Feedback-Assisted Iterative Learning Model Predictive Control with Nonlinear Fuzzy Model

Iterative learning control (ILC), due to its advantage of requiring less system knowledge, can serve as a feedforward signal in system control. ILC can be combined with model predictive control (MPC) to constitute a feedforward-feedback configuration. In this scheme, ILC provides most of the control signal and copes with the repetitive disturbances. MPC provides the supplementary control for regulation purpose and also for nonrepeating disturbance rejection. Considering the nonlinear industrial process, this paper establishes the plant nonlinear fuzzy model to constitute the fuzzy model-based feedback-assisted ILC. The integrated control strategy can achieve wide-range operation and good tracking performance. The performance of the feedback-assisted ILC is illustrated by a steam-boiler system.


Introduction
Ever since the original work on iterative learning control (ILC) [1], it has been researched and applied widely in industrial robotics [2,3] and manufacturing process [4] as a learning-type strategy.The objective of ILC is to use information obtained from past iteration runs to track a predefined reference and reject repetitive disturbances iteratively.The major advantage of ILC over traditional control algorithms is that it requires less computational efforts and less exact knowledge of the system.Thus it has drawn extensive attention in recent years [5][6][7].
However, ILC can be considered as an open-loop control scheme that is only capable of rejecting repetitive disturbances and cannot deal with nonrepeating disturbances efficiently.For this, ILC is often combined with a well-designed feedback controller.In this scheme, the tracking performance is guaranteed by ILC, which acts as a feedforward signal and provides most of control signals.The conventional feedback control path is meant to be supplementary to guarantee the system robustness to noises and disturbances.
The quadratically optimal ILC (Q-ILC) [8] was early constituted to be a model-based ILC, since it adopted a quadratic performance criterion to obtain an optimal learning gain.Its property provides a chance to incorporate feedback model predictive control (MPC) into ILC in order to improve the capability of rejecting disturbances.
Recently, various advanced MPC-ILC schemes have been proposed.Lee et al. [9] proposed the batch MPC (BMPC) to incorporate real-time feedback into Q-ILC to handle nonrepeating disturbances.By combining and modifying the existing Q-ILC and BMPC, Chin et al. [10] proposed a two-stage ILC framework to ensure that ILC control was unaffected by real-time disturbances.
In existing nonlinear MPC-ILC schemes, the plant model was usually expanded in Taylor's series, which was only valid in a normal operation point.Since the nonlinear industry process dynamics may vary in a large operating range, the model errors would inevitably introduce additional perturbations and deteriorate the control effect.
One way to deal with such difficulty is to develop the high accuracy plant model.In this paper, the Takagi-Sugeno (T-S) fuzzy model [11] is utilized to approximate the nonlinear system.The T-S fuzzy model, adopting the experts' experience, has the characteristic of simple structure and strong approximation ability.It is composed of a number of local 2 Mathematical Problems in Engineering linear models.The parameters of each local model are passed through the membership functions to generate an overall nonlinear model.Thus the proposed controller is a plantwide controller rather than a linear controller.With higher model exactness, the iteration process can converge to the predefined trajectory more quickly and tracking performance can also be improved.
In power generation industrial process, the steam-boiler generations in power plant should meet the load demand of power system under wide-range operation.The processes are complex systems with strong nonlinearity and subject to external disturbances.Since the dynamic of the steam-boiler system in power plant can be depicted as load dependent, T-S fuzzy model was established to represent the system dynamic.Thus the feedforward-(FF-) feedback (FB) control strategy is particularly suitable for controlling the system [12].
The aim of this paper is to derive a nonlinear FF-FB ILC based on fuzzy modeling technique.Section 2 describes the overall FF-FB control scheme.Section 3 presents the feedforward ILC control strategy.Section 4 incorporates MPC into ILC to formulate the FF-FB scheme.A simulation on the steam-boiler system is presented in Section 5 to illustrate the performance of the proposed feedback-assisted ILC.Finally, the conclusion is drawn in Section 6.

The Overall Control Law
Generally, the FF-FB strategy in a repetitive system can be expressed as where  denotes the iteration number and  denotes the current time,  ff  () represents the feedforward control signal in the th iteration at time ,  fb  () represents the feedback control signal in the th iteration at time , and   () is the overall control signal that will be applied to the process.The FF-FB controller structure is illustrated in Figure 1.
In Figure 1,  ff  () is the feedforward portion which is provided by ILC updating law. fb  () is the feedback portion which is a nonlinear MPC control signal.  () is the total control signal in the th iteration.  is the desired output reference,   () is the error in the th iteration, and   is the plant output in the th iteration.

