Receding horizon control: An overview and some extensions for constrained control of disturbed nonlinear systems

Receding horizon control is a common control concept dealing with control approaches, where their parameters are updated frequently along the time axis by using process informations in the past. Various methods of receding horizon control have been proposed, under which also optimization based receding horizon control methods, that is often known as model predictive control (MPC).

Review RECEDING HORIZON CONTROL: AN OVERVIEW AND SOME EXTENSIONS FOR CONSTRAINED CONTROL OF DISTURBED NONLINEAR SYSTEMS (INVITED PAPER) Nguyen Doan Phuoc1,*, Tran Duc Thuan2 Abstract: Receding horizon control is a common control concept dealing with control approaches, where their parameters are updated frequently along the time axis by using process informations in the past Various methods of receding horizon control have been proposed, under which also optimization based receding horizon control methods, that is often known as model predictive control (MPC) This paper gives a rough overview of MPC methods together with their main advantage as well as disadvantage From this point, the paper proposes a nonlinear receding horizon control strategy which can be applied to constrained output tracking control by output feedback for a wide range of various nonlinear objects, which are perturbed additionally by system disturbances All output feedback control methods corresponding to this proposed strategy are established based on piecewise linear quadratic optimizing subjected to required constraints for state feedback control and then combined with either a suitable system state observation EKF/UKF for noise filtering or a disturbance attenuationt unit, to become a conformed output feedback receding horizon controller Keywords: MPC, EKF/UKF, Adaptive control, Tracking control, Receding horizon control, Constrained optimization, LQR INTRODUCTION Receding horizon control with its well known representation named model predictive control (MPC), is an advanced method of process control, which has been applied successfully in industry since many decades ago [1] The MPC uses the mathematical process model to predict future changes of process dynamic from measured system states at the current time instant These predictive future changes of process dynamic will be then calculated to hold process outputs close to desired values, while honoring constraints on both process state and process inputs Fig.1 illustrates the principle structure of MPC with three main components in it: the prediction model, the objective function and a suitable optimization algorithm Optimization algorithm {w k } e k i y k i Objective function Predictive model uk Controlled subject y k xk Figure 1: Basic structure of a closed loop control system using MPC Since the very complicatedness of output prediction y k i at the current time instant k during the control horizon  i  N , and moreover, for a possibility of the usage of an Journal of Military Science and Technology, Special Issue, No 48A, - 2017 Electronics and Automation appropriate constrained optimization algorithm afterward, the application range of MPC in practice is restricted initially on discrete time linear systems describing by the discrete state model: x k 1  Ax k  Bu k , x k  Rn , u k  Rm (1)  r y k  C x k , y k  R or by discrete transfer function [2]: b  b z 1    bm z m G (z )  1 (2) a0  a1z    an z n While fast of all real processes in practice are not linear, the application of MPC requires obligatory a linearization of the process model over a small operating range This causes obviously an undesired effect on system performance To avoid this effect by linearizing, some nonlinear approaches are proposed in [3] However, this technique for nonlinear model predictive control (NMPC) requires additionally a penalty function for objective function in order to guarantee the stability of the closed system Unfortunately the question how to choose this penalty function suitably is still open, even till today To overcome these all circumstances, the moving LQRs/LQGs along the time axis looks to be a promising remedy and which is the main content of the extension, which is proposed in this paper CONVENTIONAL MODEL PREDICTIVE CONTROL METHODS FOR DISCRETE TIME LINEAR SYSTEMS: A ROUGH OVERVIEW MPC is based on iterative, finite horizon optimization of a process model At the current time instant tk  kT , where T is sampling time, the current process states x k are measured and together with past outputs y k  j , j  1, 2,  , M a cost minimizing control strategy is solved via a numerical optimization algorithm for a relatively short time horizon [k , k  N ) in the future to obtain future inputs u k i , i  0,1,  , N Only the first input value u k of them is sent to process, then the calculations are repeated starting from now current states x k 1 , yielding a new control u k 1 Nowaday, there are many basic MPC methods are available, such as [2]:  Model algorithmic control and Dynamic matrix control,  Generalized predictive control,  State feedback MPC and they are all classified mainly by predictive model and optimization algorithm to be used in it 2.1 Model algorithmic control (MAC) The MAC uses the impuls response of SISO process (single input-single output): (3) {gk }   1 G (z ) for output prediction, where {}  denotes the z-transformation With this model, the process