Modeling, Identification and Contol, Vol., No., 1, pp. 5, ISSN 189 18 Discete LQ optimal contol with integal action: A simple contolle on incemental fom fo MIMO systems David Di Ruscio Telemak Univesity College, Faculty of Technology, N-918 Posgunn, Noway. Fax: +7 5 57 5 1, Tel: +7 5 57 51 68, E-mail: David.Di.Ruscio@hit.no Abstact A simple Linea Quadatic (LQ) optimal contolle of velocity (incemental) fom with appoximately the same popeties as a conventional PID contolle of velocity fom is pesented, i.e. integal action. The poposed optimal contolle is insensitive to slowly vaying system and measuement tends and has the ability of stabilizing any linea dynamic system unde weak assumptions such as the stabilizability of the system and the detectability of the system seen fom the pefomance index. Keywods: MIMO systems, optimal contolle, integal action, PI contolle, Kalman filte, system identification 1. Intoduction The famous Linea Quadatic (LQ) optimal contolle fo linea Multiple Input and Multiple Output (MIMO) systems (se. e.g. Kwakenaak and Sivan (197) and Andeson and Mooe (1989)), has some emakable popeties due to the guaanteed nominal stability of the closed loop contolled system (unde weak conditions such as the stabilizability of the system and the detectability of the system seen fom the objective). On the othe hand, this LQ optimal contolle has not attained the position it deseves. One eason fo this is pobably that it has been difficult to compae the LQ optimal contolle with a standad PID contolle which has eceived a geat deal of attention owing to its simplicity and its pactical applications. In this pape we will show how we may use the standad LQ optimal contolle to design stabilizing contolles fo MIMO systems with appoximately the same popeties that a PI contolle has on velocity (incemental) fom. The poposed contolle is emakably simple and it has almost the same stuctue and popeties as a standad PID contolle, i.e., the contolle has integal action. The poposed contolle also has the popeties of stabilizing any detectable and stabilizable linea MIMO system. Hence, the esulting contolle may be used as a fist choice contolle fo contolling a linea system. The main contibutions of this pape ae itemized as follows: A LQ optimal contolle fo discete time systems with appoximately the same popeties as a standad PID contolle is poposed. The poposed LQ optimal contolle may be used fo finding stabilizing contolles with integal action fo complex MIMO systems. The poposed LQ optimal contolle is insensitive to constant o slowly vaying pocess and measuement noise. The poposed contolle is suitable fo contolling doi:1.17/mic.1..1 c 1 Nowegian Society of Automatic Contol
Modeling, Identification and Contol non-linea systems when a linea state space model is available. This linea model may be the esult of lineaizing a non-linea model o the esult of system identification. The poposed LQ optimal contolle is illustated on fou non-linea pocess models, e.g. the quaduple tank pocess Johansson (). The poposed LQ optimal contolle is in this pape also illustated with pactical expeiments on a quaduple tank laboatoy pocess. System identification is used to identify a state space model on innovations fom (The Kalman filte) and used in the design of the poposed LQ contolle. This stategy may be viewed as a model fee LQ optimal contolle stategy because only pocess data ae used. The main diffeences of the poposed MIMO PI LQ optimal contolle and a conventional PI contolle is that the optimal contolle consists of both output and state feedback, while a conventional PI contolle consists of output feedback and is suitable only fo decentalized Single Input Single Output (SISO) systems. Howeve, the same stategy as used in this pape in ode to develop the poposed simple MIMO LQ optimal PI contolle will be used to