Linear regression model of the system Consider a general representation of a linear, timeinvariant:
y (t ) = P( z ,θ )u (t ) + υ (t )
(6)
where P ( z , θ ) is the plant to be controlled in transfer function
form with its unknown parameters denoted by θ , u(t) and y(t) are the input and output of the plant, respectively, υ (t ) is the effect of process noise, and z is the forward shift operator. The linear regression model of the system can be shown as:
y (t ) = ϕ (t ) T θ + υ (t )
(7)
where θ and ϕ (t ) are column vectors containing the model parameters and the regressors, respectively. For M sets of input–output measurements, the unbiased least squares estimate of the model parameters can be computed using the expression
θˆ = (Φ *M Φ M ) Φ *M Y −1
In the work of [4] and [5] an iterative scheme of identification and control design is elaborated, utilizing the optimal H∞ control-design method of [6] and [7] respectively.
( J ( P0 , C i −1 ) − J ( Pˆi , C i −1 )
Figure 1. The iterative algotirhm for identification and control
A METHODOLOGY FOR CONTROL RRELEVANT IDENTIFICATION The proposed algorithm for iterative control relevant identification is shown in Fig. 1. Each step of the algorithm will be described next.
(8)
where ∗ denotes complex conjugate transpose. The covariance matrix associated with these parameter estimates is given by the following expression: −1 Cov(θˆ) = (Φ *M Φ M ) σ υ2
(9)
It is shown in [9] that for a process which can be described by (4), the probability of the true system lies inside the ellipsoid:
(θ − θˆ)T Γ(θ − θˆ) = 1
(10)
where
Γ=
Cov(θˆ) −1
(11)
α
is (1 − χ α ( d )) at the α level. 2
By making the assumption that the true parameters will always lie inside a certain confidence region of the parameter estimates, one can then obtain a hard bound for the parameters:
{
}
θ ∈ Θ = θ : (θ − θˆ)T Γ(θ − θˆ) ≤ 1
(12)
In this paper, the statistical bound at the 99% probability level will be used to define this hard bound [10].
ISCCSP 2008, Malta, 12-14 March 2008
B.
517
Choosing the repressors, the FSF model The FSF model which is given in [11, 12] is:
θˆws = arg max J ( Pˆws (θ ), C ) θ ∈Θ
y (t ) = P ( z , θ )u (t ) N −1 2
∑
=
k =−
where P (e response
ωk =
2πk N
P ( e jω k )
N −1 2
jω k
s.t. 1 1 − z −N u (t ) N 1 − e jωk z −1
, k = 0, ±1,..., ±
2- Design a controller for the worst obtained model
at
N −1 radians (referred to as 2
the FSF frequencies) and also represent the parameters of the FSF model, and N is an estimate of the process settling time Ts, where N =
Ts and ∆t is the sampling interval. For ∆t
decreasing the number of repressors, the reduced order FSF model can be used from (13) by replacing N by n, with the assumption
that
P(e jωk )
is
negligible
}
(16)
θ ∈ Θ = θ : (θ − θˆ) T Γ(θ − θˆ) ≤ 1
(13)
) denote the discrete-time process frequency evaluated
{
∞
for
C = arg min J ( Pˆws , C ) C
∞
(17)
The H∞ controller design for the discrete time system is described in [14]. IV.
SIMULATION AND RESAULTS
In this section, we used the proposed algorithm for iterative identification and control of continues flow bioreactor for penicillin production. The model of bioreactor is described in [15]. A schematic diagram of a continuous bioreactor is shown in Figure 2. From a chemical engineering point of view it can be viewed as a continuous stirred tank reactor (CSTR) with well-mixed contents and constant volume. The dilution rate (D) is available as manipulated inputs.
n −1 N −1 < k < . [11] 2 2 The reduced order FSF can be formulated in the linear regression form of (4) with the parameter vector defined as [13]: f (t ) 0 1 f (t ) − f (t ) 1 . ϕ (t ) = . . n −1 2 f (t ) n −1 − f (t ) 2 P(0) 2π j P (e N ) 2π −j P (e N ) . θ = . . (n −1)π j N P ( e ) ( n −1)π −j P(e N )
f (t ) r =
1 N
1 − z−N 2πr j N
u (t )
(14)
z −1 n −1 r = 0,±1,...,± 2 1− e
Figure 2. Schematic diagram of a continuous bioreactor
(15)
C. Optimal controller design Our objective is to design a controller for the worst model obtained from identification. So this problem consists of two parts: 1- Finding the worst model of the system (according to the performance measure)
The cell mass concentration (X) and substrate concentration S are the process state variables, and we assume that X or S are available for controller design. Although this assumption is rarely satisfied in practice, these state variables can often be estimated from secondary variables such as oxygen consumption rate and carbon dioxide production rate. The discrete time model of the bioreactor is (with sampling time= 1 min):
s( k + 1) =
1 ((1 − D (k ))s (k ) − rs (k ) 1 + u (k ) + D (k )(S in ( k + 1) + S in (k ) ))
(18)
x(k + 1) = rx (k ) + (1 − D(k )) x (k )
(19)
p(k + 1) = rp (k ) − (1 − D(k ) ) p(k )
(20)
518
ISCCSP 2008, Malta, 12-14 March 2008
,
rs (k) = v s (k)x(k)
v s (k) =
,
rp (k) = v p(k)x(k)
= µ 0 e − Kp ( k )
µ
1.6
v p (k)
(21)
y p (k)
1.2
1
v p = v0
,
output of the model output of the system
1.4
Substrate Concentration
rx (k) = µ(k)x(k)
Where x is biomass concentration, s is substrate concentration, p is product concentration, µ is the specific growth rate, v is the specific substrate consumption rate, D is dilution rate (i.e. influent flow rate/volume), and Sin is the influent substrate concentration.
