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Stable Model Predictive Tracking Control for Boiler-Turbine Coordinated Control System
Stable Model Predictive Tracking Control for Boiler-Turbine Coordinated Control System Xiao Wu*, Jiong Shen*, Yiguo Li*, Kwang Y. Lee** * Department of Energy Information and Automation, Southeast University, Nanjing, 210096, China (e-mail: [email protected]; [email protected]; [email protected]). ** Department of Electrical and Computer Engineering, Baylor University, One Bear Place #97356, Waco, TX 76798-7356, USA (e-mail: [email protected]) Abstract: In this paper, a stable model predictive tracking controller (SMPTC) is designed based on the piecewise linear model to control a boiler-turbine system for a wide-range operation with unmeasured state and significant disturbance or modeling mismatch. It is shown that the proposed controller is constructed by the combination of a state and disturbance observer, a steady-state targets calculator as well as a stable model predictive controller, which can be obtained efficiently by solving a set of linear matrix inequalities. The main advantage of the proposed SMPTC is that it can guarantee the stability of the closed-loop model predictive control system with input constraints, while obtaining an offset-free tracking performance in an optimal way. Simulation results demonstrate the advantage and effectiveness of the proposed controller. Keywords: Boiler-turbine coordinated system, model predictive control, piecewise linear model, offsetfree tracking, linear matrix inequality. 1. INTRODUCTION Boiler-turbine coordinated control system is the most important part of the fossil-fuel power plant which is used to convert fuel in chemical energy to mechanical energy and then to electrical energy. For a typical boiler-turbine control system, the primary task is to regulate the power output to meet the demand of the grid, while maintaining the drum pressure, steam temperature and water level within some tolerances to keep the power plant to operate in a safe condition. However, control of such system is challenging due to its characteristics such as multi-variables, nonlinear, timevarying, strong coupling among variables, large time-delays, etc. In addition, to meet the load demand for electric power at all times, and at constant voltage and frequency, large-scale power plants should be able to change the load frequently in a wide range and operate in multiple operating regimes. For this reason, the nonlinear behaviour of the system becomes more significant during the transitions between operating points, and the conventional proportional-integral-derivative (PID) controllers with fixed gains at a nominal operating point cannot satisfy the control requirement. Therefore, in order to further improve the performance and enhance plant efficiency, the boiler-turbine unit has been studied widely in the literatures using various control techniques. In Dimeo & Lee (1995), genetic algorithm is used to optimize the traditional controller through adaptively regulating controller parameters during changes in operating points; and while the resulting controller achieves a good performance, the optimization procedure is time consuming.
In Park & Lee (1996) and Wen, Marquez, Chen & Liu (2005), robust controller such as LQG/LTR or H ∞ is designed based on a linear model around one operating point and thus cannot satisfy the requirement of the wide-range operation. Multimodel strategy which uses a combination of several linear models to approximate the nonlinear system provides a good way to solve the problems of wide-range operation and heavy computational burden on nonlinear optimization. Based on the multi-model method, a gain scheduled optimal controller is presented for the boiler-turbine system (Chen & Shamma, 2004), and an H ∞ controller is proposed, which receives good tracking and disturbance rejection properties (Wu, Nguang, Shen, Liu & Li, 2010). However, the close-loop stability that is more desirable for industrial processes such as power plant cannot be guaranteed with all of the aforementioned controllers. Model Predictive Control (MPC) uses an explicit model to predict the future behavior of the plant and solves a constrained optimization problem on-line to obtain the optimal control sequence (Mayne, Rawlings & Rao, 2000). Since MPC can guarantee the stability of the closed-loop system and deal with the input constraint in an optimal way, it has become a popular strategy in process control (Rawlings, 2000). By adopting an infinite horizon objective function and constructing Lyapunov function to find its upper bound (Kothare, Balakrishnan & Morari, 1996; Lu & Arkun, 2000; Zhang, Feng & Lv, 2007), stable MPCs are also designed and used for boiler-turbine system based on the multi-model system, such as piecewise linear (PWL) model or TakagiSugeno (TS) fuzzy model (Wu, Shen & Li, 2010; Wu, Shen & Li, 2011). However, these MPCs as well as some of the
aforementioned control strategies are all developed with the assumption that all state variables can be measured, which makes the modeling requirement strict and usually cannot be achieved. In addition, in the situation where modeling mismatch and disturbance exist, tracking offset occurs for these controllers which greatly affect their performance.
Remark 2.2: To ensure the detectability of the state and disturbance variables, we assume the nominal system of (1)(2) is detectable and
By extending the results in Wu, Shen & Li (2010; 2011) this paper presents a stable model predictive tracking controller (SMPTC) for boiler-turbine coordinated system by using the switchable piecewise linear models. An observer is first developed to estimate the unmeasured state variables and disturbances, and then steady-state targets are determined to remove the effect of disturbances and modeling mismatch to guarantee the offset-free performance. With the estimation of state variables and disturbances as well as the steady-state targets, a stable MPC is built at the last. By solving a set of linear matrix inequalities (LMIs), the resulting controller can guarantee both the stability of the closed-loop predictive control system and the input constraints while achieving an offset-free tracking control in an optimal way.
(Pannocchia, 2003 & 2004; Muske & Badgwell, 2002).
2. STABLE MODEL PREDICTIVE TRACKING CONTROL BASED ON PIECEWISE LINEAR MODEL 2.1 Piecewise Linear Model Suppose a nonlinear discrete-time system can be represented as the following PWL model: m
x(k + 1) = ∑ Si ( z (k ))( Ai x(k ) + Bi u (k ) + Bid d (k ))
(1)
i =1
m
y (k ) = ∑ Si ( z (k ))(Ci x(k ) + Di u (k ) + Ei + Cid d (k ))
(2)
i =1
z (k )
where
is
a
with Si ( z (k )) ∈ {0,1} ,
measurable m
∑ Si ( z (k )) = 1
switching
⎡ I − Ai rank ⎢ ⎣ Ci
(5)
2.2 State and Disturbance Observer Since the state variables and disturbances are unmeasured, an observer is first designed. Consider the following augmented multi-model observer:
⎡ xˆ ( k + 1) ⎤ ⎡ Az ⎢ˆ ⎥=⎢ ⎣ d ( k + 1) ⎦ ⎣ 0
⎡ L1z ⎤ Bzd ⎤ ⎡ xˆ ( k ) ⎤ ⎡ Bz ⎤ ⎥ ⎢ ˆ ⎥ + ⎢ ⎥ u ( k ) + ⎢ 2 ⎥ [ yˆ ( k ) − y ( k ) ] I ⎦ ⎣ d (k ) ⎦ ⎣ 0 ⎦ ⎣ Lz ⎦
yˆ (k ) = Cz xˆ (k ) + Dz u (k ) + Ez + Czd dˆ (k )
(6)
Following the method in Feng (2006), we can construct a stable multi-model observer if there exist matrices H , Gi , and a symmetric positive definite matrix X , such that the following LMI problem is feasible:
⎡ HT + H − X *⎤ (7) ⎥>0 ⎢ aug aug X⎦ ⎣ HAi + Gi Ci where the augmented matrices are defined by ⎡ A Bid ⎤ aug d Aiaug = ⎢ i ⎥ and Ci = ⎡⎣Ci Ci ⎤⎦ , and “ * ” in the 0 I ⎣ ⎦ matrix stands for the corresponding terms of the symmetric matrix. Then the observer gains can be determined by
variable
Li = H −1Gi = ⎡⎣ L1iT
. x(k ) ∈ R , u ( k ) ∈ R , n
− Bid ⎤ ⎥ = n + nd Cid ⎦
l
T
L2i T ⎤⎦ , i = 1, 2...m .
