Adaptive Control System Part 6 pps

5.6 Conclusions In this chapter we have presented a new uni®ed switching control basedapproach to adaptive stabilization of parametrically uncertain discrete-timesystems.. 1995 `Robust A

(a) Suppose at time …t ‡ 1†, (5.89) is violated This can occur in one of two

ways which we consider separately:

(ii)

z…t ‡ 1† ˆ z…t ‡ 1†i t < …
kx…t†k ‡ c0† …5:93†

In this case

z…t ‡ 1†i< …
kx…t†k ‡ c0† …5:94† for all i ˆ j1; j2; ; k f g In either case, we see that if (5.91) is violated at

time t, then

from which the result follows

(b) First, we note that the control is well de®ned, that is, St is never empty

iwhich contains the true plant, which is always an element of St

Next, we note that although we cannot guarantee that we converge to thecorrect control, from (a) we know (5.85) is satis®ed all but a ®nite number oftimes

Since ITis a stabilizing inclusion, then by de®nition the states and all signalswill be bounded

Furthermore, since It is a stabilizing inclusion, there exist 0; 0 and

 2 …0; 1† such that if the inclusion (5.89) is satis®ed, for t 2 ‰t0; t0‡ T†, then

kx…t0‡ T†k  0 Tkx…t0†k ‡ 0 …5:96† (Note that if this is not the case, then from the de®nition, Itis not a stabilizinginclusion.) Also, there exist  and  such that when (5.89) is violated:

…1 ‡ 0 ‡ …0 †2†0…1 ‡ 0† , provided (5.89) is not violated more than

twice in the interval ‰t0; t0‡ T†, then

The desired result follows from (a) since we know that there are at most

N ˆ ‰log2…s†Š times at which (5.89) is violated.

Case 2: L > 1

Suppose that we do not know a single C such that It is a stabilizing inclusion,and CB is of known sign, then using ®nite covering ideas [8], as in Remark 4.3let

[L

lˆ1

lˆ[Llˆ1

[s l

mˆ1

l

where for each l, we know Cl;
l; cl

0 such that It is a stabilizing inclusion on

At this point, one might be tempted to apply localization, as previously

indices, Sl, become empty Unfortunately, this procedure cannot be

guaran-ldoes not contain the true plant, Itneed not be

a stabilizing inclusion, and so divergence of the states may occur withoutviolating (5.89) To alleviate this problem, we use the exponential stabilityresult, (5.90), in our subsequent development

Algorithm D

We initialize t…i† ˆ 0; R0ˆ f1; 2; ; Lg and take any l02 R0

l, with the following additional2steps: If

at any time

kx…t†k >  t t…i† kx…t…i††k ‡ …5:102†

lfrom Theorem 4.1), then weset Slˆ fg If at any time t; Sl becomes empty, we set Rtˆ Rt 1 flg;

t…i† ˆ t, and we take a new l from Rt

2In fact, we can localize simultaneously within other Oi; i 6ˆ l: however, for

simplicity and brevity we analyse only the case where we localize in one set at a time

With these modi®cations, it is clear that Theorem 4.1 can be extended tocover this case as well:

Corollary 4.1 The control algorithm (5.86)±(5.88) with the above tions applied to a plant with decomposition as in (5.101) satis®es:

modi®ca-(a) There are no more than: L 1 ‡PLlˆ1 blog2…sl†c instants such that

jz…t ‡ 1†ltj 
ltkx…t†k ‡ c0;lt …5:103†

(where lt denotes the value of l at time t)

(b) All signals in the closed loop are bounded In particular, there existconstants ;  < 1;  2 …0; 1† such that for any t0; T > 0

kx…t0‡ T†k  Tkx…t0†k ‡  …5:104†

Proof Follows from Theorem 4.1

5.4.1 Localization in the presence of unknown disturbance

In the previous section the problem of indirect localization based switchingcontrol for linear uncertain plants was considered assuming that the level of the

generalized exogenous disturbance …t† was known This is equivalent to knowing some upper bound on …t† The ¯exibility of the proposed adaptive

scheme allows for simple extension covering the case of exogenous bances of unknown magnitude This can be done in the way similar to thatconsidered in Section 5.3.2 Omitting the details we just make the followinguseful observation The control law described by Algorithms C and D is wellde®ned, that is, Rt6ˆ fg for all t  t0 if cl

distur-0 sup tt0jClE…t†j; 8l ˆ 1; ; L.This is the key point allowing us to construct an algorithm of on-lineidenti®cation of the parameters cl

0; l ˆ 1; ; L.