output yk i in the future  i  N will be predicted as follows: yk i    g j uk i  j j 0    j i 1 g j uk i  j  i  g j uk i  j j 0  ci  i  g j uk i  j (4) j 0 for all i  1, 2,  , N , where ul  if l  and N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review ci    j i 1 g j uk i  j  k i  j i 1 g j uk i  j  gi 1uk 1  gi  2uk     gi k u Next, for output tracking purpose ek  yk  wk  , where {wk } is the desired trajectory, the following objective function belonged to current horizon [k , k  N ) will be used: N J k   eTk 1Qk ek i  uTk 1Rk u k i  i 0   (5) where Qk , Rk are two arbitrarily chosen symmetric positive definite matrices With the symbols:  0  g0  wk   yk   uk   c0            g g0   w y u c w   k 1  , y   k 1  , p   k 1  , c    , G                                wk N   yk N   uk N   cN   gN gN 1  g  all N predictive outputs (4) are rewritten in y  G p  c and therefore (5) becomes:  T 1 GTQk c  w  J k  pT GTQkG  Rk p  c  w  QkG p   p which implies: p*   GTQkG  Rk   for unconstrained case, or p*  arg J k pP by using an appropriate constrained optimization method introduced in [4], for constrained circumstance Finally, only the first value uk  1,0,  ,0  p* of them is implemented to the process At the next time instant k  the whole calculating steps above are repeated again for determining the new control signal u k 1 with the prediction horizon moving forward The following algorithm presents this iterative working performance of MAC Algorithm 1: MAC Set k : 0, u0  Choose arbitrarily N  Determine G Choose appropriately two symmetric positive definite matrices Qk , Rk Calculate ci , i  0,1,  , N and determine the vector c * Determine the optimal solution p and the element uk of it Send uk to the process for a while of sampling time T , then set k : k  and go back to the step It is immediately recognizable from this algorithm, that MAC is an open loop controller Therefore it is very sensible with disturbances and can be applied only for stable processes 2.2 Dynamic matrix control (DMA) On contrary to MAC, the DMA uses step response {hk } instead of (3) for output prediction: Journal of Military Science and Technology, Special Issue, No 48A, - 2017 Electronics and Automation  yk i   h j uk i  j  j 1 k i   j i 1   j i 1 h j uk i  j  h j uk i  j  i  h j uk i  j j 0 i  h j uk i  j j 0  di  i  h j uk i  j for i  0,1,  , N (6) j 0 with uk  uk  uk 1 and di  k i  j i 1 h j uk i  j  hi 1uk 1  hi  uk     hi k u0 (7) Therefore the DMA algorithm is completely equivalent to MAC as follows: Algorithm 2: DMA Set k : 0, u0  Choose arbitrarily N  Determine  h0  h H      hN  0   0      h0  Choose appropriately two symmetric positive definite matrices Qk , Rk Calculate di , i  0,1,  , N given in (7) and determine the vector h0  hN 1 T d  d ,d1 ,  ,dN  T as well as y  y k , yk 1 ,  , yk N  with yk i , i  0,1,  , N given in (6) Determine the constrained optimal solution p*  arg J k pP with  Jk  y  w T  T Qk y  w  pT Rk p and w  w k , wk 1 ,  , wk N    Send uk  1,0,  ,0  p* to the process for a while of sampling time T , then set k : k  and go back to step The same as MAC, the DMA algorithm given above is an open loop controller It is therefore very sensible with system disturbances and can be applied for stable processes only 2.3 Generalized predictive control (GPC) In GPC, the transfer function (2) of a process with an integral unit in it, will be used for output prediction Such a process has the mathematical model in form of difference function as follows: (8) A(z 1 )yk  B (z 1 )uk where z  j x k  x k  j and A(z 1 )  a  a1z 1    an z n , B (z 1 )  b0  b1z 1    bm z m Denote Ei (z 1 ), Fi (z 1 ), i  0,1,  , N the solutions of Diophaltine equations:  Ei (z 1 )A(z 1 )  z i Fi (z 1 ), i  0,1,  , N and N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review Gi (z 1 )  Ei (z 1 )B (z 1 ) the equation (8) will be rewritten in: yk i  n m i 1 j 0 j 0  fi , j yk  j   gi , j uk i  j 1 where fi , j , gi , j are the parameters of Ei (z 1 ), Gi (z 1 ), i  0,1,  , N It means: Fi (z 1 )  fi ,0  f j ,1z 1    fi ,n z n Gi (z 1 )  gi ,0  gi ,1z 1    gi ,m i 1z  (m i 1) T Hence, all predictive outputs y  y k 1 , yk  ,  , yk N  will be performed in: y  E1 p  E u b  Fy b (9) with    g1,0  uk   uk 1   yk          g 2,1 g 2,0   uk 1  uk   yk 1      p ,u  ,y  ,E    b    b                    uk N 1   uk m   yk n   gN ,N 1 gN ,N   gN ,0  g1,2  g1,m   g1,1  f1,0 f1,1  f1,n      g 2,2 g 2,3  g 2,m 1  f2,0 f2,1  f2,n  E2   , F                   gN ,N gN ,N 1  gN ,m N 1   fN ,0 fN ,1  fN ,n  By substituting all predictive outputs of Eq (9) in the objective function (5) it is obtained: J k  pT G1TQkG1  Rk p  2bTQkG1 p  bTQk b    where  T  b  E ub  Fyb  w and w  w k 1 , wk  ,  , wk N  which implies: Algorithm 3: GPC Choose arbitrarily N  Determine Ei (z 1 ), Fi (z 1 ), i  1,  , N and E1 , E , F Set k  0, ub  0, yb  Choose appropriately two symmetric positive definite matrices Qk , Rk Measure the current output yk Rearrange ub , yb Determine the vector b Determine the constrained optimal solution p*  arg J k pP * Send uk  1,0,  ,0  p to the process for a while of sampling time T , then set k : k  and go back to step It is recognizable from this algorithm, that GPC is an output feedback controller Therefore it is robust with output constant disturbances and can be applied also for unstable processes The GPC algorithm can be easily reperformed for MIMO systems Such a version of GPC is already proposed in [2] Journal