NON-LINEAR SYSTEM CONTROL

Transcription

1 Non-linear Proce Control NON-LINEAR SYSTEM CONTROL. Introduction In practice, all phyical procee exhibit ome non-linear behaviour. Furthermore, when a proce how trong non-linear behaviour, a linear model may be inadequate; a non-linear model will be more realitic. Unfortunately, thi improved cloene to reality i attained at a cot: the convenience and implicity offered by the linear model i acrificed. Since many procee exhibit only mildly non-linear dynamic behaviour, linear model can reaonably approximate them. However, even for uch ytem, the behaviour over a wide range of operating condition will be noticeably non-linear, and no ingle linear model i able to repreent uch behaviour adequately. The iue of analying the dynamic behaviour of procee repreented by non-linear model i the main concern in thi topic; thi i motivated by the fact that ituation do arie when it i undeirable to neglect the inherent nonlinearitie of a proce. The objective here i to provide an introduction of how we can expect to carry out dynamic analyi and control when the proce model are non-linear. The nature of non-linear problem i that they are not a eaily amenable to the general complete treatment which are poible with linear ytem. To keep non-linear dynamic analyi on a realitic and practical level uually involve trade-off between accuracy and implicity. There are two baic propertie that characterize the behavior of a linear ytem: Superpoition: the repone of a proce to input I i R and the repone to I i R, then according to the principle of uperpoition, the repone to I+ I i R+ R if the ytem i linear. In general, thi principle tate that the repone of a linear ytem to a um of N input i the ame a the um of the repone to the individual input. A a reult of thi principle, for example, the repone of a linear ytem to tep change of magnitude A i exactly the ame a A time the ytem' unit tep repone. Independence of Dynamic Repone Character and Proce Condition The dynamic character of a linear proce repone to an input change i independent of the pecific operating condition. I.e, identical input change implemented at different operating teady-tate condition will give rie to output change of identical magnitude and dynamic character. The principle of uperpoition alo mean that from the ame initial teady-tate condition, the output change oberved in repone to a certain poitive input change will be a perfect mirror image of the output oberved in repone to a negative input change of the ame magnitude. According to thi property, the tep repone of a linear ytem i the ame regardle of the actual initial value of the output or input variable, or, the direction of the input change, the "tep down" giving rie to a perfect mirror image of the "tep up" repone. A nonlinear ytem doe not exhibit any of thee propertie: KNU/EECS/ELEC83500

2 Non-linear Proce Control the repone to a um of input i not equal to a um of the individual repone; the repone to a tep change of magnitude A i not equal to A time the unit tep repone; the magnitude and dynamic character of the tep repone are dependent on the initial teady-tate operating condition; In particular, the ultimate repone of the nonlinear ytem to a inuoidal input function i not a perfect inuoidal output function. The model equation that will repreent the dynamic behavior of uch ytem adequately mut necearily contain nonlinear term; and immediately, the convenience of being able to completely characterize the ytem' dynamic behavior by the linear tranfer function i lot.. Method of Dynamic Analyi of Nonlinear Sytem Since the proce model are almot alway nonlinear, the following major quetion we wih to ak ourelve are: How do we analyze the dynamic behavior of procee when they are repreented by nonlinear model? I there a technique for analyzing the dynamic behavior of nonlinear ytem that can be generalized to cover a wide variety of uch ytem? The anwer to thee quetion form the bai of the material covered in thi Section. The rigorou analytical approach. Numerical analyi 3. Approximate linearization 4. Exact linearization by variable tranformation Each of thee method i briefly dicued below: Rigorou Analytical Technique It i well known that very few nonlinear problem can be olved formally, primarily becaue there i no general theory for the analytical cloed-form olution of nonlinear equation. Neverthele, there are a few analytical method for analyzing the dynamic behavior of nonlinear ytem; for example, claical method and the technique of Liapunov, Hopf, Golubitky, etc. Thee technique will allow one to characterize qualitatively the dynamic of nonlinear ytem. However, thee approache find limited ue in proce control practice; thu a detailed dicuion of their alient feature ha been excluded from the context. Perhap, with further reearch, the analytical treatment of nonlinear ytem will evolve to a tate where it will be more ueful for proce control. Numerical Solution and Computer Simulation The firt tep in the analyi of a ytem dynamic behavior i the olution of the modeling equation; and the model, are uually in the form of differential equation (ordinary or partial). For linear ytem, analytical olution are poible. For nonlinear ytem, intead of having an analytical expreion that repreent the behavior of the proce output over time, we make ue of numerical value for the proce output at pecific point of time. KNU/EECS/ELEC83500

