Predictive Control of a Laboratory Time Delay Process Experiment

Transcription

1 Print ISSN:3 6; Online ISSN: DOI:0478/itc Preictive Control of a aboratory ime Delay Process Experiment S Enev Key Wors: Moel preictive control; time elay process; experimental results Abstract he paper presents the esign an implementation of a Moel Preictive Control (MPC) scheme of a laboratory heatexchange process with a significant time elay in the input-output path he optimization problem formulation is given an an MPC control algorithm is esigne achieving integral properties Details relate to the practical implementation of the control law are iscusse an the first experimental results are presente Introuction he presence of a time elay in the process to be controlle maes the esign problem more ifficult an reuces the achievable performance within the classical feebac loop structure he Moel Preictive Control [] is inherently an appropriate option when the control of such processes is consiere since its funamental mechanism of control elaboration relies on preiction of the process behavior which in principle counteracts the elay Moreover the time elays are easily incorporate an hanle in iscrete-time escriptions a fact that correspons well to MPC being inherently iscrete-time Nowaays linear MPC in its many variants may be calle a mature inustrial technology in the process control fiel with well-evelope theoretical basis an methoology that enable the control system characterization with respect to stability an performance Following the ever increasing computational power of moern computers the application areas of MPC grow in number an are beginning to brige the milli- an micro-secon sampling perio range [] (where a goo overview of the current an future perspectives challenges an research irections can also be foun) hese avances practically support the iea to transfer MPC algorithms towars an into the lower level control loop of the plant control system hierarchy an to tae avantage of their capabilities In this paper a linear MPC scheme is implemente which controls a laboratory heat-exchange process with a significant time elay in the input-output path Base on a iscrete-time moel of the process a formulation of the optimization problem with respect to the control input changes is evelope an an output feebac MPC algorithm achieving integral action is esigne Some experimental results are presente along with etails concerning the implementation of the control algorithm Process Description an Mathematical Moel Structure he laboratory heat-exchanger is epicte in figure he heate flui being water passes through an electrical flow heater with a constant flow rate an then enters a long coile pipeline he water temperature at the pipeline exit point is the controlle variable he transportation process in the pipeline introuces a significant time elay in the system input-output path he control input is the uty cycle - μ in the PWM scheme use to moulate the power to the electric heater ie the fraction of the maximal heating power available he main isturbance is the temperature of the inflow stream - θ in Within the existing installation it is observe that after a certain perio of operation the thermal capacity of the structure begins to affect the outlet temperature Besies with the pipeline not being well insulate the outlet temperature is affecte by the heat transfer towars/from the environment when there is a significant eviation from the Figure 03 9

2 ambient temperature Finally the variations of the flow rate will significantly affect the performance of the control system since the flow rate itself is a major factor etermining the ynamic properties of the process hough the constant consiere is set by the manually operate valve at the piping input for moeling an throughout the experiments such variations exist ue to the pressure variations at the installation input he outlet temperature is measure by a 4-0 ma temperature transmitter using a Pt00 resistor as a sensor an a temperature inicator is available for the inlet temperature he PWM control is implemente with the help of a Soli State Relay (SSR) A set of open loop experiments on the process were realize in orer to acquire ata for ientification purposes It was foun that for a constant flow rate the process has a linear steay-state characteristic an quite similar transient curves throughout the entire operating omain Figure shows the process responses to a series of two step changes of the control input with the flow rate set at f = 5 l/min he parameters of the secon-orer plus time elay moels were fitte to the actual responses an the following parameters were obtaine () W( p) θ ( p) e μ ( p) ( p )( p ) ou t( h) = = μ τ p transfer function moel + + with () μ = 085 C / % = 60 s = 40 s τ = 95 s for the output temperature θ out ; () μ = 03 C / % = 40 s = 40 s τ = 30 s for θ h the temperature at the heater exit point he responses of the moels obtaine are shown on the same graph with the processes (figure ) he firstprinciple moeling consierations suggest that the entry point of the isturbance in the system is at the input so the same transfer functions can be consiere but with a steaystate gain equal to one ( = ) Figure 3 MPC Convex QP Problem Formulation with Respect to the Control Input Changes In this paragraph the formulation of the optimization problem is escribe he unerlying process moel use is of the following iscrete-time escription: () x + = Ax + bu + b y = Cx ( ) ( ) ( ) ( ) ( ) ( ) where x () R n y () R u () R () R A R n n b R n b R n C R n s () enotes the value of the signal s(t)at the sampling instant t = 0 an 0 - the sampling perio he input structure is chosen accoring to the process moel A basic requirement an esign specification towars a control system is the ability to achieve offset-free control for constant reference in the presence of unnown constant isturbances an/or moel mismatch proviing stability wo basic approaches exist to achieve an integral action in MPC he first approach introuces an estimate isturbance (possibly a vector) in the escription in which the steaystate values (input an state) are assume epenent [34] he secon approach use here [] is base on the following reformulation of the process moel in which the ynamics are escribe in terms of the changes as follows: Δ x = AΔx + bδ u + b Δ ( + ) ( ) ( ) ( ) y = y + CAΔ x + CbΔ u + Cb Δ ( + ) ( ) ( ) ( ) ( ) where Δ x( ) = x( ) x ( ) Δ u( ) = u( ) u( ) Δ ( ) = ( ) ( ) he moel is expresse in the following form: x( + ) = Ax( ) + bδ u( ) + bδ( ) (3) y( ) = Cx ( ) where