of Military Science and Technology, Special Issue, No 48A, - 2017 Electronics and Automation 2.4 State feedback MPC for linear systems The state feedback MPC for LTI systems uses the given model in (1) with an additive integral unit in it:   x k 1  Ax k  Bu k  Ax k  Bu k 1  B u k  z k 1  Az k  B u k (10)  u k  u k 1  u k for state prediction, where  x    A B   B  zk   k , A   , B     I  I   u k 1  and  is the null matrix The alternative model (10) has an integral unit in it, because the  matrix A with  B   zI  A m det zI  A  det     z  1 det  zI  A (z  1)I    has m eigenvalues z  This integral behaviour of prediction model guarantees that the steady error of a stable closed loop system will be definitely zero Together with prediction model (10) the system output y k is rewritten in:   y k  C x k  C ,   z k  C z k where C  C ,   (11)   and now, from prediction model (10) and (11) it is obtained:       y k i  CAi x k  CAi 1Bu k    CABu k i 2  CBu k i 1 which deduces: y  Dz k  Fu (12) where    y k 1   CA   CB      uk          y    , D    , F       , u      y    N     N 1    N   u   k N 1  CA  CA B CA B  CB   k N  Finally, the subtitution of (12) into objective function (5) implies:  Jk  y  w T  Qk y  w  uT Rk u   T T T  uT FTQk F  Rk u   Dz k  w  Qk Fu   Dz k  w  Qk  Dz k  w    with w  col w k 1 , w k  ,  , w k N  and correspondingly, the following algorithm performs desired state feedback MPC by summarizing all caculations given above Algorithm 4: State feedback linear MPC    Set k : 0, u 1  Choose arbitrarily N  Determine A, B ,C , D, F Choose appropriately two symmetric positive definite matrices Qk , Rk Measure the current states x k Determine u *  arg J k pP * Send u k   I , ,  ,   u to the process for a while of sampling time T , then set k : k  and go back to step N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review 2.5 Output feedback MPC for linear systems In model predictive controllers that consits only of linear models, the superposition principle of linear control theory enables an opportunity to convert the state feedback controller to output feedback one by using additionally a state observer This state  observer has a purpose to produce approximately process state x k and then the state feedback controller uses this observed states instead of the real state x k measured from process Fig.2 illustrates this separation principle output feedback control strategy w State feedback controller  x u System noises Output disturbance Controlled plan y State observer Figure 2: Using state obsever to convert a state feedback controller to an appriopriate output feedback one Algorithm 5: Output feedback linear MPC  Set k : 0, u 1  0, y 1  Choose arbitrarily N  and an initial process state x    Determine A, B ,C , D, F Choose appropriately two symmetric positive definite matrices Qk , Rk  Set x k  x k Determine u *  arg J k pP Send u k   I , ,  ,   u * to the process for a while of sampling time T  Measure the output y k from process Set k : k  and estimate x k by using an appropriate observer Go back to the step MODEL PREDICTIVE CONTROL FOR PERTURBED NONLINEAR DISCRETE TIME SYSTEMS Consider a nonlinear system, which is described generally in: x k 1  f (x k , u k ,  k )  y k  g (x k ,  k )  d k where both functions f (), g () are assumed to be smooth in x k and u k , as well (13) T x k   x1[k ] ,  , x n [k ] is the vector of all system states at the current time instant tk  kT , where T is the sampling time, and T T u k  u1[k ] ,  , um [k ] , y k  y1[k ] ,  , yr [k ] are vectors of inputs and outputs signals respectively at the same time instant Both  k ,  k are white noises, which could propagate nonlinearity in system, and d k is a vector of slow disturbances, which can be seen obviously as the model errors Journal of Military Science and Technology, Special Issue, No 48A, - 2017 Electronics and Automation The here regarded control problem for the given nonlinear system (13) above is an output feedback controller u k (x k ) to design, which is subjected to the input constraint u k U  R m , so that its output vector y k will be convergence asymptotically to any desired output vector w k , and this tracking control performance will not be affected by white noises  k ,  k and by system errors d k For solving the above formulated tracking control problem this paper proposes the control concept with three following steps to be carried out: Replace approximately the original model (13) by a set of infinite of LTI models H k , k  0,1, as depicted in Fig.3 This set of infinite of LTI models H k will be called in this paper the moving horizon predictive model of the original nonlinear system (13) Then each of those LTI models will be used subsequently at the time instant tk , k  0,1, , together with moving finite control horizon [tk ,tk N ] along the time axis toward, to design correspondingly a state feedback controller uk (x k ) subjected to the constraint uk U for tracking control the original nonlinear system (13) during the current time interval [tk ,tk 1 ) , where tk 1  tk  T and T is the sampling time of the system (13) Replace the states x k in the above obtained state feedback controller u k (x k ) by  observed states x k , which is received from an applying extended (EKF) or unscented  Kalman filer (UKF), to obtain an output feedback controller u k (x k ) 3.1 Receding horizon LTI predictive model If all noises  k ,  k and disturbance d k in (13) are