DYNAMIC MODELS FOR CONTROLLER DESIGN

DYNAMIC MODELS FOR CONTROLLER DESIGN M.T. Tham (996,999) Dept. of Chemical and Proce Engineering Newcatle upon Tyne, NE 7RU, UK.. INTRODUCTION The problem of deigning a good control ytem i baically that of matching the dynamic characteritic of a proce by thoe of the controller. In other word, if the dynamic of the proce and the characteritic of the diturbance affecting it are known, then the controller that will provide the deired cloed loop performance can be deigned. Modern approache to controller deign therefore preuppoe that a uitable decription (model) of the proce to be controlled i available. The model could be in the form of tatitical relationhip; continuou time differential equation; difference equation; continuou or dicrete tranfer function; time-erie; even qualitative decription uch a qualitative tranfer function and rule baed model. However, due to the difficultie involved in developing phyical-chemical model, relatively imple, lumped parameter, input-output (black-box) model are often ued. With few exception, current proce control ytem are alo implemented on ampled data device. Thi et of note will therefore focu only on the relevant model form that can be ued for the ynthei of dicrete model baed controller, namely tranfer function and timeerie model.. DISCRETE-TIME TRANSFER FUNCTION MODELS. Single-input Single-output (SISO) Linear Sytem Conider a continuou time proce with the Laplace tranfer function: Y () U() = G () = B() e A () T d () relating the effect of input U() on output Y(). A ( ) and B ( ) are polynomial in the Laplace operator '', decribing repectively, the numerator and denominator dynamic of the of 8

proce. T d i the time delay. Upon introduction of a zero-order-hold (ZOH) and ampling, the equivalent dicrete tranfer function i: { (). ()} Z ZOH G Yz () = = HG() z = U() z Bzz () Az () k () where k i the time delay of the dicretied proce, expreed a an integer multiple of the ampling interval and Az ( ) and Bz ( ) are polynomial in z -. That i: Td k = int + (3a) T ( ) N A Az () = + az + az +! + a z, N = deg Az () (3b) N A A ( ) N B Bz () = b + bz + bz +! + a z, N = deg Bz () (3c) N B It i eay to convert Eq.() into a difference equation, for the purpoe of imulation ay. Simply introduce y(t) and u(t) in place of Y(z) and U(z) repectively, and regard z - a a time hift operator. Hence, the difference equation form of Eq.() i written a, Azyt ()() = Bzut ()( k) N A N B or (! N A ) (! N B ) + az + az + + a z yt () = b + bz + bz + + b z ut ( k) B or yt () = ayt ( ) ayt ( )! a yt ( N ) + NA b u( t k) + b u( t k ) + b u( t k ) +! + b u( t k N ) A NB B (4) Here, 't' denote the time index, and the operation of z -n on a time variable x(t) reult in n z x() t = x( t n) (5) that i, x(t) i hifted n ample interval back in time. It i important to remember that we are working with tranfer function, where all initial condition are aumed to be zero. In other word, the time indexed variable y(t) and u(t) are actually deviation variable, i.e. Appendix and Appendix contain introductory material on ampled-data-ytem and z-tranform In ome publication, q - i ued a the time-hift operator intead of z - to avoid confuion between the complex z operator in the dicrete z-domain. of 8

yt () = Yt () Y( ) and ut () = Ut () U( ) where Y(t) and U(t) are the meaured output and input ignal while Y( ) and U( ) are ome correponding compatible teady-tate: a teady input U( ) will caue the output to ettle at Y( ). Thi ha ignificant implication not only from the modelling point of view but alo from the parameter etimation perpective a will be dicued in detail later. It i alo important to bear in mind that theoretically, tranfer-function are capable of repreenting linear-ytem only. Intead of expreing the dicrete tranfer function a a ratio of two polynomial, we can alo write Yz () U() z k Bzz () k = = z β () z (6) Az () where β(z) i another polynomial in z -, the reult of dividing B(z) by A(z). In thi cae, the difference equation relating y(t) to u(t) become yt () = β ut ( k) + β ut ( k ) + β ut ( k ) +! (7) Thu, the output i dependent only on a weighted um of previou input. Depending on the nature of B(z) and A(z), the right-hand-ide of Eq.(7) may be an infinite erie in u. Thi repreentation i ueful becaue if u(t-k) i a unit impule, then the coefficient of β(z) define the magnitude of the reultant output equence. Conider the following example yt ( ) = 5. ut ( 3) + 7. ut ( 4) + 4. ut ( 5) + 3. ut ( 6) +. ut ( 7 ) (8) which ha a delay of 3 ample interval. Figure how what happen when it i ubject to an unit impule input (dahed line) at ample number. After the paage of the time delay, i.e. at ample number 5, the ytem output i.5. At ample 6, the output i.7, and o on. It can be een that the output trajectory matche exactly the weight on the input equence of Eq.(8). What thi mean i that if the proce wa ubject to a unit impule, then the coefficient of β(z) could be determined from the output meaurement. 3 of 8

