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

Abstract

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.

1. 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 [5–7].

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 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 plant-wide 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.

2. The Overall Control Law

Generally, the FF-FB strategy in a repetitive system can be expressed as 𝑢_𝑘(𝑡)=𝑢^ff_𝑘(𝑡)+𝑢^fb_𝑘(𝑡), (1) where 𝑘 denotes the iteration number and 𝑡 denotes the current time, 𝑢^ff_𝑘(𝑡) represents the feedforward control signal in the 𝑘th iteration at time 𝑡, 𝑢^fb_𝑘(𝑡) represents the feedback control signal in the 𝑘th iteration at time 𝑡, and 𝑢_𝑘(𝑡) is the overall control signal that will be applied to the process. The FF-FB controller structure is illustrated in Figure 1.

Figure 1 

The control scheme for the feedback-assisted ILC controller.

In Figure 1, 𝑢^ff_𝑘(𝑡) is the feedforward portion which is provided by ILC updating law. 𝑢^fb_𝑘(𝑡) is the feedback portion which is a nonlinear MPC control signal. 𝑢_𝑘(𝑡) is the total control signal in the 𝑘th iteration. 𝑦_𝑑 is the desired output reference, 𝑒_𝑘(𝑡) is the error in the 𝑘th iteration, and 𝑦_𝑘 is the plant output in the 𝑘th iteration.

3. The ILC Control Law

A nonlinear multi-input multi-output (MIMO) discrete-time system is written in the form of 𝑦=𝑓(𝑢,𝑑), (2) where 𝑓 denotes the nonlinear relationship between the input 𝑢, the disturbances 𝑑, and the output 𝑦.

Define the following sequences 𝑢, 𝑦, and 𝑑 as 𝑢=^Δ𝑢^𝑇(0)𝑢^𝑇(1)⋯𝑢^𝑇(𝑁−1)^𝑇∈𝑅^𝑛𝑢𝑁,𝑦=^Δ𝑦^𝑇(1)𝑦^𝑇(2)⋯𝑦^𝑇(𝑁)^𝑇∈𝑅^𝑛𝑦𝑁,𝑑=^Δ𝑑^𝑇(1)𝑑^𝑇(2)⋯𝑑^𝑇(𝑁)^𝑇∈𝑅^𝑛𝑦𝑁, (3) where the run length is fixed with 𝑁 sample steps and the dimensions of input and output are 𝑛_𝑢 and 𝑛_𝑦, respectively.

The nonlinear mapping 𝑓 can be presented by the T-S fuzzy model: 𝑅^𝑖∶If𝑧₁is𝐴_𝑖1and⋯and𝑧_𝑝is𝐴_𝑖𝑝,then𝑦^𝑖=𝐺^𝑖𝑢−𝑝^𝑖,𝑖=1,2,…,𝑟, (4) where 𝑟 is the number of the rules, 𝑅^𝑖 denotes the 𝑖th rule, 𝑧₁⋯𝑧_𝑝 are variables of the premise of implication, 𝐴_𝑖1⋯𝐴_𝑖𝑝 are fuzzy sets in the premise, 𝑝^𝑖 denotes the total effects of disturbances and bias errors in the 𝑖th subarea, and 𝐺^𝑖 is impulse response coefficient matrix of the 𝑖th submodel.

The FF-FB scheme can be written as 𝑢=𝑢^ff+𝑢^fb. (5)

Combining (4) and (5), the input-output relationship can be rewritten as 𝑅^𝑖∶If𝑧₁is𝐴_𝑖1and⋯and𝑧_𝑝is𝐴_𝑖𝑝,then𝑦^𝑖=𝐺^𝑖𝑢^ff+𝑢^fb−𝑝^𝑖,𝑖=1,2,…,𝑟. (6)

Then 𝑦^𝑖 can be decomposed as 𝑦^𝑖 and 𝑦^𝑖. Consider 𝑦^𝑖=𝐺^𝑖𝑢^ff−𝑝^𝑖,𝑦^𝑖=𝐺^𝑖𝑢^fb=𝐺^𝑖𝑢, (7)(8) where 𝑦^𝑖 is the suboutput of feedforward ILC control law in the 𝑖th subarea and 𝑦^𝑖 is the suboutput of feedback MPC control law in the 𝑖th subarea. 𝑢 denotes 𝑢^fb.

