On the Effect of the Lambda Parameter on Performance and Stability Indexes in DMC Control

Transcription

1 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// On the Effect of the Labda Paraeter on Perforance and Stability Indexes in DMC Control JOSE MANUEL LOPEZ-GUEDE, ANA BOYANO, IGOR ANSOATEGUI, EKAITZ ZULUETA JOSE ANTONIO RAMOS-HERNANZ Deartent of systes Enineerin and Autoatics Deartent of Mechanics University of the Basque Country (UPV/EHU) Nieves Cano, 6, Vitoria-Gasteiz SPAIN Abstract: - One of the ore outandin araeters that define the erforance and stability of a concrete tye of Model Predictive Control (MPC) naed Dynaic Matrix Control (DMC) is the λ araeter. The objective of the aer is to analyze i effect considerin a set of indexes defined to easure the characteristics of the outut of an instable syste when it is controlled by a DMC controller. A secific syste has been chosen to ake such analysis because it has deonstrated to be instable usin a discretized PID controller tuned throuh the Zieler-Nichols ethod. A total of, tes have been erfored to assess the effect of the λ araeter, obtainin as conclusion suorted by fiures that it is a outandin araeter. Key-Words: - Model Predictive Control, MPC, Dynaic Matrix Control, DMC, Advanced Control, Stability Introduction Dynaic Matrix Control (DMC) is a articular tye of Model Predictive Control (MPC), which is a faily of advanced control scheas. In the literature, this tye of advanced controllers have been used and coared with PID controllers [] showin a ood behavior. Based on revious works [][7], our research rou has been workin with these controllers obtainin accurate neuronal ileentations [4] even for ulti-aent systes []. However, we have not yet discussed the effect of the λ araeter of the DMC controller because we have always suosed a fixed ileentation of the redictive controller, which defines a concrete λ value. The ain objective of this aer is to study the effect of the λ araeter in the erforance of a odel redictive controller. The aer is structured as follows. In the second section, we recall soe basic conce of MPC and DMC to address the iortance and the role of the λ araeter in such control schee. In the third section, we describe the exeriental desin which we have carried out, detailin the indexes which we have used to describe the erforance of the DMC controllers, the syste and the workin oint in which it has been workin to assess the effect of chanes in the λ araeter and the tested values for that araeter. In the fourth section, we discuss the exeriental resul that we have obtained; enueratin the articular effect caused in each of the erforance indexes. Finally, the last section rovides our conclusions. Backround This section reviews soe basic conce about Model Predictive Control (MPC) as a eneral technique, and about a concrete technique called Dynaic Matrix Control (DMC).. Model Predictive Control MPC is an advanced control technique used to deal with systes that are not controllable usin classic control scheas. This kind of controllers works like the huan brain in the sense that instead of usin the ast error between the outut of the syste and the desired value, it controls the syste redictin the value of the outut in a short tie, so the syste outut is as closer as ossible to i desired value for these oen. Predictive Control is not a concrete technique. It is a set of techniques that have several coon characteristics: there is a world odel which is used to redict the syste outut fro the actual oent until sales, an objective function that ust be iniized and a control law ISSN: Volue, 7

2 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// that iniizes the objective function. The redictive controllers follow these stes: - Each salin tie, throuh the syste odel, the controller calculates the syste outut fro now until salin ties (rediction horizon), which deends on the future control sinals that the controller will enerate. - A set of control sinals is calculated otiizin the objective function to be used alon salin ties (control horizon). - In each salin tie only the first of the set of control sinals is used, and in the next salin tie, all the rocess is reeated aain. The concet of Predictive Control is a set of techniques that share certain characteristics, and the enineer has liberty to choose in each of the. So, there are several tyes of redictive controllers. These coon characteristics are the followin: - There is a lant odel, and there can be used a ste resonse odel, an iulse ste resonse odel, a transfer function, etc. - There is an objective function that the controller has to otiize. - There is a control law to iniize the objective function. To learn ore about Predictive Control in eneral and about diverse redictive control aloriths, see [-3][6][][8][9].. Dynaic Matrix Control It is a concrete MPC alorith that fixes each of the three characteristics that we have seen reviously as we will see below. To learn ore about Dynaic Matrix Control, see [-3][6][]... Syste Model The lant odel used by DMC alorith is the ste resonse odel. This odel uses the i coefficien that are the outut of the lineal syste when it is excited usin a ste. To reduce the nuber of coefficien we assue that the subsyste is stable and the outut does not chane after soe salin tie k. The exression of the outut of the syste is iven throuh Eq. (). k i= ( t i) y( t) = u () i.. Prediction Model Usin the ste resonse odel to odel the syste and aintainin the hyothesis that erturbations over the subsyste are constan, it is ossible to calculate a rediction at the instant t of the outut until the instant ( t + ) under the effect of control actions. The rediction is iven by the Eq. (): y ˆ = G u + f () bein ŷ the rediction of the outut, G a atrix which contains the syste dynaics and f the free resonse of the syste. In Eq. (3) we show the diensions of the atrix and vectors involved in Eq. (). yˆ yˆ yˆ = yˆ ( t + t) ( t + t) ( t + t) u u = u u( t) ( t + ) ( t + ) G = f f f = f ( t,) ( t,) ( t, ) + x (3) In Eq. (4) we describe how the free resonse of the syste f ( t, k) is calculated: f N ( t, k) = y ( t) + ( ) u( t i) i= k+ i i (4)..3 Control Law The derivation of the control law is based on the existence of an objective function, which uses the future outu rediction odel that we have described before. As objective function we used the described by Eq. (). J = j= j= [ yˆ ( t + j t) w( t + j) ] + λ [ u( t + j ) ] () We have to iniize the difference between the reference and the outut rediction alon a rediction horizon with the control actions enerated in the control horizon, odulatin the rouhness in the variation of the aniulated variables usin the λ araeter. Miniizin the objective function J described in Eq. () we obtain ISSN: Volue, 7