3 Non-linear Proce Control 3 Note that the recurive calculation required by thi procedure are bet carried out with the aid of a digital computer. There are many pecial ubroutine available for the numerical integration of differential equation, o that detail of thi procedure are uually handled by the oftware. Variable Tranformation On the bai of the fact that linear ytem are eaier to analyze than nonlinear one, it eem attractive to conider the following propoition: If a ytem i nonlinear in it original variable i it poible that it will be linear in ome tranformation of thee original variable? If thi were poible, the general trategy for dynamic analyi would then involve firt carrying out thi tranformation, then performing dynamic analyi on the linear, tranformed verion, and finally tranforming the reult back to the original variable. Linearization In the contet of proce dynamic and control, linearization i a term ued in general for the proce by which a nonlinear ytem i approximated by a linear proce model. The mot popular technique for obtaining the linear approximation i baed on Taylor erie expanion of the nonlinear apect of the proce model. Summary: The firt two approache are clearly of limited control application. The technique of numerical analyi, even though traightforward, provide limited information becaue it i uually impoible to infer anything of a general form from the numerical repone of a proce ytem. It provide pecific numerical anwer to pecific problem; it i not poible to obtain general olution in term of arbitrary parameter and unpecified input in order to undertand the proce behavior in a more general fahion, a wa the cae with linear ytem. The quetion now i: can we ever achieve thi deirable objective of being able to analyze, in a general fahion, the behavior of an arbitrary nonlinear ytem, in repone to unpecified input? The anwer i ye: but at a cot. Oberve that the nonlinear proce model can be linearized around a particular teady-tate value, and with thi linearized approximation of the true nonlinear model, we can obtain fairly general reult regarding the proce behavior, but thee will only be approximate, never completely repreenting true behavior accurately.. Nonlinear Control The objective here i to examine the main apect of the rapidly developing ubject of nonlinear proce control, and to provide a ummary of the variou alternative technique available for deigning control ytem, including thoe that explicitly recognize the nonlinearitie of the proce. Control ytem may be carried out for nonlinear ytem along the line indicated by any of the following four cheme. Local Linearization Thi involve linearizing the modeling equation around a teady-tate operating condition and applying linear control ytem deign reult. It i obviou that the controller performance will deteriorate a the proce move further away from the teady tate around which the model wa linearized, but quite often, adequate controller can be deigned thi way. KNU/EECS/ELEC83500

4 Non-linear Proce Control 4 Local Linearization with Adaptation Thi cheme eek to improve on Scheme by recognizing the preence of nonlinearitie and conequently providing the controller a mean for ytematically adapting whenever the adequacy of the linear approximation become quetionable. Exact Linearization by Variable Tranformation Thi i a cheme by which a ytem model that i nonlinear in it original variable i converted into one that i exactly linear in a different et of variable by the ue of appropriate nonlinear tranformation; controller deign may then be carried out for the tranformed ytem with greater facility ince it i now linear. "Special Purpoe" Procedure Thee are uually cutom made for pecific procee, or pecific type of nonlinearitie. Some of the factor that typically determine which of thee approach one hould adopt are now ummarized below: The pecific nature of the nonlinear control problem at hand; the nature of the proce, and the control ytem performance objective. The amount of time available for carrying out the deign. The type of hardware available for implementing the controller. The availability and quality of the proce model.. Linearization While there may be an extenive undertanding of the behavior of non-linear procee, atifactory method for their control are till evolving. The prevalent approach to date ha been to ue a model of the proce linearied about a teady-tate operating point to deign a linear controller uch a the claical PID algorithm. Thi i only an approximate olution. The General Linearization Problem Conider the general nonlinear proce model: dx = f ( xu, ) (.) y = h( x) where f (,) i an arbitrary nonlinear function of the two variable, x, the proce tate variable, and u the proce input; h() i another nonlinear function relating the proce output, y to the proce tate variable x. The linearized approximation of thi very general nonlinear model (.) may now be obtained by carrying out a Taylor erie expanion of the nonlinear function around the point ( x, u ), Thi give: KNU/EECS/ELEC83500

5 Non-linear Proce Control 5 dx f f = f( x, u) + ( x x) + ( u u) + higher order term x u ( x ) ( x, u ) ( x, u ) h y = h( x) + ( x x) + higher order term x Ignoring the higher order term now give the linear approximation: where dx = f ( x, u) + a( x, u) ( x x) + b( x, u) ( u u) y = h( x ) + c x ( x x ) ( ) (, ) a x u (, ) b x u ( ) c x f = x f = u h = x ( x, u ) ( x, u ) ( x ) (.) It i cutomary to expre the equation in term of deviation variable: x% = ( x x) u% = ( u u) y% = ( y y ) = y h( x ) If addition to thi, the linearization point ( x, u ) i choen to be a teady-tate operating condition, then oberve from the definition of a teady tate that both dx and f ( x, u ) will be zero. (.) then become dx% = ax% + bu% y% = cx% where for implicity, the argument have been dropped from a, b, and c. A tranform-domain tranfer function model may now be obtained by the uual procedure; the reult i: cx ( ) bx (, u) y %() = u %() (.4) a( x, u) With the tranfer function a indicated in the quare bracket. Thi tranfer function hould provide an approximate linear model valid in a region cloe to ( x, u ) The principle involved in obtaining approximate linear model by linearization may now be ummarized a follow: (.3) KNU/EECS/ELEC83500

