Typical Process Control System. Lecture 8. ExampleMIMO-system: ADistillation Column. Multivariable transfer functions

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1 Lecture 8 Typical Proce Control Sytem Today lecture: Multivariable ytem... Tranfer function for MIMO-ytem vehicle power network proce control indutry Limitation due to untable multivariable zero Decentralized/decoupled control by pairing of ignal Short warning on integral action in parallel ytem Baed on material from KJ Åtröm and A Rantzer See alo Lecture note and [G&L, Ch. and ExampleMIMO-ytem: ADitillation Column Example: Ditillation column: raw oil inerted at bottom different petro-chemical ubcomponent extracted [ Y () Y 2 () = 5+ e e e e 8 6+ e 4 4+ e 5 }{{} P() U () U 2 () U 3 () Multivariable tranfer function d n r F e Σ C u Σ x P Σ Order matter!! y Output: y = top draw compoition y 2 = ide draw compoition Input: u = top draw flowrate u 2 = ide draw flowrate u 3 = bottom temperature control input X()=PCF R()+P D() PC [N()+X() [I+PCX()=PCF R()+P D() PC N() X()=[I+PC (PCF R()+P D() PC N()) Notice that[i+pc i generally not the ame a[i+cp. Senitivity function for MIMO-ytem Output enitivity function S=(I+PC) Input enitivity function (I+CP) Complementary enitivity function T=(I+PC) PC -minute problem: G?? G?? G?? Find the tranfer function above in the block diagram on the previou lide. (Extra: What are the dimenion?) Someuefulmath relation Notice the following identitie: (i) [I+PC P=P[I+CP (ii) C[I+PC =[I+CP C (iii) T=P[I+CP C=PC[I+PC =[I+PC PC (iv) S+T=I Proof: The firt equality follow by multiplication on both ide with (I+PC) from the left and with(i+cp) from the right. LHS(i):[I+PC[I+PC P[I+CP=I [P+PCP=[I+PCP RHS(i):[I+PCP[I+CP [I+CP=[I+PCP I=[I+PCP Puh through and keep track of order Lecture 8 Limitation due to untable zero Today lecture: Multivariable ytem... Tranfer function for MIMO-ytem Limitation due to untable multivariable zero Decentralized/decoupled control by pairing of ignal Short warning on integral action in parallel ytem For a multivariable ytem with quare tranfer matrixp(), i.e. the ame number of input and output, the zero can be defined a the pole ofp(). The following theorem capture the influence of an untable zero: Theorem LetW S () be table and lets()=[i+p()c() be the enitivity function of a table cloed loop ytem. Then,the pecification W S S i impoible to atify unle W S (z) for every untable zerozofp().

2 Non-minimum phae MIMO Sytem Example [G&L, Ch Conider a feedback ytemy()=(i+pc) R() with the multivariable proce Computing the determinant detp()= [ 2 P()= (+) 2 3 (+2)(+) = + (+) 2 (+2) how that the proce ha an untable zero at=, which will limit the achievable performance. See lecture note for detail of the following lide (checking three different controller) The controller Example controller C ()= [ K (+) K (+) give the diagonal loop tranfer matrix 3K 2(+.5) (+2) 2K 2 (+.5) (+) [ K ( +) (+2) P()C ()= K 2 (+.5)( +) (+)(+2) Hence the ytem i decoupled into to calar loop, each with an untable zero at=that limit the bandwidth. The cloed loop tep repone are hown in Figure. Step repone uing controller Example controller Step Repone The controller C 2 ()= [ K (+) K 2 K (+) K Step Repone Figure: Cloed loop tep repone with decoupling controllerc () for the two outputy (olid) andy 2 (dahed). The upper plot i for a reference tep fory. The lower plot i for a reference tep fory 2. give the diagonal loop tranfer matrix P()C 2 ()= [ K ( +) (+2) K 2 (5+7) (+2)(+) 2K 2 + Now the decoupling i only partial: Outputy 2 i not affected byr. Moreover, there i no untable zero that