Scalable, Reconfigurable Model Predictive Control for

Scalable, Reconfigurable Model Predictive Control for Building Heating Systems Edward O’Dwyer, Marcin Cychowski, Kostas Kouramas, Luciano De Tommasi and Gordon Lightbody

Abstract— Inefficient design and operation of building heating systems can have a large impact on global energy consumption. Traditional heuristic approaches often supply excess heat and cannot adapt to faults and changes in the building and heating system. Model Predictive Control (MPC) based strategies can incorporate future building usage and weather conditions to achieve more efficient heating. While MPC can produce an improved performance over standard strategies, many approaches taken in the literature are not easily scalable and do not allow for intuitive reconfiguration. Two possible MPC strategies for control of a building heating system are designed and compared here. In the first strategy, the thermal comfort of the occupants of the building is balanced with the energy use in a single objective function. In the second strategy, a lexicographic, multi-objective formulation is used to split the competing goals of energy reduction and thermal comfort. The strategies are assessed in a validated simulation platform in terms of energy efficiency, comfort performance, scalability and reconfigurability in times of system changes or faults.

I. I NTRODUCTION In 2011, the services and households sector was responsible for approximately 35% of global energy consumption [1]. One of the main contributors to this energy use was the heating and cooling of buildings. Inefficient design and operation of building heating systems can have a large, undesirable impact on overall energy consumption. Model Predictive Control (MPC) has been referred to as a potential alternative control methodology for control of heating systems [2]. By exploiting the slow thermal dynamics and using weather forecasts and predictions of building energy demands, energy consumption can be reduced without compromising the comfort of the buildings occupants. Given the wide variation of possible building structures and energy demands, commissioning effort can prove to be a significant obstacle as the set-up and tuning will be system specific. Furthermore, if faults or changes occur in the system, reconfiguration of the formulation may be required in order to achieve a desired level of performance. For an MPC strategy Research supported by United Technologies Research Centre Ireland (UTRC-I) and the Irish Research Council (IRC) Edward ODwyer is with the Department of Electrical and Electronic Engineering, University College Cork, Ireland (phone: 086-0528385; email: [email protected]) Marcin Cycowski is with United Technologies Research Centre Ireland (email: [email protected]) Konstantinos Kouramas is with United Technologies Research Centre Ireland (email: [email protected]) Luciano De Tommasi is with United Technologies Research Centre Ireland (email: [email protected]) Gordon Lightbody is with the Department of Electrical and Electronic Engineering, University College Cork, Ireland (email: [email protected])

to be scalable and applicable to a wide range of buildings, the set-up, tuning and reconfiguration of the formulation must be intuitive. In this paper, two MPC strategies are developed to achieve improved performance with regard to energy consumption and comfort criteria satisfaction. The strategies are designed to be scalable in terms of set-up and reconfigurability. The first strategy includes the energy use and thermal comfort (in the form of temperature set-point deviation) in a single objective function. The balance between energy and comfort is tuned based on the steady-state set-point deviation taken from the analytic solution of the unconstrained optimization problem. The second strategy separates the comfort and energy problems into two different objectives. These are solved by a lexicographic multi-objective optimization in which the energy input is only minimized if particular comfort constraints are satisfied. The strategies are compared in terms of performance and commissioning effort. Reduced-order models are derived from data to represent the thermal dynamics of the building. The strategies are tested and analysed in a simulation platform which has been designed to replicate the thermal dynamics of a real building, specifically, the Nimbus Centre at Cork Institute of Technology (CIT), Cork. Section II of this paper introduces the simulation platform used and the real building on which it is based. Section III outlines both the single objective and the multi-objective strategies investigated. Section IV examines the effect of system faults on the strategies, along with the reconfiguration requirements to maintain performance. II. S IMULATION P LATFORM A. The Nimbus Centre The Nimbus Centre at Cork Institute of Technology (CIT) Cork is a two-story building with an interior area of 1868m2 and with office and laboratory space for approximately 80 people. The building is fitted with wireless temperature and occupancy sensors. The heating system in the Nimbus Centre is comprised of mixing valves on each floor and thermostatically controlled radiator on-off valves in each zone in the building. Under standard weather-compensated control, the mixing valves are designed to supply water to the radiators at a temperature which varies linearly with the external temperature. The zone level valves operate to maintain the zone temperature inside a 1o C band, remaining open until the temperature in the zone exceeds a threshold of 0.5o C above the set-point before closing and remaining closed until the temperature falls below

a low threshold, 0.5o C below the set-point. The boiler in the system switches on and off in such a way as to keep the header temperature in a band around a specified set-point.