Impule repone.9.8.7.7 Input & Output.6.5.4.5.4.3.3... 5 5 5 3 35 4 45 Sample Number Figure. Impule repone of ytem given by Eq.(8) The repone of the ytem when it i ubject to a unit tep input (dahed line) i hown in Fig.. Step-repone.8.6.6.9 Input & Output.4..8.6.4..5. 5 5 5 3 35 4 45 Sample Number Figure. Step repone of ytem given by Eq.(8) In thi cae, we ee that the output repone trajectory i given by ucceive convolution of the coefficient of Eq.(8). That i, at ample number 5, the output i at.5. It then move to (β +β =.5+.7 =.) followed by (β +β +β =.5+.7+.4 =.6) and o on. 4 of 8

Although model with tructure given by Eq.() are more compact, Eq.(6) i more veratile. Since they are eaily parameteried by imple experiment, they can be ued to model ytem with 'irregular' repone behaviour.. Multi-input Single-output Repreentation Both Eq.() and (6) can be extended to include the effect of other determinitic input on the proce output. For example, if the proce i affected by a load diturbance, v(t), then Eq.() can be modified to k B () z z B () z z Yz () = HGz (). Uz () + GD (). z Vz () = U() z + A () z A () z l V() z (9a) which can then be expreed a k l AzYz () () = z BzU () () z + z DzV () () z (9b) 'l' i the delay between the diturbance and the output, again expreed a an integer multiple of the ample interval and D(z) i another polynomial in z - However, unlike 'k' the delay between the manipulated input and output which ha a minimum value of due to the preence of the ZOH, l. Thi i becaue there i no hold device aociated with the diturbance input. If there are more input, they can be included in a imilar manner. Likewie, Eq.(6) can be modified to include input other than the manipulated variable, e.g. k l Yz () = z β() zuz () + z γ (). z Vz () () where γ(z) i a polynomial in z -. A with continuou time ytem, a et of tranfer function relationhip can be combined in vector-matrix form to decribed multivariable ytem, i.e. ytem with more than one output and output. 3. TIME-SERIES MODELS Next we conider the time-erie approach to modelling time dependent behaviour which ha root in tatitical forecating. There are numerou time erie model tructure and thee are decribed below: 5 of 8

3. Auto-Regreive (AR) Model One of the implet form of a time-erie i one where the current value of a time variable i aumed to be a function of it pat value only. The form of the expreion i, Ay() t =ξ() t () ξ(t) i a 'chance' component ued to account for uncertainty in the relationhip and aumed to be identically ditributed with zero mean and finite variance, σ. y i therefore alo a random equence. A i a monic polynomial (with leading coefficient ) in the time hift operator which in time erie convention i written uually a q - intead of our 'practical' interpretation of z - thu far. However, to avoid confuion, from hereon, we will not make explicit reference to the argument of the polynomial preented in equation. Now, expanion of Eq.() give: yt () = ayt ( ) a yt ( )! a yt ( N ) + ξ () t () N A Thu, the current output i a weighted um of previou output value. Thu thi type of repreentation i called an auto-regreive (AR) model becaue it form how a regreion of a time variable with itelf at different time intant. In Eq.(), N A define the oldet output value that ha a ignificant influence on the current output and the repreentation i termed an AR(N A ) model, or auto-regreive model of order N A. Compare thi with the difference equation, Eq. (4), where N A define the order of the underlying tranfer function. A 3. Moving-Average (MA) Model Suppoe we have an AR() model yt () = φyt ( ) + ξ() t (3) Time hifting one unit back, we get yt ( ) = φyt ( ) + ξ ( t ) (4) Subtituting Eq.(4) into Eq.(3), we obtain yt () = ξ() t + φξ( t ) + φ yt ( ) (5) Repeated ubtitution for pat value of y will eventually yield an expreion of the form, 6 of 8

