AN APPROACH TO TOTAL STATE CONTROL IN BATCH PROCESSES H. Marcela Moscoso-Vasquez, Gloria M. Monsalve-Bravo and Hernan Alvarez Departamento de Procesos y Energía, Facultad de Minas Universidad Nacional de Colombia Sede Medellín, Medellín Colombia Abstract The purpose of this work is to analyze the possibilities of total state control in batch processes using multivariable control strategies. For this, a review of control strategies for batch processes is made, analyzing their advantages and limitations. It is found that the most common approaches for this kind of processes are Input-Output strategies which do not guarantee an adequate performance of the control system due to the inherent transient character of batch processes. A new approach for batch processes control based on the process state controlability and observability is proposed, using Hankel matrix for determining the dynamics hierarchy as a tool for multivariable control of these processes. This represents a challenge since Hankel matrix is a tool for linear processes and batch processes linearization is not possible because a process operating point does not exist. To overcome this, an extension for using Hankel matrix in batch processes is developed. The proposed multivariable control strategy for achieving total state control in batch processes is based both on plantwide control structures and the idea of regulating only the main process dynamics through coordination of the other process dynamics. It is found that total state control represents an appropriate approach for batch process control and Hankel matrix represents the best way to establish a hierarchy between process dynamics and selecting the main one. The proposed strategy is applied to a penicillin fermentation process achieving the regulation of all state variables using as references the optimal state profiles for this process. Keywords: Total state control, phenomenological-based model, dynamics hierarchy, batch processes, distributed control structure. Introduction In Batch Processes (BP) raw materials are loaded in predefined amounts and they are transformed through a specific sequence of activities (known as recipe) by a given period. In this way, a determined amount of a specific product is obtained after a given time [1]. The same behavior is shared by semi-batch processes, with the exception that the raw materials are loaded continuously during the batch. This sort of operation is highly used in specialty chemicals industry [2] due to its flexibility and versatility, and it also represents the natural way to scale-up processes from the laboratory to the industrial plant [3]. Despite the antiquity of BP, the control issue has been a problematic area [1]. The literature presents many particular cases (not general problem solutions) and the studies on this topic have been limited to proposing a control strategy and evaluating its performance in simulation, as can be seen from [1], [2], [4], []. Additionally, there are no tools to evaluate properties as controllability and stability in BP, highlighting the lack of tools for designing control strategies for this kind of processes. In order to consider batch process characteristics, and aiming to a methodology for controlling BP, an approach to the control issue is proposed based on the Total State Control paradigm and on a classification of the process dynamics and their collaboration for achieving process control.
The paper is organized as follows. First, relevant characteristics of batch processes and the control strategies applied to these processes are discussed. Then, the proposed control scheme is described, detailing the establishment of the dynamics hierarchy and the methodology for designing the regulatory and supervisory levels of the control structure. Finally, the proposed control scheme is applied to a fed batch fermentor for penicillin production. Control Strategies for Batch Processes Before considering control strategies for BP, it is important to distinguish between discontinuous (including batch and semi-batch processes) and Continuous Processes (CP). As mentioned before, the main differences between these kinds of processes are: (i) non-stationary operating point in BP and (ii) end-product quality (run-end output) is the real control objective in BP [2]. The main characteristics of batch and semi-batch processes which imply a challenge for process control engineers are: Dynamic Operating Point. In BP input and output variables change with time [6]. In discontinuous operations, chemical transformations proceed from an initial state to a very different final state [3]. As a consequence of these time-varying characteristics, it does not exist a single operating point around which the control system is designed as in the case of continuous processes [2]. Irreversible Behavior. In discontinuous processes product properties and some state variables (related with product quality) depend on their history, which makes impossible to take corrective actions on the process once the product is out of quality specifications [2; 3]. In other words, the future quality (run-end output) depends, and is actually controlled by the past evolution of the state variables during the batch. Nonlinear Behavior. Nonlinearities might originate from various sources, the two major ones being: (i) the nonlinear dependency of the reaction rates on concentrations (often) and on temperature (always), and (ii) the nonlinear relationship between the heat transferred from the reactor to the jacket [3]. While continuous processes also exhibit nonlinearities, they can be easily treated because CP have a