ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling imprecisions and external disturbances: A comparison with type-I servo controller Soheil Ghabraei, Hamed Moradi n, Gholamreza Vossoughi Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 7 March 2015 Accepted 26 April 2015 This paper was recommended for publication by Oscar Camacho
To guarantee the safety and efficient performance of the power plant, a robust controller for the boiler– turbine unit is needed. In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to control a nonlinear multi-input multi-output (MIMO) model of industrial boiler–turbine unit, in the presence of unknown bounded uncertainties and external disturbances. To overcome the coupled nonlinearities and investigate the zero dynamics, input–output linearization is performed, and then the new decoupled inputs are derived. To tackle the uncertainties and external disturbances, appropriate adaption laws are introduced. For constructing the RASMC, suitable sliding surface is considered. To guarantee the sliding motion occurrence, appropriate control laws are constructed. Then the robustness and stability of the proposed RASMC is proved via Lyapunov stability theory. To compare the performance of the purposed RASMC with traditional control schemes, a type-I servo controller is designed. To evaluate the performance of the proposed control schemes, simulation studies on nonlinear MIMO dynamic system in the presence of high frequency bounded uncertainties and external disturbances are conducted and compared. Comparison of the results reveals the superiority of proposed RASMC over the traditional control schemes. RAMSC acts efficiently in disturbance rejection and keeping the system behavior in desirable tracking objectives, without the existence of unstable quasi-periodic solutions. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Boiler–turbine Nonlinear uncertain model Adaptive sliding mode control Servo control Robust performance Periodic orbit
1. Introduction Dramatic growth of industries and the world population in the past few decades, especially in developing countries, leads to the more need and consumption of electricity. Thermal power plants have significant role in supplying electricity, but their construction needs the vast investment and their operation has economic and environmental expenses. Boiler–turbine unit is the primary component of the power plant and its performance affects the whole power plant performance as well as its safety. An automatic control system is needed to achieve the desired performance and efficiency as well as the safety in the boiler– turbine unit. Controller must be able to maintain the mechanical energy generation in balance with electrical energy which is demanded by the power grid. Moreover, the controller must be able to keep the drum water level in its safe range and prevent over-heating of drum components or flooding of the steam lines
n
Corresponding author. Tel.: þ 98 21 66165545; fax: þ 98 21 66000021. E-mail addresses: [email protected], [email protected] (H. Moradi).
and consequently prevent the emergency shut downs and reduce the costly overhaul periods. To achieve this aim, an appropriate controller is required which must be able to regulate the variables such as the steam pressure, electric power and drum water level in desired operating points with acceptable tolerances. In addition, the controller must be able to provide satisfactory tracking of the desired command references to increase or decrease the power generation due to the power grid's demand changes. However, the physical constraints exerted on the actuators must be satisfied by the control signals. These constraints can be the magnitude and saturation rate for the control valves of the fuel, steam and feed-water flows. Moreover, drum water must be maintained in its safe range and proportional to the system operating points, during operation of boiler–turbine unit to prevent expensive damages to the hardware [1,2]. The need to increase the reliability and accuracy in the boiler– turbine unit motivates many researchers to study the boiler–turbine complex nonlinear multi-input multi-output dynamics. Many efforts have been done in recent decades to achieve more precise models of boiler–turbine dynamics. Among the early works, simplification of nonlinear models [3], dynamic modeling based on parameter estimation [4], system identification using neural networks [5] and
http://dx.doi.org/10.1016/j.isatra.2015.04.010 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
2
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
modeling based on data logs [6] have been done. In the recent studies, simple dynamic modeling and stability analysis of a steam boiler drum [7], development of various simulation packages for steam plants with natural and controlled recirculation (e.g.,[8]), cold start-up condition model for heat recovery steam generators [9] and using a computational model for the analysis and minimizing the fuel and environmental costs of a 310 MW fuel oil fired boiler have been investigated [10]. In addition, dynamic nonlinear modeling of power plants through neural networks (e.g., [11]), nonlinear model of boiler–turbine unit based on data logs and parameter estimation [12] and identification of boiler–turbine system based on a T–S fuzzy method [13] have been presented. Recently, using fuzzy-neural network methods, modeling of a 1000 MW power plant including an ultra-supercritical boiler system [14] and numerical simulations of a smallscale biomass boiler unit have been investigated [15]. Various control methods have been implemented on boiler or boiler–turbine units to improve the performance and reliability. To achieve this aim, many classical approaches have been used such as, PID-based control using iterative feedback tuning methodology [16], polynomial based H 1 robust control [17], improving boiler unit performance via optimum minimum-order observer [18], approximate feedback linearization [19], and its comparison with gain scheduling [20] and classical control of a combined cycle gas turbine plant using firefly algorithm [21] have been studied. Furthermore, the performance of traditional control approaches are compared with the performance of the sliding mode approach [1,22]. Moreover, intelligent based approaches such as hierarchical optimization using fuzzy-model predictive control [23], fuzzypredictive control based on GA [24], adaptive control with fuzzyinterpolated model [25], PID- fuzzy based control systems [26] and fuzzy sliding mode control [27] have been studied. Also, other methods including cascade control of superheated steam temperature with neuro-PID controller [28], predictive control via fuzzy clustering and subspace methods [29], nonlinear genetic algorithm (GA) [30], data-driven modeling [31] and local model networks [32] have been investigated. As other control strategies including the optimal control [33] and trajectory tracking via linear algebraic techniques [34] have been developed. In more recent researches which are based on nonlinear models, suppression of harmonic perturbations and bifurcation control [35,36], multi-objective control with inclusion of actuators' limitations [37], and adaptive backstepping-based nonlinear control [38] have been presented. Nonetheless, most of the aforementioned control approaches are based on the linear models or linearized models of the boiler– turbine dynamics or work on the basis of the input–output data without modeling system dynamics. Furthermore, most of the mentioned works focused on boiler–turbine dynamics without modeling imprecisions and external disturbances. However, in practice it is hard to determine the exact model of the boiler–turbine dynamics, therefore dealing with the modeling imprecision is unavoidable. Adaptive control is a reliable approach to overcome modeling imprecisions and external disturbances. Adaptive control schemes are widely used and its performance is validated in many engineering challenging problems such as chaos synchronization [39], control of DC motor [40], hard disk driver [41] and electro-hydraulic system [42] and many other problems available in literature. Combination of adaptive scheme with sliding mode approach leads to the more efficient and robust controllers. The sliding mode control [43] (SMC) which is also known as a variable structure control, is a powerful and robust tool for the problem of maintaining stability and consistent performance of high-order nonlinear complex dynamic systems. The SMC provides several advantages such as fast responses and easy realization but the
main advantage of SMC over the traditional control approaches is its robustness against the modeling imprecisions and external disturbances. Although the SMC approach merits the excellent robustness property, the pure SMC is accompanying several disadvantages such as large control signal efforts and chattering phenomenon. The performance of SMC approach is improvable by using appropriate adaption laws which undertakes the unknown uncertainty bounds and tune the control law parameters [44]. In this article, the design of a robust adaptive sliding mode controller (RASMC) for a multivariable nonlinear model of boiler– turbine in spite of modeling imprecisions and external disturbances is presented. To overcome the coupled nonlinearities between the control inputs and plant states in the model dynamics, input–output feedback linearization is performed and then the new decoupled inputs are driven. Unknown bounded uncertainties and external disturbances with the same frequencies, which are known as the worst off-design cases, are supposed in the dynamic model of the boiler–turbine. In addition, a type-I servo controller is designed based on linearized model around the median operating point #4. Then, simulation studies for nonlinear multi-input multi-output (MIMO) are conducted to compare and justify the performance and efficiency of the discussed control schemes.
2. Nonlinear multivariable model of an industrial boiler– turbine unit The schematic view of a water-tube boiler is shown in Fig. 1a in which the preheated water is fed into the steam drum and flows through the down-comers into the mud drum. Water is heated and changed into the saturation condition by passing through the risers. The saturated mixer of steam and water enters the steam drum, where the steam is separated from water and flows into the primary and secondary super-heaters. Then, the steam is more heated and is fed into the header. The steam temperature is regulated by a spray attemperator between two super-heaters; where the low temperature water is mixed with the steam. The boiler–turbine model used in this paper was presented by Bell and Astrom, which is a third-order MIMO nonlinear system and its parameters are obtained through physical modeling, system identification and model simplification. This model of boiler–turbine unit has been guided by plant experiments in Sweden and Australia
Fig. 1. (a) A schematic view of the boiler–turbine components and its (b) multivariable model including the input–output variables.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
[3], which is described as 8 9=8 _ > > < x1 ¼ α1 u2 x1 þ α2 u1 α3 u3 9=8 x_ 2 ¼ ðβ1 u2 β2 Þx1 β3 x2 > > : x_ ¼ ½γ u ðγ u γ Þx =γ 3 1 3 2 2 3 1 4
(
y1 ¼ x1
ð1Þ
y2 ¼ x2
where y1 , y2 and x3 are the drum pressure ðkg f =cm2 Þ, electric power (MW) and fluid density ðkg=m3 Þ, respectively (see Fig. 1b). Coefficients αi ; βi ; γ j i ¼ 1::3; j ¼ 1::4 are given in Table 1. Drum water level (L) can be obtained through the following complex equation: L ¼ 0:05ð0:13073x3 þ 100acs þ qe =9 67:975Þ ð1 0:001538x3 Þð0:8x1 25:6Þ ; acs ¼ x3 ð1:0394 0:0012304x1 Þ qe ¼ ð0:854u2 0:147Þx1 þ 45:59u1 2:514u3 2:096
ð2Þ
3
the linearizability index, which is related to the smallest integer that at least one of the control inputs uk explicitly appears inyλi i , P where λ ¼ m i ¼ 1 λi is defined as the relative degree of the nonlinear system. By using the Lie derivative notation, the derivatives can be expressed as [43] yλi i ¼ Lλf i hi þ
r X k¼1
Lλgik Lλf i 1 hi uk
ð5Þ
where the Lie Derivative Lλgik Lλf i 1 hi a 0 for at least one k in a neighborhood Ωi of the operating point x0 . The Lie derivatives are defined as follows: Lf hi ¼
∂hi ∂h f þ ::: þ i f n ∂x1 1 ∂xn
; Lλf hi ¼ Lf Lλf 1 hi
; Lg k h i ¼
∂hi ðxÞ g ∂x k
ð6Þ
where acs is the steam quality and qe ðkg=sÞ is the evaporation rate. Control inputs and their constraint are presented as follows:
By applying procedure which is discussed above for each output, yields
0 r ui r 1; 0:007 r u_ 1 r 0:007; 2 r u_ 2 r 0:02; 0:05 r u_ 3 r 0:05
h iT yλ11 :::yλ1m ¼ ½Lλf 1 h1 ðxÞ:::Lλf m hm ðxÞT þ EðxÞu
ði ¼ 1; 2; 3Þ
ð3Þ
where u1 ; u2 and u3 are the valves position for the fuel, steam and feed-water flows, respectively. Some typical operating points of the Bell and Astrom model are given in Table 2; where the median model coincides with the operating point #4 [3].