3 A 0 b b x x A b b n+ [ n] n+ n+ n+ n+ ( ) Δ ( ) y( ) R R R R CA Cb Cb n+ [ n] R C 0 C he following so-calle preiction horizon vectors are efine: y ( ) [ ( ) ( ) ( )] N y + y + y + N R a preiction horizon output vector; y ( ) [ ( ) ( ) ( )] N R yr + yr + yr + N R a preiction horizon reference signal vector; Δ u = [ Δu Δu Δu ] R N a preiction ( ) ( ) ( + ) ( + N ) horizon input change vector with Δ u( + i ) = u( + i ) u( + i ) ; Δ ( ) [ ( ) ( ) ( )] N = Δ Δ + Δ + N R a preiction horizon isturbance change vector where N enotes the preiction horizon s ( + i ) being the preicte future value of the signal s(t) at the sampling instant t = 0 Remar Δ u( ) is the control input change to be applie to the plant at t = 0 ( zero-orer hol is assume at the plant input so that the control is ept constant uring each sampling interval) etermine as a result of the optimization proceure performe at t = 0 he objective function is expresse in a preiction horizon vector form as: (4) J ( ) = ( yr( ) y( ) ) Q( yr( ) y( ) ) + Δu( ) PΔu ( ) where Q = iag (q i ) R N N i = N an P = iag (p i ) R N N i = N weight coefficients We have the following: (5) y () = MΔu () + Gx () +M Δ () where Cb CAb Cb 0 0 M = CA b CAb Cb 0 N N CA b CA b Cb Cb CAb Cb M = CA b CAb Cb 0 N N CA b CA b Cb G CA CA CA 3 = CA N N N N n M M R G R Remar If no apriori information is available a constant isturbance is usually assume uring the preiction horizon that is ( + i ) = ( ) i= N which in turn leas to Δ (+i ) = 0 i = N hus all elements of the vector Δ () except for probably the first one are equal to zero (if unmeasure isturbance is consiere the last term in (5) may not be consiere) he objective function is now rewritten as (6) J ( ) = ( yr ( ) MΔu( ) f( ) ) Q( yr( ) MΔu( ) f( ) ) + Δu( ) PΔ u( ) = ( )( ) ( ) ( ( ) ( )) ( ) ( ( ) ( ) ) = Δ u M QM+ P Δ u + f yr QMΔ u + yr f Q ( yr( ) f( ) ) where f () = Gx () + M Δ () he last term can be remove from the expression since it is no function of the optimization variables et enotes the control horizon hen the following is obtaine: Δu( ) Δ u( ) = 0 Δu [ N ] () = [Δu ( ) Δu ( +) Δu ( +-) ] R an the objective function can be rewritten as J ( ) = Δ u( ) Λ M QM + P ΛΔ u( ) + ( f( ) yr( ) ) QMΛΔu ( ) (7) ( ) I[ ] where Λ = 0 I [ N ] [ ] the ientity matrix with the respective imensions he Hessian an the graient of the objective function are (8) J () = Λ (M QM + P)Λ Δu () +(f () y R() ) QMΛ J () = Λ (M QM + P)Λ It is easily shown that the Hessian Λ (M QM + P)Λ is a symmetric positive efinite matrix hus J ( ) is a convex function of Δu () on R Expressing the objective function in terms of the control input changes has also the following avantages: Problem size reuction the optimization problem is on R = 3 6(7) typically Equality constraints appearing when < N (which is practically always the case) are eliminate that is they are incorporate in the objective function (an aitional useful feature as a result of the problem size reuction) 03 3