single operating point around which the process can be linearized, which is not possible to do in discontinuous processes [1]. Constrained Operation. In real processes, all manipulated variables are physically constrained. Also, certain output variables may need to be constrained for safety or stability reasons [6]. Furthermore, due to the wide operating range of batch processes, it is rarely possible to design and operate a batch process away from constraints, as is typically done for continuous processes [2]. Regarding control strategies for BP it is possible to identify two types of control strategies in discontinuous processes. The first one is the one-time control strategies, in which the controller calculates the control law in real time during the batch run [2]. This type of control includes: adaptive control and model predictive control strategies, and almost all this strategies are recognized as tracking control strategies. The second one, is the run-to-run control strategies, in this case the controller calculates the control law before starting the batch run from a previous batch information. This type of control includes: Iterative Learning Control (ILC) and Run-To-Run (R2R) Control [7]. As mentioned before, BP control literature presents many particular cases (nothing generalizable). Also, a plethora of studies on this topic have been limited to proposing a control strategy and evaluating its performance in simulation, some examples could be viewed in the works addressed above. In addition, a less prolific field is the dynamics characterization of BP from the control theory, because there are no
tools to evaluate properties such as controllability and stability in BP which highlights the lack of tools to design control strategies for these processes. Total State Control Strategy for Batch Processes as Plantwide Control It is frequently found in BP that not enough control loops are installed over the process. Only those measurable variables are controlled regardless if those variables can or cannot be directly associated with product quality, which places BP control under critical situation regarding the required end-product quality. Until today, product quality has been exclusively associated to process control objectives or process outputs. However, this Input Output approach represents a partial view of product quality given its limitations originated from partial measurement of the process state. Given the sequential nature of BP that can be seen as transformation stages the product undergoes, each stage can be considered as independent process equipment so then the single batch process is analogue to a process plant composed by this individual process equipment with high energy and mass interactions. As such, BP can be controlled using plant-wide control strategies as they are stated for CP plants. In plant wide control the mostly used approach is the decentralized control strategy in which each controller acts individually without communicating with the other controllers [8]. This is the strategy used when a set of the individual dynamics are controlled or only the product quality property is controlled. Other strategies, i.e. distributed control approach, involve communication among controllers giving a collaborative or cooperative action between controllers to achieve control of the whole plant [9]. The control strategy herein proposed involves (i) the establishment of a dynamics hierarchy, which allows the classification of process variables in Main or Critical Dynamics (MaD) and Secondary or Noncritical Dynamics (SeD), and (ii) a collaborative integration of the SeD controllers to guarantee the behavior of the MaD through single control loops for each SeD, which allows reaching Total State Control (TSC) for BP by controlling only the SeD. On Fig. 1 the methodology for obtaining the control strategy is illustrated. The control structure consists of two control layers: (i) regulatory layer which deals with the SeDs control, and (ii) supervisory layer which defines the set points for the regulatory controllers. These set points are obtained by minimizing the deviation on the MaD over the batch time using as manipulated variables the set points for the SeDs controllers. Figure 1. Methodology for Total State Control in Batch Processes. Once all the interesting process state variables are determined, a TSC strategy is being used. This strategy acts over the process with complete knowledge about MaD and SeD dynamic behaviors, and over them the strategy applies enough control loops in order to maintain such dynamics under control, taking into consideration that inside the MaD and SeD group of process state variables exist enough variables to determine product quality. However, it is known that total state measurement is not easily achieved, especially in chemical processes in which most sensors are not available or cannot be acquired