ð7Þ
that the new decoupled inputs can be obtained as ν ¼ EðxÞu ) u ¼ EðxÞ 1 ν
ð8Þ
which yields simpler form of m equations as
3. Control via input–output linearization
h iT yλ11 :::yλ1m ¼ ½Lλf 1 h1 ðxÞ:::Lλf m hm ðxÞT þ ν
The general schematic of the control system is shown in Fig. 2a. In this section to decouple the nonlinear coupled dynamics between the system inputs and the states and to investigate the system zero dynamics, input–output linearization approach is implemented. Consider a square MIMO system which has the same number of inputs and outputs as
The necessary and sufficient condition for the existence of a transformation linearizing the system completely from the input– output point of view is that the relative degree, λ be equal to the system order. In the case of λ o n, the nonlinear system can only be partially linearized and its stability is dependent to the stability of system's zero dynamics [45].
x_ ¼ f ðxÞ þGðxÞu yi ¼ hi ðxÞ i ¼ 1; :::; n
ð4Þ
where x is the n 1 state vector, u is the r 1 control input vector, y is the m 1 vector of the system outputs, f and h are the sufficiently smooth vector fields in ℜn and G is n r matrix whose columns are smooth vector fields g i (in this paper m ¼ r ¼ 3). Input–Output linearization uses the full state feedback to globally linearize the nonlinear dynamics of the selected controlled outputs. Input–output linearization is achieved by differentiating each of the output channels yi a sufficient number of times until a control input component appears in the resulting equation. Let λi Table 1 Dynamic coefficients of the boiler–turbine model by Bell and Astrom [3,12]. α1 ¼ 0:0018 α2 ¼ 0:9 α3 ¼ 0:15
β1 ¼ 0:073 β2 ¼ 0:016 β3 ¼ 0:1
γ 1 ¼ 141 γ 2 ¼ 1:1 γ 3 ¼ 0:19 γ 4 ¼ 85
3.1. Input–output linearization control of multivariable model of boiler–turbine dynamics T By assuming the y1 ; y2 ; y3 ¼ ½x1 ; x2 ; x3 T as the system outputs and ½u1 ; u2 ; u3 T as the control inputs, applying the input–output linearization procedure is straightforward. The relative degree of P the system,λ ¼ 3i ¼ 1 λi ¼ 3 equals to the system order n ¼ 3. Therefore the nonlinear model can be linearized completely and the closed-loop system will have no zero dynamics. By presenting the system dynamics as x_ ¼ AðxÞ þ EðxÞu 2 6 AðxÞ ¼ 4
#2
#3
#4
#5
#6
#7
x01
75.6
86.4
97.2
108
118.8
129.6
140.4
x02
15.27
36.65
50.52
66.65
85.06
105.8
128.9
x03
299.6
342.4
385.2
428
470.8
513.6
556.4
u01
0.156
0.209
0.271
0.34
0.418
0.505
0.6
u02
0.483
0.552
0.621
0.69
0.759
0.828
0.897
u03
0.183
0.256
0.34
0.433
0.543
0.663
0.793
0.97
0.65
0.32
0
0.32
0.64
0.98
y03
3
0 9=8 0:1x2 0:016x1 0:19 85 x1
0:9 6 EðxÞ ¼ 6 4 0 0
#1
ð10Þ
where
2
Table 2 Typical operating points of the Bell and Astrom model [3,12].
ð9Þ
7 5;
9=8
0:0018x1 9=8
0:073x1
1:1 85 x1
0:15 0 141 85
3 7 7; 5
9=8 EðxÞ ¼ 92637x1 850000
ð11Þ
By using Eq. (8), one has x_ ¼ AðxÞ þ ν
ð12Þ
The new decupled inputs are realizable if and only if the determinant of the matrix EðxÞ is nonzero. According to Eq. (11), the determinant of EðxÞ is zero if x1 ¼ 0, which never occurs in boiler–turbine operating range. Therefore, the new decupled inputs are realizable in all operating conditions.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Fig. 2. (a) Structure of the hybrid control system and its components used in the boiler–turbine unit. (b) Structure of the type-I servo control system including the integrator for tracking objectives [48].
4. Design of robust adaptive sliding mode controller (RASMC) To achieve the aim of designing RASMC, the error between the system states and the desired trajectories is defined as e ¼ x xdes
ð13Þ
Then, by subtracting the Eq. (12) from xdes and adding the bounded model imprecisions and disturbances, the error dynamics is obtained as 3 2 3 2 2 3 2 3 0 x_ des1 ν1 e_ 1 6 e_ 7 6 0:1x 0:016x9=8 7 6 ν 7 6 x_ 7 2 4 25¼4 1 5 þ 4 2 5 4 des2 5 0:19x1 =85 x_ des3 ν3 e_ 3 3 2 3 2 Δf 1 ðx; u; tÞ d1 ðtÞ 7 6 7 6 þ 4 Δf 2 ðx; u; tÞ 5 þ 4 d2 ðtÞ 5 ð14Þ Δf 3 ðx; u; tÞ d3 ðtÞ Assumption 1. Since the boiler–turbine unit's states are operating in a bounded subspace of ℜ3 , it is rational to suppose the model imprecisions Δf i ðx; u; tÞ are bounded. Therefore, there exist suitable positive constants θi ; i ¼ 1; 2; 3 such that Δf ðx; u; tÞ r θi ; i ¼ 1; 2; 3 ð15Þ
control laws must be designed to satisfy the reaching condition si s_ i o 0. Reaching condition makes the sliding surfaces an invariant set. It means that once the system's trajectories reach the sliding surface, they will remain on it for subsequent times. To achieve this aim, control inputs must be designed to include a continuous control law (equivalent control) and discontinuous control law (including the sign function). The equivalent control law manipulates the system trajectories on the sliding surface and the sign function handles the uncertainties. To ensure the achieving to the mentioned properties of SMC, the control law is proposed as νi ¼ f i ðxÞ þ x_ des ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ; θ^ i ¼ φi jsi j
θ^ i ð0Þ ¼ θi0 40;
ρ^ i ¼ ψ i jsi j
i ¼ 1; 2; 3
ð18Þ
ρ^ i ð0Þ ¼ ρi0 4 0
ð19Þ
where φi and ψ i are the positive gains of adaptation determining the adaptive process. θ^ i and ρ^ i are the adaption parameters, θi0 and ρi0 are the initial values for them and λ0 is a small positive constant. Now the convergence of the system trajectories to the sliding surface si ¼ 0 is proved as follows: First, a positive definite Lyapunov function candidate is considered as inspiration by [46] as follows:
i
VðtÞ ¼
3 2 γ 2 1X γ s2i þ i θ^ i θi þ i ρ^ i ρi 2i¼1 φi ψi
ð20Þ
Assumption 2. In general, it is conventional to assume that the external disturbances are norm-bounded in C 1 , i.e., di ðtÞ r ρi ; i ¼ 1; 2; 3 ð16Þ
By taking Lyapunov function candidate derivative with respect to time, we have
To design the robust adaptive sliding mode control, a suitable sliding surface with desired behavior must be chosen. The siding surface suitable for the application is designed as
3 X γ _ γ _ ¼ si s_ i þ i θ^ i θi θ^ i þ i ρ^ i ρi ρ_^ i ð21Þ VðtÞ ψi e_ i from Eq. (14), we have By using si ¼ γφi iei and substituting i¼1
si ¼ γ i ei ;
_ ¼ VðtÞ
i ¼ 1; 2; 3
ð17Þ
where the parameter γ i must be positive. In the next step, suitable
3 X γ i si f i ðxÞ þ dðtÞ þ Δf ðx; u; tÞ þ νi i¼1
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 3. Time response of the state variables after implantation of the robust adaptive sliding mode controller (RASMC) in the presence of uncertainties and disturbances (a) in tracking of desired set-paths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding periodic orbits of state variables. (c) Corresponding variation of the drum water level.
γ γ _ x_ desi þ i θ^ i θi θ^ i : þ i ρ_^ i ρi ρ_^ i φi ψi
ð22Þ
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Fig. 4. Tracking errors of the RASMC in the presence of uncertainties and disturbances.
By using control law of Eq. (18) and adaption laws by Eq. (19), one has 3 h X
_ ¼ VðtÞ
γ i si dðtÞ þ Δf ðx; u; tÞ ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ
i¼1
þ
γi ^ γ θi θi φi jsi j þ i ρ^ i ρi ψ i jsi j φi ψi
ð23Þ
By using Assumptions 1 and 2, we can obtain 3 h X γ i jsi j θi þ ρi γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ
_ r VðtÞ
Fig. 5. Time response of the adaption parameters (a) θ^ i and (b) ρ^ i ; i ¼ 1; 2; 3.
i¼1
γ γ þ i θ^ i θi φi jsi j þ i ρ^ i ρi ψ i jsi j φi ψi
ð24Þ
Barbalat's lemma, one finds lim wðλÞ ¼ lim μjsj ¼ 0
t-1
It is clear that 3 h X _ r γ i jsi j θi þ ρi γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ þ γ i θ^ i θi jsi j VðtÞ
t-1
ð31Þ
since μ is greater than zero, implies that s ¼ 0. Hence the proof is completed. □
i¼1 3 h i X γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ þ γ i θ^ i jsi j þ γ i ρ^ i jsi j þ γ i ρ^ i ρi jsi j ¼
5. Servo system design for tracking of desired performance
i¼1
ð25Þ By substituting jsi j=si instead ofsignðsi Þ, one can obtain _ r VðtÞ
3 X
3 X γ i jsi jλ0 ¼ μi jsi j ¼ μjsj
i¼1
ð26Þ
i¼1
Lemma (Barbalat's lemma). [43,47]. If w : ℜ-ℜ is a uniformly continuous function for t 4 0 and if the limit of the integral Z t lim wðλÞ:dλ ð27Þ
t-1
0
exists and is finite, then lim wðλÞ ¼ 0
t-1
_ r VðtÞ
γ i jsi jλ0 ¼
i¼1
3 X
μi jsi j ¼ μjsj ¼ wðtÞ
τ_ ¼ r γ ¼ r Px
ð29Þ
and by combining Eqs. (32) and (33), one has
x_ ~ 0 þ B½u ~ 0 þ Bu ~ þ B~ 0 r ¼ Ax ^ r ; y ¼ C~ x0 ¼ Ax τ_
i¼1
Now by integrating Eq. (29) from zero to t yields: Z t wðλÞ:dλ Vð0Þ Z VðtÞ þ 0
where y is a m 1 vector and γ ¼ Px is a p 1 vector representing the outputs that are required to follow a p 1 command input vector ðrÞ, and Q is the remained part of C. Due to lack of integrator in the plant dynamics, a type-I servo system is designed. The basic principle of the design of a type-I servo system is to insert an integrator in the feed-forward path between the error signal and the plant [48,49], as shown in Fig. 2b. By defining the new state variables as
ð28Þ
by assuming wðtÞ ¼ μjsj Z 0, Eq. (26) becomes 3 X
For designing the type-I servo controller system, boiler–turbine dynamics is linearized around the median operating point #4, as follows: 0 x_ ¼ Ax þ Bu ; y ¼ Cx ¼ P Q x ð32Þ
ð30Þ
since dVðtÞ=dt r 0, V ð0Þ VðtÞ Z 0 is positive and finite and that Rt concludes lim 0 wðλÞ:dλ exists and is finite. Thus, according to
where
x B A 0 B~ ¼ A~ ¼ x0 ¼ τ 0 P 0
0 B B^ ¼ C~ ¼ C 0 B~0 ¼ I I
ð33Þ
ð34Þ
ð35Þ
t-1
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
where m is the number of input variables (in this case, m ¼ 6). Introducing the modified controllability matrix as h C b^ 1 b^ 2 :: b^ m ⋮ A~ b^ 1 A~ b^ 2 :: i nm nm nm ð41Þ b^ 1 A~ b^ 2 ::A~ b^ m A~ b^ m ⋮:::::⋮A~ ^ A regular type-I servo where bi are the columns of matrix B. controller basis for the C is developed as: h C^ ¼ b^ 1 A~ b^ 1 :: A~ r1 1 b^ 1 ⋮ b^ 2 A~ b^ 2 :: A~ r2 1 b^ 2 i rm 1 b^ m ⋮:::::⋮b^ m A~ b^ m ::A~ ð42Þ
Fig. 6. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 3a (when RASMC is used).
