Noise Modelling and MPC Tuning for Systems with Infrequent Step Disturbances

Transcription

1 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, Noise Modelling and MPC Tuning for Systems with Infrequent Step Disturbances Jakob K. Huusom Niels K. Poulsen Sten B. Jørgensen John B. Jørgensen Department of Chemical and Biochemical Engineering, Technical niversity of Denmark, Building 9. DK - 8 Lyngby, Denmark ( jkh@kt.dtu.dk, sbj@kt.dtu.dk) Department of Informatics and Mathematical Modelling, Technical niversity of Denmark, Building. DK - 8 Lyngby, Denmark ( nkp@imm.dtu.dk, jbj@imm.dtu.dk) Abstract: In this paper, an offset-free SISO MPC implementation based on an ARX model of the system dynamics is investigated. Special emphasis is directed to achieving good closed loop performance for systems which may be step wised perturbed by a sustained, unmeasured disturbance. Hence a noise model which expresses the behaviour of this non-stationary noise process is sought. Tuning of the ARX-based MPC implementation is discussed and illustrated in a simulation example. Guidelines for tuning of the free parameters are presented. Keywords: Model Predictive Control, Autoregressive models, Noise models, Controller tuning. INTRODCTION Model Predictive Control (MPC) is a state of the art control technology which utilizes a model of the system to predict the process output over some future horizon and solve a quadratic optimization problem with the control signal as decision variables. Early achievements and industrial implementations of Model Prediction Control include IDCOM and Dynamic Matrix Control [Richalet et al., 978, Cutler and Ramaker, 98]. These algorithms were based on step or impulse response models. More general linear input-output model structures were used in Generalized Predictive Control [Clarke et al., 987], but an interest in MPC implementations based on state space models were created by the seminal paper Muske and Rawlings [99]. The state space approach provides a unified framework for discussion of the various predictive control algorithms and is well suited for stability analysis [Mayne et al., ]. Inthispapertheplantisrepresentedbythelinear,discrete time, single input/single output ARX model (). This model class is selected based on a system identification argument. This class is linear in the parameters and the parameter estimation problem is convex. A(q )y(t) = B(q )u(t)+ε(t) (a) where A and B are polynomials of order n in the backward shift operator q and ε(t) N iid (,σ ). A(q ) = +a q +a q + +a n q n B(q ) = b q +b q + +b n q n (b) (c) Tuning of the free parameters in a MPC implementation in order to achieve good closed loop performance is far fromtrivial[qinandbadgwell,,garrigaandsoroush, 8]. Typical tuning parameters are the prediction horizon and the weight matrices in the performance cost function which balance the relative penalty between tracking errors and the size of the control moves. Other tuning parameters can be related to the implementation ensuring offset-free tracking and finally the reference trajectory for the outputs [Maciejowski, ]. In several applications and particularly in chemical engineering a process can be perturbed by an unmeasured step disturbance. Examples are refineries and cement industries where the composition of the crude oil or raw minerals may change significantly when feed is changed from one source to another. In biochemical production the problem may appear in continues downstream processing where the feed comes from batch processes. The purpose of this contribution is to present a simple ARX-model based MPC implementation for SISO systems which guaranties offset-free tracking and rejection of unmeasured non-zero mean disturbances. Tuning rules are developed for the free parameters in this implementation in order to balance both fast disturbance rejection versus noise sensitivity and the variance of the closed loop input and output signals. This paper is organized as follows; in Sec. the ARX-model based MPC is introduced and in Sec. noise models are proposed in order to ensure offset-free tracking and good closed loop behavior. Sec. 4 illustrates the tuning in a simulation example. Conclusions are presented in Sec. 5.. ARX-MODEL BASED MPC The ARX model () may be realized as a stationary state space model in innovation form x k+ = Ax k +Bu k +Kε k (a) y k = Cx k +ε k (b) with the matrices (A,B,K,C) in observer canonical form Copyright by the International Federation of Automatic Control (IFAC) 6

