Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 347 ,1 , , () () T kx k y s p k y xk y s p k V N xk S u I y Q N xk S u I y ΔΔ ⎡ ⎤⎡ ⎤ =+− +− ⎣ ⎦⎣ ⎦   (9) The term corresponding to the infinite horizon error on the system output in (7) can be written as follows ()() ,2 , , 1 (|) (|) T ks p k y s p k j VykpjkyQykpjky ∞ = =++− ++− ∑ () () ,2 , 1 , (|)( )(|) (|)( )(|) T sd ks p k y j sd sp k VxkmkpjmxkmkyQ xk mk p j mxk mk y Ψ Ψ ∞ = =+++−+− +++− +− ∑ (10) where, (|)() sss k xk mk xk B u Δ +=+  , ss s m BB B ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎣ ⎦     , () ( ) . /1/ T TT mnu k uukk ukmk ΔΔ Δ ⎡⎤ =+−∈ℜ ⎣⎦  Also, (|) () dmdd k xkmk Fxk B u Δ += +  , 12dmdmd d BFBFB B −− ⎡ ⎤ = ⎣ ⎦   ()() j pj m p mF ΨΨ +− = − (11) In order to force V k,2 to be bounded, we include the following constraint in the control problem , (|) 0 s sp k xk mk y + −= or , () 0 ss kspk xk B u y Δ + −=  With the above equation and (11), Eq. (10) becomes ()() ,2 1 ()(|) ()(|) T jj dd ky j V p mFx k m k Q p mFx k m k ΨΨ ∞ = =− + − + ∑ ( ) ( ) ,2 () () T md d md d kkdk V Fxk Bu QFxk Bu ΔΔ =+ +  where ()() 1 () () T jj dy j Q p mF Q p mF ΨΨ ∞ = =− − ∑ Finally, the infinite term corresponding to the error on the input along the infinite horizon in (7) can be written as follows ()() ,3 , , 1 (|) (|) T kdeskudesk j V ukjk u Qukjk u ∞ = =+− +− ∑ (12) Robust Control, Theory and Applications 348 Then, it is clear that in order to force (12) to be bounded one needs the inclusion of the following constraint , (|) 0 des k uk m k u + −= or , (1) 0 T uk desk uk I u u Δ − +−=  (13) where T unu nu m II I ⎡⎤ ⎢⎥ = ⎢⎥ ⎣⎦      Then, assuming that (13) is satisfied, (12) can be written as follows ( ) ( ) ,3 , , (1) (1) T k u kudesk uu kudesk V Iuk M u Iu Q Iuk M u Iu ΔΔ =−+− −+−   where 00 0 ; nu nu nu uuu m nu nu nu I II M Qdia g QQ II I ⎡⎤ ⎢⎥ ⎛⎞ ⎢⎥ ⎜⎟ == ⎢⎥ ⎜⎟ ⎝⎠ ⎢⎥ ⎣⎦           Now, taking into account the proposed terminal constraints, the control cost defined in (7) can be written as follows ()() ()() ,, ,, () () () () (1) (1) . T kx kyspkyx kyspk T md d md d kd k T T u k udesk u u k udesk k k V N xk S u I y Q N xk S u I y Fx k B u Q Fx k B u Iuk M u Iu Q Iuk M u Iu uR u ΔΔ ΔΔ ΔΔΔΔ ⎡⎤⎡⎤ =+− +− ⎣⎦⎣⎦ ++ + +−+− −+− +      To formulate the IHMPC with zone control and input target for the time delayed nominal system, it is convenient to consider the output set point as an additional decision variable of the control problem and the controller results from the solution to the following optimization problem: , , min 2 kspk TT kkk f k uy VuHucu Δ Δ ΔΔ =+ subject to , (1) 0 T uk desk uk I u u Δ − +−=  (14) , () 0 ss kspk xk B u y Δ + −=  (15) min , maxsp k yyy ≤ ≤ (16) max max (|) 0,1,,1uukjku j m ΔΔ Δ − ≤+≤ = − min max 0 (1) ( |) ; 0,1,, 1 j i uuk ukiku j m Δ = ≤ −+ + ≤ = − ∑  Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 349 where TdTdT yd u HSQSBQB MQMR = +++     () () ()( ) ( 1) T TTT dTmTd T fxy d desuu cxkNQSxkFQBuk u IQM=+ +−−    Constraints (14) and (15) are terminal constraints, and they mean that both, the input and the integrating component of the output errors will be null at the end of the control horizon m. Constraint (16), on the other hand, forces the new decision variable y sp,k to be inside the zone given by y min and y max . So, as y sp,k is a set point variable, constraint (16) means that the effective output set point of the proposed controller is now the complete feasible zone. Notice that if the output bounds are settled so that the upper bound equals the lower bound, then the problem becomes the traditional set point tracking problem. 