i =1
y (k ) ∈ R p are state, input and output variables, respectively,
and d (k ) ∈ R nd is a disturbance term including all the modeling mismatch and unmeasured disturbances, but it would reach an unknown steady-state value asymptotically: that is: lim d (k ) = d . The matrices Ai , Bi , Bid , Ci , Di , Ei , Cid k →∞
are the parameters of the local model for i = 1, 2...m . Model (1)-(2) can be rewritten into the global model form:
x(k + 1) = Az x(k ) + Bz u (k ) + Bzd d (k ) d z
y ( k ) = C z x (k ) + Dz u (k ) + E z + C d ( k )
(3) (4)
m
2.3 Steady-State Targets Calculator For a given set of desired input and output set points, the goal of the SMPTC is first to find the steady-state targets and then design the MPC to track the steady-state targets. To remove the effect of unknown disturbances and modeling mismatch, a steady-state targets calculator (SSTC) is used in (Pannocchia, 2003 & 2004; Muske & Badgwell, 2002), achieving an offset-free tracking control. In this paper an SSTC is considered based on the PWL model, where a steady-state target ( xt , ut ) can be obtained by solving the following quadratic programming at each sampling time k :
where Az = ∑ Si ( z (k )) Ai and other matrices are defined
min(ut − uref )T Rs (ut − uref )
similarly.
⎡ I − Az s.t. ⎢ ⎣ Cz
i =1
Remark 2.1: Without the disturbance term, model (1)-(2) or (3)-(4) is a nominal PWL system which can be obtained using the Taylor’s series approximation or other modeling method. To simplify the stability design of MPC, an affine term is omitted in (1) and (3).
xt , ut
umin
⎤ Bzd dˆ (k ) − Bz ⎤ ⎡ xt ⎤ ⎡ ⎥ =⎢ ⎥ ⎢ ⎥ Dz ⎦ ⎣ut ⎦ ⎢ yref − Ez − C zd dˆ (k ) ⎥ ⎣ ⎦ ≤ ut ≤ umax
(8)
In which uref and yref are desired input and output set points, respectively, Rs = R > 0 is a symmetric weighting matrix, T s
and umin , umax are respectively the lower and upper limits of the input. Remark 2.3: With the augmented observer and SSTC, an offset-free tracking performance can be achieved. We omit the proof here due to the space limitation, and one can refer to (Pannocchia, 2003 & 2004; Muske & Badgwell, 2002) for more details. Remark 2.4: Optimization problem (8) may be infeasible due to the stringent constraints; in that situation, we suggest to use soft constraints on the input, and change (8) to:
min(ut − uref )T Rs (ut − uref ) + ζ 1T Q1ζ 1 + ζ 2T Q2ζ 2 xt ,ut
⎤ Bzd dˆ (k ) − Bz ⎤ ⎡ xt ⎤ ⎡ ⎢ ⎥ = ⎢ ⎥ ⎥ d Dz ⎦ ⎣ut ⎦ ⎢ yref − Ez − C z dˆ (k ) ⎥ ⎣ ⎦ + ζ 1 ≤ ut ≤ umax + ζ 2
⎡ I − Az s.t. ⎢ ⎣ Cz umin
(9)
However, if (8) keeps infeasible all the time, it means that the set points yref cannot be tracked without offset. Once the steady-state target ( xt , ut ) is obtained, we have:
xt = Az xt + Bz ut + Bzd dˆ (k )
(10)
Then, (3) can be changed to
x(k + 1) − xt = Az ( x(k ) − xt ) + Bz (u (k ) − ut ) + B ( d ( k ) − dˆ ( k )) d z
(11)
Noting that the disturbance estimation error is bounded with the help of the observer, we now use the nominal model of (11) for prediction:
x (k + s + 1| k ) = Az x (k + s | k ) + Bz u (k + s | k )
(12)
where x = x − xt and u = u − ut .
control variable u (k | k ) , matrices Yi , G and symmetric positive definite matrix S , such that the following LMI problem is feasible:
min
γ ,u ( k |k ),Yi ,G , S
γ
s.t.(15) − (18)
(14)
then, through minimizing the upper bound of the infinite horizon objective function (13), the proposed predictive controller will achieve an offset-free tracking performance in an optimal way while guaranteeing the stability of the closedloop system:
⎡ G + GT − S ⎢ ⎢( Ai G + BiYi ) ⎢ Q01/2 G ⎢ 1/2 ⎢⎣ R0 Yi
*⎤ ⎥ 0⎥ i = 1, 2...m (15) >0 0 γI 0 ⎥ ⎥ 0 0 γ I ⎥⎦ 1 * * * *⎤ ⎡ ⎢ ⎥ ⎢ A (k ) xˆ (k ) + B (k )u (k | k ) S 0 ⎥ 0 0 t z ⎢ z ⎥ 2 ⎢ ⎥ γI ⎢ 0 0 0 ⎥ > 0 (16) Q01/ 2 xˆt (k ) 2 ⎢ ⎥ ⎢ 0 0 γI 0⎥ R01/2 u (k | k ) ⎢ ⎥ ⎢ S ⎥ 0 0 0 Az (k )ω ⎢ ⎥ 2⎦ ⎣ (umin − ut (k )) ≤ u (k | k ) ≤ (umax − ut (k )) (17) ⎡U ⎢Y T ⎣ i
* S
* 0
Yi ⎤ >0 i = 1, 2,..., m G + GT − S ⎥⎦
(18)
in which xˆt (k ) = xˆ (k ) − xt (k ) , ω is the upper bound of the state estimation error | x |≤ ω , and diagonal matrix U with elements: U pp = min((umax , p − ut , p (k )) 2 ,(umin, p − ut , p (k ))2 ),
p = 1, 2,..., l. Proof: See Appendix A.