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, A = a a a n B = b b n a n K = a. a n C = [ ] The optimal predictions in the stationary state space model in innovation form () is based on computation of the innovations ε k = y k ŷ k k () using the measurement y k at time k and the one-stepahead prediction, ŷ k k = Cˆx k k. The one-step-ahead prediction of the states and outputs are ˆx k+ k = Aˆx k k +Bu k k +Kε k (4a) ŷ k+ k = Cˆx k+ k (4b) and similarly the (j + )-step-ahead (j ) predictions are ˆx k++j k = Aˆx k+j k +Bu k+j k j =,...,N (5a) ŷ k++j k = Cˆx k++j k j =,...,N (5b) The l -based constrained predictive controller uses an objective function of the form φ = N (ŷ k++j k r k++j ) +ρ u k+j k (6) j= which obviously depends on the control variables, hence the optimal control problem is min u k+j k φ = φ({u k+j k } N j= ) s.t. (4),(5) (7) u min u k+j k u max j N u min u k+j k u max j N with u k+j k = u k+j k u k+j k (j N), u k k = û k k, and N = {,,...,N }. The optimal solution is denoted {û k+j k } N j=. The constrained optimal control problem (7) can be converted into a standard convex quadratic program, see e.g. Huusom et al. []. Closed loop stability will be achieved through the choice of a sufficiently long prediction horizon rather than including terminal constraints [Rawlings and Mayne, 9]. Any industrial implementation will suffer from using an approximation of the plant in the controller, so our aim is focused on good closed loop performance rather than proving nominal stability. The ARXMPC will therefore handle any system which can be reasonably approximated by an ARX model.. Offset-free ARXMPC Since one objective of the control is to ensure offset-free tracking, it is proposed to identify an ARX model of the form () from a set of input/output plant-data and base the control implementation on the model A(q )y(t) = B(q )u(t)+η(t) (8) where η represent a noise sequence of some sort which is notnecessarilywhite.themotivationbehindthisreformulation is that the noise to the true process is assumed to contain both the random sequence ε(t) and a deterministic term d(t) which exhibits a series of infrequent stepwise changes. Therefore this term can be assumed constant over a limited time horizon. The challenge is to provide a reasonable noise model for η which reflects the process conditions and produce good closed loop performance for the controller. Hence a noise model is chosen based on the process characteristics rather than identifying the noise structure as part of the system identification. The real noise structure for the process can be modelled as η(t) = d(t)+e(t), e(t) N iid (,σe) (9) e(t) being the model representation of ε(t) and d(t) the stepwise deterministic change. In case the linear noise model () is used with a fixed parameterization of the polynomials G(q ) and F(q ) this can be exploited in the identification of the polynomials A(q ) and B(q ). η(t) = F(q ) G(q ) ν(t), ν(t) N iid(,σν) () where ν is white noise. The plant data {Y,} should be filtered through the inverse of the noise model prior to the parameter estimation to give {Ỹ,Ũ} which is then used for the identification of the ARX model. Hence the best fitwillalsoreflecttheactualmodelusedforthepredictions in the MPC. The system (8) with the linear noise model can be realized as an ARMAX model Ā(q )y(t) = B(q )u(t)+c(q )e(t) where Ā(q ) = G(q )A(q ) B(q ) = G(q )B(q ) (a) (b) (c) C(q ) = F(q ) (d) When transforming the ARMAX model () to the state space form () for the MPC, the A(q ) and B(q ) are substituted by Ā(q ) and B(q ). The coefficients in the C(q ) polynomial are included when forming K matrix K T = [c a c a... c n a n ] (). Analysis of the closed loop variance The performance cost function in (6) express the trade off between the tracking error and the aggressiveness of the controller through the parameter ρ. In reality the interesting question is: How is the mapping from ρ to the actual closed loop performance, such as the input and output variance? This mapping depends on the system itself and the model which is used in the controller. It is convenient to investigate the infinite horizon solution to the optimal control problem (7), excluding constraints. This is the LQG control analogue to the MPC, and has the advantage of resulting in an explicit expression for the system states, the input and the state estimates. It is important to exploit the unique correlation between measurement and process noise imposed by the innovation form [Åström and Wittenmark, 997, Kailath et al., ]. Furthermore a distinction needs to be made between the true system, assumed here to be the ARX-model, (), and the model used for control, (), which is different due to inclusion of the noise model.. A NOISE MODEL FOR ARXMPC In this section noise models for η in (8) are proposed which aims at providing both offset-free tracking for the 7