4.1 Enlarging the feasible region The set of constraints added to the optimization problem in the last section may produce a severe reduction in the feasible region of the resulting controller. Specifically, since the input increments are usually bounded, the terminal constraints frequently result in infeasible problems, which means that it is not possible for the controller to achieve the constraints in m time steps, given that m is frequently small to reduce the computational cost. A possible solution to this problem is to incorporate slack variables in the terminal constraints. So, assuming that the slack variables are unconstrained, it is possible to guarantee that the control problem will be feasible. Besides, these slack variables must be penalized in the cost function with large weights to assure the constraint violation will be minimized by the control actions. Thus, the cost function can be written as follows ()() ()() ()() ()() ,, ,, 0 ,, ,, 1 1 ,, ,, 0 ,, ,, (|) (|) (|) (|) (|) (|) (|) (|) p T kspkykyspkyk j T sp k y k y sp k y k j m T des k u k u des k u k j T des k u k u des k u k V ykjk y Qykjk y yk p j k y Q yk p j k y uk j k u Q uk j k u uk m j k u Q uk m j k u δδ δδ δδ δδ = ∞ = − = =+−− +−− +++−− ++−−+ ++−− +−− + ++− − ++− − ∑ ∑ ∑ 0 1 ,,, , 0 (|) (|) j m TTT yk y yk uk u uk j uk j k R uk j k S S ΔΔδδδδ ∞ = − = + ++ ++ + ∑ ∑ (17) where , y u SS are positive definite matrices of appropriate dimension and ,, , ny nu yk uk δδ ∈ℜ ∈ℜ are the slack variables (new decision variables) that eliminate any infeasibility of the control problem. Following the same steps as in the controller where slacks are not considered, it can be shown that the cost defined in (17) will be bounded if the following constraints are included in the control problem: ,, () 0 ss kspkyk xk B u y Δδ + −−=  Robust Control, Theory and Applications 350 ,, (1) 0 T uk desk uk uk I u u Δδ − +−−=  (18) In this case, the cost defined in (17) can be reduced to the following quadratic function 11 12 13 14 , 21 22 23 ,,, , 31 32 33 41 44 , , ,1 ,2 ,3 ,4 , , 0 0 00 2 k sp k TT T T kkspkykuk yk uk k sp k ffff yk uk u HHHH y HHH Vuy HHH HH u y cccc c Δ Δδδ δ δ Δ δ δ ⎡ ⎤ ⎡⎤ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎡⎤ = + ⎢ ⎥ ⎣⎦ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎣⎦ ⎣ ⎦ ⎡⎤ ⎢⎥ ⎢⎥ ⎡⎤ ++ ⎢⎥ ⎣⎦ ⎢⎥ ⎢⎥ ⎣⎦ where 11 () TdTdT ydu HSQSBQBMQMR = +++     12 21 TT yy HH SQI==−   , 13 31 TT yy HH SQI==−   , 14 41 TT uu HH MQI==−   22 T uuu HIQI=   , 23 32 TT yyy HHIQI==   , 33 T yyy y HIQIS = +   , 44 T uuu u HIQIS = +   24 42 34 43 0 TT HHHH = === () ,1 () ()( ) ( 1) T TT d TmT d T fxy dm desuu cxkNQSxkFQBuk uIQM=+ +−−    ,2 () TT f x yy cxkNQI=−   , ,3 () TT f x yy cxkNQI=−   () ,4 , (1) T T fdeskuuu cukuIQI=− − −   ()() ,, () () ()( ) () ( 1) ( 1) T TT d TmT md T xyx d desk uuu desk cxkNQNxk xk F QFxk uk u IQIuk u=+ +−− −−   Then, the nominally stable MPC controller with guaranteed feasibility for the case of output zone control of time delayed systems with input targets results from the solution to the following optimization problem: Problem P1 , ,, ,, , min kspk yk uk k uy V Δ δδ subject to: max max (|) 0,1,,1uukjku j m Δ ΔΔ − ≤+≤ = − Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 351 min max 0 (1) ( |) ; 0,1,, 1 j i uuk ukiku j m Δ = ≤ −+ + ≤ = − ∑  min , maxsp k yyy ≤ ≤ (19) ( ) ,, ,, () 0 ( |) 0 ss s kspkyk spkyk xk B u y xk mk y Δδ δ + −−= + −−=  ( ) ,, ,, (1) 0 ( 1|) 0 T u k des k u k des k u k uk I u u uk m k u Δδ δ − +−−= +−−−=  It must be noted that the use of slack variables is not only convenient to avoid dynamic feasibility problems, but also to prevent stationary feasibility problems. Stationary feasibility problems are usually produced by the supervisory optimization level shown in the control structure defined in Figure 1. In such a case, for instance, the slack variable , y k δ allows the predicted output to be different from the set point variable ,s p k y at steady state (notice that only ,s p k y is constrained to be inside the desired zone). So, the slacked problem formulation allows the system output to remain outside the desired zone, if no stationary feasible solution can be found. It can be shown that the controller produced through the solution of problem P1 results in a stable closed loop system for the nominal system. However, the aim here is to extend this formulation to the case of multi model uncertainty. 5. Robust MPC with zone control and input target In the model formulation presented in (1) and (2) for the time delayed system, uncertainty concentrates not only on matrices F, B s and B d as in the system without time delay, but also on matrix n y nu θ × ∈ℜ that contains all the time delays between the system inputs and outputs. Observe that the step response coefficients S 1 ,…,S p+1 , which appears in the input matrix and (1)p Ψ + , which appears in the state matrix of the model defined in (1) and (2) are also uncertain, but can be computed from F, B s , B d and θ . Now, considering the multi- model uncertainty, assume that each model is designated by a set of parameters defined as { } ,,, sd nnnnn BBF Θ θ = , 1, ,nL = . Also, assume that in this case ,, max ( , ) n ijn p ij m θ >+ (this condition guarantees that the state vector of all models have the same dimension). Then, for each model n Θ , we can define a cost function as follows ()() ()() ()() ,, ,, 0 ,, ,, 1 1 ,, ,, 0 () ( |) () () ( |) () () (|)()() (|)()() (|) (|) (| p T kn n spkn ykn yn spkn ykn j T n spkn ykn yn spkn ykn j m T des k u k u des k u k j V ykjk y Qykjk y yk p jk y Qyk p jk y uk jk u Q uk jk u uk m j ΘΘδΘΘδΘ ΘδΘ ΘδΘ δδ = ∞ = − = =+− − +− − + ++− − ++− − ++−− +−− +++ ∑ ∑ ∑ ()() ,, ,, 0 1 ,,,, 0 )(|) ( |) ( |) () () T des k u k u des k u k j m TT T yk n y yk n uk u uk j ku Qukmjku uk j k R uk j k S S δδ ΔΔδΘδΘδδ ∞ = − = −− ++−− ++ ++ + ∑ ∑ (20) Robust Control, Theory and Applications 352 Following the same steps as in case of the nominal system, we can conclude that the cost defined in (20) will be bounded if the control actions, set points and slack variables are such that (18) is satisfied and ,, () () () ()0 ss nkspkn ykn xk B u y ΘΔ Θ δ Θ + −−=  Then, if these conditions are satisfied, (20) can be written as follows ( ) () ()() ,, ,, () () () () () ( ) ( ) ( ) ( ) () () () ()() () () (1) T kn x n k yspkn yykn y xnkyspknyykn T md d md d nmnkdnnmnk ukude VNxkSuIy I Q Nxk S u Iy I F xkB uQ F xkB u Iuk M u Iu ΘΘΔΘδΘ ΘΔ Θ δ Θ ΘΘΔΘΘΘΔ Δ =+ − − +− − ++ + +−+−      ()() ,, ,, ,,,, (1) () () T sk u uk u u k u desk u uk TT T kkyknyyknukuuk IQIukMuIuI uRu S S δΔδ Δ Δ δΘ δΘ δ δ −−+−− ++ +    (21) or 11 12 13 14 , 21 22 23 ,,, , 31 32 33 41 44 , , ,1 ,2 ,3 ,4 , , () () () () () 0 () () () () () 0 00 () 2() () k nnn sp k n n