TSupply(t) 1

Tflow (t)

B. The Simulation Platform 0

Mixing Valve

Fig. 2.

1 TReturn(t)

Block diagram of first order mixing valve model with PI controller

25

20

Nimbus Simulation Platform 15 11/03

TSupply(t)

Tflow(t)

for one room (the technical support office). Historical data was taken from the building from a seven day period in March 2013. The external temperature from these dates was used as an input to the simulation platform. Solar radiation recorded for the period was also included. While differences between the real data and the simulation platform are visible due to significant unmeasured disturbances in the real building, the dominant dynamics are captured by the simulation platform.

Zone Temperature (°C)

A simulation model of the Nimbus building’s thermal dynamics was developed in Simulink, to allow for testing and analysis of the control strategy. As detailed information of the Nimbus structure was available, a physics-based model was established using first principal heat transfer laws and material properties. A common technique for modelling the thermal dynamics of a building is to represent it as an RC circuit [3]. In this case, each wall (internal and external), floor and ceiling was modelled by a 2-capacitance, 3-resistance system and each room by a single capacitor [4]. The resulting model was made up of 341 states with 30 inputs (the heat supplied to each of the 30 zones by the radiators). The external temperature, solar heat gains and internal heat gains (from occupants and equipment) were included as disturbances to the system. The heating system was simulated in the same platform in Simulink. Limitations in the system (e.g. flow rates and temperature restrictions) as seen in the real Nimbus building were also included. Fig. 1 shows a schematic of the heating system model. The mixing valves are modelled in the simulation

M(t)

1 1+pτ

PI

SP

12/03

13/03

14/03

15/03

16/03

17/03

18/03

Date (2013)

Zone-level radiator valves

Fig. 3. Comparison of temperatures taken from the Technical Support Office (ground floor) during the period from the 11/03/13 to the 18/03/13 in the Nimbus building (dashed line) and the simulation platform

Boiler Zone 1

Header Fig. 1.

Zone 2

TReturn(t) Schematic of Nimbus heating system model

platform with first-order dynamics under PI control. Supply water from the boiler (TSupply ) and colder return water from all the radiators (TReturn ) is mixed to attain a desired radiator supply temperature (Tf low ). The PI controller manipulates the valve position which determines the amount of return water used. Where M represents the fraction of supply water allowed through the mixing valve: Tf low (t) =M (t) TSupply (t) + (1 − M (t)) TReturn (t)

(1)

The PI controlled mixing valve is represented by the block diagram in Fig. 2, where p is the differentiation operator. The parameter τ and the proportional and integral gains of the PI controller are tuned to achieve a response that matches the dynamics of the real system. Fig. 3 shows a comparison between the thermal dynamics of the simulation platform and those of the real Nimbus building

III. S TRATEGIES A. Optimization Models In the building energy domain, by considering predictions of weather, occupancy, and building thermal dynamics, MPC is a promising alternative to more traditional strategies, such as trial and error based scheduling, weather-compensation and PI control. The merits of decentralized and centralized frameworks for MPC based building energy control strategies have been studied [5]. In this paper, the models used in the optimization are derived from data taken from the simulation platform. First order models of each zone are used with dynamic coupling between zones neglected. Different zones are linked in the optimization only through the heating system. A linear discrete time state-space model is used: T Zone (k + 1) = AT Zone (k) + Bu(k) + Ev(k)

(2)

where T Zone (k) denotes the system state vector (in this case, a vector of n zone temperatures), u (k) denotes a vector of inputs at the k th sample and v(k) represents a vector of predicted measurable disturbances. A = diag(a1 . . . an ), B = diag(b1 . . . bn ) and E = diag(e1 . . . en ) are matrices comprising the derived coefficients of n zone models.