6 Non-linear Proce Control 6 Identify the function reponible for the nonlinearity in the ytem model. Expand the nonlinear function a a Taylor erie around a teady tate, and truncate after the firt-order term. Reintroduce the linearized function into the model; implify, and expre the reulting model in term of deviation variable. The application of thee principle to pecific problem i uually much impler can be demontrated by example. Example: linearization of a nonlinear model involving a nonlinear function of a ingle variable. Conider a Liquid Level Sytem hown a in Fig... Fig.. Liquid Level Sytem Material balance equation to Fig.. yield But dh A F F where c i flow reitance. Combining (.5) and (.6), we have F i = (.5) = c h (.6) dh A F c h i = (.7) Thi i a nonlinear equation. To linearie thi equation, one can ue Taylor erie. That i, around h=h, f ( h) = h = h ( h) + ( h) ( h h) + higher order term (.8) The approximation i hown a in Fig.. KNU/EECS/ELEC83500

7 Non-linear Proce Control 7 Fig.. Nonlinear Function and Linearized Approximation Subtitute (.8) into (.7) and noting that the teady tate flow we have where F i = ch ( ) dy τ = Ku y (.9) ( h ) τ = A ; u = Fi Fi; y = h h c From (.9), we get K Y() = U() τ + The linearied ytem i given in Fig..3. (.0) Fig..3 Linearied Liquid level ytem (about h=h ) Linear controller can now be applied to the linearied ytem of Fig..3. Fig..4 how a feedback loop where a Proportional and Integral (PI) controller control the lineraied liquid level ytem of Fig..3. Notice that the linearied ytem i an approximation ince it i derived for a particular level h=h. If the level change, K and τ will change with it a well. Fig..4 Feedback Control of (approximately) linearied ytem KNU/EECS/ELEC83500

8 Non-linear Proce Control 8 where The characteritic equation can be put a c r ggc = (.) + gg + gg c = 0 K g = ; gc = Kc( + ) + τ τ ( + KKc) KKc + + = 0 τ ττ Auming the cloed loop pole to be at the location -±j, which correpond to the root of the characteritic equation, for a linearied ytem with K=.38 and τ= 0.59, we have K c = 0.58 and τ I = 0.9. Example: Linearization of a nonlinear model involving a nonlinear function of two variable. The dynamic behavior of the liquid level h in the conical torage tank ytem hown in Figure.5 can be hown to be repreented below, where now the cro ection area of the tank i given by: o that the tank model become: vhere α and β are parameter defined by: Rh A= πr = π H I c I dh α F i 3/ = β h (.) h H α =, β = cα π R Here, F i i, again, the inlet flowrate, the manipulated variable. Obtain an approximate linear model for thi ytem. KNU/EECS/ELEC83500

9 Non-linear Proce Control 9 Fig..5 Conical torage tank ytem Solution The proce model (.) ha two type of nonlinear function: F i h -, a product of two function, 3/ and h. We hall have to linearize each of thee function eparately around the teady tate ( h, F ). i The linearization of f ( hf, ) i = Fh i proceed a follow: f f f( h, F) = f( h, F ) + ( h h ) + ( F F ) + higher order term i i i i h ( h, F ) Fi ( h, F ) i i whereupon carrying out the indicated operation now give: f hf = f h F Fh h h + h F F (.3) 3 (, i) (, i ) i ( ) ( i i ) if we ignore the higher order term. The tep involved in linearizing the econd nonlinear term are no different from thoe illutrated in the previou example; the reult i: 5 3/ 3/ 3 h = ( h ) ( h ) ( h h ) (.4) We may now introduce thee expreion in place of the correponding nonlinear term in (.). / Recalling that under teady-tate condition αfi = βh, and introducing the deviation variable y = ( h h ) and u = ( Fi Fi), the approximate linear model i obtained upon further implification a: dy τ + y = Ku (.5) where the teady-tate gain, and time contant aociated with thi approximate linear model, are given by: and α K = h = h β c τ = β / / If we deire an approximate tranform-domain tranfer function model, Laplace tranformation again give (.0). Note that the approximate linear model for the conical tank ha the ame proce gain, K, a for the cylindrical tank; but the time contant i a much tronger function of the liquid level h. Some important point to note about the reult of thee two illutrative example are the following: 5/ h KNU/EECS/ELEC83500