TT T T kn k spkn ykn uk yk n n uk k sp k n fnf f f yk n uk u HHHH y HHH Vuy HHH HH u y cccc Δ ΘΘΘ Θ Θ ΘΔ ΘδΘδ δΘ Θ δ Δ Θ Θ δΘ δ ⎡⎤ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎡⎤ = ⎢⎥ ⎣⎦ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎡ ⎡⎤ + ⎣⎦ () n c Θ ⎤ ⎢⎥ ⎢⎥ + ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ 11 () () ()(()) ()() TdTdT nnyn ndnn u HSQSBQBMQMR ΘΘ Θ Θ ΘΘ = +++     12 21 () TT nyy HH S QI Θ ==−   , 13 31 () TT nyy HH S QI Θ ==−   , 14 41 TT uu HH MQI==−   22 T y yy HIQI=   , 23 32 TT y yy HHIQI==   , 33 T y yy HIQI=   24 42 34 43 0 TT HHHH = === () ,1 () ( ) ()(( )) ( ) ( 1) T TT d T mT d T fxyn nxdn desuu cxkNQS xkF QB uk uIQM ΘΘΘ =+ +−−    ,2 () TT fxyy cxkNQI=−   , ,3 () TT fxyy cxkNQI=−   () ,4 (1) T T fdesuuu cukuIQI=− − −   Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 353 ()() ,, () () ()(( )) ( ) () (1) (1) TT d T mT md xyx n xd n T T des k u u u des k cxkNQNxk xk F QF xk uk u I QI uk u ΘΘ = ++ +−− −−    Then, the robust MPC for the system with time delay and multi-model uncertainty is obtained from the solution to the following problem: Problem P2 ,,, ,(),(), 1, , min ( ) kspk n yk n uk kN uy nL V ΔΘδΘδ Θ = (22) subject to max max (|) 0,1,,1uukjku j m ΔΔ Δ − ≤+≤ = − min max 0 (1) ( |) ; 0,1,, 1 j i uuk ukiku j m Δ = ≤ −+ + ≤ = − ∑  min , max () ; 1,, sp k n yy y nL Θ ≤ ≤= ,, () () () ()0; 1,, ss nkspkn ykn xk B u y nL ΘΔ Θ δ Θ +−−==   (23) ,, (1) 0 T uk desk uk uk I u u Δδ − +−−=  (24) ( ) ( ) ,,, ,,, , ( ), , ( ), , ( ), , ( ), , 1, , kkyknukspknn kkyknukspknn Vu y Vu y nL Δ δ Θδ ΘΘ Δ δ Θδ ΘΘ ≤=     (25) where, assuming that ( ) *** 1,1 ,1,1 ,(),,() ks p knuk y kn uy ΔΘδδΘ −− −− is the optimal solution to Problem P2 at time step k-1, we define ** (| 1) ( 2| 1) 0 T TT k uukk ukmk ΔΔ Δ ⎡⎤ =− +−− ⎣⎦   ; * ,,1 () () s p kn s p kn yy Θ Θ − =  and ,uk δ  such that ,, (1) 0 T uk desk uk uk I u u Δδ − +−−=   (26) and define , () y kn δ Θ  such that ,, () () () ()0 ss nkspkn ykn xk B u y ΘΔ Θ δ Θ + −−=   (27) In (20), N Θ corresponds to the nominal or most probable model of the system. Remark 1: The cost to be minimized in problem P2 corresponds to the nominal model. However, constraints (23) and (24) are imposed considering the estimated state of each model n Θ Ω ∈ . Constraint (25) is a non-increasing cost constraint that assures the convergence of the true state cost to zero. Remark 2: The introduction of L set-point variables allows the simultaneous zeroing of all the output slack variables. In that case, whenever possible, the set-point variable ( ) ,s p kn y Θ Robust Control, Theory and Applications 354 will be equal to the output prediction at steady state (represented by ( ) s n xkm+ ), and so the corresponding output penalization will be removed from the cost. As a result, the controller gains some flexibility that allows achieving the other control objectives. Remark 3: Note that by hypothesis, one of the observers is based on the actual plant model, and if the initial and the final steady states are known, then the estimated state () ˆ T xk will be equal to the actual plant state at each time k. Remark 4: Conditions (26) and (27) are used to update the pseudo variables of constraint (25), by taking into account the current state estimation ( ) ˆ s n xk for each of the models lying in Ω , and the last value of the input target. One important feature that should have a constrained controller is the recursive feasibility (i.e. if