negligeable, then from (13) the corresponding nominal model is obtained: x k 1  f (x k , u k )  y k  g (x k ) (14) Since the smooth property, both function vectors f (), g () of the nominal model (14) can be now approximated at the previous time instant tk 1 and during time interval [tk 1 ,tk ) afterwards as follows: f f f (x k , u k )  f (x k 1 , u k 1 )  x k  x k 1    u k  u k 1  x x ,u u x ,u k 1 k 1 k 1 k 1  Ak x k  Bk u k  d k g (x k )  g (x k 1 )  g x x k  x k 1   C k x k  h k x k 1 where Ak  f x , Bk  x k 1 ,u k 1 f u , Ck  x k 1 ,u k 1 g x x k 1 (15) d k  f (x k 1 , u k 1 )  Ak x k 1  Bk u k 1 and h k  g (x k 1 )  C k x k 1 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review are all now determined at the current time instant k , because all past system values x k 1 , u k 1 are already known For the controller design hereafter both vectors d k , h k in (15) will be considered as constant during the whole current control horizon [k , k  N ) Hence, it is deduced: x k 1  Ak x k  Bk u k  d k Hk :  (16) y k  C k x k  h k and this model will be used afterward for the prediction of system outputs y k i in the current prediction horizon  i  N the current predictive horizon the next predictive horizon tk tk 1 tk  NT t H k H k 1 Figure 3: Using infinite number of LTI system models instead of nonlinear one 3.2 State feedback controller At the current time instant k and based on the already measured system states x k , all predictive system states x k i ,  i  N can be now obtained from the LTI predictive model (16) as follows: y k i  C k Aki x k  C k Aki 1Bk u k    C k Aki  2Bk u k 1  C k Bk u k i 1    C k Aki 1    Ak  I d k  h k Now, if all predictive output vectors are rewritten as a mergence vector: y  col y k 1 , y k  ,  , y k N   then it is obtained: y  Fp d (17) where:  C k Ak    uk   2   u  C A  , E   k k  , p   k 1            u  C AN   k N 1   C k Bk  k k   C k Bk  C AB C k Bk k k k F      N 1 N 2 C k Ak Bk C k Ak Bk       (18)  1  d  Ex k     with  i  C k Aki 1    Ak  I d k  h k      N   It is easily to recognize, that the predictive mergence vector y given in (17) depends only on all inputs p in the future associated in the current horizon Journal of Military Science and Technology, Special Issue, No 48A, - 2017 Electronics and Automation With the expression (17) of obtained predictive outputs y k i ,  i  N , all tracking errors during the current control horizon will be deduced as follows:  e  w y  w  Fp d  (19) where: w  col w k 1 , w k 1 ,  , w k N  (20) is the mergence desired output values during the same control horizon Next, according to the output tracking purpose y k  w k or e  associated with the current control horizon, the mergence input vector p would be determined by minimizing the following objective function: J k  eTQk e  pT Rk p (21) where Qk , Rk are any arbitrarily chosen symmetric positive matrices This objective function is clearly equivalent with: T J k  w  F p  d  Qk w  F p  d   pT Rk p           T T  pT F TQk F  Rk p  w  d  Qk F p  w  d  Qk w  d  which is obtained by replacing (19) into (21), or:   T J k/  pT F TQk F  Rk p  w  d  Qk F p (22) T since the last term w  d  Qk w  d  is independent on p Easily to see that the obtained objective function (22), which is to be minimized, is quadratic Hence for solving this optimization problem subjected to the constraint p  P with:  P  p  col u k ,  , u k N 1   RNm u k U  (23) or: p*  arg J k/ (p ) (24) pP it is obviously [4]:  the QP method could be used, if the constraint U is linear (described by linear inequations), or  the SQP method is an appropriate one, if the constraint U is nonlinear For unconstrained case it is:  p*  FTQk F  Rk  1 FTQk w  d  (25) Finally, the control value u k for the original perturbed nonlinear system (14) is then getting from the received optimal solution p* of the optimization problem (24) as follows: u k   I , ,  ,   p* (26) and this control value u k , which is clearly dependent on current system states x k and therefore will be denoted by uk (x k ) , is only valid during the short current sampling time 10 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review interval kT  t  (k  1)T For receiving the next control value u k 1 at the next time instant k  all calculation steps above including (15),(18),(19)-(26) have to be repeated The following algorithm summarizes respectively these calculation steps to present this recursive working performance of the proposed state feedback controller Algorithm 6: State feedback nonlinear MPC Set k : Choose arbitrarily N  0, u 1 , x 1 Choose appropriately two symmetric positive definite matrices Qk , Rk Measure the current state vector x k from system and then determine the matrices Ak , Bk , C k given in (15), F , E given in (18) Calculate all vectors d k , h k , w , d given in (15), (18) and (20) Determine the solution p* of the optimal problem (24) subjected to the constraint (23) by using either the QP or SQP method In unconstraint case it is determined by using the equation (25) Determine the control signal u k given in (26) and sent it to the original system (14) for a while of the sampling time interval T Set k : k  and go back to the step 3.3 Using EKF or UKF for state observation To convert the state feedback controller uk (x k ) presented in Algorithm correlatively into an output feedback controller based on separation principle it is needed a suitable state observer In this paper the EKF and UKF will be used