fomulate a simple Model Pedictive Contolle (MPC) with integal action in a upcoming pape. See e.g. Maciejowski () fo the MPC contolle. The est of the pape is oganized as follows. In Section the optimal contol poblem is defined. In Section the poblem solution and the poposed LQ optimal contolle with integal action fo MIMO systems ae pesented. In Section 5 the optimal contolle is compaed with the conventional PI contolle and the main diffeences and similaities ae pointed out. In Section 6 the poposed LQ optimal contolle with integal action is demonstated on the poblem of contolling some systems descibed with non-linea models, e.g. the quaduple tank pocess Johansson () as well as thee othe examples. In Section 7 the poposed LQ optimal contolle with integal action is illustated in a pactical expeiment on the quaduple tank pocess. The fist pinciples model is also compaed with system identification based models. Some conclusions follow in Section 8. In Appendix A a MAT- LAB m-file scipt is povided fo the easy application of the poposed method fo LQ optimal contolle with integal action.. Poblem fomulation Given a pocess model x k+1 Ax k + Bu k + v, (1) y k Dx k + w, () whee x k R n is the state vecto, u k R is the contol input vecto, y k R m is the output (measuement) vecto, and A, B and D ae known system matices of appopiate dimensions. The distubances v and w ae both unknown, i.e., v is an unknown constant o a slowly vaying pocess distubance, and w is an unknown constant o a slowly vaying measuement noise vecto. Note that the vaiables u k and y k in the model Eqs. (1) and () ae the actual input and output vaiable, espectively. Futhemoe, note that the model Eqs. (1) and () may aise fom lineaizing non-linea models aound some nominal steady state and input vaiables, o fom system identification based on tended vaiables. Hence, in these cases, the extenal noise vaiables v and w ae known, but the esulting contol algoithm to be pesented in this pape is insensitive to these noise vaiables. Futhemoe the system and the measuements may be influenced by difts and in these cases the noise vaiables v and w may be unknown and slowly vaying. Hence, the model Eqs. (1) and () is a ealistic model. We will study the LQ optimal contolle when it is subjected to the following scala pefomance index, + 1 J i 1 xt N Sx N N 1 ki ((y k ) T Q(y k ) + u T k P u k), () whee u k u k u k 1 is the contol incement (deviation) and is a efeence signal and S, Q and P ae symmetic positive semi-definite weighting matices of appopiate dimensions, i is the initial time and often i fo simplicity of notation. The efeence vecto may be a time vaiant but fo easons of simplicity of the poblem solution, we put k. The efeence is teated as constant o slowly vaying in the design phase of the LQ optimal contolle with integal action fo MIMO systems. Fo lage o infinite pediction hoizons N o when S is chosen as the solution to the Riccati equation of the poblem, then Eq. () is equivalent to using the index J i 1 ((y k ) T Q(y k ) + u T k P u k ). () ki 6
. Poblem solution Di Ruscio, Discete LQ optimal contol with integal action In ode to solve the LQ optimal contol poblem we need a model which is independent of the unknown distubances v and w in Eqs. (1) and (). Fo the sake of geneality we will focus on state space modeling. Fom the state Equation (1) we have x k+1 A x k + B u k, (5) whee x k x k x k 1. Fom the measuement equation () we have y k y k 1 + D x k. (6) (15) is ob- Augmenting (5) with (6) gives the state space model and whee the feedback matix G in eq. tained as xk+1 A n m xk B + u y k D I m m y k, (7) k 1 m y k x D I k m m. (8) y k 1 The pefomance index () with and the state space model (7) and (8) define a standad LQ contol poblem. If is a non-zeo constant efeence then the measuements equation (8) can be witten as y k y k 1 + D