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at 1 t À d T Pt À 1t À d  1 1 À t 2 À et 2 1 1 À tÀtQt ! at 1 t À d T Pt À 1t À d À et 2 1 1 À tQt ! at 1 t À d T Pt À 1t À d À et 2 1 1 À 1 et 2 ! at 1 t À d T Pt À 1t À d À et 2 1 0 et 2 ! À 0 À 1 0 at 1 t À d T Pt À 1t À d et 2 À 0 À 1 0 f t 2 1 t À d T Pt À 1t À d 2:18 where the fact that atet 2 f tet!f t 2 has been used. Therefore, following the same arguments in [20], [21], [23], the results in Lemma 4.1 are thus proved. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)±(v) in Lemma 4.1, the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated " " 1 is small enough, which are summarized in the following theorem. Theorem 4.1 The direct adaptive control system satisfying assumptions (A1)± (A4) with the adaptive controller described in equations (2.7), (2.9), (2.13)± (2.16) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. In this approach, we have eliminated the requirement for the knowledge of the parameters of the upper bounding function on the modelling uncertainties. But the requirement for the knowledge of the lower bound on the leading coecient of the parameter vector, i.e. assumption (A4) is still there. In the next section, the technique of the parameter correction procedure will be combined with the algorithm developed in this section to ensure the least prior knowledge on the plant. That is, only assumptions (A1)±(A3) are needed. Adaptive Control Systems 31 2.5 Robust adaptive control with least prior knowledge The following modi®ed least squares algorithm will be used for robust parameter estimation: t t À 1at Pt À 1t À d 1 t À d T Pt À 1t À d " et PtPt À 1Àat Pt À 1t À dt À d T Pt À 1 1 t À d T Pt À 1t À d 2:19 PÀ1k 0 I; k 0 > 0 and the parameter estimate is then corrected [24] as " t tPtt2:20 where " et " ytÀ " t À 1 T t À d2:21 the vector t is described in Figure 2.1 where pt is the ®rst column of the covariance matrix Pt, and the term at is now de®ned as follows: at 0if " etj 2 tQt f 1=2 tQt 1=2 ; " et= " et otherwise @ : with 0 <<1, 2 0 1 À ; 0 > 1, and Qtt À 1 T Pt À 1t À d 2 21À1 t À d T Pt À1t Àd sup 0 t jjxjj 2 1 45 T sup 0 t jjxjj 2 1 45 32 An algorithm for robust adaptive control with less prior knowledge t t pt kptk "j 1 tj kptk À2j 1 tj À"kptk t0 Figure 2.1 Parameter correction vector And tis calculated by t Ct T sup 0 t jjxjj 2 1 P R Q S 2:22 Ct Ct À 1 at 1 À 1 t À d T Pt À 1t À d sup 0 t jjxjj 2 1 P R Q S ; >0 2:23 where Ct T " 1 " 2 with zero initial condition. It should be noted that " 1 and " 2 will be always positive and non-decreasing. Remark 5.1 The prediction error " et is used in the modi®ed least squares algorithm to ensure that the estimator property (iii) in the following lemma can be established. The properties of the above modi®ed least squares parameter estimator are summarized in the following lemma. Lemma 5.1 If the plant satis®es the assumptions (A1)±(A3), the least squares algorithm (2.19)±(2.23) has the following properties: (i) t is bounded, and jj tÀ t À 1jj P l 2 . (ii) Ct is bounded and non-decreasing, thus converges (iii) f 1=2 tQt 1=2 ; " et 2 1 t À d T Pt À 1t À d P l 2 (iv) jjptjj j 1 tj > b min where b min j 1 j mx1; jj à jj with à de®ned such that à tPt à (v) j " 1 tj > 1 À " 3 " b min (vi) " t is bounded, and jj " tÀ " t À 1jj P l 2 Proof De®ne a Lyapunov function candidate Vt 1 1 2 ~ t T Pt À1 ~ t ~ Ct 1 T À1 ~ Ct 1 2:24 Adaptive Control Systems 33 where ~ t tÀ à , ~ Ct 1 Ct 1À" 1 " 2 T . Noting that " et " ytÀ " t À 1 T t À d " ytÀ t À 1t À dÀt À 1 T Pt À 1t À d etÀt À 1 T Pt À 1t À d2:25 Then, the dierence of the Lyapunov function candidate becomes Vt 1ÀVt at 1 t À d T Pt À 1t À d  1 t À d T Pt À 1t À d 1 1 Àatt À d T Pt À 1t À d ÂtÀt À1 T Pt À 1t À d 2 À " et 2 ! at ~ Ct T 1 À 1 t À d T Pt À 1t À d sup 0 t jjxjj 2 1 P R Q S at 2 1 À 2 1 t À d T Pt À 1t À d 2  sup 0 t jjxjj 2 1 P R Q S T sup 0 t jjxjj 2 1 P R Q S at 1 t À d T Pt À 1t À d  1 1 À tÀt À 1 T Pt À 1t À d 2 À " et 2 ! 