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, controller and an appropriate noise modelling. A first trivial case is η k = e k () which means that (8) is equal to the identified ARX model () and offset-free tracking is not possible due to the unknown and non-zero term d. An alternative choice is to adopt the noise model from GPC and consider the process noise as integrated white noise [Bitmead et al., 99]. η k = q e k (4) this would effectively mean that we model the constant plus random noise as q e(t) = d+ε(t) e k = ( q )ε k = ε k ε k The benefit from this model is that a constant nonzero mean contribution can be removed since d does not change between samples. When integrated noise is used as the noise model in the process model which is implemented in the controller, offset-free tracking can be obtained at the expense of an increased variance. Modelling the noise as integrated white noise in the ARX-model (8) means that F(q ) = and G(q ) = q in the ARMAX description (). A third, and the alternative of interest, is to combine () and (4) such the noise model becomes an addition of a Gaussian signal and a drift term. η k = q w k +v k (5) Bothw k andv k areassumedzero-meangaussianprocesses with variance σw and σv. By an appropriate selection of the variances of w(t) and v(t) the statistical properties of the real noise signal, with a Gaussian and a deterministic signal with stepwise changes, can be approximated. The model (5) can be simplified to the expression η k = q q e k (6) where is a parameter in the interval [;] and e t is a zero-mean Gaussian process with variance σe. sing the identified ARX model combined with the noise model, (6), means that F(q ) = q and G(q ) = q in the ARMAX description (). It is seen that for equal to the previously described noise model (4) appears. = will give pole/zero cancelation and the basic noise model for a true ARX model, (), appears. These two models are therefore the limits for the combined model (5). The values of and σe are given by = ( ) +4κ (7a) κ σe = σ v (7b) where κ = σ v/σ w expresses the ratio between the contributions from w t and v t. A derivation of these equations are provided in appendix and a curve showing as function of the ratio σ v/σ w is drawn in Fig.. Knowledge of the frequency of the steps in the disturbance to a process can be translated into a value for σw which means that an optimal value of can be selected if σv is estimated. This log(σ /σw v ) Fig.. as function of the ratio between the variances σ v and σ w from equation (5) wouldreflectasituationwherethedisturbanceisknownto enter the system due to scheduled changes to the process. Due to the integrator this noise model will still remove the undesired effect of a constant disturbance q e(t) = d+ε(t) q ( q )e k = ( q )ε k = ε k ε k This model is still characterized by the integrator, but the noise model is extended with an adjustable stable zero to provide some flexibility. This flexibility balances the speed at which a step disturbance can be rejected versus how noisy this estimate is. To investigate this trade off the optimal one step ahead predictor for the noise process (6) canbewritten,exploitingthearmastructure.asaresult the following process can be deduced for the disturbance estimate as function of the actual disturbance in (9) ˆη k+ k = q η k (8) In Fig. (a) the effect, has on the variance of the disturbance estimate, based on (8) is shown. The resulting effect on the closed loop output and actuator variance will be affected by the system dynamics and the controller tuning. It is seen that the price of a dead-beat observer is that the disturbance estimate is as noisy as the output measurement itself. In Fig. (b) the speed of the observer is depicted as function of, where the steep is given as the number of time unit it takes the observer to estimate 6%, 95% or 99% of the change from a step disturbance. It is seen from these plots in Fig. that a value of =.7 seems to be well suited. For this value of, 95% of an unmeasured disturbance is estimated within less than samplesandthevarianceofthedisturbanceestimateisless than % of the process noise. Higher values of means a significantly slower disturbance estimation and for lower values the variance of the estimate is increasing without giving a much faster estimator. Filtered integrated white noise as in (6) or other filters have been used widely in early MPC scheems such as GPC or IMC when the integrated white noise model (4) was consideredtooaggressiveorwhenamodelofthearimax structure had been identified Garcia et al. [989], Lee et al. [994]. A more general noise model than (6) includes an observer polynomial T(q ) Åström and Wittenmark [997]. 8