the optimization problem is feasible at a given time step, it should remain feasible at any subsequent time step). The following lemma shows how the proposed controller achieves this property. Lemma. If problem P2 is feasible at time step k, it will remain feasible at any subsequent time step k+j, j=1,2,… Proof: Assume that the output zones remain fixed, and also assume that () ( ) ** * . |1| T TT mnu k uukk ukmk ΔΔ Δ ⎡⎤ =+−∈ℜ ⎣⎦  , (28) ( ) ( ) ** ,1 , ,, s p ks p kL yy Θ Θ  , ( ) ( ) ** ,1 , ,, y k y kL δ ΘδΘ  and * , uk δ (29) correspond to the optimal solution to problem P2 at time k. Consider now the pseudo variables () ( () 1,11 ,1 ,,,, kspk spk L uy y ΔΘ Θ ++ +    ( ) ,1 1 , , yk δΘ +  () ) ,1 ,1 , yk L uk δΘδ ++  where () ( ) ** 1 1| 1| 0 T TT k uukk ukmk ΔΔ Δ + ⎡ ⎤ =+ +− ⎣ ⎦   (30) ( ) ( ) * ,1 , ,1,, sp k n sp k n yy nL ΘΘ + ==   , (31) Also, the slacks ,1uk δ +  and ( ) ,1 y kn δ Θ +  are such that 1,,1 () 0 T uk deskuk uk I u u Δδ ++ + −− =   (32) and ( ) ( ) ( ) 1,1 ,1 ˆ (1) 0, 1, , ss nnkspknykn xk B u y nL ΘΔ Θ δ Θ ++ + ++ − − = =   (33) We can show that the solution defined through (30) to (33) represent a feasible solution to problem P2 at time k+1, which proves the recursive feasibility. This means that if problem P2 is feasible at time step k, then, it will remain feasible at all the successive time steps k+1, k+2, …  Now, the convergence of the closed loop system with the robust controller resulting from the later optimization problem can be stated as follows: Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 355 Theorem. Suppose that the undisturbed system starts at a known steady state and one of the state observers is based on the actual model of the plant. Consider also that the input target is moved to a new value, or the boundaries of the output zones are modified. Then, if condition (3) is satisfied for each model n Θ Ω ∈ , the cost function of the undisturbed true system in closed loop with the controller defined through the solution to problem P2 will converge to zero. Proof: Suppose that, at time k the uncertain system starts from a steady state corresponding to output ( ) ss y k y = and input ( ) 1 ss uk u−= . We have already shown that, with the model structure considered in (1) and (2), the model states corresponding to this initial steady state can be represented as follows: () ˆ 0, 1, , nssssss p xk y y y n L ⎡⎤ ⎢⎥ == ⎢⎥ ⎢⎥ ⎣⎦   and consequently, ( ) ( ) ˆˆ ,0,1,, sd nssn xk y xk n L=== . At time k, the cost corresponding to the solution defined in (28) and (29) for the true model is given by () () () () { () () () ()() } () () **** *** ,, ,, 0 **** ,, ,, 1 ****** ,,,, 0 (|) (|) (|) (|) (|) (|) T kT T spkT ykT yT spkT ykT j T des k u k u des k u k m TT T yk T y yk T uk u uk j V ykjky Qykjky uk jk u Q uk jk u uk jk Ruk jk S S ΘΘδΘθδθ δδ ΔΔδΘδΘδδ ∞ = − = =+−− +−− ++−− +−− ++ ++ + ∑ ∑ (34) At time step k+1, the cost corresponding to the pseudo variables defined in (30) to (33) for the true model is given by () () () () { () () () ()() } () () 1 ****** ,, ,, 0 **** ,, ,, 1 ****** ,,,, 0 (1|) (1|) (1/) (1/) (1|)(1|) kT T T spkT ykT yT spkT ykT j T des k u k u des k u k m TTT yk T y yk T uk u uk j V ykjky Qykjky uk j k u Q uk j k u uk j k Ruk j k S S Θ ΘδΘ ΘδΘ δδ ΔΔδΘδΘδδ + ∞ = − = = ++ − − ++ − − +++−− ++−− +++ +++ + ∑ ∑  (35) Observe that, since the same input sequence is used and the current estimated state corresponding to the actual model of