2at tÀt 1 À 1 t À d T Pt À 1t À d at 2 1 À 2 1 t À d T Pt À 1t À d 2  sup 0 t jjxjj 2 1 P R Q S T sup 0 t jjxjj 2 1 P R Q S 34 An algorithm for robust adaptive control with less prior knowledge at 1 t À d T Pt À 1t À d  2 1 À t 2 À 2 1 À t À 1 T Pt À 1t À d 2 À " et 2 ! 2at tÀt 1 À 1 t À d T Pt À 1t À d at 2 1À 2 1t À d T Pt À1t Àd 2 sup 0 t jjxjj 2 1 P R Q S T sup 0 t jjxjj 2 1 P R Q S at 1 t À d T Pt À 1t À d  2 1 À t 2 À " et 2 2 1 À tÀtQt ! at 1 t À d T Pt À 1t À d À " et 2 2 1 À tQt ! at 1 t À d T Pt À 1t À d À " et 2 2 1 À 1 " et 2 ! at 1 t À d T Pt À 1t À d À " et 2 1 0 " et 2 ! À 0 À 1 0 at " et 2 1 t À d T Pt À 1t À d À 0 À 1 0 f 1=2 tQt 1=2 ; " et 2 1 t À d T Pt À 1t À d 2:26 where the fact that atet 2 f tet!f t 2 has been used. Therefore, following the same arguments in [26], [11], [21], the results (i)±(iii) in Lemma 5.1 are thus proved. The properties (iv)±(vi) in the lemma can also be obtained directly from the results in [23]. If using the same adaptive control law as in equation (2.11), then with the parameter estimation properties (i)±(vi), the global stability and convergence results of the new adaptive control system can be established as in [26], [11] as long as the estimated " " 1 is small enough, which are summarized in the following theorem. Adaptive Control Systems 35 Theorem 5.1 The direct adaptive control system satisfying assumptions (A1)± (A3) with the adaptive controller described in equations (2.19)±(2.23) and (2.11) is globally stable in the sense that all the signals in the loop remain bounded. 2.6 Simulation example In this section, one numerical example is presented to demonstrate the performance of the proposed algorithm. A fourth order plant is given by the transfer function as GsG n sG u s with G n s 5s 2 ss 1 as a nominal part, and G u s 229 s 2 30s 229 as the unmodelled dynamics. With the sampling period T 0:1 second, we have the following corre- sponding discrete-time model Gq À1 0:09784q À1 0:1206q À2 À 0:1414q À3 À 0:01878q À4 1 À 2:3422q À1 1:0788q À2 À 0:4906q À3 0:04505q À4 The reference model is chosen as G m s 1 0:2s 1 whose corresponding discrete-time model is G m q À1 0:3935q À1 1 À 0:6065q À1 We have chosen k 0 1, and 00:6000 T . If no dead zone is used, the simulation results are divergent. If using the algorithm developed in this chapter with 10 À5 , the simulation results are shown as in Figure 2.2, where (a) represents the system output yt and reference model output y à t, (b) is the control signal ut, (c) is the estimated parameter 1 , and (d) denotes the estimated bounding parameters " 1 and " 2 . In order to demonstrate the eect of the update rate parameter , the following simulation with 1:4  10 À5 was also conducted. The result is shown in Figure 2.3. The steady state values of the several important parameters and the tracking error in both cases are summarized in Table 2.1. 