CHBE507 LECTURE II MPC Revisited. Professor Dae Ryook Yang

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1 CHBE507 LECURE II MPC Revisited Professor Dae Ryook Yang Fall 2013 Dept. of Chemical and Biological Engineering Korea University Korea University III -1

2 Process Models ransfer function models Fixed order and structure Parametric: few parameters to identify Need very high order model for unusual behavior Convolution models Continuous form y() t h( ) u( t) d Discrete form t 0 k y( k) h( i) u( ki) i0 Impulse response Many parameters, but easily obtained from the step or impulse response Korea University III -2

3 Step Response Model From open-loop step test Sampling time: t Step response coefficients: a i Read the values of the unit step response FSR model Finite step response (FSR) y a ( u 1, k 0) k k k Using superposition principle for arbitrary input changes u u u u where u u u y y y y y k 0 1 k i i i 1 k 0 k u k u k u 0 1 k1 y a u a u au 0 k 0 k1 1 1 k1 Korea University III -3

4 After t t, the step response reaches steady state at least 99% y y au y y a u au y y a u a u au y y a u a u a u au y y a u a u a u a u au y y a u a u a u a u a u au n y y au ( a a, i) n 0 i ni i i1 (FSR Model) If there is a delay, the FSR coefficients during the delay will be zero. Korea University III -4

5 Impulse Response Model Impulse response coefficients h a a ( i 1,2,, ) h i i i1 0 0 y y au y a ( u u ) n 0 i ni 0 i ni ni1 i1 i1 y ( au au ) ( a u a u ) ( a u a u ) ( a u a u ) 0 1 n1 1 n2 2 n2 2 n3 n 1 n 0 n 0 n 1 0 y au ( a a ) u ( a a ) u ( a a ) u 0 1 n1 2 1 n2 n n1 1 n n ( 1 0) n1 ( 2 1) n2 ( n n1) 1 y a a u a a u a a u y y hu ( h 0, i) n 0 i ni i i1 (FIR Model) Korea University III -5

6 Horizons Matrix Form of the Predictive Model Model horizon: (number of model coefficients) Control horizon: U (number of control moves) Prediction horizon: V (number of predictions in the future) y1 a u0 y a a 0 0 u y 3 a2 a1 0 u 2 y a a a a u V V V1 V2 VU1 U1 y Au A: Dynamic matrix Korea University III -6

7 From the FIR model n 0 i n i i1 y y hu Single-Step Prediction n1 0 i n 1 i i1 y y hu yn 1 yn hi u (Recursive prediction) n1i i1 Corrected prediction based on the measurement Assume the error between the model prediction and the measurement will present in the future with same magnitude * yn 1 yn 1 yn yn ( yn is the current measurement) y y y y y hu * n1 n1 ( n n) n i n1i i1 Korea University III -7

8 Multi-Step Prediction From the single-step prediction (j-step prediction) y y hu ( j 1,2,, V) y y y y ( y is not available if j>1) n j n j1 i n ji * * i1 n j n j n j1 n j1 n j1 * * n j n j1 i n ji i1 y y hu ( j 1,2,, V) Matrix form when V U Dynamic Matrix, A y a u y P * n1 1 n n 1 * a 2 2 a1 0 0 u n n 1 yn P y 2 * y a 3 2 a1 0 u n n2 yn P 3 * y a n V V av 1 av2 a VU1 u nu1 yn P V Korea University III -8

9 where i P S ( i 1,2,, V) i j1 j l n jl l1 j S hu ( i 1,2,, V) S j : the incremental effect of the past (previously implemented) movements of input on the (n+j)-th future output prediction (where n is current time) P i : the projection which includes future prediction of y based on all previously implemented input changes. P i and S j depend only on past input changes. If the past information is known, then the future input changes will affect the future outputs and the future outputs can be adjusted by carefully selecting the future inputs. Korea University III -9

10 Currently, n is current time and y n is measured. * n1 n i n 1 i= 1 n n i n 1 i 1 n n i n 1 i i1 i2 i2 y y h u h u y h u a u y h u y y hu =( h h) u hu hu y hu * * n2 n1 i n2i 2 1 n 1 n1 i n2i n i n1i i1 i3 i2 * * n3 n3 i n3i i1 y y h u =( h h h) u ( h h) u hu hu y hu hu n 2 1 n1 1 n2 i n2i n i n1i i n2i i4 i2 i3 * * nv nv1 i nv1i= V n V1 n1 VU1 nu1 i1 y y h u a u a u a u y hu hu hu n i nvi i n2i i n1i iv1 i3 i2 a u a u a u y hu V n V1 n1 VU1 nu1 n i n ji j1 ij1 V Depend on only future Depend on only past Korea University III -10

11 Objective Controller Design Method (DMC) Minimize errors between future set points and predictions * rn 1 y n1 * rn2 yn2 E r( Au ynep) AuE * rn V yn V Closed-loop prediction error based only on current and future where rn 1 yn P control action 1 rn 2 yn P2 E Open-loop prediction error based only on past control rn V ynpv action Solution ( ) * 1 A u E 0 u A E Some inverse of A Korea University III -11