4 Ultimately this leas to an unconstraine optimization problem Inequality constraints are introuce in the optimization problem to account for the limite power availability an the physically possible in the experiment process control inputs he control input must satisfy the following constraints (9) 0 u () u max hey are written in a stanar form in terms of the control input changes as (0) ϕ ϕ + i u + Δu 0 i = + i u + Δu u 0 i = i( ) ( ) ( j) j= i( ) ( ) ( j) max j= We have + i + i ϕi( ) = ϕi( ) ϕi( ) = u( ) + Δu( j) umax u( ) Δ u( j) i = j= j= he sets SΔ ( ) = { Δu( ) 0[ N] u( ) [ N] umax} (u () being efine in a similar way to Δu () with respect to the control inputs rather than to the control input changes) are convex since it can be shown that they are obtaine through convexity preserving transformations from the sets S u() = {u () 0 [N] u () [N] u max } which are obviously convex hus the overall problem is a convex optimization problem In orer to incorporate the inequality constraints into the objective function a logarithmic barrier function [5] is constructe as φ( ) = φi( ) i= where φ i() = ln( ϕ i() ) ln( ϕ i() ) = ln(ϕ i() ) It is easily shown that ϕ i( ) are concave an positive on S u() = {u () S u() u () hu max } from which it follows that the functions φ i() are convex on S u() an ( S Δ( ) efine in the same way but on S u( ) ) Hence φ () is convex We have φi( ) = ϕi( ) ϕ ; i( ) φ = ϕ ϕ ϕ ; i( ) i( ) i( ) i( ) ϕi( ) ϕi( ) ( ) φi( ) i= ; φ = φ( ) φi( ) i= = ; S Δ ( ) + i i( ) i( ) i( ) i( ) 0 [ i] i( ) u( j) umax u( ) j= i elements ϕ = ψ ψ ψ R ψ = Δ + ; [ i i] ϕi( ) = S = [ i i] 0[ i i] 0[ i i] S 0 R he graient an the Hessian operators are efine as follows: f f f f n f = f R x R f R x x x x an n f f f n n f f = f x x x x x x xn n R he output an state constraints expresse using the unerlying preiction moel can be introuce in the same way to the problem An approximate problem is formulate in which a barrier term is ae to the objective function accounting for the inequality constraints: () min J ( ) = J( ) + rφ( ) We also have ~ J () = J () + r φ () ; ~ J () = J () + r φ () As r 0 the approximate problem goes to an inequality constraine problem As it can be seen the approximate problem is an unconstraine convex optimization problem he MPC is introuce solving the above formulate unconstraine optimization problem at each sampling instant opt yieling the optimal sequence Δu () From the optimal sequence the first element is applie as a change in the control input to the process As seen from the formulation the term f () is upate at each sampling instant which requires state an isturbance information 4 Extension of the Basic Algorithm an Implementation Since information about all states is require as an initial conition for the optimization proceure the overall control law is augmente by a uenberger type of an observer base on moel () () x ˆ ˆ ˆ ( + ) = Ax( ) + bδu ( ) ( Cx ( ) ξ( ) ) where ξ () is the vector of measure variables ξ () [θ out() Δθ h() ] [y () Δθ h() ] with Δθ h() =θ h() θ h(-) he output matrix of the observer is augmente by a secon C n+ row an efine as C R C he observer gain θ matrix R n+ is chosen so that A C is Hurwitz (a close-loop observer is require since as a consequence of 3 03