for this purpose The EKF is useful for nonlinear systems, where noises  k ,  k propagate linearity: x k 1  f (x k , u k )   k (27)  y k  g (x k )   k There are many versions of EKF and all of them have been detailed in [5] The version used in this paper is as follows: x k ()  f  x k 1 ( ), u k 1   Pk ()  Fk 1Pk 1 ( )FkT1    (28)  Kk  Pk ()GkT (Gk Pk ()GkT   ) 1  Pk ( )   I  KkGk  Pk ()  x k ( )  x k ()  K k y k  Gk x k () for k  1,2, , where ,  are the covariance matrices of  k ,  k , and  Gk  g (x k ) x k , Fk 1  x k x k (  )   f (x k , u k ) x k , x k 1 (  ),u k 1  as well the initial values x  x (), P0 () are arbitrarily chosen The obtained value x k () from each loop above k is the observed state vector at the current time instant  tk  kT and will be denoted with x k  x k () On the contrary to EKF, the UKF is then used for state observation of perturbed nonlinear systems: Journal of Military Science and Technology, Special Issue, No 48A, - 2017 11 Electronics and Automation x k 1  f (x k , u k ,  k )  y k  g (x k ,  k ) when noises  k ,  k propagate nonlinearity and invisibly in system model Detailed  calculations steps of UKF to obtain x k ware already presented in [6], [7] including all iterative operations as follows:  x ak 1 if i    caculate Xia,k 1  x a   Si if i  1,  , N k 1   a  x k 1   Si if i  N  1,  , 2N   a  with    (N  ) and Si is the i -th column of S  Pk 1  a a x   rearrange Xi ,k 1 , i  0,1,  , 2N as Xi ,k 1  col Xi ,k 1 , Xi ,k 1 , Xi ,k 1  x Xi ,k k 1  f Xix,k 1 , Xi,k 1 ,u k 1 , i  0,1,  , 2N  2N    2N (i ) x   T  (i ) x x x k   wm Xi ,k k 1 and Px k   wc Xi ,k k 1  x k Xi ,k k 1  x k i 0 i 0     (0)  with wm  , wc(0)   1    N  N   (i ) as well with wm  wc(i )  , i  1, 2,  , 2N and    (N  )  N  2(N   )  Yi ,k k 1  g Xix,k 1 , Xi,k 1 ,u k 1 , i  0,1,  , 2N     2N (i ) y k   wm Yi ,k k 1 i 0  2N    T (i ) Py k   wc Yi ,k k 1  y k Yi ,k k 1  y k i 0  (29)  2N T Px y   wc(i ) Xix,k k 1  xk Yi ,k k 1  y  k  k k i 0    1   T Kk  Px k y k Py k , x k  x k  Kk (y k  y k ) and Px k  Px k  Kk Py k Kk  for all k  0,1,  The obtained value x k from each loop is the observed state vector at                  the current time instant tk  kT The initial values used for (29) are:    0,    1,   0, N , x [0], x [1],  , x [2N ] and are chosen arbitrarily     x  E {x }, Px  E (x  x )(x  x )T    x a0  E {x a0 }  E {col (x , , 0)}   12 P0a E (x a0   P  x a a a T  x )(x  x )        P      P  N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review where Px , P , P are the matrices of dimension n , v , s as the vectors x k ,  k ,  k respectively 3.4 Improving the steady performance of closed loop system with integral control Since the minimizing of J k/ given in (22) occurs only over a finite horizon [k , k  N ] , the desired tracking performance e k  w k  y k  may not be satisfied Therefore, to guarantee that the tracking error ek always tends asymptotically to zero, an integral will be added to the proposed state feedback controller presented in Algorithm With two new variables x k  x k  x k 1 , u k  u k  u k 1 the LTI predictive model H k given in equation (16) is changed to:   z k 1  Ak z k  Bk u k H k/ :   y k  C k z k where  x k    Ak     Bk   zk   ,A  ,B  ,C  C  y  k C I  k    k  k   r   k  k 1  (30) Ir  (31) and I r is the r  r identity matrix This new LTI predictive model (30) has obviously an integral in it, because with:     I  Ak r det(I n r  Ak )  det  n   (  1) det(I n  Ak )  C (   1) I k r   it has r eigenvalues   Accordingly, Algorithm will be changed to: Algorithm 7: State feedback integral control Set k : Choose arbitrarily N  0, u 1, x 1 Choose appropriately two symmetric positive definite matrices Qk , Rk Measure the current vectors x k , y k 1 from system and then determine the matrices    Ak , Bk , C k given in (15), Ak , Bk , C k given in (31), the vector w given in (20) and then F , E as follows:    C k Bk               2  N C k Bk     C k Ak Bk E  col C k Ak , C k Ak ,  , C k Ak , F   (32)         C AN 1B C A N  2B  C B  k k k k k k  k k * Determine the solution p of the optimal problem (24), where the vector x k of   objective function J k/ (p ) in (22) is replaced by z k  col (x k , y k 1 ) If the optimal problem is subjected to the constraint (23), then either the QP or SQP method is used to determine p* Otherwise, for unconstraint case it is determined by using the equation (25) with z k instead of x k Determine the control signal u k as follows: Journal of Military Science and Technology, Special Issue, No 48A, - 2017 13 Electronics and Automation u k  u k 1  u k  u k 1   I m , ,  ,   p* and sent it to the original system (16) for a while of the sampling time interval T Set k : k  and go back to the step 3.5 Output feedback control algorithm The following output feedback controller based on separation principle, as depicted in Fig.2, is established by combining the already given state feedback controller (Algorithm or Algorithm 7) and a suitable state observer, such as EKF given in (28) or UKF presented in (29) Algorithm 8: Output feedback control  Set k : Choose arbitrarily N  0, u 1 , x 1 Choose appropriately two symmetric positive definite matrices Qk , Rk  Set x k  xk 1 and then determine the matrices Ak , Bk , C k given in (15), F , E given in (18) and the vector w given in (20) * Determine p of the problem (24) subjected to the constraint (23) by using either the QP or SQP method In unconstraint case it is determined with the