by economic constraints. As for TSC total knowledge of process state is required, the need for state observers is highlighted as they allow the estimation of non-measurable state variables from the most suitable set of available measurements In order to overcome the issues previously mentioned regarding control of product quality in BP, a dynamics classification of state variables is proposed. Such classification, or dynamics hierarchy allows the classification of process variables in (i) Main Dynamic (MaD) constituted by one dynamic behavior that relates both process characteristics and process objective and (ii) Secondary Dynamics (SeD). Since TSC follows the line of controlling all states in a process, in this work the proposed approach has the explicit requirement of controlling all SeD in a way that MaD and therefore product quality are taken to pre-established values during batch operation. Therefore, to have total control over MaD and SeD is the ideal process condition and is what this work aims to. The process dynamics hierarchy is established by an extension of the discrete control tool called Hankel matrix which has been widely used in model reduction, systems identification, digital filter design [], and recently in controllers design for establishing Input Output pairings in continuous processes [11]. This methodology uses a Phenomenological-Based Semiphysical Model (PBSM) for representing the process and Hankel matrix for: (i) representing the dynamic behavior of the process in input state terms, (ii) determining the effect of the input variables as a whole over each state variable, constituted by the State Impactability Index (SII) of each state variable; (iii) Determining the effect of each input variable over state variables as a whole, constituted by the Input Impactability Index (III); and (iv) selecting as MaD the state variable with the highest SII, and establishing states and inputs hierarchies to create control loops (design of the regulatory layer), by pairing the state variable with the highest SII with the input variable with the highest III. However, Hankel Matrix is a tool used in linear processes [12] and therefore cannot be used since linearization of BP is not possible given the inexistence of a unique process operating point. An extension for using this tool in BP is developed considering a piecewise linearization of the process. For this procedure, the operation trajectory of the process (given by (1) and (2)) is used for its linearization as shown in (3) and (4). ẋ = f (x(t),u(t)), x() = x (1) δẋ = Aδx(t) + Bδu(t) (3) y = h(x(t)) (2) δy = Cδx(t) (4) where A = [ f x ]x n,u n, B = [ ] f u x n,u n and C = [ ] h x x n,u n are the Jacobian matrices of the system described by (1) and (2). Further, taking into account that: δx(t) = x(t) x n () δu(t) = u(t) u n (6) δy(t) = y(t) y n (7) where x n, y n and u n are state and output trajectories and input nominal values respectively. Hankel Matrix is then used for establishing the dynamics hierarchy by means of its Singular Value Decomposition (SVD) as shown in (). These are computed as the eigenvalues of the matrix obtained by multiplying the observability (O b ) and controllability (C o ) matrixes using 8. Then, the State Impactability Index (SII k ) and Input Impactability Index (III k ) are computed using 9 and 11 respectively. H = O b C o (8) SII k = m n 1 U 2 k+n j,i (9) i=1 σ 2 i j= H = USV T () III k = m n 1 V 2 i,k+n j (11) i=1 σ 2 i j=
where the matrices U and V consist of the orthonormalized eigenvectors of HH T and H T H respectively. The diagonal elements of S are the non-negative square roots of the eigenvalues of H T H and they are called singular values σ i. SII k and III k represent the impactability of process manipulated inputs u i as a whole over a k-th given state x k and the impactability of a k-th given process manipulated input u k over process states x i as a whole, respectively. According to this the MaD is the x k with the highest impactability in the process. A Brief Illustrative Example In this section the TSC strategy it is applied to a penicillin fermentor studied by Banga et al. [13], where an optimal control problem is solved to maximize the total amount of penicillin produced. In this work, the optimal profiles found in [13] are used as reference trajectories for the control system. Dynamics Hierarchy for the Penicillin Process The dynamics hierarchy of the process is presented in Fig. 3, where it is worth noticing that it changes during the batch run, being (substrate concentration) the MaD during the first 2 hours and after that (biomass concentration) becomes the MaD. Thus, the SeD are (biomass concentration), x 2 (product concentration) and x 4 (reactor volume), during the first 2 hours, and then x 2, and x 4 become the SeD. This means that at batch starting point the substrate concentration is the dominant dynamics since it regulates biomass growth (inhibited at high substrate concentrations) and therefore penicillin production. Then, when the biomass has grown, its concentration becomes the dominant dynamics since it regulates penicillin production and substrate consumption. In addition, the change on the behavior of the SII can be explained as inhibition by high substrate concentration during the first 2 hours (see Fig. 2). State Variables 3 2 2 x 2 x 4 SII 1-1 -2 x 2 x 4-3 -4 Figure 2. State variable profiles, where, x 2, and units are g/l and x 4 units are L. Figure 3. State impactability index (SII) for the penicillin process. Regulatory layer According to Alvarez and Espinosa [11] the control loop design involves the selection of the manipulated variable for controlling each state variable. In this case, the available control actions are the substrate input flow (u 1 ) and concentration (u 2 ). For this, the state variable with the highest SII k is paired with the input variable with the highest III k. Here is important to note that only the SeD are considered for the control loops pairing since the MaD is controlled by coordination of the SeD controllers. According to this, the SII is computed again for all the SeD except the reactor volume (x 4 ) since it does not change in a significant amount during the batch (see Fig. 2) and thus can be considered constant. On Table 1, the pairing of variables for each stage of the batch considering the change on the MaD is presented.