The state-feedback control law to be used is 0 ; K^ ¼ k kτ u ¼ kx þ kτ τ ¼ K^ x τ
ð36Þ
According to Eqs. (1) and (34), since the controllability matrix
B A C^ ¼ 0 P is of rank 6, dynamic system is completely state controllable [49]. Using the similarity transformation T~ as x0 ¼ T~ s, Eq. (34) is represented as s_ χ ¼ A^ P sχ þ B^ P uχ
1 A^ P ¼ T~ A~ T~ ;
;
1 B^ P ¼ T~ B^
ð37Þ
where the new state vector is denoted by sχ . Using the transformations in control input as uχ ¼ Θ 1 ζ χ ;
T~ ¼
h
ζ χ ¼ ηχ Θ2 sχ
ð38Þ
j where each column, A~ b^ i ; i ¼ 1; ::; m; j ¼ 0; ::; m, is independent from its previous columns. Inverse of C^ given by Eq. (42) is displayed as (½ 0 stands for the transpose of ½ quantity): h 0 i0 1 0 0 0 0 0 C^ ¼ q11 :: q1r1 ⋮ p21 :: q2r2 ⋮ ::::: ⋮ qm1 :: qmrm
ð43Þ Similarity transformation T~ is defined as [49] Considering again Eq. (39) and constructing the feedback control law as ηχ ¼ Ωsχ , yields s_ χ ¼ Ad sχ ;
Ad ¼ AP BP Ω
where Ad is the desired state matrix including the coefficients representing the desired closed loop poles ðsI Ad ¼ ðs λ1 Þðs λ2 Þ:::ðs λn ÞÞ; having the general form of AP as given by Eq. (40). Considering Eq. (38) and similarity transformation xχ ¼ T~ sχ , yields the feedback control law of the system as: uχ ¼ Kxχ 1 K ¼ Θ1 ½Ω þ Θ2 T~
Θ1 ¼ ðB0P B^ P Þ 1 ; BP ¼ B^ P Θ1
½A1 6 6 0 AP ¼ 6 6 4 0 2
0
60 6 6 ½Ai ¼ 6 6 6 40 0
0 ½A2
::: :::
0
:::
:
1
0
:::
0
1
::: :
0
0
:::
0
0
:::
3
0 7 0 7 7 7 5 ½Am
2
nn
½B1 6 6 0 ; BP ¼ 6 6 4 0
2 3 0 607 07 6 7 7 6 7 7 7 7 ; ½Bi ¼ 6 6:7 7 6 7 7 15 4:5 0 r r 1 ri 1 i i 0
ð44Þ
Θ2 ¼ B0P ðAP A^ P Þ;
Ω ¼ B0P ðAP Ad Þ
ð47Þ
ð39Þ
where ηχ is the new control input vector and AP ; BP has the general canonical form with elements of ½Ai ri ri , ½Bi ri 1 , i ¼ 1; 2; ::; m and Pm i ¼ 1 r i ¼ n as [49] 2
i0 1
follows
s_ χ ¼ AP sχ þ BP ηχ AP ¼ A^ P B^ P Θ1 Θ2 ;
ð46Þ
where Θ1 ; Θ2 and Ω are obtained using Eqs. (38), (39) and (45) as
0 0 ~ ::: q0 A~ r1 1 ⋮ q0 q0 A~ :: q0 A~ r2 1 ⋮ ::: ⋮ q0 ~ ~ rm 1 q01r1 q01r1 A; 1r 1 2r 2 2r 2 2r 2 mr m qmr m A ::: qmr m A
Eq. (37) is described as
ð45Þ
0 ½B2
::: :::
0
:::
:
3
0 7 0 7 7 7 5 ½Bm
nm
3
ð40Þ
6. Simulation studies and performance evaluation of the proposed control schemes In this section, simulations for the realistic nonlinear model of boiler turbine dynamics in the presence of modeling imprecisions as well as disturbances are performed to investigate the efficiency and effectiveness of the proposed control schemes. As a worst case assumption and for investigation the occurrence of chattering phenomenon in the control inputs, it is assumed that the uncertainties and disturbances have the same high frequencies. For convenience in numerical studies, the plant modeling imprecisions as well as disturbances are assumed to be as follows: Δx_ 1 ¼ Δf 1 ðx; u; tÞ þ d1 ðtÞ ¼ 0:5 cos ð0:5tÞ Δx_ 2 ¼ Δf 2 ðx; u; tÞ þ d2 ðtÞ ¼ 0:5 cos ð0:5tÞ Δx_ 3 ¼ Δf 3 ðx; u; tÞ þ d3 ðtÞ ¼ 0:5 cos ð0:75tÞ
ð48Þ
where the imprecisions and disturbances can be modeled by any bounded function based on the mentioned Assumptions 1 and 2. Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
8
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. Time response of the state variables after implantation of the type-I servo controller in the presence of uncertainties and disturbances (a) in tracking of desired setpaths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding periodic orbits of state variables. (c) Corresponding variation of the drum water level.
6.1. Parameters assignment and simulations of the robust adaptive sliding mode controller In this section, simulation studies are conducted for the robust adaptive sliding mode controller to validate and demonstrate the
effectiveness of the proposed RASMC. For solving the nonlinear differential equations, MATLAB ode45 solver is used. For the RASMC control law which is described by Eq. (18) associated with Eq. (19), the positive constant parameters are chosen as, φi ¼ 1:25 10 5 ; ψ i ¼ 1:25 10 5 ; λ0 ¼ 10 3 . Initial condition of the adaption
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
parameters are chosen as: θ0 ¼ 0:05; ρ0 ¼ 0:05. The other positive constant parameters are chosen as γ i ¼ 2. It is well-known that the magnitude of the chattering is proportional to the sign function gain. Therefore, to have the smaller control signals and eliminate the chatter phenomenon, the parametersφi , ψ i , λ0 and the initial values of the adaption parameters φi and ψ i must be chosen small positive values. In that case, the adaptive law finds the best control gains which leads to the less control signal inputs and chatter elimination. To avoid the occurrence of chatter phenomenon as it is possible, the sign function which is a rigid switcher and may lead to the chatter is replaced with the sat function as ( si Z 10 signðsi Þ satðsi Þ ¼ tan hð0:02si Þ si o 10 Since it is impossible to get the zero error absolutely, the update of adaptive parameters will be continued. It means that the adaptive parameters tend to the infinity as the time tends to the infinity. This leads to the large sat function gains which may cause the large control efforts and chattering. To overcome this difficulty, as a tradeoff between the accuracy and performance, the parameters adaption is terminated if si r 2 and it will be resumed if it exceeds this threshold. Simulation results for the implementation of robust adaptive sliding mode controller are presented in Figs. 3–6. Fig. 3a shows the variation of drum pressure, electric output and fluid density in regulating around the median operating point #4 with initial conditionx0 ¼ ½80; 50; 390T . Then, its behavior is studied in tracking the desired sequence of ramp inputs from operating point #4 to #7, then to #1 and finally to #5. Accordingly, at the beginning of the plots, the state variables are regulated around the median operating point #4 after about 500 s without any overshoot. Thereafter, there is almost perfect control in tracking objectives between various operating points (during the interval [500–7000 s]). Corresponding periodic orbits of sate variable are displayed in Fig. 3b after implementation of the RASMC. It is observed that the RAMSC acts efficiently in disturbance rejection and keeping the system dynamics behavior in a periodic manner (rather than the existence of unstable quasi-periodic solutions). It should be noticed that in Fig. 3b, the centers of narrow ellipses coincide with the operating points #7, #1 and #5 of the boiler–turbine unit, as listed in Table 2. Fig. 3c displays the variation of the drum water level, which is absolutely stable and desirable without any oscillations and jumps. Fig. 4 demonstrates the variation of the errors where it is obvious that the tracking error is limited in an acceptable bound. Time response of the adaption parameters θ^ i ; ρ^ i ; i ¼ 1; 2; 3 are displayed in Fig. 5a and b, respectively, which are used as the sat function gains. Obviously, all adaptation parameters converge to the some positive constants, which are almost the most suitable sign function gains. Moreover, Fig. 6 demonstrates the required variation of the fuel, steam and feed-water valves (as the control inputs) for the tracking objectives of Fig. 3a. As it is observed, control inputs have no chattering, oscillation and saturation. Also, their steady values coincide with the corresponding operation point's values, listed in Table 2. These features imply that they are feasible in practical applications. 6.2. Type-I servo controller parameters assignment and simulation To have an appropriate performance of the boiler–turbine, and after several trial and error assignments, desired closed-loop poles are assigned as λ1 ¼ 0:006; λ2 ¼ 0:01; λ3 ¼ 0:014; λ4 ¼ 0:016; λ5 ¼ 0:018; λ6 ¼ 0:012 Finding transformation T~ and matrices Ad ; Θ1 ; Θ2 and Ω (as discussed in Section 5), and using Eqs. (36) and (46), feedback gain
Fig. 8. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 7a (when the servo type-I controller is used).
matrix K^ is determined as 2 0:0325 0:0028 0:0022 6 0 K^ ¼ 4 0:0049 0:0054 0:0001 0:0045 0:0133
0:0002
0
0
0
0
0
0
0
0:0001
3 7 5
ð49Þ
Type-I servo controller simulations are presented in Figs. 7 and 8. Fig. 7a illustrates the variation of drum pressure, electric output and fluid density in regulating around the median operating point #4 with initial condition x0 ¼ ½80; 50; 390T . Then, their behavior is studied in tracking the desired sequence of ramp inputs from operating point #4 to #7, then to #1 and finally to #5. Accordingly, at the beginning of the plots, the state variables are regulated around the median operating point #4 after about 200 s with a slight overshoot, which is acceptable. Also, according to Fig. 7a, the performance of the type-I servo controller is acceptable for the drum pressure x1 and the fluid density x3 . But it is disappointing in tracking objective for the electric power x2 ; while its stability is important for the power grid. There are several jumps in system response around the operating point #7. The periodic orbits of the state variables are displayed in Fig. 7b after implementation of the type-I servo controller. According to Fig. 7b, it is obvious that the type-I servo controller cannot guarantee the suppression of quasi-periodic solutions for the electric power around the operating point #7 (the irregular orbits of second plot of Fig. 7b confirms this fact). The corresponding variation of drum water level is shown in Fig. 7c in which there are undesirable oscillations around the operating point #7. Fig. 8 displays the required variation of the fuel, steam and feed-water valves (as the control inputs) for the tracking objectives of Fig. 7a. It is apparent that the control inputs are associated with the jumps and oscillations, especially around the operating point #7. Also, chattering is obvious in the fuel flow rate, u1 .
7. Comparison of the performance of robust adaptive sliding mode controller and type-I servo controller To investigate the effectiveness of proposed control approaches, the performance of the robust adaptive controller and type-I servo controller are compared in Figs. 9 and 10. Fig. 9a illustrates the time response of the drum pressure, electric power and fluid density for the both of RASMC and type-I servo controller. Although the RASMC has slower response in regulation rather than the type-I servo controller (due to process of tuning gains of sat function), it provides almost perfect behavior in tracking objectives. Furthermore, the response of the RASMC is without any significant jumps rather than poor performance of the type-I servo in tracking objective for the electric power ðx2 Þ; which its stability is crucial for the power grid. Moreover,
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
10
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 10. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 9a when the RASMC (black line) or type-I servo controller (blue line) is used. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
oscillations. The sliding mode control is well-known for its accompanying chatter phenomenon, but by using the appropriate adaptation laws and sat function, this phenomenon has been eliminated.