limit the rate of repone iny 2! The cloed loop tep repone fork =,K 2 = are hown in Figure 2. Step repone uing controller 2 Example controller Step Repone The controller C 3 ()= [ K 2 (+.5) K (+2) 2K K 2 (+.5) (+) Step Repone Figure: Cloed loop tep repone with controllerc 2 () for the two outputy (olid) andy 2 (dahed). The right half plane zero doe not prevent a faty 2 -repone tor 2 but at the price of a imultaneou undeired repone iny. give the diagonal loop tranfer matrix P()C 3 ()= [ K (5+7) (+)(+2) 2K K 2 ( +)(+.5) + (+) 2 (+2) In thi caey i decoupled fromr 2 and can repond arbitrarily fat for high value ofk, at the expene of bad behavior iny 2. Step repone fork =,K 2 = are hown in Figure 3. Step repone uing controller 3 Example ummary.5 Step Repone Step Repone To ummarize, the example how that even though a multivariable untable zero alway give a performance limitation, it i poible to influence where the effect hould how up Figure: Cloed loop tep repone with controllerc 3 () for the two outputy (olid) andy 2 (dahed). The right half plane zero doe not prevent a faty -repone tor but at the price of a imultaneou undeired repone iny 2. 2

3 Lecture 8 Interaction ofsimpleloop Today lecture: Multivariable ytem... y p C u y Proce Tranfer function for MIMO-ytem Limitation due to untable multivariable zero Decentralized/decoupled control by pairing of ignal Short warning on integral action in parallel ytem y p2 u 2 C 2 y 2 Y ()=p ()U ()+p 2U 2() Y 2()=p 2()U ()+p 22U 2(), What happen when the controller are tuned individually? Roenbrock Example There i a nice collection of linear multivariable ytem with intereting propertie. Here i one of them 2 P()= Very benign ubytem (compare with example in [G&L, Ch.). The tranmiion zero are given by detp()= ( ) = +3 (+) 2 (+3) =. Difficult to control the ytem with gain croover frequencie larger than ω c =.5. An Example ControllerC i a PI controller with gaink =,k i =, and the C 2 i a proportional controller with gaink 2 =,.8, and.6. y u The econd controller ha a major impact on the firt loop! Analyi RGA / Britol Relative Gain 2 Y ()= (+) 2U ()+ (+) 2U 2() Y 2 ()= (+) 2U ()+ (+) 2U 2(). P-control of econd loopu 2 ()= k 2 Y 2 () give Y ()= cl ()U k 2 ()= (+) 2 ( k 2 ) U (). The gaink 2 in the econd loop ha a ignificant effect on the dynamic in the firt loop. The tatic gain i cl ()= k 2 +k 2. Notice that the gain decreae with increaingk 2 and become negative fork 2 >. RGA / Britol Relative Gain Conider the firt loopu y when the econd loop i in perfect control (y 2 =) Y ()=p ()U ()+p 2 U 2 () =p 2 ()U ()+p 22 U 2 (). EliminatingU 2 () from the firt equation give Y ()= p ()p 22 () p 2 ()p 2 () U (). p 22 () The ratio of the tatic gain of loop when the econd loop i open and cloed i λ= p ()p 22 () p ()p 22 () p 2 ()p 2 (). Parameter λ i called Britol interaction index A imple way of meauring interaction baed on tatic propertie Edgar H. Britol, "On a new meaure of interaction for multivariable proce control", [IEEE TAC (967) pp Idea: What i effect of control of one loop on the teady tate gain of another loop? Conider one loop when the other loop i under perfect control Y ()=p ()U ()+p 2 U 2 () =p 2 ()U ()+p 22 U 2 (). Many Loop Aumeninput andnoutput. Pick an input output pair and relabel o that the input iy, let the remaining output be y 2 =. Let the input beu 2 and the remaining input beu. y =p u +p 2 u 2 =p 2 u +p 22 u 2 Solving fory give y =(p 2 p p 2 p p 2 22)u 2, r 2 = (p 2 p p 2 p 22) Compare P= p p 2, P = p 2 p 22 (p 2 p p 2 p 22) The relative gain array ir=p. P T 3