the plant is equal to the actual state, then the predicted state and output trajectory will be the same as the optimal predicted trajectories at time step k. That is, for any 1j ≥ , we have ( ) ( ) |1 | TT xk j kxk j k++= + Robust Control, Theory and Applications 356 and ( ) ( ) |1 | TT y k j k y k j k++= + In addition, for the true model we have ( ) ( ) * ,1 , y kT y kT δ ΘδΘ + =  and * ,1 , uk uk δ δ + =  . However, the first of these equalities is not true for the other models, as for these models we have ( ) ( ) ˆ 1| 1 1| , for nn nT xk k xk k Θ Θ ++≠ + ≠ . Now, subtracting (35) from (34) we have () () ( ) () () ( ) ( ) () () ( ) ()() () () ******* 1,, ,, ****** ,, ,, || (|) (|) T k T k T T spkT ykT yT spkT ykT T T des k u k u des k u k VV ykky Qykky ukk u Q ukk u u k Ru k ΘΘ ΘδΘ ΘδΘ δδΔΔ + −=−− −− +−− −−+  and, from constraint (25), the following relation is obtained ( ) ( ) * 11 kTkT VV Θ Θ ++ ≤  , which finally implies () () ( ) () () ( ) () () () ( ) ()() () () ** * * * * * * 1,, ,, ****** ,, ,, || (|) (|) T k T k T T spkT ykT yT spkT ykT T T des k u k u des k u k VV ykky Qykky ukk u Q ukk u uk Ruk ΘΘ ΘδΘ ΘδΘ δδΔΔ + −≥−− −− +−− −−+ (36) Since the right hand side of (36) is positive definite, the successive values of the cost will be strictly decreasing and for a large enough time k , we will have () () ( ) ** 1 0 TT kk VV ΘΘ + − = , which proves the convergence of the cost. The convergence of ( ) * kT V Θ means that, at steady state, the following relations should hold ( ) () () ** * ,, | TTT sp k y k ykk y Θ δΘ −= ** ,, (|) des k u k ukk u δ −= ( ) * 0uk Δ = At steady state, the state is such that () () () () () () ˆ () ˆ () 0 ˆ n s n d n yk y k yk p xk y k xk y k xk ⎡⎤ ⎡ ⎤ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ + == ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦   where ( ) y k is the actual plant output. Note that the state component ( ) ˆ d n xk is null as it corresponds to the stable modes of the system and the input increment is null at steady state. Then, constraint (23) can be written as follows: [...]... Controlled outputs and set points for the FCC subsystem with modified zones Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 369 250 u1 200 150 100 0 10 20 30 40 50 60 70 80 90 100 60 70 80 90 100 time (min) 100 u2 80 60 40 0 10 20 30 40 50 time (min) Fig 11 Manipulated inputs for the FCC subsystem with modified output zones 7 10 8 x 10 3 x 10 9 2.5 8 7 2... Chemical Engineering Research and Design, 84 (A11), 104 1 -105 0 González A H & Odloak D (2009) Stable MPC with zone control Journal of Process Control, 19, 110- 122 González A H.; Odloak D & Marchetti J L (2009) Robust Model Predictive Control with zone control IET Control Theory Appl., 3, (1), 121–135 González A H.; Odloak D & Marchetti J L (2007) Extended robust predictive control of integrating systems... Input feasible sets when the second input target is changed 366 Robust Control, Theory and Applications y1 550 500 0 50 100 150 100 150 100 150 time (min) y2 700 650 600 0 50 time (min) y3 700 600 500 0 50 time (min) Fig 6 Controlled outputs and set points for the FCC subsystem with modified input target 240 u1 220 200 180 160 0 50 100 150 100 150 time (min) 80 u2 70 60 50 0 50 time (min) Fig 7 Manipulated... the FCC system umin 75 25 umax 250 101 Robust Model Predictive Control for Time Delayed Systems with Optimizing Targets and Zone Control 363 Before starting the detailed