𝐺^𝑖 has the following lower-block triangular structure: 𝐺^𝑖=𝑔^𝑖_1,0𝑔0⋯0_𝑖2,0𝑔^𝑖_2,1𝑔⋯0⋮⋮⋱0_𝑖𝑁,0⋯⋯𝑔^𝑖_{𝑁,𝑁−1}∈𝑅^{𝑛𝑦𝑁×𝑛𝑢𝑁}, (9) where 𝑔^𝑖_𝑝𝑞∈𝑅^{𝑛𝑦×𝑛𝑢} is the impulse response coefficient matrix which is attained by applying unit pulse inputs at time 𝑞 to outputs at time 𝑝 independently.

The integral system output 𝑦 can be formulated in the following form: ∑𝑦=^𝑟𝑖=1𝑤_𝑖(𝑧)𝑦^𝑖∑^𝑟_𝑖=1𝑤_𝑖(𝑧), (10) where 𝑧=𝑧₁,…,𝑧_𝑝,𝑤_𝑖(𝑧)=^𝑝_𝑗=1𝐴_𝑖𝑗(𝑧). (11)

Let ℎ_𝑖(𝑧)=𝑤_𝑖(𝑧)∑^𝑟_𝑖=1𝑤_𝑖(𝑧), (12) where ℎ_𝑖(𝑧)=ℎ_𝑖(𝑧(1))0⋯00ℎ_𝑖(𝑧(2⋱))⋯000⋯ℎ_𝑖(𝑧(𝑁))∈𝑅^{𝑛𝑦𝑁×𝑛𝑦𝑁}. (13)

According to (10)–(13), ∑𝑦=^𝑟𝑖=1𝑤_𝑖(𝑧)𝑦^𝑖∑^𝑟_𝑖=1𝑤_𝑖(𝑧)=^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝑝^𝑖. (14)

Define the weighted sum of disturbances and errors as 𝐷=^𝑟_𝑖=1ℎ_𝑖(𝑧)𝑝^𝑖. (15)

Let 𝑦_𝑑 represent the predefined output; the error trajectory 𝑒 can be expressed as 𝑒=𝑦_𝑑−𝑦=𝑦_𝑑−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff+𝐷. (16)

Generally, 𝐷 have both deterministic and stochastic components. In a repetitive system, it is convenient to express 𝐷 as the output of a linear stochastic system: 𝐷_𝑘+1=𝐷_𝑘+𝑤_𝑘,𝐷_𝑘=𝐷_𝑘+𝑣_𝑘, (17) where 𝑘 is the iteration number, 𝐷_𝑘 can be considered as the part of 𝐷_𝑘 that will also appear in the next iteration run, and both 𝑤_𝑘 and 𝑣_𝑘 are zero-mean independent and identically distributed (i.i.d.) sequences in the 𝑘th iteration with covariant matrices 𝑅_𝑤 and 𝑅_𝑣, respectively.

Define that 𝑒̃ is the part of 𝑒 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: 𝑒_𝑘=𝑦_𝑑−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff_𝑘+𝐷_𝑘+𝑣_𝑘,𝑒_𝑘̃=𝑦_𝑑−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff_𝑘+𝐷_𝑘,𝑒_𝑘+1=𝑦_𝑑−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff_𝑘+1+𝐷_𝑘+1+𝑣_𝑘+1,𝑒̃_𝑘+1=𝑦_𝑑−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑢^ff_𝑘+1+𝐷_𝑘+1. (18)

Combining (17) with (18) can result in 𝑒̃_𝑘+1=𝑒_𝑘̃−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖Δ𝑢^ff_𝑘+1+𝑤_𝑘,𝑒_𝑘=𝑒_𝑘̃+𝑣_𝑘, (19) where Δ𝑢^ff_𝑘+1=𝑢^ff_𝑘+1−𝑢^ff_𝑘, 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: 𝑢^ff_𝑘=𝑢^ff_𝑘−1+Δ𝑢^ff_𝑘. (20)

Δ𝑢^ff_𝑘 is determined by minimizing the following quadratic criterion: min_{Δ𝑢ff𝑘}12𝑒̃^𝑇_{𝑘|𝑘−1}²_𝑆+Δ𝑢^ff_𝑘²_𝑇. (21)

According to (19), (20) is subject to 𝑒̃_{𝑘|𝑘−1}=𝑒̃_{𝑘−1|𝑘−1}−^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖Δ𝑢^ff_𝑘,𝑒̃_{𝑘−1|𝑘−1}=𝑒̃_{𝑘−1|𝑘−2}+𝐾𝑒_𝑘−1−𝑒̃_{𝑘−1|𝑘−2}, (22) where 𝑒̃_𝑛∣𝑚 is an estimation of 𝑒_𝑛̃ that can be interpreted as using the information of 𝑒̃ in the 𝑚th iteration to estimate the behavior of 𝑒̃ in the 𝑛th iteration; 𝑆 and 𝑇are positive-definite matrices; 𝐾 is the Kalman gain.