10 Non-linear Proce Control 0 In each cae, the approximate tranfer function model i of the firt-order proce, with the time contant, and teady-tate gain value dependent on the pecific teady tate around which the ytem model wa linearized. Uing uch approximate tranfer function model will give approximate reult which are good in a mall neighborhood around the initial teady tate; farther away from thi teady tate, the accuracy of the approximate reult become poorer. Uing uch approximate model, we are able to get a general (even if not 00% accurate) idea about the peed and magnitude of repone to expect from any arbitrary nonlinear ytem. Numerical analyi can give more accurate information, but only about the pecific repone to a pecific input, tarting from a pecific operating condition for a pecific et of parameter. Let u conclude with a comparion of the repone of the nonlinear model and an approximate linear model for the conical tank of the Example. Suppoe that α =, β =, c = 0.5 o that the nonlinear model i: dh F i 3/ = h h while the approximate linear model ha the tranfer function which in the time domain i 4h Y() = U() h + 5/ / 5/ dh = + h / { h h() t 4h [ Fi Fi] } The tep repone (up and down) for thee model i hown in Figure.6 for tep of two different magnitude. The repone to a ine wave input of two different amplitude i hown in Figure.7. For mall magnitude tep input and mall amplitude ine wave, the linear model i a good approximation of the nonlinear model; For the large amplitude input the linear model prediction deviate omewhat from the nonlinear model behavior. The ditinct difference in nonlinear repone for a tep up and tep down. KNU/EECS/ELEC83500

11 Non-linear Proce Control Fig..6 Repone of model (.5)/(.) for tep input up/down F i =± 0.05 and F i =± 0.5 Fig..7 Repone of model (.5)/(.) to ine wave input amplitude A = 0.05, 0.5. However, in ome ituation thi may be inadequate (e.g. for the control of highly non-linear procee or batch ytem) o the development of nonlinear controller ha featured prominently in proce control in the lat decade. In the following Section, we will give an overview of ome of the more popular non-linear methodologie that have been propoed for ue within the proce control indutrie including, Adaptive Control approache; Globally Lineariing Control (GLC) and Generic Model Control (GMC). 3. Adaptive Control A controller deigned for the ytem in the level control example in lat ection uing the approximate tranfer function model will function acceptably a long a the level, h, i maintained at, or cloe to, the teady-tate value h. Under thee condition (h -h ) will be mall enough to make the linear approximation adequate. However, if the level i required to change over a wide range, the farther away from h the level get, the poorer the linear approximation. KNU/EECS/ELEC83500

12 Non-linear Proce Control Oberve alo from the approximate tranfer function model in (.0) that the apparent teadytate gain and time contant are dependent on the teady tate around which the linearization wa carried out. Thi implie that in going from one teady tate to the other, the approximate proce parameter will change. The main problem with applying the claical approach under thee circumtance i that it ignore the fact that the characteritic of the approximate model mut change a the proce move away from h if the approximate model i to remain reaonably accurate. The immediate implication i that any controller deigned on the bai of thi changing approximate model mut alo have it parameter adjuted if it i to remain effective. In the adaptive control cheme, the controller parameter are adjuted (in an automatic fahion) to keep up with the change in the proce characteritic. We know intuitively that, if properly deigned, thi cheme will be a ignificant improvement over the claical cheme. There are variou type of adaptive control cheme, differing mainly in the way the controller parameter are adjuted. The three mot popular cheme are: cheduled adaptive control, model reference adaptive control, and elf-tuning controller. We hall dicu each of thee in the following ubection. 3. Scheduled Adaptive Control A cheduled adaptive control cheme i one in which, a a reult of a priori knowledge and eay quantification of what i reponible for the change in the proce characteritic, the commenurate change required in the controller parameter are programmed (or cheduled) ahead of time. Thi type of adaptive control, ometime referred to a gain cheduling, i illutrated by the block diagram in Fig. 3., which i an improvement over that of Fig..3. Fig. 3. Scheduled Adaptive Controller Continuing with the example of liquid level ytem, Fig..3 repreent control of linearied approximation of the original non-linear ytem. The approximation i in the ene that the proce tranfer function g only hold good for a particular level of the liquid in the tank. If the level h change dratically, the tranfer function g may fail to repreent the original ytem even approximately. Then there i a need to change the tranfer function g to account for the change in level h. One way of doing thi i through Scheduled adaptive controller hown in Fig. 3.. From (.0), it i een that both K and τ are affected by the liquid level h. The effect on K i more prominent ince it affect the ytem gain. The effect on the time contant i le apparent and not o prominent. Thu we only compenate for the effect of liquid level h on K. Thi i done in a way that the overall ytem gain KK c i contant. Notice that, from (.0), ( h ) K = c KNU/EECS/ELEC83500