2.3 Stable Model Predictive Tracking Control Based on Piecewise Linear Model
3. THE BOILER-TURBINE SYSTEM MODEL
Considering the infinite horizon objective function: ∞
J 0∞ (k ) = ∑ [ x (k + s | k )T Q0 x (k + s | k ) + s =0
(13)
u (k + s | k ) R0 u (k + s | k )] T
where Q0 = Q0T > 0, R0 = R0T > 0 are symmetric weighting matrices of states and control moves, respectively, we present the following main result: Theorem 1: For the discrete PWL system (1) under input constraint: umin, p ≤ u p (k + s | k ) ≤ umax , p , s ≥ 0, p =1, 2,..., l ,
with the augmented observer (6) determined by LMIs (7) and a feasible SSTC (8) at sampling time k , if there exist
The boiler-turbine system used in this paper represents the behavior of a 160MW oil-fired power plant and has already been modeled as a third order nonlinear model. The controllable inputs into the system are valve actuator positions that control the mass flow of fuel, represented as u1, steam to the turbine, u2, and feedwater to the drum, u3. All valve position variables are constrained to lie in the interval [0, 1]. The measurable outputs of the model are electric power, E in MW, drum steam pressure, P in kg/cm2, and the drum water level deviation from a set level, L in meters. The state variables are electric power, E, drum steam pressure, P, and steam-water density, ρf. The dynamics of this particular power plant were recorded and formulated into mathematical model by Bell and Åström (1987) and are shown below in a summarized form, to be used as a model in our control system:
dP = 0.9u1 − 0.0018u2 P9/8 − 0.15u3 (19) dt dE = ((0.73u2 − 0.16) P 9/8 − E ) / 10 (20) dt dρf = (141u3 − (1.1u2 0.19) P) / 85 (21) dt Using the solution for ρf, the drum water level can be calculated using the following equations: qe = (0.85u2 − 0.14) P + 45.59u1 − 2.51u3 − 2.09 (22)
α s = (1 / ρ f − 0.0015) / (1 / (0.8P − 25.6) − 0.0015) (23) L = 50(0.13ρ f + 60α s + 0.11qe − 65.5)
(24)
where αs is the steam quality and qe is the evaporation rate in kg/s. Remark 3.1: This model has been widely studied using different controllers. Among them, a large number of papers directly design controllers based on state feedback with assumptions that all state variables can be measured (Wu, Nguang, Shen,Liu & Li, 2010; Wu, Shen & Li, 2010; Wu, Shen & Li, 2011; Keshavarz, Barkhordari & Motlagh, 2010). However, it is obvious that state variable ρf cannot be measured. 4. SIMULATION RESULTS We choose the drum steam pressure P as the switching signal and divide the system into “low,” “middle” and “high” load operating regions. Corresponding local models are then established through linearizing the original system at 80%, 100% and 120% operating points by taking Taylor’s series approximation and discretizing them under sampling time 1s. The resulting PWL model has the same structure as with (1)(2) by omitting an affine term in (1) and adding the disturbance term. We set:
0 0 ⎤ ⎡ 0.02 ⎢ 0.03 0 ⎥⎥ ，C1d = C2d = C3d = 03×3 B =B =B =⎢ 0 0 0.01⎦⎥ ⎣⎢ 0 We now apply the proposed SMPTC to the boiler-turbine coordinated system and consider the case of load change in a wide range as well as significant disturbances and modeling mismatch. The control mission is tracking the expected operating points of drum pressure and output power while maintaining the drum water level. We assume that at t=25s, both the pressure and power output desired operating points change in step from 70% to 130%, and at t=100s, all parameters in the original nonlinear model (19)-(21) change to their 70% values for disturbances and modeling mismatch. The setpoint of drum water level deviation is maintained at zero. With controller parameters: d 1
d 2
d 3
Q0 = I 3×3 ; R0 = 100 I 3×3 ; Rs = I 3×3 ; ω = [ 0.1 0.1 5] , T
the simulation results are shown in Fig. 1 and Fig. 2.
Fig. 1. SMPTC output performance of the boiler-turbine unit (solid line: proposed model predictive tracking controller; dashed line: controller in Wu, Shen & Li (2011), dotted line: reference.). From the control results, we can see that when the load is increased, the drum pressure and power output respond rapidly, and then approaches to the expected operating points, simultaneously, and the drum water level fluctuates within an acceptable range and gradually returns to zero. Even in the case of significant modeling mismatch, the offset-free performance can be achieved. The proposed stable MPC can control the boiler-turbine coordinated system effectively. Another stable MPC based on TS fuzzy model (Wu, Shen & Li, 2011) is used for comparison. In Wu, Shen & Li (2011), all state variables are assumed to be measurable and a stable MPC is developed based on state feedback law with one-step free input variables. However, due to the effect of modeling mismatch and disturbances, the tracking error was persistent and the controller was not able to give a satisfactory performance. ACKNOWLEGEMENT This work was supported in parts by the National Natural Science Foundation of China (NSFC) under Grant 51076027 and Grant 51036002, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant
20090092110051, the Scientific Research Foundation of Graduate School of Southeast University under grant ybjj1120, and the U.S. National Science Foundation under grant ECCS 0801440.