x k. (9) The state and output equations (7) and (8) can then be ewitten as x k+1 { }}{ xk+1 y k à x k {}} {{ }}{ A n m xk D I m m y k 1 + B { }} { B u k, (1) m Hence, the state space model given by Eqs. (1) and (1) with the pefomance index given by Eq. (1) define a standad LQ optimal contol poblem. We hee assume P >, the pai (Ã, B) is stabilizable and that the pai (C, Ã) is detectable whee C is the squae oot matix of Q such that Q C T C, in ode fo an optimal solution to exist. The solution to the LQ optimal contol poblem, i.e. minimizing the pefomance index (1) with espect to the contol deviation u k subject to the state Eq. (1), is given by the state feedback u k G x k, (15) G (P + B T 1 R B) BT RÃ, (16) whee R is the positive solution to the discete time algebaic Riccati equation R Q + ÃT Rà ÃT R B(P + B T R B) 1 BT Rà Q + G T P G + (à + BG) T R(à + BG), (17) whee the last fomulation of the Riccati equation is known as the Joseph s stable vesion which ensues symmety of the solution R. The solution to the LQ optimal contol poblem, Eqs.(16) and (17) is well known in the liteatue, see e.g. Andeson and Mooe (1989) p. 5 o Lemma 11..1 in Södestöm (199) p. 91. Now fom eq. (15) we find the following contolle on incemental (velocity) fom ỹ k {}}{ y k x k D {{ }}{ }}{ xk D Im m y k 1. (11) The state space model (1) and (11) with the pefomance index () define a standad LQ optimal contol poblem. Hence, we have a stictly pope state space model of the fom x k+1 à x k + B u k, (1) Note that the index () yields J i 1 1 ỹ k D x k (1) (ỹk T Qỹ k + u T k P u k ) ki ( x T Q x k k + u T k P u k ), (1) ki whee the weighting matix is Q D T Q D. G {}}{ {}} { xk u k G1 G, (18) y k 1 which can be ewitten as u k u k 1 + u k o as u k u k 1 + G 1 x k + G (y k 1 k ), (19) whee we ae putting k in eq. (18) to obtain the poposed contolle eq. (19). The esulting contolle eq. (19) has an appealing stuctue vey simila to a PI contolle on velocity fom. See Sec. 5 fo compaison. Possible constaints ae handled as with conventional PI contolles on velocity (incemental) fom, e.g. as in Åstöm and Hägglund (1995) p.8. Notice also that it is simple to limit the ate of change u k of the contol signal, and the contol signal u k, using the poposed LQ contolle in eqs. (18) and (19). A MATLAB m-file scipt fo computing the LQ optimal feedback matices G 1 and G with the model matices A, B, D and the weighting matices Q and P as aguments is povided in Appendix A. x k 7
Modeling, Identification and Contol The weighting matices Q and P > ae usually chosen by some tial and eo pocedue fo acceptable esponses and pefomance. The weighting matices may often be chosen as simple diagonal matices, e.g. as P I and Q qi whee I is the identity matix and q > a tuning paamete. See Sec. 6 fo illustating examples. The LQ optimal contolle (18) gives a zeo steady state eo, i.e. y in steady state, since the closed loop system is stable owing to the popeties of the LQ optimal contolle (assuming the contol vaiables ae not satuated, i.e. the contol vaiables ae within allowed bounds). Notice that it is possible to use Q R (n+m) (n+m) in Eq. (1) diectly as a weighting matix in ode to incease the degee of feedom in tuning the LQ optimal contolle feedback matices in Eq. (19). But owing to easons of simplicity, we popose the stategy as pesented.. State obseve fo the state deviation The states ae seldom measued in pactice. In this case we can use a state obseve o Kalman filte, Jazwinski (1989), Södestöm (199), to define the deviation state x k. Howeve, anothe solution is to define x k in tems of some past and known outputs..., y k 1, y k and some known inputs..., u k 1 and the model matices A, B and D. A common solution to the poblem of estimating the state, x k, in a model of the fom (1) and () in which the noise is coloed, is to include a andom walk (integato) in