The solution to the above is Δ𝑢^ff_𝑘=^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑇𝑆^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖+𝑇⁻¹×^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑇𝑆𝑒̃_{𝑘−1|𝑘−1}. (23)

Consequently, the feedforward control signal at time 𝑡 is 𝑢^ff_𝑘(𝑡)=𝑢^ff_𝑘−1(𝑡)+Δ𝑢^ff_𝑘(𝑡)=𝑢^ff_𝑘−1(𝑡)+𝐿𝑒̃_{𝑘−1|𝑘−1}, (24) where 𝐿=0_{}𝑛𝑢×𝑛𝑢𝑡}⋯1_{}𝑛𝑢×𝑛𝑢}⋯0_{}𝑛𝑢×()𝑛𝑁−𝑡−1𝑢}⋅^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑇𝑆^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖+𝑇⁻¹×^𝑟_𝑖=1ℎ_𝑖(𝑧)𝐺^𝑖𝑇𝑆. (25)

4. 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: 𝜆^𝑡1_𝑡2=𝜆𝑡₁𝜆𝑡₁⋮𝜆𝑡+1₂. (26)

The process model (8) is used to deduce the revised MPC feedback control law. At time 𝑡, 𝑦^𝑖_𝑘(𝑡)=𝑔^𝑖_𝑡𝑢_𝑘⁰_𝑡−1, (27) where 𝑔^𝑖_𝑡 denotes nonzero elements in 𝑡th row vector of 𝐺^𝑖.

At time 𝑡 in MPC, we partitioned the control signal as two parts to predict the output at time 𝑡+𝑗(𝑗=1,…,𝑚): 𝑦^𝑖_𝑘𝑡+𝑗|𝑡=𝑔^𝑖1_𝑡+𝑗𝑢_𝑘⁰_𝑡−1+𝑔^𝑖2_𝑡+𝑗𝑢_𝑘^𝑡_{𝑡+𝑗−1}, (28) where 𝑔^𝑖1_𝑡+𝑗=𝑔^𝑖_𝑡+𝑗,0𝑔^𝑖_𝑡+𝑗,1⋯𝑔^𝑖_{𝑡+𝑗,𝑡−1},𝑔^𝑖2_𝑡+𝑗=𝑔^𝑖_{𝑡+𝑗,𝑡}𝑔^𝑖_{𝑡+𝑗,𝑡+1}⋯𝑔^𝑖_{𝑡+𝑗,𝑡+𝑗−1}. (29)

Considering the predictive horizon is 𝑚, (29) can be expanded as 𝑦^𝑖_𝑘^𝑡+1_𝑡+𝑚𝑡=𝐺_𝑡^𝑖1𝑢_𝑘⁰_𝑡−1+𝐺_𝑡^𝑖2𝑢_𝑘^𝑡_{𝑡+𝑚−1}, (30) where 𝐺_𝑡^𝑖1=𝑔^𝑖_𝑡+1,0𝑔^𝑖_𝑡+1,1⋯𝑔^𝑖_{𝑡+1,𝑡−1}𝑔^𝑖_𝑡+2,0𝑔^𝑖_𝑡+2,1⋯𝑔^𝑖_{𝑡+2,𝑡−1}𝑔⋮⋮⋱⋮_{𝑖𝑡+𝑚,0}𝑔^𝑖_𝑡+𝑚,1⋯𝑔^𝑖_{𝑡+𝑚,𝑡−1},𝐺_𝑡^𝑖2=𝑔^𝑖_𝑡+1,𝑡𝑔0⋯0_{𝑖𝑡+2,𝑡}𝑔^𝑖_{𝑡+2,𝑡+1}𝑔⋯0⋮⋮⋱⋮_{𝑖𝑡+𝑚,𝑡}𝑔^𝑖_{𝑡+𝑚,𝑡+1}⋯𝑔^𝑖_{𝑡+𝑚,𝑡+𝑚−1}. (31)