5.5 Simulation examples

Extensive simulations conducted for a wide range of LTI, LTV and nonlinearsystems demonstrate the rapid falsi®cation capabilities of the proposedmethod We summarize some interesting features of the localization techniqueobserved in simulations which are of great practical importance

(i) Falsi®cation capabilities of the algorithm of localization do not appear to

be sensitive to the switching index update rule One potential implication

of this observation is as follows If not otherwise speci®ed any choice of anew switching index is admissible and will most likely lead to goodtransient performance;

(ii) The speed of localization does not appear to be closely related to the total

number of ®xed controllers obtained as a result of decomposition Thepractical implication of this observation (combined with the quadraticconducted in a straightforward way employing, for example, a uniform

‰ 0:1; 0:1Š, and b1…t† and b2…t† are uncertain parameters We deal with two

cases which correspond to constant parameters and large-size jumps in thevalues of the parameters

Case 1: Constant parameters

The a priori uncertainty bounds are given by

b1…t† 2 ‰ 1:6; 0:15Š [ ‰0:15; 1:6Š; b2…t† 2 ‰ 2; 1Š [ ‰1; 2Š …5:106†

subsets with their centres iˆ …b1i; b2i†; i ˆ 1; ; 600 corresponding to

0 50 100

−5 0 5

b1i2 f 1:6; 1:5; ; 0:3; 0:2; 0:2; 0:3; ; 1:5; 1:6g

b2i2 f 2; 1:9; ; 1:1; 1; 1; 1:1; ; 1:9; 2g

respectively

Figures 5.4(a)±(c) illustrate the case where  is constant The switching

sequence fi…1†; i…2†; g depicted in Figure 5.4(a) indicates a remarkable speed

of localization

Case 2: Parameter jumps

The results of localization on the ®nite set fi g600iˆ1 are presented in Figures5.5(a)±(e) Random abrupt changes in the values of the plant parameters occurevery 7 steps In both cases above the algorithm of localization in Section 5.2 isused However, in the latter case the algorithm of localization is appropriately

modi®ed Namely, I…t† is updated as follows

I…t† ˆ I…t 1† \ ^I…t† if I…t 1† \ ^I…t† 6ˆ f g

Once (or if) the switching controller, based on (5.107) has falsi®ed every index

in the localization set it disregards all the previous measurements, and theprocess of localization continues (see [40] for details) In the example above apole placement technique was used to compute the set of the controller gains

fKig600iˆ1 The poles of the nominal closed loop system were chosen to be (0,0.07, 0.1)

Example 5.2 Here we present an example of indirect localization considered

in Section 5.4 The model of a third order unstable discrete-time system is givenby

y…t ‡ 1† ˆ a1y…t† ‡ a2y…t 1† ‡ a3y…t 2† ‡ u…t† ‡ …t† …5:108†

where a1; a2; a3 are unknown constant parameters, and …t† ˆ 0sin…0:9t†

represents exogenous disturbance The a priori uncertainty bounds are given by

the vector C and the stabilizing set I as prescribed in Section 5.4, we obtain

Kiˆ …k1i; k2i; k3i† Each element of the gain vector kij, i 2 f1; 2; 3g,

j 2 f1; ; 256g takes values in the sets (5.111), (5.112), (5.113), respectively.