36 An algorithm for robust adaptive control with less prior knowledge Adaptive Control Systems 37 Output and refe rence E stimate d the ta1 Time in seconds Time in seconds Time in seconds Time in seconds Contr ol Est. b ound ing param eters Figure 2.2 Robust adaptive control with 10 À 5 Output and re ferenc e Estimated the ta1 Contro l Est. bo unding p aramet ers Time in secondsTime in seconds Time in seconds Time in seconds Figure 2.3 Robust adaptive control with 1:4  10 À5 It can be observed from the above simulation results that the algorithm developed in this chapter can guarantee the stability of the adaptive system in the presence of the modelling uncertainties, and the smaller tracking error could be achieved with smaller update rate parameter . Most importantly, the knowledge of the parameters " 1 and " 2 of the upper bounding function and the knowledge of the leading coecient of the param- eter vector 1 are not required a priori. 2.7 Conclusions In this chapter, a new robust discrete-time direct adaptive control algorithm is proposed with respect to a class of unmodelled dynamics and bounded disturbances. Dead zone is indeed used but no knowledge of the parameters of the upper bounding function on the unmodelled dynamics and disturbances is required a priori. Another feature of the algorithm is that a correction procedure is employed in the least squares estimation algorithm so that no knowledge of the lower bound on the leading coecient of the plant numerator polynomial is required to achieve the singularity free adaptive control law. The global stability and convergence results of the algorithm are established. References [1] Rohrs, C., Valavani, L., Athans, M. and Stein, G. (1985). `Robustness of Adaptive Control Algorithms in the Presence of Unmodelled Dynamics', IEEE Trans. Automat. Contr., Vol. AC-30, 881±889. [2] Egardt, B. (1979). Stability of Adaptive Controllers, Lecture Notes in Control and Information Sciences, New York, Springer Verlag. [3] Ortega, R. and Tang, Y. (1989). `Robustness of Adaptive Controllers ± A Survey', Automatica, Vol. 25, 651±677. [4] Ydstie, B. E. (1989). `Stability of Discrete MRAC-revisited', Systems and Control Letters, Vol. 13, 429±438. 38 An algorithm for robust adaptive control with less prior knowledge Table 2.1 Steady state values 10 À5 1:4 Â10 À5 " 1 1.1898 Â10 À3 1.4682 Â10 À3 " 2 0.3509 Â10 À3 0.4467 Â10 À3 1 0.5649 0.5833 jy Ày à j 0.0179 0.07587 [5] Naik, S. M., Kumar, P. R., Ydstie, B. E. (1992). `Robust Continuous-time Adaptive Control by Parameter Projection', IEEE Trans. Automat. Contr., Vol. AC-37, No. 2, 182±197. [6] Praly, L. (1983). `Robustness of Model Reference Adaptive Control', Proc. 3rd Yale Workshop on Application of Adaptive System Theory, New Haven, Connecticut. [7] Praly, L. (1987). `Unmodelled Dynamics and Robustness of Adaptive Controllers', presented at the Workshop on Linear Robust and Adaptive Control, Oaxaca, Mexico. [8] Petersen, B. B. and Narendra, K. S. (1982). `Bounded Error Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-27, 1161±1168. [9] Samson, C. (1983). `Stability Analysis of Adaptively Controlled System Subject to Bounded Disturbances', Automatica, Vol. 19, 81±86. [10] Egardt, B. (1980). `Global Stability of Adaptive Control Systems with Disturbances', Proc. JACC, San Francisco, CA. [11] Middleton, R. H., Goodwin, G. C., Hill, D. J. and Mayne, D. Q. (1988). `Design Issues in Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-33, 50±58. [12] Kreisselmeier, G. and Anderson, B. D. O. (1986). `Robust Model Reference Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-31, 