4 Britol Relative Gain Array (RGA) LetP() be ann n matrix of tranfer function. The relative gain array i Λ=P(). P T () The product. i element-by-element product (Schur or Hadamard product, ame notation in matlab). Propertie (A. B) T =A T. B T P diagonal or triangular give Λ=I Not effected by diagonal caling Inight and ue A meaure of tatic interaction for quare ytem which tell how the gain in one loop i influenced by perfect feedback on all other loop Dimenion free. Row and column um are. Negative element correpond to ign reveral due to feedback of other loop Pairing... Pairing When deigning complex ytem loop by loop we mut decide what meaurement hould be ued a input for each controller. Thi i called the pairing problem. The choice can be governed by phyic but the relative gain can alo be ued Conider the previou example P()= 2, P ()= 2 Λ=P(). P T ()= 2, 2 Negative ign indicate the ign reveral found previouly Better to ue revere pairing, i.e. letu 2 controly Step Repone with Revere Pairing Conider P()= 2 (+) 2 (+) 2 (+) 2 (+) 2 Introducing the feedbacku = k 2 y 2 give Y ()= 2 cl ()U k 2 2()= (+) 2 ( k 2 ) U 2(), Zero frequency gain decreae from 2 to whenk 2 range from to. Dicu how dynamic change withk 2! Ue rootlocu! y u ( U 2 = + ) (Y p Y ) u = k 2 y 2 withk 2 =,.8, and.6. Summary for2 2 Sytem (RGA) λ=no interaction λ=cloed loop gainu y i zero. Pairu andy 2 intead <λ<cloed loop gainu y i larger than open loop gain. Interaction tronget for λ= λ>cloed loop gainu y i maller than open loop gain. Interaction increae with increaing λ. Very difficult to control both loop independently if λ i very large. λ<the cloed loop gainu y ha different ign than the open loop gain. Opening or cloing the econd loop ha dramatic effect. The loop are counteracting each other. Such pairing hould be avoided for decentralized control and the loop hould be controlled jointly a a multivariable ytem. Extra: Singular Value LetA be ank n matrix whoe element are complex variable. The ingular value decompotion of the matrix i A=U ΣV where denote tranpoe and complex conjugation,u andv are unitary matrice (UU =IandVV =Ii. The matrix Σ i ak n matrix uch that Σ ii = σ i and all other element are zero. The element σ i are called ingular value. The larget σ=max i σ i and mallet σ=min i σ ingular value are of particular interet. The number σ/σ i called the condition number. The ingular value are the quare root of the eigenvalue ofa A. Example: A real2 2 matrix can be written a A= coθ inθ σ coθ 2 inθ 2 inθ coθ σ 2 inθ 2 coθ 2 Extra: Singular DecompoitionA=U ΣV Extra: Interaction Analyi The columnu i ofu repreent the output direction The columnv i ofv repreent the input direction We haveav=u Σ, orav i = σ i u i. An input in the direction v i thu give the output σ i u i, i.e. in the directionu i Since the vectoru i andv i are of unit length the gain ofa for the inputu i i σ i The larget gain i σ=max i σ i There are efficient numerical algorithm vd in Matlab Singular value can be applied to nonquare matrice A natural way to define gain for matricea and tranfer function matriceg() Av gain=max v v = σ(a), gain=max σ(g(iω)) ω Conider a ytem with the caled zero frequency gain y u y 2 = u y 3 Relative gain array Λ= Singular value: σ =.683, σ 2 =.434 and σ 3 =.97. Condition number κ=66. Only two output can be controlled in practice. What variable hould be choen? u 3 4