The results of simulation with 0 ˆ 0:1, a1 ˆ 1:1, a2ˆ 0:7, a3ˆ 1:4, are

presented in Figure 5.6(a)±(b) Algorithm C has been used for this study

5.6 Conclusions

In this chapter we have presented a new uni®ed switching control basedapproach to adaptive stabilization of parametrically uncertain discrete-timesystems Our approach is based on a localization method which is conceptuallydi€erent from the existing switching adaptive schemes and relies on on-linesimultaneous falsi®cation of incorrect controllers It allows slow parameterdrifting, infrequent large parameter jumps and unknown bound on exogenousdisturbance The unique feature of localization based switching adaptivecontrol distinguishing it from conventional adaptive switching controllers isits rapid model falsi®cation capabilities In the LTI case this is manifested inthe ability of the switching controller to quickly converge to a suitablestabilizing controller We believe that the approach presented in this chapter

is the ®rst design of a falsi®cation based switching controller which isapplicable to a wide class of linear time-invariant and time-varying systemsand which exhibits good transient performance

Appendix A

Proof of Theorem 3.1 First we note that it follows from Lemma 3.1 and theswitching index update rule (5.37) that the total number of switchings made by

the controller is ®nite Let ft1; t2; ; tl g be a ®nite set of switching instants By

virtue of (5.31)±(5.33) the behaviour of the closed loop system between any two

consecutive switching instants ts; tj; 1  s; j  l; tj ts is described by

x…t ‡ 1† ˆ …A…† ‡ B…†K i…ts† †x…t† ‡ E……t† ‡ …t††

ˆ …A… i…ts† † ‡ B… i…ts† †K i…ts† †x…t† ‡ E …t† …5:114† where j …t†j  r i…ts† jj…t†jj ‡ ‡ …t†.

Therefore, taking into account the structure of the parameter dependent

matrices A…† and B…†, namely the fact that only the last rows of A…† and

B…† depend on  the last equation can be rewritten as

x…t ‡ 1† ˆ …A… i…ts† ‡
…t†† ‡ B… i…ts† ‡
…t††K i…ts† †x…t† ‡ E ^…t† …5:115† for some
…t† : jj
…t†jj  r i…ts† ‡ q and j^…t†j  ‡ …t† This is a direct

consequence of the fact that the last equation in (5.114) can be rewritten as

system subject to small parametric perturbations
…t† and bounded turbance ^…t† and this property holds regardless of the possible evolution of

dis-the plant parameters This is dis-the key point making dis-the rest of dis-the prooftransparent

Assume temporarily that …t†  0, then it follows from (5.115), (5.116) that jjx…ts‡ 1†jjHts  Ptsjjx…ts†jjHts ‡ ^ts …5:117† jjx…ts‡ 2†jjHts  P2

Since i…t† 2 …It0†; K i…t† 2 fKigL

iˆ1 for all t 2 N; i…t† 2 It0 there exist constants

0 < M0 0ˆ max jj jjEjj < 1 such that

Having denoted M1ˆ …M0M=†l; M2…† ˆ ^Ml…† < 1 we obtain (5.42) To

conclude the proof we note that the result above can be easily extended to the

case …t† 6ˆ 0, provided that the `size' of unmodelled dynamics " is suciently small Indeed, let …t† 6ˆ 0 First, we note that due to the term …t† in the

algorithm of localization (5.33)±(5.37) the process of localization cannot bedisrupted by the presence of small unmodelled dynamics In view of (A5),(5.117)±(5.126) it is easy to show that provided that " is suciently small

the states to the residual set (if ^Ml…† > 0) can be easily established Indeed, in

this case it is always possible to specify a suciently large integer T such that

…M1T‡ M"† < 1 This, in turn, trivially implies stability The ®nite number

of the controller switchings follows directly from the switching index updaterule (5.37) This also implies the ®nite convergence of switching; however, it is

quite dicult, in general, to put an upper bound on tl This obviously does nota€ect the stability properties of the closed loop.

Appendix B

Proof of Theorem 3.4 First we note that the property …1† follows directly

from the structure of the algorithm of localization (5.66) It is straightforward

to verify that relations (5.60)±(5.65) guarantee that the sequence of localization

sets I…t† is well de®ned.