B. Single Objective Formulation 1) Formulation: The first strategy presented here uses a single objective formulation in which a weighted combination of set-point deviation and energy consumption is minimized. The mixing valves on each floor of the building are used to control the temperature of the water flowing to the radiators, while the heat flowing to each zone of the building is controlled by the zone level valves. The header temperature set-point is also controlled to allow for the boiler efficiency to be included in the formulation. In this strategy, the zone level on-off valves follow a pulsewidth modulation (PWM) signal, which, when averaged over one sample period, are open for a percentage of time equal to the duty cycle. This type of strategy is useful as it allows for a valve that is binary in nature to be represented by a continuous variable, thus avoiding the need for mixed integer optimization (this is particularly important due to non-linearities introduced by including the boiler efficiency in the optimization [6]). The problem can be formulated as follows: (Tf∗lSP , d∗ ) = arg

min

T ,d,Tf lsp

H X

+uin (i) Ruin (i)






1 dj (k) Tf lsp (k) − TZonej (k) η(k) TZonej (k) = Tsp (k) + Tj (k)

(3)

(4)

H X

λi ∇hi (T∗` , u∗` ) (10)

i=1

(11) =0

(12)

µj ≥ 0

(13)

where f (T∗` , u∗` ) =

H X


T` (i)T q` T` (i) + u` (i)T r` u` (i)

hi (T∗` , u∗` ) = T` (k + i|k) − Tsp + ai` TZone (k)

   ψ= 

a`H−2 b`

···

e` a` e` .. .

0 e` .. .

··· ··· .. .

aH−2 e` `

···

(8)

aH−1 e` `

(9)

∀i = 1, 2 . . . H

where d is the duty cycle of the PWM signal sent to the zone level valves, T is a vector of zone temperature setpoint deviations, Q = diag(q1 . . . qn ) is a matrix of tuning parameters, uin is a vector of heat energy inputs to each zone of the building, R is the identity matrix, Tsp is the zone temperature set-point, Tf lsp is the flow temperature set-point sent to the mixing valves, TExt is the external temperature, Theader is the header temperature, η is the boiler efficiency (calculated as a quadratic function of the header temperature based on the boiler efficiency curves), H is the prediction horizon and n is the number of zones. 2) Scalability: The single objective is made up of a combination of two goals: to reduce energy consumption and maintain comfort. By the nature of the application, these two goals will often conflict with each other, resulting in a tradeoff between energy and comfort. This trade-off is subjective and as such, any scalable strategy should be designed to be adjustable and easy to set up. The balance between energy and comfort is determined in this strategy by the values in the Q matrix. This paper

(17) ··· ··· .. .

aH−1 b` `

TZonej (k) ≤ Tf lsp (k) ≤ Theader (k)

(15) (16)

gH+i = u` (i) − uM AX (i)  b` 0  a` b` b `  ζ= .. ..  . .

(7)

∀j = 1, 2...n

(14)

i=1

0 ≤ dj (k) ≤ 1

∀k = 1, 2...H

µj ∇gj (u∗` ) −

j=1

gj (u∗` ) ≤ 0 hi (T∗` , u∗` )

(5)

TZonej (k + 1) = aj TZonej (k) + bj dj (k) Tf lsp (k)
−TZonej (k) + ej TExt (k) (6)

η(k) = f (Theader )

2H X

gi = −u` (i)

s.t. uinj (k) =

∇f (T∗` , u∗` ) = −

+ ζi u` (k) + ψi T Ext

T (i)T QT (i)

i=1

T

proposes a method in which the steady-state temperature setpoint deviation is taken as a performance indicator. This can be selected and the Q tuning matrix can be calculated accordingly to achieve it. For the `th zone, the temperature set-point deviation can be found as the solution that satisfies the following KKT conditions:

 0 0     b`  0 0     e`

(18)

(19)

∀j = 1, 2 . . . 2H

By neglecting inequality constraints, a solution for the notional first element of the set-point deviation (T ) can be found in terms of the tuning parameter q` in the following form: T∗` (k) =α1 (q` )T` (k − 1) + α2 (q` )Tsp + α3 (q` )TExt (k)

(20)

where α1 , α2 and α3 are H th order functions of q` . If a perfect plant model is assumed and the disturbance term is held constant, the following function can be solved to find q` : T`SS =

α2 (q` )Tsp + α3 (q` )TExt (k) 1 − α1 (q` )

(21)