yt () = () t + ( t ) + + t ( ) t ξ φξ " φ ξ + φ ξ() + φ t y( ) (6) Thu, in thi cae, y(t) i re-expreed a a weighted um (moving average) of all pat ξ and it initial value y(). If the initial value i zero, then Eq.(6) can be written more compactly a yt () = Cξ() t (7) where C i a monic polynomial C = + c z + c z + " + c z N C N C Such a repreentation i known a a Moving Average (MA) time erie model of order N C, or an MA(N C ) model. 3.3 Auto-Regreive Moving Average (ARMA) Model The AR and MA repreentation can be combined. For example, if we combine an AR() model yt () + φyt ( ) = ξ() t and an MA() model yt () = ξ() t + θξ( t ) φ θ φ we get yt () + yt ( ) = () t + ( t ) + z yt () z () t = + ξ ξ θ ξ which can be expreed in the more general form Ay() t = Cξ() t (8) Equation (8) i called an Auto-Regreive Moving Average (ARMA) model. If A and C have order N A and N C repectively, then Eq.(8) i an ARMA(N A,N C ) model. ARMA model are ueful in that they can provide for impler repreentation when the erie i an aggregate of everal impler erie. 3.4 Auto-Regreive Moving Average with Exogenou Input (ARMAX) Model The AR, MA and ARMA repreentation involve random equence y and ξ.. In the control context however, action are often taken to influence proce behaviour. To include the effect 7 of 8

of uch exogenou input to the ytem in a time erie model, the ARMA model can be further extended to, Ay() t = Bu( t k) + Cξ () t (9) Again, B i a polynomial in the time-hift operator and u i the manipulated or exogenou input equence. Equation (9) decribe an Auto-Regreive Moving Average with exogenou input (ARMAX) ytem. Note the imilarity between Eq.(9) and the difference equation of Eq.(4). However, Eq. (9) ha an extra term to decribe random effect. Thu the ARMAX model could be regarded a the tochatic equivalent of dicrete tranfer function model. Equation (9) i alo called the Controlled Auto-Regreive Moving Average (CARMA) model, to tate it explicit link to proce control ituation. 3.5 Auto-Regreive Integrated Moving Average with Exogenou Input (ARIMAX) In all the previou time-erie model, ξ(t) wa aumed to be an identically ditributed random equence. Paing thi through the C polynomial of the moving average component enable erially correlated random effect to be modelled. In practice however, random diturbance are rarely that well behaved. Mot of the time, proce noie how drifting characteritic. 8 of 8

6 4 Noie - -4-6 -8-3 4 5 6 7 8 9 Sample Number Random Integrated Filtered Integ. & Filtered Figure 3. Different type of random noie equence Figure 3 how the behaviour of 4 different type of random equence. A normally ditributed random equence, ξ(t), i hown by the continuou line. Paing it through C = +.95z -, reult in the 'filtered' or erially correlated equence (denoted ). The equence that exhibit drifting characteritic are hown by the lower two plot. Thee were generated repectively by the integrated random equence, ξ() t (dahed plot), and integrated and filtered equence, Cξ( t) (denoted ). C i defined a above and = z. Thu to enable the modelling of drifting non-tationary behaviour, the ARMAX model can be modified to ξ() Ay() t = Bu( t k) + C t () which i called the Auto-Regreive Integrated Moving Average with exogenou input (ARIMAX) time erie model. A more recent terminology i to decribe Eq. () a a Controlled Auto-Regreive Integrated Moving Average (CARIMA) model. 9 of 8

