In Remark 2.4 it has been mentioned that the choice of the gains for the parameter update laws in model reference adaptive control is not trivial. High adaptation rates will lead to high-frequency oscillations of the system and the estimated parameters as shown in [44, 94, 145]. On the other hand, slow adaptation rates will lead to high tracking errors at the beginning of adaptation. In order to improve this transient behavior in a systematic way, the closed-loop reference (CRM) model has been introduced and examined in [44].
For the control scheme given in Table 2.1 the CRM is defined as
˙xref = Arefxref + Brefr− Lref (xp− xref) , (2.56) where Lref ∈ Rn×n is a gain matrix for the tracking error e = xp− xref. As can be seen in (2.56), the tracking error e = xp − xref is fed back to the reference model, so that a closed-loop system with feedback-gain Lref results . Using the same procedure as in Section 2.2.1 with a CRM leads to the closed-loop equation
˙e = (Aref + Lref) e + Bpλ (K˜xTxp+ ˜krr− ˜θTnlfnl(xp)) (2.57) of the tracking error dynamics. Stability of MRAC with CRM can be proven the same way as for MRAC with an open-loop reference model, if the Lyapunov equation
(Aref + Lref)T P + P (Aref + Lref) = −Q for Q = QT >0 (2.58) is satisfied. Furthermore, the properties stated in Theorem 2.1 also apply to state-feedback MRAC with CRM.
As the main benefit of a CRM it is shown in [44] for linear plants that the choices Γx = γcIn×n, γr = γc,
Lref = −Aref − g In×n, Q= g In×n ⇒ P = 1
2In×n,
(2.59)
for the estimation gains and the CRM-gain with γc>0 and g = γc>0 lead to significant improvement of the transient closed-loop behavior in the sense of a reduced square integral
of the tracking error and the parameter derivatives with respect to time. It is also shown that the closed-loop response, as well as the parameter estimation, are less oscillatory. In [94] an alternative procedure is shown to tune Lref which is similar to observer tuning.
Two different interpretations on how the CRM improves closed-loop performance can be given. The first one follows from equation (2.57) and the computation of Lref in (2.59).
It can be seen that the feedback of e to the reference model shifts the reference state xref
in the direction of the plant state xp. Hence, not only the closed-loop state approaches the reference state, but also the other way round. This, as it is stated in [44], reduces the burden of tracking on the adaptive system. The second interpretation is adopted from arguments of observer-based control. From equation (2.57) it follows that Lref can be chosen such that the stable dynamics of the tracking error is faster than that of the open-loop reference model. Hence, the tracking error is reduced faster than the closed-loop system is desired to respond to the reference signal r. This reduces the influence of the transients of e on the closed-loop system. A similar argument stems from observer based control where the dynamics of the observer is demanded to be faster than that of the closed-loop system in order to reduce the influence of the observer error transients on the closed-loop system (see e.g. [60, 100]).
Application of state-feedback MRAC with CRM is very similar to MRAC with open-loop reference model in Table 2.1. It is only necessary to replace the reference model equation and the Lyapunov equation by (2.57) and (2.58), respectively. Calculation of Lref can then be done by (2.59). Similar results for output-feedback MRAC with CRM can be found in [43].
Simulation Example 2.4. In order to compare MRAC for state-feedback with and without closed-loop reference model, the plant Gex1 from Example 2.1 is considered. The initial estimates of the parameters, as well as the requirements on the closed-loop are also adopted from Example 2.1. The remaining tuning parameters have been chosen to be Q = I2×2, Γx = Γnl = 200 I2×2 and γr = 200. Note that the estimation gains are 2000 times larger than in Example 2.1. The closed-loop reference model becomes
˙xref =
has been computed according to (2.59) with g = 200. The simulation results are shown in Figure 2.7, where the results for MRAC with standard reference model are depicted in the left column.
A first difference of the CRM can be seen in the system response at the beginning of adaptation in the first graph of Figure 2.7. The responses of the CRM and the closed-loop system do not differ, even before adaptation is started at t = 10s. The reason for that is the feedback of xp to the reference model in (2.4) which pulls the reference trajectories closer to the plant trajectories. Another difference can be observed for the controlled input of the plant, which is shown in the second graph. For the standard
0 10 20 30
systemresponse
without CRM
desired response plant output
0 5 10 15
−500 0 500
controlledinput
9.8 10.5
180 185 190 195
0 10 20
systemresponse
output tracking error
0 50 100 150 200
−40
−20 0 20 40
time in s
controllerparameter
kx,1 kx,2 kr ˆθnl,1 ˆθnl,2
with CRM
reference model output plant output
0 5 10 15
9.8 10.5
180 185 190 195
output tracking error
0 50 100 150 200
time in s
Figure 2.7: Comparison of MRAC with and without CRM. First and second graph: Sys-tem response and controlled input at beginning of adaptation. Third graph: Closed-loop response and output tracking error after 180s of adaptation. Fourth graph: Estimations of controller parameters.
MRAC strong oscillations occur, which are even stronger than in Example 2.1 due to the higher estimation gain. In contrast to that, the controlled input of MRAC with CRM shows no oscillations at all. This is a great improvement, since for technical systems strong oscillation can be critical for actuators or other wear parts of the system. Finally, the last graph of Figure 2.7 shows that parameter estimation with CRM is quicker in this example and results in less oscillatory peaks of the estimated parameters. After adaptation is switched off at t = 190s the scaled tracking error ¯ey = 20 (xp,1− xref,1) is smaller for the closed-loop system with CRM as can be seen in the third graph of Figure 2.7. Note that the the scaled tracking error for MRAC with CRM as well as for standard MRAC has been computed with xref,1 of the constant reference model (2.14) in order to guarantee comparability.