The ILC Control Law
A nonlinear multi-input multi-output (MIMO) discrete-time system is written in the form of where  denotes the nonlinear relationship between the input , the disturbances , and the output .
Define the following sequences , , and  as where the run length is fixed with  sample steps and the dimensions of input and output are   and   , respectively.The nonlinear mapping  can be presented by the T-S fuzzy model: where  is the number of the rules,   denotes the th rule,  1 ⋅ ⋅ ⋅   are variables of the premise of implication,  1 ⋅ ⋅ ⋅   are fuzzy sets in the premise,   denotes the total effects of disturbances and bias errors in the th subarea, and   is impulse response coefficient matrix of the th submodel.
The FF-FB scheme can be written as Combining ( 4) and ( 5), the input-output relationship can be rewritten as Then   can be decomposed as   and ŷ .Consider where   is the suboutput of feedforward ILC control law in the th subarea and ŷ is the suboutput of feedback MPC control law in the th subarea.û denotes  fb .  has the following lower-block triangular structure: where    ∈    ×  is the impulse response coefficient matrix which is attained by applying unit pulse inputs at time  to outputs at time  independently.
The integral system output  can be formulated in the following form:  where Let where According to ( 10)-( 13), Define the weighted sum of disturbances and errors as Let   represent the predefined output; the error trajectory  can be expressed as Generally,  have both deterministic and stochastic components.In a repetitive system, it is convenient to express  as the output of a linear stochastic system: where  is the iteration number,   can be considered as the part of   that will also appear in the next iteration run, and both   and V  are zero-mean independent and identically distributed (i.i.d.) sequences in the th iteration with covariant matrices   and  V , respectively.
Define that ẽ is the part of  that will also appear in the next iteration.According to ( 16) and ( 17), we can formulate the expression of the tracking error sequences for two coherent iterations: Mathematical Problems in Engineering Combining (17) with (18) can result in where Δ ff +1 =  ff +1 −  ff  , Δ represents a difference operator in terms of the iteration index.
When disturbances and noises in a specific iteration are significant, the following observer-based algorithm is adopted: Δ ff  is determined by minimizing the following quadratic criterion: min According to (19), ( 20) is subject to where ẽ| is an estimation of ẽ that can be interpreted as using the information of ẽ in the th iteration to estimate the behavior of ẽ in the th iteration;  and are positive-definite matrices;  is the Kalman gain.The solution to the above is Consequently, the feedforward control signal at time  is where (25)

The Feedback-Assisted Control Law
In order to effectively reject nonrepeating disturbances in the current run, it is necessary to combine MPC with the above ILC to deduce the overall FF-FB control law.For convenience, the following notation is introduced: . . .
The process model ( 8) is used to deduce the revised MPC feedback control law.At time , where    denotes nonzero elements in tth row vector of   .At time  in MPC, we partitioned the control signal as two parts to predict the output at time  +  ( = 1, . . ., ): where Considering the predictive horizon is , (29) can be expanded as where Similar to ( 14), the integral system output ŷ (| +1 + |) can be expressed as where The where  and  are positive-definite matrices with appropriate dimensions.
According to (36), the feedback control signal can be obtained: Only the first element û () from û (|  +−1 ) is used in the process at the current time  as the feedback control signal.
According to (24) and (37), the overall control law is obtained as In industrial applications, various constraints are imposed on the input magnitude and the input change rate (in terms of time index), which are given as the following linear inequalities: where   () =   () −   ( − 1),  represents a difference operator in terms of the time index and  min and  min represent the lower limits of   () and   (), respectively;  max and  max represent the upper limits of   () and   (), respectively.
In the FF-FB control scheme, the ILC feedforward signal  ff  is known before the th iteration.Define Through some straightforward manipulation, the constraints imposed on û (| where The constraints can be put in the general form of where In conditional MPC method, the constraints imposed on the practical system are often solved by a quadratic program (QP).The standard QP problem can be expressed as follows: where The QP problem is often solved by active set method and interior point method.In this case, the QP is solved in MATLAB by the function named quadprog for convenience.The first element which served as the optimal feedback control signal is applied to the process at the current time.