Fig. 2. SMPTC input performance of the boiler-turbine unit (solid line: proposed model predictive tracking controller; dashed line: controller in Wu, Shen & Li (2011).). REFERENCES Bell, R. D. and Åström, K. J. (1987). Dynamic Models for Boiler-Turbine-Alternator Units: Data Logs and Paramter Estimation for 160 MW Unit, TRFT-3192. Lund Institute of Technology, Lund, Sweden. Chen, P. C. and Shamma, J. S. (2004). Gain-scheduled l1 optimal control for boiler-turbine dynamics with actuator saturation, Journal of Process Control, vol. 14, pp. 263277. Dimeo, R. and Lee, K. Y. (1995). Boiler-turbine control system design using a genetic algorithm, IEEE Transactions on Energy Conversion, vol. 10(4), pp. 752759. Feng, G. (2006). A survey on analysis and design of modelbased fuzzy control systems, IEEE Transactions on Fuzzy Systems, vol. 14(5), pp. 676-697. Keshavarz, M., Barkhordari, Y., and Jahed-Motlagh, M. R. (2010). Piecewise affine modeling and control of a
boiler–turbine unit, Applied Thermal Engineering, vol. 30, pp. 781–791. Kothare, M. V., Balakrishnan, V., and Morari, M. (1996). Robust constrained model predictive control using linear matrix inequalities, Automatica, vol. 32, no. 10, pp. 1361–1379. Lu, Y. and Arkun Y. (2000). Quasi-min-max MPC algorithms for LPV systems, Automatica, vol. 36, pp. 527–540. Mayne, D. Q., Rawings, J. B., and Rao, C. V. (2000). Constrained model predictive control: Stability and optimality, Automatica, vol. 36, pp. 789–814. Muske, K. R. and Badgwell, Thomas A. (2002). Disturbance modeling for offset-free linear model predictive control, Journal of Process Control, vol 12, pp. 617-632. Pannocchia, G. (2003). Robust disturbance modeling for model predictive control with application to multivariable ill-conditioned processes, Journal of Process Control, vol. 13, pp. 693-701. Pannocchia, G. (2004). Robust model predictive control with guaranteed setpoint trackingJournal of Process Control, vol 14, pp. 927-937. Park, Y. M. and Lee, K. Y. (1996). An auxiliary LQG/LTR robust controller for cogeneration plants, IEEE Transactions on Energy Conversion, vol. 11(2), pp. 407413. Rawlings, J. B. (2000). Tutorial overview of model predictive control, IEEE Contr. Syst. Mag., vol. 20, no. 3, pp. 38– 52. Wen, T., Marquez, H. J., Chen, T., and Liu, J. (2005). Analysis and control of a nonlinear boiler-turbine unit, Journal of Process Control, vol. 15(8), pp.883-891. Wu, J., Nguang, S.K., Shen, J., Liu, G. and Li, Y. G. (2010). Robust H infinite tracking control of boiler–turbine systems, ISA Transactions, vol. 49, pp. 369-375. Wu, X., Shen, J., and Li, Y. (2010). Control of boiler-turbine coordinated system using multiple-model predictive approach, 2010 8th IEEE International Conference on Control and Automation, ICCA. Wu, X., Shen, J., and Li, Y. G. (2011). Stable model predictive control based on TS fuzzy model with application to boiler-turbine coordinated system, Proceedings of the Chinese Society of Electrical Engineering, vol. 31(11), pp. 106-112. Zhang, T., Feng, G., and Lv, J. (2007). Fuzzy Constrained Min-Max Model Predictive Control Based on Piecewise Lyapunov Functions, IEEE Trans. Fuzzy Systems, vol.15, no.4, pp. 686-698. Appendix A. PROOF OF THEOREM 1 Part 1 (Minimizing the upper bound of infinite horizon objective function): For the infinite horizon objective function (13), divide it into two parts (Lu & Arkun, 2000):
J 0∞ (k ) = x (k | k )T Q0 x (k | k ) + u (k | k )T R0 u (k | k ) + J1∞ (k ) Suppose a common Lyapunov function V ( x) = xT S −1 x
(A1)
(A2)
satisfies:
2 xˆt (k )T Q0 xˆt (k ) + 2( Az (k )ω )T S −1 ( Az (k )ω )
V ( x (k + s + 1| k )) − V ( x (k + s | k )) ≤ −[ x (k + s | k )T T
× Q0 x (k + s | k ) + u (k + s | k ) R0 u ( k + s | k )]
(A3)
+2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T S −1 ( Az (k ) xˆt (k )
(A10)
+ Bz (k )u (k | k )) + u (k | k ) R0 u (k | k ) ≤ γ T
Summing (A3) from s = 1 to s = ∞ , and with x (∞ | k ) = 0 and V ( x (∞ | k )) = 0 , we get:
Then minimizing the upper bound of J 0∞ (k ) is equivalent to the minimization of γ , subject to (A10).
J1∞ (k ) ≤ V ( x (k + 1| k )) = x (k + 1| k )T S −1 x (k + 1| k )
(A4)
By defining S −1 = γ −1 S −1 ,and using Schur complements (Kothare, Balakrishnan & Morari, 1996), (A10) can be expressed as (16).
(A5)
Part 2(Stability constraint): With the state feedback control law:
Thus we can get the upper bound of J 0∞ (k ) :
J 0∞ (k ) ≤ x (k | k )T Q0 x (k | k ) + u (k | k )T R0 u ( k | k ) + x (k + 1| k )T S −1 x (k + 1| k )
However, in (A5), x (k | k ) = x (k ) = x(k ) − xt (k ) is an unmeasured variable, so we rewrite it as:
x (k | k ) = xˆ (k ) + x (k ) − xt (k ) = xˆt (k ) + x (k )
(A6)
Then the first term of the right hand of (A5) can be rewritten as:
≤ 2 xˆt (k ) Q0 xˆt (k ) + 2 x (k ) Q0 x (k ) T
(A7)
≤ 2 xˆt (k )T Q0 xˆt (k ) + C
At sampling time k , since the current switching variable z (k ) is assumed to be available, the current model { Az (k ), Bz (k )} can be obtained, and the third term of the right hand of (A5) can be rewritten as:
x (k + 1| k )T S −1 x (k + 1| k ) = ( Az (k ) x (k ) + Bz (k )u (k | k ))T
×S −1 ( Az (k ) x (k ) + Bz (k )u (k | k )) = ( Az (k ) xˆt (k ) + Bz (k )u (k | k ) + Az (k ) x (k )) ×S ( Az (k ) xˆt (k ) + Bz (k )u (k | k ) + Az (k ) x (k ))
T
≤ 2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T × S −1 ( Az (k ) xˆt (k ) + Bz (k )u (k | k )) + 2( Az (k ) x (k ))T × S −1 ( Az (k ) x ( k )) ≤ 2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T × S −1 ( Az (k ) xˆt (k ) (A8) + Bz (k )u (k | k )) + 2( Az (k )ω )T × S −1 ( Az (k )ω ) Substituting (A7) and (A8) into (A5), we can get:
J 0∞ (k ) ≤ 2 xˆt (k )T Q0 xˆt (k ) + 2( Az (k )ω )T ×S −1 ( Az (k )ω ) + 2( Az (k ) xˆt (k ) + Bz (k )u (k ))T ×S −1 ( Az (k ) xˆt (k ) + Bz (k )u (k )) + u (k | k )T R0u (k | k ) + C Define a scalar γ and suppose:
(A9)
(A12)
Substituting (A11),(A12) into (A3) and noting that (A13)
then the stability constraint (A3) is satisfied if the following holds:
GT + G − S − ( Az G + BzYz )T S −1 ( Az G + Bz Yz ) −
where C = 2ω T Q0ω is a constant.