3.6 General Time-Serie Model Repreentation From the previou dicuion, we note that we can contruct time erie from everal uberie and in a variety of configuration. In proce modelling, the tak then i to determine the mot appropriate time-erie tructure and etimating the parameter of the polynomial. An even more general time erie repreentation i given below. B yt A ut k C () t () = ( ) + ξ D n () Here the contribution of u and ξ to y are clearly delineated. For a noie free ytem, then Eq.() reduce to the difference equation of Eq.(4). What i intereting i the econd term of the expreion. C, D and n, can be ued together or in variou combination to model all practical tochatic effect. 3.7 Non-linear Time Serie Model (NARMAX) Thu far, we have only conidered time erie model of linear ytem. Time erie can alo be ued to model non-linear ytem. For intance, Ayt ( ) = But ( k) + But ( k) + But 3 ( k) yt ( ) + Cξ ( t) () Equation () i an example of an NARMAX (Non-linear ARMAX) model. If appropriate, we can add more cro-product and higher order term. Generally, if the equence y(t) i dependent on it pat value, ome input u and tochatic component ξ, we can write, ( ) yt ( ) = f ut ( τ), yt ( τ), ξ( t τ) τ + ξ( t) (3) where f(.) i ome functional. From the practical point of view, there i baically no retriction on how f(.) i realied. The non-linear model could therefore be explicitly written a a NARMAX model a in Eq.(), or the time-erie repreented indirectly by an Artificial Neural Network for example. 4. SUMMARY While dicrete tranfer function repreentation have a direct correpondence to the underlying continuou ytem, time-erie are inherently dicrete in nature. In thi ene, it i eay to include tochatic component into the ytem decription a random effect are of 8

'better' poed in the dicrete domain. For example, an integrated-noie equence in the continuou time domain ha infinite variance, and theoretically, control of thi proce i impoible. On the other hand, the incluion of integrated moving average in time erie doe not incur thi limitation. Neverthele, tranfer function model and time erie model are related, and although not identical, they are often treated in the ame manner when in ytem analyi and controller deign. In developing a decription of the proce, the tructure of the model i firt pecified. A uitable numerical technique i then ued to determine the parameter of the model from input-output data gathered from the proce. Tranfer function and time-erie model (including NARMAX tructure) are linear-in-the-parameter and therefore can be parameteried by popular leat-quare baed algorithm. For controller deign however, a 'control affine' (linear in the manipulated input, u(t)) repreentation would be ideal. The control problem then become trivial. Otherwie, numerical earch procedure will have to be invoked to generate an appropriate control ignal. of 8

APPENDIX : SAMPLED DATA SYSTEMS Becaue of the nature of digital device, ignal from plant have to be converted into a uitable form before it can be tranferred for proceing by a computer. Similarly, ignal generated by a computer mut be preented in a form uitable for receipt by the plant. The important piece of hardware that achieve thee tak are the: ampler analog-to-digital converter (ADC) digital-to-analog converter (DAC) ignal hold device The Sampler The ampler i eentially a witch, operating uually at fixed interval of time. When the 'witch' cloe, it grab or ample the output of the tranmitting device. It then tranfer the ampled ignal to a receiver. The ampler can operate on both continuou or dicrete ignal. Thu if the ource ignal i continuou, the output of the ampler i a erie of pule, and the magnitude of each pule i equal to the magnitude of the continuou ignal at the intant of ampling a hown in the figure below. CONTINUOUS SIGNAL SAMPLING AT INTERVALS SAMPLED SIGNAL ADC and DAC ADC convert ampled voltage or current ignal to their binary equivalent while DAC convert binary ignal to continuou ignal uch a voltage or current. Thee converter provide the interface between a computer and the external environment. of 8

Signal Hold Device The output of a ampler i a train of pule, regardle of whether the ource i continuou or dicrete. Thu the output of a computer after digital-to-analog converion i alo a train of pule. If thi i a control ignal, then unle the device receiving thi ignal, ay a pump or valve, ha integration capabilitie, then the proce will be driven by pule. Thi i obviouly not acceptable. So, in proce control application, the ignal from the DAC i alway 'held' uing hardware known a ignal hold device. The mot common i the Zero-Order-Hold, where each pule i held until the next pule come along, that i: COMPUTER SAMPLING AT INTERVALS SAMPLED SIGNAL ZERO- ORDER HOLD HELD SIGNAL Reult of Zero-Order-Hold 3 of 8

APPENDIX : THE Z-TRANSFORM The z-tranform i the mot commonly ued tool for the analyi of ampled data ytem. Suppoe we ample a continuou time variable f(t). Becaue the ampled ignal exit only at the ampling intant, the equence of pule can be repreented mathematically a: f () t for n=,,,... f ( nt ) = otherwie Again, to implify notation, let the ampled equence be denoted by f tranform of f(t) i defined a: t F () = f() te dt Since the ampled ignal f it: Since f * * t F () = f () t e dt * ( t). Now the Laplace * ( t) i a ubet of f(t), we can alo apply the Laplace tranform to * ( t) only exit at ampling intant, thi mean that we can replace the integral with a ummation, that i, Defining: z * * t F () = f () t e dt = f ( nt ) e n= e =, then nt F * nt n ( ) = f ( nt ) e = f ( nt ) z = F( z) n= n= Thi i the definition of the z-tranform of a continuou time ignal f(t) ampled with a ampling interval of, i.e. { } Z f () t = F() z = f ( nt ) z n= n 4 of 8