ode to estimate the non-zeo pat of the noise, v k in addition to the state estimate x k. This is necessay in ode fo the innovations to become white. One can theeafte fom the state deviation x k x k x k 1 which is needed in the contol algoithm. Howeve, anothe moe simple solution in this situation is to design an obseve fo the deviation model (5) and (6). This gives a state obseve fo x k of the fom x k+1 A x k + B u k + K(y k y k 1 D x k ), () whee the initial estimate x should be specified. A natual choice is x. System identification, e.g. the subspace system identification method DSR in Di Ruscio (9) may also be used to diectly identify the model matices A, B, D and the Kalman filte gain K needed in the obseve eq. (). The DSR identified model matices A, B and D may then also be used to develop the poposed LQ contolle eq. (19). This model fee LQ optimal contolle stategy is implemented on a pactical laboatoy pocess and illustated with expeimental esults in Sec. 7. Using a state obseve in connection with the optimal contolle eq. (19) leads to a Linea Quadatic Gausian (LQG) contolle, (see e.g. Ch. 11 in Södestöm (199)). We ae awae of the possible obustness poblems with LQG contolles as demonstated in, Doyle (1978). Howeve, this possible poblem is also involved in the common and widely used MPC contolles. 5. Connection with the PI contolle In this section we compae the stuctue of the poposed LQ contolle eq. (19) with a PI contolle on velocity (incemental) fom. A conventional PI contolle can be witten as 1 + s u K p ( y) K p ( y) + K p 1 ( y). (1) s s Defining the PI contolle state z, as z 1 ( y). () s Hence, the PI contolle can in continuous time be witten as ż y, () u K p ( y) + K p z. () A discete fomulation of the PI contolle is then z k+1 z k h( y k ), (5) u k K p ( y k ) + K p z k, (6) whee h is the sampling inteval. A deviation fomulation of the PI contolle is then found as follows u k u k 1 K p ( y k ) + Kp z k (K p ( y k 1 ) + Kp z k 1 ) K p (y k y k 1 ) + Kp (z k z k 1 ). (7) Fom (5) we have that z k z k 1 h( y k 1 ). Substituting this into (7) gives u k u k 1 + G 1 (y k y k 1 ) + G (y k 1 ). (8) whee G 1 K p, G K p h. (9) 8
Di Ruscio, Discete LQ optimal contol with integal action Futhemoe, using that y k Dx k + w gives u k u k 1 + G 1 D x k + G (y k 1 ). () The above discussion shows that the PI contolle is exactly of the same stuctue as the LQ optimal contolle (19). The diffeence is that the optimal contolle takes feedback fom the deviation state vecto x k x k x k 1 while the PI-contolle only uses feedback fom the output deviation y k D x k. 6. Numeical examples Example 6.1 (Quaduple tank pocess) Conside the quaduple tank pocess, Johansson (), with the non-linea state space model deived fom mass balances and Benoulli s/toicelli s law. By equating the potential enegy and kinetic enegy, i.e. mgh 1 mv and solving fo the velocity we obtain v gh. Multiplying with the aea, a, of the outlet hole of the tank we obtain the volumetic flowate, q, out of the tank as q av a gh. Hence, a mass balance of the fou tank pocess gives the state space model A 1 ẋ 1 a 1 gx1 + a gx + γ 1 k 1 u 1, (1) A ẋ a gx + a gx + γ k u, () A ẋ a gx + (1 γ )k u, () A ẋ a gx + (1 γ 1 )k 1 u 1, () whee A i i 1,..., is the coss-section aea of tank i, a i i 1,..., is the coss-section aea of the outlet pipe of tank i. The flow k 1 u 1 fom pump 1 may be divided into a flow γ 1 k 1 u 1 into tank 1 and a flow (1 γ 1 )k 1 u 1 to tank, i.e. such that the flow fom pump numbe 1 is k 1 u 1 γ 1 k 1 u 1 + (1 γ 1 )k 1 u 1. Similaly, the flow k u fom the second pump may be divided into a flow γ k u into tank and a flow (1 γ )k u into tank. Hee γ 1 and γ ae fixed paametes. The system is non-minimum phase when choosing these paametes such that, < γ 1 +γ < 1, and the system is minimum phase when, 1 < γ 1 + γ <. The numeical