equation (25) Determine u k given in (26) and sent to the system (14) for a while of the sampling time interval T   Measure the output y k and then calculate x k  x k () with Eq (28) or x k with Eq (29) Set k : k  and go back to the step To design the output feedback integral controller based on separation principle for original systems (14) the EKF or UKF can not be directly applied Since the state vector z k  col (x k , y k 1 ) of the integral LTI predictive model H k/ given in (30) contains the system output y k  g (x k ,  )  d k in it, which is still disturbed d k , and EKF/UKF can k filter Gaussian noises  k ,  k only, the disturbance d k still remains in z k , even when the  system state vector x k  x k has been observed ideally by EKF/UKF Therefore, together with EKF/UKF for filtering  k ,  k from x k an additional d k eliminator from y k to  obtain the undisturbed system output y k must be needed    Denote the undisturbed output with y k  y k  d k , where d k is the mean of d k over a certain horizon M , then is can be estimated averagely as follows:  M 1   y k  yk  d k  yk  y k i  g (x k i )  M i 0   (33) Algorithm 9: Output feedback integral control    Set k : Choose arbitrarily initial values u 1 , z 1 , x 1 , x 2 , y 1 and N  M  Choose appropriately two symmetric positive definite matrices Qk , Rk     Set x k  xk 1 and then determine the matrices Ak , Bk , C k given in (15), Ak , Bk , C k given in (31), the vector w given in (20) and then F , E given in (32) 14 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review Determine the solution p* of the optimal problem (24), where the vector x k of  objective function J k/ (p ) in (22) is replaced by z k  col (x k , y k 1 ) If the optimal problem is subjected to the constraint (23), then either the QP or SQP method is used to determine p* Otherwise, for unconstraint case it is determined by using the equation (25) with z k instead of x k Determine the control signal u k as follows: u *k  u *k 1  u *k  u *k 1   I m , ,  ,   p* and sent to the original system (14) for a while of the sampling time interval T  Measure the output y k and then calculate y k with the equation (33)  *  Send u k , y k to EKF (28) or UKF (29) for observation of x k Set k : k  and go back to the step RECEDING HORIZON CONTROL WITH INFINITE HORIZON FOR CONTINUOUS TIME PERTURBED NONLINEAR SYSTEMS For a possible application of proposed controllers given above (Algorithm 6-9) to control continuous time nonlinear systems, which is perturbed additionally by  ,  : x  f (x , u )    y  g (x )   where: f (x , u )  col  f1 (x , u ),  , fn (x , u )  and g (x )  col  g1 (x ),  , gm (x )  (34) are state and input dependent function vectors respectively, which are assumed to be smooth in x and in u , it is needed the discretization of (34) with an appropriately chosen sampling time T to obtain a suitably approximated discrete model given in (27) However, since the discrete model obtained by discretizing could not reflect all intersample behaviors of the real system, which may be cause a number of critical event in practical applications, for example the smaller T is chosen, the better this approximation for signals will be, but unfortunately the smaller T is, the more dynamical information of (34) will be lost, any control method to control the system (34) directly in continuous time domain is always preferential, and the control approach proposed below is one of them 4.1 Receding horizon LTI model Consider the system (34) at the current time instant tk and suppose that the system x  x (tk ) state vector k at this time instant is measurable, as well as tk  tk 1   with a x ,u very small positive value  Then in the neighbourhood of k k 1 , it means with x (  tk ) , u (  tk 1 ) for      , the nonlinear system (34) can be approximated by: f (x , u )  f (x k , u k 1 )  Ak  x  x k   Bk u  u k 1   Ak x  Bk u  d k where Ak  f x , Bk  x k ,u k 1 f u and d k  f (x k , u k 1 )  Ak x k  Bk u k 1 (35) x k ,u k 1 Journal of Military Science and Technology, Special Issue, No 48A, - 2017 15 Electronics and Automation Therefore, the original nonlinear state equation in (34) can be now replaced accordingly during the same time interval [tk ,tk 1 ) by a linear model:  x  f (x , u )  Ak x  Bk u  d k   k  where  k is an estimation of disturbance  (t ), tk  t  tk 1 It is clearly that all matrices  Ak , Bk and vectors d k ,  k are determined, since x k  x (tk ) at the current time instant tk are measureable and u k 1  u (tk 1 ) at the previous time instant is already known In analogy the output equation of (34) can be replaced appropriately during the same time interval [tk ,tk 1 ) by:  y  g (x )    C k x  ek   k with Ck  g x e k  g (x k )  C k x k , (36) xk  and  k is an average estimation of  (t ) over the time interval tk  t  tk 1 Hence, the original perturbed nonlinear model (34) can be now replaced accordingly during the current time interval [tk ,tk 1 ) by the following determined LTI model:  x  Ak x  Bk u  d k   k Hk :  (37)  y  C k x  ek   k Each model (37) can be replaced the original original model (34) only during the appropriate time interval [tk ,tk 1 ) and all of them together with k  0,1, will be called hereafter the receding horizon LTI models as already depicted in Fig.3 4.2 Disturbances estimation Next, under the assumption, that the disturbances  , are altering slowly, they could be estimated obviously from the past information of system (34) as follows: