1 Adaptive Control. 1.1 Indirect case:

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1 Adative Control Adative control is the attet to redesign the controller while online, by looking at its erforance and changing its dynaic in an autoatic way. Adative control is that feedback law that looks at the rocess and the erforance of the controller and reshaes the controller closed loo behavior autonoously. Controller Plant Adatation Procedure Figure. Adative Control as Feedback Law Adative Control is ade by cobining. online araeter estiator based on current easureents and control actions 2. control law, that recalculates the controller based on those araeters There are two ain odalities of adative control. We discussed the below.. Indirect case: In this case, the estiated araeters refer to the lant, and the controller is designed based on those araeters. For exale, an energy consution coefficient is estiated; a linearized version of the lant is roduced, which is then used in a ole laceent ethod to calculate the gains of the final controller. The ain roble is to ake sure that the controller behaves well in cases when the estiation of the lant is not good, like for exale, during transients. In rincile, the indirect ethod is alicable to all tyes of lants.

2 Controller Plant K(t) K=F() (t) Paraeter Estiator Figure 2. Indirect Method.2 Direct case Now the lant odel is araeterized based on the controller araeters, which are then estiated directly without interediate stes. Exale is the self tuning PID loo. The sae robles as in the indirect case can be encountered here. This ethod can be safely alied only to iniu hase systes. Controller Plant K(t) Paraeter Estiator efigure 3. Direct Method

3 .3 Design of Online Paraeter Estiator Paraeter estiation is at the very core of adative control, in articular obvious in the indirect ethod. Paraeter estiation devices are constructed using one of the following ethods..3. Sensitivity ethod: This ethod is related to of the sensitivity equation, where the araeter is estiated by using a noinal trajectory and its sensitivity with resect to that araeter x( t, λ) x( t, λ0) S( t)( λ λ0) λ λ0 + S( t) ( x( t, λ) x( t, λ0)) Whereas the seudo inverse (least squares algorith) of the sensitivity atrix ust be calculated in ost cases as the atrix will not be invertible..3.2 Gradient ethod or MIT Rule This ethod is driven by the idea of iniizing the square of the rediction error. 2 e J ( K) = 0.5e ( K) & = γ e Since the araeters are unknown, the ain roble is the calculation of the artial derivative in real tie. One well known aroach to address this is the so called MIT rule. The basic set u is shown in below Figure 4. MIT rule setu The lant is k Z (s), where k is unknown aart fro its sign, and Z (s) is a known stable transfer function. The basis of identification is that an adjustable ositive known gain k c (t) is introduced as shown, and the outut of the uer branch is coared to the outut of k Z (s) when the sae driving signal is alied; where k is a known gain. The MIT rule is the rule of adjustent for kc. The idea is to use gradient descendent to adjust kc(.), which in this case leads to

4 k & = g( y y ) y c The rule is aealing due to its silicity but it often leads to instabilities. The echanis is one where the fast dynaics of the adatation destabilizes the closed loo syste (based on the controller that used the estiated araeter). In alications, it is iortant to understand which the tie constants are in the syste to avoid self inflicted oscillations. Indeed, selecting different values for learning rate and or excitation ay lead to stable or unstable systes. In other words, if the excitation or the adatation rates are too fast, the syste can becoe unstable. Exale. Let Z = s + r( t) = sin( ω t) k k = = 2 Alying the MIT rule we obtain the following closed loo syste, where the araeter kc in Figure 4 is calculated as follows k & = g( y y ) y c With the following syste as signal generator y& y& = y = y + k c + k k r r Note in the following 2 ictures how slow learning rate allows stability of the adatation, while fast learning rate renders the syste unstable.

5 Figure 5. MIT Rule. Slow adatation rate. Figure 6. MIT Rule. Fast adatation rate

6 .3.3 Extended Kalan Filter The so called Extended Kalan Filter is a related alternative. Kalan filters are norally used for state estiation. However, using a device called state augentation that converts the araeters into states with no dynaics but with known rocess noise covariance it becoes ossible to aly this technique to this case..see below for details x& = f ( x, u, ) + ω & = 0 + ω 2 y = h( x, ) + η E( ω ω ) = Q E( ω ω ) = Q E( ηη) = R Figure 7. Extended Kalan Filter When does this algorith work?. No corehensive theory. 2. Equivalent to the controllability of a nonlinear syste by outut feedback 3. Detectability and stabilizability of the linearizations are necessary but not suffcient conditions Is the data required always available?

7 . Main roble is selecting Q, Q and R. They are tuning araeters. 2. Soe adative (self-tuning) ethods treat the syste as a nonlinear lant to be stabilized via Q and R. Higher order filters are used when strong nonlinearities render EKF useless.4 Observability and Persistency of Inforation. Bursting Phenoena. One ust be aware of the fact that trying to identify ore araeters than what the inforation available can deliver ay also lead to instabilities. For instance, we have a henoenon known as bursting henoena, see below: instability is observed when the signals becoe steady. After a furious hase, the syste goes back to steady state again. Figure 8. Bursting This reason behind this behavior is that when the signals are near zero the adative algorith does not have enough inforation in order to identify all araeters siultaneously. This leads to divergence of the estiates. The consequence of that is a faulty controller (because it is based on those araeters) rendering the lant unstable. Once the signals becoe rich enough though, the adative algorith begins to deliver good estiates of the araeters. Now, just in tie, the controller begins to erfor well, leading to lant recovery. Eventually, the roble aears again. Clearly, in industrial alications bursting ust not occur