Thu, the z-tranform i merely the Laplace Tranform of a ampled data equence and a uch, inherit many of the propertie of the Laplace Tranform. Propertie Some of the more important propertie of the z-tranform are a follow: a) Linearity The z-tranform i a linear tranform. That i, given contant a and b and time variable f(t) and g(t): { } { } { } Z af () t + bg() t = az f () t + bz g() t = af() z + bg() z a) z-tranform of time delay If f(t-k ) i f(t) delayed by k ampling interval (k i an integer), and f(t) = for t <, then the z-tranform of f(t-k ) i given by: { } k Z f ( t kt ) = z F( z) a) Final Value Theorem Thi theorem allow the calculation of the final value of a z-tranformed function and i tated a: lim f ( t) = lim( z ) F( z) t z 5 of 8

Relationhip with the -plane -plane z-plane z-tranform The mapping from the -plane to the z-plane i accomplihed through the relationhip z= e Thi function map the whole of the left ide of the -plane to a unit circle on the z-plane a hown in the above figure. In the Laplace domain, ytem are table if they do no poe pole on the right half of the -plane. In the cae of ampled data ytem, they are table if they do not poe pole that lie outide the unit-circle in the z-plane. Example z-tranform a) z-tranform of a unit tep function t < The time domain repreentation of a unit tep function i: f ()= t t Thu { } Z f () t = F() z = f ( nt ) z n= n That i, F z f f T z f T z f nt z n () = () + ( ) + ( ) + " + ( ) + " n = + z + z + " + z + " which i an infinite erie, illutrating that the z-tranform operate on an infinite equence. Fortunately, there i alo a 'cloed-form' equivalent, a it can be hown that 6 of 8

z n z = + z + z + " + z + " = z and thi i the form that i alway preented in z-tranform table. b) z-tranform of an exponential decay The time function in thi cae i: f () t = exp( t / τ) n Thu F( z) = exp( nt / τ) z = + exp( T / τ) z + exp( T / τ) z + " n= which i another infinite erie. Again the cloed-form olution i available and can be verified by long diviion to be: z Fz () = exp( = T / τ) z z exp( T / τ) Inverion of z-tranform Like Laplace tranform, z-tranform can be inverted back into the time domain. Given a tranfer function, we can either apply long diviion to obtain the erie form of the ampled ignal or make ue of the table. The firt i imple but tediou and the reult may not be uitable for further analyi. Thu table are often ued. In thi cae, the tranfer function i factored into lower order component uing partial fraction expanion, and table are ued to look up the correponding time function of each component. The final reult i obtained by adding up thee individual time function. However, due to the nature of the problem, it i not often that z-tranform function need to be converted back to the time domain. Given the range of imulation tool available nowaday, it i uually impler to imulate the repone of the dicrete ytem to enable viualiation of repone characteritic. Block Diagram Manipulation The manipulation of block diagram of ampled data ytem i very imilar to that for block diagram of ytem expreed in the Laplace domain. However, becaue of the preence of 7 of 8

ampler, ome extra rule have to be adhered to. Conider a continuou proce under digital control via the claical feedback framework. The detailed block diagram of thi ytem including the preence of the ampler i, W() W(z) Σ E(z) U(z) Y() T D(z) ZOH() G p () CONTROLLER PROCESS Y(z) Note that the ampler determine the type of ignal propagating through the ytem. Hence the poition of the ampler are important. In particular, note that the tranfer function between Y(z) and U(z) for the following two ytem are different. Sytem A U(z) F() G() Y(z) Here, the relationhip between Y(z) and U(z) i given by: { } { } Y() z = Z F() Z G() U() z = F() z G() z U() z Sytem B U(z) F() G() Y(z) With thi ytem, the relationhip between Y(z) and U(z) i given by: { } Yz () = ZFG () () Uz () = FGzUz () () Note that in general, Z{ F() } Z{ G() } Z{ F() G() } 8 of 8