x  x k 1   (tk )   (tk 1 )  x (tk 1 )  f  x (tk 1 ), u (tk 1 )   k  f (x k 1 , u k 1 ) :  k (38)  and   (tk )   (tk 1 )  y (tk 1 )  g (x k 1 )  y k 1  g (x k 1 ) :  k  (39)  and both values  k ,  k will be considered hereafter for controller design as constants during the whole time interval t  tk for the determination of u k at tk 4.3 State feedback control In the following, the obtained LTI model (37) will be used to design the state feedback controller u (x ) based on linear quadratic variation technique to control the perturbed nonlinear system (34) during an appropriate time interval [tk ,tk 1 ) The obtained optimal controller, which is obviously also valid only during the next time interval [tk 1 ,tk  ) , will be denoted by k , k  0,1, as illustrated in Fig.4 The merged controller from them:   k , tk 1  t  tk  , k  0,1, (40) for all time domain t , will be called the receding horizon controller Consistently, the purpose of this receding horizon controller  is the asymptotical convergence to zero of 16 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review tracking error ek  w (tk )  y (tk ) of closed loop system for all k , where w (t ) is the desired output With (40) the designing of  can be now replaced by determining of all instant controllers k , k  0,1, In order to avoid tracking errors w (tk 1 )  y (tk 1 ) by designing of k , which could be remaining from previous control time instant tk 1 , an alternative desired value r k for the current time instant tk given below will be used instead of the original w (tk ) : r k  w (tk )  w (tk 1 )  y (tk 1 )  LQR (41) k   k tk Ak , Bk ,Qk ,   Rk ,d k ,  , k tk  tk 1 k 1 t tk 3 k Figure 4: To demonstrate the idea of receding horizon controller design with infinite horizon for contiuous time perturbed nonlinear systems Now, for a possible usage of optimal variation technique to design the controller k , such that the outputs y of linear time invariant system (37) converge asymptotically to desired output r k , it is required firstly this tracking problem to be converted correspondingly in a stabilizing control problem Signify the steady state of closed loop system of (37) after tracking phase with x s [k ] and us [k ] , then this steady state must be satisfied:  0  Ak x s [k ]  Bk u s [k ]  d k   k   r k  C k x s [k ]  ek   k which implies immediately:  1  x s [k ]   Ak Bk   d k   k         us [k ]  C k    r k  ek   k  if the matrix  A Bk  Fk   k  C k   (42) is invertible (it is easy to recognize that for the invertibility of matrix Fk the number of inputs u and of output y must be coincided.) Then, with the new symbols: z  x  x s [k ], v  u  us [k ] the tracking control problem of (37) to an alternative desired value r k will be converted correspondingly in the stabilization problem of following nominal system: z  Ak z  Bk v (43) Journal of Military Science and Technology, Special Issue, No 48A, - 2017 17 Electronics and Automation For optimal stabilizing this above obtained nominal system the following costfunction could be used: Jk   T T  z Qk z  v Rk v  dt  20 (44) where Qk , Rk are two arbitrarily chosen symmetric positive definite matrices Thence: v  Rk1BkT Lk z  k z (45) which is obtained based on the continuous time variation technique, where the symmetric positive definite matrix Lk is obtained by solving the algebraic Riccati equation: Lk Bk Rk1BkT Lk  AkT Lk  Lk Ak  Qk (46) and it is equivalent to: u (x )  v  u s [k ]   k z  u s [k ]   k  x  x s [k ]  u s [k ]  Rk1BkT Lk  x  x s [k ]  u s [k ] (47) The obtained value u (x ) above is sent subsequently to the system (34) as control signal for a while of tk 1  t  tk  To receive the next control value for the next time interval tk   t  tk 3 all calculation steps above have to be repeated 4.4 Opportunity to control with input constraints In the constrained circumstance u U with:  U  u  (u1 ,  , um uimin  ui  uimax , i  1,  , m  where the solution u of (47) may not be satisfied, some constrained optimization methods proposed in [4] could be applied to determine u instead of (47) However, since two matrices Qk , Rk in cost-functions J k , k  0,1, of proposed approach are arbitrarily selected, it arises here an opportunity to select them so appropriately that together with the forward movement of the control horizon along the time axis, the obtained control vector u with (47) will satisfy additionally the required input constraint u U More precisely, it is immediately recognizable from the cost-function (44) that:  the bigger the rate Rk Qk is chosen, the smaller u will be  the smaller Qk is selected, the smaller is v which will be used hereby in order to update suitably Qk , Rk along the time axis for satisfying the required constraint Note that not any solution v obtained from (45) to stabilizing the nominal system (43) could guarantee definitely the satisfaction of the required constraint u U However, based on the obviousness: lim u  lim Rk  Rk   R 1 T k Bk Lk x  x s [k ]  us [k ]  us [k ] it could be needed for satisfying this unavoidable constraint u U an assumption, that us [k ] U satisfies for all k 18 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” Review 4.5 State feedback control algorithm For a convenient implementation of proposed approach, the following algorithm has been established, which summarizes completely all calculation steps (35)-(47) given above Algorithm 10: State feedback receding horizon control continuous time perturbed nonlinear