−1
= ( Az + BzYz G −1 ) x (k + s | k ) (G − S )T S −1 (G − S ) > 0 ⇒ GT S −1G ≥ GT + G − S
x (k | k )T Q0 x (k | k ) = ( xˆt (k ) + x (k ))T Q0 ( xˆt (k ) + x (k ))
(A11)
the nominal predictive closed-loop PWL system (12) can be described as:
x (k + s + 1| k ) = Az x (k + s | k ) + Bz Yz G −1 x (k + s | k )
with x (k ) the estimation error between x(k ) and xˆ (k )
T
u (k + s | k ) = Yz G −1 x (k + s | k ), s > 0
GT Q0γ −1G − YzT R0γ −1Yz > 0
(A14)
which can be expressed by the LMIs (15) and since the common Lyapunov function (A2) satisfies ΔV < 0 for all time, the system can be guaranteed to be stable in the sense of Lyapunov. Part 3(Input constraint): Since we split the input into free variables and future control moves determined by state feedback law, we must constrain them accordingly. For the free variables, we directly constrain them by the peak bound (17). For the future control moves, by using the approach in Kothare, Balakrishnan & Morari (1996) and considering (A13), the constraint is satisfied if the LMIs in (18) are feasible and the proof is completed.
In Park & Lee (1996) and Wen, Marquez, Chen & Liu (2005), robust controller such as LQG/LTR or H ∞ is designed based on a linear model around one operating point and thus cannot satisfy the requirement of the wide-range operation. Multimodel strategy which uses a combination of several linear models to approximate the nonlinear system provides a good way to solve the problems of wide-range operation and heavy computational burden on nonlinear optimization. Based on the multi-model method, a gain scheduled optimal controller is presented for the boiler-turbine system (Chen & Shamma, 2004), and an H ∞ controller is proposed, which receives good tracking and disturbance rejection properties (Wu, Nguang, Shen, Liu & Li, 2010). However, the close-loop stability that is more desirable for industrial processes such as power plant cannot be guaranteed with all of the aforementioned controllers. Model Predictive Control (MPC) uses an explicit model to predict the future behavior of the plant and solves a constrained optimization problem on-line to obtain the optimal control sequence (Mayne, Rawlings & Rao, 2000). Since MPC can guarantee the stability of the closed-loop system and deal with the input constraint in an optimal way, it has become a popular strategy in process control (Rawlings, 2000). By adopting an infinite horizon objective function and constructing Lyapunov function to find its upper bound (Kothare, Balakrishnan & Morari, 1996; Lu & Arkun, 2000; Zhang, Feng & Lv, 2007), stable MPCs are also designed and used for boiler-turbine system based on the multi-model system, such as piecewise linear (PWL) model or TakagiSugeno (TS) fuzzy model (Wu, Shen & Li, 2010; Wu, Shen & Li, 2011). However, these MPCs as well as some of the
aforementioned control strategies are all developed with the assumption that all state variables can be measured, which makes the modeling requirement strict and usually cannot be achieved. In addition, in the situation where modeling mismatch and disturbance exist, tracking offset occurs for these controllers which greatly affect their performance.
Remark 2.2: To ensure the detectability of the state and disturbance variables, we assume the nominal system of (1)(2) is detectable and
By extending the results in Wu, Shen & Li (2010; 2011) this paper presents a stable model predictive tracking controller (SMPTC) for boiler-turbine coordinated system by using the switchable piecewise linear models. An observer is first developed to estimate the unmeasured state variables and disturbances, and then steady-state targets are determined to remove the effect of disturbances and modeling mismatch to guarantee the offset-free performance. With the estimation of state variables and disturbances as well as the steady-state targets, a stable MPC is built at the last. By solving a set of linear matrix inequalities (LMIs), the resulting controller can guarantee both the stability of the closed-loop predictive control system and the input constraints while achieving an offset-free tracking control in an optimal way.
(Pannocchia, 2003 & 2004; Muske & Badgwell, 2002).
2. STABLE MODEL PREDICTIVE TRACKING CONTROL BASED ON PIECEWISE LINEAR MODEL 2.1 Piecewise Linear Model Suppose a nonlinear discrete-time system can be represented as the following PWL model: m
x(k + 1) = ∑ Si ( z (k ))( Ai x(k ) + Bi u (k ) + Bid d (k ))
(1)
i =1
m
y (k ) = ∑ Si ( z (k ))(Ci x(k ) + Di u (k ) + Ei + Cid d (k ))
(2)
i =1
z (k )
where
is
a
with Si ( z (k )) ∈ {0,1} ,
measurable m
∑ Si ( z (k )) = 1
switching
⎡ I − Ai rank ⎢ ⎣ Ci
(5)
2.2 State and Disturbance Observer Since the state variables and disturbances are unmeasured, an observer is first designed. Consider the following augmented multi-model observer:
⎡ xˆ ( k + 1) ⎤ ⎡ Az ⎢ˆ ⎥=⎢ ⎣ d ( k + 1) ⎦ ⎣ 0
⎡ L1z ⎤ Bzd ⎤ ⎡ xˆ ( k ) ⎤ ⎡ Bz ⎤ ⎥ ⎢ ˆ ⎥ + ⎢ ⎥ u ( k ) + ⎢ 2 ⎥ [ yˆ ( k ) − y ( k ) ] I ⎦ ⎣ d (k ) ⎦ ⎣ 0 ⎦ ⎣ Lz ⎦
yˆ (k ) = Cz xˆ (k ) + Dz u (k ) + Ez + Czd dˆ (k )
(6)
Following the method in Feng (2006), we can construct a stable multi-model observer if there exist matrices H , Gi , and a symmetric positive definite matrix X , such that the following LMI problem is feasible:
⎡ HT + H − X *⎤ (7) ⎥>0 ⎢ aug aug X⎦ ⎣ HAi + Gi Ci where the augmented matrices are defined by ⎡ A Bid ⎤ aug d Aiaug = ⎢ i ⎥ and Ci = ⎡⎣Ci Ci ⎤⎦ , and “ * ” in the 0 I ⎣ ⎦ matrix stands for the corresponding terms of the symmetric matrix. Then the observer gains can be determined by
variable
Li = H −1Gi = ⎡⎣ L1iT
. x(k ) ∈ R , u ( k ) ∈ R , n
− Bid ⎤ ⎥ = n + nd Cid ⎦
l
T
L2i T ⎤⎦ , i = 1, 2...m .