Table 1. Input Output pairings for the process MaD MaD u 1 u 1 x 2 u 2 x 2 u 2 Each controller implemented is a Proportional one tuned for tracking, instead of the traditional Model Predictive Controller (MPC) used in distributed control strategies [14], so they follow the trajectories determined by the optimization problem described in the next section. It is worth clarifying that if enough control inputs for all MaD and SeD are available, the pairing analysis is performed considering the MaD as well. Supervisory Level In the supervisory layer the set points for the regulatory layer (SeD) controllers are determined by minimizing the deviation on the MaD over the batch time using as variables the set points for the SeD controllers. The objective function for this optimization is stated as shown by (12), where f (x,u 1,u 2 ) represents the model of the process and x 3 is the reference trajectory for the MaD, which corresponds to the state profile for determined by Banga et al. [13]. min x,x 3 (12) 2 s.t. (t) 14 (t) 2 x 4 (t) x(k + 1) = x(k) + f (x,u 1,u 2 ) The trajectory obtained for both MaD is presented on Fig. 4 and Fig.. Under these conditions, the state profiles obtained are the same as those obtained by Banga et al. [13], using a different substrate concentration on the feed and the same feed flowrate (See Fig. 7 and Fig. 6) which leads to a higher product production. The advantage of using the TSC approach herein introduced allows the process to reject disturbances while remaining on an optimal trajectory for the MaD, that allow the maximum penicillin production possible. Additionally, with the proposed control strategy, dynamic characteristics of BP are considered since both Input Output pairings and the control objective change considering the variations on the dynamic behavior of the batch. Substrate Concentration [kg/h] 2 2, sp Substrate Concentration [kg/m 3 ] 3 3 2 2, sp Figure 4. trajectory for the closed loop system. Figure. trajectory for the closed loop system.
-1 Flowrate [kg/h] -2-3 u 1 u 1 (Banga et al. 2) Figure 6. Substrate feedrate trajectories. Substrate Concentration [kg/h] 4 4 3 3 u 2 u 2 (Banga et al. 2) 2 2 4 6 8 12 14 Figure 7. Trajectories for substrate concentration. A comparison between the state profiles obtained by the proposed approach (Fig. 8) and optimized by Banga et al. [13] (Fig. 9) shows that a higher penicillin production (x 2 ) can be achieved by means of the proposed TSC approach, which can be explained from the higher biomass concentration ( ) obtained, having a greater potential for penicillin production and reducing the effects of substrate inhibition during the first hours of the batch. 3 3 x 2 x 4 3 2 x 2 x 4 State Variables 2 2 State Variables 2 Figure 8. State profiles obtained by the proposed TSC approach. Figure 9. et al. [13] State profiles optimized by Banga Conclusions In this work a multivariable control strategy for batch processes is proposed. One of the main contributions of this work is the usage of an index to determine a dynamics hierarchy and using it for establishing control objectives which was an important issue in batch processes. This hierarchy allows the achievement of total state control of the process by model based coordination of non critical state variables to achieve the regulation of the main dynamic of the process. The proposed two layer control structure consists of a regulatory layer which deals with direct control of the process, and a supervisory layer which optimizes the set points of the non critical state variables. The main contribution of this work is the achievement controlling all the states of the process by coordination of SeD to achieve the regulation of MaD, instead of only controlling one variable during the batch. Additionally, the design of the control loops is made considering the controllability and observability of the process (SII and III), and therefore its dynamic characteristics, instead of stationary tools (as RGA) for an inherently transient process. Finally, the proposed approach represents a paradigm
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