3 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// the followin exression, which roduces control actions, althouh in t only one of the is used: t [( G G + λi ) G ( )] t w f u = (6) After this brief introduction, we can understand the role of the λ araeter and intuit which iht be the effect on controller erforance due to chanes in that araeter. 3 Exeriental Desin In this section we rovide the exeriental desin of the study which we have carried out to assess the effect of the λ araeter in DMC controllers erforance. First, we detail the definition of the erforance indexes that we have used. Then, the syste whose resonse will be analyzed is secified in a otivated way. We also discuss the workin oint of the controlled syste. Finally, we secify the set of exerien that we have carried out to obtain sinificant resul. 3. Perforance Indexes Definition To assess the effect of the λ araeter of the DMC controller we have defined several indexes. These indexes are focused on the erforance reached at the outut of the syste when it is controlled by a MPC controller when the reference is a unitary ste of arbitrary frequency. The indexes which we have used to easure the erforance are described in Table. On the other hand, Fi. shows a rahical reresentation of those indexes. 3. Syste and Workin Point We have analyzed the discretized syste described by Eq. (7). This is a quite sile but interestin syste because it shows an unstable resonse when the reference is a unitary ste and it is controlled by a discretized Proortional-Interal- Derivative (PID) controller in a closed loo confiuration, which has been tuned by eans of the Zieler-Nichols ethod. That unstable resonse is shown in Fi.. To deterine the workin oint of the syste, i.e., the frequency of the unitary ste sinal which is at the inut of the closed loo syste as reference, we have used the Bode diara shown in Fi. 3. Based on that diara, we have chosen a frequency of 3 sale ties because it is a frequency at which the iven ain is reresentative of any other frequencies for that syste. G ( z) = z. (7) Table. Sybol and descrition of several erforance indexes used in the exeriental desin Sybol se M t s Descrition Mean squared error between the reference and the syste outut. Overshoot of the outut over the reference, exressed as a ercentae. Usually, it takes lace on the first risin of the sinal. Tie elased between the outut oes fro % to 9% of the reference value. Fi.. Grahical reresentation of the erforance indexes % 9% 8% % % t Fi.. Unstable resonse of the syste when it is controlled by a discretized PID controller M 3. x R*H/(+R*H ISSN: Volue, 7

4 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// To deterine the workin oint of the syste, i.e., the frequency of the unitary ste sinal which is at the inut of the closed loo syste as reference, we have used the Bode diara shown in Fi. 3. Based on that diara, we have chosen a frequency of 3 sale ties because it is a frequency at which the iven ain is reresentative of any other frequencies for that syste. Fi.3. Bode diara used to deterine the workin oint of the syste - -4 Fi.4. se with λ=. Phase (de) Manitude (db) Bode Diara Syste: H Frequency (rad/sec):. Manitude (db): Fi.. se with λ= Frequency (rad/sec) 3.3 Sensitivity Analysis As stated before, the ain objective of this aer is to analyze the effect of the λ araeter in DMC controllers. So, we have studied the behavior of the reviously secified syste at that workin oint when it is controlled by DMC controllers varyin the value of the λ araeter. The values which have been chosen are., and. 4 Exeriental Resul In this section we rovide several resul reardin to the erforance indexes which we introduced in the section devoted to the exeriental desin, when the araeter sensitivity secified in that section is carried out Fi.6. se with λ= 4. Effect on se value In this subsection we are oin to enuerate the effec on the se erforance index values, takin into consideration Fis The followin are the ain effec: - In eneral, we can see that with increasin values of the λ araeter, the value of the se index becoes hiher for each cobination of (, ) araeters values. - The se index value ranes fro about -7 with λ=. to about - with λ= ISSN: Volue, 7