values fo the above paametes, as well as nominal values fo the states and contol inputs, ae chosen as pesented in Johansson (). The tank pocess is studied in a numbe of papes, see e.g. Gatzke et al. () whee Intenal Model Contol (IMC) and Dynamic Matix Contol (DMC) wee used to contol the tank pocess. Hee we use the poposed LQ optimal contolle with integal action as pesented in Sec.. The esults afte using the LQ optimal contolle, eq. (19), in ode to contol the non-linea model eqs. (1)- () ae pesented in Figues 1 and. The weighting matices wee chosen simply as P I and Q I fo both the minimum and non-minimum phase cases. 5.5 5.5.5 u 1.5 1 7.5 7 6.5 : Level tank 1 6 1 Time s.5.5 u.5 1 6.9 6.8 6.7 6.6 6.5 6. y : Level tank 1 Time s Figue 1: Simulation of the quaduple tank pocess and the minimum phase case in Example 6.1 with LQ optimal contol with integal action. 5.5.5.5 7.6 7. 7. u 1 1 7 6.8 6.6 6. : Level tank 1 6. 1 Time s 7 6 5 7.5 6.5 u 1 8 7 y : Level tank 6 1 Time s Figue : Simulation of the quaduple tank pocess and the non-minimum phase case in Example 6.1 with LQ optimal contol with integal action. Example 6. (Isothemal chemical eacto) A chemical isothemal eacto with a eaction sa k B, which can be modeled as ẋ 1 u 1 V (u x 1 ) skx 1, (5) ẋ u 1 V x + kx 1, (6) whee V 1, k 1 and s. u 1 is the flow ate, u the feed concentation, V the volume and k a eaction velocity constant. The states x 1 and x ae the mola compositions of the substances A and B espectively. The steady state contol vaiables u s 1 1 and u s 1 give the steady states x s 1.851 and x s.79. Lineaizing aound the steady state gives the linea model ẋ A c x + B c u, (7) y y 9
Modeling, Identification and Contol whee x x x s and u u u s and A c f1 us 1 V skxs 1 kx 1 us 1 V B c u s u s 1 V V xs V f 1 x 1 x f f x 1 x f f1 u 1 f 1 u f u 1 u x s,u s 1.16 1.78 1. x s,u s 1. 1..79. (9) A discete time model is obtained by using a zeo ode hold on the input and a sampling inteval h.1, i.e., (8), 1.5 1. 1. 1. 1.1.856.855.85.85.85.851 u 1 1 1.85 1 Time s 1. 1.998.996.99.99.99.988 1.7.7.71 u.69 1 Time s Figue : Simulation of the chemical eacto in Example 6. with LQ-optimal contol. Example 6. (Van de Vusse chemical eacto).7 y whee x k+1 Ax k + Bu k + v, () A e Ach B A 1 c (e Ach I)B c D 1 1 Choosing a LQ citeion J i 1 with P y k Dx k, (1).87.15.98.9.96.6.8, (), (), v x s Ax s Bu s. () N ((y k ) T Q(y k ) + u T k P u k ), (5) ki 1 1 gives the LQ optimal contol whee 5, Q 1, (6) u k u k 1 + G 1 x k + G (y k 1 ), (7) G 1 G 15.75 55.7 1.971 6.588.769 6.19.69.75, (8). (9) Simulation esults afte changes in the efeence signal ae illustated in Figue. A chemical isothemal eacto (Van de Vusse) is studied in this example. The elationship fom the feed flow ate u into the eacto to the concentation of the poduct y at the outlet of the eacto is modeled by the following non-linea state space model. x 1 k 1 x 1 k x 1 + (v x 1 )u, (5) x k 1 x 1 k x x u, (51) y x, (5) whee the eaction ate coefficients ae given by k 1 5, k 1, k 1. The concentation of the bypoduct into the eacto, v, is teated as an unknown constant o slowly vaying distubance with nominal value v s 1. Choosing a steady state contol u s 5 gives the steady states x s 1.5 and y s x s 1. A lineaized model aound the steady state is given by ẋ A c x + B c u + C c v, (5) whee x x x s, u u u s and v v v s, and A c f1 x 1 f 1 x f f x 1 x k1 k x s 1 us k 1 k u B c f1 u f u v s x s 1 x s x s,v s x s,u s 7.5 1 15 5 15 (5),. (55) Notice that C c is computed simila as B c but is not needed. A discete time model is obtained by using a