5 the problem formulation A has a pole equal to ) hen the optimization problem solve at each sampling instant is constructe using the estimate values x ˆ ( ) that is the term f () in the objective function (7) an its graient (8) are calculate with the estimates No information of the isturbance has been use in the control law It shoul be note that if the reference signal for t = ( + ) 0 is nown at the sampling instant t = ( ) 0 then it is possible to etermine the control input (solve the optimization problem) to be applie in the interval 0 t < (+) 0 in the interval ( ) 0 t < 0 since all the require information is available at t = ( ) 0 his possibility is not exploite in the current implementation he optimization problem (0) is solve by implementing Newton s metho an bactracing the line search [5] with a fixe value of r (though not avise in [5] satisfying results were foun uring the preliminary tests) he value of r was set to 0-8 an Newton s algorithm suboptimality tolerance estimate by Newton s ecrement was set to 0-7 he overall MPC algorithm was implemente in Matlab using the capabilities of the Data Acquisition oolbox to interface the USB DAQ evice NI 6009 use to realize the physical interface to the process he iscrete-time preiction moel of the process is obtaine by iscretizing assuming zero-orer hols at the inputs phase-variable canonical form representation of the elay-free part (the rational part) of () hen the time elay is accounte in the moel by introucing n = t / 0 aitional states at the system output such as f f x j(+) = x j+() j = n an xn ( + ) = y( ) with y ( ) the output of the elay-free part of the process moel hus the following state space matrices were obtaine an inserte in () (we have n = n + accoring to the efinitions in Section 3): A = R a a a a b b = R 0 0 b b b b ( n + ) C ( n+ ) ( n + ) ( n + ) 0 = R 0 where a a F 0 b b H e ( [ ] ) a / a H = F = / / ( ) b b = H I F μ / + 5 Some Experimental Results For the experimental results iscusse in this paragraph the following values of the parameters were set: N = 0 = 4 Q = iag(q i ) q i = i i = N P = iag(p i ) p i = 000 i = N he mean time neee for solving the optimization problem on the computer configuration use was less than 00 s A sampling perio of 0 s was selecte he PWM perio was set to s thus having 5 PWM perios per a sampling perio Several experiments were conucte with the implemente moel preictive controller an some of the obtaine transients are shown in figure 3 For the experiment shown in figure 3a the process moel was constructe using the rational part of () with parameters given in () (thus using a transfer function relative to the output temperature) he number of the aitional states was set to 9 ie n = 9 (an equivalent elay of 90 s) he secon row in the observer matrix was efine as Cθ [ n 3] 0 0 [ 5] thus accounting for the 30 s elay foun in () he coefficients in the observer gain matrix were set by a trial an error manner an the resulting close-loop ynamics were etermine by the following poles: an the rest of the poles being equal to 0 For the experiment shown in figure 3b the process moel was constructe using the rational part of () with parameters given in () he number of the aitional states was set to 0 ie n = 0 (an equivalent elay of 00 s) he same observer output matrix was use he observer gains were set to attribute analogous close-loop ynamics As seen on both graphs a zero offset is inee achieve an the control input rives the outlet temperature to its reference with satisfactory transient performance Slow variations of the inlet temperature within the range C were observe uring the experiments As expecte the inuce effects are compensate by the control law an any eviations are harly visible on the curves Besies the first transient in both series iffers performance-wise from the following ones which is probably ue to the initial convergence of the observer estimates Further experiments with more aequate initialization of the observer estimates an optimize observer close-loop ynamics are planne 6 Conclusion An MPC scheme was implemente in orer to control a laboratory heat-exchange process experiment with 03 33

6 34 03 Figure 3 ransient responses (a)

7 Figure 3 ransient responses (b) 03 35

8 a significant transportation time elay he time elay is reflecte in the preiction moel by augmenting the state vector with the respective number of aitional elaye states hen a reformulation of the escription in terms of the control an state changes is use to attribute an integral action to the overall control law which incorporates also a uenberger observer in orer to estimate the non-measure state variables First experiments with the controller propose were conucte which have confirme the expecte performance Zero offset control was achieve without using isturbance information in the elaboration of the control signal References Maciejowsi J Preictive Control with Constraints Prentice Hall 00 Morari M Preicting the Future of Moel Preictive Control Systems an Process Control Centennial Session 008 Annual AIChE Mtg Maeer U Augmente Moels in Estimation an Control DSc Dissertation EH Zurich 00 No Faanes A S Sogesta Offset-Free racing of Moel Preictive Control with Moel Mismatch: Experimental Results In Eng Chem Res 44 () DOI:00/ie0494y 5 Boy St Vanenberghe Convex Optimization Cambrige University Press 004 Manuscript receive on 03 Stanislav Enev (born 980) receive the MSc egree in Electrical Engineering in 004 an the PhD egree in Inustrial Automation in 009 from the echnical University Sofia French anguage Department of Electrical Engineering Since March 008 he is with the Department of Inustrial Automation (FA) echnical University Sofia as an Assistant Professor His current research interests are in nonlinear rive an electromechanical systems control theory an applications moel-preictive an aaptive process control Contacts: echnical University Sofia Faculty of Automation Department of Inustrial Automation enev@tu-sofiabg 36 03