5 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// - For each order of anitude that the λ araeter increases, the se index value increases in the sae easure. - As the λ araeter value increases, hiher values are needed in the (, ) araeters cobination to aintain the se index value relatively low, but always within the reviously referred increase of order. - The worst resul are roduced when araeter is very low, (indeendently of araeter), but if λ araeter increases, the resul are relatively better if is low. 4. Effect on M values In this subsection we suarize the effect on the M erforance index values, takin into consideration Fis. 7-9, bein these the ain effec: - Even for the case of very hih values of the λ araeter, for ost controller structures, quite sall overshoot values are obtained, always less than %. - Even with oderate values of the λ araeter, if the controller has very sall values of araeter, the overshoot increases, reachin even 3% if the araeter value is very hih. - In the case of λ=, for virtually all structures, M the easure value is less than zero because it is not reached the reference value. Particularly saller is the alitude reached by the outut with the structure = y =. - For oderate values of the λ araeter, it sees that there are values for which the value of the araeter is irrelevant: with any of the the eak occurs once that the reference has raised, and for other values, with any values of it takes or 3 sale ties to occur the eak. Fi.7. M with λ=. 3 3 M Fi.8. M with λ= 8 M Fi.9. M with λ= 4.3 Effect on t s value In this subsection we exose the effect on the t s erforance index values, takin in consideration Fis. -. The followin are the ain effec: - As the value of the λ araeter increases, the value of chanes in the control sinal is slower and this is erceived at the outut, so that the ascent seed is slower, increasin the aount of tie to ove fro % to 9% of the reference value. - - M ISSN: Volue, 7

6 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// - Values rane fro sale ties (vertical rise) of alost all structures with λ=. to sale ties of the ost of the structures with λ=. - Excet for very hih values of the λ araeter, we can see that in eneral, the structures with very sall values of the araeter are not aroriated, but is articularly bad when the values are hih. Fi.. t s with λ=. Conclusion In this aer we have studied the effect of the λ araeter on DMC controllers erforance. We have introduced the MPC and DMC controllers to frae the iortance of that araeter. In the exeriental desin we have secified the erforance indexes which we have taken into account and the values used to the λ araeter. Once we have carried out a nuber of exerien, we have discussed the ain resul reardin to each of the indexes. We have tested 4 structures of DMC controllers for each value of λ araeter, and as suary, with the secificities that we have shown throuh fiures, we can conclude that a lower value of the λ araeter resul in a better erforance of the DMC controllers. References: [] Arahal, M.R., Berenuel, M., Caacho, E.F. Neural identification alied to redictive control of a solar lant, Control Enineerin Practice, Vol. 6, 98, [] Caacho, E.F., Bordons, C. Model Predictive Control, Sriner-Verla, London, 4. [3] Caacho, E. F., Bordons, C. Model Predictive Control in the Process Industry. Sriner- Verla, London, 9. [4] Loez-Guede, J.M., Graña, M., Zulueta, E., Barabones, O. Econoical Ileentation of Control Loos for Multi-robot Systes. th International Conference, ICONIP 8, Aucland, New Zealand. [] Loez-Guede, J.M., Zulueta E., Graña, M., D Anjou, A. Neuronal Ileentation of Predictive Controllers. Hybrid Artificial Intellience Systes, HAIS, Sriner Berlin / Heidelber, Vol. 677,, 3-3. [6] Maciejowski, J. M. Predictive Control with Constrain, Prentice Hall, London, Fi.. t s with λ= Fi.. t s with λ= ISSN: Volue, 7

7 Jose Manuel Loez-Guede et al. International Journal of Control Systes and Robotics htt:// [7] McKinstry, J.L., Edelan, G.M., Krichar, J.L. A cerebellar odel for redictive otor control tested in a brain-based device, Proceedins of the National Acadey of Sciences of The United States of Aerica, Vol. 3, No 9, 6, [8] Rawlins, J.B. Tutorial: Model Predictive Control Technoloy. Proceedins of the Aerican Control Conference, San Dieo, California, 99, [9] Soeterboek, R. Predictive control. A unified aroach, Prentice Hall, 9. [] Sunan, H., Kok T., Ton L. Alied Predictive Control, Sriner-Verla, London,. [] Voicu, M., Lazär, C., Schönberer, F., Pästravanu, O., Ifri, S. Predictive Control vs. PID Control of Theral Treatent Processes. Control Enineerin Solution: a Practical Aroach, 9, ISSN: Volue, 7