u Di Ruscio, Discete LQ optimal contol with integal action zeo ode hold on the input and a sampling inteval h., i.e., whee x k+1 Ax k + Bu k + v, (56) A e Ach y k Dx k, (57).7788.779.7788 B A 1 c (e Ach I)B c.1.11, (58), (59) D 1, v : x s Ax s Bu s + C(v v s ). (6) Notice that C is computed simila as B but is not needed because the LQ optimal contolle is independent of the constant distubance v in the state Eq. (56) (assuming a constant o slowly vaying distubance in the eacto). Choosing a LQ citeion with J i 1 (Q(y k ) + P u k), (61) ki gives the LQ optimal contol whee P 1, Q 5, (6) u k u k 1 + G 1 x k + G (y k 1 ), (6) G 1.61 8.5791, G.581. (6) Simulation esults afte changes in the efeence signal ae illustated in Figue. Example 6. (Distillation column) One advantage of the pesented LQ optimal contol is that the contol is designed in discete time. Continuous pocesses with slow dominant dynamics ae often contolled with a digital/discete contolle. If the sampling time is lage then a continuous time contolle design may give poo esults when used as a discete contolle. We will hee illustate the simple discete time LQ optimal contol design fo a distillation column. Conside a distillation column with eight tays and a elative volatility α.99. Let the contol vaiable u 1 R be the eflux to the column and u V be the flow ate of vapo in the column. The composition of the top poduct x 8 x D and the composition of the bottom poduct x 1 x B ae teated as measued output vaiables. The feed flow ate F and the composition x F of the light poduct in F ae both teated as unknown constant o slowly vaying distubances. The continuous non-linea model with n 8 states is fist lineaized aound the steady state opeating point R s, V s.5, F s 1 and x s F.5. This gives a continuous time linea model of the fom x A c x + B c u + C c v, (65) y D x. (66) This model is then discetized with a sample inteval of h 5 min. This gives a discete time model of the fom Choosing a LQ citeion J i 1 x k+1 Ax k + Bu k + v, (67) y k Dx k + w. (68) N ((y k ) T Q(y k ) + u T k P u k ), (69) ki 1 Chemical eacto esponses: LQ optimal contol.5.1.15..5..5 with P 1 1 1, Q 5 1, (7) x 1.5 1.5 1. 1.5.1.15..5..5 gives the LQ optimal contol u k u k 1 + G 1 x k + G (y k 1 ), (71) yx 1. 1.8 y k k whee.5.1.15..5..5 Continuous time: t. Figue : Simulation of the chemical eacto in Example 6. with LQ-optimal contol. 1.899.9.961.187 G 1.. 1.7.559 1.5158.99 6.9 5.8.669.99.1887.67, (7) 1
Modeling, Identification and Contol G 1.8 9.5.666 17.189. (7) The linea contolle (71)-(7) on the deviation fom is used in this example to contol the non-linea distillation column model with eight states. If the state vecto is not available, then we may use a state obseve o compute an expession fo x k fom some past inputs and outputs, e.g. as in the Extended Model Pedictive Contol (EMPC) algoithm, Di Ruscio and Foss (1998). Simulation esults afte changes in the efeence signal ae illustated in Figue 5. x 1 x B u 1 R.5.5.8.6... Bottom composition x 1 and efeence 1.8 5 1 15 5.8.6.. x 1 1 Column eflux u 1 5 1 15 5 Time min x 8 x D u V.975.97.965.96 Top composition x 8 and efeence.955 5 1 15 5..8.6 Column steam flow u. 5 1 15 5 Time min Figue 5: Simulation of the distillation column in Example 6. with LQ optimal contol. 7. Expeimental esults on a quaduple tank pocess The esults fom pactical expeiments on a quaduple tank laboatoy pocess will be pesented in this section. The sampling ate in all expeiments is one second. We stated with empty tanks in all expeiments. Hence, this may be viewed as a test fo obustness fo unknown non-lineaities when using the poposed LQ contolle. The quaduple tank pocess setup esults in a non-minimum phase behavio. The expeiments ae descibed in the following items. 