To prove …2† consider ®rst the case ˆ 0 It is clear that

\t

kˆs…t†

if mink2‰s…t†;tŠ f…k ‡ 1†g   for all t > t0 Since, according to (5.64), the

estimate …t† is updated only if (5.129) does not hold, and taking into account

the discrete nature of updating expressed by (5.65) we conclude that

sup

tt0

Let > 0 Then it is easy to see that the arguments above remain valid for

any ®nite interval of time ‰s…t†; s…t† ‡ td †, provided that the rate of parameter variations is suciently small, namely,  q=td To conclude the proof of

(5.130) it suces to note that the estimate …t† in (5.64) does not change if

t s…t†  td

Proof of statements …5:3†; …5:4† follows closely those of Theorem 3.1 Here

we present a brief sketch of the proof Consider a ®nite time interval

T ˆ ‰s…t†; s…t† ‡ td†; l < td < 1 Let …s…t††  , then the total number of switchings s made by the controller over T satis®es the condition s  l if  q=td Therefore, the states are bounded by (5.126) with t0replaced by s…t†.

Moreover, (5.126) is valid for any time interval T ˆ ‰s…t†; s…t† ‡ t†; t> td suchthat

of the closed loop system Let  be unknown, then for any …t0 † > 0 the inequality (5.126) can be possibly violated no more than …‰=Š ‡ 1† times.

Relying on this fact and using standard arguments exponential stability of theclosed loop system is easily established

[5] Byrnes, C I., Lin, W and Ghosh, B K (1993) `Stabilization of Discrete-TimeNon-linear Systems by Smooth State Feedback', Systems and Control Letters, 21,255±263.

[6] Chang, M and Davison, E J (1995) `Robust Adaptive Stabilization of UnknownMIMO Systems using Switching Control', Proceedings of the 34th Conference onDecision Control, 1732±1737

[7] Etxebarria, V and De La Sen, M (1995) `Adaptive Control Based on SpecialCompensation Methods for Time-varying Systems Subject to BoundedDisturbances', Int J Control, 61, 3, 667±699

[8] Fu, M and Barmish, B R (1986) `Adaptive Stabilization of Linear Systems viaSwitching Control', IEEE Trans Auto Contr., AC-31, 12, 1097±1103

[9] Fu, M and Barmish, B R (1988) `Adaptive Stabilization of Linear Systems withSingular Perturbations', Proc IFAC Workshop on Robust Adaptive Control,Newcastle, Australia

[10] Fu, M (1996) `Minimum Switching Control for Adaptive Tracking', Proceedings25th IEEE Conference on Decision and Control, Kobe, Japan, 3749±3754.[11] Furuta, K (1990) `Sliding Mode Control of a Discrete System', Systems andControl Letters, 14, 145±152

[12] Goodwin, G C and Sin, K S (1984) Adaptive Filtering Prediction and Control.Prentice-Hall, Englewood Cli€s, N.J

[13] Hocherman-Frommer, J., Kulkarni, S R and Ramadge, P (1995) `SupervisedSwitched Control Based on Output Prediction Errors', Proc 34th Conference onDecision and Control, 2316±2317

[14] Hocherman-Frommer, S K J and Ramadge, P (1993) `Controller SwitchingBased on Output Prediction Errors, preprint, Department of ElectricalEngineering, Princeton University

[15] Haber, R and Unbehauen, H (1990) `Structure Identi®cation of NonlinearDynamic Systems ± A Survey on Input/Output Approaches', Automatica, 26, 4,651-677

[16] Ioannou, P A and Sun, J (1996) Robust Adaptive Control Prentice-Hall.[17] Kreisselmeier, G (1986) `Adaptive Control of a Class of Slowly Time-varyingPlants', Systems and Control Letters, 8, 97±103

[18] Kung, M C and Womack, B F (1983) `Stability Analysis of a Discrete-TimeAdaptive Control Algorithm Having a Polynomial Input', IEEE Trans Auto.Contr., 28, 1110±1112

[19] Lai, W and Cook, P (1995) `A Discrete-time Universal Regulator', Int J ofControl, 62, 17±32.