analysis of the properties of the proposed robust controller, we find it useful to justify the need for a robust controller for this specific system We compare, the performance of the proposed robust controller defined through Problem... the controller uses model Θ 3 to calculate the output predictions However, the proposed robust controller performs quite well and is able to bring the three outputs to their zones y1 550 500 0 5 10 15 20 25 30 35 40 45 50 30 35 40 45 50 30 35 40 45 50 time (min) y2 800 600 400 0 5 10 15 20 25 time (min) y3 100 0 500 0 0 5 10 15 20 25 time (min) Fig 2 Controlled outputs for the nominal (- - -) and robust. .. that, we cannot find an input that, 364 Robust Control, Theory and Applications taking into account the gains of all the models and all the estimated states, satisfies the output constraints u1 250 200 150 0 5 10 15 20 25 30 35 40 45 50 30 35 40 45 50 time (min) 100 u2 80 60 40 20 0 5 10 15 20 25 time (min) Fig 3 Manipulated inputs for the nominal (- - -) and robust (⎯⎯) MPC ϑu (θ 2 ) ϑu (θ1 ) ϑu (θ3... develop 372 Robust Control, Theory and Applications robust control strategies which maximize the overall reliability of controlled structures Robust control design of systems with parametric uncertainties have also been studied by Mengali and Pieracci (2000); Crespo and Kenny (2005) These works are meaningful in improving the reliability of uncertain controlled systems, and it has been shown that the use... approach and to achieve a balance between reliability and performance /control- cost in design of uncertain systems In fact, traditional probabilistic reliability methods have ever been utilized as measures of stability, robustness, and active control effectiveness of uncertain structural systems by Spencer et al (1992,1994); Breitung et al (1998) and Venini & Mariani (1999) to develop 372 Robust Control, Theory. .. respectively when Sy = 10 3 * diag ( 1 1 1 ) and Su = 10 3 * diag ( 1 1 ) The final stationary value of the input is u= [155 84], which represents a the closest feasible input value to the target udes Finally, Figure 12 shows the control cost of 368 Robust Control, Theory and Applications the two simulated time periods Observe that in the last period of time (from 51min to 100 min) the true cost... to problem P2 defined by: Δu ( k / k ) = udes , k − u ( k − 1 ) = −δ u , k (40) and ( ) ysp , k (θ n ) = y k − Bs (θn ) δ u , k n=1,…,L (41) Now, consider the cost function defined in (21), written for time step k and the control move defined in (40) and the output set point defined in (41): 358 Robust Control, Theory and Applications ( ) T Vk (Θn ) = I y y( k ) − S1 (Θn )δ u , k − I y ysp , k (Θn ) . u ( ) 1 c u ϑ θ ( ) 2 c u ϑ θ ( ) 3 c u ϑ θ Robust Control, Theory and Applications 366 0 50 100 150 500 550 y1 time (min) 0 50 100 150 600 650 700 y2 time (min) 0 50 100 150 500 600 700 y3 time ( min ) Fig. 6. Controlled. functions: Robust Control, Theory and Applications 362 () () 24 3 6 1 2 65 0.4515 0.2033 2.9846 1 1.7187 1 0.1886 3.8087 1.5 20 1 17.7347 10. 8348 1 1.7455 6.1355 9 .108 5 1 10. 9088 1 ss s s ss ee ss se e G s ss ee ss Θ − − − − −− ⎡ ⎤ ⎢ ⎥ ++ ⎢ ⎥ ⎢ ⎥ − = ⎢ ⎥ + ++ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ++ ⎢ ⎥ ⎣ ⎦ ,. | TT xk j kxk j k++= + Robust Control, Theory and Applications 356 and ( ) ( ) |1 | TT y k j k y k j k++= + In addition, for the true model we have ( ) ( ) * ,1 , y kT y kT δ ΘδΘ + =  and * ,1 , uk