8. Conclusion Consistent and reliable performance of the boiler–turbine unit is crucial for the safety and efficiency of the power plant. Controller must be able to regulate the output variables around the operating points as well as track the desired trajectories to switch into the other desired operating points in a balance with the power grid demand. To achieve this aim, a robust adaptive sliding mode controller (RASMC) is proposed to control a nonlinear multi-input multi-output (MIMO) model of the industrial boiler–turbine unit, in the presence of unknown bounded uncertainties and external disturbances. To tackle the uncertainties and external disturbances, appropriate adaption laws are introduced. To guarantee the sliding motion occurrence, suitable control laws are constructed. Then the robustness and stability of the proposed RASMC is proved via Lyapunov stability theory. To compare the performance of the proposed RASMC with traditional control schemes, a type-I servo controller is designed. Simulations reveal the superiority of the RASMC over the type-I servo controller through the following general conclusions:
In regulating around the median operating point #4, both Fig. 9. Time response of the state variables after implantation of the robust adaptive sliding mode controller (RASMC) (black line) and type-I servo controller (blue line) in the presence of uncertainties and disturbances (a) in tracking of desired set-paths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding variation of the drum water level. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the behavior of the drum water level for both control approaches is compared in Fig. 9b. It indicates the better performance of the RASMC. Drum water level response of the servo controller is oscillating around the operating point #7 which may leads to the flooding of the steam lines and consequently emergency shutdown. Fig. 10 demonstrates that chatter phenomenon is significantly eliminated in corresponding variation of the fuel, steam and feedwater valves when RASMC is used. Conversely, the control inputs of the type-I servo controller are associated with the jumps and
controllers have an acceptable performance but RASMC is slower due to the process of tuning adaption parameters. In tracking objectives, the performance of the RASMC is excellent rather than the weak performance and instability of responses in type-I servo controller (especially, poor behavior is observed for the electric power; which affects adversely the power grid). Chattering phenomenon is significantly eliminated in control inputs of the RASMC. At the same time, the control inputs of the type-I servo controller are associated with the jumps and oscillations. After implementation of the RASMC, the variation of the drum water level is stable without jumps and oscillations. Conversely, the variation of the drum water level in type-I servo controller has oscillations and jumps which reduce the safety and reliability of the boiler–turbine unit (due to possible flooding of the steam lines and consequently the emergency shutdown). RASMC acts efficiently in disturbance rejection and keeping the system dynamics behavior in a periodic manner; rather than
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
the existence of unstable quasi-periodic solutions which was observed in type-I servo controller.
References [1] Moradi H, Bakhtiari-Nejad F, Saffar-Avval M. Robust control of an industrial boiler system; a comparison between two approaches: Sliding mode control & H1 technique. Energy Convers Manag 2009;50:1401–10. [2] Chen P-C, Shamma. JS. Gain-scheduled ℓ1 -optimal control for boiler-turbine dynamics with actuator saturation. J Process Control 2004;14:263–77. [3] Bell R, Åström KJ. Dynamic models for boiler-turbine-alternator units: data logs and parameter estimation for a 160 MW unit, Technical Report. Sweden: Lund Institute of Technology, Department of Automatic Control; 1987. [4] K Lo, P Zeng, E Marchand, A Pinkerton. Modelling and state estimation of power plant steam turbines. In: generation, transmission and distribution, IEE Proceedings; 1990. p. 80–94. [5] Murty, VV, Sreedhar, R, Fernandez, B, Masada, GY, Hill, AS Boiler system identification using sparse neural networks. In: Proceedings of the advances in robust and nonlinear control systems, ASME winter meeting.Louisiana, USA: New Orleans Article. [6] Donate P, Moiola J. Model of a once-through boiler for dynamic studies. Lat Am App Res 1994;24:159–66. [7] Bracco S, Troilo M, Trucco A. A simple dynamic model and stability analysis of a steam boiler drum. Proc Inst Mech Eng Part A: J Power Energy 2009;223:809–20. [8] Aranda, E, Frye,.M, Chunjiang, Q Model development, state estimation, and controller design of a nonlinear utility boiler system. In: Proceedings of the IEEE international conference on industrial technology (ICIT’08). Chengdu, China: Sichuan University. [9] Sindareh-Esfahani P, Habibi-Siyahposh E, Saffar-Avval M, Ghaffari A, BakhtiariNejad F. Cold start-up condition model for heat recovery steam generators. Appl Therm Eng 2014;65:502–12. [10] Kouprianov V, Kaewboonsong W. Modeling the effects of operating conditions on fuel and environmental costs for a 310 MW boiler firing fuel oil. Energy Convers Manag 2004;45:1–14. [11] Kouadri A, Namoun A, Zelmat M. Modelling the nonlinear dynamic behaviour of a boiler–turbine system using a radial basis function neural network. Int J Robust Nonlinear Control 2014;24:1873–86. [12] Åström KJ, Bell RD. Drum-boiler dynamics. Automatica 2000;36:363–78. [13] Li C, Zhou J, Li Q, An X, Xiang X. A new T–S fuzzy-modeling approach to identify a boiler–turbine system. Expert Syst Appl 2010;37:2214–21. [14] Liu X, Kong X, Hou G, Wang J. Modeling of a 1000 MW power plant ultra super-critical boiler system using fuzzy-neural network methods. Energy Convers Manag 2013;65:518–27. [15] Collazo J, Porteiro J, Míguez J, Granada E, Gómez M. Numerical simulation of a small-scale biomass boiler. Energy Convers Manag 2012;64:87–96. [16] Zhang S, Taft CW, Bentsman J, Hussey A, Petrus B. Simultaneous gains tuning in boiler/turbine PID-based controller clusters using iterative feedback tuning methodology. ISA Trans 2012;51:609–21. [17] Maher RA, Mohammed IA, Ibraheem IK. Polynomial based H1 robust governor for load frequency control in steam turbine power systems. Int J Electr Power Energy Syst 2014;57:311–7. [18] Moradi H, Bakhtiari-Nejad F. Improving boiler unit performance using an optimum robust minimum-order observer. Energy Convers Manag 2011;52:1728–40. [19] Yu T, Chan K, Tong J, Zhou B, Li D. Coordinated robust nonlinear boiler-turbinegenerator control systems via approximate dynamic feedback linearization. J Process Control 2010;20:365–74. [20] Moradi H, Alasty A, Saffar-Avval M, Bakhtiari-Nejad F. Multivariable control of an industrial boiler–turbine unit with nonlinear model: a comparison between gain scheduling and feedback linearization approaches. Sci Iran 2013;20:1485–98. [21] Saikia LC, Sahu SK. Automatic generation control of a combined cycle gas turbine plant with classical controllers using Firefly Algorithm. Int J Electr Power Energy Syst 2013;53:27–33.
11
[22] Moradi H, Saffar-Avval M, Bakhtiari-Nejad F. Sliding mode control of drum water level in an industrial boiler unit with time varying parameters: a comparison with H1 robust control approach. J Process Control 2012;22:1844–55. [23] Wu X, Shen J, Li Y, Lee KY. Hierarchical optimization of boiler–turbine unit using fuzzy stable model predictive control. Control Eng Pract 2014;30:112–23. [24] Li Y, Shen J, Lee KY, Liu X. Offset-free fuzzy model predictive control of a boiler–turbine system based on genetic algorithm. Simul Model Pract Theory 2012;26:77–95. [25] Moon U-C, Lee KY. An adaptive dynamic matrix control with fuzzyinterpolated step-response model for a drum-type boiler-turbine system. Energy Convers IEEE Trans 2011;26:393–401. [26] Li, XF, Zhang, W Coordinated control of Fossil-Fuel power plant based on the fuzzy PID control. In: Proceedings of the IEEE conference on decision and control. Hawaii, USA: Maui; 2012. Article # 6427049, p. 3080–85. [27] Thangavelusamy D, Ponnusamy L. Elimination of chattering using fuzzy sliding mode controller for drum boiler turbine system. J Control Eng Appl Inform 2013;15:78–85. [28] Zhang J, Zhang F, Ren M, Hou G, Fang F. Cascade control of superheated steam temperature with neuro-PID controller. ISA Trans 2012;51:778–85. [29] Wu X, Shen J, Li Y, Lee. KY. Data-driven modeling and predictive control for boiler–turbine unit using fuzzy clustering and subspace methods. ISA Trans 2014;53:699–708. [30] Wu J, Shen J, Krug M, Nguang SK, Li Y. GA-based nonlinear predictive switching control for a boiler-turbine system. J Control Theory Appl 2012;10:100–6. [31] Wu X, Shen J, Li Y, Lee. KY. Data-driven modeling and predictive control for boiler–turbine unit. Energy Convers IEEE Trans 2013;28:470–81. [32] Liu X, Kong X. Nonlinear fuzzy model predictive iterative learning control for drum-type boiler–turbine system. J Process Control 2013;23:1023–40. [33] Yang, S, Qian, C Real-time optimal control of a boiler–turbine system using pseudospectral methods. In: Proceedings of the 19th annual joint isa powidepri controls and instrumentation conference and 52nd ISA POWID symposium. Illinois, USA: Rosemont. Article #4. [34] S Godoy, G Scaglia, S Romoli, R Suvire, O Ortiz. Trajectory tracking of boilerturbine. In: Proceedings of the IEEE biennial congress of Argentina (ARGENCON).; 2014. p. 159–164. [35] Moradi H, Vossoughi G, Alasty A. Suppression of harmonic perturbations and bifurcation control in tracking objectives of a boiler–turbine unit in power grid. Nonlinear Dyn 2014;76:1693–709. [36] Moradi H, Alasty A, Vossoughi G. Nonlinear dynamics and control of bifurcation to regulate the performance of a boiler–turbine unit. Energy Convers Manag 2013;68:105–13. [37] Chen P-C. Multi-objective control of nonlinear boiler–turbine dynamics with actuator magnitude and rate constraints. ISA Trans 2013;52:115–28. [38] Fang F, Wei. L. Backstepping-based nonlinear adaptive control for coal-fired utility boiler–turbine units. Appl Energy 2011;88:814–24. [39] Li R, Chen W. Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems. Nonlinear Dyn 2014;76:785–95. [40] Yao J, Jiao Z, Ma D. Adaptive robust control of DC motors with extended state observer. IEEE Transactions on Industrial Electronics 2014;61:3630–7. [41] Liu L, Li D-J, Liu Y-J. Adaptive control for a class of nonlinear systems and application to hard disk drives. J Vib Control 2014;20:153–60. [42] Guan C, Pan S. Adaptive sliding mode control of electro-hydraulic system with nonlinear unknown parameters. Control Eng Pract 2008;16:1275–84. [43] Slotine J-JE, Li W. Applied nonlinear control. New Jersey: Prentice-Hall Englewood Cliffs; 1991. [44] Slotine J-J, Coetsee J. Adaptive sliding controller synthesis for non-linear systems. Int J Control 1986;43:1631–51. [45] FERNÁNDEZ R B, Hedrick JK. Control of multivariable non-linear systems by the sliding mode method. Int J Control 1987;46:1019–40. [46] Krstic M, Kanellakopoulos I, Kokotovic PV. Nonlinear and adaptive control design. Wiley; 1995. [47] Khalil HK. Nonlinear systems. 3rd ed. New Jersey: Prentice hall; 2002. [48] Ogata K. Modern control engineering. 5th ed.. New Jersey: Prentice Hall; 2010. [49] D'Azzo J, Houpis C. Linear control system analysis and design: conventional and modern. McGraw-Hill; 1995.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling imprecisions and external disturbances: A comparison with type-I servo controller Soheil Ghabraei, Hamed Moradi n, Gholamreza Vossoughi Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 7 March 2015 Accepted 26 April 2015 This paper was recommended for publication by Oscar Camacho
To guarantee the safety and efficient performance of the power plant, a robust controller for the boiler– turbine unit is needed. In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to control a nonlinear multi-input multi-output (MIMO) model of industrial boiler–turbine unit, in the presence of unknown bounded uncertainties and external disturbances. To overcome the coupled nonlinearities and investigate the zero dynamics, input–output linearization is performed, and then the new decoupled inputs are derived. To tackle the uncertainties and external disturbances, appropriate adaption laws are introduced. For constructing the RASMC, suitable sliding surface is considered. To guarantee the sliding motion occurrence, appropriate control laws are constructed. Then the robustness and stability of the proposed RASMC is proved via Lyapunov stability theory. To compare the performance of the purposed RASMC with traditional control schemes, a type-I servo controller is designed. To evaluate the performance of the proposed control schemes, simulation studies on nonlinear MIMO dynamic system in the presence of high frequency bounded uncertainties and external disturbances are conducted and compared. Comparison of the results reveals the superiority of proposed RASMC over the traditional control schemes. RAMSC acts efficiently in disturbance rejection and keeping the system behavior in desirable tracking objectives, without the existence of unstable quasi-periodic solutions. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Boiler–turbine Nonlinear uncertain model Adaptive sliding mode control Servo control Robust performance Periodic orbit
1. Introduction Dramatic growth of industries and the world population in the past few decades, especially in developing countries, leads to the more need and consumption of electricity. Thermal power plants have significant role in supplying electricity, but their construction needs the vast investment and their operation has economic and environmental expenses. Boiler–turbine unit is the primary component of the power plant and its performance affects the whole power plant performance as well as its safety. An automatic control system is needed to achieve the desired performance and efficiency as well as the safety in the boiler– turbine unit. Controller must be able to maintain the mechanical energy generation in balance with electrical energy which is demanded by the power grid. Moreover, the controller must be able to keep the drum water level in its safe range and prevent over-heating of drum components or flooding of the steam lines
n
Corresponding author. Tel.: þ 98 21 66165545; fax: þ 98 21 66000021. E-mail addresses: [email protected], [email protected] (H. Moradi).