i =1
y (k ) ∈ R p are state, input and output variables, respectively,
and d (k ) ∈ R nd is a disturbance term including all the modeling mismatch and unmeasured disturbances, but it would reach an unknown steady-state value asymptotically: that is: lim d (k ) = d . The matrices Ai , Bi , Bid , Ci , Di , Ei , Cid k →∞
are the parameters of the local model for i = 1, 2...m . Model (1)-(2) can be rewritten into the global model form:
x(k + 1) = Az x(k ) + Bz u (k ) + Bzd d (k ) d z
y ( k ) = C z x (k ) + Dz u (k ) + E z + C d ( k )
(3) (4)
m
2.3 Steady-State Targets Calculator For a given set of desired input and output set points, the goal of the SMPTC is first to find the steady-state targets and then design the MPC to track the steady-state targets. To remove the effect of unknown disturbances and modeling mismatch, a steady-state targets calculator (SSTC) is used in (Pannocchia, 2003 & 2004; Muske & Badgwell, 2002), achieving an offset-free tracking control. In this paper an SSTC is considered based on the PWL model, where a steady-state target ( xt , ut ) can be obtained by solving the following quadratic programming at each sampling time k :
where Az = ∑ Si ( z (k )) Ai and other matrices are defined
min(ut − uref )T Rs (ut − uref )
similarly.
⎡ I − Az s.t. ⎢ ⎣ Cz
i =1
Remark 2.1: Without the disturbance term, model (1)-(2) or (3)-(4) is a nominal PWL system which can be obtained using the Taylor’s series approximation or other modeling method. To simplify the stability design of MPC, an affine term is omitted in (1) and (3).
xt , ut
umin
⎤ Bzd dˆ (k ) − Bz ⎤ ⎡ xt ⎤ ⎡ ⎥ =⎢ ⎥ ⎢ ⎥ Dz ⎦ ⎣ut ⎦ ⎢ yref − Ez − C zd dˆ (k ) ⎥ ⎣ ⎦ ≤ ut ≤ umax
(8)
In which uref and yref are desired input and output set points, respectively, Rs = R > 0 is a symmetric weighting matrix, T s
and umin , umax are respectively the lower and upper limits of the input. Remark 2.3: With the augmented observer and SSTC, an offset-free tracking performance can be achieved. We omit the proof here due to the space limitation, and one can refer to (Pannocchia, 2003 & 2004; Muske & Badgwell, 2002) for more details. Remark 2.4: Optimization problem (8) may be infeasible due to the stringent constraints; in that situation, we suggest to use soft constraints on the input, and change (8) to:
min(ut − uref )T Rs (ut − uref ) + ζ 1T Q1ζ 1 + ζ 2T Q2ζ 2 xt ,ut
⎤ Bzd dˆ (k ) − Bz ⎤ ⎡ xt ⎤ ⎡ ⎢ ⎥ = ⎢ ⎥ ⎥ d Dz ⎦ ⎣ut ⎦ ⎢ yref − Ez − C z dˆ (k ) ⎥ ⎣ ⎦ + ζ 1 ≤ ut ≤ umax + ζ 2
⎡ I − Az s.t. ⎢ ⎣ Cz umin
(9)
However, if (8) keeps infeasible all the time, it means that the set points yref cannot be tracked without offset. Once the steady-state target ( xt , ut ) is obtained, we have:
xt = Az xt + Bz ut + Bzd dˆ (k )
(10)
Then, (3) can be changed to
x(k + 1) − xt = Az ( x(k ) − xt ) + Bz (u (k ) − ut ) + B ( d ( k ) − dˆ ( k )) d z
(11)
Noting that the disturbance estimation error is bounded with the help of the observer, we now use the nominal model of (11) for prediction:
x (k + s + 1| k ) = Az x (k + s | k ) + Bz u (k + s | k )
(12)
where x = x − xt and u = u − ut .
control variable u (k | k ) , matrices Yi , G and symmetric positive definite matrix S , such that the following LMI problem is feasible:
min
γ ,u ( k |k ),Yi ,G , S
γ
s.t.(15) − (18)
(14)
then, through minimizing the upper bound of the infinite horizon objective function (13), the proposed predictive controller will achieve an offset-free tracking performance in an optimal way while guaranteeing the stability of the closedloop system:
⎡ G + GT − S ⎢ ⎢( Ai G + BiYi ) ⎢ Q01/2 G ⎢ 1/2 ⎢⎣ R0 Yi
*⎤ ⎥ 0⎥ i = 1, 2...m (15) >0 0 γI 0 ⎥ ⎥ 0 0 γ I ⎥⎦ 1 * * * *⎤ ⎡ ⎢ ⎥ ⎢ A (k ) xˆ (k ) + B (k )u (k | k ) S 0 ⎥ 0 0 t z ⎢ z ⎥ 2 ⎢ ⎥ γI ⎢ 0 0 0 ⎥ > 0 (16) Q01/ 2 xˆt (k ) 2 ⎢ ⎥ ⎢ 0 0 γI 0⎥ R01/2 u (k | k ) ⎢ ⎥ ⎢ S ⎥ 0 0 0 Az (k )ω ⎢ ⎥ 2⎦ ⎣ (umin − ut (k )) ≤ u (k | k ) ≤ (umax − ut (k )) (17) ⎡U ⎢Y T ⎣ i
* S
* 0
Yi ⎤ >0 i = 1, 2,..., m G + GT − S ⎥⎦
(18)
in which xˆt (k ) = xˆ (k ) − xt (k ) , ω is the upper bound of the state estimation error | x |≤ ω , and diagonal matrix U with elements: U pp = min((umax , p − ut , p (k )) 2 ,(umin, p − ut , p (k ))2 ),
p = 1, 2,..., l. Proof: See Appendix A.