13 Non-linear Proce Control 3 So, to keep K K c = K 0 (a contant), K c K K c = 0 = 0 K ( h ) Thi mean that the controller gain K c ha to be adjuted inverely a the quare root of the liquid level h. Thi i what will be done by the block repreented a Parameter adjutment in Fig. 3.. More generally, if K = K(t), that i, the proce gain change with time, then to keep the product KK c contant it i neceary to adjut K c inverely with repect to K(t), i.e., K c / K(t) K(t) itelf can be determined periodically by inerting a ignal u into the proce and monitoring the proce output y. In practice, it i often poible to find meaured variable that correlate well with change in proce dynamic. Thee variable can be ued to change the controller parameter, uing a precalculated chedule. The controller parameter are computed off-line for everal operating condition and tored in memory. Gain cheduling can alo be baed on nonlinear tranformation uch that the tranformed ytem doe not depend on the operating condition. It i difficult to give general rule. Each cae mut be treated individually. The key quetion i to determine the auxiliary variable to be ued a cheduling variable. It i neceary to have a good inight into the dynamic of the proce if gain cheduling i going to be ued. The controller can be automatically tuned at different operating point and the reulting tuning parameter can be aved and a chedule created. Example: Fig. 3. how different valve charactretic, if the cheduling variable in a level control ytem i the opening of a control valve, then a chedule may look a in Table 3.: Fig. 3. Different valve charactretic Table 3. Scheduled control for Valve Valve opening Kp Ti Td Example: The nonlinearity of the valve i aumed to be: KNU/EECS/ELEC83500

14 Non-linear Proce Control 4 v= f( u) = u 4 Fig Gain Schedule for Nonlinear actuator. Let f be an approximation of the invere of the valve characteritic. To compenate for the non-linearity, the output of the controller fed through thi function before it i applied to the valve. Thi give the relation: where c i the output of the PI controller. v= f u = f f c ( ) ( ( )) 4 Aume that f ( u) = u i approximated by two line a hown in Fig One from (0,0) to (.3,3) and the other from (.3, 3) to (,6). Then we have: Fig. 3.4 A Crude Approximation of Valve Characteritic ˆ 0.433c 0 c 3 f () c = c c 6 Simulation Reult KNU/EECS/ELEC83500

15 Non-linear Proce Control 5 Fig. 3.5 Output Repone without Gain Scheduling Fig. 3.6 Output Repone with Gain Scheduling 3. Model Reference Adaptive Controller (MRAC) The key component of the MRAC cheme i the reference model that conit of a reaonable cloed-loop model of how the proce hould repond to a et-point change. Thi could be a imple a a reference trajectory, or it could be a more detailed cloed-loop model. The reference model output i compared with the actual proce output and the oberved error ε m i ued to drive ome adaptation cheme to caue the controller parameter to be adjuted o a to reduce ε m to zero. The adaptation cheme could be ome control parameter optimization algorithm that reduce the integral quared value of ε m or ome other procedure. Thi i an adaptive control technique where the performance pecification are given in term of a model. The model repreent the ideal repone of the proce to a command ignal. The controller ha two loop: The inner loop, which i an ordinary feedback loop coniting of the proce and the controller. The outer loop, which adjut the controller parameter in uch a way that the error e= y y m i mall (not trivial) KNU/EECS/ELEC83500

16 Non-linear Proce Control 6 Approache: Gradient approach Tracking error: Introduce the cot function J: Fig. 3.7 Model Reference Adaptive Control Scheme e= y y m (3.a) J θ = (3.b) ( ) e Where θ i a vector of controller parameter. Change the parameter in the direction of the negative gradient of e dθ dj de = γ = γe (3.c) where e/ θ i called the enitivity derivative. It indicate how the error i influenced by the adjutable parameter θ. Example: MRAS of a fit order ytem Proce: Model: Controller: Cloed loop ytem: dy dy ay bu = + m = a y + b u m m m c u = θ u θ y c KNU/EECS/ELEC83500

17 Non-linear Proce Control 7 dy = ay + bu = ay + b( θuc θy) = ( a + bθ) y + bθu Ideal controller parameter for perfect model-following: Derivation of adaptive law Error: where Senitivity derivative: Approximate 0 bm 0 am a θ = ; θ = b b e= y y m bθ y = u + a + bθ e b = θ + + u θ + + c a bθ e b θ b = u c = ( + a+ bθ ) a bθ + a+ bθ + a m ( ) ( ) then the MIT rule dθ/ = - γ' e e/ θ, may be written a follow where γ = γ' b / a m Block diagram dθ b γ a a a dθ b γ a γ y e θ y e a a ' m = γ uc e θ = uc e + m + m ' m = = + m + m c y c KNU/EECS/ELEC83500

18 Non-linear Proce Control 8 Fig. 3.8 Model Reference Adaptive Control for the example Simulation: a =, b = 0.5, a m = b m =, uc : quare wave with period 0 and value and 0. KNU/EECS/ELEC83500