While this approach results in an intuitive way of determining tuning parameters, mismatches between the plant and the zone models as well as unmeasured disturbances in the building will affect actual set-point deviation. C. Lexicographic Formulation 1) Formulation: To overcome the tuning issues arising from the opposing nature of the energy and comfort targets, the objective function can be separated into two or more individual

objectives. A widely used approach is to minimize a weighted sum of the different objectives. This is equivalent to the single objective approach outlined above. Lexicographic multi-objective MPC however, solves the different objectives in a prioritized order [7], [8]. The solutions of higher priority optimizations are taken as constraints in lower priority optimizations. Solutions to lower priority objectives will only be considered if they result in an equivalent or lower cost in all higher priority objectives. In the strategy proposed here, achieving and maintaining a temperature that lies within a specific comfort band around the set-point in all zones is taken as the highest priority. Minimization of energy supplied to the building is taken as the next priority. As a result, the energy supplied is only reduced if doing so does not drive any of the zone temperatures outside the specified comfort band. The third priority is to determine header temperature set-points that maximize boiler efficiency. As the optimization problem only requires the zone temperatures to fall within a specified comfort band, it is not necessary to include the zone level valves in the problem. These valves can then be used as thermostatic valves in the same way as the standard strategy Allowing them to operate autonomously from the central control strategy can be desirable, as safe operation is ensured [9]. The highest priority optimization seeks to minimize the largest zone temperature deviation below a specific comfort threshold (Tsp − TBand ) at each time-step as follows: Tf∗lsp1 = arg

min

T1 ,Tf lsp1

J1

(22)

s.t. J1 =

(33)

TZonej (k) ≤ Tf lsp2 (k) ≤ Tf lM AX (k) ∀k = 1, 2...H

(34)

∀j = 1, 2...n

The third optimization priority seeks to maximize the boiler efficiency by reducing the header temperature set-point. ∗ Theader = arg sp

min

Theadersp

J3

(35)

s.t. J3 =

H X

−η(i)

(36)

i=1

J1 (T3 ) ≤ J1 (T2 )

(37)

J2 (Tf lsp3 ) ≤ J2 (Tf lsp2 )

(38)

TZonej (k + 1) = aj TZonej + bj Tf lsp3 (k)
−TZonej (k) + ej TExt (k)

(39)

T3 (k) ≥ Tsp (k + 1) − TZonej (k + 1) − TBand

(40)

T3 (k) ≥ 0

(41)

TZonej (k) ≤ Tf lsp3 (k) ≤ Theadersp (k)

(42)

η(k) = f (Theadersp (k))

(43)

∆TheaderM IN ≤ ∆Theadersp (k) ≤ ∆TheaderM AX

(44)

∀k = 1, 2...H

∀j = 1, 2...n

2) Scalability: The lexicographical approach to the multiobjective problem removes the need for tuning parameters, resulting in a scalable strategy. Adjusting the level of comfort can be achieved by changing the temperature band (TBand ). D. Results

H X

T1 (i)

(23)

i=1

TZonej (k + 1) = aj TZonej (k) + bj Tf lsp1 (k)
−TZonej (k) + ej TExt (k)

(24)

T1 (k) ≥ Tsp (k + 1) − TZonej (k + 1) − TBand

(25)

T1 (k) ≥ 0

(26)

TZonej (k) ≤ Tf lsp1 (k) ≤ Tf lowM AX (k) ∀k = 1, 2...H

(27)

∀j = 1, 2...n

The second priority seeks to minimize the energy supplied to the building without increasing the temperature set-point deviation calculated in the higher priority optimization. Tf∗lsp2 = arg

min

T2 ,Tf lsp2

J2

(28)

s.t. J2 =

T2 (k) ≥ 0

H X

Tf lsp2 (i)

(29)

J1 (T2 ) ≤ J1 (T1 )

(30)

TZonej (k + 1) = aj TZonej + bj Tf lsp2 (k)
−TZonej (k) + ej TExt (k)

(31)

T2 (k) ≥ Tsp (k + 1) − TZonej (k + 1) − TBand

(32)

i=1

The performances of the two MPC strategies were evaluated under normal operational circumstances using the simulation platform. They were compared to the standard weathercompensation strategy used in the real Nimbus building in terms of energy and comfort over a three day period in which the building was occupied between 08:00 and 17:00. Additional disturbances are included in the simulation platform in the form of internal and solar gains as well as noise on the weather forecast. The standard weather-compensation strategy determines the radiator flow temperature set-point (Tf lsp ) based on the external temperature (TExt ) with the heating system turned on for four hours in the morning, two hours in the afternoon and 30 minutes in the evening. The two MPC approaches use a sample-time of 15 minutes and a prediction horizon of 10 time-steps. The single objective MPC formulation is tuned to produce a steady-state set-point temperature deviation in all zones during occupancy hours of 1o C (though disturbances can be seen to affect this). The lower acceptable temperature bound of the multi-objective approach is chosen at 1.5o C below the set-point in all zones. Fig. 4 and Fig. 5 show the ground floor zone temperatures of the single objective and multi-objective formulations respectively. Fig. 6 shows the energy consumption associated with each strategy. Fig. 7 shows the accumulated deviation from the comfort set-point measured across all zones of the building for

