Partial control controlling a subset of the outputs y for which there is a control objective y 1 - (temporary) uncontrolled outputs, can be controlled at some higher level. y 2 – (locally) measured and controlled output The inputs u is diveded u 2 –inputs used for controlling y 2 u 1 – remaining inputs which may be used for controlling y 1
Applications of partial control Sequential control of decentralized controllers Sequential design of convensional cascade control True partial control Indirect partial control Mes. y 1 Control objec. for y 2 Decent. controlyes cascadeyesno partialnoyes indirectno
Transfer function for y1 and y2 y= Gu y 1 = G 11 u 1 +G 12 u 2 +Gd 1 d y 2 = G 21 u 1 + G 22 u 2 +Gd 2 d Feedback control u 2 = K 2 (r 2 -y 2 ) =>u 2 and y 2 can be eliminated y 1 =(G 11 -G 12 K 2 (I+G 22 K 2 )-1 u 1 + (Gd 1 -G 21 K 2 (I+G 22 K 2 )-1 Gd 2 )d+ (G 12 K 2 (I+G 22 K 2 )-1 r 2
Control objectives for y 2 Fast control of y2 => y2 =r2 By substitution P u is the effect of u 1 on y 2 P d is the partial disturbance gain P r is reference gain
Hierarchical decomposition by sequential design First implement a local lower level control or an inner loop for controlling the output y 2 Objectives: Simpel or on-line tuning of lower-layer controllers Longer sampling intervals for hiegher layers Simpel models for design of higher-layer control systems To stabilize lower layers such that manual control is possible
Selecting u 2 and y 2 The lower layer must quickly implement the setpoints computed by the higher layers control of y 2 using u 2 schould provide local disturbance rejection control of y 2 using u 2 schouldnot impose unnecessary control limitations on the remaining control problem
Distilation column u=[L V D B V T ] T L = reflux V = boilup D= destilate B=bottom flow V T = overhead vapour y= [y d x b M D M B p] T y D = top composition x B = bottom composition M D =condenser holdup M B = reboiler holdup p= pressure 5 X 5 control problem
Stabilizing the distilation column Locally measured and controlled outputs are loops for level and pressure, they interacts weakly y 2 =[M D M B p] T The levels(tanks) has an inlet and two outlets ther are several possible inputs. One can be: u 2 = [D B V T ] T Three SISO loops can be designed
Composition control Teporary uncontrolled inputs y 1 = [y D x B ] T u 1 =[L V ] T Y D can be controlled using L X B can be controlled using V The choise of u 2 made u 1 usefull for controlling y 1
”True” Partial control y 1 is left uncontrolled when control of y 2 indirectly gives acceptable control of y 1. y 1 must be left uncontrolled if the effects of all disturbances on y 1 are less than 1 in magnitude for all frequencies. y 1 must be left uncontrolled if the effects of all reference changes in the controlled outputs (y 2 ) on y 1, are less than 1 in magnitude for all frequencies.
y= Guy 1 = G 11 u 1 +G 12 u 2 +Gd 1 d y 2 = G 21 u 1 + G 22 u 2 +Gd 2 d The effect of a disturbance d k on an uncontrolled output y i is Select the unused input u j such that j’th row in G -1 G d has small elements = keep the input constant if its desired change is small. Select the uncontrolled output y i an unused u j such that the ji’th element in G -1 is large = Keep an output uncontrolled if it is insentive to changes in the unused input with the other outputs controlled.
Simpel distilation column 2 X 2 control problem u=[L V ] T L = reflux V = boilup y= [y d x b ] T y D = top composition x B = bottom composition Disturbances d=[F z F ] F= feed flowrate z F =feed composition
Control problems in the destilation proces Difficult to control the two outputs idenpendently due to strong interactions y 1 is left uncontrolled To improve performance of y 1 (y D composition), u 1 (reflux L)is adjusted by a feedforward from the disturbance – not from y 1 measurements.
Distillation matrices Eksample: At steady state we have The row elements in G -1 G d are similar i magnitude as are the elements in G -1 => The values of P d are quite similar In cases the elements in Pd is much smaller than i Gd => control of one output reduces the effect of the distrubance of the uncontrolled output.
Effect of disturbance d 1 on output y 1 Control y 2 using u 2 with y 1 uncontrolled and worst disturbance d 1 Curve 1:open loop disturbance gain Curve 2: partial disturbance gain Low frequency gains > 1
Reduction of disturbance effect on y1 using feed forward Measure d1 (F) and adjust u1 (L) as a ratio controller curve 3 Unsecure measurements (20%) reduce the effect curve 4 The static FF reacts to fast – a filter is introduced Curve 5 Reduction of the disturbance at all frequences