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, var(d est ) units (samples) τ ~ 6 % τ ~ 95 % 5τ ~ 99 % (a) Variance of disturbance estimate (b) Speed of observer Fig.. The variance of the disturbance estimate as function of in the noise model (6) assuming a unit variance for the measurement y(t) are given in figure a). In figure b) the speed in which the observer can estimate 6 %, 95 % or 99 % of a deterministic step change in terms of time units assuming the sample time is one unit. η = T(q ) q e k (9) The tuning of the parameters in this polynomial has in GPC mainly been based on robustness issues toward under modelled dynamics leading to a degree of the observer polynomial as high as for A-polynomial [Clarke and Mohtadi, 989, Camacho and Bordons, 4]. ˆdk =.5 =.7 =.9 4. A SIMLATION EXAMPLE It is of interest to see how the MPC implementations performgiventhedifferentnoisemodels,bothwithrespect to rejection of zero mean random noise and for the case whereastepdisturbanceentersthesystem.forthesystem with noise model () or (4) the MPC is based on the estimated ARX model or the ARX model with integrated white noise respectively. The two implementations are in short referred to as the ARXMPC and the ARXMPC. For the noise model (6) the implementation will be referred to as the Extended ARXMPC (E ARXMPC) duetotheadditionoftheonelagmovingaveragedynamics in the noise model. For this model =.7 will be used based on the analysed trade off between speed of convergence and the variance of the disturbance estimate in Sec. The only free parameter is the weight ρ in the cost function (6). It specifies the relative penalty on the control move compared to the tracking error. In the following a series of closed loop simulations with different MPC control implementations will be performed and compared in terms of performance on a numerical example. The example will use the same ARX-model as the true system and for the model in the MPC. The parameters for the model () are A(q ) =.4q +.5q.6q (a) B(q ) =.5q (b) σ =. (c) This model has a pole in.9 and a set of complex poles in.75±.7i. The total simulation horizon is 5 samples. Between time 5 and a step is introduced in the reference and between 5 and an unmeasured step disturbance is acting on the system. The input will be Fig. 4. The disturbance estimate from closed loop simulation with E ARXMPC tuned with ρ =. and {.5,.7,.9}. Two steps are induced in the reference signal and an unmeasured step disturbance of.4 is added between time 5 and. constrainedbetweenu [ ;]butthecontrolmoveisleft unconstrained.inallthreesimulations thevalueofρis.. The closed loop response is shown on Fig.. The random noise sequence used is kept the same for all runs in order to compare performance. It is seen that the modification of the ARX model used by the controller is necessary in order to estimate and reject a constant unmeasured disturbance. It is furthermore seen that introducing a pure integrator in the ARXMPC renders the input more aggressive compared to the ARX-MPC implementation. The extension in the E ARXMPC produces offset-free trackingbutgivesalessaggressivecontrolcomparedtothe pure integrator. This is due to the trade off in selecting an appropriate value of c.f. Fig. The trade off in selecting is illustrated in Fig. 4 where the disturbance estimate is plotted for a set of closed loop MPC simulation with the E ARXMPC. It is clearly seen that a fast estimator leads to a noisier estimate. Hence this implementation is the preferred and will be investigated further for tuning of ρ in the performance cost function. 9

5 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, ARXMPC ARXMPC E ARXMPC (a) Deterministic sim ARXMPC (b) Deterministic sim ARXMPC (c) Deterministic sim E ARXMPC (d) Noisy sim (e) Noisy sim (f) Noisy sim. Fig.. Closed loop simulation of all three MPC implementations with ρ =. and =.7. Two steps are induced in the reference signal and an unmeasured step disturbance of.4 is added between time 5 and.,,6 MPCsim LQGcal,,6 MPCsim LQGcal..6 MPCsim LQGcal,,8,,8..8,4, (a) ARX (b) ARX (c) E ARX( =.7) Fig. 5. Pareto plot of closed loop input and output variance when the ARX, ARX and E ARX( =.7) model are used in the control implementation. The results are presented based on long MPC simulations (-) and calculated based on a LQG control loop (x) for ρ {.,.5,.,.5,.,.5,,5}. 4. Tuning of ρ in the performance cost function This weight balances the relative penalty on the tracking error versus the magnitude of the control move, implicitly it gives the bandwidth of the controller. By increasing the weight on the control move one achieve a less aggressive control at the expense of larger tracking errors. A penalty on the control or control move is desired since the minimum variance controller is known to have a small stability margin and hence poor robustness to modelling errors. When selecting an appropriate value for the penalty, one is ultimately interesting in low input and output variance for the closed loop in the case where neither the reference is changing nor disturbances affect the system. Hence for a long simulation horizon ( samples) the closed loop input and output variance is recorded