0.8
0.6
0.4
0.2
0
-0.2
-0.4
The proposed algorithm is applied on this system and the control signal, step response of the mode and the system are shown for iteration 1 and 10.
0
5
10
15
20
25
30
1.2 control signal
1
0.8
0.6
Dilution Rate
1.8 output of the model output of the system
0.4
1.6 0.2
Substrate Concentration
1.4
0
1.2
1
-0.2
0
5
10
15
20
25
30
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
Figure 4. Top: Step response of the model (solid line) and step response of the system (dashed line), bottom: control signal at tenth iteration.
1 control signal
9
0.9 8
0.8 7
0.6
6
0.5
5 J
Dilution Rate
0.7
0.4
4
0.3
3
0.2
2
0.1 1
0
0
5
10
15
20
25
30
0
1
2
3
4
5
6
7
8
9
10
iterat ion
Figure 3. Top: Step response of the model (solid line) and step response of the system (dashed line), bottom: control signal at first iteration.
V. CONCLUSIONS This paper presents a framework for designing control relevant system identification experiments to obtain process models for use in a worst case optimal controller design. The problem of designing a control relevant identification for a closed-loop identification experiment is formulated. An iterative procedure for designing a control relevant identification experiment is also proposed. At each step of iteration, an optimal controller is designed for the worst case model that obtained from identification task. The proposed framework uses simple model for identification task and an optimal H∞ controller will design. This method is also applied on a continuous stream bioreactor and the result was quite satisfactory.
Figure 5.
Achieved γ in controller design
REFERENCES [1]
[2]
[3]
[4]
B.L. Cooley, and J.H. Lee, “Control-relevant experiment design for multivariable systems described by expansions in orthonormal bases,” Automatica,vol. 37, pp. 273-281, 2001. D.S. Bayard, Y. Yam, and E. Mettler, “A criterion for joint optimization of identification and robust control. IEEE Trans. on Automatic Control, vol. 37 , pp. 986-991, 1992. M. Gevers, “A decade ofprogress in iterative process control design: From theory to practice. Journal of Process Control,” vol. 12, pp. 519– 531, 2002. R. J. P. Schrama, and P. M. J. Van den Hof, “An iterative scheme for identification and control design based on coprime factorizations,” In Proc. Am. Control Conf, Chicago, pp. 2842-2846, 1992.
ISCCSP 2008, Malta, 12-14 March 2008 [5]
R. J. P. Schrama, and H. Bosgra, “Adaptive performance enhancement by iterative identification and control design,” Int. J Adnptive Control & Signal Proc.,vol. 7, pp. 475-481, 1993. [6] P. M. Bangers, and . H. Bosgra, “Low order robust H∞ controller synthesis,” Proc. 29th IEEE Conf Decision and Control, Honolulu, HI, pp. 194-199, 1990. [7] D. McFarlane, and K. Glover, Robust Controlfer Design Using Normalized Coprime Factor Plant Descriptions. Springer, Berlin, 1990. [8] P.M.J. Van den Hof, and R.J.P. Schrama, “Identification and control, closed-loop issues,” Automatica, vol. 31, pp. 1751–1770, 1995. [9] L. Ljung, System identification, theory for the user.second edition, prentice hall, New Jersey, 1999. [10] N.K. Dinata, and W.R. Cluett, “Control relevant identification for robust optimal control,” Automatica, vol. 41, pp. 1349-57, 2005.
519 [11] L. Wang, and W.R. Cluett, “Frequency-sampling filters: An improved model structure for step-response identification,” Automatica,vol. 33, pp. 939–944, 1997. [12] L. Wang, and W.R. Cluett, From plant data to process control: Ideas for process identification and PID design, Taylor & Francis, London, 2000. [13] A.D. Kalafatis, L. Wang, and W.R. Cluett, “Identification of timevarying pH processes using sinusoidal signals,” Automatica, vol. 41, pp. 685-691, 2005. [14] M. Saeki, “ Methods of solving a polynomial equation for an H∞ optimal control problem for a single-input single-output discrete-time system,” IEEE Trans. on Automatic Control, vol. 34, pp. 166-8, 1989. [15] B. Dahhou, G. Roux, I. Queinnec, and J.B. Pourciel, “Adaptive pole placement control of a continuous fermentation process,” International Journal of Systems Science, vol. 22, pp. 2625-38, 1991.