19 Non-linear Proce Control Self-tuning Adaptive Controller Self-tuning adaptive controller uch a the elf-tuning regulator differ from the model-reference adaptive controller in baic principle. The elf-tuning controller, illutrated by the block diagram in Figure 3.8, make ue of the proce input and output to etimate recurively, on-line, the parameter of an approximate proce model. Thu a the actual non-linear proce change operating region or change with time, an approximate linear model i continuouly updated with new parameter. The updated model i then ued in a prepecified control ytem deign procedure to generate updated controller parameter. The controller could be a PID controller or ome of the model-baed controller. Fig Self-Tuning Control Scheme Since the model etimated determine the effectivene of the controller, the mot eential feature of the elf-tuning controller i reliable and robut model identification. Thi require a good parameter etimation algorithm and procedure for enuring adequate dynamic experimental deign. Example: Proce i aumed to have the tranfer function of the form, ay, α Ke g () = τ + Let the controller be a PI controller of the form g = ( ) c K + c τ (3.) The controller parameter are typically et according to a tuning formula, uch a (Minimum ITAE tuning rule for et-point tracking), τ ( ) 0.96 Kc = K α α ( ) = τ τ τ I Notice that the controller parameter K c and τ I are adjuted in term of the proce parameter K, α, and τ. Since the proce model parameter are continuouly changing a the proce operating (load) condition are changed, the parameter etimator will periodically update the I (3.3) KNU/EECS/ELEC83500

20 Non-linear Proce Control 0 parameter K, α, and τ. Baed on thi updated parameter value, the controller etting K c and τ I are recalculated uing (3.3). The new controller parameter are then adjuted on the controller a hown in Fig Comparion of Adaptive Controller There i a clear differentiation in the philoophy behind each of the three adaptive controller dicued above. Let u ue an analogy to illutrate. When parent end a child off to college, they uually try to provide them with adaptive controller to help them deal with the unknown ituation they will counter out on their own. We can eaily recognize three type of the adaptive controller in ue: Scheduled Adaptive Control i ued by parent who ue a lit of if thi occur, do thi intance. Thi mean they aume they can anticipate and enumerate all the ituation that can occur and pecify programmed remedie. No thinking or deciion making i expected of their children. Thu cheduled adaptive control will work when all poibilitie are known in advance. Model-Reference Adaptive Control i employed by parent when they have a good role model (uch a an older ibling) available. They only tell their child to emulate the behavior of the role model; i.e., do whatever they do in the ame ituation. In thi cae there i ome limited deciion making on the part of the child, but it i contrained to deciding how bet to emulate their role model. So long a uch emulation i poible, then MRAC work well. Self-tuning Adaptive Control i the trategy recommended by parent, have a little more confidence in the thinking ability of their children. While they do not recommend complete independence of action, they ak that the child carefully evaluate the ituation baed on the data at hand (model building) and then baed on their judgment of the ituation, take action baed on fixed principle. Self-tuning adaptive control allow the mot freedom to the controller and thu provide the broadet range of adaptation to unknown ituation; however, it i alo the mot dangerou, potentially untable, form of adaptive control becaue poor model identification (poor aement of ituation) can lead to untable ytem and proce runaway. All thee type of adaptive controller are commercially available and have enjoyed a certain level of indutrial acceptance. By their very nature, adaptive controller can be applied to any ytem for which the proce parameter vary with time; they are alo applicable to other ytem for which thee parameter variation are not due to nonlinearitie per e, but to otherphyical phenomena uch a fouling in heat exchanger, catalyt decay in catalytic reactor, etc. 4. Variable Tranformation The idea of variable tranformation can be exploited for nonlinear controller deign in the following manner: Obtain an equivalent linear ytem by variable tranformation. Deign a linear controller for the tranformed ytem, in term of the new variable. KNU/EECS/ELEC83500

21 Non-linear Proce Control Implement the controller in term of the original variable of the nonlinear ytem (uch a controller will be nonlinear). Several different method have been reported in the literature for carrying out uch variable tranformation for non-linear ytem and utilizing the reult for controller deign. The following i a imple illutration 4. Global Liberalization Let the proce be decribed by dx f ( xu, ) = (4.) We can alway eparate the RHS of (4.) into two term where the firt term depend only on x a in (4.): Conider a new variable z(x). f ( xu, ) = cf( x) + c f( xu, ) (4.) dz dz dx = (4.3) dx That i, uing (4.3), (4.) can be written a If z(x) i choen uch that then (4.4) become, That i, where dz dz dz = c f( x) + c f( x, u) (4.4) dx dx dz z = ; f( x) 0 (4.5) dx f ( x) dz czf( xu, ) = cz + (4.6) f( x) dz cz (, ) cv xu = + (4.7) zf( x, u) vxu (, ) = f ( x) Notice that in (4.6), v i the new input and (4.6) i linear in the new variable z (becaue coefficient of z in the firt term i a contant c ). Tranformation of old input variable u into new input variable v i given by (4.7). One can alo expre, uing (4.7), u in term of v a u = q( x, v) (4.8) for ome function q. KNU/EECS/ELEC83500