for the unconstrained MPC implementations with the different models for a range of value of ρ. These are presented in the Pareto plots in Fig. 5. It is reasonable to use the unconstrained controller in this analysis since the process should be operated in a regime which limit activation of the constraints [Loeblein and Perkins, 999]. In Fig. 5 the closed loop input/output variances are also computed using the corresponding LQG controller. It is seen on the figure that this type of plot is very useful when selecting an appropriate value of the penalty ρ. It is further more seen that it is sufficient to base this analysis on the calculated variances from the LQG controller since this is practically identical to the unconstrained MPC with a long horizon. Since the expression for the variances for the LQG controller is explicit, this formulation can be used in an optimization to selection of cost function weight given some objective e.g. minimize the sum of the input

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, X:.446 Y:.846 log().5.5 X:.6 Y:.74 dlog()/dlog() (a) Pareto plot variance 5 4 log() (b) Pareto plot log. of var log() (c) First derivative Fig. 6. Pareto plot of closed loop input and output variance for the E ARXMPC with =.7. The preferred tuning ρ =.5 is found based on minimum of the plot of the derivative, and the corresponding point in the Pareto plots are indicated. and output variance. For the E ARXMPC with =.7 the closed loop input and output variances are calculated based on a 5 values of ρ in the interval [ 7 ; ] from the LQG formulation. This basicly spans the region from minimum variance control to no feedback. The results is plottet in Fig. 6 as the Pareto plot, but also the Pareto plot of the logarithm to the variances are plotted and the fist derivative of this curve. The point where the curve for the logarithm of the variances has the steepest negative slope is identified as the minimum of the derivative curve. A value of ρ =.5 is found for this location. The corresponding location is indicated on the variance curves. It is seen that this point gives a good balance between input and output variance. In Fig. 7 the closed loop response of the system using this tuning is plotted. It is seen compared to the tuning with ρ =. on Fig. (f) that the input is much less aggressive but the output variance is increased. Which tuning is to be preferred depends on the actual application but the method illustrated here may serve as a good guess for an initial tuning Fig. 7. Closed loop simulation with E ARXMPC with ρ =.5 and =.7. Two steps are induced in the reference signal and an unmeasured step disturbance of.4 is added between time 5 and. 5. CONCLSIONS A MPC implementation based on a ARX-model representationofthesystemhavebeenpresented.offset-freetracking is achieved based on choosing an appropriate noise model, which is motivated by systems which experience infrequent unmeasured step disturbances. It is found that modelling the noise as integrated noise from a moving average process of one lag is a reasonable representation. The free parameter,, in the noise model can be selected basedonknowledgeofthenoisevarianceandthefrequency of the step wise disturbances. In case this frequency is unknown, it is shown that =.7 provides a good balance between fast converge and low noise sensitivity of the disturbance estimate. For a given choice of in the noise model, the remaining tuning parameter in the MPC is the penalty in the performance cost function, ρ, which balances the relative penalty on the tracking error versus the control move. It is illustrated how Pareto curves of the input and output variance can be used to tune ρ to find an appropriate trade off. In general the initial tuning of this weight may conveniently be based on the corresponding LQG problem. ACKNOWLEDGEMENTS The first author gratefully acknowledges the Danish Council for Independent Research, Technology and Production Sciences(FTP)forfundingthroughgrandno REFERENCES Karl Johan Åström and Björn Wittenmark. Computer Controllerd Systems: Theory and Design. Information and systems science series. Prentice Hall,. edition, 997. Robert R. Bitmead, Michel Gevers, and Vincent Wertz. Adaptive Optimal Control. The Thinking Man s GPC. Prentice Hall, Sydney, Australia, 99. Eduardo F. Camacho and Carlos Bordons. Model Predictive Control. Advanced Textbooks in Control and Signal Processing. Springer, ed. edition, 4. David W. Clarke and C. Mohtadi. Properties of generalized predictive control. Automatica, 5(6): , 989. D.W. Clarke, C. Mohtadi, and P. S. Tuffs. Generalized predictive control - part i. the basic algorithm. Automatica, ():7 48, 987. C. Cutler and B. Ramaker. Dynamic matrix control - a computer control algorithm. In Proceedings of the Joint Automatic Control Conference, 98.