Similar to (14), the integral system output 𝑦_𝑘(|^𝑡+1_𝑡+𝑚|𝑡) can be expressed as 𝑦_𝑘^𝑡+1_𝑡+𝑚𝑡=^𝑟_𝑖=1𝐻^𝑖_𝑡𝑦^𝑖_𝑘^𝑡+1_𝑡+𝑚𝑡, (32) where 𝐻^𝑖_𝑡=ℎ_𝑖(𝑧(𝑡+1))⋯000ℎ_𝑖(𝑧(⋱𝑡+2))⋯000⋯ℎ_𝑖(𝑧(𝑡+𝑚))∈𝑅^{𝑛𝑦𝑚×𝑛𝑦𝑚}. (33)

The tracking error of the predictive model is defined as 𝑒̂_𝑘^𝑡+1_𝑡+𝑚𝑡=𝑦_𝑑^𝑡+1_𝑡+𝑚−𝑦_𝑘^𝑡+1_𝑡+𝑚𝑡=𝑓_𝑡−^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2𝑢_𝑘^𝑡_{𝑡+𝑚−1}, (34) where 𝑓_𝑡=𝑦_𝑑^𝑡+1_𝑡+𝑚−^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖1𝑢^𝑖_𝑘⁰_𝑡−1. (35)

Define the cost function of MPC as follows: 𝑢min_{𝑘𝑡𝑡+𝑚−1}𝐽_𝑘(𝑡)=12𝑒̂_𝑘^𝑡+1_𝑡+𝑚𝑡²_𝑄+𝑢_𝑘^𝑡_{𝑡+𝑚−1}²_𝑅, (36) where 𝑄 and 𝑅 are positive-definite matrices with appropriate dimensions.

According to (36), the feedback control signal can be obtained: 𝑢_𝑘^𝑡_{𝑡+𝑚−1}=^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2𝑇𝑄^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2+𝑅⁻¹×^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2𝑇𝑄𝑓_𝑡. (37)

Only the first element 𝑢_𝑘(𝑡) from 𝑢_𝑘(|^𝑡_{𝑡+𝑚−1}) is used in the process at the current time 𝑡 as the feedback control signal.

According to (24) and (37), the overall control law is obtained as 𝑢_𝑘(𝑡)=𝑢^ff_𝑘(𝑡)+𝑢^fb_𝑘(𝑡)=𝑢^ff_𝑘−1(𝑡)+𝐿𝑒̃_{𝑘−1|𝑘−1}+𝑢_𝑘(𝑡). (38)

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: 𝑢^min≤𝑢_𝑘(𝑡)≤𝑢^max,𝛿𝑢^min≤𝛿𝑢_𝑘(𝑡)≤𝛿𝑢^max, (39) where 𝛿𝑢_𝑘(𝑡)=𝑢_𝑘(𝑡)−𝑢_𝑘(𝑡−1), 𝛿 represents a difference operator in terms of the time index and 𝑢^min and 𝛿𝑢^min represent the lower limits of 𝑢_𝑘(𝑡) and 𝛿𝑢_𝑘(𝑡), respectively; 𝑢^max and 𝛿𝑢^max represent the upper limits of 𝑢_𝑘(𝑡) and 𝛿𝑢_𝑘(𝑡), respectively.

In the FF-FB control scheme, the ILC feedforward signal 𝑢^ff_𝑘 is known before the 𝑘th iteration.

Define 𝑈^min=𝑢^min𝑇⋯𝑢^min𝑇𝑇∈𝑅^𝑛𝑢𝑚,𝑈^max=𝑢^max𝑇⋯𝑢^max𝑇𝑇∈𝑅^𝑛𝑢𝑚. (40)

Through some straightforward manipulation, the constraints imposed on 𝑢_𝑘(|^𝑡_{𝑡+𝑚−1}) and 𝛿𝑢_𝑘(|^𝑡_{𝑡+𝑚−1}) can be expressed as 𝑈^min−𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}≤𝑢_𝑘^𝑡_{𝑡+𝑚−1}≤𝑈^max−𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1},𝛿𝑈^min−𝛿𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}+𝐶₂≤𝐶₁𝑢_𝑘^𝑡_{𝑡+𝑚−1}≤𝛿𝑈^max−𝛿𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}+𝐶₂, (41) where 𝐶₁=⋱𝐼𝐼⋯𝐼𝐼0𝐼⋯𝐼𝐼𝐼𝐼⋯0𝐼∈𝑅^{𝑛𝑢𝑚×𝑛𝑢𝑚},𝐶₂=𝑢_𝑘()0⋮0𝑡−1∈𝑅^𝑛𝑢𝑚. (42)