20 19 18 17

1200

Weather Compensation Single Objective Multi−Objective

1000 800 600 400 200 0 20/04

21/04

22/04

23/04

15 14

Set−Point

13 21/04

22/04

23/04

Date (2013)

Fig. 4.

Zone Temperatures (°C)

1400

Date (2013)

20/04

20 19 18 17 16 15 14 13

Set−Point 21/04

22/04

23/04

Date (2013)

Fig. 5.

Fig. 7. Cumulative deviation from set-point across the entire building weather compensation vs. MPC strategy. 1o C deviation from the set-point for 1 hour = 1o C.hr

Ground floor zone temperatures (single objective strategy)

20/04

Energy Consumption (kWh)

1600

16

Ground floor zone temperatures (multi-objective strategy)

600

500

400

300

200

Standard Single Objective Multi−Objective

100

0 20/04

21/04

22/04

23/04

Date (2013)

Fig. 6.

Comparison of energy consumption

The single objective and lexicographic strategies improve the cumulative set-point deviation when compared to the standard weather compensation by approximately 19% and 7% respectively. Energy consumption is reduced by 11% and 8% respectively. By lowering the desired steady-state set-point deviation for the single objective strategy, or increasing the temperature band of the multi-objective strategy, it would be possible to reduce energy at the expense of comfort. While the single-objective approach achieves the best performance, the tuning requirement and need to control individual zone level valves explicitly make it the least scalable option. IV. R ECONFIGURATION Fault tolerant properties of MPC controllers have been widely studied [10], [11]. In many cases, if the nature and magnitude of a fault is known (though this may not be trivial), the underlying process model can be changed to incorporate the fault and continue operation. It may be necessary when

this happens to reconfigure the objective function to obtain a performance consistent with the pre-faulted conditions. In the case of the single objective formulation outlined in this paper, an altered optimization model cost function that has not been retuned will result in a different steady-state set-point deviation. To maintain the same level of comfort as before the fault occurred, the tuning parameters must be recalculated as described in Section III. Again as the effect of unmeasured noise is neglected in the calculation of the tuning parameters, this may not be accurate. The multi-objective formulation does not need to be retuned as the highest priority will still be to push the zone temperatures into the same comfort band. Furthermore, it would possible to lower the priority of faulted zone objectives to prevent faults from propagating through the building. The impact of faults on the different strategies is compared here. As fault detection is outside the scope of this paper it is assumed that information on the fault is available. A. Results To illustrate the reconfiguration capabilities of the MPC strategies, a scenario is simulated in which a fault in two zone level valves results in the heat supplied to the two zones being reduced by 50%. This fault occurs after one day of normal operation and continues for the remaining two days. It is assumed that the time and nature of the fault is known. Using the standard weather-compensation strategy, there is no mechanism by which action can be taken to reduce the impact of the fault. Fig. 8 shows the fault increases the cumulative set-point deviation in the building by 16%. Set−Point Deviation (°C.hr)

Zone Temperatures (°C)

21

Set−Point Deviation (°C.hr)

each of the strategies. This is taken as a discomfort metric and is measured in o C.hr.

2500

Not Faulted Faulted 2000

1500

1000

500

0 02/04

03/04

04/04

05/04

Date (2013)

Fig. 8. Cumulative set-point deviation in building with two faulted zones (weather-compensated control)

Fig. 9 and Fig. 10 show that when using the single objective

Zone Temperatures (°C)

1200

1000

Not Faulted Faulted

800

600

400

200

0 02/04

03/04

04/04

05/04

Date (2013)

Fig. 12. Cumulative set-point deviation in building with two faulted zones (lexicographical formulation)

20

18

16

14

12

10

Not Faulted Faulted − Not Reconfigured Faulted − Reconfigured

8

02/04

03/04

04/04

05/04

Date (2013)

Fig. 9. Effect of reconfiguration of cost function in single objective strategy on zone temperature in faulted zone

Set−Point Deviation (°C.hr)

Set−Point Deviation (°C.hr)

formulation, even if the underlying models of the zones are updated to include the effects of the reduced heat supply, the setpoint deviation in the zones is greatly increased (in this case by 15%) unless the cost function is reconfigured. By appropriately adjusting the tuning parameters (by the same method outlined in Section III, with updated model parameters), the control strategy can achieve a level of comfort performance which is almost identical to the pre-faulted conditions.