systems Choose arbitrarily two symmetric positive matrices Q , R and their update factors    1,   Select a sufficiently small moving distance   along time axis for    the control horizon Set x  0, u  0, y  and t    Measure the current state and output x , y and determine A, B ,d ,  , , r as follows: A f x , B f g    , d  f (x , u )  Ax  Bu , e  g (x )  C x u x          (x  x )   f (x , u ),   y  C x , r  w (t )  w (t   )  y   x ,u u , C  x ,u Determine: A B  F   C   If F is singular, then go back to the step 2) Calculate:  d    xs  1   F      us  r e   Calculate LBR 1BT L  AT L  LA  Q and u  R 1BT L (x  x s )  u s If u U then set R :  R and go back to the step 5) Send u to the controlled object (34) for a while of     Set x  x , u  u , y  y , Q : Q and t : t   Then go back to the step 2) 4.6 Output feedback control The output feedback controller will be obtained based on separation principle by combining the state feedback controller given in Algorithm 10 above and an appropriate state observer introduced in [5],[6] and [7] CONCLUSIONS A few basic receding horizon controllers have been presented in this paper Futhermore, the paper has also proposed some extension of them to control continuous time perturbed nonlinear systems by output feedback The control performance of all proposed extensions have been verified by simulation in [9] for steam boiler systems, in [10] for a quardrotor and in [11] for an inverted pendulum And these simulation results showed that proposed receding horizon controllers have meet completely the expectation [1] [2] [3] [4] REFERENCES Maciejowski, M.J (2011): Predictive control with constrains Prentice Hall Camacho,E and Bordons,C (1999): Model predictive control Springer LarsGrüne, JürgenPannek (2011) Nonlinear predictive control: Theory and Algorithms Springer-Verlag, London, 2011 Nocedal,J and Wright,S.J (1996): Numerical Optimization Springer-New York Journal of Military Science and Technology, Special Issue, No 48A, - 2017 19 Electronics and Automation [5] Grewal, M.S and Andrews, A.P.(2001): Kalman filtering: Theory and Practice using MatLab John Wiley & Sons [6] Julier, S.J and Uhlmann, J.K (2004): Unscented filtering and nonlinear estimation Proceedings of the IEEE, 92(3), pp.401-422 [7] Rambabu Kandepu, Bjarne Foss, Lars Imsland (2008) Applying the unscented Kalman filter for nonlinear state estimation Journal of Process Control 18 (2008) 753–768 [8] Findeisen,R.; Imsland,L.; Allgöwer,F and Foss, B.A (2003): Output-feedback nonlinear model predictive control using high-gain observers in original coordinates Int J of Robust and Nonlinear Control, vol 13, no 3-4, pp 211-227, 2003 [9] Phuoc N.D.; Anh,L.D.; Thanh,V.T.; Hung,P.V and Quynh,H.D (2016): Robust Output Tracking Control with Constraints for Nonlinear Systems based on Piecewise Linear Quadratic Optimization and its Perspective for Practical Application Proceedings of Workshop on Vietnamese-German technology cooperation and cultural exchange VDZ, pp.57-67 [10] Phuoc N.D.; Thang L.V and Truong B.D (2015): Optimal receding horizon control for a quardrotor Proceedings of VCCA-2015, pp.2-6 Thai Nguyen (2015) [11] Quỳnh, H.Đ (2016): Điều khiển bám tín hiệu cho hệ lắc ngược điều khiển tối ưu đoạn Hội thảo khoa học, Khoa Điện Đại học Kỹ thuật Công nghiệp Thái Nguyên TÓM TẮT ĐIỀU KHIỂN TRƯỢT DỌC TỪNG ĐOẠN TRÊN TRỤC THỜI GIAN: VÀI NÉT TỔNG QUAN VÀ MỘT SỐ MỞ RỘNG ĐỂ ĐIỀU KHIỂN CÓ RÀNG BUỘC CÁC HỆ PHI TUYẾN BỊ NHIỄU TÁC ĐỘNG (BÀI BÁO MỜI) Điều khiển trượt dọc trục thời gian khái niệm nhóm phương pháp điều khiển mà tham số điều khiển chỉnh định theo thời gian dựa thông tin khứ đối tượng điếu khiển Cho tới thời điểm có nhiều phương pháp điều khiển cụ thể cơng bố, đại diện tiêu biểu phương pháp xây dựng tối ưu hóa, gọi điều khiển dự báo theo mơ hình (MPC) Bài báo cung cấp nhìn tổng quan sơ lược phương pháp điều khiển với ưu nhược điểm chúng Cũng từ báo giới thiệu số dạng mở rộng điều khiển trượt dọc trục thời gian phản hồi đầu để áp dụng cho lớp hệ phi tuyến bị nhiễu tác động Tất phương pháp điều khiển liên quan tới điều khiển tuyến tính tối ưu phản hồi trạng thái có tham số biến đổi theo thời gian để có khả thỏa mãn thêm điều kiện ràng buộc đặt ra, kết hợp quan sát trạng thái thích hợp để lọc nhiễu với ước lượng nhiễu điều khiển tách thành phần bất định mơ hình, giúp trở thành điều khiển phản hồi đầu tương ứng Từ khóa: MPC, EKF/UKF, Điều khiển thích nghi, điều khiển bám; Điều khiển trượt dọc trục thời gian, Tối ưu hóa có ràng buộc, LQR Received date, 10th April 2017 Revised manuscript, 25th April 2017 Published on 26th April 2017 Author affiliations: Department of Automatic Control, Hanoi University of Science and Technology; Academy of Military Science and Technology; * Corresponding author: phuoc.nguyendoan899@gmail.com 20 N.D.Phuoc, Tr.D.Thuan, “Receding horizon control: an overview and some extensions…” ... process for a while of sampling time T , then set k : k  and go back to step N.D.Phuoc, Tr.D.Thuan, Receding horizon control: an overview and some extensions ” Review 2.5 Output feedback MPC for. .. of Science and Technology; Academy of Military Science and Technology; * Corresponding author: phuoc.nguyendoan899@gmail.com 20 N.D.Phuoc, Tr.D.Thuan, Receding horizon control: an overview and. .. Bk u k 1 and h k  g (x k 1 )  C k x k 1 N.D.Phuoc, Tr.D.Thuan, Receding horizon control: an overview and some extensions ” Review are all now determined at the current time instant k , because