The constraints can be put in the general form of 𝑀𝑢_𝑘^𝑡_{𝑡+𝑚−1}≤ℜ, (43) where 𝐼𝐶𝑀=−𝐼₁−𝐶₁,𝑈ℜ=^max−𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}−𝑈^min+𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}𝛿𝑈^max−𝛿𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}+𝐶₂−𝛿𝑈^min+𝛿𝑢^ff_𝑘^𝑡_{𝑡+𝑚−1}−𝐶₂. (44)

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: min_𝜃1𝐽=2𝜃^𝑇Φ𝜃+𝜃^𝑇𝜙,s.t.𝑀𝜃≤ℜ, (45)(46) where (45) denotes the objective function and (46) denotes the constraints, the decision variable is 𝜃, Φ, 𝜙, 𝑀, and ℜ are compatible matrices and vectors, and Φ is supposed to be a symmetric and positive definite matrix.

Combining (34), (36), and (43), a QP problem is constituted: 𝑢min_{𝑘𝑡𝑡+𝑚−1}𝐽_𝑘(𝑡)=12𝑢^𝑇_𝑘^𝑡_{𝑡+𝑚−1}Φ𝑢_𝑘^𝑡_{𝑡+𝑚−1}+𝑢^𝑇_𝑘^𝑡_{𝑡+𝑚−1}𝑢𝜙,s.t.𝑀_𝑘^𝑡_{𝑡+𝑚−1}≤ℜ, (47) where Φ=(∑^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2)^𝑇𝑄(∑^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2)+𝑅, 𝜙=−^𝑟_𝑖=1𝐻^𝑖_𝑡𝐺_𝑡^𝑖2𝑇𝑄𝑓_𝑡. (48)

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.

5. 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]: 𝑥̇₁=−0.0018𝑢₂𝑥₁^9/8+0.9𝑢₁−0.15𝑢₃,𝑥̇₂=0.073𝑢₂−0.016𝑥₁^9/8−0.1𝑥₂,𝑥̇₃=141𝑢₃−1.1𝑢₂−0.19𝑥₁,𝑦85₁=𝑥₁,𝑦₂=𝑥₂,𝑦₃=0.050.13073𝑥₃+100𝛼_cs+𝑞_𝑒9,𝛼−67.975_cs=1−0.001538𝑥₃0.8𝑥₁−25.6𝑥₃1.0394−0.0012304𝑥₁,𝑞_𝑒=0.845𝑢₂−0.147𝑥₁+45.59𝑢₁−2.514𝑢₃−2.096, (49) where 𝑥₁, 𝑥₂, and 𝑥₃ are drum steam pressure (kg/cm²), electric power (WM), and steam-water fluid density in the drum, respectively, the output 𝑦₃ is the drum water level (cm) obtained using two algebraic calculations 𝛼_cs and 𝑞_𝑒 which are the steam quality and the evaporation rate (kg/s), and the inputs 𝑢₁, 𝑢₂, and 𝑢₃ 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 −0.007≤𝛿𝑢₁(𝑡)≤0.007,−2.0≤𝛿𝑢₂(𝑡)≤0.02,−0.05≤𝛿𝑢₃(𝑡)≤0.05. (50)