12 If U=V and A is invertible, 1 ua E If U<V (A is not invertible), A + :Left pseudoinverse of A 1 u( A A) A E K E Optimization concept min( J EE) min( AuE) ( AuE) J 2 A ( AuE) 2( A AuA E) 0 u 1 u( A A) A E min J ( EWE uwu) 1 2 c It gives no steady-state offset since it has integral action. A + A=I: identity matrix AA + : idempotent matrix (BB=B) J 2 AW1( AuE) 2W2u2(( AWA 1 W2) uawe 1 ) 0 u u( A WAW ) A WE Korea University III -12

13 Adjustable parameters of MPC (uning parameters) Weighting matrices If W 1 >>W 2, the most important objective is to minimize error of the process outputs and inputs will move quite freely. If W 1 <<W 2, the most important objective is to minimize the input movements and controller cares much less the errors. (almost no control) Otherwise, it depends on the relative size of the weighting matrices. If W 1 >W 2, aggressive action will be taken to reduce the error. If W 1 <W 2, conservative action will be taken to reduce the input movements while reduce the error if the action is not too aggressive. he W 2 is called input penalty or input move suppression factor. ypically, use W 1 =I and W 2 =f 2 I and adjust f. If a different weighting for outputs or inputs is required, use diagonal matrix as the weighting matrix. Korea University III -13

14 Horizons Model horizon () Select such that t (open-loop settling time) is typically 20 to 70. Prediction horizon (V) Increasing V results in more conservative control action, a stabilizing effect, and more computational burden. An important tuning parameter Control horizon (U) Suitable first guess is to choose U so that Ut t 60 he larger the value of U is, the more computation time is required. oo large a value of U results in excessive control action Smaller value of U leads to a robust controller that is relatively insensitive to model error. Korea University III -14

15 2x2 case EAuE where E [ E ; E ] A A A General case A A MIMO Extension u[ u ; u ] 1 2 Extend the vectors and matrices in the same manner. If the MPC is formulated in a different form such as statespace model, different form of MIMO extension is more convenient. Korea University III -15

16 Constraints Handling Formulate and solve the MPC in an optimization framework min J ( EWEuWu) subject to 1 2 L U u u u L U y y y and other constraints Solve this optimization problem in QP DMC by DMCC used LP Korea University III -16

17 Identification of Models FSR or FIR models: use step or pulse test Assume operation at steady state Make change in input u (or u) If u is too small, output change may not noticeable If u is too large, linearity may not hold Measure output at regular intervals t he should be chosen so that is 20-70,typically 40. Perform multiple experiments and average them and additional experiments for verification High frequency information may not be accurate for step test. Ideal pulse is hard to implement. t Korea University III -17

18 Least Squares Identification Get the output using PRBS (Pseudo Random Binary Signal) u [ u1 u2 u M ] y [ y1 y2 y M ] Get the FIR model y hu h u h u k 1 k1 2 k2 N kn Minimize the error between measurements dk yk y k y1 u0 u 1 u1 N h1 d1 y 2 u1 u0 u 2 N h 1 d 2 ym um1 um2 umn hn d M d y Uh and output, min min( ) ( ) ( ) h 1 dd y Uh y Uh h U U U y h Korea University III -18

19 Discussions Random input testing, if appropriately designed, gives better models than the step or pulse testing does since it can equally excite low to high frequency dynamics of the process. If U U is singular, the inverse doesn't exist and identification fails. (Need persistent excitation condition) When the number of coefficients is large, U U can be easily singular (or nearly singular). o avoid the numerical, a regularization term is added the the cost function. (ridge regression) min[( ) ( ) ] ( ) h 1 y Uh y Uh h h h U U I U y Korea University III -19

20 Data reatments he data need to be processed before they are used in identification. Spike/Outlier Removal Check plots of data and remove obvious outliers (e.g., that are impossible with respect to surrounding data points). Fill in by interpolation. After modeling, plot of actual vs. predicted output (using measured input and modeling equations) may suggest additional outliers. Remove and redo modeling, if necessary. But don't remove data unless there is a clear justification. Korea University III -20

21 Bias Removal and Normalization Compute the data average and subtract it to create deviation variables, i.e., M y k ( y k yref )/ cy where y ref y 1 i / M i M uk ( uk uref )/ cu where u ref u / i 1 i M Use the given steady-state values of the variables instead to compute the deviation variables, i.e., yk ( yk yss) / cy and uk ( uk uss) / cu where y ss and u ss represent a priori given steady-state values of the process output and input respectively. he input/output data can be biased by the nonzero steady state and also by load disturbance effects. o remove the (timevarying) bias, differencing can be performed for the input/output data. y ( y y ) / c and u ( u u ) / c k k k1 y k k k1 u Identification for yk and uk In all cases, the process data are conditioned by scaling before using in identification. Korea University III -21

22 Prefiltering If the data contain too much frequency components over an undesired range and/or if we want to obtain a model that fits well the data over a certain frequency range, data prefiltering (via digital filters) can be done. Korea University III -22