127±133. [13] Kreisselmeier, G. and Narendra, K. S. (1982). `Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-27, 1169±1175. [14] Iounnou, P. A. (1984). `Robust Adaptive Control', Proc. Amer. Contr. Conf.,San Diego, CA. [15] Ioannou, P. and Kokotovic, P. V. (1984). `Robust Redesign of Adaptive Control', IEEE Trans. Automat. Contr., Vol. AC-29, 202±211. [16] Iounnou, P. A. (1986). `Robust Adaptive Controller with Zero Residual Tracking Error', IEEE Trans. Automat. Contr., Vol. AC-31, 773±776. [17] Anderson, B. D. O. (1981). `Exponential Convergence and Persistent Excitation', Proc. 20th IEEE Conf. Decision Contr., San Diego, CA. [18] Narendra, K. S. and Annaswamy, A. M. (1989). Stable Adaptive Systems, Prentice- Hall, NJ. [19] Feng, G. and Palaniswami, M. (1994). `Robust Direct Adaptive Controllers with a New Normalization Technique', IEEE Trans. Automat. Contr., Vol. 39, 2330±2334. [20] Goodwin, G. C. and Sin, K. S. (1981) `Adaptive Control of Nonminimum Phase Systems', IEEE Trans. Automat. Contr., Vol. AC-26, 478±483. [21] Feng, G. and Palaniswami, M. (1992). `A Stable Implementation of the Internal Model Principle', IEEE Trans. Automat. Contr., Vol. AC-37, 1220±1225. [22] Feng, G., Palaniswami, M. and Zhu, Y. (1992). `Stability of Rate Constrained Robust Pole Placement Adaptive Control Systems', Systems and Control Letters, Vol. 18, 99±107. [23] Lazono-Leal, R. and Goodwin, G. C. (1985). `A Globally Convergent Adaptive Pole Placement Algorithm without a Persistency of Excitation Requirement', IEEE Trans. Automat. Contr., Vol. AC-30, 795±799. Adaptive Control Systems 39 [24] Lazono-Leal, R., Dion, J. and Dugard, L. (1993). `Singularity Free Adaptive Pole Placement Using Periodic Controllers', IEEE Trans. Automat. Contr., Vol. AC-38, 104±108. [25] Lazono-Leal, R. and Collado, J. (1989). `Adaptive Control for Systems with Bounded Disturbances', IEEE Trans. Automat. Contr., Vol. AC-34, 225±228. [26] Goodwin, G. C. and Sin, K. S. (1984). Adaptive Filtering, Prediction and Control, Prentice-Hall, NJ. [27] Lazono-Leal, R. (1989). `Robust Adaptive Regulation without Persistent Excitation', IEEE Trans. Automat. Contr., Vol. AC-34, 1260±1267. 40 An algorithm for robust adaptive control with less prior knowledge [...]... Verlag [3] Ortega, R and Tang, Y (1989) `Robustness of Adaptive Controllers ± A Survey', Automatica, Vol 25, 651±677 [4] Ydstie, B E (1989) `Stability of Discrete MRAC-revisited', Systems and Control Letters, Vol 13, 429± 438 Adaptive Control Systems 39 [5] Naik, S M., Kumar, P R., Ydstie, B E (1992) `Robust Continuous-time Adaptive Control by Parameter Projection', IEEE Trans Automat Contr., Vol AC -37 ,...  10 À5 37 38 An algorithm for robust adaptive control with less prior knowledge Table 2.1 Steady state values 10À5 "1 "2 1 j y À yÃj 1:4  10À5 1.1898  10 3 0 .35 09  10 3 0.5649 0.0179 1.4682  10 3 0.4467  10 3 0.5 833 0.07587 It can be observed from the above simulation results that the algorithm developed in this chapter can guarantee the stability of the adaptive system in the... [6] Praly, L (19 83) `Robustness of Model Reference Adaptive Control' , Proc 3rd Yale Workshop on Application of Adaptive System Theory, New Haven, Connecticut [7] Praly, L (1987) `Unmodelled Dynamics and Robustness of Adaptive Controllers', presented at the Workshop on Linear Robust and Adaptive Control, Oaxaca, Mexico [8] Petersen, B B and Narendra, K S (1982) `Bounded Error Adaptive Control' , IEEE Trans... (1994) `Robust Direct Adaptive