and consequently prevent the emergency shut downs and reduce the costly overhaul periods. To achieve this aim, an appropriate controller is required which must be able to regulate the variables such as the steam pressure, electric power and drum water level in desired operating points with acceptable tolerances. In addition, the controller must be able to provide satisfactory tracking of the desired command references to increase or decrease the power generation due to the power grid's demand changes. However, the physical constraints exerted on the actuators must be satisfied by the control signals. These constraints can be the magnitude and saturation rate for the control valves of the fuel, steam and feed-water flows. Moreover, drum water must be maintained in its safe range and proportional to the system operating points, during operation of boiler–turbine unit to prevent expensive damages to the hardware [1,2]. The need to increase the reliability and accuracy in the boiler– turbine unit motivates many researchers to study the boiler–turbine complex nonlinear multi-input multi-output dynamics. Many efforts have been done in recent decades to achieve more precise models of boiler–turbine dynamics. Among the early works, simplification of nonlinear models [3], dynamic modeling based on parameter estimation [4], system identification using neural networks [5] and
http://dx.doi.org/10.1016/j.isatra.2015.04.010 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
2
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
modeling based on data logs [6] have been done. In the recent studies, simple dynamic modeling and stability analysis of a steam boiler drum [7], development of various simulation packages for steam plants with natural and controlled recirculation (e.g.,[8]), cold start-up condition model for heat recovery steam generators [9] and using a computational model for the analysis and minimizing the fuel and environmental costs of a 310 MW fuel oil fired boiler have been investigated [10]. In addition, dynamic nonlinear modeling of power plants through neural networks (e.g., [11]), nonlinear model of boiler–turbine unit based on data logs and parameter estimation [12] and identification of boiler–turbine system based on a T–S fuzzy method [13] have been presented. Recently, using fuzzy-neural network methods, modeling of a 1000 MW power plant including an ultra-supercritical boiler system [14] and numerical simulations of a smallscale biomass boiler unit have been investigated [15]. Various control methods have been implemented on boiler or boiler–turbine units to improve the performance and reliability. To achieve this aim, many classical approaches have been used such as, PID-based control using iterative feedback tuning methodology [16], polynomial based H 1 robust control [17], improving boiler unit performance via optimum minimum-order observer [18], approximate feedback linearization [19], and its comparison with gain scheduling [20] and classical control of a combined cycle gas turbine plant using firefly algorithm [21] have been studied. Furthermore, the performance of traditional control approaches are compared with the performance of the sliding mode approach [1,22]. Moreover, intelligent based approaches such as hierarchical optimization using fuzzy-model predictive control [23], fuzzypredictive control based on GA [24], adaptive control with fuzzyinterpolated model [25], PID- fuzzy based control systems [26] and fuzzy sliding mode control [27] have been studied. Also, other methods including cascade control of superheated steam temperature with neuro-PID controller [28], predictive control via fuzzy clustering and subspace methods [29], nonlinear genetic algorithm (GA) [30], data-driven modeling [31] and local model networks [32] have been investigated. As other control strategies including the optimal control [33] and trajectory tracking via linear algebraic techniques [34] have been developed. In more recent researches which are based on nonlinear models, suppression of harmonic perturbations and bifurcation control [35,36], multi-objective control with inclusion of actuators' limitations [37], and adaptive backstepping-based nonlinear control [38] have been presented. Nonetheless, most of the aforementioned control approaches are based on the linear models or linearized models of the boiler– turbine dynamics or work on the basis of the input–output data without modeling system dynamics. Furthermore, most of the mentioned works focused on boiler–turbine dynamics without modeling imprecisions and external disturbances. However, in practice it is hard to determine the exact model of the boiler–turbine dynamics, therefore dealing with the modeling imprecision is unavoidable. Adaptive control is a reliable approach to overcome modeling imprecisions and external disturbances. Adaptive control schemes are widely used and its performance is validated in many engineering challenging problems such as chaos synchronization [39], control of DC motor [40], hard disk driver [41] and electro-hydraulic system [42] and many other problems available in literature. Combination of adaptive scheme with sliding mode approach leads to the more efficient and robust controllers. The sliding mode control [43] (SMC) which is also known as a variable structure control, is a powerful and robust tool for the problem of maintaining stability and consistent performance of high-order nonlinear complex dynamic systems. The SMC provides several advantages such as fast responses and easy realization but the
main advantage of SMC over the traditional control approaches is its robustness against the modeling imprecisions and external disturbances. Although the SMC approach merits the excellent robustness property, the pure SMC is accompanying several disadvantages such as large control signal efforts and chattering phenomenon. The performance of SMC approach is improvable by using appropriate adaption laws which undertakes the unknown uncertainty bounds and tune the control law parameters [44]. In this article, the design of a robust adaptive sliding mode controller (RASMC) for a multivariable nonlinear model of boiler– turbine in spite of modeling imprecisions and external disturbances is presented. To overcome the coupled nonlinearities between the control inputs and plant states in the model dynamics, input–output feedback linearization is performed and then the new decoupled inputs are driven. Unknown bounded uncertainties and external disturbances with the same frequencies, which are known as the worst off-design cases, are supposed in the dynamic model of the boiler–turbine. In addition, a type-I servo controller is designed based on linearized model around the median operating point #4. Then, simulation studies for nonlinear multi-input multi-output (MIMO) are conducted to compare and justify the performance and efficiency of the discussed control schemes.
2. Nonlinear multivariable model of an industrial boiler– turbine unit The schematic view of a water-tube boiler is shown in Fig. 1a in which the preheated water is fed into the steam drum and flows through the down-comers into the mud drum. Water is heated and changed into the saturation condition by passing through the risers. The saturated mixer of steam and water enters the steam drum, where the steam is separated from water and flows into the primary and secondary super-heaters. Then, the steam is more heated and is fed into the header. The steam temperature is regulated by a spray attemperator between two super-heaters; where the low temperature water is mixed with the steam. The boiler–turbine model used in this paper was presented by Bell and Astrom, which is a third-order MIMO nonlinear system and its parameters are obtained through physical modeling, system identification and model simplification. This model of boiler–turbine unit has been guided by plant experiments in Sweden and Australia
Fig. 1. (a) A schematic view of the boiler–turbine components and its (b) multivariable model including the input–output variables.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
[3], which is described as 8 9=8 _ > > < x1 ¼ α1 u2 x1 þ α2 u1 α3 u3 9=8 x_ 2 ¼ ðβ1 u2 β2 Þx1 β3 x2 > > : x_ ¼ ½γ u ðγ u γ Þx =γ 3 1 3 2 2 3 1 4
(
y1 ¼ x1
ð1Þ
y2 ¼ x2
where y1 , y2 and x3 are the drum pressure ðkg f =cm2 Þ, electric power (MW) and fluid density ðkg=m3 Þ, respectively (see Fig. 1b). Coefficients αi ; βi ; γ j i ¼ 1::3; j ¼ 1::4 are given in Table 1. Drum water level (L) can be obtained through the following complex equation: L ¼ 0:05ð0:13073x3 þ 100acs þ qe =9 67:975Þ ð1 0:001538x3 Þð0:8x1 25:6Þ ; acs ¼ x3 ð1:0394 0:0012304x1 Þ qe ¼ ð0:854u2 0:147Þx1 þ 45:59u1 2:514u3 2:096
ð2Þ
3
the linearizability index, which is related to the smallest integer that at least one of the control inputs uk explicitly appears inyλi i , P where λ ¼ m i ¼ 1 λi is defined as the relative degree of the nonlinear system. By using the Lie derivative notation, the derivatives can be expressed as [43] yλi i ¼ Lλf i hi þ
r X k¼1
Lλgik Lλf i 1 hi uk
ð5Þ
where the Lie Derivative Lλgik Lλf i 1 hi a 0 for at least one k in a neighborhood Ωi of the operating point x0 . The Lie derivatives are defined as follows: Lf hi ¼
∂hi ∂h f þ ::: þ i f n ∂x1 1 ∂xn
; Lλf hi ¼ Lf Lλf 1 hi
; Lg k h i ¼
∂hi ðxÞ g ∂x k
ð6Þ
where acs is the steam quality and qe ðkg=sÞ is the evaporation rate. Control inputs and their constraint are presented as follows:
By applying procedure which is discussed above for each output, yields
0 r ui r 1; 0:007 r u_ 1 r 0:007; 2 r u_ 2 r 0:02; 0:05 r u_ 3 r 0:05
h iT yλ11 :::yλ1m ¼ ½Lλf 1 h1 ðxÞ:::Lλf m hm ðxÞT þ EðxÞu
ði ¼ 1; 2; 3Þ
ð3Þ
where u1 ; u2 and u3 are the valves position for the fuel, steam and feed-water flows, respectively. Some typical operating points of the Bell and Astrom model are given in Table 2; where the median model coincides with the operating point #4 [3].