1. An open loop input expeiment is designed as illustated in Fig (6) and the coesponding outputs, i.e. the levels in the two lowe tanks, also illustated in Fig (7).. The input and output data ae collected into data matices U R N, and Y R N whee the numbe of samples is N 559. The fist x 8 N ID fist samples wee used fo identification. Hence, the last 159 samples may be used fo validation of the identified state space models. The data wee also centeed befoe use in the identification methods.. A Fist Pinciples (FP) model, vey simila to the one pesented in Example 6.1, wee fitted to the pocess as well as believed possible. Using the input expeiment as illustated in Fig. (6) gave the simulated outputs as illustated in Fig. (7). The Pediction Eo (PE) citeion evaluated fo the validation data was V fp 7.57.. The MATLAB IDENT Toolbox system identification function pem.m whee used to identify a n ode state space model. The simulated outputs ae illustated in Fig. (7). The PE citeion evaluated fo the validation data was V PEM.8. 5. The subspace system identification method, Di Ruscio (9) was used. The best DSR model with n states wee found with paametes L and J 9. The simulated outputs ae illustated in Fig. (7). The PE citeion evaluated fo the validation data was V DSR.7. 6. Two SISO PI contolles wee tuned by using the model based tuning method in Di Ruscio (1). The model used was the DSR model. The expeimental esults using this decentalized contol stategy ae illustated in Figs. (8) and (9). 7. The LQ optimal contol stategy eq. (19) was implemented. The Kalman filte identified by the DSR method was used to identify the pesent state deviation x k x k x k 1 needed in the contolle. The expeimental esults using this LQ optimal contolle with integal action stategy ae illustated in Figs. (8) and (9). The conclusions dawn fom this expeimental esults ae discussed in the following. Inteestingly the identified state space models, both fom PEM and DSR, fit the eal data bette than the FP model. Hee the simulated output, i.e. the behavio fom the input u, to the output y, is used in ode to calculate the PE citeion. The esults using the FP model, the PEM model and the DSR model ae V fp 7.57, V PEM.8 and V DSR.7, espectively. Inteestingly the DSR model fit the validation data slightly bette than the PEM model. Based on this conclusion we ae using the identified DSR model fo both tuning the PI contolles and fo use in the LQ optimal contolle with integal action stategy eq. (19). The deteministic pat of the
Di Ruscio, Discete LQ optimal contol with integal action model, i.e. x k+1 Ax k + Bu k and y k Dx k, was used to tune the PI contolle stategy (by fist using the RGA paiing stategy, Bistol (1966), Skogestad and Postlethwaite (1996)), as well as fo the calculation of the feedback matices G 1 and G. Futhemoe the DSR identified Kalman filte gain matix K was used in the Kalman filte on deviation fom eq. (), fo estimating the deviation states x k needed in eq. (19). As we see fom Figs. (8) and (9) the LQ stategy woks vey well compaed to the PI contolle stategy. This is justified by compaing the Integated Absolute (IAE) indices. The DSR model with the LQ optimal contolle in Eq. (19) gave IAE indices 1.689 and 1.9 fo level one and two, espectively, and fo the PI contolles.7 and.511 fo level one and two, espectively. It is also woth mentioning that it is vey difficult to tune PI contolles fo this pocess due to the non-minimum phase behavio of the pocess..... 5 1 15 5 5 5 5.... 5 1 15 5 5 5 5 Samples u 1 V u V Figue 6: Open loop system identification input expeiment, i.e. the volt input to the pumps. 8. Concluding emaks A simple LQ optimal contolle with integal action on velocity (incemental) fom fo MIMO systems is poposed. The poposed LQ contolle is demonstated to wok well on fou simulation examples. Futhemoe, pactical expeiments show that the poposed LQ contolle woks well on a quaduple tank laboatoy pocess. Acknowledgment The autho acknowledges M Danuskha Dodampe Gamag who did most of the pactical expeiments in Sec. 7 of this pape. 1 5 5 1 vs samples eal ds pem fp 15 1 5 6 1 5 5 1 h vs samples h eal h ds h pem h fp 15 1 5 6 Samples Figue 7: This figue illustates the eal