Controllers with a New Normalization Technique', IEEE Trans Automat Contr., Vol 39 , 233 0± 233 4 [20] Goodwin, G C and Sin, K S (1981) `Adaptive Control of Nonminimum Phase Systems', IEEE Trans Automat Contr., Vol AC-26, 478±4 83 [21] Feng, G and Palaniswami, M (1992) `A Stable Implementation of the Internal Model Principle', IEEE Trans Automat Contr., Vol AC -37 , 1220±1225 [22]... Samson, C (19 83) `Stability Analysis of Adaptively Controlled System Subject to Bounded Disturbances', Automatica, Vol 19, 81±86 [10] Egardt, B (1980) `Global Stability of Adaptive Control Systems with Disturbances', Proc JACC, San Francisco, CA [11] Middleton, R H., Goodwin, G C., Hill, D J and Mayne, D Q (1988) `Design Issues in Adaptive Control' , IEEE Trans Automat Contr., Vol AC -33 , 50±58 [12]... Reference Adaptive Control' , IEEE Trans Automat Contr., Vol AC -31 , 127± 133 [ 13] Kreisselmeier, G and Narendra, K S (1982) `Stable Model Reference Adaptive Control in the Presence of Bounded Disturbances', IEEE Trans Automat Contr., Vol AC-27, 1169±1175 [14] Iounnou, P A (1984) `Robust Adaptive Control' , Proc Amer Contr Conf., San Diego, CA [15] Ioannou, P and Kokotovic, P V (1984) `Robust Redesign of Adaptive. .. Control Output and reference Adaptive Control Systems Time in seconds Estimated theta1 Est bounding parameters Time in seconds Time in seconds Time in seconds Control Output and reference Figure 2.2 Robust adaptive control with 10 À5 Time in seconds Estimated theta1 Est bounding parameters Time in seconds Time in seconds Time in seconds Figure 2 .3 Robust adaptive control with 1:4  10 À5 37 ... Placement Adaptive Control Systems', Systems and Control Letters, Vol 18, 99±107 [ 23] Lazono-Leal, R and Goodwin, G C (1985) `A Globally Convergent Adaptive Pole Placement Algorithm without a Persistency of Excitation Requirement', IEEE Trans Automat Contr., Vol AC -30 , 795±799 40 An algorithm for robust adaptive control with less prior knowledge [24] Lazono-Leal, R., Dion, J and Dugard, L (19 93) `Singularity... or matrix (3) k
Át kI sup t j
Á
j: denotes the truncated LI norm of the argument function or vector (4) kP
skI : denotes the HI norm of the transfer function P
s The chapter is organized as follows In Section 3. 2, we give the plant Adaptive Control Systems 43 description, control objective and then derive the MRAC based error model In Section 3. 3, the adaptive variable structure controller... `Singularity Free Adaptive Pole Placement Using Periodic Controllers', IEEE Trans Automat Contr., Vol AC -38 , 104±108 [25] Lazono-Leal, R and Collado, J (1989) `Adaptive Control for Systems with Bounded Disturbances', IEEE Trans Automat Contr., Vol AC -34 , 225±228 [26] Goodwin, G C and Sin, K S (1984) Adaptive Filtering, Prediction and Control, Prentice-Hall, NJ [27] Lazono-Leal, R (1989) `Robust Adaptive Regulation . Automat. Contr., Vol. 39 , 233 0± 233 4. [20] Goodwin, G. C. and Sin, K. S. (1981) `Adaptive Control of Nonminimum Phase Systems', IEEE Trans. Automat. Contr., Vol. AC-26, 478±4 83. [21] Feng, G Â10 3 1.4682 Â10 3 " 2 0 .35 09 Â10 3 0.4467 Â10 3 1 0.5649 0.5 833 jy Ày à j 0.0179 0.07587 [5] Naik, S. M., Kumar, P. R., Ydstie, B. E. (1992). `Robust Continuous-time Adaptive Control by. Automat. Contr., Vol. AC -30 , 795±799. Adaptive Control Systems 39 [24] Lazono-Leal, R., Dion, J. and Dugard, L. (19 93) . `Singularity Free Adaptive Pole Placement Using Periodic Controllers', IEEE
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