ð7Þ
that the new decoupled inputs can be obtained as ν ¼ EðxÞu ) u ¼ EðxÞ 1 ν
ð8Þ
which yields simpler form of m equations as
3. Control via input–output linearization
h iT yλ11 :::yλ1m ¼ ½Lλf 1 h1 ðxÞ:::Lλf m hm ðxÞT þ ν
The general schematic of the control system is shown in Fig. 2a. In this section to decouple the nonlinear coupled dynamics between the system inputs and the states and to investigate the system zero dynamics, input–output linearization approach is implemented. Consider a square MIMO system which has the same number of inputs and outputs as
The necessary and sufficient condition for the existence of a transformation linearizing the system completely from the input– output point of view is that the relative degree, λ be equal to the system order. In the case of λ o n, the nonlinear system can only be partially linearized and its stability is dependent to the stability of system's zero dynamics [45].
x_ ¼ f ðxÞ þGðxÞu yi ¼ hi ðxÞ i ¼ 1; :::; n
ð4Þ
where x is the n 1 state vector, u is the r 1 control input vector, y is the m 1 vector of the system outputs, f and h are the sufficiently smooth vector fields in ℜn and G is n r matrix whose columns are smooth vector fields g i (in this paper m ¼ r ¼ 3). Input–Output linearization uses the full state feedback to globally linearize the nonlinear dynamics of the selected controlled outputs. Input–output linearization is achieved by differentiating each of the output channels yi a sufficient number of times until a control input component appears in the resulting equation. Let λi Table 1 Dynamic coefficients of the boiler–turbine model by Bell and Astrom [3,12]. α1 ¼ 0:0018 α2 ¼ 0:9 α3 ¼ 0:15
β1 ¼ 0:073 β2 ¼ 0:016 β3 ¼ 0:1
γ 1 ¼ 141 γ 2 ¼ 1:1 γ 3 ¼ 0:19 γ 4 ¼ 85
3.1. Input–output linearization control of multivariable model of boiler–turbine dynamics T By assuming the y1 ; y2 ; y3 ¼ ½x1 ; x2 ; x3 T as the system outputs and ½u1 ; u2 ; u3 T as the control inputs, applying the input–output linearization procedure is straightforward. The relative degree of P the system,λ ¼ 3i ¼ 1 λi ¼ 3 equals to the system order n ¼ 3. Therefore the nonlinear model can be linearized completely and the closed-loop system will have no zero dynamics. By presenting the system dynamics as x_ ¼ AðxÞ þ EðxÞu 2 6 AðxÞ ¼ 4
#2
#3
#4
#5
#6
#7
x01
75.6
86.4
97.2
108
118.8
129.6
140.4
x02
15.27
36.65
50.52
66.65
85.06
105.8
128.9
x03
299.6
342.4
385.2
428
470.8
513.6
556.4
u01
0.156
0.209
0.271
0.34
0.418
0.505
0.6
u02
0.483
0.552
0.621
0.69
0.759
0.828
0.897
u03
0.183
0.256
0.34
0.433
0.543
0.663
0.793
0.97
0.65
0.32
0
0.32
0.64
0.98
y03
3
0 9=8 0:1x2 0:016x1 0:19 85 x1
0:9 6 EðxÞ ¼ 6 4 0 0
#1
ð10Þ
where
2
Table 2 Typical operating points of the Bell and Astrom model [3,12].
ð9Þ
7 5;
9=8
0:0018x1 9=8
0:073x1
1:1 85 x1
0:15 0 141 85
3 7 7; 5
9=8 EðxÞ ¼ 92637x1 850000
ð11Þ
By using Eq. (8), one has x_ ¼ AðxÞ þ ν
ð12Þ
The new decupled inputs are realizable if and only if the determinant of the matrix EðxÞ is nonzero. According to Eq. (11), the determinant of EðxÞ is zero if x1 ¼ 0, which never occurs in boiler–turbine operating range. Therefore, the new decupled inputs are realizable in all operating conditions.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
Fig. 2. (a) Structure of the hybrid control system and its components used in the boiler–turbine unit. (b) Structure of the type-I servo control system including the integrator for tracking objectives [48].
4. Design of robust adaptive sliding mode controller (RASMC) To achieve the aim of designing RASMC, the error between the system states and the desired trajectories is defined as e ¼ x xdes
ð13Þ
Then, by subtracting the Eq. (12) from xdes and adding the bounded model imprecisions and disturbances, the error dynamics is obtained as 3 2 3 2 2 3 2 3 0 x_ des1 ν1 e_ 1 6 e_ 7 6 0:1x 0:016x9=8 7 6 ν 7 6 x_ 7 2 4 25¼4 1 5 þ 4 2 5 4 des2 5 0:19x1 =85 x_ des3 ν3 e_ 3 3 2 3 2 Δf 1 ðx; u; tÞ d1 ðtÞ 7 6 7 6 þ 4 Δf 2 ðx; u; tÞ 5 þ 4 d2 ðtÞ 5 ð14Þ Δf 3 ðx; u; tÞ d3 ðtÞ Assumption 1. Since the boiler–turbine unit's states are operating in a bounded subspace of ℜ3 , it is rational to suppose the model imprecisions Δf i ðx; u; tÞ are bounded. Therefore, there exist suitable positive constants θi ; i ¼ 1; 2; 3 such that Δf ðx; u; tÞ r θi ; i ¼ 1; 2; 3 ð15Þ
control laws must be designed to satisfy the reaching condition si s_ i o 0. Reaching condition makes the sliding surfaces an invariant set. It means that once the system's trajectories reach the sliding surface, they will remain on it for subsequent times. To achieve this aim, control inputs must be designed to include a continuous control law (equivalent control) and discontinuous control law (including the sign function). The equivalent control law manipulates the system trajectories on the sliding surface and the sign function handles the uncertainties. To ensure the achieving to the mentioned properties of SMC, the control law is proposed as νi ¼ f i ðxÞ þ x_ des ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ; θ^ i ¼ φi jsi j
θ^ i ð0Þ ¼ θi0 40;
ρ^ i ¼ ψ i jsi j
i ¼ 1; 2; 3
ð18Þ
ρ^ i ð0Þ ¼ ρi0 4 0
ð19Þ
where φi and ψ i are the positive gains of adaptation determining the adaptive process. θ^ i and ρ^ i are the adaption parameters, θi0 and ρi0 are the initial values for them and λ0 is a small positive constant. Now the convergence of the system trajectories to the sliding surface si ¼ 0 is proved as follows: First, a positive definite Lyapunov function candidate is considered as inspiration by [46] as follows:
i
VðtÞ ¼
3 2 γ 2 1X γ s2i þ i θ^ i θi þ i ρ^ i ρi 2i¼1 φi ψi
ð20Þ
Assumption 2. In general, it is conventional to assume that the external disturbances are norm-bounded in C 1 , i.e., di ðtÞ r ρi ; i ¼ 1; 2; 3 ð16Þ
By taking Lyapunov function candidate derivative with respect to time, we have
To design the robust adaptive sliding mode control, a suitable sliding surface with desired behavior must be chosen. The siding surface suitable for the application is designed as
3 X γ _ γ _ ¼ si s_ i þ i θ^ i θi θ^ i þ i ρ^ i ρi ρ_^ i ð21Þ VðtÞ ψi e_ i from Eq. (14), we have By using si ¼ γφi iei and substituting i¼1
si ¼ γ i ei ;
_ ¼ VðtÞ
i ¼ 1; 2; 3
ð17Þ
where the parameter γ i must be positive. In the next step, suitable
3 X γ i si f i ðxÞ þ dðtÞ þ Δf ðx; u; tÞ þ νi i¼1
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 3. Time response of the state variables after implantation of the robust adaptive sliding mode controller (RASMC) in the presence of uncertainties and disturbances (a) in tracking of desired set-paths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding periodic orbits of state variables. (c) Corresponding variation of the drum water level.
γ γ _ x_ desi þ i θ^ i θi θ^ i : þ i ρ_^ i ρi ρ_^ i φi ψi
ð22Þ
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Fig. 4. Tracking errors of the RASMC in the presence of uncertainties and disturbances.
By using control law of Eq. (18) and adaption laws by Eq. (19), one has 3 h X
_ ¼ VðtÞ
γ i si dðtÞ þ Δf ðx; u; tÞ ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ
i¼1
þ
γi ^ γ θi θi φi jsi j þ i ρ^ i ρi ψ i jsi j φi ψi
ð23Þ
By using Assumptions 1 and 2, we can obtain 3 h X γ i jsi j θi þ ρi γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ
_ r VðtÞ
Fig. 5. Time response of the adaption parameters (a) θ^ i and (b) ρ^ i ; i ¼ 1; 2; 3.
i¼1
γ γ þ i θ^ i θi φi jsi j þ i ρ^ i ρi ψ i jsi j φi ψi
ð24Þ
Barbalat's lemma, one finds lim wðλÞ ¼ lim μjsj ¼ 0
t-1
It is clear that 3 h X _ r γ i jsi j θi þ ρi γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ þ γ i θ^ i θi jsi j VðtÞ
t-1
ð31Þ
since μ is greater than zero, implies that s ¼ 0. Hence the proof is completed. □
i¼1 3 h i X γ i si ðθ^ i þ ρ^ i þ λ0 Þsignðsi Þ þ γ i θ^ i jsi j þ γ i ρ^ i jsi j þ γ i ρ^ i ρi jsi j ¼
5. Servo system design for tracking of desired performance
i¼1
ð25Þ By substituting jsi j=si instead ofsignðsi Þ, one can obtain _ r VðtÞ
3 X
3 X γ i jsi jλ0 ¼ μi jsi j ¼ μjsj
i¼1
ð26Þ
i¼1
Lemma (Barbalat's lemma). [43,47]. If w : ℜ-ℜ is a uniformly continuous function for t 4 0 and if the limit of the integral Z t lim wðλÞ:dλ ð27Þ
t-1
0
exists and is finite, then lim wðλÞ ¼ 0
t-1
_ r VðtÞ
γ i jsi jλ0 ¼
i¼1
3 X
μi jsi j ¼ μjsj ¼ wðtÞ
τ_ ¼ r γ ¼ r Px
ð29Þ
and by combining Eqs. (32) and (33), one has
x_ ~ 0 þ B½u ~ 0 þ Bu ~ þ B~ 0 r ¼ Ax ^ r ; y ¼ C~ x0 ¼ Ax τ_
i¼1
Now by integrating Eq. (29) from zero to t yields: Z t wðλÞ:dλ Vð0Þ Z VðtÞ þ 0
where y is a m 1 vector and γ ¼ Px is a p 1 vector representing the outputs that are required to follow a p 1 command input vector ðrÞ, and Q is the remained part of C. Due to lack of integrator in the plant dynamics, a type-I servo system is designed. The basic principle of the design of a type-I servo system is to insert an integrator in the feed-forward path between the error signal and the plant [48,49], as shown in Fig. 2b. By defining the new state variables as
ð28Þ
by assuming wðtÞ ¼ μjsj Z 0, Eq. (26) becomes 3 X
For designing the type-I servo controller system, boiler–turbine dynamics is linearized around the median operating point #4, as follows: 0 x_ ¼ Ax þ Bu ; y ¼ Cx ¼ P Q x ð32Þ
ð30Þ
since dVðtÞ=dt r 0, V ð0Þ VðtÞ Z 0 is positive and finite and that Rt concludes lim 0 wðλÞ:dλ exists and is finite. Thus, according to
where
x B A 0 B~ ¼ A~ ¼ x0 ¼ τ 0 P 0
0 B B^ ¼ C~ ¼ C 0 B~0 ¼ I I
ð33Þ
ð34Þ
ð35Þ
t-1
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
where m is the number of input variables (in this case, m ¼ 6). Introducing the modified controllability matrix as h C b^ 1 b^ 2 :: b^ m ⋮ A~ b^ 1 A~ b^ 2 :: i nm nm nm ð41Þ b^ 1 A~ b^ 2 ::A~ b^ m A~ b^ m ⋮:::::⋮A~ ^ A regular type-I servo where bi are the columns of matrix B. controller basis for the C is developed as: h C^ ¼ b^ 1 A~ b^ 1 :: A~ r1 1 b^ 1 ⋮ b^ 2 A~ b^ 2 :: A~ r2 1 b^ 2 i rm 1 b^ m ⋮:::::⋮b^ m A~ b^ m ::A~ ð42Þ
Fig. 6. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 3a (when RASMC is used).