Case Study
The proposed feedback-assisted ILC has been simulated on a 160 MW drum-type boiler-turbine-generator plant represented by a third order MIMO coupling nonlinear model [13]: where  1 ,  2 , and  3 are drum steam pressure (kg/cm 2 ), electric power (WM), and steam-water fluid density in the drum, respectively, the output  3 is the drum water level (cm) obtained using two algebraic calculations  cs and   which are the steam quality and the evaporation rate (kg/s), and the inputs  1 ,  2 , and  3 are normalized positions of valve actuators that control the mass flow rates of fuel, steam to the turbine, and feedwater to the drum, respectively.The sampling period is 1 second.The constraints imposed on input change rates are Though the model shows strong nonlinearity, one major characteristic of the boiler-turbine-generator is that the plant dynamic changes with load within the whole operating range.This property motivates many researchers to establish the fuzzy model by incorporating the human operators' experience [14].Choose the operating points to be (120, 40, 0), (135, 90, 0.5), and (150, 140, 1), which can be defined as low (L), medium (M), and high (H), respectively.This division also uses the experience of the operators, who regard a load of 40 MW as low, 90 MW as medium, and 140 MW as high.Therefore the rules can be constructed as follows: where the fuzzy sets are illustrated in Figures 2, 3, and 4. The sampling step  = 1200.The system is initially steady in the following states  = (100, 50, 449.5),  = (100, 50, 0), and  = (0.271, 0.604, 0.336).
Simulations are first made under some fixed value of electric load demands to demonstrate the learning ability of Mathematical Problems in Engineering  Figures 5-7 represent the outputs in the 1st, 4th, 8th, 10th, 15th, and 20th iteration, respectively.In order to make a comparison, the reference trajectories are also included in each group curve.It is obvious that the outputs of steam pressure, electric power, and drum water level converge to their desired references perfectly after several iterations.With the increase of the iteration run, the tracking property of the proposed strategy is improved.It shows that the algorithm can be applied for power plant control to attain wide-range operation.
The comparing results between the proposed feedbackassisted ILC and the pure ILC are shown in Figures 8, 9, 10, 11, 12, and 13 for the 8th iteration, under the case  1 = 120,  2 = 100,  3 = 0.It can be seen that the proposed feedbackassisted ILC achieves a better tracking property over pure ILC for agreeing with the predefined steam pressure, electric power, and drum water level.In order to make a total comparison, define the sum of squared errors (SSE) between the reference trajectory and the actual output: The qualitative difference is analyzed under the case  1 = 120.Comparison is then made between the proposed feedback-assisted ILC and the pure feedback MPC.The feedback MPC adopts the same algorithm as depicted above.Figure 14 shows the comparing result of the tracking performance with respect to the iterations.It is clearly seen that although the proposed control scheme has a larger SSE in the initial iterations than the MPC, it converges to a smaller SSE boundary after the 5th iteration.Meanwhile, the feedback MPC almost has the same tracking performance for the system, due to the fact that it cannot learn from the previous runs.The repetitive errors exist from the 1st iteration and its tracking property cannot be improved in the iteration process [15].In this case, because of the plant model mismatch, the controller leaves repetitive tracking errors so the feedback MPC loses its efficacy for this kind of system.

Conclusion
In this paper, a feedback-assisted iterative learning controller is derived to track the predefined trajectory for a steamboiler nonlinear system whose dynamic can be presented by a T-S fuzzy model.MPC is appropriately incorporated into ILC as a feedback controller to eliminate the realtime random disturbances.Because of the using T-S model, the proposed FF-FB control strategy can attain wide-range load-following operation.The feedforward ILC control signal drives the process close to the desired reference.Meanwhile, the feedback MPC control signal compensates for uncertainties and disturbances left by the ILC control signal.The simulation results demonstrate that the performance can be improved significantly under the proposed strategy than pure ILC and MPC.

Figure 1 :
Figure 1: The control scheme for the feedback-assisted ILC controller.
Figures 5,6,and 7 show the simulation results.Figures5-7represent the outputs in the 1st, 4th, 8th, 10th, 15th, and 20th iteration, respectively.In order to make a comparison, the reference trajectories are also included in each group curve.It is obvious that the outputs of steam pressure, electric power, and drum water level converge to their desired references perfectly after several iterations.With the increase of the iteration run, the tracking property of the proposed strategy is improved.It shows that the algorithm can be applied for power plant control to attain wide-range operation.The comparing results between the proposed feedbackassisted ILC and the pure ILC are shown in Figures8, 9, 10, 11, 12, and 13 for the 8th iteration, under the case  1 = 120,  2 = 100,  3 = 0.It can be seen that the proposed feedbackassisted ILC achieves a better tracking property over pure ILC for agreeing with the predefined steam pressure, electric power, and drum water level.Figures11-13show the total control signal   , feedforward signal  ff  , and feedback signal  fb  of the three input signals, respectively.The actual input signal is in accordance with the expected behaviour.The feedforward signal provides most of control signals and the feedback control signal compensates for uncertainties and disturbances.

Figure 4 :
Figure 4: The fuzzy set of drum level.
tracking error of the predictive model is defined as