2000 1800 1600

Not Faulted Faulted − Not Reconfigured Faulted − Reconfigured

1400 1200

faults or changes in the heating system. They are tested in a validated simulation platform, designed to simulate the thermal dynamics of a real building, namely, the Nimbus Centre. The single objective strategy can achieve improved control when compared to weather compensation with a high level of flexibility, though this comes at the expense of tuning parameters which are sensitive to disturbance and model mismatch. The cost function must be retuned when faults occur in the system. A lexicographic formulation removes the need for tuning and reconfiguration. Though the controllability of individual zones is reduced in comparison to the single objective formulation, the overall simplicity and robustness of the strategy make it a viable alternative.

1000

VI. ACKNOWLEDGEMENTS

800 600 400 200 0 02/04

03/04

04/04

05/04

Date (2013)

Fig. 10. Cumulative set-point deviation in building with two faulted zones (single-objective formulation)

Zone Temperatures (°C)

It can be seen in Fig. 11 and Fig. 12 that the multi-objective formulation achieves a cumulative set-point deviation that is slightly lower than the pre-faulted conditions after the fault occurs. This is achieved without adjusting the cost function.

20

18

16

14

12

Not Faulted Faulted

10

02/04

03/04

04/04

05/04

Date (2013)

Fig. 11. Effect of fault on zone temperature in faulted zone in Lexicographical strategy

V. C ONCLUSION Two MPC strategies (a single objective strategy and a lexicographic multi-objective strategy) are developed in this paper for control of a building heating system. The strategies are formulated in such a way as to be easily set up and adjusted as well as being easy to reconfigure in times of

This research was co-funded by the Irish Research Council (IRC) and United Technologies Research Centre Ireland (UTRCI). R EFERENCES [1] International Energy Agency, “Key World Energy Statistics,” tech. rep., 2013. [2] F. Luk´asˇ, v. Jan, and P. Samuel, “Model Predictive Control of Buildings: The Efficient Way of Heating,” in IEEE International Conference on Control Applications, (Yokohama), pp. 1922–1926, 2010. [3] Y. Ma, A. Kelman, A. Daly, and F. Borrelli, “Predictive control for energy efficient buildings with thermal storage,” IEEE control systems magazine, vol. 32, no. February, pp. 44–64, 2012. [4] E. O’Dwyer, K. Kouramas, M. Cychowski, and G. Lightbody, “A Hierarchical Model-based Predictive Control Strategy for Building Heating Systems,” in 25th IET Irish Signals & Systems Conference 2014, (Limerick), pp. 298–303, 2014. [5] V. Chandan and A. Alleyne, “Optimal partitioning for the decentralized thermal control of buildings,” IEEE Transactions on Control Systems Technology, vol. 21, no. 5, pp. 1756–1770, 2013. [6] A. W¨achter and L. Biegler, “On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming,” Mathematical programming, vol. 106, pp. 25–57, Apr. 2006. [7] E. Kerrigan and J. Maciejowski, “Designing model predictive controllers with prioritised constraints and objectives,” IEEE International Symposium on Computer Aided Control System Design, pp. 33–38, 2002. [8] T. Miksch and A. Gambier, “Fault-tolerant control by using lexicographic multi-objective optimization,” 8th Asian Control Conference (ASCC), pp. 1078–1083, 2011. [9] J. Maciejowski, Predictive Control: With Constraints. UK: PrenticeHall, 2002. [10] J. Maciejowski, “Fault-tolerant aspects of MPC,” IEE Two-Day Workshop on Model Predictive Control: Techniques and Applications, no. 1, pp. 1–4, 1999. [11] I. Hwang, S. Kim, Y. Kim, and C. Seah, “A survey of fault detection, isolation, and reconfiguration methods,” IEEE Transactions on Control Systems Technology, vol. 18, no. 3, pp. 636–653, 2010.