The state-feedback control law to be used is 0 ; K^ ¼ k kτ u ¼ kx þ kτ τ ¼ K^ x τ
ð36Þ
According to Eqs. (1) and (34), since the controllability matrix
B A C^ ¼ 0 P is of rank 6, dynamic system is completely state controllable [49]. Using the similarity transformation T~ as x0 ¼ T~ s, Eq. (34) is represented as s_ χ ¼ A^ P sχ þ B^ P uχ
1 A^ P ¼ T~ A~ T~ ;
;
1 B^ P ¼ T~ B^
ð37Þ
where the new state vector is denoted by sχ . Using the transformations in control input as uχ ¼ Θ 1 ζ χ ;
T~ ¼
h
ζ χ ¼ ηχ Θ2 sχ
ð38Þ
j where each column, A~ b^ i ; i ¼ 1; ::; m; j ¼ 0; ::; m, is independent from its previous columns. Inverse of C^ given by Eq. (42) is displayed as (½ 0 stands for the transpose of ½ quantity): h 0 i0 1 0 0 0 0 0 C^ ¼ q11 :: q1r1 ⋮ p21 :: q2r2 ⋮ ::::: ⋮ qm1 :: qmrm
ð43Þ Similarity transformation T~ is defined as [49] Considering again Eq. (39) and constructing the feedback control law as ηχ ¼ Ωsχ , yields s_ χ ¼ Ad sχ ;
Ad ¼ AP BP Ω
where Ad is the desired state matrix including the coefficients representing the desired closed loop poles ðsI Ad ¼ ðs λ1 Þðs λ2 Þ:::ðs λn ÞÞ; having the general form of AP as given by Eq. (40). Considering Eq. (38) and similarity transformation xχ ¼ T~ sχ , yields the feedback control law of the system as: uχ ¼ Kxχ 1 K ¼ Θ1 ½Ω þ Θ2 T~
Θ1 ¼ ðB0P B^ P Þ 1 ; BP ¼ B^ P Θ1
½A1 6 6 0 AP ¼ 6 6 4 0 2
0
60 6 6 ½Ai ¼ 6 6 6 40 0
0 ½A2
::: :::
0
:::
:
1
0
:::
0
1
::: :
0
0
:::
0
0
:::
3
0 7 0 7 7 7 5 ½Am
2
nn
½B1 6 6 0 ; BP ¼ 6 6 4 0
2 3 0 607 07 6 7 7 6 7 7 7 7 ; ½Bi ¼ 6 6:7 7 6 7 7 15 4:5 0 r r 1 ri 1 i i 0
ð44Þ
Θ2 ¼ B0P ðAP A^ P Þ;
Ω ¼ B0P ðAP Ad Þ
ð47Þ
ð39Þ
where ηχ is the new control input vector and AP ; BP has the general canonical form with elements of ½Ai ri ri , ½Bi ri 1 , i ¼ 1; 2; ::; m and Pm i ¼ 1 r i ¼ n as [49] 2
i0 1
follows
s_ χ ¼ AP sχ þ BP ηχ AP ¼ A^ P B^ P Θ1 Θ2 ;
ð46Þ
where Θ1 ; Θ2 and Ω are obtained using Eqs. (38), (39) and (45) as
0 0 ~ ::: q0 A~ r1 1 ⋮ q0 q0 A~ :: q0 A~ r2 1 ⋮ ::: ⋮ q0 ~ ~ rm 1 q01r1 q01r1 A; 1r 1 2r 2 2r 2 2r 2 mr m qmr m A ::: qmr m A
Eq. (37) is described as
ð45Þ
0 ½B2
::: :::
0
:::
:
3
0 7 0 7 7 7 5 ½Bm
nm
3
ð40Þ
6. Simulation studies and performance evaluation of the proposed control schemes In this section, simulations for the realistic nonlinear model of boiler turbine dynamics in the presence of modeling imprecisions as well as disturbances are performed to investigate the efficiency and effectiveness of the proposed control schemes. As a worst case assumption and for investigation the occurrence of chattering phenomenon in the control inputs, it is assumed that the uncertainties and disturbances have the same high frequencies. For convenience in numerical studies, the plant modeling imprecisions as well as disturbances are assumed to be as follows: Δx_ 1 ¼ Δf 1 ðx; u; tÞ þ d1 ðtÞ ¼ 0:5 cos ð0:5tÞ Δx_ 2 ¼ Δf 2 ðx; u; tÞ þ d2 ðtÞ ¼ 0:5 cos ð0:5tÞ Δx_ 3 ¼ Δf 3 ðx; u; tÞ þ d3 ðtÞ ¼ 0:5 cos ð0:75tÞ
ð48Þ
where the imprecisions and disturbances can be modeled by any bounded function based on the mentioned Assumptions 1 and 2. Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
8
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. Time response of the state variables after implantation of the type-I servo controller in the presence of uncertainties and disturbances (a) in tracking of desired setpaths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding periodic orbits of state variables. (c) Corresponding variation of the drum water level.
6.1. Parameters assignment and simulations of the robust adaptive sliding mode controller In this section, simulation studies are conducted for the robust adaptive sliding mode controller to validate and demonstrate the
effectiveness of the proposed RASMC. For solving the nonlinear differential equations, MATLAB ode45 solver is used. For the RASMC control law which is described by Eq. (18) associated with Eq. (19), the positive constant parameters are chosen as, φi ¼ 1:25 10 5 ; ψ i ¼ 1:25 10 5 ; λ0 ¼ 10 3 . Initial condition of the adaption
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
parameters are chosen as: θ0 ¼ 0:05; ρ0 ¼ 0:05. The other positive constant parameters are chosen as γ i ¼ 2. It is well-known that the magnitude of the chattering is proportional to the sign function gain. Therefore, to have the smaller control signals and eliminate the chatter phenomenon, the parametersφi , ψ i , λ0 and the initial values of the adaption parameters φi and ψ i must be chosen small positive values. In that case, the adaptive law finds the best control gains which leads to the less control signal inputs and chatter elimination. To avoid the occurrence of chatter phenomenon as it is possible, the sign function which is a rigid switcher and may lead to the chatter is replaced with the sat function as ( si Z 10 signðsi Þ satðsi Þ ¼ tan hð0:02si Þ si o 10 Since it is impossible to get the zero error absolutely, the update of adaptive parameters will be continued. It means that the adaptive parameters tend to the infinity as the time tends to the infinity. This leads to the large sat function gains which may cause the large control efforts and chattering. To overcome this difficulty, as a tradeoff between the accuracy and performance, the parameters adaption is terminated if si r 2 and it will be resumed if it exceeds this threshold. Simulation results for the implementation of robust adaptive sliding mode controller are presented in Figs. 3–6. Fig. 3a shows the variation of drum pressure, electric output and fluid density in regulating around the median operating point #4 with initial conditionx0 ¼ ½80; 50; 390T . Then, its behavior is studied in tracking the desired sequence of ramp inputs from operating point #4 to #7, then to #1 and finally to #5. Accordingly, at the beginning of the plots, the state variables are regulated around the median operating point #4 after about 500 s without any overshoot. Thereafter, there is almost perfect control in tracking objectives between various operating points (during the interval [500–7000 s]). Corresponding periodic orbits of sate variable are displayed in Fig. 3b after implementation of the RASMC. It is observed that the RAMSC acts efficiently in disturbance rejection and keeping the system dynamics behavior in a periodic manner (rather than the existence of unstable quasi-periodic solutions). It should be noticed that in Fig. 3b, the centers of narrow ellipses coincide with the operating points #7, #1 and #5 of the boiler–turbine unit, as listed in Table 2. Fig. 3c displays the variation of the drum water level, which is absolutely stable and desirable without any oscillations and jumps. Fig. 4 demonstrates the variation of the errors where it is obvious that the tracking error is limited in an acceptable bound. Time response of the adaption parameters θ^ i ; ρ^ i ; i ¼ 1; 2; 3 are displayed in Fig. 5a and b, respectively, which are used as the sat function gains. Obviously, all adaptation parameters converge to the some positive constants, which are almost the most suitable sign function gains. Moreover, Fig. 6 demonstrates the required variation of the fuel, steam and feed-water valves (as the control inputs) for the tracking objectives of Fig. 3a. As it is observed, control inputs have no chattering, oscillation and saturation. Also, their steady values coincide with the corresponding operation point's values, listed in Table 2. These features imply that they are feasible in practical applications. 6.2. Type-I servo controller parameters assignment and simulation To have an appropriate performance of the boiler–turbine, and after several trial and error assignments, desired closed-loop poles are assigned as λ1 ¼ 0:006; λ2 ¼ 0:01; λ3 ¼ 0:014; λ4 ¼ 0:016; λ5 ¼ 0:018; λ6 ¼ 0:012 Finding transformation T~ and matrices Ad ; Θ1 ; Θ2 and Ω (as discussed in Section 5), and using Eqs. (36) and (46), feedback gain
Fig. 8. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 7a (when the servo type-I controller is used).
matrix K^ is determined as 2 0:0325 0:0028 0:0022 6 0 K^ ¼ 4 0:0049 0:0054 0:0001 0:0045 0:0133
0:0002
0
0
0
0
0
0
0
0:0001
3 7 5
ð49Þ
Type-I servo controller simulations are presented in Figs. 7 and 8. Fig. 7a illustrates the variation of drum pressure, electric output and fluid density in regulating around the median operating point #4 with initial condition x0 ¼ ½80; 50; 390T . Then, their behavior is studied in tracking the desired sequence of ramp inputs from operating point #4 to #7, then to #1 and finally to #5. Accordingly, at the beginning of the plots, the state variables are regulated around the median operating point #4 after about 200 s with a slight overshoot, which is acceptable. Also, according to Fig. 7a, the performance of the type-I servo controller is acceptable for the drum pressure x1 and the fluid density x3 . But it is disappointing in tracking objective for the electric power x2 ; while its stability is important for the power grid. There are several jumps in system response around the operating point #7. The periodic orbits of the state variables are displayed in Fig. 7b after implementation of the type-I servo controller. According to Fig. 7b, it is obvious that the type-I servo controller cannot guarantee the suppression of quasi-periodic solutions for the electric power around the operating point #7 (the irregular orbits of second plot of Fig. 7b confirms this fact). The corresponding variation of drum water level is shown in Fig. 7c in which there are undesirable oscillations around the operating point #7. Fig. 8 displays the required variation of the fuel, steam and feed-water valves (as the control inputs) for the tracking objectives of Fig. 7a. It is apparent that the control inputs are associated with the jumps and oscillations, especially around the operating point #7. Also, chattering is obvious in the fuel flow rate, u1 .
7. Comparison of the performance of robust adaptive sliding mode controller and type-I servo controller To investigate the effectiveness of proposed control approaches, the performance of the robust adaptive controller and type-I servo controller are compared in Figs. 9 and 10. Fig. 9a illustrates the time response of the drum pressure, electric power and fluid density for the both of RASMC and type-I servo controller. Although the RASMC has slower response in regulation rather than the type-I servo controller (due to process of tuning gains of sat function), it provides almost perfect behavior in tracking objectives. Furthermore, the response of the RASMC is without any significant jumps rather than poor performance of the type-I servo in tracking objective for the electric power ðx2 Þ; which its stability is crucial for the power grid. Moreover,
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
10
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 10. Required variation of the fuel, steam and feed-water valves (as the control inputs) for the objectives of Fig. 9a when the RASMC (black line) or type-I servo controller (blue line) is used. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
oscillations. The sliding mode control is well-known for its accompanying chatter phenomenon, but by using the appropriate adaptation laws and sat function, this phenomenon has been eliminated.