measuements of the level in the two lowe tanks as well as the coesponding simulated outputs of the system identification models, fom DSR and PEM, as well as the simulated outputs fom the fist pinciples model. A. MATLAB scipt fo computation function G1,G,At,Bt,Dt,R... dlqdu_pi(a,b,d,q,rw); % DLQDU_PI syntax % G1,G,At,Bt,Dtdlqdu_pi(A,B,D,Q,R); % Pupose % Compute LQ-optimal feedback matices % G1 and G in the contolle % uu+g1*(x-x_old)+g*(y_old-); % On input % A,B,D- discete state space model matices. % Q - Weighting matix fo the output y_k. % R - Weighting matix fo the contol % incement, Delta u_ku_k-u_(k-1). % On output % G1 and G - Matices in LQ contolle % At, Bt, Dt - Matices in augmented model % Make augmented state space model % matices. nxsize(a,1); nusize(b,); nysize(d,1); AtA,zeos(nx,ny);D,eye(ny,ny); BtB;zeos(ny,nu); DtD,eye(ny,ny); QtDt *Q*Dt; % Solve Riccati-equation % and compute feedback matix. K,Rdlq(At,Bt,Qt,Rw); G-K; G1G(:,1:nx); GG(:,nx+1:nx+ny); % END dlqdu_pi
Modeling, Identification and Contol Refeences 18 16 1 Level in tank 1: Refeence and outputs using PI and poposed LQ contolle Andeson, B. D. O. and Mooe, J. B. Optimal Contol: Linea Quadatic Methods. Pentice-Hall Intenational Editions, 1989. 1 1 8 6 5 1 15 5 5 Samples Figue 8: Quaduple tank pocess. Level in tank one. Illustating the efeence and the outputs fom the pocess contolled by two single loop PI contolles, and the poposed LQ optimal contolle with integal action. The LQ contolle was constucted by using the DSR method fo system identification. The DSR model was used to identify a Kalman filte fo the system. The states wee estimated with this Kalman filte and the deteministic pat of the model was used to design the contolle. h 18 16 1 1 1 8 6 Level in tank : Refeence and outputs using PI and poposed LQ contolle 5 1 15 5 5 Samples Figue 9: Quaduple tank pocess. Level in tank two. Illustating the efeence and the outputs fom the pocess contolled by two single loop PI contolles, and the poposed LQ optimal contolle with integal action. The LQ contolle whee constucted by using the DSR method fo system identification. The DSR model was used to identify a Kalman filte fo the system. The states wee estimated with this Kalman filte and the deteministic pat of the model was used to design the contolle. PI PI LQ 1 LQ Åstöm, K. and Hägglund, T. PID Contolles: Theoy, Design, and Tuning. Instument Society of Ameica, 1995. Bistol, E. H. On a new measue of inteactions fo multivaiable pocess contol. Tansactions on Automatic Contol, 1966. 11(11):11 11. doi:1.119/tac.1966.19866. Di Ruscio, D. Closed and Open Loop Subspace System Identification of the Kalman Filte. Modeling, Identification and Contol, 9. ():71 86. doi:1.17/mic.9... Di Ruscio, D. On Tuning PI Contolles fo Integating Plus Time Delay Systems. Modeling, Identification and Contol, 1. 1():15 16. doi:1.17/mic.1... Di Ruscio, D. and Foss, B. On model based pedictive contol. In The 5th IFAC Symposium on Dynamics and Contol of Pocess Systems. 1998. Doyle, J. C. Guaanteed Magins fo LQG Regulatos. IEEE Tansactions on Automatic Contol, 1978. ():756 757. doi:1.119/tac.1978.11181. Gatzke, E. P., Meadows, E. S., Wang, C., and Doyle, F. J. I. Model based contol of a fou-tank system. Computes and Chemical Engineeing,. ():15 159. doi:1.116/s98-15()555- X. Jazwinski, A. H. Stochastic pocesses and filteing theoy. Academic Pess, 1989. Johansson, K. H. Inteaction bounds in multivaiable contol systems. Automatica,. 8():15 151. doi:1.116/s5-198(1)85-. Kwakenaak, H. and Sivan, R. Linea Optimal Contol Systems. Wiley-Intescience, 197. Maciejowski, J. Pedictive contol: with constaints. Peason Education,. Skogestad, S. and Postlethwaite, I. Multivaiable Feedback Contol: Analysis and Design. Wiley, 1996. Södestöm, T. Discete-time Stochastic Systems: Estimation and Contol. Pentice Hall, 199.