22 Non-linear Proce Control To find the function z(x), integrate (4.5) to get z = g( x) = e Alternatively, one can define, in place of (4.5), z(x) a dx f ( x ) dz = ; f ( x ) 0 (4.9) dx f ( x) Uing (4.9), (4.) can be expreed a dz f( xu, ) = c+ c (4.0) f ( x) (4.0) can be put a dz c (, ) c v x u = + (4.) (4.) i linear in the new variable z and v i the new input. A before, new input i expreed a v = Or, f( xu, ) f ( x) u = q'( x, v) (4.) for ome function q. Integrating (4.9), z(x) i obtained a dx z = g'( x) = f ( x) (4.3) KNU/EECS/ELEC83500

23 Non-linear Proce Control 3 Fig. 4.. Global Linearized Control Example: Application to level Control in a Cylindrical Tank (4.) can be written a dx cx cu = + c where, c = and c A = A. Conider a change of variable z(x).then, dz dz dx = dx Subtituting (4.3) into (4.3), dz dz dz = c x + cu (4.4) dx dx Put dz =, x 0 (4.5) dx x Then (4.4) become dz c c v where u vxu (, ) = x Integrating (4.5), = + (4.6) u can alo be expreed a z = x (4.7) KNU/EECS/ELEC83500

24 Non-linear Proce Control 4 u = vx (4.8) Since (4.6) i linear, one can ue a fixed controller, uch a, PID controller to control the tranformed ytem. We now wih to apply thee reult to a pecific flow ytem cylindrical pipe for which A =.06 cm (I.D. i 5.3 cm) and, from previou experimentation, it i known that the dependence of the outflowrate on the liquid level in the pipe i given by: The control ytem i implemented a follow: / 3 F 0.76 l out = A h cm / Obtain the actual meaurement x (in thi cae h) from the proce. Calculate z from (4.7); i.e., take the quare root, and multiply by. From the ame expreion, given the deired value for x, obtain the deired value for z; i.e., Z = x. d d The PI controller (tuned for the linear, tranformed ytem (4.)) receive z and z d and precribe v(t), the control action required for the linear peudo-ytem. The actual flowrate implemented on the real, nonlinear flow ytem i obtained from (4.8); a control law nonlinear in the original proce variable x. It i intereting to note that the tranformation controller bae the precribed inflowrate on a product of the control action for the linear peudo- ytem and the quare root of the liquid level in the tank. The phyical implication of thi are particularly important: the nonlinearity in the flow ytem i caued by the fact that the outflow i proportional to the quare root of the liquid level in the tank; the tranformation controller i een to take advantage of thi fact by adopting the indicated trategy. The conventional PI control trategy will bae the inlet flowrate on the actual liquid level in the tank; the tranformation controller bae it action on the product of the output of a linear controller, and the quare root of the liquid level. Uing controller parameter K c = 90 and KI = Kc τ I = 0. (for the PI controller tuned for the linear peudo-ytem), the repone of the example flow ytem to a et-point change in the liquid level from 40 to 75 cm, operating under the nonlinear tranformation controller, i hown in the dahed line in Figure 4.. A conventional PI controller uing the ame controller parameter reulted in an almot untable ytem upon reducing the controller parameter to K c = 9 and KI = Kc τ I = the ytem repone to the ame et-point change under conventional PI control (i.e., deciion on the inlet flowrate are taken uing direct level meaurement) i hown in the olid line. A we would expect, the tranformation controller perform better than the conventional PI controller. KNU/EECS/ELEC83500

25 Non-linear Proce Control 5 Fig. 4. Repone to et-point change Application of the nonlinear tranformation trategy i retricted by the conideration: The nonlinear tranformation are not alway eay to find (for ome general approache). The ucce of the controller deign i dependent on having a good linear proce modelomething that i unuual in practice. Concluion: It i not a univerally applicable control ytem deign procedure. In general, the conceptual bai for GLC lie in explicitly relating the ytem output to it manipulated input in term of an ordinary differential equation. The output i ucceively differentiated with repect to time until the reulting derivative i a linear function of the input ignal. Differential geometric concept are then applied in developing the control law. 5. Generic Model Control Generic Model Control (GMC) ue a model of the proce in formulating the control law. The deign framework i imilar to other model baed approache uch a Dahlin algorithm and IMC. However, rather than adopting a claical approach of comparing the trajectory of the proce output againt a deired trajectory, GMC define the performance objective in term of the time derivative of the proce output, i.e minimiing the difference between the deired derivative of the proce output and the actual derivative. Generally the following control affine, non-linear tate pace decription i ued, x& = f( x) + g( x) u y = h( x) where f(x) and g(x) are vector field, i.e. they are vector valued function of a vector and h(x) i a calar field, i.e. a calar valued function of a vector, x. In term of a general ytem of equation thi can be expreed a: dx = f( x, Lxn) + g( x, Lxn) u M dxn = fn( x, Lxn) + gn( x, Lxn) u y = h( x Lx ), n (5.) KNU/EECS/ELEC83500