8. Conclusion Consistent and reliable performance of the boiler–turbine unit is crucial for the safety and efficiency of the power plant. Controller must be able to regulate the output variables around the operating points as well as track the desired trajectories to switch into the other desired operating points in a balance with the power grid demand. To achieve this aim, a robust adaptive sliding mode controller (RASMC) is proposed to control a nonlinear multi-input multi-output (MIMO) model of the industrial boiler–turbine unit, in the presence of unknown bounded uncertainties and external disturbances. To tackle the uncertainties and external disturbances, appropriate adaption laws are introduced. To guarantee the sliding motion occurrence, suitable control laws are constructed. Then the robustness and stability of the proposed RASMC is proved via Lyapunov stability theory. To compare the performance of the proposed RASMC with traditional control schemes, a type-I servo controller is designed. Simulations reveal the superiority of the RASMC over the type-I servo controller through the following general conclusions:
In regulating around the median operating point #4, both Fig. 9. Time response of the state variables after implantation of the robust adaptive sliding mode controller (RASMC) (black line) and type-I servo controller (blue line) in the presence of uncertainties and disturbances (a) in tracking of desired set-paths between the operating points # 4 to #7, then to #1 and finally to #5. (b) Corresponding variation of the drum water level. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the behavior of the drum water level for both control approaches is compared in Fig. 9b. It indicates the better performance of the RASMC. Drum water level response of the servo controller is oscillating around the operating point #7 which may leads to the flooding of the steam lines and consequently emergency shutdown. Fig. 10 demonstrates that chatter phenomenon is significantly eliminated in corresponding variation of the fuel, steam and feedwater valves when RASMC is used. Conversely, the control inputs of the type-I servo controller are associated with the jumps and
controllers have an acceptable performance but RASMC is slower due to the process of tuning adaption parameters. In tracking objectives, the performance of the RASMC is excellent rather than the weak performance and instability of responses in type-I servo controller (especially, poor behavior is observed for the electric power; which affects adversely the power grid). Chattering phenomenon is significantly eliminated in control inputs of the RASMC. At the same time, the control inputs of the type-I servo controller are associated with the jumps and oscillations. After implementation of the RASMC, the variation of the drum water level is stable without jumps and oscillations. Conversely, the variation of the drum water level in type-I servo controller has oscillations and jumps which reduce the safety and reliability of the boiler–turbine unit (due to possible flooding of the steam lines and consequently the emergency shutdown). RASMC acts efficiently in disturbance rejection and keeping the system dynamics behavior in a periodic manner; rather than
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
S. Ghabraei et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
the existence of unstable quasi-periodic solutions which was observed in type-I servo controller.
References [1] Moradi H, Bakhtiari-Nejad F, Saffar-Avval M. Robust control of an industrial boiler system; a comparison between two approaches: Sliding mode control & H1 technique. Energy Convers Manag 2009;50:1401–10. [2] Chen P-C, Shamma. JS. Gain-scheduled ℓ1 -optimal control for boiler-turbine dynamics with actuator saturation. J Process Control 2004;14:263–77. [3] Bell R, Åström KJ. Dynamic models for boiler-turbine-alternator units: data logs and parameter estimation for a 160 MW unit, Technical Report. Sweden: Lund Institute of Technology, Department of Automatic Control; 1987. [4] K Lo, P Zeng, E Marchand, A Pinkerton. Modelling and state estimation of power plant steam turbines. In: generation, transmission and distribution, IEE Proceedings; 1990. p. 80–94. [5] Murty, VV, Sreedhar, R, Fernandez, B, Masada, GY, Hill, AS Boiler system identification using sparse neural networks. In: Proceedings of the advances in robust and nonlinear control systems, ASME winter meeting.Louisiana, USA: New Orleans Article. [6] Donate P, Moiola J. Model of a once-through boiler for dynamic studies. Lat Am App Res 1994;24:159–66. [7] Bracco S, Troilo M, Trucco A. A simple dynamic model and stability analysis of a steam boiler drum. Proc Inst Mech Eng Part A: J Power Energy 2009;223:809–20. [8] Aranda, E, Frye,.M, Chunjiang, Q Model development, state estimation, and controller design of a nonlinear utility boiler system. In: Proceedings of the IEEE international conference on industrial technology (ICIT’08). Chengdu, China: Sichuan University. [9] Sindareh-Esfahani P, Habibi-Siyahposh E, Saffar-Avval M, Ghaffari A, BakhtiariNejad F. Cold start-up condition model for heat recovery steam generators. Appl Therm Eng 2014;65:502–12. [10] Kouprianov V, Kaewboonsong W. Modeling the effects of operating conditions on fuel and environmental costs for a 310 MW boiler firing fuel oil. Energy Convers Manag 2004;45:1–14. [11] Kouadri A, Namoun A, Zelmat M. Modelling the nonlinear dynamic behaviour of a boiler–turbine system using a radial basis function neural network. Int J Robust Nonlinear Control 2014;24:1873–86. [12] Åström KJ, Bell RD. Drum-boiler dynamics. Automatica 2000;36:363–78. [13] Li C, Zhou J, Li Q, An X, Xiang X. A new T–S fuzzy-modeling approach to identify a boiler–turbine system. Expert Syst Appl 2010;37:2214–21. [14] Liu X, Kong X, Hou G, Wang J. Modeling of a 1000 MW power plant ultra super-critical boiler system using fuzzy-neural network methods. Energy Convers Manag 2013;65:518–27. [15] Collazo J, Porteiro J, Míguez J, Granada E, Gómez M. Numerical simulation of a small-scale biomass boiler. Energy Convers Manag 2012;64:87–96. [16] Zhang S, Taft CW, Bentsman J, Hussey A, Petrus B. Simultaneous gains tuning in boiler/turbine PID-based controller clusters using iterative feedback tuning methodology. ISA Trans 2012;51:609–21. [17] Maher RA, Mohammed IA, Ibraheem IK. Polynomial based H1 robust governor for load frequency control in steam turbine power systems. Int J Electr Power Energy Syst 2014;57:311–7. [18] Moradi H, Bakhtiari-Nejad F. Improving boiler unit performance using an optimum robust minimum-order observer. Energy Convers Manag 2011;52:1728–40. [19] Yu T, Chan K, Tong J, Zhou B, Li D. Coordinated robust nonlinear boiler-turbinegenerator control systems via approximate dynamic feedback linearization. J Process Control 2010;20:365–74. [20] Moradi H, Alasty A, Saffar-Avval M, Bakhtiari-Nejad F. Multivariable control of an industrial boiler–turbine unit with nonlinear model: a comparison between gain scheduling and feedback linearization approaches. Sci Iran 2013;20:1485–98. [21] Saikia LC, Sahu SK. Automatic generation control of a combined cycle gas turbine plant with classical controllers using Firefly Algorithm. Int J Electr Power Energy Syst 2013;53:27–33.
11
[22] Moradi H, Saffar-Avval M, Bakhtiari-Nejad F. Sliding mode control of drum water level in an industrial boiler unit with time varying parameters: a comparison with H1 robust control approach. J Process Control 2012;22:1844–55. [23] Wu X, Shen J, Li Y, Lee KY. Hierarchical optimization of boiler–turbine unit using fuzzy stable model predictive control. Control Eng Pract 2014;30:112–23. [24] Li Y, Shen J, Lee KY, Liu X. Offset-free fuzzy model predictive control of a boiler–turbine system based on genetic algorithm. Simul Model Pract Theory 2012;26:77–95. [25] Moon U-C, Lee KY. An adaptive dynamic matrix control with fuzzyinterpolated step-response model for a drum-type boiler-turbine system. Energy Convers IEEE Trans 2011;26:393–401. [26] Li, XF, Zhang, W Coordinated control of Fossil-Fuel power plant based on the fuzzy PID control. In: Proceedings of the IEEE conference on decision and control. Hawaii, USA: Maui; 2012. Article # 6427049, p. 3080–85. [27] Thangavelusamy D, Ponnusamy L. Elimination of chattering using fuzzy sliding mode controller for drum boiler turbine system. J Control Eng Appl Inform 2013;15:78–85. [28] Zhang J, Zhang F, Ren M, Hou G, Fang F. Cascade control of superheated steam temperature with neuro-PID controller. ISA Trans 2012;51:778–85. [29] Wu X, Shen J, Li Y, Lee. KY. Data-driven modeling and predictive control for boiler–turbine unit using fuzzy clustering and subspace methods. ISA Trans 2014;53:699–708. [30] Wu J, Shen J, Krug M, Nguang SK, Li Y. GA-based nonlinear predictive switching control for a boiler-turbine system. J Control Theory Appl 2012;10:100–6. [31] Wu X, Shen J, Li Y, Lee. KY. Data-driven modeling and predictive control for boiler–turbine unit. Energy Convers IEEE Trans 2013;28:470–81. [32] Liu X, Kong X. Nonlinear fuzzy model predictive iterative learning control for drum-type boiler–turbine system. J Process Control 2013;23:1023–40. [33] Yang, S, Qian, C Real-time optimal control of a boiler–turbine system using pseudospectral methods. In: Proceedings of the 19th annual joint isa powidepri controls and instrumentation conference and 52nd ISA POWID symposium. Illinois, USA: Rosemont. Article #4. [34] S Godoy, G Scaglia, S Romoli, R Suvire, O Ortiz. Trajectory tracking of boilerturbine. In: Proceedings of the IEEE biennial congress of Argentina (ARGENCON).; 2014. p. 159–164. [35] Moradi H, Vossoughi G, Alasty A. Suppression of harmonic perturbations and bifurcation control in tracking objectives of a boiler–turbine unit in power grid. Nonlinear Dyn 2014;76:1693–709. [36] Moradi H, Alasty A, Vossoughi G. Nonlinear dynamics and control of bifurcation to regulate the performance of a boiler–turbine unit. Energy Convers Manag 2013;68:105–13. [37] Chen P-C. Multi-objective control of nonlinear boiler–turbine dynamics with actuator magnitude and rate constraints. ISA Trans 2013;52:115–28. [38] Fang F, Wei. L. Backstepping-based nonlinear adaptive control for coal-fired utility boiler–turbine units. Appl Energy 2011;88:814–24. [39] Li R, Chen W. Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems. Nonlinear Dyn 2014;76:785–95. [40] Yao J, Jiao Z, Ma D. Adaptive robust control of DC motors with extended state observer. IEEE Transactions on Industrial Electronics 2014;61:3630–7. [41] Liu L, Li D-J, Liu Y-J. Adaptive control for a class of nonlinear systems and application to hard disk drives. J Vib Control 2014;20:153–60. [42] Guan C, Pan S. Adaptive sliding mode control of electro-hydraulic system with nonlinear unknown parameters. Control Eng Pract 2008;16:1275–84. [43] Slotine J-JE, Li W. Applied nonlinear control. New Jersey: Prentice-Hall Englewood Cliffs; 1991. [44] Slotine J-J, Coetsee J. Adaptive sliding controller synthesis for non-linear systems. Int J Control 1986;43:1631–51. [45] FERNÁNDEZ R B, Hedrick JK. Control of multivariable non-linear systems by the sliding mode method. Int J Control 1987;46:1019–40. [46] Krstic M, Kanellakopoulos I, Kokotovic PV. Nonlinear and adaptive control design. Wiley; 1995. [47] Khalil HK. Nonlinear systems. 3rd ed. New Jersey: Prentice hall; 2002. [48] Ogata K. Modern control engineering. 5th ed.. New Jersey: Prentice Hall; 2010. [49] D'Azzo J, Houpis C. Linear control system analysis and design: conventional and modern. McGraw-Hill; 1995.
Please cite this article as: Ghabraei S, et al. Multivariable robust adaptive sliding mode control of an industrial boiler–turbine in the presence of modeling.... ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.04.010i
