12. APPLICATION OF DUAL CONTROLLERS TO A LABORATORY SCALE
12.2. Experimental Setup and Results
12.2.2. Comparison of Standard and Dual Control
The parameters of the discrete transfer function, eq. (12.36), were preliminarily estimated (Patra, et al., 1994). This was necessary for the determination of the discrete- time closed-loop model according to eqs. (12.2) and (12.3). For a nominal plant model according to eq. (12.34), the continuous-time parameters are estimated asa=3.02 sec−1, b=6.84, where the motor time constant and overall gain parameter are obtained by simple measurements; the third order discrete-time model in eq. (12.36) has the parameters given in Table 12.1 which are obtained after z-transformation for the sampling time 0.2 sec applying a zero-order hold.
12.2. Experimental Setup and Results 143
Table 12.1. Parameters of the model according to eq. (12.36) after preliminary identification
Numerator Denominator
b1 7.89⋅10−3 a1 -2.5466
b2 2.73⋅10−2 a2 2.0932
b3 5.84⋅10−3 a3 -0.5466
This model is nonminimum phase and unstable, and the parameters are changing depending on the setpoint. According to eqs. (12.30) and (12.32), the closed-loop model was chosen to give the pole locations specified in Table 12.2.
Table 12.2.Closed-loop pole locations in thez-domain for the experiment
Real part Imaginary part
0.829 0.829 0.72 0.45 0.2
0.2 -0.2 0 0 0
Obviously the controller given by eq. (12.1) has the polynomial orders nS=2 and
R =2
n . Hence, there are five parameters to be estimated. The startup-parameters for the adaptive controller are chosen as follows:
, ] 10 10 10 10 10 [ diag ) 0 (
, ] 4 . 1 51 . 0 8 . 2 7 . 7 8 . 5 [ ) 0
ˆ( T
=
−
−
= P
p
, 0 for 0 ) ( and 0 ) ( , 0001 . 0 , 1 .
2=0 η= uk = y k = k≤
σξ
where diag[ã] denotes a diagonal matrix with the indicated diagonal elements. The con- troller parameter values for the considered plant model according to eq. (12.36) and Ta- ble 12.1 can be calculated by eqs. (12.32) and (12.33) and the initial values in pˆ(0) have been chosen arbitrarily. These parameters are assumed to be unknown, and they depend on the operating point of the system.
Fig. 12.5 shows the results of the indirect adaptive pole-placement controller of Keuchel et al. (1987). In contrast to Keuchel et al. (1987), who used a robust and an LQ optimal pole-placement design, the pole positions according to Table 12.1 have been
chosen for the experiment plotted in Fig. 12.5. Besides the parameters, the manipulated and controlled signals of the experiment, Fig. 12.5 also shows the prediction error
)
~( ) ˆ~ ( ) ˆ( )
(k yk T k k
e = − p m ,
where pˆ~ and m~ contain the estimated plant parameters and input/output signals for direct estimation of the plant, respectively. The controlled variable y(t) in Fig. 12.5 exhibits the typical problems of indirect adaptive control schemes based on the CE as- sumption during the phases of adaptation. There is a large overshoot (more than 200 %) during the adaptation period, and the convergence of parameters is slow mainly due to
10
5 15
0.02 0.0 -0.02 0
e
t
5 10 15
0 1.0 0.0 -1.0
u
t
10
5 15
1.02.0 -1.00.0 0
y w
t
5 10 15
0 5.0 0.0 -5.0
p^
t
(sec) Fig. 12.5.Indirect adaptive pole-placement control
the missing excitation since the normalized manipulated signal reaches the normalized saturation limits, u(k) =1.
Fig. 12.6 shows the results for the direct adaptive pole-placement control law based on the standard CE assumption. The adaptation transients die out significantly faster than in the case of indirect control, but there is still a 40 % overshoot after the first setpoint step. Here and in all further diagrams
) ( ) ˆ ( ) 1 ( )
(k y k T k k
e = + −p m
is the equation error as introduced in eq. (12.11).
Figs. 12.7 and 12.8 display the results for the same experiment applying the pro- posed direct adaptive dual control strategy with different settings of the tuning parameter
12.2. Experimental Setup and Results 145
ηfor cautious and dual control action (η=0 and η=0.0001, respectively). The results obtained by the direct dual control approach as illustrated in Fig. 12.8 show better control performance at the beginning of the adaptation because of the cautious and probing prop- erties. The adaptive control based on the CE assumption (Fig. 12.6) gives a non- negligible overshoot at the beginning of the adaptation, because the estimated controller parameters are far from the desired values and the accuracy of the estimates is not taken into account. The presence of the term representing the covariance of the estimates in eq. (12.25) for the direct dual control (Fig 12.8) avoids this undesired large control de- viation. It is necessary to point out that cautious controllers normally provide good results (see Fig. 12.7) but have not found a broad practical application up to now, because they lead to slow adaptation and sometimes to the "turn off" effect (Wittenmark, 1995) when the estimation process in the adaptive system is interrupted. Only the combination of cautious control with optimal excitation, as in eq. (12.29), provides acceptable control performance as can be seen from Fig. 12.8.
10
5 15
0.5 0.0 -0.50
w y
t
5 10 15
0 1.0 0.0 -1.0
u
t 10
5 15
0.02 0.0 -0.02 0
e
t
5 10 15
0 5.0
0.0 -5.0
p^
t
(sec) Fig. 12.6.Direct adaptive pole-placement control
10
5 15
0.02 0.0 -0.02 0
e
t
5 10 15
0 1.0 0.0 -1.0
u
t
10
5 15
1.0 2.0 0.0 -1.00
w y
t
5 10 15
0 5.0 0.0 -5.0
p^
t
(sec) Fig. 12.7.Direct cautious adaptive pole-placement control
10
5 15
2.0 0.0 -2.0 0
e
t
5 10 15
0 1.0 0.0 -1.0
u
t
10
5 15
0.5 0.0 -0.50
w y
t t
5 10 15
0 5.0 0.0 -5.0
p^
(sec) Fig. 12.8.Direct dual pole-placement control
12.2. Experimental Setup and Results 147
It should be emphasized that the direct dual approach presented above may be extended to other types of adaptive pole-placement controllers, for example, to the direct adaptive pole-placement controller suggested by Elliott (1982), as discussed in Chapter 6.
Contrary to innovational dual control (Milito et al., 1982), the proposed dual control according to eqs. (12.25) and (12.29) uses the parameterθ to define the amplitude of the excitation. This clear physical interpretation of the controller parameter θ makes direct dual control even more attractive for applications. The pole-zero placement system con- sidered here is characterized by a small number of parameters to be estimated. The coef- ficients of Bm(z−1) and Am(z−1) can be obtained easily from standard polynomials of the continuous-time case provided in literature (Unbehauen, 1985), and the selection of an appropriate sampling time.