[20] Martensson, B (1985) `The Order of Any Stabilizing Regulator is Sucient a prioriInformation for Adaptive Stabilizing', Systems and Control Letters, 6, 2, 87±91.[21] Middleton, R H and Goodwin, G C (1988) `Adaptive Control of Time-varyingLinear Systems', IEEE Trans Auto Contr., 33, 1, 150±155

[22] Middleton, R H., Goodwin, G C., Hill, D J and Mayne, D Q (1988) `DesignIssues in Adaptive Control', IEEE Trans Auto Contr., 33, 1, 50±80

[23] Miller, D and Davison, E J (1989) `An Adaptive Controller Which ProvidesLyapunov Stability', IEEE Trans Auto Contr., 34, 599-609

[24] Morse, A S (1982) `Recent Problems in Parametric Adaptive Control', Proc.CNRS Colloquium on Development and Utilization of Mathematical Models inAutomatic Control., Belle-Isle, France, 733±740

[25] Morse, A S (1993) `Supervisory Control of Families of Linear Set-pointControllers', Proceedings of the 32nd Conference on Decision and Control,1055±1060

[26] Morse, A S (1995) `Supervisory Control of Families of Linear Set-pointControllers ± Part 2: Robustness', Proceedings of the 34th Conference on DecisionControl, 1750±1760

[27] Morse, A S., Mayne, D Q and Goodwin, G C (1992) `Applications of HysteresisSwitching in Parameter Adaptive Control', IEEE Trans Auto Contr., 37, 9,1343±1354

[28] Mudgett, D and Morse, A (1985) `Adaptive Stabilization of Linear Systems withUnknown High-frequency Gains', IEEE Trans Auto Contr., 30, 549±554.[29] Narendra, K S and Balakrishnan, J (1994) `Intelligent Control Using Fixed andAdaptive Models', Proceedings of the 33th Conference on Decision Control.[30] Narendra, K S and Annaswamy, A M (1989) Stable Adaptive Systems Prentice-Hall

[31] Nussbaum, R D (1983) `Some Remarks on a Conjecture in Parameter AdaptiveControl', Systems and Control Letters, 3, 243±246

[32] Peng, H J and Chen, B S (1995) `Adaptive Control of Linear Unstructured varying Systems', Int J Control., 62, 3, 527±555

Time-[33] Praly, L., Marino, R and Kanellakopoulos, I (eds) (1992) `Special Issue onAdaptive Nonlinear Control', International Journal of Adaptive Control andSignal Processing, 6

[34] Rohrs, C E., Valavani, E., Athans, M and Stein, G (1985) `Robustness ofContinuous-time Adaptive Control Algorithms in the Presence of UnmodelledDynamics', IEEE Trans Auto Contr., AC-30, 9, 881±889

[35] Tsakalis, K S and Ioannou, P A (1987) `Adaptive Control of Linear varying Plants', Automatica, 23, 459±468

[36] Tsakalis, K S and Ioannou, P A (1989) `Adaptive Control of Linear varying Plants: A New Model Reference Controller Structure', IEEE Trans Auto.Contr., 34, 10, 1038±1046

Time-[37] Weller, S R and Goodwin, G C (1992) `Hysteresis Switching Adaptive Control

of Linear Multivariable Systems', Proceedings of the 31st Conference on Decisionand Control, 1731±1736

[ 16] Ioannou, P A and Sun, J (19 96) Robust Adaptive Control Prentice-Hall.[17] Kreisselmeier, G (19 86) `Adaptive. .. Parameter AdaptiveControl'', Systems and Control Letters, 3, 243±2 46

[32] Peng, H J and Chen, B S (1995) `Adaptive Control of Linear Unstructured varying Systems'', Int J Control. , 62 , 3, 527±555... Discrete-TimeNon-linear Systems by Smooth State Feedback'', Systems and Control Letters, 21,255± 263 .

[6] Chang, M and Davison, E J (1995) `Robust Adaptive Stabilization of UnknownMIMO Systems using Switching Control'' ,