5 Extra: Interaction Analyi We havey=usv T. How to pick two input output pair SV T = U= The matrixsv T how thatu andu 2 are obviou choice of input. A far a the output are concerned. We have two choicey,y 3 or y 2,y 3 (angle between row). Notice thaty,y 2 i not a good choice becaue the correponding row ofus are almot parallel. The ingular value are Selectiony,y 3 u,u 2 Condition number κ=.5 dition number κ=.45 Selectiony 2,y 3 u,u 2 Con Λ= Λ= The Quadruple Tank γ γ Tank 3 (A) Tank 4 (B) γ 2 y 3 y 4 Tank Tank 2 Pump (BP) (A2) y (B2) y 2 Pump 2 (AP) u u 2 γ 2 Interaction Can bebeneficial P()= p () p 2 () = p 2 () p 22 () The relative gain array Tranmiion zero (+)(+2) 6 (+)(+2) R=, (+)(+2) 2. (+)(+2) detp()= ( )( 2)+6 (+) 2 (+2) 2 = (+) 2 (+2) 2 Difficult to control individual loop fat becaue of the zero at =. Since there are no multivariable zero in the RHP the multivariable ytem can eaily be controlled fat but thi ytem i not robut to loop break. Tranfer Function oflinearized Model Tranfer function fromu,u 2 toy,y 2 γ c ( γ 2 )c +T (+T )(+T 3 ) P()= ( γ )c 2 γ 2 c 2 (+T 2 )(+T 4 ) +T 2 Tranmiion zero (+T 3 )(+T 4 ) ( γ )( γ 2 ) γ detp()= γ 2 (+T )(+T 2 )(+T 3 )(+T 4 ) No interaction of γ = γ 2 = Minimum phae if γ + γ 2 2 Nonminimum phae if< γ + γ 2. Intuition? Relative Gain Array Zero frequency gain matrix P()= γ c ( γ 2 )c ( γ )c 2 γ 2 c 2 The relative gain array P()= λ λ where λ= γ γ 2 γ + γ 2 No interaction for γ = γ 2 = Severe interaction if γ + γ 2 < v C D w D 2 λ λ FindD andd 2 o that the controller ee a diagonal plant : D 2 PD = Then we can ue a "decentralized" controllercwith ame block-diagonal tructure. u P y Decoupling Simple idea: Find a compenator o that the ytem appear to be without coupling ("block-diagonal tranfer function matrix"). Many verion here we will conider Input decouplingq=pd Output decouplingq=d2 P both Q=D2 PD but many different method including Conventional (Feedforward) Invere (Feedback) Static Important to conider windup, manual control and mode witche. Keep the decentralized philoophy Tranfer function The Quadruple Tank γ c +T ( γ 2 )c (+T )(+T 3 ) P()= ( γ )c 2 γ 2 c 2 (+T 2 )(+T 4 ) +T 2 Relative gain array R=P(). P() = λ λ λ λ where λ= γ γ 2 γ + γ 2 Recall RHP zero if γ + γ 2 <. Phyical interpretation! 5

6 Decoupling - FlightControl Lecture 8 Longitudinal Lateral Today lecture: Multivariable ytem... May be good to decouple interaction to output, but you hould alo be careful not to wate control action to trange decoupling!! Tranfer function for MIMO-ytem Limitation due to untable multivariable zero Decentralized/decoupled control by pairing of ignal Short warning on integral action in parallel ytem Sytem with Parallel Actuation Gearbox C A w ω p C 2 A 2 Motor drive for papermachine and rolling mill Train with everal motor or everal coupled train Power ytem J dω dt +Dω=M +M 2 M L, Proportional control M =M +K (ω p ω) M 2 =M 2 +K 2 (ω p ω) A Prototype Example C A w p The proportional gain tell how the load i ditributed C 2 A 2 Gearbox J dω dt +(D+K +K 2 )ω=m +M 2 M L +(K +K 2 )ω p. A firt order ytem with time contantt=j/(d+k +K 2 ) Dicu repone peed, damping and teady tate ω= ω = K +K 2 D+K +K 2 ω p + M +M 2 M L D+K +K 2. ω Integral Action? Power Sytem - Maive Parallellim What if we intead ue two PI-controller? WARNING!!! C A w p Prototype for lack of controllability and obervability! C 2 A 2 Gearbox ω Edion experience Two generator with governor having integral action Many generator upply power to the net. Frequency control Voltage control Iochronou governor (integral action) and governor with peed-drop (no integral action) Summary All real ytem are coupled Multivariable zero - limitation Never forget proce redeign Relative gain array and ingular value give inight Why decouple Simple ytem. SISO deign, tuning and operation can be ued What i lot? Parallel ytem One integrator only! Next lecture: Multivariable deign LQ/LQG 6