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: 𝑅¹₁∶𝑦¹₁(𝑡)=108+0.9792𝑦₁(𝑡−1)+9.5039𝑢₁(𝑡−1)−298.1253𝑢₂(𝑡−1)+115.7860𝑢₃(),𝑅𝑡−1₂₁∶𝑦²₁(𝑡)=4.1736+0.9682𝑦₁(𝑡−1)+8.9645𝑢₁(𝑡−1)−4.2498𝑢₂(𝑡−1)−1.5497𝑢₃(),𝑅𝑡−1₃₁∶𝑦³₁(𝑡)=4.8814+0.9688𝑦₁(𝑡−1)+8.9645𝑢₁(𝑡−1)−5.2957𝑢₂(𝑡−1)−1.4789𝑢₃(),𝑅𝑡−1₁₂∶𝑦¹₂(𝑡)=−33.7364+0.1868𝑦₁(𝑡−1)+0.0491𝑦₂(𝑡−1)+104.9749𝑢₂(),𝑅𝑡−1₂₂∶𝑦²₂(𝑡)=−120.6358+0.7299𝑦₁(𝑡−1)−0.3592𝑦₂(𝑡−1)+209.4179𝑢₂(),𝑅𝑡−1₁₂∶𝑦¹₂(𝑡)=−170.5567+0.8565𝑦₁(𝑡−1)+0.0195𝑦₂(𝑡−1)+201.3431𝑢₂(),𝑅𝑡−1₁₃∶𝑦¹₃(𝑡)=−0.3727+0.2428𝑦₃(𝑡−1)−0.8475𝑢₁(𝑡−1)+0.7170𝑢₂(𝑡−1)+0.8352𝑢₃(),𝑅𝑡−1₂₃∶𝑦²₃(𝑡)=0.2704+0.1171𝑦₃(𝑡−1)−0.1904𝑢₁(𝑡−1)+0.1894𝑢₂(𝑡−1)+0.0370𝑢₃(),𝑅𝑡−1₃₃∶𝑦³₃(𝑡)=−0.0650+0.3692𝑦₃(𝑡−1)−0.2430𝑢₁(𝑡−1)+0.5564𝑢₂(𝑡−1)−0.2019𝑢₃(𝑡−1), (51) where the fuzzy sets are illustrated in Figures 2, 3, and 4.

Figure 2 

The fuzzy set of steam pressure.

Figure 3 

The fuzzy set of electric power.

Figure 4 

The fuzzy set of drum level.

The sampling step 𝑁=1200. The system is initially steady in the following states 𝑋=(100,50,449.5), 𝑌=(100,50,0), and 𝑈=(0.271,0.604,0.336).

The turning parameters are as follows: the covariance matrices of stochastic disturbances are 𝑅_𝑤=𝑅_𝑣=0.05𝐼; the control and predictive horizons are 𝑚=𝑝=20; 𝑆, 𝑇, 𝑄, and 𝑅 are chosen as 0.1𝐼, 0.8𝐼, 𝐼, and 0.025𝐼, respectively; 𝑈₀=(0.271,0.604,0.336) is the input in the 1st iteration.

Simulations are first made under some fixed value of electric load demands to demonstrate the learning ability of the proposed feedback-assisted ILC. The references of steam pressure, electric power, and drum water level are 𝑦₁=110,𝑦₂=80,𝑦₃𝑦=0,0≤𝑡≤400,₁=120,𝑦₂=100,𝑦₃𝑦=0,40≤𝑡≤800,₁=130,𝑦₂=120,𝑦₃=0,800≤𝑡≤1200. (52)

Figures 5, 6, and 7 show the simulation results.

Figure 5 

The steam pressure (kg/cm2).

Figure 6 

The electric power (MW).

Figure 7 

The drum water level (cm).

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 feedback-assisted ILC and the pure ILC are shown in Figures 8, 9, 10, 11, 12, and 13 for the 8th iteration, under the case 𝑦₁=120, 𝑦₂=100, 𝑦₃=0. It can be seen that the proposed feedback-assisted ILC achieves a better tracking property over pure ILC for agreeing with the predefined steam pressure, electric power, and drum water level. Figures 11–13 show the total control signal 𝑢_𝑖, feedforward signal 𝑢^ff_𝑖, and feedback signal 𝑢^fb_𝑖 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 8 

The steam pressure (kg/cm2).

Figure 9 

The electric power (MW).

Figure 10 

The drum water level (cm).

Figure 11 

The control efforts of fuel valve.

Figure 12 

The control efforts of steam valve.

Figure 13 

The control efforts of feedwater valve.

In order to make a total comparison, define the sum of squared errors (SSE) between the reference trajectory and the actual output: SSE(𝑘)=^𝑁_𝑡=1𝑦_𝑑(𝑡)−𝑦_𝑘(𝑡)²,𝑘∈[1,20]. (53)

The qualitative difference is analyzed under the case 𝑦₁=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.

Figure 14 

Comparison of tracking performance between the feedback-assisted ILC and feedback MPC.

6. Conclusion

In this paper, a feedback-assisted iterative learning controller is derived to track the predefined trajectory for a steam-boiler 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 real-time 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grants 60974051 and 61273144 and by Natural Science Foundation of Beijing under Grant 4122071.