26 Non-linear Proce Control 6 The model i termed control affine or, control linear, becaue the manipulated input appear linearly - it imply multiplie g(x). Chooe the following deired trajectory: y& = α ( y y) + α ( y y) (5.) d d d where α and α are deign contant (to be pecified) and y d i the etpoint. Thi i the form of a PI controller. In order to deign a controller o that the ytem follow the trajectory defined by equation (5.) a cloely a poible, the following performance index i pecified: t 0 J = e& = ( y& y& ) (5.3) d i.e. minimie the error quared over a pecified time horizon. In order to obtain y& from equation (5.) the chain rule mut be ued, Note: dy dx i a 'row' vector while i a column vector given by, dx therefore dh dy dy dx = (5.4) dx dx = x& = f( x) + g( x) u y = h( x) dx i alo a 'row' vector and, dy dh = [ f ( x ) + g ( x ) u ] dx dh dh = f ( x) + g( x) u dx dx Uing equation (5.6) the performance index, equation (5.3), i minimied when e & = 0, i.e. Thi control law ha two deirable propertie: It utilie a proce model. (5.5) (5.6) dh dh α( yd y) + α ( yd y) f( x) + g( x) u = 0 dx dx (5.7) dh u() t = ( yd y) ( yd y) f( x) dh α + α gx ( ) dx (5.7) dx any inaccuracie in the proce model (i.e. plant model mimatch) will be compenated for by the integral term within the control law. KNU/EECS/ELEC83500

27 Non-linear Proce Control 7 Example: Conider the following continuou tirred tank exothermic reactor a hown in Fig. 5., Figure 5.. A Continuou Stirred Tank Reactor (CSTR) The inlet concentration of the reactant and product are A i and R i (mol/l) repectively while the outlet concentration i A o and R o (mol/l). The inlet temperature i T i (K) while the outlet temperature i T o (K). The concentration of both reactant and product in the CSTR are taken to be proce tate, along with the reactor outlet temperature. The dynamic behaviour of thee tate i decribed by three nonlinear-coupled differential equation: dao Ai Ao = ka o + kr o τ dro Ri Ro = + ka o kr o τ dt H ( T T ) = ( ka kr ) + o R i o o o ρcp τ where the reaction rate k and k depend on the reactor temperature via the Arrheniu relationhip E E k = cexp, k = cexp RTo RTo and E (Cal), E (Cal), R (Cal/mol k), A i (mol/l), ρ (kg/l), C p (Cal/kg K), τ(), = time contant of the veel and H R (Cal/mol) are normal ytem parameter. The control objective i to regulate reactor temperature (T o ) uing inlet feed temperature (T i ). Thi proce i evidently highly non-linear due to the inter-relationhip of the tate, and particularly the exponential dependence of each tate on the reactor temperature. The differential equation can be re-arranged into the generic vector repreentation of the proce with the following vector and calar field: State Vector: x = ( A, R, T ) T o o o f vector field: f ( x) = ( f( x), f( x), f3( x)) T KNU/EECS/ELEC83500

28 Non-linear Proce Control 8 Ai Ao f( x) = kao + kro τ Ri Ro f( x) = + kao kro τ H To f ( x) = ( k A k Ro) τ R 3 o ρcp T g vector field: gx ( ) = ( g( x), g( x), g3( x)) = 0 0 τ, u = Ti h functtion: With the CSTR, the h calar field decribing the output function i imply the tate itelf, i.e. y= h(x) = T o, therefore dy dx 3 = = In other word, the GMC controller i given by, T y& f ( x) g ( x) u (5.8) HR To T i α( yd y) + α ( yd y) ( kao kro) + = 0 ρcp τ τ (5.9) and olving for the manipulated input, yield, HR T o Ti = τ α( yd y) + α ( yd y) ( kao kro) ρcp τ (5.0) Taking Laplace tranform of equation (5.) give: α + y () α = y () d α + + α α (5.) Thi ideal cloed loop repone i not the ame a a claical nd order ytem. However, it i poible to plot normalied repone of the ytem veru normalied time with the damping ratio of a econd order ytem a a parameter. Thi allow the election of the hape of repone a well a an appropriate cloed loop time contant. Thi then allow the pecification of α and α. KNU/EECS/ELEC83500