2.3 Stable Model Predictive Tracking Control Based on Piecewise Linear Model
3. THE BOILER-TURBINE SYSTEM MODEL
Considering the infinite horizon objective function: ∞
J 0∞ (k ) = ∑ [ x (k + s | k )T Q0 x (k + s | k ) + s =0
(13)
u (k + s | k ) R0 u (k + s | k )] T
where Q0 = Q0T > 0, R0 = R0T > 0 are symmetric weighting matrices of states and control moves, respectively, we present the following main result: Theorem 1: For the discrete PWL system (1) under input constraint: umin, p ≤ u p (k + s | k ) ≤ umax , p , s ≥ 0, p =1, 2,..., l ,
with the augmented observer (6) determined by LMIs (7) and a feasible SSTC (8) at sampling time k , if there exist
The boiler-turbine system used in this paper represents the behavior of a 160MW oil-fired power plant and has already been modeled as a third order nonlinear model. The controllable inputs into the system are valve actuator positions that control the mass flow of fuel, represented as u1, steam to the turbine, u2, and feedwater to the drum, u3. All valve position variables are constrained to lie in the interval [0, 1]. The measurable outputs of the model are electric power, E in MW, drum steam pressure, P in kg/cm2, and the drum water level deviation from a set level, L in meters. The state variables are electric power, E, drum steam pressure, P, and steam-water density, ρf. The dynamics of this particular power plant were recorded and formulated into mathematical model by Bell and Åström (1987) and are shown below in a summarized form, to be used as a model in our control system:
dP = 0.9u1 − 0.0018u2 P9/8 − 0.15u3 (19) dt dE = ((0.73u2 − 0.16) P 9/8 − E ) / 10 (20) dt dρf = (141u3 − (1.1u2 0.19) P) / 85 (21) dt Using the solution for ρf, the drum water level can be calculated using the following equations: qe = (0.85u2 − 0.14) P + 45.59u1 − 2.51u3 − 2.09 (22)
α s = (1 / ρ f − 0.0015) / (1 / (0.8P − 25.6) − 0.0015) (23) L = 50(0.13ρ f + 60α s + 0.11qe − 65.5)
(24)
where αs is the steam quality and qe is the evaporation rate in kg/s. Remark 3.1: This model has been widely studied using different controllers. Among them, a large number of papers directly design controllers based on state feedback with assumptions that all state variables can be measured (Wu, Nguang, Shen,Liu & Li, 2010; Wu, Shen & Li, 2010; Wu, Shen & Li, 2011; Keshavarz, Barkhordari & Motlagh, 2010). However, it is obvious that state variable ρf cannot be measured. 4. SIMULATION RESULTS We choose the drum steam pressure P as the switching signal and divide the system into “low,” “middle” and “high” load operating regions. Corresponding local models are then established through linearizing the original system at 80%, 100% and 120% operating points by taking Taylor’s series approximation and discretizing them under sampling time 1s. The resulting PWL model has the same structure as with (1)(2) by omitting an affine term in (1) and adding the disturbance term. We set:
0 0 ⎤ ⎡ 0.02 ⎢ 0.03 0 ⎥⎥ ，C1d = C2d = C3d = 03×3 B =B =B =⎢ 0 0 0.01⎦⎥ ⎣⎢ 0 We now apply the proposed SMPTC to the boiler-turbine coordinated system and consider the case of load change in a wide range as well as significant disturbances and modeling mismatch. The control mission is tracking the expected operating points of drum pressure and output power while maintaining the drum water level. We assume that at t=25s, both the pressure and power output desired operating points change in step from 70% to 130%, and at t=100s, all parameters in the original nonlinear model (19)-(21) change to their 70% values for disturbances and modeling mismatch. The setpoint of drum water level deviation is maintained at zero. With controller parameters: d 1
d 2
d 3
Q0 = I 3×3 ; R0 = 100 I 3×3 ; Rs = I 3×3 ; ω = [ 0.1 0.1 5] , T
the simulation results are shown in Fig. 1 and Fig. 2.
Fig. 1. SMPTC output performance of the boiler-turbine unit (solid line: proposed model predictive tracking controller; dashed line: controller in Wu, Shen & Li (2011), dotted line: reference.). From the control results, we can see that when the load is increased, the drum pressure and power output respond rapidly, and then approaches to the expected operating points, simultaneously, and the drum water level fluctuates within an acceptable range and gradually returns to zero. Even in the case of significant modeling mismatch, the offset-free performance can be achieved. The proposed stable MPC can control the boiler-turbine coordinated system effectively. Another stable MPC based on TS fuzzy model (Wu, Shen & Li, 2011) is used for comparison. In Wu, Shen & Li (2011), all state variables are assumed to be measurable and a stable MPC is developed based on state feedback law with one-step free input variables. However, due to the effect of modeling mismatch and disturbances, the tracking error was persistent and the controller was not able to give a satisfactory performance. ACKNOWLEGEMENT This work was supported in parts by the National Natural Science Foundation of China (NSFC) under Grant 51076027 and Grant 51036002, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant
20090092110051, the Scientific Research Foundation of Graduate School of Southeast University under grant ybjj1120, and the U.S. National Science Foundation under grant ECCS 0801440.
Fig. 2. SMPTC input performance of the boiler-turbine unit (solid line: proposed model predictive tracking controller; dashed line: controller in Wu, Shen & Li (2011).). REFERENCES Bell, R. D. and Åström, K. J. (1987). Dynamic Models for Boiler-Turbine-Alternator Units: Data Logs and Paramter Estimation for 160 MW Unit, TRFT-3192. Lund Institute of Technology, Lund, Sweden. Chen, P. C. and Shamma, J. S. (2004). Gain-scheduled l1 optimal control for boiler-turbine dynamics with actuator saturation, Journal of Process Control, vol. 14, pp. 263277. Dimeo, R. and Lee, K. Y. (1995). Boiler-turbine control system design using a genetic algorithm, IEEE Transactions on Energy Conversion, vol. 10(4), pp. 752759. Feng, G. (2006). A survey on analysis and design of modelbased fuzzy control systems, IEEE Transactions on Fuzzy Systems, vol. 14(5), pp. 676-697. Keshavarz, M., Barkhordari, Y., and Jahed-Motlagh, M. R. (2010). Piecewise affine modeling and control of a