7 Preprints of the 8th IFAC World Congress Milano (Italy) August 8 - September, Carlos E. Garcia, David M. Prett, and Manfred Morari. Modelpredictivecontrol:Theoryandpractice-asurvey. Automatica, 5():5 48, 989. Jorge L. Garriga and Masoud Soroush. Model predictive controller tuning via eigenvalue placement. In Proceedings of the American Control Conference, pages 49 44, 8. JakobKjøbstedHuusom,NielsKjølstadPoulsen,StenBay Jørgensen, and John Bagterp Jørgensen. Tuning of methods for offset free mpc based on arx model representations. In Proceedings of the American Control Conference - ACC, pages 55 6,. T. Kailath, A. H. Sayed, and B. Hassibi. Linear Estimation. Prentice Hall information and system sciences series. Prentice Hall,. Jay H. Lee, Manfred Morari, and Carlos E. Garcia. Statespace interpretation of model predictive control. Automatica, (4):77 77, 994. C. Loeblein and J. D Perkins. Structual design for online process optimization: I. dynamic economics of mpc. AIChE Journal, 45(5):8 9, 999. Jan M. Maciejowski. Predictive Control with Constraints. Prentice Hall,. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 6(6):789 84,. Kenneth R. Muske and James B. Rawlings. Model predictive control with linear models. AIChE Journal, 9(): 6 87, 99. S. Joe Qin and Thomas A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, (7):7 764,. James B. Rawlings and David Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, 9. J. Richalet, A. Rault, J. L. Testud, and J. Papon. Model predictive heuristic control: Application to industrial processes. Automatica, 4(5):4 48, 978. q w t +v t = q q e t w t +( q )v t = q e t w t +v t v t = e t e t (A.) Equality of the variance and the co-variance of equation (A.) result in the following set of coupled equations σ w +σ v = (+ )σ e σv = σe from (A.4b) it can be seen that (A.4a) (A.4b) σe = σ v (A.5) hence eliminating σe from (A.4a) and solving for gives σ w +σ v = (+ ) σ v σ v (σ w +σ v)+σ v = and the roots are = σ w +σv ± (σw +σv) 4σv 4 σv = σ w +σv ± σw(+4 4 σ v/σw) σv = σ w +σv ±σw +4 σv/σ w σv Only solution of [;] is of interest, i.e. only the minus sign is feasible. The feasible solution to (A.4) is = σ e = σ v ( +4 σ v σw ) σ w σ v (A.6a) (A.6b) Appendix A. DERIVATION OF FNCTIONAL EXPRESSION FOR The results in Sec. related to the noise model (5) is based on the following lemma: Lemma A.. Let w t N iid (,σ w), v t N iid (,σ v) and e t N iid (,σ e) be Gaussian processes where w t and v t is uncorreleted, then q w t +v t = q q e t (A.) and the variance of e t and the parameter are related to the variance of w t and v t as: ( ) = +4 σ v σw σw σv (A.a) σ e = σ v (A.b) Proof: For the equality (A.) to hold, the variance and the covariance on both sides of the equality sign are the same. The equality is rewritten as