boiler–turbine unit, Applied Thermal Engineering, vol. 30, pp. 781–791. Kothare, M. V., Balakrishnan, V., and Morari, M. (1996). Robust constrained model predictive control using linear matrix inequalities, Automatica, vol. 32, no. 10, pp. 1361–1379. Lu, Y. and Arkun Y. (2000). Quasi-min-max MPC algorithms for LPV systems, Automatica, vol. 36, pp. 527–540. Mayne, D. Q., Rawings, J. B., and Rao, C. V. (2000). Constrained model predictive control: Stability and optimality, Automatica, vol. 36, pp. 789–814. Muske, K. R. and Badgwell, Thomas A. (2002). Disturbance modeling for offset-free linear model predictive control, Journal of Process Control, vol 12, pp. 617-632. Pannocchia, G. (2003). Robust disturbance modeling for model predictive control with application to multivariable ill-conditioned processes, Journal of Process Control, vol. 13, pp. 693-701. Pannocchia, G. (2004). Robust model predictive control with guaranteed setpoint trackingJournal of Process Control, vol 14, pp. 927-937. Park, Y. M. and Lee, K. Y. (1996). An auxiliary LQG/LTR robust controller for cogeneration plants, IEEE Transactions on Energy Conversion, vol. 11(2), pp. 407413. Rawlings, J. B. (2000). Tutorial overview of model predictive control, IEEE Contr. Syst. Mag., vol. 20, no. 3, pp. 38– 52. Wen, T., Marquez, H. J., Chen, T., and Liu, J. (2005). Analysis and control of a nonlinear boiler-turbine unit, Journal of Process Control, vol. 15(8), pp.883-891. Wu, J., Nguang, S.K., Shen, J., Liu, G. and Li, Y. G. (2010). Robust H infinite tracking control of boiler–turbine systems, ISA Transactions, vol. 49, pp. 369-375. Wu, X., Shen, J., and Li, Y. (2010). Control of boiler-turbine coordinated system using multiple-model predictive approach, 2010 8th IEEE International Conference on Control and Automation, ICCA. Wu, X., Shen, J., and Li, Y. G. (2011). Stable model predictive control based on TS fuzzy model with application to boiler-turbine coordinated system, Proceedings of the Chinese Society of Electrical Engineering, vol. 31(11), pp. 106-112. Zhang, T., Feng, G., and Lv, J. (2007). Fuzzy Constrained Min-Max Model Predictive Control Based on Piecewise Lyapunov Functions, IEEE Trans. Fuzzy Systems, vol.15, no.4, pp. 686-698. Appendix A. PROOF OF THEOREM 1 Part 1 (Minimizing the upper bound of infinite horizon objective function): For the infinite horizon objective function (13), divide it into two parts (Lu & Arkun, 2000):
J 0∞ (k ) = x (k | k )T Q0 x (k | k ) + u (k | k )T R0 u (k | k ) + J1∞ (k ) Suppose a common Lyapunov function V ( x) = xT S −1 x
(A1)
(A2)
satisfies:
2 xˆt (k )T Q0 xˆt (k ) + 2( Az (k )ω )T S −1 ( Az (k )ω )
V ( x (k + s + 1| k )) − V ( x (k + s | k )) ≤ −[ x (k + s | k )T T
× Q0 x (k + s | k ) + u (k + s | k ) R0 u ( k + s | k )]
(A3)
+2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T S −1 ( Az (k ) xˆt (k )
(A10)
+ Bz (k )u (k | k )) + u (k | k ) R0 u (k | k ) ≤ γ T
Summing (A3) from s = 1 to s = ∞ , and with x (∞ | k ) = 0 and V ( x (∞ | k )) = 0 , we get:
Then minimizing the upper bound of J 0∞ (k ) is equivalent to the minimization of γ , subject to (A10).
J1∞ (k ) ≤ V ( x (k + 1| k )) = x (k + 1| k )T S −1 x (k + 1| k )
(A4)
By defining S −1 = γ −1 S −1 ,and using Schur complements (Kothare, Balakrishnan & Morari, 1996), (A10) can be expressed as (16).
(A5)
Part 2(Stability constraint): With the state feedback control law:
Thus we can get the upper bound of J 0∞ (k ) :
J 0∞ (k ) ≤ x (k | k )T Q0 x (k | k ) + u (k | k )T R0 u ( k | k ) + x (k + 1| k )T S −1 x (k + 1| k )
However, in (A5), x (k | k ) = x (k ) = x(k ) − xt (k ) is an unmeasured variable, so we rewrite it as:
x (k | k ) = xˆ (k ) + x (k ) − xt (k ) = xˆt (k ) + x (k )
(A6)
Then the first term of the right hand of (A5) can be rewritten as:
≤ 2 xˆt (k ) Q0 xˆt (k ) + 2 x (k ) Q0 x (k ) T
(A7)
≤ 2 xˆt (k )T Q0 xˆt (k ) + C
At sampling time k , since the current switching variable z (k ) is assumed to be available, the current model { Az (k ), Bz (k )} can be obtained, and the third term of the right hand of (A5) can be rewritten as:
x (k + 1| k )T S −1 x (k + 1| k ) = ( Az (k ) x (k ) + Bz (k )u (k | k ))T
×S −1 ( Az (k ) x (k ) + Bz (k )u (k | k )) = ( Az (k ) xˆt (k ) + Bz (k )u (k | k ) + Az (k ) x (k )) ×S ( Az (k ) xˆt (k ) + Bz (k )u (k | k ) + Az (k ) x (k ))
T
≤ 2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T × S −1 ( Az (k ) xˆt (k ) + Bz (k )u (k | k )) + 2( Az (k ) x (k ))T × S −1 ( Az (k ) x ( k )) ≤ 2( Az (k ) xˆt (k ) + Bz (k )u (k | k ))T × S −1 ( Az (k ) xˆt (k ) (A8) + Bz (k )u (k | k )) + 2( Az (k )ω )T × S −1 ( Az (k )ω ) Substituting (A7) and (A8) into (A5), we can get:
J 0∞ (k ) ≤ 2 xˆt (k )T Q0 xˆt (k ) + 2( Az (k )ω )T ×S −1 ( Az (k )ω ) + 2( Az (k ) xˆt (k ) + Bz (k )u (k ))T ×S −1 ( Az (k ) xˆt (k ) + Bz (k )u (k )) + u (k | k )T R0u (k | k ) + C Define a scalar γ and suppose:
(A9)
(A12)
Substituting (A11),(A12) into (A3) and noting that (A13)
then the stability constraint (A3) is satisfied if the following holds:
GT + G − S − ( Az G + BzYz )T S −1 ( Az G + Bz Yz ) −
where C = 2ω T Q0ω is a constant.
−1
= ( Az + BzYz G −1 ) x (k + s | k ) (G − S )T S −1 (G − S ) > 0 ⇒ GT S −1G ≥ GT + G − S
x (k | k )T Q0 x (k | k ) = ( xˆt (k ) + x (k ))T Q0 ( xˆt (k ) + x (k ))
(A11)
the nominal predictive closed-loop PWL system (12) can be described as:
x (k + s + 1| k ) = Az x (k + s | k ) + Bz Yz G −1 x (k + s | k )
with x (k ) the estimation error between x(k ) and xˆ (k )
T
u (k + s | k ) = Yz G −1 x (k + s | k ), s > 0
GT Q0γ −1G − YzT R0γ −1Yz > 0
(A14)
which can be expressed by the LMIs (15) and since the common Lyapunov function (A2) satisfies ΔV < 0 for all time, the system can be guaranteed to be stable in the sense of Lyapunov. Part 3(Input constraint): Since we split the input into free variables and future control moves determined by state feedback law, we must constrain them accordingly. For the free variables, we directly constrain them by the peak bound (17). For the future control moves, by using the approach in Kothare, Balakrishnan & Morari (1996) and considering (A13), the constraint is satisfied if the LMIs in (18) are feasible and the proof is completed.